Linear Periodic Systems:

Robustness Analysis and

Sampled-Data Control

Michael William Cantoni

St John’s College Cambridge

�

A dissertation submitted for the degree of Doctor of Philosophy

March, 1998

Abstract

The primary purpose of this dissertation is to present various results pertaining to the study of continuous-time systems that exhibit linear, periodically-time-varying (LPTV) behaviour. The original motivation for this work stems from the periodic nature of many sampled-data (SD) controlsystems and a desire to characterise sensible frameworks for design and robustness analysis of such systems.

Towards the development of a sensible framework for SD, control-system design, a new notion of frequency response is established for LPTV systems. It is defined in terms of the average-power of the asymptotic response to sinusoidal signals of a single frequency and can be characterised in terms of the singular values of a frequency-dependent, finite-dimensional matrix. The performance indicating properties of this new notion of frequency response are intuitive and it is used to derive bounds that facilitate the design of parametric weights employed in a new H^{∞}-loopshaping based procedure proposed for SD, control-system development.

Quantitative and qualitative results that characterise the uncertainty types to which LPTV, closed-loop systems can be desensitised are established in terms of the gap metric, which is a measure of the distance between the graphs of two systems. To this end, a formula for the directedgap between two LPTV systems is obtained in terms of an optimisation over H^{∞}. This plays a crucial role in the development of the main quantitative robustness result, which identifies the largest gap-ball of LPTV plants, centred at a nominal plant, that a nominal, stabilising controller is guaranteed to stabilise. A similar result that deals with simultaneous gap-perturbations to the plant and controller is also given. Qualitatively, it is established that the topology induced by the gap metric on quite a general class of LPTV systems, is the weakest with respect to which closed-loop performance varies continuously and closed-loop stability is a robust property. All of the results accommodate infinite-dimensional input/output spaces and apply to the special case of LPTV, SD control-systems.

The existence of particular representations of a class of LPTV systems is central to the framework used to obtain robustness results. Specifically, it is shown that the graph of a stabilisable, LPTV system can be expressed as the range and kernel of stable, LPTV systems that are respectively, left and right invertible by stable, LPTV systems. These representations of the graph resemble the coprime-factor representations known to exist for linear, time-invariant systems and lead to a useful characterisation of closed-loop stability. This in turn, yields a Youla-style parameterisation of stabilising controllers.

A numerical procedure is developed for computing the gap, to any desired accuracy, between LPTV systems that admit stabilisable and detectable, state-space realisations. This involves determining the existence of a solution to a related linear, shift-invariant, discrete-time, full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem, for which computationally-tractable necessary and sufficient conditions are obtained.

To complete this work, a new, compact derivation of state-space formula for H^{∞}, SD synthesis is presented. Related numerical issues are discussed and it is shown how to restructure the formulae derived for numerical robustness.

Acknowledgements

I would like to take this opportunity to express my gratitude to the many people who have provided help and encouragement over the time leading up to and during the development of this work.

First and foremost, I would like to thank my supervisor, Professor Keith Glover. His great insight, guidance and encouragement have been invaluable over the last three years. I am also very grateful to Dr Malcolm Smith and Dr Glenn Vinnicombe, who have both given freely of their time to discuss many problems. In particular Glenn, who has had a significant influence on my understanding of robust control theory. My undergraduate supervisor, Professor Kok Lay Teo, also deserves mention for first introducing me to the interesting mathematics used in control theory. Thanks to the proof-reading skill of Richard Ford, the number of typos in this dissertation has been significantly reduced.

My contemporaries in the Control Group and other friends in Cambridge have made my experience here very memorable. In particular, Alex, Brian, Edward, Gavin, George, Giles, Johannes, Mihai, Nelson, Richard F., Richard L., Richard W., Sanjay, Teddy and Tim. Special thanks to Giles and Sanjay for helping me settle in when I first arrived in Cambridge. Thanks also to my relatives in Italy for providing a summer refuge away from Cambridge.

On a more personal note, I am indebted to my parents, brother and grandparents for their constant love, encouragement and guidance. Finally, my heartfelt thanks to Şaduman for her love and understanding.

This work has been supported financially by the Gledden Studentship of the University of Western Australia, and the Committee of Vice-Chancellors and Principals of the Universities of the United Kingdom in the form of an Overseas Research Student Award. Travel grants have been gratefully received from St. John’s College and the Department of Engineering of the University of Cambridge.

As required by University Statute, I hereby declare that this dissertation is not substantially the same as any that I have submitted for a degree at any other University, is the result of my own work, and includes nothing which is the outcome of work done in collaboration.

Michael Cantoni St. John’s College

Cambridge March 12, 1998.

Contents

Abstract i

Acknowledgements ii

Notation vi

1 Introduction 1 1.1 Motivation and Background 1 1.2 Overview of Contents 3

2 Preliminaries 6 2.1 Introduction 6 2.2 Functional Analysis 6

2.2.1 Topological, Metric, Vector (Linear) and Hilbert Spaces 6 2.2.2 Operator Theory 11 2.2.3 Analytic Functions 15

2.3 Signals and Systems Theory 16 2.3.1 Signals 16 2.3.2 Systems 20 2.3.3 Closed-Loop Systems 24

2.4 LSI State-Space Systems and Discrete-Time Synthesis 27 2.4.1 Reachability, Observability, Stabilisability and Detectability 28 2.4.2 Stability 31 2.4.3 Quadratic-Regulator and Full-Information ℓ_{ℤ^{+}}^{2}(·)-Induced-Norm Synthesis 32

2.5 SD Control-Systems and State-Space Time-Lifting 40

3 Frequency-Domain Analysis and SD Control Design 49 3.1 Introduction 49 3.2 The Average-Power Gain Matrix 50

3.2.1 Frequency Response 50 3.2.2 Properties of the Average-Power Gain Matrix 54

3.3 A Framework for SD Closed-Loop Design 58 3.3.1 H^{∞}-Loopshaping for LTI Systems 58 3.3.2 H^{∞}-Loopshaping Based Design for SD Control-System Development 61

3.4 Summary 66

4 Robustness Analysis of LPTV Closed-Loop Systems 68 4.1 Introduction 68 4.2 Representations of LPTV Systems 71

4.2.1 LPTV Systems and LSI Equivalents 71 4.2.2 LSI Subspaces 72 4.2.3 Strong-Right/Left Representations and Closed-Loop Stability 74

4.3 The Gap Metric 83 4.4 Quantitative Robustness Analysis 89

4.4.1 Plant Perturbations 89 4.4.2 Simultaneous Plant and Controller Perturbations 96

4.5 The Graph (Gap) Topology 98 4.6 Robustness of SD Closed-Loop Systems 100 4.7 Summary 101

5 Computing the Gap 102 5.1 Introduction 102 5.2 The Gap Metric and Full-Information Synthesis 103 5.3 Normalised Right-Coprime Factorisations 104

5.3.1 LSI State-Space Systems 106 5.3.2 LPTV State-Space Systems 109

5.4 Computing the Directed Gap 112 5.4.1 A Necessary and Sufficient Condition for Bounding the Directed Gap 112 5.4.2 Computing À, R̀ and Q̀ in Theorem 5.6 115 5.4.3 Testing Condition (iii) in Theorem 5.6 117

5.5 Summary 124

6 H^{∞} SD Synthesis and Related Numerical Issues 126 6.1 Introduction 126 6.2 State-Space Formulae for the SD-Equivalent Generalised-Plant 128 6.3 Re-Structuring for Numerical Robustness 133 6.4 Comments Concerning Assumptions (A1) and (A2) 137 6.5 Summary 139

7 Concluding Remarks 140 7.1 Contributions 140 7.2 Directions for Future Research 141

A Appendix to Chapter 2 143 A.1 Proof of Proposition 2.20 143 A.2 Proof of Proposition 2.22 145

B Appendix to Chapter 3 152 B.1 Proof of Lemma 3.9 152 B.2 Computing the Average-Power Gain Matrix for SD Systems 155

C Appendix to Chapter 5 160 C.1 Proof of Lemma 5.9 160 C.2 Definition of Matrices Used in Lemma 5.9 171

D Appendix to Chapter 6 174 D.1 Proof of Lemma 6.5 174

Bibliography 177

Notation

\underline{Symbols}

ℝ (ℝ^{+})  set of real numbers (non-negative reals)
ℤ (ℤ^{+})  set of integers (non-negative integers)
ℂ  set of complex numbers
𝔻  open unit disc in the complex plane
𝕋  unit circle in the complex plane
𝔼  complement of the open unit disc; 𝔼:=ℂ − 𝔻
ℍ  the interval [0, h) of the real line
] · [  the integer part of a real number
∈  is an element of
⊂  is a subset of
∪  union of two sets
∩  intersection of two sets
×  Cartesian product of two sets
:=  left-hand side is defined by right-hand side
=:  right-hand side is defined by left-hand side
⇔, ⇕  if and only if
𝒜,ℱ  this font is used to denote general sets
B, W  this font is used to denote general Banach Spaces
ℋ, 𝒰  this font is used to denote general Hilbert spaces
∥ · ∥_{𝒱}  norm on the vector space 𝒱
⟨·, ·⟩_{ℋ}  inner product on the Hilbert space ℋ
ℋ ⊖ 𝒰 (or 𝒰^{⊥})  orthogonal complement of 𝒰 in ℋ
𝒰 ⊕ 𝒴  orthogonal direct sum of 𝒰 and 𝒴
Π_{𝒰}  orthogonal projection onto 𝒰
Π_{𝒢∥ℱ}  parallel projection onto 𝒢 along ℱ
B_{𝒰→𝒴}  the set of bounded, linear operators mapping 𝒰 to 𝒴
𝒟_{·}  domain of a linear operator
ℛ_{·}  range of a linear operator
𝒦_{·}  kernel of a linear operator
𝒢_{·} (𝒢·^{′})  graph (inverse-graph) of a linear operator
·|_{ℋ}  restriction of an operator to ℋ
·^{−1}  inverse of an operator
·^{∗}  adjoint of a bounded, linear operator
spec(·)  spectrum of a bounded, linear operator
rad(·)  spectral radius of a bounded, linear operator
σ(·)  singular values of a bounded, linear operator
\overline{σ}\{·\}  maximum singular value of a compact operator
\underline{σ}\{·\}  minimum singular value of a compact operator

Symbols ctd.

𝒴_{𝕊}(B)  vector space of B-valued signals on the time interval of definition 𝕊
P_{τ}  the projection that truncates a signal to zero after t = τ
U_{τ}  unilateral shift by τ on 𝒴_{ℝ^{+}}(·)
S  unilateral shift on 𝒴_{ℤ^{+}}(·)

L_{ℝ^{+}}^{2}(ℋ) the Hilbert space of ℋ-valued, (Lebesgue) square-integrable signals in 𝒴_{ℝ^{+}}(ℋ) (i.e. the finite-energy, continuous-time signals)

L_{ℝ^{+}}^{2,e}(ℋ)  the extended space corresponding to L_{ℝ^{+}}^{2}(ℋ)

ℓ_{ℤ^{+}}^{2}(ℋ) the Hilbert space of ℋ-valued, square-summable functions in 𝒴_{ℤ^{+}}(ℋ) (i.e. the finite-energy, discrete-time signals)

ℓ_{ℤ^{+}}^{2,e}(ℋ)  the extended space corresponding to ℓ_{ℤ^{+}}^{2}(ℋ)

H_{𝔻}^{2}(ℋ) the Hardy space of ℋ-valued functions of a complex variable, that are analytic in 𝔻 and (Lebesgue) square-integrable on 𝕋

Z  the Z-Transform (isomorphism between ℓ_{ℤ^{+}}^{2}(ℋ) and H_{𝔻}^{2}(ℋ))

H_{𝔻}^{∞}(B_{𝒰→𝒴} ) the Hardy space of B_{𝒰→𝒴} -valued functions of a complex variable, that are analytic and bounded in 𝔻

∥ · ∥_{∞}  the norm on H_{𝔻}^{∞}(B_{𝒰→𝒴} )
M_{·}  the multiplication operator with symbol ·

L_{ℍ}^{2}(ℋ) the Hilbert space of (Lebesgue) square-integrable, ℋ-valued functions defined on the finite-horizon ℍ

W the time-lifting operator (isomorphism between L_{ℝ^{+}}^{2}(ℋ) and ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(ℋ)))

A_{D} and D_{A}  ideal, sampling device and zero-order, hold device respectively

𝒫_{𝒰→𝒴} the set of causal, linear, periodically-time-varying, continuous-time systems, which at each point in time map 𝒰 to 𝒴

𝒫_{𝒰→𝒴}^{e} the subset of systems in 𝒫_{𝒰→𝒴} that are causally extendible to (time) locally, Lipschitz-continuous systems on extended space

𝒫_{𝒰→𝒴}^{e,sc}  the subset of systems in 𝒫_{𝒰→𝒴}^{e} that are strongly causal

𝒟_{𝒰→𝒴} the set of causal, linear, shift-invariant, discrete-time systems, which at each point in time map 𝒰 to 𝒴

P, A, u, y  this font is used to denote anything associated with discrete-time systems
·^{⊸}  is used to denote that · is in L_{ℍ}^{2}(·)
ˆ·  is used to denote frequency-domain transfer operator

\underline{Acronyms}

LPTV  linear periodically-time-varying
LSI  linear shift-invariant (discrete-time)
LTI  linear time-invariant (continuous-time)
SD  sampled-data
TPBVP  two-point, boundary-value problem

Introduction

1.1 Motivation and Background

Control theory is the study of dynamical systems and techniques that allow their behaviour to be modified by manipulating accessible inputs. The auxiliary system used to determine the actions required to achieve a desired change in a system’s behaviour is called a controller and as a whole, the original system and controller combined is called a control system. Given an exact representation of a system (including any disturbances acting upon it) and an achievable desired behaviour, it is possible to design a controller to achieve the desired change in behaviour without using measurable information.^{1} This is referred to as open-loop control. The problem with this approach is that a system is most readily represented by a mathematical model, which can at best, only approximate the real behaviour of a physical system. The notion of uncertainty arising from the mismatch between physical processes (the system to be controlled, disturbances acting on it and the controller itself) and the mathematical models used to represent them was central to the development of feedback control theory, in which measurable information is exploited in the control strategy. An important property of closed-loop (or feedback) control-systems is that the effects of uncertainty arising in this manner can be significantly reduced. The study of uncertainty types to which closed-loop systems can be desensitised and the development of suitable mathematical frameworks to exploit this property, is called robust control theory.

Classifying systems according to way their behaviour varies with time (for example, invariantly or periodically) imparts additional structure to the mathematical frameworks used to study them. It is for this reason that linear, time-invariant (LTI) systems are well-understood and robust control theory relatively mature for this class of systems [Zam81, Fra86, ZDG95]. The work presented in this dissertation arises from a desire to characterise sensible frameworks for design and robustness analysis of closed-loop systems that exhibit linear, periodically-time-varying (LPTV) behaviour. The primary motivation for this stems from the wide-spread use of digital hardware (computers) to implement control strategies for continuous-time systems, which invariably leads to time-varying, closed-loop behaviour, that is often periodic in nature. Since digital hardware can only process discrete-time information, any digital controller employed to control a continuous-time system in closed-loop incorporates a sampling device at the input, which is used to convert continuous-time signals into discrete-time signals, and a hold device at the output, which is used to convert discretetime signals into continuous-time signals. Systems like this, in which there is a hybrid mixture of continuous-time and discrete-time signals, are called sampled-data (SD) systems. In practice, it is common for the sampling device to operate at a fixed rate (period) and for the hold device to be synchronised with the sampler. In this situation, even if the discrete-time control-strategy implemented in digital hardware does not exhibit time-varying behaviour, the controller as a whole does. This is because the sampling process is periodically time-varying in the following sense: given a continuous-time input u to a (fixed-rate) sampling device and corresponding discrete-time output y, the output of the sampler in response to a time-shifted version of u corresponds to a time-shifted version of y if and only if the time-shift is an integer multiple of the sampling period. As further motivation, it is interesting to note that SD systems are not the only periodically-time-varying systems encountered in engineering and science. For example, given a time-invariant, nonlinear model of a system and a desired time-periodic trajectory, a linearisation about this gives rise to an LPTV model. Such models arise naturally for systems that involve reciprocating (rotating) masses and in celestial mechanics, for example.

1Embedded in the notion of achievable behaviour is the assumption that such a controller corresponds exactly to a physically realisable system.

The origins of robust control theory lie with the Small Gain Theorem [Zam66] and the seminal work of Zames [Zam81]. Further advances, in terms of qualitative results characterising the uncertainty types to which closed-loop systems can be desensitised, were made with the definition of the graph topology, in which two systems are considered similar (or close) if any reasonable controller for the first achieves similar closed-loop behaviour with the second [ZES80, VSF83, Vid84]. Quantitative results followed for LTI systems with the development of several metrics that induce the graph topology for such systems: the gap metric [ZES80, Geo88]; the graph metric [Vid84]; and the ν-gap metric [Vin93a]. Of specific interest in this dissertation is the gap metric, which for LTI systems, as shown in [GS90], is intimately related to the notion of coprime-factor uncertainty [VK86, MG90]. This relationship is particularly important, because it is known that the mathematical framework used to study robustness to coprime-factor uncertainty has useful performance-indicating properties. These properties were exploited in [MG90] in the development of a robust, controlsystem, design procedure for LTI systems, using notions of classical, frequency-domain loopshaping. The main goal of the work presented in this dissertation is to establish results analogous to some of those mentioned above for LPTV systems.

The equivalence of LPTV, continuous-time systems to linear, shift-invariant (LSI), discrete-time systems (cf. [BPFT91, BP92]) is of fundamental importance in this dissertation. By virtue of this equivalence, although the analysis is somewhat more involved (with tractable technical difficulties that arise concerning causality and infinite-dimensionality), periodic time-variation can be exploited in a similar manner to time-invariance. This yields powerful results for LPTV systems that are analogous to many results known to hold for LTI systems. Specifically, using the gap metric, qualitative and quantitative results are obtained by which the robustness of LPTV, closed-loop systems can be characterised. To this end, the existence of particular range and kernel representations of the graph of a stabilisable, LPTV system is established. These representations resemble the coprime-factor representations known to exist for certain classes of LTI systems [Vid85]. Towards developing a framework for design, a new notion of frequency response is developed for LPTV systems. This new notion of frequency response can be used to derive frequency-by-frequency bounds on performance-indicating, closed-loop operators, which facilitate the design of parametric weights employed in a new H^{∞}-loopshaping based procedure proposed for SD, control-system development.

Mathematical analysis is of very little use from an engineering perspective unless computational procedures exist to exploit it for specific classes of systems. Hence the wide-spread use of statespace tools in robust control theory [DGKF89, ZDG95, GL95]. As such, a considerable component of this dissertation is devoted to developing numerical algorithms for automated analysis and synthesis. More specifically: a computational procedure is developed for calculating the new notion of frequency response introduced; it also is shown how to compute the gap between any two LPTV systems that admit stabilisable and detectable, state-space realisations; finally, new formulae are derived for H^{∞}, SD synthesis, with discussion of numerical issues that arise and how they should be handled.

1.2 Overview of Contents

Chapter 2: In this chapter, material is collected to facilitate understanding of the main component of this dissertation. In Section 2.2, the underlying mathematical structure of all signal and system spaces is defined. General signal and system theory is discussed in Section 2.3. In Section 2.4, state-space tools used in the development of numerical algorithms are presented. Finally, fundamental issues involved in stability and induced-norm analysis of SD control-systems are briefly discussed in Section 2.5.

Chapter 3: A new notion of frequency response for LPTV systems is developed in this chapter. It is defined in terms of the average-power of the asymptotic response of an LPTV system to sinusoids of a single frequency. The performance-indicating properties of this new, frequencydomain, analysis tool are intuitive and it is used to derive frequency-by-frequency bounds on particular performance-indicating, closed-loop operator-gains, which facilitate the design of parametric weights used in a new H^{∞}-loopshaping based design procedure proposed for SD, control-system development.

Chapter 4: The primary purpose of this chapter is to present qualitative and quantitative results that characterise the uncertainty types to which LPTV, closed-loop systems can be desensitised. These results are obtained in terms of the gap metric, which is a measure of the distance between the graphs of two systems. Towards establishing the desired robustness results, so-called strong-right and strong-left representations of the graphs of stabilisable, LPTV systems are shown to exist. It is established that these representations can be used to characterise closed-loop stability, which leads to a Youla-style parameterisation of stabilising controllers. A formula for the directed-gap between two LPTV systems is then derived. The formula obtained is essentially a generalisation of that in [Geo88] for LTI systems and it is crucial to the development of the robustness results obtained. Specifically, the largest gap-ball of LPTV plants, centred at a nominal plant, that a nominal, stabilising controller is guaranteed to stabilise, is identified. Further to this, the maximally-tolerable, simultaneous gap-perturbation to both the nominal plant and controller is characterised. Qualitatively, it is shown that the topology induced by the gap metric on a general class of LPTV systems, is the weakest with respect to which closed-loop stability is a robust property and closedloop performance varies continuously. That is, the gap metric on LPTV systems induces the graph topology. All of the results derived accommodate infinite-dimensional, input/output spaces and importantly, apply to the special case of LPTV, SD control-systems.

Chapter 5: In this chapter a numerical procedure is developed for computing the gap (to any desired accuracy) between LPTV systems that admit stabilisable and detectable, state-space realisations. Much of the chapter is devoted to obtaining finite-dimensional and hence, computationallytractable, necessary and sufficient conditions for the existence of a solution to a related LSI, fullinformation, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem, used to establish bounds on the directed gap. Two-point, boundary-value problems (TPBVPs) are used extensively in this chapter, to characterise particular combinations of finite-horizon integral-operators that arise in this approach.

Chapter 6: A new, compact derivation of state-space formulae is presented in this chapter for H^{∞}, SD synthesis. Numerical issues are discussed and it is shown how to restructure the formulae obtained to give a numerically robust procedure for computing the state-space matrices of the socalled SD-equivalent generalised-plant.

Chapter 7: In this chapter the main contributions of this work are summarised and potential directions for future research identified.

Preliminaries

2.1 Introduction

The purpose of this chapter is to collect preliminary results that will facilitate the understanding of material presented in the main body of this dissertation. The basic structure of the chapter is as follows: fundamental results from functional analysis, which are used throughout, are given in Section 2.2; in Section 2.3, the signal and system spaces considered in this dissertation are defined and fundamental results concerning the stability of closed-loop systems presented; state-space tools for LSI system analysis and synthesis are presented in Section 2.4; finally, fundamental results concerning stability and closed-loop, induced-norm analysis for SD control-systems expressed in linear-fractional form are presented in Section 2.5.

2.2 Functional Analysis

In this section the underlying mathematical structure of all signal and system spaces considered in this dissertation is defined. This includes notation to be used throughout. For further details on the material presented here, the reader is referred to [Kre89b, Bol90, KF75, Sut93, DS88, Rud91, Hal82].

2.2.1 Topological, Metric, Vector (Linear) and Hilbert Spaces

A set 𝒜 is a collection of elements with a common property P. Given another property P_{1}, the set 𝒜_{1} :=\{x ∈ 𝒜 : x has property P_{1}\} is said to be a \underline{subset} of 𝒜. It is standard to denote this by 𝒜_{1} ⊂ 𝒜. Two sets 𝒜_{1} and 𝒜_{2} are said to be equal, denoted 𝒜_{1} = 𝒜_{2}, if 𝒜_{1} ⊂ 𝒜_{2} and 𝒜_{2} ⊂ 𝒜_{1}. The notation x ∈ 𝒜 is used to mean that x is an element of 𝒜 and x ∉ 𝒜 to mean that x is not an element of 𝒜. Given two subsets 𝒜_{1} :={x ∈ 𝒜 : x has property P_{1}} and 𝒜_{2} :=\{x ∈ 𝒜 : x has property P_{2}\} of a set 𝒜, the \underline{union} of 𝒜_{1} and 𝒜_{2} is defined to be

𝒜_{1} ∪ 𝒜_{2} :=\{x ∈ 𝒜 : x has property P_{1} \underline{or} property P_{2}\}

and the \underline{intersection} to be

𝒜_{1} ∩ 𝒜_{2} :=\{x ∈ 𝒜 : x has property P_{1} \underline{and} property P_{2}\}.

The \underline{Cartesian product} of two sets 𝒜_{1} and 𝒜_{2} is defined by

𝒜_{1} × 𝒜_{2} := \{[ \begin{matrix}x \\ y \end{matrix} ] : x ∈ 𝒜_{1}, y ∈ 𝒜_{2}\} .

In the sequel, ℤ denotes the set of integers, ℝ the set of real numbers and ℂ the set of ordered pairs {(σ, ω) : σ, ω ∈ ℝ}. Elements of ℂ are usually called complex numbers and are expressed as z = σ + jω, where j^{2} := − 1. The symbol ℤ^{+} denotes the subset of non-negative integers, ℝ^{+} the subset of non-negative reals, [τ_{1}, τ_{2}) :={t ∈ ℝ : τ_{1} ≤ t < τ_{2}} and [τ_{1}, τ_{2}] :={t ∈ ℝ : τ_{1} ≤ t ≤ τ_{2}}. For notational convenience, given a real number h > 0, the symbol ℍ denotes the interval [0, h). The open unit disc in ℂ is denoted by 𝔻:={z ∈ ℂ : |z| < 1}, 𝕋:={z ∈ ℂ : |z| = 1} denotes the unit circle and 𝔼:=ℂ − 𝔻 denotes the exterior of the open unit disc.

A \underline{mapping} (or function) between two set 𝒜_{1} and 𝒜_{2}, denoted by f : 𝒜_{1} → 𝒜_{2}, relates each element of 𝒜_{1} to a single element of 𝒜_{2}. The mapping f is said to be \underline{injective} (or one-to-one) if for every y ∈ ℱ :={x ∈ 𝒜_{2} : x = f(u), for some u ∈ 𝒜_{1}} there is a unique u ∈ 𝒜_{1} that satisfies y = f(u). The mapping is said to be \underline{surjective} (or onto) if 𝒜_{2} = ℱ and it is called \underline{bijective}, if it is both injective and surjective. If f is injective then f^{−1} : ℱ → 𝒜_{1}, defined to relate each element y ∈ ℱ to the elements u ∈ 𝒜_{1} which satisfy y = f(u), is well defined as a mapping in that it relates each element in ℱ to a single element in 𝒜_{1}. f^{−1} is called the (set theoretic) \underline{inverse} of f and if f is bijective, then f^{−1} is a well-defined mapping on the whole of 𝒜_{2}.

For a given set 𝒜 and mapping f : 𝒜 → ℝ, a number α ∈ ℝ is said to be an \underline{upper bound} of ℱ :=\{x ∈ ℝ : x = f(a), for some a ∈ 𝒜 \} if f(a) ≤ α for all a ∈ 𝒜. An upper bound \underline{α} ∈ ℝ of ℱ is the least upper bound or \underline{supremum}, denoted sup_{a∈𝒜} f(a), in the sense that any number strictly less than \underline{α} is not an upper bound of ℱ, if and only if for all ϵ > 0 there exists an a ∈ 𝒜 such that f(a) > α − ϵ. Similarly, a number β ∈ ℝ is said to be a \underline{lower bound} of ℱ if β ≤ f(a) for all a ∈ 𝒜. A lower bound \overline{β} ∈ ℝ of ℱ is the greatest lower bound or \underline{infimum}, denoted inf_{a∈𝒜} f(a), in the sense that any number strictly greater that \overline{β} is not a lower bound of ℱ, if and only if for all ϵ > 0 there exists an a ∈ 𝒜 such that f(a) < \overline{β} + ϵ.

A \underline{topology} τ on a set 𝒜 is a collection of subsets of 𝒜, called \underline{open sets}, such that the empty set ∅ ∈ τ, 𝒜 ∈ τ and arbitrary unions and finite intersections of sets in τ are in τ. A subset 𝒜_{1} ⊂ 𝒜 is said to be a closed set (with respect to τ) if its complement

𝒜_{1}^{c} :={x ∈ 𝒜 : x ∉ 𝒜_{1}},

is a set in τ. That is, if its complement is open. A topology τ on 𝒜 is said to be \underline{Hausdorff} if for any two elements x, y ∈ 𝒜, there are two open sets 𝒰_{x} and 𝒰_{y} with empty intersection, such that x ∈ 𝒰_{x} and y ∈ 𝒰_{y}. That is, it is possible to distinguish between any two points in a set equipped with a Hausdorff topology. Given two topologies τ_{1} and τ_{2} on a set 𝒜, τ_{1} is said to be \underline{weaker} than τ_{2} if all sets in τ_{1} are contained in τ_{2}.

It is often convenient to specify a topology by giving a basis for it. A \underline{basis} for a topology τ, is a sub-collection σ of subsets of 𝒜 such that σ ⊂ τ and every set in τ is a union of sets in σ. A given collection σ of subsets of 𝒜 is a basis for a topology on 𝒜 if and only if every element of 𝒜 is in some set in σ and for any 𝒜_{1}, 𝒜_{2} ∈ σ, 𝒜_{1} ∩ 𝒜_{2} is a union of sets in σ.

A topological space is defined to be a set equipped with a topology. Given a set 𝒜 and a topology τ on 𝒜, 𝔗(𝒜 , τ) denotes the corresponding topological space. If the topology τ is Hausdorff then 𝔗(𝒜 , τ) is called a Hausdorff space. Given two topological spaces 𝔗(𝒜_{1}, τ_{1}) and 𝔗(𝒜_{2}, τ_{2}), the \underline{product topology} on the Cartesian product 𝒜_{1} × 𝒜_{2} is the topology generated by the basis {𝒰_{1} × 𝒰_{2} : 𝒰_{1} ∈ τ_{1}, 𝒰_{2} ∈ τ_{2}}. Thus, a subset 𝒜 ⊂ 𝒜_{1} × 𝒜_{2} is an open set in the product topology if and only if for every [ \begin{matrix}x \\ y \end{matrix} ]∈ 𝒜 there exist open sets 𝒰_{1} ∈ τ_{1} and 𝒰_{2} ∈ τ_{2} such that

[ \begin{matrix}x \\ y \end{matrix} ]∈ 𝒰_{1} × 𝒰_{2} ⊂ 𝒜.

A subset 𝒩 (x) ⊂ 𝒜 is said to be a \underline{neighbourhood} of a point x in a topological space 𝔗(𝒜 , τ), if there exists an open set 𝒰 ∈ τ such that x ∈ 𝒰 ⊂ 𝒩 (x). A sequence {x_{k}}_{k=0}^{∞} in a topological space 𝔗(𝒜 , τ) is said to \underline{converge} to a point x ∈ 𝔗(𝒜 , τ), denoted x_{k} → x (or lim_{k→∞} x_{k} = x), if for every neighbourhood 𝒩 (x) of x there exists a positive integer K such that x_{k} ∈ 𝒩 (x) for all k ≥ K. If 𝔗(𝒜 , τ) is a Hausdorff space then any convergent sequence converges to a unique point. Given two topological spaces 𝔗_{1}(𝒜_{1}, τ_{1}) and 𝔗_{2}(𝒜_{2}, τ_{2}), a mapping f : 𝒜_{1} → 𝒜_{2} is called \underline{continuous} (with respect to τ_{1} and τ_{2}) if 𝒴 ∈ τ_{2} implies that f^{−1}(𝒴 ) ∈ τ_{1}, where f^{−1}(𝒴 ) denotes the inverse image of 𝒴 under f.

Given a set 𝒜, a mapping d : 𝒜 × 𝒜 → ℝ^{+} is called a \underline{metric} on 𝒜 if:

(M1) for x, y ∈ 𝒜, d(x, y) = 0 ⇔ x = y; (M2) d(x, y) = d(y, x) for all x, y ∈ 𝒜;

(M3) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ 𝒜.

A \underline{metric space} is defined to be a set equipped with a metric. Given a set 𝒜 and a metric d on 𝒜, 𝒳 (𝒜 , d) denotes the corresponding metric space. For a metric space 𝒳 (𝒜 , d) the open balls

ℬ_{d}(x, r) :={y ∈ 𝒳 (𝒜 , d) : d(x, y) < r} (x ∈ 𝒳 (𝒜 , d), r > 0),

form a basis for a topology on 𝒳 (𝒜 , d). This topology is said to be induced by the metric d and is called the \underline{metric topology}. As such, every metric space can be considered to be a topological space. In fact, every metric space equipped with the metric topology is Hausdorff. In a metric space, convergence (with respect to the metric topology) of a sequence {x_{k}}_{k=0}^{∞} ⊂ 𝒳 (𝒜 , d) can be expressed in the (probably) more familiar terms: x_{k} → x if and only if for all ϵ > 0 there exists a positive integer K such that d(x, x_{k}) < ϵ for all k ≥ K. Similarly, a function f : 𝒜_{1} → 𝒜_{2}, where 𝒜_{1} is equipped with the metric d_{1} and 𝒜_{2} with d_{2}, is continuous if and only if for every x ∈ 𝒜_{1} and ϵ > 0 there exists a δ > 0 such that d_{2}(f(x), f(u)) < ϵ for all u ∈ 𝒜_{1} that satisfy d_{1}(x, u) < δ. A sequence \{x_{k}\}_{k=0}^{∞} ⊂ 𝒳 (𝒜 , d) is said to be \underline{Cauchy}, if for for every ϵ > 0 there exists a positive integer N such that d(x_{m}, x_{n}) < ϵ for all m, n > N. Note that every convergent sequence in a metric space is Cauchy. If every Cauchy sequence in a metric space 𝒳 (𝒜 , d) is convergent to a point in 𝒳 (𝒜 , d), then 𝒳 (𝒜 , d) is called a \underline{complete} metric space.

A \underline{vector (or linear) space} 𝒱 over 𝔽 (where 𝔽 denotes either ℝ or ℂ) is a set of elements called \underline{vectors}, for which two operations called \underline{vector addition} and \underline{scalar multiplication} are defined with the following algebraic properties:

(V1) For all vectors x, y and z in 𝒱 there corresponds a vector x+y ∈ 𝒱 , where the correspondence is such that: x + y = y + x and x + (y + z) = (x + y) + z; 𝒱 contains a unique zero vector 0 such that x + 0 = x for all x ∈ 𝒱 ; and for every x ∈ 𝒱 there corresponds a unique vector −x ∈ 𝒱 such that x − x:=x + (−x) = 0.

(V2) For each x ∈ 𝒱 and α ∈ 𝔽, there is a corresponding vector αx ∈ 𝒱 , where the correspondence is such that: 1x = x, where 1 denotes the unit in 𝔽; α(βx) = (αβ)x (where β ∈ 𝔽); and for all x, y ∈ 𝒱 , α(x + y) = αx + αy and (α + β)x = αx + βx.

A subset 𝒱_{1} ⊂ 𝒱 is called a (linear) \underline{subspace} of 𝒱 if it is closed under the vector addition and scalar multiplication operations defined on 𝒱 . A norm on a vector space 𝒱 , denoted by ∥ · ∥_{𝒱} , is a mapping from 𝒱 to ℝ^{+}, which for all x, y ∈ 𝒱 satisfies:

(N1) ∥x∥_{𝒱} = 0 ⇔ x = 0; (N2) ∥αx∥_{𝒱} = |α| · ∥x∥_{𝒱} for all α ∈ 𝔽;

(N3) ∥x + y∥_{𝒱} ≤ ∥x∥_{𝒱} + ∥y∥_{𝒱} .

Property (N3) is often called the \underline{triangle inequality}. A vector space 𝒱 equipped with a norm ∥·∥_{𝒱} , is called a \underline{normed space}. The norm on a normed space 𝒱 induces the following metric:

d_{𝒱} (x, y) :=∥x − y∥_{𝒱} for all x, y ∈ 𝒱 .

If 𝒱 is complete with respect to d_{𝒱} then it is called a \underline{Banach} space. In the sequel the Computer Modern Caligraphic font (B, W for example) is used to identify a Banach space. A subspace 𝒱_{1} of a normed space 𝒱 is closed (with respect to the topology induced by d_{𝒱} , also known as the \underline{norm topology}) if every sequence in 𝒱_{1} that converges in 𝒱 , converges to a point in 𝒱_{1}. A closed subspace of a Banach space is therefore complete.

An \underline{inner product} on a vector space 𝒱 over 𝔽, denoted by ⟨·, ·⟩_{𝒱} , is a mapping from the Cartesian product 𝒱 × 𝒱 to 𝔽 such that for all x, y, z ∈ 𝒱 and each μ ∈ 𝔽, the following hold:

(IP1) ⟨x, y⟩_{𝒱} = \overline{⟨y, x⟩}_{𝒱} ;

(IP2) ⟨μx, y⟩_{𝒱} = μ⟨x, y⟩_{𝒱} ;

(IP3) ⟨x + y, z⟩_{𝒱} = ⟨x, z⟩_{𝒱} + ⟨y, z⟩_{𝒱} ;

(IP4) ⟨x, x⟩_{𝒱} ≥ 0 and ⟨x, x⟩_{𝒱} = 0 ⇔ x = 0,

where the over-bar in (IP1) denotes complex conjugation. An inner product induces the following norm on 𝒱 : ∥x∥_{𝒱} :=\sqrt{⟨x, x⟩_{𝒱}} for all x ∈ 𝒱 . A vector space equipped with an inner product is called an inner-product space and can be considered a normed space (and hence, a metric and topological space). If an inner-product space is complete with respect to the metric induced by the inner product (via the norm defined above), then it is called a \underline{Hilbert} space. In the sequel, the Euler Script font (ℋ, 𝒰 for example) is used to identify a Hilbert space, unless it is a special Hilbert space such as those define in Sub-Section 2.3.1.

Let ℋ be a Hilbert space and 𝒰 and 𝒴 be subspaces of ℋ. The \underline{orthogonal complement} of 𝒰 in ℋ is defined to be the subspace

ℋ ⊖ 𝒰:={x ∈ ℋ : ⟨u, x⟩_{ℋ} = 0 for all u ∈ 𝒰}.

When it is clear from context what ℋ is, the short hand 𝒰^{⊥} is used to denote ℋ ⊖ 𝒰. If 𝒰 is a closed subspace then each x ∈ ℋ can be uniquely decomposed as x = u + x_{1} where u ∈ 𝒰 and x_{1} ∈ 𝒰^{⊥}. Bearing this in mind, the mapping of each x ∈ ℋ to the corresponding u ∈ 𝒰 is called an \underline{orthogonal projection} and is denoted by the symbol Π_{𝒰}. In the sequel, the symbol ⊕ denotes the orthogonal direct sum, by which 𝒴 ⊕ 𝒰 is identified with the Cartesian product 𝒴 × 𝒰 and has inner product defined for all y_{1}, y_{2} ∈ 𝒴 and u_{1}, u_{2} ∈ 𝒰 by

⟨[ \begin{matrix}y_{1} \\ u_{1} \end{matrix} ], [ \begin{matrix}y_{2} \\ u_{2} \end{matrix} ]⟩_{𝒴⊕𝒰} :=⟨y_{1}, y_{2}⟩_{𝒴} + ⟨u_{1}, u_{2}⟩_{𝒰} .

The notation ℋ^{n} is used to denote the Hilbert space ℋ^{n} := \underbrace{ℋ ⊕ ℋ ⊕ ⋯ ⊕ ℋ}_{n times}.

A set of vectors {e_{k}}_{k∈𝒦} ⊂ ℋ (where 𝒦 is a (possibly uncountable) index set) is said to be \underline{orthonormal} if ∥e_{k}∥_{ℋ} = 1 and ⟨e_{k}, e_{l}⟩_{ℋ} = 0 for k ≠ l in 𝒦 . If the only vector in ℋ orthogonal to all e_{k}’s is the zero vector, then such a set is called a \underline{complete, orthonormal basis} for ℋ. Given an orthonormal basis {e_{k}}_{k∈𝒦} for a Hilbert space ℋ, any vector x ∈ ℋ can be expressed as

x = ∑_{k∈𝒦} ⟨x, e_{k}⟩_{ℋ} e_{k}.

The set {⟨x, e_{k}⟩_{ℋ} }_{k∈𝒦} uniquely (with respect to the basis {e_{k}}_{k∈𝒦} ) represents x. Given an x and y in ℋ with expansions ∑_{k∈𝒦} α_{k}e_{k} and ∑_{k∈𝒦} β_{k}e_{k} respectively, then

⟨x, y⟩_{ℋ} = ∑_{k∈𝒦} α_{k}\overline{β}_{k}.

Furthermore, if {α_{k}}_{k∈𝒦} is a subset of ℂ such that ∑_{k∈𝒦} |α_{k}|^{2} < ∞, then ∑_{k∈𝒦} α_{k}e_{k} is in ℋ. In the sequel, any non-specific, Hilbert space considered is assumed to have countable basis. Such Hilbert spaces are said to be \underline{separable}.

2.2.2 Operator Theory

Given two vector spaces 𝒱 and 𝒴 over 𝔽 (where 𝔽 denotes either ℝ or ℂ), a mapping P : 𝒱 → 𝒴 is called an \underline{operator} and it is said to be \underline{linear} if for every v_{1}, v_{2} ∈ 𝒱 and α, β ∈ 𝔽,

P (αv_{1} + βv_{1}) = αP v_{1} + βP v_{2}.

All operators considered in the sequel are linear and almost all are defined on Hilbert spaces. Let 𝒰 and 𝒴 be two Hilbert spaces and P : 𝒟_{P} ⊂ 𝒰 → 𝒴 be a linear operator where

𝒟_{P} :={u ∈ 𝒰 : P u ∈ 𝒴}

is called the \underline{domain} of P, which is a subspace of 𝒰. If ℋ is a subspace of 𝒟_{P} then P |_{ℋ} : ℋ → 𝒴 denotes the \underline{restriction} of P to ℋ and maps vectors from ℋ to 𝒴 as P does. The \underline{range} of P is defined to be

ℛ_{P} :={P u : u ∈ 𝒟_{P} }

and the \underline{kernel} to be

𝒦_{P} :={u ∈ 𝒟_{P} : P u = 0}.

A linear operator P : 𝒟_{P} ⊂ 𝒰 → 𝒴 is said to be \underline{bounded} if the induced norm

∥P ∥:= sup_{\begin{matrix}u∈𝒟_{P} \\ u≠0 \end{matrix}} \frac{∥P u∥_{𝒴}}{∥u∥_{𝒰}} < ∞.

B_{𝒰→𝒴} is used to denote the set of all bounded, linear operators P : 𝒰 → 𝒴 and it is a Banach space when equipped with the induced norm.

The graph of a linear operator P : 𝒟_{P} ⊂ 𝒰 → 𝒴 is defined to be the set of all bounded input-output pairs

𝒢_{P} :=[ \begin{matrix}I \\ P \end{matrix} ]𝒟_{P} ⊂ 𝒰 ⊕ 𝒴,

which since P is linear, is a subspace of 𝒰 ⊕ 𝒴. For notational convenience the inverse graph is defined to be 𝒢P^{′} :=[ \begin{matrix}0 & I \\ I & 0 \end{matrix} ]𝒢_{P}. Note that any linear subspace 𝒢 ⊂ 𝒰 ⊕ 𝒴 corresponds to the graph

of a linear operator if and only if [ \begin{matrix}0 \\ y \end{matrix} ]∈ 𝒢 ⇒ y = 0. The graph is a very useful characterisation of an operator and is used extensively in the sequel. An operator P is said to be \underline{closed} if its graph is closed with respect to the product topology on 𝒰⊕𝒴. A useful result is the Closed Graph Theorem, which states that a closed, linear operator P : 𝒟_{P} ⊂ 𝒰 → 𝒴 is bounded if 𝒟_{P} is closed with respect to the norm topology on 𝒰. Furthermore, if it is known that a linear operator P : 𝒟_{P} ⊂ 𝒰 → 𝒴 is bounded and that 𝒟_{P} is closed, then 𝒢_{P} is closed with respect to the product topology on 𝒰 ⊕ 𝒴.

Let 𝒢 and ℱ be two (linear) subspaces of a Hilbert space ℋ that are closed with respect to the norm topology on ℋ. 𝒢 and ℱ are said to induce a \underline{co-ordinatisation} of ℋ if 𝒢 ∩ ℱ = \{0\} and 𝒢 + ℱ = ℋ. Given a co-ordinatisation {𝒢, ℱ} of ℋ, any x ∈ ℋ can be uniquely decomposed as the sum x = g + f where g ∈ 𝒢 and f ∈ ℱ (cf. [DGS93]). With this decomposition in mind, the operator Π_{𝒢∥ℱ} : ℋ → 𝒢 defined by

Π_{𝒢∥ℱ} := x ↦ g,

is called the \underline{parallel projection} onto 𝒢 along ℱ. Correspondingly, Π_{ℱ∥𝒢} := x ↦ f is called the parallel projection onto ℱ along 𝒢. Since 𝒢 and ℱ are closed, linear subspaces, it follows that the graphs of the parallel projection operators are linear and closed. Hence, by the Closed Graph Theorem, the parallel projections are bounded. Note also, that ∥Π_{·∥·}∥ ≥ 1.

Given a bounded, linear operator P ∈ B_{𝒰→𝒴} , there exists a unique operator P^{∗} ∈ B_{𝒴→𝒰} such that for all u ∈ 𝒰 and y ∈ 𝒴

⟨P u, y⟩_{𝒴} = ⟨u, P^{∗}y⟩_{𝒰} .

The operator P^{∗} is called the \underline{adjoint} operator and ∥P ∥^{2} = ∥P^{∗}∥^{2} = ∥P^{∗}P ∥. A useful result is that 𝒦_{P} = (ℛ_{P^{∗}})^{⊥}. A linear operator P ∈ B_{𝒰→𝒰} is called self-adjoint if P = P^{∗}. Such an operator is said to be \underline{positive definite} (respectively \underline{positive semidefinite}), denoted P > 0 (respectively P ≥ 0), if there exists a real number α > 0 such that for all u ∈ 𝒰, ⟨u, P u⟩_{𝒰} ≥ α⟨u, u⟩_{𝒰} (respectively ⟨u, P u⟩_{𝒰} ≥ 0). Similarly, a self-adjoint operator P ∈ B_{𝒰→𝒴} is said to be \underline{negative definite} (respectively \underline{negative semidefinite}), denoted P < 0 (respectively P ≤ 0), if −P > 0 (respectively −P ≥ 0). A self-adjoint operator P is said to be \underline{idempotent} if P^{2} = P.

An operator P ∈ B_{𝒰→𝒴} , where 𝒰 and 𝒴 are Hilbert spaces, is said to be an \underline{isometry} if ⟨P u, P u⟩_{𝒴} = ⟨u, u⟩_{𝒰} for all u ∈ 𝒰 and \underline{unitary} if ℛ_{P} = 𝒴 and P is an isometry. P ∈ B_{𝒰→𝒴} is called a \underline{co-isometry} if ℛ_{P} = 𝒴 and P^{∗} is an isometry. Two Hilbert spaces 𝒰 and 𝒴 are said to be \underline{isomorphic} if there exists a bijective operator Y ∈ B_{𝒰→𝒴} , called an \underline{isomorphism}, such that

⟨u_{1}, u_{2}⟩_{𝒰} = ⟨Y u_{1}, Y u_{2}⟩_{𝒴}

for all u_{1}, u_{2} ∈ 𝒰. Two operators P_{1} and P_{2} defined on Hilbert spaces are said to be \underline{equivalent} if there exists an isomorphism Y such that P_{1}Y = Y P_{2}.

Given a Hilbert spaces 𝒰 and an operator P : 𝒟_{P} ⊂ 𝒰 → 𝒰 that is injective (or equivalently 𝒦_{P} = {0}), then P has a well defined, set-theoretic inverse defined on ℛ_{P}. The inverse operator is denoted by P^{−1} and the following equalities hold: P^{−1}P = I|_{𝒟_{P}} and P P^{−1} = I|_{ℛ_{P}} , where I denotes the identity map (that is, the operator which maps each element of a vector space to itself). It follows by the Open Mapping Theorem (cf. [Bol90, pg 80] for example), that if P ∈ B_{𝒰→𝒰} is bijective then P^{−1} ∈ B_{𝒰→𝒰} . In the sequel, an operator P ∈ B_{𝒰→𝒰} is said to be (boundedly) \underline{invertible} if P^{−1} ∈ B_{𝒰→𝒰} . The \underline{spectrum} of an operator P ∈ B_{𝒰→𝒰} is defined to be the set

spec(P) :={λ ∈ ℂ : (λI − P) is not invertible in B_{𝒰→𝒰} }

and the \underline{point spectrum} to be the subset of spec(P) for which (λI − P) is not injective. The point spectrum is a set of discrete points in ℂ and its elements are often called the \underline{eigenvalues} of P. If 𝒰 or 𝒴 is finite-dimensional (in the sense that there is a finitely-countable, complete basis for them), then all λ ∈ spec(P) are eigenvalues. The spectral radius of a linear operator P ∈ B_{𝒰→𝒴} is defined by

rad(P) := sup_{λ∈spec(P)} |λ| = lim_{n→∞} ∥P^{n}∥^{\frac{1}{n}} ≤ ∥P ∥.

If P is self-adjoint, then its spectrum is real and rad(P) = ∥P ∥.

A bounded, linear operator P ∈ B_{𝒰→𝒴} is said to be \underline{compact} if for every bounded sequence \{u_{k}\}_{k=0}^{∞} in 𝒰 there is a subsequence {u_{k_{n}}}_{n=0}^{∞}, such that P u_{k_{n}} converges in 𝒴 when equipped with the norm topology. If 𝒰 is finite-dimensional then P is compact. A compact operator P ∈ B_{𝒰→𝒴} has the following important properties:

(C1) for any R ∈ B_{𝒴→ℋ} , RP ∈ B_{𝒰→ℋ} is compact and for any R ∈ B_{ℋ→𝒰} , P R ∈ B_{ℋ→𝒴} is compact;

(C2) the eigenvalues of P constitute a countable set;

(C3) the adjoint operator P^{∗} is compact.

Another important property of compact operators is that every non-zero spectral value of a compact operator is an eigenvalue and the only possible accumulation point^{1} of the spectrum is 0.

A number σ ∈ ℝ^{+} is said to be a \underline{singular value} of a bounded, linear operator P ∈ B_{𝒰→𝒴} if σ^{2} is an eigenvalue of P^{∗}P ∈ B_{𝒰→𝒰} . The set of singular values of an operator P ∈ B_{𝒰→𝒴} is denoted by

σ(P) :={σ ∈ ℝ^{+} : σ^{2} is an eigenvalue of P^{∗}P }.

If P ∈ B_{𝒰→𝒴} is compact then so is P^{∗}P and hence, the spectrum of P^{∗}P is completely characterised by the countable set of singular values σ(P). Then since P^{∗}P is self-adjoint, it follows that given a compact operator P ∈ B_{𝒰→𝒴}

∥P ∥ = \sqrt{∥P^{∗}P ∥} = \sqrt{rad(P^{∗}P)} = \overline{σ}\{P \},

where

\overline{σ}\{P \}:= max_{σ∈σ(P)} σ

is called the \underline{maximum singular-value}. The \underline{minimum singular-value} of a compact, bounded, linear operator is defined similarly by

\underline{σ}\{P \}:= min_{σ∈σ(P)} σ.

1A point x in a metric space 𝒳 (𝒜 , d) is said to be an accumulation point of a subset ℳ ⊂ 𝒳 (𝒜 , d) if every neighbourhood of x in the metric topology contains at least one point in ℳ that is distinct from x.

2.2.3 Analytic Functions

In this short sub-section the notion of an analytic function is defined and a few properties of analytic functions assembled. For further details on this material the reader is referred to [Ahl79, Rud86, Hof62].

A norm can be defined on all sets considered in this sub-section and as such convergence and continuity (although not explicitly stated) is with respect to the norm topology on the sets concerned. Let 𝕌 be a connected,^{2} open subset of the complex plane and consider a function f : 𝕌 → W, where W is a Banach space. If for a given point z_{0} ∈ 𝕌 there exists an f_{0} ∈ W such that f(z_{i}) converges to (the same) f_{0} for every sequence {z_{i}}_{i=0}^{∞} ⊂ 𝕌 that converges to z_{0}, then the limit below is said to exist

lim_{z→z_{0}} f(z) :=f_{0}.

If at a point z_{0} ∈ 𝕌,

lim_{z→z_{0}} \frac{f(z) − f(z_{0})}{z − z_{0}}

exists, then f(z) is said to be \underline{differentiable} at z_{0}. A function f : 𝕌 → W is said to be analytic in a connected, open subset 𝕌_{1} ⊂ 𝕌 if f is differentiable at all points in 𝕌_{1}. If f is analytic in 𝕌, then it is also continuous on 𝕌 and can be expressed in terms of a power series in the following sense: for every open ball ℬ_{ℂ}(z_{0}, r) :={z ∈ ℂ : |z − z_{0}| < r} ⊂ 𝕌, there corresponds a power series

∑_{n=0}^{∞} c_{n}(z − z_{0})^{n} ({c_{n}}_{n=0}^{∞} ⊂ W),

which converges^{3} to f(z) for all z ∈ ℬ_{ℂ}(z_{0}, r). This property is often used to define when a function is analytic.

Using Cauchy’s Integral Formula, it can be shown that any function f : 𝕌 → W that is analytic in 𝕌 satisfies the mean value property

f(z) = \frac{1}{2π} ∫_{0}^{2π} f(z + re^{jθ}) dθ,

for all real r > 0 such that ℬ_{ℂ}(z, r) ⊂ 𝕌. Using this and the triangle inequality, it follows that for any function f : 𝕌 → W which is analytic in 𝕌,

∥f(z)∥_{W} ≤ \frac{1}{2π} ∫_{0}^{2π} ∥f(z + re^{jθ})∥_{W} dθ.

2A subset 𝕌 ⊂ ℂ is said to be connected if 𝕌 and ∅ are the only subsets of 𝕌 that are both open and closed.

3Convergence of the series is taken to mean lim_{N→∞} ∑_{n=0}^{N} c_{n}(z − z_{0})^{n} exists.

Furthermore, since for such a function the norm ∥f(z)∥_{W} is continuous on any open subset of 𝕌, it follows that ∥f(z)∥_{W} satisfies a \underline{maximum principle} on any connected, open subset of 𝕌_{1} ⊂ 𝕌 in the following sense: ∥f(z)∥_{W} has no maximum on 𝕌_{1} or it is constant on 𝕌_{1} (cf. [Ahl79, pg. 135] for example).

2.3 Signals and Systems Theory

In this section the signal and system spaces considered in this dissertation are defined. Much of the material presented in this section is taken from the books [Wil71, DV75, FS82, FF90, SNF70].

2.3.1 Signals

In systems theory a \underline{signal} space is usually defined to be a vector space of functions mapping a time interval of definition (for example ℝ^{+} or ℤ^{+}) to a Banach space. A \underline{system} is then defined to be an operator mapping between signal spaces. Given a Banach space B and a time interval of definition 𝕊, let 𝒴_{𝕊}(B) denote the vector space of B-valued functions defined on 𝕊. That is, 𝒴_{𝕊}(B) :=\{f : f : 𝕊 → B\}. Functions in 𝒴_{ℤ^{+}}(B) are often called \underline{discrete-time} signals and functions in 𝒴_{ℝ^{+}}(B) \underline{continuous-time} signals. As it stands, 𝒴_{𝕊}(B) has little structure and would be difficult to work with. So it is standard to introduce a norm on 𝒴_{𝕊}(B), which gives rise to a normed (linear) subspace W :={f ∈ 𝒴_{𝕊}(B) : ∥f∥_{W} < ∞}. Often, the norm is selected so that W is a Banach space or Hilbert space. Given such a W, it is generally useful to define the extension of W. Extended spaces play an important role in systems theory and are particularly important in the study of causal operators and the well-posedness of closed-loop systems [Wil71]. Let P_{τ} denote the projection defined for f ∈ 𝒴_{𝕊}(B) by

(P_{τ} f)(t) := \{ \begin{matrix}f(t) & for t ≤ τ, t ∈ 𝕊 \\ 0 & otherwise \end{matrix} .

Then for a Banach space W ⊂ 𝒴_{𝕊}(B), the \underline{extended space} W^{e} is defined to be

W^{e} :={f ∈ 𝒴_{𝕊}(B) : P_{τ} f ∈ W for all finite τ ∈ 𝕊}.

It is important to note that although W^{e} is a vector space it is not normed. In this dissertation, attention is restricted to the continuous-time signal-space L_{ℝ^{+}}^{2}(ℋ) and the discrete-time signalspace ℓ_{ℤ^{+}}^{2}(ℋ) (and their respective extensions) defined below.

Let ℋ be a Hilbert space. The Hilbert space of ℋ-valued functions f ∈ 𝒴_{ℝ^{+}}(ℋ) that satisfy

∥f∥_{L_{ℝ^{+}}^{2}(ℋ)}^{2} := ∫_{0}^{∞} ⟨f(t), f(t)⟩_{ℋ} dt < ∞,

is denoted by L_{ℝ^{+}}^{2}(ℋ). The inner-product on L_{ℝ^{+}}^{2}(ℋ) is defined to be

⟨f, g⟩_{L_{ℝ^{+}}^{2}(ℋ)} := ∫_{0}^{∞} ⟨f(t), g(t)⟩_{ℋ} dt,

which induces the norm used in the definition. The symbol U_{τ} denotes the unilateral shift by τ on 𝒴_{ℝ^{+}}(ℋ) and for all f ∈ 𝒴_{ℝ^{+}}(ℋ) is defined by

(U_{τ} f)(t) := \{ \begin{matrix}f(t − τ) & for t ≥ τ \\ 0 & for 0 ≤ t < τ \end{matrix} ,

which is an isometry on L_{ℝ^{+}}^{2}(ℋ). The Hilbert space L_{ℝ^{+}}^{2}(ℋ) is a continuous-time signal-space and corresponds to continuous-time signals with finite energy. L_{ℝ^{+}}^{2,e}(ℋ) is used to denoted the associated extended space.

The Hilbert space ℓ_{ℤ^{+}}^{2}(ℋ) consists of all ℋ-valued functions f ∈ 𝒴_{ℤ^{+}}(ℋ) that satisfy

∥f∥_{ℓ_{ℤ^{+}}^{2}(ℋ)}^{2} := ∑_{k=0}^{∞} ⟨f_{k}, f_{k}⟩_{ℋ} < ∞.

The inner-product on ℓ_{ℤ^{+}}^{2}(ℋ) is defined to be

⟨f, g⟩_{ℓ_{ℤ^{+}}^{2}(ℋ)} := ∑_{k=0}^{∞} ⟨f_{k}, g_{k}⟩_{ℋ} ,

which induces the norm used in the definition. The symbol S denotes the unilateral shift on 𝒴_{ℤ^{+}}(ℋ) and for all f ∈ 𝒴_{ℤ^{+}}(ℋ) is defined by

(Sf)_{k} := \{ \begin{matrix}f_{(k−1)} & for k ≥ 1 \\ 0 & for k = 0 \end{matrix} ,

which is an isometry on ℓ_{ℤ^{+}}^{2}(ℋ). The Hilbert space ℓ_{ℤ^{+}}^{2}(ℋ) is a discrete-time signal-space and corresponds to discrete-time signals with finite energy. ℓ_{ℤ^{+}}^{2,e}(ℋ) is used to denote the associated extended space. Lower-case Euler characters (for example f, x) are used throughout to denote discrete-time signals.

In SD control-systems the following elementary operators are used to map continuous-time signals to discrete-time (sampled) signals and vice-versa: given an arbitrary Hilbert space ℋ and a real number h > 0, the ideal, periodic (period h) sampling-device (or analogue-to-digital converter) A_{D} : 𝒴_{ℝ^{+}}(ℋ) → 𝒴_{ℤ^{+}}(ℋ), is defined by

f_{k} :=(A_{D}f)_{k} :=f(kh);

the zeroth-order hold-device (or digital-to-analogue converter) D_{A} : 𝒴_{ℤ^{+}}(ℋ) → 𝒴_{ℝ^{+}}(ℋ), is defined by

f(t) :=(D_{A}f)(t) :=f_{k} for kh ≤ t < (k + 1)h.

In systems theory it is often convenient to work with spaces that are isomorphic to the original time-domain signal-space concerned. To this end, let L_{𝕋}^{2}(ℋ) denote the Hilbert space of functions f : 𝕋 → ℋ that satisfy

∥f∥_{L_{𝕋}^{2}(ℋ)}^{2} :=\frac{1}{2π} ∫_{0}^{2π} ⟨f(e^{jω}), f(e^{jω})⟩_{ℋ} dω < ∞.

The inner-product that induces this norm is

⟨f, g⟩_{L_{𝕋}^{2}(ℋ)} :=\frac{1}{2π} ∫_{0}^{2π} ⟨f(e^{jω}), g(e^{jω})⟩_{ℋ} dω

and is it taken to be the inner-product on L_{𝕋}^{2}(ℋ). Note that any f ∈ L_{𝕋}^{2}(ℋ) has Fourier series expansion

f(e^{jω}) = ∑_{k=−∞}^{∞} e^{jkω}f_{k},

where each f_{k} ∈ ℋ and ⟨f, f⟩_{L_{𝕋}^{2}(ℋ)} = ∑_{k=−∞}^{∞}⟨f_{k}, f_{k}⟩_{ℋ} , the set {f_{k}}_{k=0}^{∞} being the set of Fourier coefficients. An important subspace of L_{𝕋}^{2}(ℋ) is denoted by L_{𝕋}^{2+}(ℋ) and consists of all functions in L_{𝕋}^{2}(ℋ) for which f_{k} = 0 when k < 0. Given any f ∈ L_{𝕋}^{2+}(ℋ), ∑_{k=0}^{∞}⟨f_{k}, f_{k}⟩_{ℋ} < ∞ by definition and consequently, it is possible to associate with f the function φ : 𝔻 → ℋ given by

φ(λ) = ∑_{k=0}^{∞} λ^{k}f_{k}.  (2.1)

Now, the Hardy space H_{𝔻}^{2}(ℋ) is defined to be the set of all ℋ-valued functions φ : 𝔻 → ℋ that are analytic in 𝔻 and satisfy the following condition:

∫_{0}^{2π} ⟨φ(re^{jω}), φ(re^{jω})⟩_{ℋ} dω (0 ≤ r < 1)  (2.2)

has finite bound independent of r. Since each φ ∈ H_{𝔻}^{2}(ℋ) is analytic, it can be expressed as a power series of the form given in equation (2.1). Given such a representation, it follows that the condition associated with equation (2.2) implies ∑_{k=0}^{∞}⟨f_{k}, f_{k}⟩_{ℋ} < ∞. Hence, every function φ(λ) ∈ H_{𝔻}^{2}(ℋ) can be associated with a function f(ω) = φ(e^{jω}) ∈ L_{𝕋}^{2+}(ℋ) and in this way, H_{𝔻}^{2}(ℋ) inherits the Hilbert-space structure of L_{𝕋}^{2+}(ℋ), embedding it in L_{𝕋}^{2}(ℋ) as a subspace.

Proposition 2.1 [SNF70, pp. 184-185] The Hilbert space ℓ_{ℤ^{+}}^{2}(ℋ) is isomorphic to H_{𝔻}^{2}(ℋ) via the Z-Transform isomorphism defined by

φ(λ) = (Zf)(λ) := ∑_{k=0}^{∞} λ^{k}f_{k},

for all λ ∈ 𝔻 and f:=k ↦ f_{k} ∈ ℓ_{ℤ^{+}}^{2}(ℋ). Z^{−1} denotes the inverse Z-Transform isomorphism.^{4} By definition of the Z-Transform isomorphism, it follows that the unilateral shift acts as multiplication by λ on H_{𝔻}^{2}(ℋ). That is, (Sφ)(λ) = λφ(λ) for φ ∈ H_{𝔻}^{2}(ℋ).

4λ = z^{−1} in the usual engineering definition of the Z-Transform.

Recall that given a real number h > 0, the symbol ℍ denotes the interval [0, h) ⊂ ℝ^{+}. L_{ℍ}^{2}(ℋ) is used to denoted the Hilbert space of ℋ-valued functions f : ℍ → ℋ that satisfy

∥f∥_{L_{ℍ}^{2}(ℋ)} := ∫_{0}^{h} ⟨f(t), f(t)⟩_{ℋ} dt < ∞.

The inner product on L_{ℍ}^{2}(ℋ) is defined by

⟨f, g⟩_{L_{ℍ}^{2}(ℋ)} := ∫_{0}^{h} ⟨f(t), g(t)⟩_{ℋ} dt,

which induces the norm used in the definition. L_{ℍ}^{2}(ℋ) is really the truncation of L_{ℝ^{+}}^{2}(ℋ) to the interval ℍ and henceforth, the symbol “^{⊸}” above a character is used to denote a vector in L_{ℍ}^{2}(ℋ) or a sequence of such vectors.

The following operator plays a pivotal role throughout this dissertation: given a real number h > 0, the bijective operator W : L_{ℝ^{+}}^{2,e}(ℋ) → ℓ_{ℤ^{+}}^{2,e}(L_{ℍ}^{2}(ℋ)) is defined by

f^{⊸}_{k}(θ) :=(Wf)_{k} = f(kh + θ) (θ ∈ ℍ).

It acts as an isomorphism between the two Hilbert spaces L_{ℝ^{+}}^{2}(ℋ) and ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(ℋ)) and is referred to as the \underline{time-lifting isomorphism}. Its inverse is denoted by W^{−1}. The importance of this operator lies in the fact that it intertwines the continuous-time, unilateral shift by h and the discrete-time, unilateral shift. That is, S^{k}W = WU_{kh} for all k ∈ ℤ^{+}. In this way, a shift by h in L_{ℝ^{+}}^{2,e}(ℋ) corresponds to a shift in ℓ_{ℤ^{+}}^{2,e}(L_{ℍ}^{2}(ℋ)) by a single time-step.

Let 𝒰 and 𝒴 be arbitrary Hilbert spaces. In the sequel, bounded operators that map the discretetime signal space ℓ_{ℤ^{+}}^{2}(𝒰) to ℓ_{ℤ^{+}}^{2}(𝒴) and commute with the unilateral shift, play a significant role. Related to these operators, is the Hardy space H_{𝔻}^{∞}(B_{𝒰→𝒴} ) of B_{𝒰→𝒴} -valued^{5} functions Φ : 𝔻 → B_{𝒰→𝒴} that are bounded and analytic in the open, unit disc 𝔻. Given a function Φ ∈ H_{𝔻}^{∞}(B_{𝒰→𝒴} ), boundary values can be defined by application of Fatou’s Theorem to Φ(λ)u for u ∈ 𝒰 (cf. [FF90, pg. 234]). The resultant boundary values Φ(e^{jω}) can be considered an extension of Φ(λ) to 𝕋 and are essentially bounded on 𝕋. ∥ · ∥_{∞} denotes the norm on H_{𝔻}^{∞}(B_{𝒰→𝒴} ), which given Φ ∈ H_{𝔻}^{∞}(B_{𝒰→𝒴} ) is defined by

∥Φ∥_{∞} := sup_{λ∈𝔻}∥Φ(λ)∥ = ess sup_{ω∈[0,2π)}∥Φ(e^{jω})∥,

5Recall that B_{𝒰→𝒴} denotes the Banach space of all bounded, linear operators mapping 𝒰 to 𝒴.

where the second equality follows from the fact that ∥Φ(λ)∥ satisfies a maximum principle on 𝔻 (cf. Sub-Section 2.2.3). An essentially-bounded, B_{𝒰→𝒴} -valued function Φ_{∂} : 𝕋 → B_{𝒰→𝒴} is said to be rigid if Φ_{∂}(e^{jω}) is an isometry almost everywhere on 𝕋. A function Φ : 𝔻 → B_{𝒰→𝒴} is said to be inner if Φ ∈ H_{𝔻}^{∞}(B_{𝒰→𝒴} ) and the function associated with its boundary values is rigid. Corresponding to each Φ ∈ H_{𝔻}^{∞}(B_{𝒰→𝒴} ) is a multiplication operator M_{Φ} : H_{𝔻}^{2}(𝒰) → H_{𝔻}^{2}(𝒴), defined by (M_{Φ}φ)(λ) :=Φ(λ)φ(λ) for all φ ∈ H_{𝔻}^{2}(𝒰) and λ ∈ 𝔻.

Recall that ℓ_{ℤ^{+}}^{2}(𝒰) and H_{𝔻}^{2}(𝒰) are isomorphic via the Z-Transform isomorphism and hence, that a linear operator F : ℓ_{ℤ^{+}}^{2}(𝒰) → ℓ_{ℤ^{+}}^{2}(𝒴) is equivalent to a linear operator ZFZ^{−1} : H_{𝔻}^{2}(𝒰) → H_{𝔻}^{2}(𝒰) (and vice-versa). Interestingly, if F is bounded and commutes with the unilateral shift, then ZFZ^{−1} can be expressed as a multiplication operator with symbol in H_{𝔻}^{∞}(B_{𝒰→𝒴} ), as is summarised in the following proposition:

Proposition 2.2 [FF90, pg. 235] Given a bounded, linear operator F : ℓ_{ℤ^{+}}^{2}(𝒰) → ℓ_{ℤ^{+}}^{2}(𝒴) that commutes with the unilateral shift, there exists a function F̂∈ H_{𝔻}^{∞}(B_{𝒰→𝒴} ) such that F = Z^{−1}MF̂Z and moreover, F is an isometry if and only if F̂is inner. Furthermore, any multiplication operator M_{Φ} with symbol Φ ∈ H_{𝔻}^{∞}(B_{𝒰→𝒴} ) is bounded and linear shift-invariant on H_{𝔻}^{2}(𝒰), with ∥M_{Φ}∥ = ∥Φ∥_{∞}.

2.3.2 Systems

A system is defined to be an operator mapping between signal spaces. In this dissertation attention is restricted to systems defined on the continuous-time signal-space L_{ℝ^{+}}^{2}(·) (or a subspace of this) and its extension L_{ℝ^{+}}^{2,e}(·), and the discrete-time signal-space ℓ_{ℤ^{+}}^{2}(·) (or a subspaces of this) and its extension ℓ_{ℤ^{+}}^{2,e}(·). As a general rule, continuous-time systems and all associated operators (for example operators associated with a state-space realisation) are denoted throughout by italic, upper-case, Roman letters (P, F for example). Discrete-time systems and all associated operators (for example operators associated with a state-space realisation) are denoted by upper-case, Euler letters (P, F for example). In what follows, let 𝒰 and 𝒴 be Hilbert spaces and for notational convenience define 𝒱:=𝒰 ⊕ 𝒴.

Two important concepts in systems theory concern the timing of input-output pairs. The first is causality, which is a fundamental property of physically realisable systems. An operator is said to be causal if the output does not depend on future values of the input. More formally, a continuoustime system P : L_{ℝ^{+}}^{2,e}(𝒰) → L_{ℝ^{+}}^{2,e}(𝒴) is said to be \underline{causal} if given any τ ∈ ℝ^{+}, P_{τ} P u_{1} = P_{τ} P u_{2} for all u_{1}, u_{2} ∈ L_{ℝ^{+}}^{2,e}(𝒰) that satisfy P_{τ} u_{1} = P_{τ} u_{2}. Further to this, P is said to be \underline{strongly causal} if it is causal and given any τ ∈ ℝ^{+}, ϵ > 0 and τ_{1} ≤ τ (τ_{1} ∈ ℝ^{+}), there exists a δ > 0 such that for all u_{1}, u_{2} ∈ L_{ℝ^{+}}^{2,e}(𝒰) with P_{τ_{1}}u_{1} = P_{τ_{1}}u_{2},

∥P_{(τ_{1}+δ)}(P u_{1} − P u_{2})∥_{L_{ℝ^{+}}^{2}(𝒴)} ≤ ϵ∥P_{(τ_{1}+δ)}(u_{1} − u_{2})∥_{L_{ℝ^{+}}^{2}(𝒰)} .

Basically, a strongly causal system cannot respond instantaneously to inputs. For many causal systems encountered in systems theory, the way their behaviour varies with time imparts additional structure to the analysis at hand. Two types of variation are exploited in this dissertation; time-invariance and periodic time-variation. Specifically, a causal, continuous-time system P : L_{ℝ^{+}}^{2,e}(𝒰) → L_{ℝ^{+}}^{2,e}(𝒴) is said to be \underline{time-invariant} if P U_{τ} = U_{τ} P for all τ ≥ 0 and P is said to be \underline{periodically time-varying} with period h, if P U_{kh} = U_{kh}P for all k ∈ ℤ^{+}. Note that as defined, a time-invariant system is also periodic. As for continuous-time systems, a discrete-time system P : ℓ_{ℤ^{+}}^{2,e}(𝒰) → ℓ_{ℤ^{+}}^{2,e}(𝒴) is said to be causal if given any i ∈ ℤ^{+}, P_{i}Pu_{1} = P_{i}Pu_{2} for all u_{1}, u_{2} ∈ ℓ_{ℤ^{+}}^{2,e}(𝒰) such that P_{i}u_{1} = P_{i}u_{2}. Furthermore, a causal, discrete-time system is said to be shift-invariant if PS^{i} = S^{i}P for all i ∈ ℤ^{+}. That is, its behaviour does not vary with time.

Any system P_{e} : L_{ℝ^{+}}^{2,e}(𝒰) → L_{ℝ^{+}}^{2,e}(𝒴) induces a system

P : 𝒟_{P} ⊂ L_{ℝ^{+}}^{2}(𝒰) → L_{ℝ^{+}}^{2}(𝒴)

by specifying 𝒟_{P} :={u ∈ L_{ℝ^{+}}^{2}(𝒰) : P_{e}u ∈ L_{ℝ^{+}}^{2}(𝒴)} and P u:=P_{e}u for all u ∈ 𝒟_{P}. When two systems are related in this way, it is said that P_{e} \underline{induces} P and that P_{e} is the \underline{extension} of P. Note that any P_{e} uniquely induces a P. However, the converse is not true in general [GS93, FS82]. Before discussing a case for which the converse is true, it is necessary to clarify the notion of causality for systems not defined on extended space. A system P : 𝒟_{P} ⊂ L_{ℝ^{+}}^{2}(𝒰) → L_{ℝ^{+}}^{2}(𝒴) is said to be causal if

𝒢_{P} ∩ [ \begin{matrix}(I − P_{τ})L_{ℝ^{+}}^{2}(𝒰) \\ L_{ℝ^{+}}^{2}(𝒴) \end{matrix} ] ⊂ (I − P_{τ})L_{ℝ^{+}}^{2}(𝒱)

for all τ ∈ ℝ^{+}. If P is linear, then this corresponds to P_{τ} 𝒢_{P} being the graph of a linear operator. Clearly, if P is induced by a causal system P_{e} : L_{ℝ^{+}}^{2,e}(𝒰) → L_{ℝ^{+}}^{2,e}(𝒴), then P is causal in the sense just defined. Now given a causal system P : L_{ℝ^{+}}^{2}(𝒰) → L_{ℝ^{+}}^{2}(𝒴) with ∥P ∥ < ∞, there is a one-to-one correspondence to a system on the extended space [FS82, pg. 179]. To see this, note that for such a P and given any τ ∈ ℝ^{+}, the input can be chosen arbitrarily over the interval [0, τ]. Then since P is causal, the output is uniquely determined over the same interval. In fact, for any causal P that satisfies

P_{τ} 𝒟_{P} = L_{[0,τ]}^{2}(𝒰) for all finite real τ > 0,  (2.3)

where L_{[0,τ]}^{2}(𝒰) :=P_{τ} L_{ℝ^{+}}^{2}(𝒰), there is a one-to-one correspondence to a system on the relevant extended space [GS93]. With this in mind, a system P is called \underline{causally extendible} if it is causal and satisfies condition (2.3). Note that all of this also holds in discrete-time with obvious notational modifications.

Further to the definition of causality for systems not defined on extended space, it is also convenient to clarify the notions of time-invariance and periodic time-variation for these systems. A causal system P : 𝒟_{P} ⊂ L_{ℝ^{+}}^{2}(𝒰) → L_{ℝ^{+}}^{2}(𝒴) is said to be time-invariant (respectively periodically timevarying) if U_{τ} 𝒢_{P} ⊂ 𝒢_{P} (respectively U_{kh}𝒢_{P} ⊂ 𝒢_{P}) for all τ ∈ ℝ^{+} (respectively k ∈ ℤ^{+}). Clearly, if P is induced by a time-invariant (respectively periodically time-varying) system P_{e} on the extended space, then P is time-invariant (respectively periodically time-varying) in the sense just defined. It goes without saying, that similar definitions hold in discrete-time. Now before proceeding to define precisely the systems considered in the sequel, it is necessary to make one more technical definition. A causal, system P : L_{ℝ^{+}}^{2,e}(𝒰) → L_{ℝ^{+}}^{2,e}(𝒴) is said to be (time) \underline{locally Lipschitz-continuous} on L_{ℝ^{+}}^{2,e}(𝒰) if and only if for all τ ∈ ℝ^{+},

sup_{\begin{matrix}u_{1},u_{2}∈L_{ℝ^{+}}^{2,e}(𝒰) \\ P_{τ} u_{1}≠P_{τ} u_{2} \end{matrix}} \frac{∥P_{τ} (P u_{1} − P u_{2})∥_{L_{ℝ^{+}}^{2}(𝒴)}}{∥P_{τ} (u_{1} − u_{2})∥_{L_{ℝ^{+}}^{2}(𝒰)}} < ∞.

A similar definition (with obvious notational modifications) holds in discrete-time. Also note that although the concepts introduced above hold for a wide class of non-linear systems, only linear systems are considered in this dissertation.

Definition 2.3 Given two Hilbert spaces 𝒰 and 𝒴, 𝒫_{𝒰→𝒴} is defined to be the set of causal, linear, periodically-time-varying (LPTV) (with period h), continuous-time systems

P : 𝒟_{P} ⊂ L_{ℝ^{+}}^{2}(𝒰) → L_{ℝ^{+}}^{2}(𝒴),

with closed graphs. Furthermore, 𝒫_{𝒰→𝒴}^{e} ⊂ 𝒫_{𝒰→𝒴} is defined to be the subset of causally extendible systems with locally-Lipschitz-continuous extension and 𝒫_{𝒰→𝒴}^{e,sc} the subset of strongly-causal systems in 𝒫_{𝒰→𝒴}^{e} . A system P ∈ 𝒫_{𝒰→𝒴} is said to be \underline{stable} if 𝒟_{P} = L_{ℝ^{+}}^{2}(𝒰), which since it is assumed that P has closed graph implies that ∥P ∥ < ∞ by the Closed Graph Theorem. In view of the discussion above, any stable system P ∈ 𝒫_{𝒰→𝒴} is an element of 𝒫_{𝒰→𝒴}^{e} . ♠

An important property of LPTV systems is that they are equivalent, via the time-lifting isomorphism (W), to linear, shift-invariant (LSI), discrete-time systems, which by virtue of their shiftinvariance are easier to work with analytically [BPFT91, BP92].

Proposition 2.4 Given an LPTV system P ∈ 𝒫_{𝒰→𝒴} , the equivalent system P:=WP W^{−1} is causal and LSI.

Proof : Note that the graph of P is simply the graph of P under the time-lifting isomorphism. That is, 𝒢_{P} = W𝒢_{P}. So since 𝒢_{P} corresponds to a linear operator it follows that 𝒢_{P} does too. That P is causal is immediate, because P_{k}𝒢_{P} = WP_{kh}𝒢_{P} for all k ∈ ℤ^{+}, which since P_{kh}𝒢_{P} corresponds to the graph of a linear operator (as P is causal), implies that P is causal. Similarly, since S^{k}W = WU_{kh} for any k ∈ ℤ^{+}, it follows that

S^{k}𝒢_{P} = WU_{kh}𝒢_{P} ⊂ W𝒢_{P} = 𝒢_{P}

and hence, that P is also shift-invariant. For notational convenience the set of causal, LSI operators is defined below:

Definition 2.5 Given two Hilbert spaces 𝒰 and 𝒴, 𝒟_{𝒰→𝒴} is defined to be the set of causal, LSI, discrete-time systems

P : 𝒟_{P} ⊂ ℓ_{ℤ^{+}}^{2}(𝒰) → ℓ_{ℤ^{+}}^{2}(𝒴).

A system P ∈ 𝒟_{𝒰→𝒴} is called \underline{stable} if 𝒟_{P} = ℓ_{ℤ^{+}}^{2}(𝒰) and ∥P∥ < ∞. ♠

By representing each signal in ℓ_{ℤ^{+}}^{2}(·) as a column vector with k^{th} entry corresponding to the signal’s value at time k (that is, with respect to the standard basis for ℓ_{ℤ^{+}}^{2}(·)), it follows by causality and shift-invariance, that every system P ∈ 𝒟_{𝒰→𝒴} has block, lower-triangular, Toeplitz structure

[ \begin{matrix}P_{[0]} & 0 & ⋯ & ⋯ \\ P_{[1]} & P_{[0]} & 0 & ⋯ \\ ⋮ & ... & ... & ... \end{matrix} ] .

This Toeplitz representation can be uniquely identified with the sequence {P_{[i]}}_{i=0}^{∞} and will be used extensively in the sequel. If P ∈ 𝒟_{𝒰→𝒴} is causally extendible to a locally-Lipschitz-continuous system P_{e} : ℓ_{ℤ^{+}}^{2,e}(𝒰) → ℓ_{ℤ^{+}}^{2,e}(𝒴), then each P_{[i]} ∈ B_{𝒰→𝒴} .^{6} Given such a system P, it is straightforward to show, using its Toeplitz representation, that P is equivalent via the Z-Transform isomorphism to a multiplication operator on (a subspace of) H_{𝔻}^{2}(𝒰) with symbol

P̂(λ) := ∑_{i=0}^{∞} P_{[i]}λ^{i},

analytically defined on some open subset of 𝔻 (it is useful to note that P_{[0]} = P̂(0)). If P is also stable, then it follows by Proposition 2.2, that P̂(λ) ∈ H_{𝔻}^{∞}(B_{𝒰→𝒴} ). Henceforth, the symbol “ˆ ” is used to denote the frequency-domain symbol of the multiplication-operator representation of an LSI system.

6If 𝒰 and 𝒴 are finite-dimensional Hilbert spaces, then all systems in 𝒟_{𝒰→𝒴} have locally-Lipschitz-continuous extension to the extend space ℓ_{ℤ^{+}}^{2,e}(·).

Remark 2.6 A system P ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} is equivalent to a system P ∈ 𝒫_{𝒰→𝒴} (via W) if and only if P_{[0]} is a causal map from L_{ℍ}^{2}(𝒰) to L_{ℍ}^{2}(𝒴). A system P ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} not satisfying this condition is equivalent (via W) to an LPTV, continuous-time system that is not locally causal in each [kh, (k + 1)h) interval of time (k ∈ ℤ^{+}). ♣

Remark 2.7 A system P ∈ 𝒫_{𝒰→𝒴}^{e} is equivalent (via W) to a system in 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} with locally-Lipschitz-continuous extension. ♣

2.3.3 Closed-Loop Systems

In this dissertation, the stability of closed-loop systems is of central concern. To define precisely what is meant by a stable closed-loop, a few preliminary, technical definitions and results are required. Let 𝒰 and 𝒴 be Hilbert spaces and consider the system shown in Figure 2.1, where

[figure: C; P; d_{1}; u_{1}; y_{1}; d_{2}; u_{2}; y_{2}; −; +; +; −]

Figure 2.1 Standard Feedback Configuration

the systems P : L_{ℝ^{+}}^{2,e}(𝒰) → L_{ℝ^{+}}^{2,e}(𝒴) and C : L_{ℝ^{+}}^{2,e}(𝒴) → L_{ℝ^{+}}^{2,e}(𝒰) are causal and locally-Lipschitz-continuous. For notational convenience, define 𝒱:=𝒰 ⊕ 𝒴. The closed-loop system is denoted by [P, C] and the functional equations describing it are

u_{1} = d_{1} − y_{2},
u_{2} = d_{2} − y_{1},
y_{1} = P u_{1},
y_{2} = Cu_{2}.

An important concept in the study of closed-loop systems is well-posedness. Well-posedness is primarily concerned with the existence and uniqueness of solutions to the functional equations describing the closed-loop, causality of the closed-loop and adequacy of mathematical models to represent physical systems. For further details and discussion of the physical significance of wellposedness, the reader is referred to the seminal work of Willems [Wil71, Chap. 4].

Definition 2.8 Given causal, locally-Lipschitz-continuous systems P and C, the closed-loop [P, C] is said to be \underline{well-posed} [Wil71, pg. 90] if

(i) The system

[ \begin{matrix}I & C \\ P & I \end{matrix} ] : [ \begin{matrix}u_{1} \\ u_{2} \end{matrix} ] ∈ L_{ℝ^{+}}^{2,e}(𝒰) × L_{ℝ^{+}}^{2,e}(𝒴) ↦ [ \begin{matrix}d_{1} \\ d_{2} \end{matrix} ] ∈ L_{ℝ^{+}}^{2,e}(𝒰) × L_{ℝ^{+}}^{2,e}(𝒴),

is bijective and hence, invertible on L_{ℝ^{+}}^{2,e}(𝒰) ⊕ L_{ℝ^{+}}^{2,e}(𝒴). That is, for any inputs d_{1} ∈ L_{ℝ^{+}}^{2,e}(𝒰) and d_{2} ∈ L_{ℝ^{+}}^{2,e}(𝒴) there exists a unique solution to the closed-loop equations.

(ii) The signals u_{1}, u_{2}, y_{1} and y_{2} depend causally on d_{1} and d_{2}. Combined with (i), this requires that the following system be causal:

H_{e}(P, C) :=[ \begin{matrix}I & C \\ P & I \end{matrix} ]^{−1} : [ \begin{matrix}d_{1} \\ d_{2} \end{matrix} ] ∈ L_{ℝ^{+}}^{2,e}(𝒰) × L_{ℝ^{+}}^{2,e}(𝒴) ↦ [ \begin{matrix}u_{1} \\ u_{2} \end{matrix} ] ∈ L_{ℝ^{+}}^{2,e}(𝒰) × L_{ℝ^{+}}^{2,e}(𝒴).

(iii) On finite intervals of time (ℝ^{+}), the signals u_{1}, u_{2}, y_{1} and y_{2} depend Lipschitz-continuously on d_{1} and d_{2}. Combined with (i) and (ii), this requires that H_{e}(P, C) be causal and locally Lipschitz-continuous.

(iv) The solution to the closed-loop equations is insensitive to very-high-frequency modelling errors, such as small transmission delays. ♠

Proposition 2.9 [Wil71, Chap. 4] A \underline{sufficient} condition for the closed-loop [P, C] to be well-posed is that either P or C be strongly causal.

Remark 2.10 In a mathematical representation of a physical, closed-loop system, it is natural to assume that at least one of the mathematical models P and C is strongly causal. To see this, observe that in closed-loop P and C transmit information to each other and that in a physical system the transmission of information cannot occur instantaneously, which is in line with the definition of strong causality. ♣

In addition to well-posedness, it is desirable that the system H(P, C) : 𝒟_{H(P,C)} → L_{ℝ^{+}}^{2}(𝒱) induced by H_{e}(P, C), satisfies 𝒟_{H(P,C)} = L_{ℝ^{+}}^{2}(𝒱). That is, an input [ \begin{matrix}d_{1} \\ d_{2} \end{matrix} ]∈ L_{ℝ^{+}}^{2}(𝒱) to the closed-loop corresponds to closed-loop outputs u_{1}, y_{2} ∈ L_{ℝ^{+}}^{2}(𝒰) and y_{1}, u_{2} ∈ L_{ℝ^{+}}^{2}(𝒴).

Definition 2.11 If for a given pair of causal, locally-Lipschitz-continuous systems P and C, the closed-loop [P, C] is well-posed and 𝒟_{H(P,C)} = L_{ℝ^{+}}^{2}(𝒱), then [P, C] is said to be \underline{stable} and P is said to be \underline{stabilised} by C (and vice-versa). A stronger condition is that [P, C] be well-posed, 𝒟_{H(P,C)} = L_{ℝ^{+}}^{2}(𝒱) and ∥H(P, C)∥ < ∞. In this case, [P, C] is said to be \underline{stable with finite gain}. If the two systems P and C are also linear, it can be shown using the Closed Graph Theorem that stable with finite gain and stable are equivalent [Wil71, pg. 117]. As such, throughout this dissertation only the term stable is used. ♠

It is now possible to characterise closed-loop stability for the systems considered in this dissertation. If in Figure 2.1, P is taken to be in 𝒫_{𝒰→𝒴}^{e} and C in 𝒫_{𝒴→𝒰}^{e,sc} , then by definition, P and C have causal extensions P_{e} and C_{e} that are locally-Lipschitz-continuous and since C_{e} is strongly causal, the closed-loop [P, C] is well-posed (cf. Proposition 2.9). With the closed-loop being well-posed, it follows by condition (i) in Definition 2.8, that the system

[ \begin{matrix}I & C \\ P & I \end{matrix} ] : 𝒟_{C} ⊕ 𝒟_{P} → 𝒢_{P} + 𝒢C^{′}

is injective onto 𝒢_{P} + 𝒢C^{′} and hence, that 𝒢_{P} ∩ 𝒢C^{′} = {0}. So on 𝒟_{H(P,C)} :=𝒢_{P} + 𝒢C^{′} it has a well-defined (set theoretic) inverse

H(P, C) :=[ \begin{matrix}I & C \\ P & I \end{matrix} ]^{−1} : 𝒟_{H(P,C)} ⊂ L_{ℝ^{+}}^{2}(𝒱) → 𝒟_{C} ⊕ 𝒟_{P} .

By the definition of stability, it is then clear that [P, C] is stable if and only if 𝒟_{H(P,C)} :=𝒢_{P} +𝒢C^{′} = L_{ℝ^{+}}^{2}(𝒱) (with 𝒢_{P} ∩ 𝒢C^{′} = {0}). Furthermore, since all of the systems involved are linear, this is equivalent to stable with finite gain. This is summarised in the following proposition, which is reasonably standard and has also appeared in [OS91, FGS93, GS93], for example.

Proposition 2.12 With P ∈ 𝒫_{𝒰→𝒴}^{e} and C ∈ 𝒫_{𝒴→𝒰}^{e,sc} , the closed-loop [P, C] is stable if and only if 𝒢_{P} and 𝒢C^{′} induce a co-ordinatisation of L_{ℝ^{+}}^{2}(𝒱). That is, if and only if 𝒢_{P} ∩ 𝒢C^{′} = {0} and

𝒢_{P} + 𝒢C^{′} = L_{ℝ^{+}}^{2}(𝒱).

Remark 2.13 Necessity of Proposition 2.12 holds for P ∈ 𝒫_{𝒰→𝒴}^{e} and C ∈ 𝒫_{𝒴→𝒰}^{e} , but sufficiency may not always hold in this case. To see why, note that while the condition {𝒢_{P} , 𝒢C^{′}} is a coordinatisation of L_{ℝ^{+}}^{2}(𝒱) is necessary and sufficient for [ \begin{matrix}I & C \\ P & I \end{matrix} ]: 𝒟_{P} ⊕ 𝒟_{C} → L_{ℝ^{+}}^{2}(𝒱) to have bounded inverse on L_{ℝ^{+}}^{2}(𝒱), it does not guarantee that the closed-loop is well-posed. However, with C strongly causal the closed-loop is always well-posed (cf. Proposition 2.9) and for technical simplicity this was assumed. Although rather mild (cf. Remark 2.10), that C be strongly causal is not the weakest assumption required for the Proposition to hold.^{7} In fact, Proposition 2.12 holds for any well-posed, plant/controller pair: for example, P ∈ 𝒫_{𝒰→𝒴}^{e,sc} and C ∈ 𝒫_{𝒴→𝒰} (cf. Proposition 2.9). ♣ Remark 2.14 Given a stable closed-loop [P, C],

7For example another sufficient condition for well-posedness is that the product of the instantaneous gains of P and C be strictly less than 1 [Wil71].

H(P, C) := [ \begin{matrix}I & C \\ P & I \end{matrix} ]^{−1} = [ \begin{matrix}(I − CP)^{−1} & −C(I − P C)^{−1} \\ −P (I − CP)^{−1} & (I − P C)^{−1} \end{matrix} ],
Π_{𝒢_{P} ∥𝒢′_{C}} = [ \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ]H(P, C) + [ \begin{matrix}0 & 0 \\ 0 & I \end{matrix} ] = [ \begin{matrix}I \\ P \end{matrix} ][ (I − CP)^{−1} −C(I − P C)^{−1} ]

and

Π_{𝒢′_{C} ∥𝒢_{P}} = [ \begin{matrix}−I & 0 \\ 0 & I \end{matrix} ]H(P, C) + [ \begin{matrix}I & 0 \\ 0 & 0 \end{matrix} ] = [ \begin{matrix}C \\ I \end{matrix} ][ −P (I − CP)^{−1} (I − P C)^{−1} ].

Since H(P, C) is causal and bounded it is causally extendible to a locally-Lipschitz-continuous operator on L_{ℝ^{+}}^{2,e}(𝒱) and it follows that H(P, C) ∈ 𝒫_{𝒱→𝒱}^{e} . Similarly, the parallel projection operators Π_{𝒢_{P} ∥𝒢′_{C}} and Π_{𝒢′_{C} ∥𝒢_{P}} are also systems in 𝒫_{𝒱→𝒱}^{e} . ♣

2.4 LSI State-Space Systems and Discrete-Time Synthesis

The purpose of this section is to assemble some basic properties of LSI, discrete-time, state-space systems and to present state-space solutions to two discrete-time, synthesis problems. All of the results presented are well-known for state-space systems that have finite-dimensional input, output and state spaces. They are perhaps less well-known however, for the case in which the state-space is importantly finite-dimensional but the input and output spaces are infinite-dimensional, which as will be seen in the sequel, arises frequently in the study of LPTV systems. Many of the ideas discussed here are fundamental to the computational procedures developed later.

Let 𝒳 be an n-dimensional (n finite) Hilbert space and 𝒰 and 𝒴 be (possibly infinite-dimensional) Hilbert spaces. Consider the discrete-time system of difference equations:

x_{k+1} = Ax_{k} + Bu_{k}; x_{0} = x_{o},  (2.4a)
y_{k} = Cx_{k} + Du_{k},  (2.4b)

where for any fixed k ∈ ℤ^{+} the state x_{k} ∈ 𝒳, the input u_{k} ∈ 𝒰, the output y_{k} ∈ 𝒴 and the initial state x_{o} ∈ 𝒳. It is assumed that the bounded, linear operators A : 𝒳 → 𝒳, B : 𝒰 → 𝒳, C : 𝒳 → 𝒴 and D : 𝒰 → 𝒴 are all independent of time. With x_{0} = 0, the system of difference equations (2.4) characterises a causal, LSI system P_{e} : ℓ_{ℤ^{+}}^{2,e}(𝒰) → ℓ_{ℤ^{+}}^{2,e}(𝒴) defined by

y_{k} :=(P_{e}u)_{k} :=C ∑_{n=0}^{k−1} A^{k−n−1}Bu_{n} + Du_{k}.  (2.5)

The system P_{e} induces (as described in Sub-Section 2.3.2) a system P : 𝒟_{P} ⊂ ℓ_{ℤ^{+}}^{2}(𝒰) → ℓ_{ℤ^{+}}^{2}(𝒴) in 𝒟_{𝒰→𝒴} and the system of difference equations (2.4), is said to be a \underline{state-space realisation} of P. It is often convenient to denote a state-space realisation such as that given in equations (2.4), by [ \begin{matrix}A & B \\ C & D \end{matrix} ]. Importantly, a state-space realisation of a system P ∈ 𝒟_{𝒰→𝒴} (if it admits one) is not unique. Suppose that P admits the state-space realisation (2.4), then given any invertible basis-transformation T : 𝒳 → 𝒳, [ \begin{matrix}T^{−1}AT & T^{−1}B \\ CT & D \end{matrix} ] is also state-space realisation of P.

2.4.1 Reachability, Observability, Stabilisability and Detectability

In this section, the well understood concepts of reachability, observability, stabilisability and detectability, which are all properties of the state-space realisation chosen to represent a system, are introduced. For further details the reader is referred to [Bro70, Kai80, Won85, ZDG95].

Given an initial condition x_{o} ∈ 𝒳 and input signal u =: k ↦ u_{k} ⊂ ℓ_{ℤ^{+}}^{2,e}(𝒰), the state of the system of difference equations (2.4) evolves as:

x_{k} = A^{k}x_{o} + ∑_{n=0}^{k−1} A^{k−n−1}Bu_{n}.

A terminal state x_{f} ∈ 𝒳 is said to be reachable if there exists a time N < ∞ and input signal u ∈ ℓ_{ℤ^{+}}^{2,e}(𝒰) such that the solution to the state equation (2.4a), satisfies x_{N} = x_{f} when x_{o} = 0. The set of reachable states is a property of the pair (A, B) and can be characterised in terms of the sequence of subspaces 𝒞_{0}, 𝒞_{1}, 𝒞_{2} … ⊂ 𝒳, defined in terms of A and B as follows:

𝒞_{p} = ⋃_{k=0}^{p} ℛ_{A^{k}B} :=ℛ_{[ _{B AB ⋯ A^{p}B} ]}, (p ∈ ℤ^{+}).

Clearly, 𝒞_{0} ⊂ 𝒞_{1} ⊂ 𝒞_{2} ⊂ … and it can be shown that the inclusions are strictly proper up to a certain index r (dependent on A and B), with equality holding thereafter. Since 𝒞_{p} ⊂ 𝒳 for all p ∈ ℤ^{+}, it follows that r < n. 𝒞_{A,B} :=𝒞_{r} is called the \underline{reachable subspace} and corresponds to the set of reachable states. It follows by definition, that A𝒞_{p} ⊂ 𝒞_{p+1} and in fact, that 𝒞_{A,B} is the smallest A-invariant subspace containing ℛ_{B}. The pair (A, B) is said to be completely reachable if 𝒞_{A,B} = 𝒳.

Given two operators B_{1}, B_{2} : 𝒰 → 𝒳 with ℛ_{B_{1}} = ℛ_{B_{2}}, it follows by definition, that 𝒞_{A,B_{1}} = 𝒞_{A,B_{2}}. In particular the pairs (A, B) and (A, BB^{∗}) have the same reachable subspace. To see this, it is sufficient to show that ℛ_{B} = ℛ_{BB}^{∗}. To this end, note that for any bounded, linear operator X, (ℛ_{X})^{⊥} = 𝒦_{X^{∗}}, but that it is not possible to write ℛ_{X} = (𝒦_{X^{∗}})^{⊥} unless the range of X is closed.

Now, since ℛ_{B} ⊂ 𝒳, which is finite-dimensional, it follows that B has closed range (since any finitedimensional subspace is closed [Kre89b]). Hence, ℛ_{B} = (𝒦_{B^{∗}})^{⊥} and ℛ_{BB^{∗}} = (𝒦_{BB}^{∗})^{⊥}. So to obtain the desired result, it is sufficient to show that 𝒦_{B^{∗}} = 𝒦_{BB^{∗}}. That 𝒦_{B^{∗}} ⊂ 𝒦_{BB^{∗}} is obvious. To see the opposite inclusion take any x ∈ 𝒦_{BB^{∗}}, by which ⟨x, BB^{∗}x⟩_{𝒳} = 0 and thus, ⟨B^{∗}x, B^{∗}x⟩_{𝒰}. That is, B^{∗}x = 0 and the desired result follows as claimed. The importance of this equivalence can be seen by noting that BB^{∗} is a finite-dimensional operator and hence, is more computationally tractable than B. For example, to test complete reachability of the pair (A, B), standard finite-dimensional tools can be used to test the equivalent condition of complete reachability of the (finite-dimensional) pair (A, BB^{∗}). The combination of infinite-dimensional but finite-rank, state-space operators in this way, arises throughout this dissertation and is fundamental to the development of numerical algorithms for automated analysis and synthesis of LPTV systems (cf. Chapters 5 and 6 for example).

Another sequence of subspaces of the state-space 𝒳, useful in the study of the system of difference equations (2.4), is now introduced. Let 𝒩_{0}, 𝒩_{1}, 𝒩_{2}, … ⊂ 𝒳 be defined in terms of A and C as follows:

𝒩_{p} = ⋂_{k=0}^{p} 𝒦_{CA^{k}}, (p ∈ ℤ^{+}).

Observe that 𝒩_{0} ⊃ 𝒩_{1} ⊃ 𝒩_{2} ⊃ …, and that the inclusions are strictly proper up to a certain index r < n (dependent on A and C), with equality holding thereafter. 𝒩_{C,A} :=𝒩_{r} is called the \underline{unobservable subspace} and it corresponds to the the subspace of initial states x_{o} ∈ 𝒳 that cannot be uniquely determined by observing {u_{k}}_{k=0}^{N} and {y_{k}}_{k=0}^{N} in equations (2.4), for some finite time N ∈ ℤ^{+}. The pair (C, A) is said to be completely observable if 𝒩_{C,A}^{⊥} = 𝒳. Since the kernel of a bounded, linear operator is always closed [Kre89b], it follows that

( ⋂_{k=0}^{p} 𝒦_{CA}_{k})^{⊥} = ⋃_{k=0}^{p} (𝒦_{CA}_{k})^{⊥} , (p ∈ ℤ^{+}).

Furthermore, since ℛ_{C^{∗}} and ℛ_{A^{∗}} are subspaces of the finite-dimensional Hilbert space 𝒳, it follows that ℛ_{(A^{∗})^{k}C^{∗}} is closed for all k ∈ ℤ^{+} and equal to (𝒦_{CA}_{k})^{⊥}. As such,

𝒩_{p}^{⊥} = ⋃_{k=0}^{p} (𝒦_{CA}_{k})^{⊥} :=ℛ_{[ _{C^{∗} A^{∗}C^{∗} … (A^{∗})^{p}C^{∗}} ]} (p ∈ ℤ^{+})

and hence, 𝒩_{C,A}^{⊥} = 𝒞_{A^{∗},C^{∗}}.

An important and very useful result is that complete reachability and observability of a statespace realisation [ \begin{matrix}A & B \\ C & D \end{matrix} ], can be characterised in terms of the eigenvalues (or modes) of A and the range and kernel of B and C respectively. This is summarised in the following proposition.

Proposition 2.15 (cf. [ZDG95] for example).

I. The following are equivalent :

(i) The pair (A, B) is completely reachable. (ii) 𝒦_{B^{∗}} ∩ 𝒦_{(λI−A)^{∗}} = {0} for all λ ∈ ℂ.

II. The following are equivalent :

(i) The pair (C, A) is completely observable. (ii) 𝒦_{C} ∩ 𝒦_{(λI−A)} = {0} for all λ ∈ ℂ.

With this result in mind, all λ ∈ ℂ such that 𝒦_{C} ∩ 𝒦_{(λI−A)} ≠ \{0\} are called \underline{unobservable modes} of the pair (C, A). Similarly, all λ ∈ ℂ such that 𝒦_{B^{∗}} ∩ 𝒦_{(λI−A)^{∗}} ≠ \{0\} are called \underline{unreachable modes} of the pair (A, B).

Two concepts closely related to reachability and observability are stabilisability and detectability. The pair (A, B) is said to be \underline{stabilisable} if there exist a bounded, linear operator F : 𝒳 → 𝒰 such that the spectrum of (A + BF) : 𝒳 → 𝒳 lies in the open unit disc 𝔻. This is often referred to as state-feedback stabilisability. Similarly, the pair (C, A) is said to be \underline{detectable} if there exists a bounded, linear operator L : 𝒴 → 𝒳 such that the spectrum of (A + LC) : 𝒳 → 𝒳 lies in 𝔻. The following proposition constitutes a useful characterisation of stabilisability and detectability.

Proposition 2.16 (cf. [ZDG95] for example)

I. The following are equivalent :

(i) The pair (A, B) is stabilisable.

(ii) 𝒦_{B^{∗}} ∩ 𝒦_{(λI−A)^{∗}} = {0} for all λ ∈ 𝔼:=ℂ − 𝔻.

II. The following are equivalent :

(i) The pair (C, A) is detectable. (ii) 𝒦_{C} ∩ 𝒦_{(λI−A)} = {0} for all λ ∈ 𝔼.

In view of this result and Proposition 2.15, it can be said that if (A, B) is stabilisable then every A-invariant subspace of 𝒳 corresponding to an eigenvalue (mode) of A in 𝔼 is reachable. Similarly, if (C, A) is detectable then every A-invariant subspace of 𝒳 corresponding to an eigenvalue (mode) of A in 𝔼 is observable. As with reachability and observability, there is also a duality between stabilisability and detectability. Specifically, (C, A) is detectable if and only if (A^{∗}, C^{∗}) is stabilisable.

Recall that a state-space realisation of a system is not unique. With this in mind, two useful canonical forms, due to Kalman, are now presented. Given a state-space realisation [ \begin{matrix}A & B \\ C & D \end{matrix} ] of a system P ∈ 𝒟_{𝒰→𝒴} , suppose that (A, B) is not completely reachable (that is 𝒞_{A,B} is not the whole of 𝒳). Let {e_{n}}_{n=0}^{r} be a basis for 𝒞_{A,B} and extend this to the whole of 𝒳. Since 𝒞_{A,B} is A-invariant and ℛ_{B} ⊂ 𝒞_{A,B}, the representations of A, B and C in this basis have the following form:

[ \begin{matrix}A_{1} & A_{2} \\ 0 & A_{3} \end{matrix} ], [ \begin{matrix}B_{1} \\ 0 \end{matrix} ] and [ C_{1} C_{2} ],

respectively. By the way the representations above are constructed, it is straightforward to see that the pair (A_{1}, B_{1}) is completely reachable and that [ \begin{matrix}A_{1} & B_{1} \\ C_{1} & D \end{matrix} ] is a state-space realisation of P. Moreover, (A, B) is stabilisable if and only if spec(A_{3}) ⊂ 𝔻. Similarly, if the pair (C_{1}, A_{1}) is not completely observable then it is possible to construct a state-space, basis transformation T such that:

T^{−1}A_{1}T = [ \begin{matrix}A_{11} & A_{12} \\ 0 & A_{13} \end{matrix} ], T^{−1}B_{1} = [ \begin{matrix}B_{11} \\ B_{12} \end{matrix} ], C_{1}T = [ C_{11} 0 ];

(C_{11}, A_{11}) is completely observable; and [ \begin{matrix}A_{11} & B_{11} \\ C_{11} & D \end{matrix} ] is a state-space realisation of P. Furthermore,

it follows that (A_{1}, C_{1}) is detectable if and only if spec(A_{13}) ⊂ 𝔻. The realisation [ \begin{matrix}A_{11} & B_{11} \\ C_{11} & D \end{matrix} ], which is completely reachable and completely observable, is called a \underline{minimal} realisation.

2.4.2 Stability

Consider the homogeneous component of the system of difference equations (2.4): x_{k+1} = Ax_{k} with x_{0} = x_{o} ∈ 𝒳. Then the state evolves as x_{k} = A^{k}x_{o}, which corresponds to a sequence in ℓ_{ℤ^{+}}^{2}(𝒳) if and only if spec(A) ⊂ 𝔻. If this is the case, then the homogeneous system is said to be stable. Now given a state-space realisation [ \begin{matrix}A & B \\ C & D \end{matrix} ] of a system P ∈ 𝒟_{𝒰→𝒴} , it can be shown [Vid78] (using the input-output description given in equation (2.5)) that if spec(A) ⊂ 𝔻, then P is input-output stable in the sense that 𝒟_{P} = ℓ_{ℤ^{+}}^{2}(𝒰) and ∥P∥ < ∞. For the converse to hold however, additional assumptions are required on the state-space realisation considered. If the state-space realisation of P is completely reachable, then every A-invariant subspace (which covers the whole of 𝒳) can be reached by an appropriate finite-energy input sequence. That is, every mode of A can be excited. If each of these subspaces is observable (that is if (C, A) is completely observable), then for P to be input-output stable it is necessary that all modes of A be stable (that is, spec(A) ⊂ 𝔻). In other words if (A, B) is completely reachable and (C, A) is completely observable then inputoutput stability implies spec(A) ⊂ 𝔻. In fact, in view of the Kalman canonical forms introduced in the previous subsection, the complete reachability and observability conditions can be relaxed to stabilisability and detectability, since under these relaxed conditions, A can be decomposed into a completely controllable and observable part and a part with spectrum contained in 𝔻 (that is, a stable part).

An important tool in state-space system-theory is the Lyapunov equation. For discrete-time, state-space systems this is also known as the Stein equation and it takes the following form:

X − A^{∗}XA = Q.

It can be shown that the solution X : 𝒳 → 𝒳 to this equation is unique provided λ_{i}\overline{λ}_{j} ≠ 1 for every λ_{i}, λ_{j} ∈ spec(A) (cf. [ZDG95] for example). The following result is well-known:

Proposition 2.17 ([ZDG95]) Let X be the unique solution to the Stein equation X−A^{∗}XA = C^{∗}C. Then spec(A) ⊂ 𝔻 if and only if X ≥ 0 and (C, A) is detectable.

Proof : (⇐) Suppose A has an eigenvalue λ ∈ 𝔼 and let x ∈ 𝒳 denote a corresponding eigenvector. Then

⟨x, Xx⟩_{𝒳} − ⟨x, A^{∗}XAx⟩_{𝒳} = (1 − |λ|^{2})⟨x, Xx⟩_{𝒳} = ⟨Cx, Cx⟩_{𝒴},

which since λ ∈ 𝔼 and X is positive-semidefinite, implies that Cx = 0. That is, λ ∈ 𝔼 must be an unobservable mode of (C, A), which contradicts detectability of (C, A). Hence, spec(A) ⊂ 𝔻.

(⇒) If spec(A) ⊂ 𝔻 then a solution to the Stein equation is given by

X = ∑_{k=0}^{∞}(CA^{k})^{∗}CA^{k} ≥ 0.

Furthermore, 𝒦_{(λI−A)} = {0} for all λ ∈ 𝔼 and hence, (C, A) does not have any unobservable modes in 𝔼.

It follows by duality that if X is the solution to X − AXA^{∗} = BB^{∗}, then spec(A) ⊂ 𝔻 if and only if X ≥ 0 and (A, B) is stabilisable.

2.4.3 Quadratic-Regulator and Full-Information ℓ_{ℤ^{+}}^{2}(·)-Induced-Norm Synthesis In this section the quadratic-regulator and ℓ_{ℤ^{+}}^{2}(·)-induced-norm, synthesis problems are considered. The results presented here are fundamental to a computational procedure developed in Chapter 5. Attention is restricted to control strategies having direct access to both the state and disturbance signals. That is, full-information control.^{8} The system to be controlled is assumed to admit a state-space realisation with finite-dimensional state-space. Specifically,

x_{k+1} = Ax_{k} + B_{1}w_{k} + B_{2}u_{k}; x_{0} = x_{0},  (2.6a)
z_{k} = Cx_{k} + D_{1}w_{k} + D_{2}u_{k},  (2.6b)

where for each fixed k ∈ ℤ^{+}, the state x_{k} ∈ 𝒳 (a finite-dimensional Hilbert space), the exogenous disturbance w_{k} ∈ 𝒲, the control input u_{k} ∈ 𝒰 and the controlled output z_{k} ∈ 𝒵 (with 𝒰, 𝒲, 𝒵 possibly infinite-dimensional, Hilbert spaces). The bounded, linear operators A, B_{1}, B_{2}, C, D_{1} and D_{2} are all assumed to be time independent.

In what follows, for k ∈ ℤ^{+}, x:= k ↦ x_{k}, u:= k ↦ u_{k}, w:=k ↦ w_{k} and z:= k ↦ z_{k}. Now, given an initial state x_{0} ∈ 𝒳 and an identically-zero, exogenous disturbance signal w (that is, w_{k} = 0 for all k ∈ ℤ^{+}), the objective of \underline{quadratic-regulator} synthesis is to define a causal, LSI, stabilising control-law u = Kx, to minimise ∥z∥_{ℓ_{ℤ^{+}}^{2}(𝒵)} .^{9} On the other hand, given a zero initial state and some real number γ > 0, the objective of full-information, ℓ_{ℤ^{+}}^{2}(·)-induced-norm synthesis is to define a causal, LSI, stabilising control-law u = K[ \begin{matrix}x \\ w \end{matrix} ], such that with the control-law in place the ℓ_{ℤ^{+}}^{2}(·)-induced norm of the system mapping w to z is bounded by γ. Clearly (A, B_{2}) stabilisable is a necessary condition for existence of the specific controllers defined for both problems and as such, it will be assumed throughout this sub-section.

Before going on to solve the synthesis problems specified above, some preliminary results concerning the discrete-time, algebraic, Riccati equation are required. The theory of discrete-time, algebraic, Riccati equations in optimal control originates in [DL70] and [Kuč72], with subsequent developments by many authors: notably, [PLS80, DGG86, BGPK85, IG91, GL95, IOW97]. The reader is referred to the books [LR95, BLW91] as general and quite comprehensive references.

Consider the following algebraic equation in X:

X = A^{∗}XA + Q − (S + B^{∗}XA)^{∗}(R + B^{∗}XB)^{−1}(S + B^{∗}XA),  (2.7)

where A ∈ B_{𝒳→𝒳} , Q^{∗} = Q ∈ B_{𝒳→𝒳} , S ∈ B_{𝒳→𝒰} , B ∈ B_{𝒰→𝒳} and R^{∗} = R∈ B_{𝒰→𝒰} . This is known as the (discrete-time) \underline{algebraic, Riccati equation}. By definition, a solution X must be a bounded, self-adjoint operator such that R + B^{∗}XBis boundedly invertible.

8For the quadratic-regulator problem there is no disturbance and hence, the controller only has access to the state.

9Here and below, stabilising means that with the control-law in place, x ∈ ℓ_{ℤ^{+}}^{2}(𝒳) and thus, that lim_{k→∞} x_{k} = 0.

Proposition 2.18 Let R be boundedly invertible and define

A_{1} :=A − BR^{−1}S, and Q_{1} :=Q − S^{∗}R^{−1}S.

Then the following algebraic (Riccati) equations in X : 𝒳 → 𝒳 have the same solutions:

X = A^{∗}XA + Q − (S + B^{∗}XA)^{∗}(R + B^{∗}XB)^{−1}(S + B^{∗}XA),
X = A_{1}^{∗}XA_{1}^{∗} + Q_{1} − A_{1}^{∗}XB(R + B^{∗}XB)^{−1}B^{∗}XA_{1},
X = A_{1}^{∗}X(I + BR^{−1}B^{∗}X)^{−1}A_{1} + Q_{1}.

Note that with 𝒳 finite-dimensional, the operators appearing in the last equation above can be grouped together so as to involve only finite-dimensional terms. That is, by defining the finitedimensional operator R_{1} :=BR^{−1}B^{∗}, the last equation can be expressed as the finite-dimensional, algebraic, Riccati equation

X = A_{1}^{∗}X(I + R_{1}^{−1}X)^{−1}A_{1} + Q_{1},

which given a method for computing R_{1} : 𝒳 → 𝒳, Q_{1} : 𝒳 → 𝒳 and A_{1} : 𝒳 → 𝒳, can be solved using standard finite-dimensional tools.

In the following proposition, what is called a \underline{stabilising} solution to the algebraic, Riccati equation (2.7) is characterised. As part of this, it is necessary to assume that R > 0. Under this assumption it is said that the algebraic, Riccati equation is \underline{sign definite} (in fact, in this case positive definite).

Remark 2.19 Note that with R > 0 there exists by definition an α > 0 such that

α^{2}⟨u, u⟩_{𝒰}^{2} ≤ ⟨u, Ru⟩_{𝒰}^{2} ≤ ⟨u, u⟩_{𝒰} ⟨Ru, Ru⟩_{𝒰} for all u ∈ 𝒰,

where the second inequality is called the Schwartz inequality. Hence, ∥Ru∥_{𝒰} ≥ α^{2}∥u∥_{𝒰} and since R is self-adjoint, it follows by [Kre89b, Theorem 9.1-2] that 0 ∉ spec(R). Consequently, R^{−1} is bounded and defined on a set dense in 𝒰. In fact R^{−1} is defined on the whole of 𝒰. Similarly, if R were negative definite, R^{−1} would be bounded and defined on the whole of 𝒰. It should be noted that this is a non-trivial fact when 𝒰 is infinite-dimensional. ♣

Proposition 2.20 Suppose that R > 0 and

[ \begin{matrix}Q & S^{∗} \\ S & R \end{matrix} ] ≥ 0.

Then, there exist a self-adjoint, positive-semidefinite solution X to the (discrete-time) algebraic, Riccati equation (2.7), such that with F:=(R + B^{∗}XB)^{−1}(S + B^{∗}XA), spec(A − BF) ⊂ 𝔻 if and only if (A, B) is stabilisable and (Q_{1}, A_{1}) has no unobservable modes on 𝕋.

Proof : For a proof see Appendix A.1.

This result is fundamental to the solution of both the quadratic-regulator and full-information, ℓ_{ℤ^{+}}^{2}(·)-induced-norm, synthesis problems discussed below.

State-Feedback Quadratic-Regulator Synthesis

Consider the component of the state-space system (2.6), mapping u to z according to

x_{k+1} = Ax_{k} + B_{2}u_{k}; x_{0} = x_{0},
z_{k} = Cx_{k} + D_{2}u_{k},

under the standing assumption D_{2}^{∗}D_{2} > 0. Given an initial state x_{0} ∈ 𝒳, recall that the objective of quadratic-regulator synthesis is to find a causal, LSI, stabilising control-law u = Kx, to minimise ∥z∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}. Since the control-law is restricted to be causal and dependent only on the state, it follows that for each k ∈ ℤ^{+}, ∥(I − P_{k−1})z∥_{ℓ_{ℤ^{+}}^{2}(𝒵)} is dependent only on x_{k} when an admissible controllaw is in place. With this in mind, the self-adjoint, positive-semidefinite operator X_{2} : 𝒳 → 𝒳 is defined to satisfy ⟨x_{k}, X_{2}x_{k}⟩_{𝒳} = ∥(I−P_{k−1})z^{⋆}∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}, where z^{⋆} denotes the output with the optimal control-law in place. In this way ∥z^{⋆}∥_{ℓ_{ℤ^{+}}^{2}(𝒵)} = ⟨x_{0}, X_{2}x_{0}⟩_{𝒳}.

In dynamic programming terms, at each time k ∈ ℤ^{+}, u_{k} is a decision variable and

⟨z_{k}, z_{k}⟩_{𝒵} + ⟨x_{k+1}, X_{2}x_{k+1}⟩_{𝒳}

is the cost at stage k. By the principle of dynamic programming, which states that a sequence {u_{k}}_{k=0}^{∞} of decisions is optimal for a problem defined over the horizon [0, ∞) if and only if for all N ∈ ℤ^{+} the control sequence {u_{k}}_{k=N}^{∞} is optimal for the problem considered over the sub-horizon [N, ∞) [Bel62], it follows that

⟨x_{k}, X_{2}x_{k}⟩_{𝒳} = min_{u_{k}∈𝒰}{⟨z_{k}, z_{k}⟩_{𝒵} + ⟨x_{k+1}, X_{2}x_{k+1}⟩_{𝒳} }.  (2.8)

This is known as Isaac’s equation and is a special form of the Hamilton-Jacobi-Bellman equation. Now note that for all k ∈ ℤ^{+},

⟨z_{k}, z_{k}⟩_{𝒵} + ⟨x_{k+1}, X_{2}x_{k+1}⟩_{𝒳}
= ⟨[ \begin{matrix}x_{k} \\ u_{k} \end{matrix} ], [ \begin{matrix}A^{∗} & C^{∗} \\ B_{2}^{∗} & D_{2}^{∗} \end{matrix} ][ \begin{matrix}X_{2} & 0 \\ 0 & I \end{matrix} ][ \begin{matrix}A & B_{2} \\ C & D_{2} \end{matrix} ][ \begin{matrix}x_{k} \\ u_{k} \end{matrix} ]⟩_{𝒳⊕𝒰}
= ⟨[ \begin{matrix}x_{k} \\ u_{k} \end{matrix} ], [ \begin{matrix}I & F_{2}^{∗} \\ 0 & I \end{matrix} ][ \begin{matrix}A^{∗}X_{2}A + C^{∗}C − F_{2}^{∗}R_{2}F_{2} & 0 \\ 0 & R_{2} \end{matrix} ][ \begin{matrix}I & 0 \\ F_{2} & I \end{matrix} ][ \begin{matrix}x_{k} \\ u_{k} \end{matrix} ]⟩_{𝒳⊕𝒰}
= ⟨[ \begin{matrix}x_{k} \\ u_{k} + F_{2}x_{k} \end{matrix} ], [ \begin{matrix}A^{∗}X_{2}A + C^{∗}C − F_{2}^{∗}R_{2}F_{2} & 0 \\ 0 & R_{2} \end{matrix} ][ \begin{matrix}x_{k} \\ u_{k} + F_{2}x_{k} \end{matrix} ]⟩_{𝒳⊕𝒰}
= ⟨x_{k}, (A^{∗}X_{2}A + C^{∗}C − F_{2}^{∗}R_{2}F_{2})x_{k}⟩_{𝒳} + ⟨u_{k} + F_{2}x_{k}, R_{2}(u_{k} + F_{2}x_{k})⟩_{𝒰} ,  (2.9)

where

F_{2} := R_{2}^{−1}(D_{2}^{∗}C + B_{2}^{∗}X_{2}A) : 𝒳 → 𝒰,
R_{2} := (D_{2}^{∗}D_{2} + B_{2}^{∗}X_{2}B_{2}) : 𝒰 → 𝒰.

Then since R_{2} > 0, it can be seen that the minimum over u_{k} ∈ 𝒰 of equation (2.9) is achieved uniquely by u_{k} = −F_{2}x_{k}. Furthermore, it follows from equation (2.8), that X_{2} must satisfy

X_{2} = A^{∗}X_{2}A + C^{∗}C − F_{2}^{∗}R_{2}F_{2}.  (2.10)

So if u_{k} = −F_{2}x_{k} is stabilising, the synthesis problem is solved.

Since (I − D_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}) is self-adjoint and idempotent it is a projection and hence, (I − D_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}) ≥ 0 (cf. [Kre89b]). It then follows that

[ \begin{matrix}C^{∗}C & C^{∗}D_{2} \\ D_{2}^{∗}C & D_{2}^{∗}D_{2} \end{matrix} ] ≥ 0.

Thus, by Proposition 2.20, (A, B_{2}) stabilisable and (C^{∗}(I−D_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗})C, A−B_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗}C) having no unobservable modes on 𝕋, is necessary and sufficient for the existence of a self-adjoint, positive-semidefinite solution X_{2} to equation (2.10) such that spec(A − BF_{2}) ⊂ 𝔻. That is, under these assumptions the control-law u_{k} = −F_{2}x_{k} is stabilising. This is summarised in the following theorem.

Proposition 2.21 Assume that D_{2}^{∗}D_{2} > 0, (A, B_{2}) is stabilisable and

𝒦[ \begin{matrix}A − e^{jθ}I & B_{2} \\ C & D_{2} \end{matrix} ] = \{0\}  (2.11)

for all θ ∈ [0, 2π). Then the unique, optimal, causal, LSI, stabilising control-law for the quadraticregulator synthesis problem is u_{k} = −F_{2}x_{k}, where

F_{2} := R_{2}^{−1}(D_{2}^{∗}C + B_{2}^{∗}X_{2}A) : 𝒳 → 𝒳,
R_{2} := (D_{2}^{∗}D_{2} + B_{2}^{∗}X_{2}B_{2}) : 𝒰 → 𝒰

and the self-adjoint, positive-semidefinite operator X_{2} : 𝒳 → 𝒳 satisfies the (discrete-time) algebraic, Riccati equation (2.10).

Proof : The proof follows from the discussion above by noting that the following are equivalent:

(i) (C^{∗}(I − D_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗})C, A − B_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗}C) has no unobservable on 𝕋; (ii)

𝒦[ \begin{matrix}A − e^{jθ}I & B_{2} \\ C & D_{2} \end{matrix} ] = \{0\}

for all θ ∈ [0, 2π).

To see this suppose that (i) holds and that there exist a non-zero [ \begin{matrix}x \\ u \end{matrix} ] ∈ 𝒳 ⊕ 𝒰 such that

[ \begin{matrix}A − e^{jθ} & B_{2} \\ C & D_{2} \end{matrix} ][ \begin{matrix}x \\ u \end{matrix} ] = 0,

for some θ ∈ [0, 2π). It then follows that u = −(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗}Cx and hence, that Ax + B_{2}u = (A − B_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗}C)x = e^{jθ}x and C^{∗}(I − D_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗})Cx = 0. This contradicts (i) and hence, (ii) must hold.

To see the converse, suppose that (ii) holds and that there exists a non-zero x ∈ 𝒳 and a θ ∈ [0, 2π) such that (A − B_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗}C)x = e^{jθ}x and C^{∗}(I − D_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗})Cx = 0. Now note that (I − D_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗}) is self-adjoint and idempotent. Hence,

C^{∗}(I − D_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗})(I − D_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗})Cx = 0,

which implies that (I − D_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗})Cx = 0. It follows then, that

[ \begin{matrix}A − e^{jθ}I & B_{2} \\ C & D_{2} \end{matrix} ][ \begin{matrix}I & 0 \\ (D_{2}^{∗}D_{2})^{−1}D_{2}^{∗}C & I \end{matrix} ][ \begin{matrix}x \\ 0 \end{matrix} ] = 0,

which contradicts (ii) and hence, (i) must hold. This completes the proof

Full-Information ℓ_{ℤ^{+}}^{2}(·)-Induced-Norm Synthesis

Consider the system of difference equations (2.6) re-written as

[ \begin{matrix}x_{k+1} \\ z_{k} \end{matrix} ] = [ \begin{matrix}A & B_{1} B_{2} \\ C & D_{1} D_{2} \end{matrix} ][ x_{k} ; \begin{matrix}w_{k} \\ u_{k} \end{matrix} ]; x_{0} = x_{0},

under the standing assumption D_{2}^{∗}D_{2} > 0. Now recall that given a real number γ > 0, the objective of full-information ℓ_{ℤ^{+}}^{2}(·)-synthesis is to obtain a causal, LSI, stabilising control-law u = K[ \begin{matrix}x \\ w \end{matrix} ], such that with it in place and with x_{o} = 0, the ℓ_{ℤ^{+}}^{2}(·)-induced norm of the operator mapping w to z is bounded by γ. Note that when x_{0} = 0, the ℓ_{ℤ^{+}}^{2}(·)-induced norm of the mapping from w to z is strictly less than γ if and only if

∥z∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}^{2} − γ^{2}∥w∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}^{2} ≤ −ϵ∥w∥_{ℓ_{ℤ^{+}}^{2}(𝒲)}^{2}

for some ϵ > 0 and all w ∈ ℓ_{ℤ^{+}}^{2}(𝒲). As such, the following theorem characterises a solution to the full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem.

Proposition 2.22 Let J:=[ \begin{matrix}I & 0 \\ 0 & −γ^{2}I \end{matrix} ] and assume that D_{2}^{∗}D_{2} > 0, (A, B_{2}) is stabilisable and for all θ ∈ [0, 2π)

𝒦[ \begin{matrix}A − e^{jθ}I & B_{2} \\ C & D_{2} \end{matrix} ] = \{0\}.

Then there exists a causal, LSI, stabilising control-law of the form u = K[ \begin{matrix}x \\ w \end{matrix} ], such that given a real number γ > 0,

∥z∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}^{2} − γ^{2}∥w∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}^{2} ≤ −ϵ∥w∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}^{2}  (2.12)

for all w ∈ ℓ_{ℤ^{+}}^{2}(𝒲)and some ϵ > 0 when x_{0} = 0, if and only if the (discrete-time) algebraic, Riccati equation

X_{∞} = A^{∗}X_{∞}A + C̃^{∗}JC̃ − L^{∗}Z^{−1}L,  (2.13)

has self-adjoint, positive-semidefinite solution X_{∞} : 𝒳 → 𝒳 such that spec(A − BZ^{−1}L) ⊂ 𝔻 and

∇:=Z_{1} − Z_{2}^{∗}Z_{3}^{−1}Z_{2} < 0,

where

Z := D̃^{∗}JD̃ + B^{∗}X_{∞}B =: [ \begin{matrix}Z_{1} & Z_{2}^{∗} \\ Z_{2} & Z_{3} \end{matrix} ] : 𝒲 ⊕ 𝒰 → 𝒲 ⊕ 𝒰,  (2.14)
L := D̃^{∗}JC̃ + B^{∗}X_{∞}A =: [ \begin{matrix}L_{1} \\ L_{2} \end{matrix} ] : 𝒳 → 𝒲 ⊕ 𝒰,  (2.15)

D̃ :=[ \begin{matrix}D_{1} & D_{2} \\ I & 0 \end{matrix} ],  C̃:=[ \begin{matrix}C \\ 0 \end{matrix} ]  and  B:=[ B_{1} B_{2} ].

In this case a causal, LSI, stabilising, full-information control-law achieving the objective is given by

u_{k}^{⋆} = Z_{3}^{−1}[ L_{2} Z_{2} ][ \begin{matrix}x_{k} \\ w_{k} \end{matrix} ].

Proof : The proof of this result follows very closely that of Green and Limebeer [GL95, Appendix B.], with essentially only notational modifications to accommodate the possibility of infinitedimensional input/output spaces. For completeness it is presented in Appendix A.2.

To complete this section it is shown that for a solution to the full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem to exist it is necessary that D̃^{∗}JD̃ : 𝒲 ⊕ 𝒰 → 𝒲 ⊕ 𝒰 be a boundedly invertible operator. In this way, it is possible to apply Proposition 2.18, by which the algebraic, Riccati equation (2.13) can be re-written in terms of finite-dimensional operators (in that they map the finite-dimensional space 𝒳 to 𝒳) only. That is, equation (2.13) can be re-written as the standard, finite-dimensional, algebraic, Riccati equation

X_{∞} = À^{∗}X_{∞}(I + R̀X_{∞})^{−1}À + Q̀,

where À:=(A − B(D̃^{∗}JD̃)^{−1}D̃^{∗}JC̃) : 𝒳 → 𝒳,

R̀:=B(D̃^{∗}JD̃)^{−1}B^{∗} : 𝒳 → 𝒳

and

Q̀ :=C̃^{∗}(I − D̃(D̃^{∗}JD̃)^{−1}D̃^{∗})C̃ : 𝒳 → 𝒳.

Lemma 2.23 If D_{2}^{∗}D_{2} > 0 and there exists a causal, LSI, stabilising control-law of the form u = K[ \begin{matrix}x \\ w \end{matrix} ], such that given a γ > 0

∥z∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}^{2} − γ^{2}∥w∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}^{2} ≤ −ϵ∥w∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}^{2}

for all w ∈ ℓ_{ℤ^{+}}^{2}(𝒲) and some ϵ > 0 when x_{0} = 0, then the operator D̃^{∗}JD̃ : 𝒲 ⊕ 𝒰 → 𝒲 ⊕ 𝒰 is boundedly invertible.

Proof : Note that

D̃^{∗}JD̃ = [ \begin{matrix}D_{1}^{∗}D_{1} − γ^{2}I & D_{1}^{∗}D_{2} \\ D_{2}^{∗}D_{1} & D_{2}^{∗}D_{2} \end{matrix} ].

Since D_{2}^{∗}D_{2} > 0, it follows as discussed in Remark 2.19, that D_{2}^{∗}D_{2} is boundedly invertible and as such, it is possible to write

D̃^{∗}JD̃ = [ \begin{matrix}I & D_{1}^{∗}D_{2}(D_{2}^{∗}D_{2})^{−1} \\ 0 & I \end{matrix} ][ \begin{matrix}Δ & 0 \\ 0 & D_{2}^{∗}D_{2} \end{matrix} ][ \begin{matrix}I & 0 \\ (D_{2}^{∗}D_{2})^{−1}D_{2}^{∗}D_{1} & I \end{matrix} ],  (2.16)

where Δ = D_{1}^{∗}(I − D_{2}(D_{2}^{∗}D_{2})^{−1}D_{2}^{∗})D_{1} − γ^{2}I. Now suppose that a stabilising, full-information control-law that satisfies (2.12) is in place and set

w_{k} = \{ \begin{matrix}0, i ≠ N \\ w̃, i = N \end{matrix}

for some N ∈ ℤ^{+} and w̃ ∈ 𝒲. Then since the control-law is causal and x_{0} = 0, x_{k} = 0 for k ≤ N, z_{k} = 0 for k < N and

−ϵ⟨w̃, w̃⟩_{𝒲} ≥ ⟨z_{N}, z_{N}⟩_{𝒵} − γ^{2}⟨w_{N} , w_{N} ⟩_{𝒲} + ∑_{k=N+1}^{∞} ⟨z_{k}, z_{k}⟩_{𝒵}
≥ ⟨z_{N}, z_{N}⟩_{𝒵} − γ^{2}⟨w_{N} , w_{N} ⟩_{𝒲}
= ⟨[ C̃ D̃ ][ \begin{matrix}& x_{N} \\ [ & \begin{matrix}w_{N} \\ u_{N} \end{matrix} & ] \end{matrix} ], [ C̃ D̃ ][ \begin{matrix}& x_{N} \\ [ & \begin{matrix}w_{N} \\ u_{N} \end{matrix} & ] \end{matrix} ]⟩_{𝒵⊕𝒲}

= ⟨[ \begin{matrix}w_{N} \\ u_{N} \end{matrix} ], D̃^{∗}JD̃[ \begin{matrix}w_{N} \\ u_{N} \end{matrix} ]⟩_{𝒲⊕𝒰}  (since x_{N} = 0)
= ⟨w̃, Δw̃⟩_{𝒲} + ⟨(u + (D_{2}^{∗}D_{2})^{−1}D_{2}^{∗}D_{1}w̃) , D_{2}^{∗}D_{2}(u + (D_{2}^{∗}D_{2})^{−1}D_{2}^{∗}D_{1}w̃) ⟩_{𝒰}
≥ ⟨w̃, Δw̃⟩_{𝒲}  (since D_{2}^{∗}D_{2} > 0.)

Since w̃ ∈ 𝒲 is arbitrary, it then follows that Δ < 0 and hence, as discussed in Remark 2.19, that Δ is boundedly invertible. As such, it is possible to conclude from equation (2.16), that D̃^{∗}JD̃ is boundedly invertible.

2.5 SD Control-Systems and State-Space Time-Lifting

The purpose of this section is to illustrate the idea of time-lifting in the context of state-space systems and to collect some results concerning stability and induced-norm analysis of SD controlsystems expressed in linear-fractional form. For further details the reader is referred to [FG88, CF90, CF91, BP92].

[figure: A_{D}; K; D_{A}; G; w(t); u(t); z(t); y(t); y_{k}; u_{k}]

Figure 2.2 SD Control System

The lower, linear-fractional configuration shown in Figure 2.2, is a convenient setup for the synthesis of (SD) control-systems. G =:[ \begin{matrix}G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix} ] is called the generalised plant and it is typically constructed from a system that is to be controlled and parametric weights to reflect closed-loop objectives. For example, consider Figure 2.1 with the controller C removed. Then the system

[ \begin{matrix}[ & \begin{matrix}d_{1} \\ d_{2} \end{matrix} & ] \\ & y_{2} \end{matrix} ] ↦ [ \begin{matrix}[ & \begin{matrix}u_{1} \\ y_{1} \end{matrix} & ] \\ & u_{2} \end{matrix} ],

is an example of a generalised plant that allows the standard closed-loop configuration to be expressed in lower, linear-fractional form. The closed-loop operator mapping w to z in Figure 2.2, is often denoted by the lower, linear-fractional transformation (LFT) F_{ℓ}(G, K), defined by

F_{ℓ}(G, K) :=G_{11} + G_{12}K(I − G_{22}K)^{−1}G_{21},

where K :=D_{A}KA_{D}. Typically, in an LFT framework, the objective of synthesis is to obtain a stabilising controller K such that some quantifiable criteria expressible in terms of F_{ℓ}(G, K) is satisfied (for example, ∥F_{ℓ}(G, K)∥ < γ for some γ > 0). Important issues concerning synthesis in this framework are presented in the rest of this section, where they are discussed in the context of SD control-systems (with an induced-norm synthesis criteria) and in terms of systems that admit state-space realisations.

Let G =:[ \begin{matrix}G_{11} & G_{12} \\ G_{21} & G_{22} \end{matrix} ] in Figure 2.2 be an LTI, continuous-time generalised-plant expressible in terms of the state-space system of differential equations

ẋ(t) = Ax(t) + B_{1}w(t) + B_{2}u(t); x(0) = 0,  (2.17a)
z(t) = C_{1}x(t) + D_{11}w(t) + D_{12}u(t),  (2.17b)
y(t) = C_{2}x(t),  (2.17c)

where x(t) ∈ ℝ^{n} is the state vector, w(t) ∈ ℝ^{m_{1}} is the disturbance input, u(t) ∈ ℝ^{m_{2}} is the control input, z(t) ∈ ℝ^{p_{1}} is the controlled output, y(t) ∈ ℝ^{p_{2}} is the measured output and A, B_{•}, C_{•} and D_{•} are all constant matrices, where “•” denotes either 1 or 2 here. Since this state-space realisation of the generalised plant is (implicitly) used for synthesis in the sequel, the standard (cf. [ZDG95]) assumption that the pair (A, B_{2}) is stabilisable and (C_{2}, A) is detectable in the sense that 𝒦_{B_{2}^{∗}} ∩ 𝒦_{(λI−A)^{∗}} = {0} and 𝒦_{C_{2}} ∩ 𝒦_{(λI−A)} = {0} for all λ ∈ {(σ + jω) ∈ ℂ : σ ∈ ℝ^{+}, ω ∈ ℝ}, is made. Let the controller K be a LSI, discrete-time system with state-space realisation, A_{D} denote an ideal, periodic (with period h) sampling-device and D_{A} a zeroth-order hold-device, which is assumed to be synchronised with A_{D}. It is well-known [CF91], that the zero-order hold device D_{A} is bounded on ℓ_{ℤ^{+}}^{2}(·) with unit norm, but that the ideal sampling device A_{D} is not bounded on the whole of L_{ℝ^{+}}^{2}(·). However, given any stable, low-pass, LTI filter F say, it can be shown that A_{D}F is bounded on L_{ℝ^{+}}^{2}(·) [CF91]. As such, a sampling device is invariably preceded by a stable, strictly-causal, LTI filter, which is used to ensure boundedness of A_{D} and to limit frequencydomain aliasing of measured information. This is not shown in Figure 2.2, because it is assumed that such a filter has already been absorbed into the generalised plant, which explains why no direct feedthrough terms from the disturbance and control inputs to the measured output appear in the state-space realisation of the generalised plant given in equations (2.17).

Analysis and synthesis of SD control-systems (cf. Figure 2.2) is complicated by the fact that the measured-output and control-input signals as “seen” by K evolve in discrete-time, whereas the disturbance and controlled outputs, by which performance is often determined, evolve in continuoustime. Compounding this, since it is assumed that the sampling and hold devices are synchronised and periodic, the closed-loop system mapping the disturbance input w to the controlled output z is LPTV. These difficulties can be alleviated however, by characterising the evolution of the generalised-plant in terms of a system of LSI, difference equations, so that tools similar to those presented in Section 2.4 for LSI, state-space systems can be used. It is important that loss of continuous-time information does not occur in this process. As such, to capture all disturbanceinput information and controlled-output information in discrete-time, it is necessary to use signals that take infinite-dimensional values at each point in time. To this end, the time-lifting isomorphism defined in Sub-Section 2.3.1 as

f^{⊸}_{k}(θ) :=(Wf)_{k} :=f(kh + θ) (θ ∈ ℍ),

is used to map the continuous-time signals to discrete-time signals. Using the time-lifting isomorphism also overcomes the difficulty imposed by periodic time-variation, since it intertwines the continuous-time and discrete-time unilateral shift as follows: WU_{kh} = S^{k}W for all k ∈ ℤ^{+} (cf. Proposition 2.4). Now, absorbing the sampling and hold devices into G and expressing the evolution of this system in terms of the state at the sampling instants kh (k ∈ ℤ^{+}), yields the LSI system of difference equations

x_{k+1} = Ax_{k} + B_{1}w^{⊸}_{k} + B_{2}u_{k};  x_{0} = 0,  (2.18a)
z^{⊸}_{k} = C_{1}x_{k} + D_{11}w^{⊸}_{k} + D_{12}u_{k},  (2.18b)
y_{k} = C_{2}x_{k},  (2.18c)

where x_{k} = x(kh), z^{⊸}_{k}(θ) = (Wz)_{k}(θ) = z(kh + θ), w^{⊸}_{k}(θ) = (Ww)_{k}(θ) = w(kh + θ), y_{k} =

(A_{D}y)_{k} = y(kh) and for all w^{⊸} ∈ L_{ℍ}^{2}(ℝ^{m_{1}}), x ∈ ℝ^{n} and u ∈ ℝ^{m_{2}},

A :ℝ^{n} → ℝ^{n},  Ax:=e^{hA}x,  (2.19)
B_{2} : ℝ^{m_{2}} → ℝ^{n},  B_{2}u:= ∫_{0}^{h} e^{(h−τ)A} dτ B_{2}u,  (2.20)
B_{1} :L_{ℍ}^{2}(ℝ^{m_{1}}) → ℝ^{n},  B_{1} w^{⊸}:= ∫_{0}^{h} e^{(h−τ)A}B_{1} w^{⊸}(τ) dτ,  (2.21)
C_{1} :ℝ^{n} → L_{ℍ}^{2}(ℝ^{p_{1}}),  (C_{1}x)(θ) :=C_{1}e^{θA}x,  (2.22)
C_{2} :ℝ^{n} → ℝ^{p_{2}},  C_{2}x:=C_{2}x,  (2.23)
D_{11} :L_{ℍ}^{2}(ℝ^{m_{1}}) → L_{ℍ}^{2}(ℝ^{p_{1}}),  (D_{11} w^{⊸})(θ) :=D_{11} w^{⊸}(θ) + ∫_{0}^{θ} C_{1}e^{(θ−τ)A}B_{1} w^{⊸}(τ) dτ  (2.24)

and

D_{12} :ℝ^{m_{2}} → L_{ℍ}^{2}(ℝ^{p_{1}}),  (D_{12}u)(θ) :=D_{12}u + ∫_{0}^{θ} C_{1}e^{(θ−τ)A} dτB_{2}u.  (2.25)

This constitutes a state-space realisation of the LSI, discrete-time system

G:=[ \begin{matrix}WG_{11}W^{−1} & WG_{12}D_{A} \\ A_{D}G_{21}W^{−1} & A_{D}G_{22}D_{A} \end{matrix} ].

Importantly, a sufficient condition for the 22-block of this state-space realisation to be stabilisable and detectable can be characterised in terms of the original continuous-time, state-space realisation of G. First though, the following technical definition is required: it is said that the \underline{sampling is not pathological} if no two eigenvalues of A, which lie on a line parallel to the imaginary axis in the complex plane, are separated by an integer multiple of the sampling frequency (\frac{2π}{h} ). That is, if the sampling is not pathological, then no two continuous-time modes are aliased to the same discrete-time mode by sampling. It can be shown that if the sampling is not pathological, then (A, B_{2}) stabilisable implies (A, B_{2}) stabilisable and (C_{2}, A) detectable implies (C_{2}, A) detectable [CF95]. As such, it is assumed henceforth that the sampling is not pathological and since (A, B_{2}, C_{2}) stabilisable and detectable is a standing assumption, it then follows that the 22-block of the state-space realisation of G given in equations (2.18) is both stabilisable and detectable.

Under the standing assumptions just mentioned, given a stabilisable and detectable state-space realisation

[ \begin{matrix}A_{K} & B_{K} \\ C_{K} & D_{K} \end{matrix} ],

of the LSI, discrete-time controller K, the closed-loop system F_{ℓ}(G, K) is said to be internally stable if and only if the homogeneous component (w^{⊸} set to zero) of the closed-loop, state-space realisation is stable. That is, if and only if

spec (A_{cl} :=[ \begin{matrix}A & 0 \\ 0 & A_{K} \end{matrix} ] + [ \begin{matrix}B_{2} & 0 \\ 0 & B_{K} \end{matrix} ][ \begin{matrix}I & 0 \\ −D_{K} & I \end{matrix} ]^{−1}[ \begin{matrix}0 & C_{K} \\ C_{2} & 0 \end{matrix} ]) ⊂ 𝔻.

(Note that for spec(A_{cl}) to lie inside the open unit disc it is necessary for both (A, B_{2}, C_{2}) and (A_{K}, B_{K}, C_{K}) to be stabilisable and detectable.) The notion of internal stability just defined, corresponds to stability of the lower (closed) loop of the linear-fractional configuration in the sense discussed in Sub-Section 2.3.3 for standard closed-loop configurations, and closed-loop, asymptotic stability to zero of the states of G and K starting from any initial states (and initial time) when w^{⊸} ≡ 0. The latter condition (involving closed-loop, asymptotic stability to zero of the plant and controller states) is often used to define internal stability for LFT configurations of general statespace systems, as is the case for the continuous-time plant with sampled-data controller here. Now since x_{k} is always equal to x(kh) and since for any time t = kh + θ (θ ∈ ℍ), the state of the original, continuous-time generalised-plant when operating in closed-loop (with w set to zero) can be expressed as

x(kh + θ) = (e^{θA} + ∫_{0}^{θ} e^{(θ−τ)A}B_{2} dτD_{K}C_{2}) x_{k} + ∫_{0}^{θ} e^{(θ−τ)A}B_{2} dτC_{K}s_{k},

where s_{k} denotes the state of the controller, it follows that under the standing assumptions mentioned above, the LSI, discrete-time, closed-loop system F_{ℓ}(G, K) is internally stable if and only if the LPTV, continuous-time, closed-loop system F_{ℓ}(G, D_{A}KA_{D}) is internally stable. Furthermore, if the LFT configuration is internally stable then the system mapping w to z is bounded and moreover, since F_{ℓ}(G, D_{A}KA_{D}) = W^{−1}F_{ℓ}(G, K)W and W is an isomorphism, ∥F_{ℓ}(G, D_{A}KA_{D})∥ = ∥F_{ℓ}(G, K)∥ (cf. also [CF95, BP92]). As such, synthesis of K to achieve internal stability of F_{ℓ}(G, D_{A}KA_{D}) and a specified bound on the closed-loop induced-norm ∥F_{ℓ}(G, D_{A}KA_{D})∥, is equivalent to synthesis of a K to achieve internal stability of F_{ℓ}(G, K) and the same bound on ∥F_{ℓ}(G, K)∥.

Synthesis of a controller K to achieve a specified bound on ∥F_{ℓ}(G, K)∥ is complicated by the fact that the input and output spaces of the closed-loop are infinite-dimensional. Fortunately, there exist methods for transforming the problem into an equivalent finite-dimensional one [BP92, Toi92, Toi93, CF95]. This is possible because the state-space operators B_{1} : L_{ℍ}^{2}(ℝ^{m_{1}}) → ℝ^{n}, C_{1} : ℝ^{n} → L_{ℍ}^{2}(ℝ^{p_{1}}) and D_{12} : ℝ^{m_{2}} → L_{ℍ}^{2}(ℝ^{p_{1}}) all have finite-rank and therefore, finite-dimensional matrix representations. However, D_{11} : L_{ℍ}^{2}(ℝ^{m_{1}}) → L_{ℍ}^{2}(ℝ^{p_{1}}) does not have finite-rank and as such, an additional transformation is required to handle this. The required transformation is an operator version of the loop-shifting principle (cf. [SLC90]) as summarised in the following lemma:

Proposition 2.24 [BP92] Consider the LSI, discrete-time generalised-plant G̃ with state-space realisation

[ \begin{matrix}Ã & B̃_{1} & B̃_{2} \\ C̃_{1} & 0 & D̃_{12} \\ C_{2} & 0 & 0 \end{matrix} ],  (2.26)

where

Ã := A + γ^{−2}B_{1}D_{11}^{∗}S^{−1}C_{1} : ℝ^{n} → ℝ^{n},  (2.27)
B̃_{1} := γ^{−1}B_{1}R^{− \frac{1}{2}} : L_{ℍ}^{2}(ℝ^{m_{1}}) → ℝ^{n},  (2.28)
B̃_{2} := γ^{−2}B_{1}D_{11}^{∗}S^{−1}D_{12} + B_{2} : ℝ^{m_{2}} → ℝ^{n},  (2.29)
C̃_{1} := γS^{− \frac{1}{2}} C_{1} : ℝ^{n} → L_{ℍ}^{2}(ℝ^{p_{1}}),  (2.30)
D̃_{12} := γS^{− \frac{1}{2}} D_{12} : ℝ^{m_{2}} → L_{ℍ}^{2}(ℝ^{p_{1}}),  (2.31)

R:=(I − γ^{−2}D_{11}^{∗}D_{11}) and S:=(I − γ^{−2}D_{11}D_{11}^{∗}). Provided ∥D_{11}∥ < γ and given an LSI, discretetime controller K, the following are equivalent:

(i) F_{ℓ}(G, K) is internally stable and ∥F_{ℓ}(G, K)∥ < γ

(ii) F_{ℓ}(G̃, K) is internally stable and ∥F_{ℓ}(G̃, K)∥ < γ.

Note that in analogy with the standard finite-dimensional H^{∞} problem, the assumption γ > ∥D_{11}∥ is required. This ensures that R and S are well defined and as noted in [Toi93], is equivalent to the condition that for fixed k ∈ ℤ^{+}, given any bounded x_{k} ∈ ℝ^{n} and u_{k} ∈ ℝ^{m_{2}},

⟨ z^{⊸}_{k}, z^{⊸}_{k}⟩_{L_{ℍ}^{2}(ℝ^{p_{1}})} − γ^{2}⟨w^{⊸}_{k}, w^{⊸}_{k}⟩_{L_{ℍ}^{2}(ℝ^{m_{1}})}

has bounded supremum over w^{⊸}_{k} ∈ L_{ℍ}^{2}(ℝ^{m_{1}}). To ensure that this assumption is not violated, a procedure for determining a bound on ∥D_{11}∥ is needed. Since D_{11} : L_{ℍ}^{2}(ℝ^{m_{1}}) → L_{ℍ}^{2}(ℝ^{p_{1}}) is not a compact operator, this is not simply a matter of bounding the maximum eigenvalue of D_{11}^{∗}D_{11} (cf. Sub-Section 2.2.2 or [Kre89b]). However, using the principle of loop-shifting again, a compact operator Ď_{11} : L_{ℍ}^{2}(ℝ^{m_{1}}) → L_{ℍ}^{2}(ℝ^{p_{1}}) can be constructed such that ∥D_{11}∥ < γ ⇔ ∥Ď_{11}∥ < γ. Given a v^{⊸} ∈ L_{ℍ}^{2}(ℝ^{m_{1}}), the action of g^{⊸} = Ď_{11} v^{⊸} can be expressed in terms of the system of differential equations

\frac{d a^{⊸}}{dθ} = Ǎa^{⊸}(θ) + B̌ v^{⊸}(θ); a^{⊸}(0) = 0,  (2.32a)
g^{⊸}(θ) = Č a^{⊸}(θ),  (2.32b)

where R:=(I − γ^{−2}D_{11}^{∗}D_{11}), S :=(I − γ^{−2}D_{11}D_{11}^{∗}),

Ǎ := A + γ^{−2}B_{1}R^{−1}D_{11}^{∗}C_{1} : ℝ^{n} → ℝ^{n},
B̌ := γ^{−1}B_{1}R^{− \frac{1}{2}} : L_{ℍ}^{2}(ℝ^{p_{1}}) → ℝ^{n}

and

Č :=γS^{− \frac{1}{2}} C_{1} : ℝ^{n} → L_{ℍ}^{2}(ℝ^{p_{1}}).

Note that provided \overline{σ}\{D_{11}\} < γ, R^{− \frac{1}{2}} and S^{− \frac{1}{2}} are well defined. Now since Ď_{11} is compact,^{10} it follows that ∥D_{11}∥ < γ if and only if \overline{σ}\{Ď_{11}\} < γ (cf. Sub-Section 2.2.2). This is why the compact operator Ď_{11} is defined the in way it is, since it is possible to characterise in a computationallytractable manner the non-zero singular values of a compact operator described by a system of differential equations such as that given in equation (2.32) (cf. Chapters 5 and 6 for details of such characterisations). Let

E :=[ \begin{matrix}−A^{∗} − γ^{−2}C_{1}^{∗}D_{11}R^{−1}B_{1}^{∗} & −C_{1}^{∗}S^{−1}C_{1} \\ γ^{−2}B_{1}R^{−1}B_{1}^{∗} & A + γ^{−2}B_{1}R^{−1}D_{11}^{∗}C_{1} \end{matrix} ],  (2.33)

and

Q(t) =: [ \begin{matrix}Q_{11}(t) & Q_{12}(t) \\ Q_{21}(t) & Q_{22}(t) \end{matrix} ]:=e^{tE}.  (2.34)

Lemma 2.25 Given a real number γ > \overline{σ}\{D_{11}\}, then γ is a singular value of Ď_{11} if and only if det(Q_{11}(h)) = 0, where det(·) denotes the determinant of a matrix (cf. [GVL89]).

Proof : The proof is essentially the same as that of Theorem 13.5.1 in [CF95]. See also Sub-Section 5.4.3 for a similar result.

Remark 2.26 Clearly, with ∥D_{11}∥ < γ the matrix Q_{11}(h) is invertible. ♣

Although it has been suppressed for notational convenience, det(Q_{11}(h)) is a function of γ. Define the function d(γ) :=det(Q_{11}) : ℂ → ℝ. Then in view of Lemma 2.25 and since Ď_{11}^{∗} Ď_{11} is a selfadjoint, compact operator, the zeros of d(γ) constitute a discrete, real set with zero being the only possible accumulation point (cf. Sub-Section 2.2.2). Furthermore, with γ ↦ μ:=1/γ, d(μ) is an analytic function in the whole of the complex plane and the zeros of d(μ) form a discrete, real set with no finite accumulation points. So with wno_{r}(d) defined to be the number of clock-wise encirclements of the origin made by d(μ) as μ traverses the standard Nyquist contour of radius r,^{11} the following corollary is a simple consequence of Lemma 2.25 and the Principle of the Argument [MH87].^{12}

Corollary 2.27 Given γ > ∥D_{11}∥,

∥D_{11}∥ < γ ⇔ ∥Ď_{11}∥ < γ ⇔ wno_{\frac{1}{γ}} (d) = 0.

10Ď_{11} is compact because it corresponds to an integral operator with no direct feed-through term [Kre89a].

11A semi-circle (centred at the origin) into the right-half plane of radius r.

12Prof. Bassam Bamieh must be acknowledged for first suggesting the use of winding numbers in this context.

Having established a method for determining a bound on ∥D_{11}∥, it is now appropriate to state the important result of [BP92], which implicitly characterises a finite-dimensional (in terms of input, output and state spaces) equivalent of G̃.

Proposition 2.28 [BP92] Given the LSI, discrete-time generalised-plant G̃ with state-space realisation (2.26), obtain the symmetric matrix-factorisations

B̃_{1}B̃_{1}^{∗} = T ∗ ; B̃_{1}[ \begin{matrix}ΣB̃_{1} & 0 \\ 0 & 0 \end{matrix} ]TB̃_{1},
[ C̃_{1}^{∗} ; D̃_{12}^{∗} ][ C̃_{1} D̃_{12} ] = T_{C̃_{1},D̃_{12}}^{∗}[ \begin{matrix}Σ_{C̃_{1},D̃_{12}} & 0 \\ 0 & 0 \end{matrix} ]T_{C̃_{1},D̃_{12}}

and define the (completely) \underline{finite-dimensional}, LSI, discrete-time generalised-plant Ğ to have statespace realisation

[ \begin{matrix}Ã & B̆_{1} & B̃_{2} \\ C̆_{1} & 0 & D̆_{12} \\ C_{2} & 0 & 0 \end{matrix} ]  (2.35)

where

B̆_{1} :=T ∗ ; B̃_{1} [ \begin{matrix}ΣB̃_{1}^{\frac{1}{2}} \\ 0 \end{matrix} ]  (2.36)

and

[ C̆_{1} D̆_{12} ]:=[ ΣC̃_{1},D̃_{12}^{\frac{1}{2}} 0 ]T_{C̃_{1},D̃_{12}}.  (2.37)

Provided γ > ∥D_{11}∥ and given a LSI, discrete-time controller K, the following are equivalent:

(i) F_{ℓ}(G̃, K) is internally stable and ∥F_{ℓ}(G̃, K)∥ < γ
(ii) F_{ℓ}(Ğ, K) is internally stable and ∥F_{ℓ}(Ğ, K)∥ < γ.

Remark 2.29 Recall that under the standing assumptions that the sampling is not pathological and that (A, B_{2}, C_{2}) is stabilisable and detectable,

F_{ℓ}(G, K) is internally stable and ∥F_{ℓ}(G, K)∥ < γ
⇕
F_{ℓ}(G, D_{A}KA_{D}) is internally stable and ∥F_{ℓ}(G, D_{A}KA_{D})∥ < γ.

So in view of Propositions 2.28 and 2.24, synthesis of an internally-stabilising, LSI, discrete-time controller K to achieve a specified bound on ∥F_{ℓ}(G, D_{A}KA_{D})∥ is equivalent to the synthesis of K to achieve internal stability for F_{ℓ}(Ğ, K) and the same bound on ∥F_{ℓ}(Ğ, K)∥. Although singular in the sense that D̆_{21} = 0, the latter induced-norm synthesis problem can be solved using the LMI framework of [GA94] or often, by using a bilinear transformation to a continuous-time problem [KHI90]. As such, Ğ is referred to as the \underline{SD-equivalent generalised-plant}. Computation of the state-space matrices for Ğ is discussed in detail in Chapter 6, including related numerical issues.

♣

Frequency-Domain Analysis and SD Control Design

3.1 Introduction

In this chapter a new frequency-domain, analysis tool is developed for studying the performance of LPTV systems. The motivation for this stems from the wide-spread use of frequency-domain, analysis tools in the important “design step” of control-system development. Development of control systems usually involves two major steps: first, a design step, in which a generalised plant is constructed from the system to be controlled and parametric weights that reflect closed-loop objectives; second, a synthesis step, in which a stabilising controller is synthesised by solving an optimisation problem involving the generalised plant obtained in the design step. These two main steps are often iterated in conjunction with an intermediate analysis step, in which tools such as those developed in Chapter 4 for example, are used. The H^{∞}-loopshaping procedure of McFarlane and Glover [MG90] for linear, time-invariant (LTI) systems, is an intuitive and reasonably wellunderstood example of such a control-system development procedure. It is based on classical loopshaping ideas in the design step, by which the singular values of the frequency-response matrix of the plant are shaped open-loop to reflect closed-loop objectives. A four-block generalised plant is then constructed from the shaped plant and an H^{∞}-optimisation is solved in the synthesis step. The ultimate goal of this chapter is to develop a similar design procedure for SD, control-system development. This first requires a suitable definition of frequency response for LPTV systems.

To date, most of the work on frequency-domain, analysis methods for LPTV systems has been presented in the context of SD systems [AI93, AHI93, YA94, GS94, YK96]. The standard definition of frequency response for LPTV systems is often expressed in terms of the induced-norm of a frequency-response operator, which characterises the asymptotic response of an LPTV system to an infinite series of sinusoidal signals of frequencies separated by integer multiples of the frequency of periodicity. Such a definition is useful for frequency-domain robustness-analysis of SD, closedloop systems [Dul94]. However, from the perspective of designing for nominal performance, it is perhaps more meaningful and certainly more intuitive to characterise the asymptotic response of a system to sinusoidal inputs of a single frequency. For example, in disturbance rejection problems it is useful to know if the “gain” of the system is small for signals with power concentrated in a specific band of frequencies.

The robustness properties of a closed-loop system developed using any given procedure are to a large extent (but not completely) governed by the structure of the generalised plant defined in the design step and the type of optimisation problem solved in the synthesis step (cf. Chapter 4 for a suitable framework in which quantitative robustness result can be obtained). Bearing this in mind, a control-system development procedure in which design for performance is based on the response of performance-indicating, closed-loop operators to sinusoids of a single frequency, is sensible provided a suitable (from the perspective of robustness) generalised-plant structure and optimisation formulation are employed. Motivated by this, in the following section the \underline{average-power gain matrix} is defined for LPTV systems. The singular values of this finite-dimensional matrix characterise the average-power gain (in input-directions corresponding to the singular vectors) of the asymptotic response of a system to sinusoidal inputs of a single frequency. In fact, when the system is LTI, the non-zero singular values of the average-power gain matrix correspond to the non-zero singular values of the standard, continuous-time, frequency-response matrix. This new frequency-domain, analysis tools is used to derive frequency-by-frequency bounds on performance-indicating, closedloop operators, which facilitate the design of weights used to construct a generalised plant in an H^{∞}-loopshaping based design procedure proposed for SD, control-system development. State-space formulae for computing the average-power gain matrix are presented for the sampled-data case in Appendix B.2.

3.2 The Average-Power Gain Matrix

In this section, a new notion of frequency response is developed for LPTV systems. It is based on the average power of the asymptotic response to sinusoidal inputs of a single frequency and hence, differs from the standard definition of frequency response for LPTV systems [YA94, YK96].

3.2.1 Frequency Response

For notational convenience, let 𝒰:=ℝ^{m} and 𝒴:=ℝ^{p}. Now consider a stable system P ∈ 𝒫_{𝒰→𝒴} (cf. Definition 2.3) with period h and recall that P is equivalent via the time-lifting isomorphism W to a stable, LSI system

P:=WP W^{−1} ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} .

Since P is a stable system, it has a locally Lipschitz-continuous extension and hence, its block-Toeplitz representation

[ \begin{matrix}P_{[0]} & 0 & ⋯ & ⋯ \\ P_{[1]} & P_{[0]} & 0 & ⋯ \\ ⋮ & ... & ... & ... \end{matrix} ]

can be uniquely identified with the sequence {P_{[i]}}_{i=0}^{∞} ⊂ B_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)}. Furthermore, by Proposition 2.2, P is equivalent via the Z-Transform isomorphism to a multiplication operator with symbol

P̂(λ) := ∑_{i=0}^{∞} P_{[i]}λ^{i} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)})

and ∥P ∥ = ∥P∥ = ∥P̂∥_{∞}. In the sequel, P̂(λ) is referred to as the transfer operator of P. For technical convenience it is assumed throughout this chapter only, that the sequence uniquely associated with the Toeplitz representation of the time-lifted equivalent of any stable system considered, for example P ∈ 𝒫_{𝒰→𝒴} here, satisfies ∑_{i=0}^{∞} ∥P_{[i]}∥ < ∞. In this way, the corresponding transfer operator P̂(λ) will have continuous extension to 𝕋. That is, P̂(e^{jωh}) is defined for all ω ∈ [0, \frac{2π}{h}).

Central to the standard (cf. [YA94, YK96]) definition of frequency response for LPTV systems is the following lemma, which characterises the asymptotic response of P to a power signal.

Lemma 3.1 [Yam94] Consider an input to P of the form

u(kh + θ) = μ^{k} u^{⊸}_{d}(θ), u^{⊸}_{d} ∈ L_{ℍ}^{2}(𝒰), |μ| ≥ 1,

for k ∈ ℤ^{+} and θ ∈ ℍ. Let y(t) = (P u)(t). Then y(t) asymptotically approaches

y_{as}(t) = y_{as}(kh + θ) = μ^{k}(P̂(μ^{−1}) u^{⊸}_{d})(θ)

as t → ∞.

Taking μ = e^{jωh} (ω ∈ [0, \frac{2π}{h})), the signal

u := Wu := k ↦ e^{jωhk} u^{⊸}_{d} (k ∈ ℤ^{+})

is simply a discrete-time sinusoid with direction u^{⊸}_{d} in the infinite-dimensional, Hilbert space L_{ℍ}^{2}(𝒰). It follows by Lemma 3.1, that the response of P to such a signal asymptotically approaches a discrete-time sinusoid of the same frequency, but new direction given by P̂(e^{−jωh}) u^{⊸}_{d}. This is by no means surprising, since P is a stable, LSI system. Note however, that unless u^{⊸}_{d} takes a particular form, the signal u_{k} = e^{jωhk} u^{⊸}_{d} does not correspond (via W) to a sinusoidal signal of a single frequency in continuous-time. In fact, since for any fixed ω_{0} ∈ [0, \frac{2π}{h}) the sequence of functions

\{h^{− \frac{1}{2}} e^{j( \frac{2πn}{h} −ω_{0})θ}u_{b}_{i}\}_{n∈ℤ, i=1,... ,m} ⊂ L_{ℍ}^{2}(𝒰),

where {u_{b}_{i}}_{i=1,... ,m} is any complete, orthonormal basis for the m-dimensional, Hilbert space 𝒰, constitutes a complete orthonormal basis for L_{ℍ}^{2}(𝒰) [Dul94, YA94],^{1} it is readily verifiable that in general, the signal u:=k ↦ e^{jωhk} u^{⊸}_{d} (k ∈ ℤ^{+}) corresponds to an infinite series of continuous-time sinusoids with frequencies separated by integer multiples of the frequency of periodicity (\frac{2π}{h}). ^{2}

The standard definition of frequency response for LPTV systems is the L_{ℍ}^{2}-induced norm of P̂(e^{jωh}) (ω ∈ [0, \frac{2π}{h})). This characterises the maximal gain over all directions in L_{ℍ}^{2}(𝒰) and as such corresponds to characterising the maximal, asymptotic gain of the system to an infinite series of continuous-time sinusoids, each separated by integer multiples of the frequency of periodicity. Although this definition is useful for studying the robustness of closed-loop stability and performance to LTI uncertainty for example [Dul94], all input-directional information is lost in the definition and its usefulness from the perspective of performance is not immediately clear. With this in mind, a new definition of frequency response is developed in what follows, in terms of the average power of the asymptotic response to sinusoidal inputs of single frequencies.

Observe that a continuous-time, sinusoidal signal

u(t) = e^{jωt}u_{d} (u_{d} ∈ 𝒰:=ℝ^{m})

of frequency ω ∈ ℝ, corresponds (via W) to a discrete-time sinusoid

u_{k} = e^{jω_{0}hk} u^{⊸}_{d},

where

u^{⊸}_{d}(θ) = e^{jωθ}u_{d} ∈ L_{ℍ}^{2}(𝒰),

1To see this simply take the Fourier series expansion of the periodic extension of any function in L_{ℍ}^{2}(𝒰) .

2Correspondingly, by Lemma 3.1, the asymptotic response to such a signal is in general an infinite-series of sinusoids with frequencies separated by integer multiples of the frequency of periodicity. By this, it can be said that such signals are asymptotically invariant for LPTV systems in the same way that sinusoids of a single frequency are asymptotically invariant for LTI systems.

ω_{0} = ω−]\frac{hω}{2π} [ (\frac{2π}{h}) and ] · [ denotes the integer part of a real number. As such, it follows by Lemma 3.1, that the asymptotic response of an LPTV system to a continuous-time, sinusoidal input of frequency ω is given by

y_{as}^{ω}(t) = y_{as}^{ω}(kh + θ) = (e^{jω_{0}h})^{k}(P̂(e^{−jω_{0}h}) u^{⊸}_{d})(θ).  (3.1)

Again, note that this corresponds to a discrete-time sinusoid with direction P̂(e^{−jω_{0}h}) u^{⊸}_{d} ∈ L_{ℍ}^{2}(𝒰), but that in general, y_{as}^{ω}(t) is not simply a continuous-time sinusoid of a single frequency. Despite this, it is possible to characterise the average power of the asymptotic response as follows:

Lemma 3.2 Let y_{as}^{ω}(t) be as defined in (3.1). Then,

∥y_{as}^{ω}∥_{pow} := \sqrt{lim_{T →∞} \frac{1}{T} ∫_{0}^{T} ⟨y_{as}^{ω}(t), y_{as}^{ω}(t)⟩_{𝒴} dt}
= \frac{1}{\sqrt{h}}∥P̂(e^{−jω_{0}h}) u^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒴)}.

Proof : Let y^{⊸} :=P̂(e^{−jω_{0}h}) u^{⊸}_{d} ∈ L_{ℍ}^{2}(𝒴). Then,

∥y_{as}^{ω}∥_{pow}^{2} := lim_{T →∞} \frac{1}{T} ∫_{0}^{T} ⟨y_{as}^{ω}(t), y_{as}^{ω}(t)⟩_{𝒴} dt
= lim_{N→∞} \frac{1}{Nh} ∑_{k=0}^{N−1} ∫_{kh}^{(k+1)h} ⟨y_{as}^{ω}(t), y_{as}^{ω}(t)⟩_{𝒴} dt
= lim_{N→∞} \frac{1}{Nh} ∑_{k=0}^{N−1} ⟨ y^{⊸}, y^{⊸}⟩_{L_{ℍ}^{2}(𝒴)}
= \frac{1}{h}∥ y^{⊸}∥_{L_{ℍ}^{2}(𝒴)}^{2}.

This completes the proof.

For each u_{d} ∈ 𝒰, let the B_{𝒰→L_{ℍ}^{2}(𝒴)} -valued function Γ_{P} (jω) : jℝ → B_{𝒰→L_{ℍ}^{2}(𝒴)} be defined by

y^{⊸} :=Γ_{P} (jω)u_{d} :=P̂(e^{−jω_{0}h})Ω(jω)u_{d} ∈ L_{ℍ}^{2}(𝒴),  (3.2)

where Ω(jω) : jℝ → B_{𝒰→L_{ℍ}^{2}(𝒰)} is the B_{𝒰→L_{ℍ}^{2}(𝒰)} -valued function defined by

u^{⊸}_{d} :=Ω(jω)u_{d} :=e^{jωθ}u_{d} ∈ L_{ℍ}^{2}(𝒰)  (3.3)

and ω_{0} :=ω − ]\frac{hω}{2π} [ \frac{2π}{h}. Then by Lemma 3.2,

∥y_{as}^{ω}∥_{pow} = \frac{1}{\sqrt{h}}\sqrt{⟨Γ_{P} (jω)u_{d}, Γ_{P} (jω)u_{d}⟩_{𝒴}} = \frac{1}{\sqrt{h}}\sqrt{⟨(Γ_{P} (jω))^{∗}Γ_{P} (jω)u_{d}, u_{d}⟩_{𝒰}} . (3.4)

For fixed ω ∈ ℝ, (Γ_{P} (jω))^{∗}Γ_{P} (jω) is a square, symmetric, positive-semidefinite matrix and hence, its square root exists. With this in mind, define

Θ_{P} (jω) :=\frac{1}{\sqrt{h}}\{(Γ_{P} (jω))^{∗}Γ_{P} (jω)\}^{\frac{1}{2}} ,  (3.5)

noting from equation (3.4), that the average power of the asymptotic response can be characterised in terms of the singular values of this matrix function of frequency. Accordingly the following definition is made:

Definition 3.3 The frequency-dependent m × m matrix

Θ_{P} (jω) :=\frac{1}{\sqrt{h}}\{(Γ_{P} (jω))^{∗}Γ_{P} (jω)\}^{\frac{1}{2}} ,

where Γ_{P} (jω) is defined in equation (3.2), is defined to be the the average-power gain matrix of the LPTV system P. ♠

Remark 3.4 Note that any LTI system F is LPTV and as such, if F is also stable, that Θ_{F} (jω) is well defined. In fact, by the way the average-power gain matrix has been defined, it follows that Θ_{F} (jω) = (F̂(jω)^{∗}F̂(jω))^{\frac{1}{2}} , where F̂(jω) denotes the standard, continuous-time, frequencyresponse matrix. That is, the non-zero singular values of F̂(jω) correspond to the non-zero singular values of Θ_{F} (jω). In forming Θ_{F} (jω) the phase information of F̂(jω) is lost but the gain information is retained. ♣

3.2.2 Properties of the Average-Power Gain Matrix

In this sub-section a few technical results are presented, which characterise some important properties of the average-power gain matix used in the sequel.

Lemma 3.5 Given a stable, LTI system F ∈ 𝒫_{𝒰→𝒴} with frequency-domain transfer-matrix F̂(jω), for any ω_{0} ∈ [0, \frac{2π}{h})

∥F̂(e^{−jω_{0}h})∥ = \overline{σ}_{max}^{ω_{0}}\{F̂\},

where F̂ is the transfer operator of the time-lifted, equivalent system F = WF W^{−1},

\overline{σ}_{max}^{ω_{0}}\{F̂\}:= max_{n∈ℤ} \overline{σ}\{F̂(jω_{n})\},  (3.6)

and ω_{n} :=(ω_{0} + n\frac{2π}{h}).

Proof : Consider any u^{⊸}_{d} ∈ L_{ℍ}^{2}(𝒰) and note that by taking a Fourier series expansion of the periodic extension u_{d}(t) = u_{d}(kh + θ) := u^{⊸}_{d}(θ), for any ω_{0} ∈ [0, 2π) the function u^{⊸}_{d}(θ) can be expressed as (cf. [YA94, Lemma 3.3])

u^{⊸}_{d}(θ) = ∑_{n∈ℤ}e^{jω_{n}θ}α_{n},

where ω_{n} = ω_{0} + n\frac{2π}{h} and

α_{n} = \frac{1}{h} ∫_{0}^{h} e^{−jω_{n}θ} u^{⊸}_{d}(θ) dθ ∈ 𝒰.

Furthermore,

∥ u^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒰)}^{2} = h ∑_{n∈ℤ}⟨α_{n}, α_{n}⟩_{𝒰} .  (3.7)

Now given an LTI system F, denote by F̂ the transfer operator of F:=WF W^{−1}. Then since F̂ is linear, it is possible to write

(F̂(e^{−jω_{0}h}) u^{⊸}_{d})(θ) = ∑_{n∈ℤ}(F̂(e^{−jω_{0}h})e^{jω_{n}τ} α_{n})(θ)  (3.8)

Also note that the asymptotic response of F to a sinusoidal input u(t) = e^{jω_{n}t}α_{n} is given by

y_{as}^{ω_{n}}(t) = e^{jω_{n}t}F̂(jω_{n})α_{n},

which via W corresponds to the discrete-time sinusoidal signal

y^{⊸}_{as,k}^{ω_{n}}(θ) :=y_{as}^{ω_{n}}(kh + θ) = (e^{jω̂h})^{k}e^{jω_{n}θ}F̂(jω_{n})α_{n}.  (3.9)

By Lemma 3.1,

y^{⊸}_{as,k}^{ω_{n}}(θ) :=y_{as}^{ω_{n}}(kh + θ) = (e^{jω̂h})^{k}(F̂(e^{−jω̂h})e^{jω_{n}τ} α_{n})(θ),

which combined with equation (3.9) implies

(F̂(e^{−jω_{0}h})e^{jω_{n}τ} α_{n})(θ) = e^{jω_{n}θ}F̂(jω_{n})α_{n}.

So from equation (3.8), it follows that

(F̂(e^{−jω_{0}h}) u^{⊸}_{d})(θ) = ∑_{n∈ℤ}e^{jω_{n}θ}F̂(jω_{n})α_{n}

and consequently, that

∥F̂(e^{−jω_{0}h}) u^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒴)}^{2} = h ∑_{n∈ℤ}⟨F̂(jω_{n})α_{n}, F̂(jω_{n})α_{n}⟩_{𝒴}
≤ h max_{n∈ℤ} \overline{σ}\{F̂(jω_{n})\}^{2} ∑_{n∈ℤ}⟨α_{n}, α_{n}⟩_{𝒰}
= max_{n∈ℤ} \overline{σ}\{F̂(jω_{n})\}^{2} ∥ u^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒰)}^{2} ,

where the last equality follows from equation (3.7). That is,

\frac{∥F̂(e^{−jω_{0}h}) u^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒴)}}{∥ u^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒰)}} ≤ max_{n∈ℤ} \overline{σ}\{F̂(jω_{n})\}.  (3.10)

In fact by taking u^{⊸}_{d}(θ) = \frac{1}{\sqrt{h}}e^{jω_{nmax}θ}u_{d}_{max}, where n_{max} :=arg max_{n∈ℤ} \overline{σ}\{F̂(jω_{n})\} and u_{d}_{max} is the singular-vector corresponding to \overline{σ}\{F̂(jω_{n_{max}})\}, equality holds in equation (3.10) and therefore,

∥F̂(e^{−jω̂h})∥ := sup_{\begin{matrix}u^{⊸}_{d}∈L_{ℍ}^{2}(𝒰) \\ u^{⊸}_{d}≠0 \end{matrix}} \frac{∥F̂(e^{−jω̂h}) u^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒴)}}{∥ u^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒰)}}
= max_{n∈ℤ} \overline{σ}\{F̂(jω_{n})\},

as claimed.

Lemma 3.6 Given a stable, LTI system F ∈ 𝒫_{𝒰→𝒳} and a stable, LPTV system P ∈ 𝒫_{𝒳→𝒴} , where 𝒳 is some finite-dimensional, Hilbert space, the system X :=P F ∈ 𝒫_{𝒰→𝒴} is stable and

\overline{σ}\{Θ_{X}(jω)\} ≤ \overline{σ}\{Θ_{P} (jω)\}\overline{σ}\{F̂(jω)\},

where Θ_{·}(jω) is the average-power gain matrix defined in Definition 3.3 and F̂ is the standard, frequency-domain transfer-matrix of F.

Proof : Since F is LTI, its asymptotic response to an input u(t) = e^{jωt}u_{d} is given by y_{as}^{ω}(t) = e^{jωt}F̂(jω)u_{d}. It then follows by the definition of Θ_{·}(ω), that the average power of the asymptotic response of X :=P F to the sinusoidal input u(t) is given by

∥y_{as}^{ω}∥_{pow} = ∥Θ_{X}(jω)u_{d}∥_{𝒰}
= ∥Θ_{P} (jω)F̂(jω)u_{d}∥_{𝒳}
≤ \overline{σ}\{Θ_{P} (jω)\}\overline{σ}\{F̂(jω)\}∥u_{d}∥_{𝒰} ,

Then since u_{d} is an arbitrary vector in 𝒰,

\overline{σ}\{Θ_{X}(jω)\} = sup_{\begin{matrix}u_{d}∈𝒰 \\ u_{d}≠0 \end{matrix}} \frac{∥Θ_{X}(jω)u_{d}∥_{𝒰}}{∥u_{d}∥_{𝒰}} ≤ \overline{σ}\{Θ_{P} (jω)\}\overline{σ}\{F̂(jω)\},

as required.

Lemma 3.7 Given a stable, LPTV system P ∈ 𝒫_{𝒰→𝒳} and a stable, LTI system F ∈ 𝒫_{𝒳→𝒴} , where 𝒳 is some finite-dimensional Hilbert space, X :=F P is a stable system in 𝒫_{𝒰→𝒴} and

\overline{σ}\{Θ_{X}(jω)\} ≤ \overline{σ}_{max}^{ω_{0}}\{F̂\}\overline{σ}\{Θ_{P} (jω)\}.

where Θ_{·}(jω) is the average-power gain matrix defined in Definition 3.3, F̂ is the standard, frequencydomain transfer-matrix of F, \overline{σ}_{max}^{ω_{0}}\{·\} is defined in equation (3.6) and ω_{0} :=ω − ]\frac{hω}{2π} [ \frac{2π}{h}.

Proof : By Lemma 3.1, the asymptotic response of P to a sinusoidal input u(t) = e^{jωt}u_{d} (u_{d} ∈ 𝒰) is given by

x_{as}^{ω}(t) = x_{as}^{ω}(kh + θ) = (e^{jω_{0}h})^{k} x^{⊸}_{d}(θ),  (3.11)

where x^{⊸}_{d}(θ) = (P̂(e^{−jω_{0}h})e^{jωτ} u_{d})(θ), ω_{0} = ω − ]\frac{hω}{2π} [ \frac{2π}{h} and P̂ is the transfer operator of the timelifted, equivalent system P:=WP W^{−1}. Moreover, by Lemma 3.2 and the definition of Θ_{P} (jω),

∥x_{as}^{ω}∥_{pow} = \frac{1}{\sqrt{h}}∥ x^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒳)} = ∥Θ_{P} (jω)u_{d}∥_{𝒰} .  (3.12)

Note from equation (3.11), that x_{as}^{ω}(t) is of the form considered in Lemma 3.1. Then since the asymptotic response of X to u(t) is the same as the asymptotic response of F to x_{as}^{ω}(t), it follows by Lemma 3.2 and the definition of Θ_{X}(jω), that

∥y_{as}^{ω}∥_{pow} = ∥Θ_{X}u_{d}∥_{𝒰} = \frac{1}{\sqrt{h}}∥F̂(e^{−jω_{0}h}) x^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒴)},  (3.13)

where F̂ is the transfer operator of the time-lifted, equivalent system F:=WF W^{−1}. Then, from equations (3.12) and (3.13),

∥y_{as}^{ω}∥_{pow} = ∥Θ_{X}u_{d}∥_{𝒰}
≤ \frac{1}{\sqrt{h}}∥F̂(e^{jω_{0}h})∥ · ∥ x^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒳)}
≤ ∥F̂(e^{jω_{0}h})∥ \overline{σ}\{Θ_{P} (jω)\}∥u_{d}∥_{𝒰} .

Since u_{d} is an arbitrary vector in 𝒰, it follows that

\overline{σ}\{Θ_{X}(jω)\} = sup_{\begin{matrix}u_{d}∈𝒰 \\ u_{d}≠0 \end{matrix}} \frac{∥Θ_{X}u_{d}∥_{𝒰}}{∥u_{d}∥_{𝒰}} ≤ ∥F̂(e^{jω_{0}h})∥ \overline{σ}\{Θ_{P} (jω)\}
= \overline{σ}_{max}^{ω_{0}}\{F̂\} \overline{σ}\{Θ_{P} (jω)\}

where \overline{σ}_{max}^{ω_{0}}\{F̂\} is defined in equation (3.6). This completes the proof.

3.3 A Framework for SD Closed-Loop Design

As mentioned previously, an intuitive procedure for the development of LTI, closed-loop systems is that of McFarlane and Glover [MG90]. It is based on classical loopshaping in the design step and an H^{∞} optimisation in the synthesis step. By virtue of the structure of the generalised plant constructed and the optimisation problem solved, robustness to coprime-factor (gap) uncertainty and the deviation from desired performance can be readily quantified for closed-loop systems designed using this framework. A brief introduction to the H^{∞}-loopshaping procedure of McFarlane and Glover is given in Sub-Section 3.3.1. The main purpose of this section is to propose a H^{∞}- loopshaping based design procedure for SD control-system development. Details of the proposed procedure are presented in Sub-Section 3.3.2, in which the new notion of frequency response introduced for LPTV systems in Section 3.2, plays an important role.

3.3.1 H^{∞}-Loopshaping for LTI Systems

The design step of McFarlane and Glover’s H^{∞}-loopshaping, design procedure is based on the notions of classical loopshaping. The basic idea in classical loopshaping is to trade off desired closedloop performance and robustness objectives by shaping open-loop gains in the frequency-domain [DFT92, ZDG95]. Consider for example, the typical closed-loop configuration shown below, where P and C are causal, LTI systems. Simple algebraic manipulation leads to the following functional

[figure: C; P; u; u_{p}; d_{i}; d; y; n; −; +; +; +; +; +; +; +; r]

Figure 3.1 Typical Closed-Loop Configuration

characterisation of the closed-loop:

y = T_{o}(r − n) + S_{o}P d_{i} + S_{o}d,  (3.14a)
r − y = S_{o}(r − d) + T_{o}n − S_{o}P d_{i},  (3.14b)
u = CS_{o}(r − n) − CS_{o}d − T_{i}d_{i},  (3.14c)
u_{p} = CS_{o}(r − n) − CS_{o}d + S_{i}d_{i},  (3.14d)

where L_{o} :=P C, L_{i} :=CP, S_{•} = (I + L_{•})^{−1} and T_{•} = I − S_{•}. Recall that the symbol “ˆ ” denotes transfer operator (matrix) of a system. Now from equation (3.14a), it can be seen that disturbance rejection at the plant output y can be achieved by making \overline{σ}\{Ŝ_{o}(jω)\} = \frac{1}{\underline{σ}\{I+L̂_{o}(jω)\}} small in frequency ranges where d has significant power and \overline{σ}\{Ŝ_{o}P̂(jω)\} = \overline{σ}\{P̂Ŝ_{i}(jω)\} small in frequency ranges where d_{i} has significant power. Similarly, from equation (3.14d), it is clear that disturbance rejection at the plant input u_{p} can be achieved by making \overline{σ}\{Ŝ_{i}(jω)\} = \frac{1}{\underline{σ}\{I+L̂_{i}(jω)\}} small in frequency ranges where d_{i} has significant power and \overline{σ}\{Ŝ_{i}Ĉ(jω)\} = \overline{σ}\{ĈŜ_{o}(jω)\} small in frequency ranges where d has significant power. Now since

\underline{σ}\{L̂_{o}(jω)\} − 1 ≤ \underline{σ}\{I + L̂_{o}(jω)\} ≤ 1 + \underline{σ}\{L̂_{o}(jω)\}

and

\underline{σ}\{L̂_{i}(jω)\} − 1 ≤ \underline{σ}\{I + L̂_{i}(jω)\} ≤ 1 + \underline{σ}\{L̂_{i}(jω)\},

it follows that if \underline{σ}\{L̂_{o}(jω)\} > 1 and \underline{σ}\{L̂_{i}(jω)\} > 1,

\frac{1}{1 + \underline{σ}\{L̂_{o}(jω)\}} ≤ \overline{σ}\{Ŝ_{o}(jω)\} ≤ \frac{1}{\underline{σ}\{L̂_{o}(jω)\} − 1}

and

\frac{1}{1 + \underline{σ}\{L̂_{i}(jω)\}} ≤ \overline{σ}\{Ŝ_{i}(jω)\} ≤ \frac{1}{\underline{σ}\{L̂_{i}(jω)\} − 1}.

Thus, disturbance rejection of d at y and d_{i} at u_{p} can be achieved by selecting a stabilising C to shape the minimal, open-loop gains \underline{σ}\{L̂_{o}(jω)\} and \underline{σ}\{L̂_{i}(jω)\} to be large in the frequency ranges where the disturbances d and d_{i} have significant power respectively. This should also be done in frequency regions (typically low-frequency) where good tracking performance is desired (cf. equation (3.14b)).

Similarly, it is desirable to design the stabilising C to achieve small maximal, open-loop gains \overline{σ}\{L̂_{o}(jω)\} and \overline{σ}\{L̂_{i}(jω)\} in frequency ranges where the sensor noise^{3} n is significant and where high-levels of control activity are not acceptable (cf. equations (3.14)). A typical loop-shape is shown in Figure 3.2. In the H^{∞}-loopshaping procedure of McFarlane and Glover the plant P is shaped open-loop and a stabilising controller synthesised to solve an H^{∞} optimisation problem. Briefly, the procedure can be summarised as follows:

(i) Using the notions of classical loopshaping, shape the plant’s frequency-response matrix P̂(jω) open-loop with stable, pre- and post-compensators Ŵ_{1}(jω) and Ŵ_{2}(jω) according to the closed-loop objectives. There should be no unstable pole/zero cancellations between P and the weights. For notational convenience the shaped plant is denoted by P_{s} :=W_{2}P W_{1}.

3This can also be used to characterise unmodelled, high-frequency dynamics.

\overline{σ}\{L̂_{•}\} \underline{σ}\{L̂_{•}\}  log ω ω_{h} ω_{l}

Figure 3.2 Typical Loop-Shape

(ii) Synthesise a stabilising controller C_{∞} to achieve some specified bound on the induced-norm of the operator mapping the disturbances w_{1} and w_{2} to z_{1} and z_{2} of the closed-loop [P_{s}, C_{∞}] shown in Figure 3.3;

∥[ \begin{matrix}w_{1} \\ w_{2} \end{matrix} ] ↦ [ \begin{matrix}z_{1} \\ z_{2} \end{matrix} ]∥ = ∥[ Ĉ_{∞} ; I ](I − P̂_{s}Ĉ_{∞})^{−1}[ P̂_{s} I ]∥_{∞} =: b_{P_{s},C_{∞}}^{−1}.

If b_{P_{s},C_{∞}} is sufficiently large (that is, > 0.25 but preferably 0.35) then continue to implement the controller. If b_{P_{s},C_{∞}} is too small then redesign the weights.

(iii) The final controller C is then constructed by absorbing the weights W_{1} and W_{2} into the controller: C :=W_{1}C_{∞}W_{2}.

[figure: C_{∞}; P_{s}; w_{1}; w_{2}; z_{2}; z_{1}; +; +; +; +]

Figure 3.3 Standard Loopshaping Design Configuration

An advantage of using the McFarlane-Glover, loopshaping, design procedure over classical loopshaping, is that the designer does not have to explicitly consider closed-loop stability. Of course, ignoring completely the fundamental objective of closed-loop stability (by not shaping the phase characteristic appropriately at cross-over for example) however, can lead to poor robustness as would be indicated by a small b_{P_{s},C_{∞}}.

Note that b_{P_{s},C_{∞}}^{−1} is a bound on all six common, performance-indicating, closed-loop operators, although tighter bounds are possible since the the closed-loop operators are not independent [Vin93b]. Furthermore, b_{P_{s},C_{∞}} can be used to indicate the level of degradation of the desired loop-shape P_{s} and bounds on performance-indicating, closed-loop operators can be expressed in terms of P_{s}, W_{1}, W_{2} and b_{P_{s},C_{∞}} only [MG90]. For example, the input and output sensitivity of the closed-loop [P, C] = [P, W_{1}C_{∞}W_{2}] are bounded by

\overline{σ}\{(I + P̂Ĉ(jω))^{−1}\} ≤ min \{b_{P_{s},C_{∞}} \overline{σ}\{M̂_{l,s}(jω)\}κ\{Ŵ_{2}(jω)\}, 1 + b_{P_{s},C_{∞}} \overline{σ}\{N̂_{r,s}(jω)\}κ\{Ŵ_{2}(jω)\}\} (3.15a)

and

\overline{σ}\{(I + ĈP̂(jω))^{−1}\} ≤ min \{1 + b_{P_{s},C_{∞}} \overline{σ}\{N̂_{l,s}(jω)\}κ\{Ŵ_{1}(jω)\}, b_{P_{s},C_{∞}} \overline{σ}\{M̂_{r,s}(jω)\}κ\{Ŵ_{1}(jω)\}\} , (3.15b)

where P_{s} = N_{r,s}M_{r,s}^{−1} = M_{l,s}^{−1}N_{l,s} are normalised, right- and left-coprime factorisations^{4} respectively and κ{·} denotes the condition number of a matrix. Since frequency-by-frequency

\overline{σ}\{M̂_{l,s}\} = \overline{σ}\{M̂_{r,s}\} = (\frac{1}{1 + \underline{σ}^{2}\{P̂_{s}\}})^{1/2} and \overline{σ}\{N̂_{l,s}\} = \overline{σ}\{N̂_{l,s}\} = (\frac{\overline{σ}^{2}\{P̂_{s}\}}{1 + \overline{σ}^{2}\{P̂_{s}\}})^{1/2} ,

bounds such as those given in equations (3.15) give the designer a feel for how the weights selected in the design step affect the final closed-loop.

The H^{∞}-loopshaping procedure of McFarlane-Glover is also appealing from the perspective of robust stability. By virtue of the generalised plant’s structure and the optimisation problem solved in the synthesis step, it is possible to quantify the level of robustness of the resultant closed-loop system to coprime-factor uncertainty [MG90], which is equivalent to gap-metric uncertainty, as shown in [GS90]. The tightest bounds on degradation in performance and the strongest possible robustness results for the H^{∞}-loopshaping framework are given in [Vin93b] in terms of the ν-gap metric, which induces the same topology on the class of LTI systems as the gap metric. For further information on H^{∞}-loopshaping for LTI systems and its application in practice, the reader is referred to [Vin98, Pap98, MG90, Hyd95].

3.3.2 H^{∞}-Loopshaping Based Design for SD Control-System Development In this sub-section, a new procedure is proposed for the development of SD control-systems. The proposed procedure is related to the H^{∞}-loopshaping procedure of McFarlane and Glover [MG90] described above, with the main difference being a slight modification of the structure of the generalised plant constructed in the design step. Recall that in the final step of the McFarlane-Glover procedure, the weights used to shape the plant open-loop are absorbed into the synthesised controller. This is not possible if the controller is constrained to be SD, because the weights cannot be absorbed directly across the sampling and hold devices. The alternative of leaving the weights on the plant side of the sampling and hold devices defeats the purpose of using SD control, although using W_{2} to shape a simple, stable, low-pass, LTI filter used to ensure boundedness of A_{D} for synthesis, is sensible (which is the strategy adopted in the formulation presented here). Importantly though, with C_{∞o} :=W_{1}C_{∞} and P_{so} :=W_{2}P,

4For LTI systems, a right-coprime factorisation N_{r,s}M_{r,s}^{−1} is said to be normalised if the frequency-domain symbol Ĝ_{r,s} =[ M̂_{r,s} ; N̂_{r,s} ] is inner (that is, if Ĝ_{r,s}(−s)^{T} Ĝ_{r,s}(s) = I) and a left-coprime factorisation M_{l,s}^{−1}N_{l,s} is said to be normalised if frequency-domain symbol Ĝ_{l,s} =[ _{−M̂_{l,s}} N̂_{l,s} ] is co-inner (that is, if Ĝ_{l,s}(s)Ĝ_{l,s}(−s)^{T} = I), where the superscript T denotes matrix transpose (without conjugation) (cf. [ZDG95] for example).

∥[ \begin{matrix}C_{∞} \\ I \end{matrix} ](I − P_{s}C_{∞})^{−1}[ I P_{s} ]∥ = ∥[ \begin{matrix}W_{1}^{−1} & 0 \\ 0 & I \end{matrix} ][ \begin{matrix}C_{∞o} \\ I \end{matrix} ](I − P_{so}C_{∞o})^{−1}[ P_{so} I ][ \begin{matrix}W_{1} & 0 \\ 0 & I \end{matrix} ]∥ .

Using this identity it follows that if W_{1} is stably invertible, then synthesising a stabilising C_{∞} to bound the first expression and absorbing W_{1} at the output and W_{2} at the input of this controller, is equivalent to synthesising a stabilising C_{∞o} to bound the second expression and absorbing the weight W_{2} at the input of the resultant controller. In the context of SD control, the latter can be viewed schematically as shown in Figure 3.4, where A_{D} is an ideal, periodic (period h) samplingdevice, D_{A} is a zeroth-order hold-device synchronised with A_{D} and F_{b} is a simple, stable, low-pass, LTI filter used to ensure bounded operation of the sampling device and to limit frequency-domain aliasing. The new design procedure for SD, control-system development is summarised below:

[figure: C_{∞}; P_{so} = W_{2}P; W_{1}; W_{1}^{−1}; C_{∞o} = C_{∞}^{sd}; D_{A}; A_{D}; F_{b}; w_{1}; z_{1}; z_{2}; w_{2}; +; +; +; +]

Figure 3.4 SD Closed-Loop Design Configuration

(i) Given an LTI plant P ∈ 𝒫_{𝒰→𝒴}^{e} , select the LTI weights W_{1} and W_{2} shown in Figure 3.4 to reflect closed-loop objectives, subject to the constraints that both W_{1} and W_{1}^{−1} are stable and that there should be no unstable pole/zero cancellations between W_{2} and P. Closed-loop performance bounds similar to those given in equation (3.15) for the LTI case, are obtained in Theorem 3.8 to facilitate the design of W_{1} and W_{2}.

(ii) Synthesise a stabilising, LSI, discrete-time controller C_{∞} to achieve some specified bound on the induced-norm of the closed-loop operator mapping the disturbances w_{1} and w_{2} to z_{1} and z_{2} of the closed-loop configuration shown in Figure 3.4;

∥[ \begin{matrix}w_{1} \\ w_{2} \end{matrix} ] ↦ [ \begin{matrix}z_{1} \\ z_{2} \end{matrix} ]∥ = ∥[ \begin{matrix}W_{1}^{−1}C_{∞}^{sd} \\ I \end{matrix} ](I − P_{so}C_{∞}^{sd})^{−1}[ P_{so}W_{1} I ]∥ =: b_{P_{s},W_{1}^{−1}C_{∞}^{sd}}^{−1}, (3.16)

where P_{so} :=W_{2}P, P_{s} = P_{so}W_{1} and C_{∞}^{sd} :=D_{A}C_{∞}A_{D}F_{b}.

(iii) Shape the simple, stable, low-pass, LTI filter F_{b} with the weight W_{2} to form the final controller C :=C_{∞}^{sd}W_{2}. Since by design there are no hidden unstable modes in W_{2}P, moving W_{2} around the loop in this manner does not affect closed-loop stability.

The synthesis step can be carried out as described in Section 2.5, by noting that the closed-loop operator mapping w_{1} and w_{2} to z_{1} and z_{2} in Figure 3.4 can be expressed in the linear-fractional form

F_{ℓ}(G, D_{A}C_{∞}A_{D}) = [ \begin{matrix}w_{1} \\ w_{2} \end{matrix} ] ↦ [ \begin{matrix}z_{1} \\ z_{2} \end{matrix} ],

where

G:= [ \begin{matrix}0 & 0 & W_{1}^{−1} \\ P_{so}W_{1} & I & P_{so} \\ F_{b}P_{so}W_{1} & F_{b} & F_{b}P_{so} \end{matrix} ] .

Explicit formulae for H^{∞}, SD synthesis and related numerical issues are discussed in Chapter 6.

As in the LTI case, it is possible to quantify the robustness of SD control-systems designed using the proposed procedure. More specifically, in [CG96] it is shown that if β ≤ b_{P_{s},W_{1}^{−1}C_{∞}^{sd}}, all LTI plants in the weighted coprime-factor ball

𝒩_{NLCF} (P_{s}, β) := {P̀ = (M_{l,s}W_{2} + Δ_{M_{l}}W_{2})^{−1}(N_{l,s}W_{1}^{−1} + Δ_{N_{l}}W_{1}^{−1}) : ∥[ Δ̂_{M_{l}} Δ̂_{N_{l}} ]∥_{∞} < β} ,

where P_{s} = M_{l,s}^{−1}N_{l,s} is a normalised, left-coprime factorisation, are stabilised by the controller C :=C_{∞}^{sd}W_{2}. Furthermore, using results developed in Chapter 4, it can be shown (cf. Corollary 4.28) that all LPTV plants in the weighted gap-ball

𝒩_{δ_{g},W_{1},W_{2}}^{e}(P_{s}, β) :={P̀ ∈ 𝒫_{𝒰→𝒴}^{e} : δ_{g}(P_{s}, W_{2}P̀W_{1}) < β}

are stabilised by C :=C_{∞}^{sd}W_{2} if and only if β ≤ b_{P_{s},W_{1}^{−1}C_{∞}^{sd}}.

Since (cf. [DGS93] and Lemma 4.21)

∥[ \begin{matrix}W^{−1}C_{∞}^{sd} \\ I \end{matrix} ](I − P_{so}C_{∞}^{sd})^{−1}[ P_{so}W_{1} I ]∥ = ∥[ \begin{matrix}P_{so}W_{1} \\ I \end{matrix} ](I − C_{∞}^{sd}P_{so})^{−1}[ W_{1}^{−1}C_{∞}^{sd} I ]∥ , (3.17)

it is possible to bound the behaviour of all six common performance-indicating, closed-loop operators. Moreover, as for the LTI case (cf. equations (3.15)), it is possible to express bounds on the the average-power gain of a subset of these operators to sinusoidal inputs of a single frequency in terms of P_{s} :=W_{2}P W_{1}, W_{1}, W_{2} and b_{P_{s},W_{1}^{−1}C_{∞}^{sd}} only. These bounds aid in the design of W_{1} and W_{2} in the first step of the proposed procedure and are summarised in the following theorem.^{5}

Theorem 3.8 Let

W_{2}S_{o} := W_{2}(I − P C)^{−1},
S_{i} := (I − CP)^{−1}

and

S_{i}C := (I − CP)^{−1}C,

where C :=C_{∞}^{sd}W_{2} = D_{A}C_{∞}A_{D}F_{b}W_{2}. Recall that these operators characterise the (weighted) tracking error, the plant-input disturbance to plant-output error and the sensor noise to plant-input error (cf. equations (3.14)). Then, the maximal, average-power gain of these performance-indicating, closed-loop operators in response to sinusoidal inputs of a single frequency can be bounded as follows:

\overline{σ}\{Θ_{W_{2}S_{o}}(jω)\} ≤ b_{P_{s},W_{1}^{−1}C_{∞}^{sd}}^{−1} \overline{σ}\{M̂_{s}^{l}(jω)\}\overline{σ}\{Ŵ_{2}(jω)\},  (3.18a)
\overline{σ}\{Θ_{S_{i}}(jω)\} ≤ b_{P_{s},W_{1}^{−1}C_{∞}^{sd}}^{−1} \frac{\overline{σ}_{max}^{ω_{0}}\{Ŵ_{1}M̂_{r,s}\}}{\underline{σ}\{Ŵ_{1}(jω)\}}  (3.18b)

(3.18c)

and

\overline{σ}\{Θ_{S_{i}C}(jω)\} ≤ b_{P_{s},W_{1}^{−1}C_{∞}^{sd}}^{−1} \overline{σ}_{max}^{ω_{0}}\{Ŵ_{1}M̂_{r,s}\}\overline{σ}\{Ŵ_{2}(jω)\},  (3.18d)

where Θ_{·} denotes the average-power gain matrix of a system (cf. Definition 3.3), \overline{σ}_{max}^{ω_{0}}\{·\} is defined in (3.6), P_{s} :=W_{2}P W_{1} = N_{r,s}(M_{r,s})^{−1} = (M_{l,s})^{−1}N_{l,s} are normalised, right- and left-coprime factorisations and ω_{0} :=ω − ]\frac{hω}{2π} [ \frac{2π}{h}.

5The operators for which bounds are given have been chosen according to their significance from a performance perspective. Such frequency-by-frequency bounds for closed-loop operators that reflect robustness are meaningless when the closed-loop is LPTV. To study robustness properties of LPTV systems in the frequency-domain, it is necessary to consider the asymptotic response of the system to infinite series of sinusoids with frequencies separated by integer multiples of the sampling frequency [Dul94]. Recall that such signals are asymptotically invariant for LPTV systems in the same way that sinusoids of single frequencies are asymptotically invariant for LTI systems.

Before proving this, the following technical result is required:

Lemma 3.9 Let 𝒲, 𝒰 and 𝒴 be finite-dimensional Hilbert spaces. Given a stable, LPTV system X ∈ 𝒫_{𝒲→𝒴} and a stable LTI system G equivalent, via the Laplace-Transform isomorphism, to a multiplication operator M_{Ĝ(s)} with symbol Ĝ(s) ∈ H_{ℂ_{+}}^{∞}(B_{𝒰→𝒲} ) that satisfies Ĝ(s)Ĝ(−s)^{T} = I (i.e. Ĝ is co-inner), where the superscript T denotes matrix transpose, then ∥XG∥ = ∥X∥. Similarly, if Ĝ(−s)^{T} Ĝ(s) = I (i.e. Ĝ is inner) then ∥GX∥ = ∥X∥.

Proof : The proof is deferred to Appendix B.1. Proof : (of Theorem 3.8) Only the bound on \overline{σ}\{Θ_{S_{i}C}(jω)\} is proved, since the other bounds follow in a similar manner. First note that by Lemma 3.9 and from equations (3.17) and (3.16),

∥[ \begin{matrix}W_{1}^{−1}C_{∞}^{sd} \\ I \end{matrix} ](I − P_{s}W_{1}^{−1}C_{∞}^{sd})^{−1}M_{l,s}^{−1}∥ =

∥M_{r,s}^{−1}(I − W_{1}^{−1}C_{sd}^{∞}P_{s})^{−1}[ W_{1}^{−1}C_{sd}^{∞} I ]∥ = b_{P_{s},W_{1}^{−1}C_{∞}^{sd}}^{−1},  (3.19)

where P_{s} :=W_{2}P W_{1} = N_{r,s}(M_{r,s})^{−1} = (M_{l,s})^{−1}N_{l,s} are normalised, right- and left-coprime factorisations respectively. Now let

Q := M_{r,s}^{−1}(I − W_{1}^{−1}C_{∞}^{sd}P_{s})^{−1}W_{1}^{−1}C_{∞}^{sd}

and u := Qy with y(t) = e^{jωt}y_{d} (y_{d} ∈ 𝒴). Defining y^{⊸}_{d}(θ) = e^{jωθ}y_{d}, it follows by Lemma 3.2 and the definition of Θ_{Q}(jω), that

∥Θ_{Q}(jω)y_{d}∥_{𝒴} = \frac{1}{\sqrt{h}}∥Q̂(e^{−jω_{0}h}) y^{⊸}_{d}∥_{L_{ℍ}^{2}(𝒰)}
≤ ∥Q̂(e^{−jω_{0}h})∥∥y_{d}∥_{𝒴},  (3.20)

where ω_{o} :=ω −]\frac{ωh}{2π}[ \frac{2π}{h} and Q̂ is the transfer operator of the LSI equivalent system Q:=WQW^{−1}. Then since

∥Q̂(e^{−jω_{0}h})∥ ≤ ∥Q∥ = ess sup_{ω∈[0,2π)}∥Q̂(e^{−jωh})∥,

it follows from equation (3.19) that

∥Q̂(e^{−jω_{0}h})∥ ≤ b_{P_{s},W_{1}^{−1}C_{∞}^{sd}}^{−1}.

So using equation (3.20),

\overline{σ}\{Θ_{Q}(jω)\}:= sup_{\begin{matrix}y_{d}∈𝒴 \\ y_{d}≠0 \end{matrix}} \frac{∥Θ_{Q}(jω)y_{d}∥_{𝒴}}{∥y_{d}∥_{𝒴}} ≤ ∥Q̂(e^{−jω_{0}h})∥ ≤ b_{P_{s},W_{1}^{−1}C_{∞}^{sd}}^{−1}.

Now note that S_{i}C = W_{1}M_{r,s}QW_{2} and hence, by Lemmas 3.6 and 3.7, that

\overline{σ}\{Θ_{S_{i}C}(jω)\} ≤ \overline{σ}_{max}^{ω_{0}}\{Ŵ_{1}M̂_{r,s}\}\overline{σ}\{Θ_{Q}(jω)\}\overline{σ}\{Ŵ_{2}(jω)\}
≤ b_{P_{s},W_{1}^{−1}C_{∞}^{sd}}^{−1} \overline{σ}_{max}^{ω_{0}}\{Ŵ_{1}M̂_{r,s}\}\overline{σ}\{Ŵ_{2}(jω)\},

as claimed. This completes the proof.

Remark 3.10 The bounds presented in Theorem 3.8 are similar to those given in [MG90] for the LTI case (cf. equations (3.15) for example). Note that the “tracking-error” bound is weighted by W_{2}. This was necessary to obtain a reasonable bound and does not pose a significant problem, since provided Ŵ_{2}(jω) is designed to roll-off at a frequency sufficiently greater than the bandwidth of the un-weighted open-loop P, the bound given in equation (3.18a) divided by \underline{σ}\{W_{2}(jω)\} will not be too far from that of the actual maximal, tracking-error gain in frequency ranges where tracking is important. Furthermore, since frequency-by-frequency

\overline{σ}\{M̂_{l,s}\} = \overline{σ}\{M̂_{r,s}\} = (\frac{1}{1 + \underline{σ}^{2}\{P̂_{s}\}})^{1/2} and \overline{σ}\{N̂_{l,s}\} = \overline{σ}\{N̂_{l,s}\} = (\frac{\overline{σ}^{2}\{P̂_{s}\}}{1 + \overline{σ}^{2}\{P̂_{s}\}})^{1/2} ,

it is possible to design Ŵ_{1}(jω) so that

sup_{ω} \overline{σ}\{Ŵ_{1}M̂_{r,s}(jω)\} ≈ 1.

As such, the bounds derived on input-disturbance and sensor-noise rejection are sensible in the frequency ranges where they are important.

It is suggested that as a first step, the weights W_{1} and W_{2} be selected to be sensible in the context of standard H^{∞} loopshaping for LTI control-system design, accounting for the simple, stable, lowpass, LTI filter used to ensure boundedness of A_{D} in the SD synthesis step. These can then be adjusted (using the bounds given in Theorem 3.8 as a guide) according to the sampling-period chosen or perhaps more simply, a sampling-period can be chosen so that these weights are sensible in the context of SD control. ♣

3.4 Summary

In this section a H^{∞}-loopshaping based design procedure has been established for SD, controlsystem development. To this end, a new notion of frequency response has been defined for LPTV systems. Specifically, a frequency-domain, analysis tool called the average-power, gain matrix has been defined, which as its name suggests, characterises the average-power gain of an LPTV system to sinusoidal inputs of a single frequency. This matrix is finite-dimensional (can be computed as described in Appendix B.2) and retains input-directional information. By the way it is defined the new, frequency-domain, analysis tool has clear performance-indicating value, although it should be pointed out that it would not be suitable for robustness analysis in the frequency-domain.

Robustness Analysis of LPTV Closed-Loop Systems

4.1 Introduction

Motivation for this chapter stems from a desire to characterise the robustness properties of SD control-systems developed using the design procedure proposed in Chapter 3. The approach taken is to study a general class of LPTV, closed-loop systems, of which the SD control-systems considered in Chapter 3 are a special case.

One of the important properties of closed-loop systems is that they can be desensitised to uncertainty. This property is often referred to as robustness. This is particularly important from the perspective of model-based design, since any mathematical model used to describe a system’s behaviour is uncertain to some degree. The question is: how to qualify and quantify the uncertainty a closed-loop system can tolerate? The answer is to define a suitable topology on the set of systems concerned and a metric to induce this topology. An appropriate topology must capture all perturbations to closed-loop components that do not cause instability, thus qualifying permissible uncertainty. Furthermore, since performance is also an important objective when designing closed-loop systems, suitable indicators of performance must vary continuously in the topology selected.

For stable systems, a topology suitable for studying robustness is the induced-norm topology, which lays the foundation for H^{∞} control-theory as developed in the seminal paper [Zam81]. A short-coming of the induced-norm topology is the inability to consider explicitly, unstable components of a given closed-loop. This led to the development of the graph topology for robustness analysis [VSF83, Vid84, ZES80]. The foundations for robustness analysis in the graph topology were established in [VSF83], where a topology deemed suitable for this was proposed in the following way. Consider the closed-loop system shown in Figure 4.1, which is said to be stable if the operator H(P, C) :=[ \begin{matrix}I & C \\ P & I \end{matrix} ]^{−1} mapping [ \begin{matrix}d_{1} \\ d_{2} \end{matrix} ] to [ \begin{matrix}u_{1} \\ u_{2} \end{matrix} ] is causal and bounded. A topology on a set of plants (or controllers) suitable for robustness analysis is one that satisfies the following properties: given any plant P in the set of interest and a stabilising controller C, there exists a neighbourhood of P (in the topology) such that all plants in this neighbourhood are also stabilised by C and the mapping P ↦ H(P, C) is continuous at P (with respect to the topology on the set of plants concerned and the induced norm topology on the closed-loop system set). That is, for any sequence {P_{i}} converging to P (with respect to a topology with these properties), all but a finite number of the plants in that sequence must be stabilised by C and H(P_{i}, C) → H(P, C) in the induced norm topology.

[figure: C; P; d_{1}; u_{1}; y_{1}; d_{2}; u_{2}; y_{2}; −; +; +; −]

Figure 4.1 Standard Closed-Loop Configuration

For certain classes of system, a topology that satisfies the properties described above can be characterised explicitly in terms of the graph representation of a system. Hence, such a topology is often called the graph topology [Vid85]. For systems known to admit fractional representations that are coprime (such as LTI systems [Vid85, Smi89] for example), a basis for the graph topology can be constructed by additively perturbing a representation of the graph derived from coprime factorisations [Vid84]. Qualitatively, two systems are “close” in the graph topology if their graphs are “close”. It is shown in [Vid84], that for LTI systems the graph topology (defined in terms of the basis just described) is the weakest with respect to which closed-loop performance varies continuously and closed-loop stability is a robust property.

The topological aspects of robustness analysis described above qualify the types of uncertainty to which closed-loop systems can be desensitised. To quantify a level of robustness requires a metric to induce the graph topology. For LTI systems a number of metrics have this property, including: the graph metric [Vid84]; the gap metric [ZES80, ES85, Geo88, GS90, OS91]; and the ν-gap metric [Vin93a]. In this chapter attention is restricted to the gap metric, which is defined below. Given two operators defined on the same ambient Hilbert spaces, the distance between them can be defined in terms of the aperture between their graphs [Kat66]. More specifically, the gap between two (closed) linear operators P_{0} and P_{1} is defined to be

δ_{g}(P_{0}, P_{1}) :=∥Π_{𝒢_{P_{0}}} − Π_{𝒢_{P_{1}}} ∥,

where Π_{𝒢_{P_{i}}} denotes the orthogonal projection onto 𝒢_{P_{i}}. This is a metric and as for any two (closed) subspaces of a Hilbert space, can be expressed as the maximum of two directed gaps δ⃗_{g}(P_{i}, P_{j}) :=∥(I − Π_{𝒢_{P_{i}}} )Π_{𝒢_{P_{j}}} ∥ [KVZ^{+}72].

All of the robustness results derived in this chapter, are obtained in terms of the gap metric. To this end, a formula is derived for the directed gap between two LPTV systems, which is expressed in terms of a particular representation of the graph and an H^{∞} optimisation. The formula obtained is essentially a generalisation of that given in [Geo88] for LTI systems and is fundamental to the framework used in the sequel to quantify and qualify robustness of LPTV, closed-loop systems. The main quantitative robust stability results presented concern robustness to perturbations measured in the gap metric and are to some extent analogous to the results of [GS90] for LTI systems and those of [FGS93] for linear, time-varying (LTV) systems, although they do not follow from either directly. In particular, problems concerning causality are not addressed in [FGS93] and even though the periodic nature of the time-variation considered here imparts a high degree of structure to the analysis, it is somewhat more involved than in the time-invariant case. Qualitatively, it is shown that the gap metric induces on a general class of LPTV systems, the weakest topology with respect to which closed-loop performance varies continuously and closed-loop stability is a robust property. All of the results obtained accommodate possibly infinite-dimensional input and output spaces and importantly, can be applied to LPTV, SD control-systems as a special case.

The existence of particular representations of the graph of a class of LPTV systems is fundamental to the robustness results obtained. Specifically, is it shown that the graph of a stabilisable, LPTV system can be expressed as the range and kernel of stable, LPTV systems that are respectively, left and right invertible by stable, LPTV systems. These so-called strong-right and strong-left representations resemble the coprime-factor representations known to exist for LTI systems [Vid85]. Given a well-posed plant/controller pair [P, C], a useful characterisation of closed-loop stability is obtained in terms of the strong-right and strong-left representations of 𝒢_{P} and 𝒢_{C}. In turn, this leads to a Youla-style parameterisation of stabilising controllers.

Briefly, the chapter is structured as follows: Section 4.2 is devoted to presenting results concerning the representation of LPTV systems and closed-loop stability; in Section 4.3, the gap metric is introduced and a formula for the directed gap between two LPTV systems is derived; in Section 4.4, quantitative robustness results are obtained in terms of the gap metric and in Section 4.5, it is shown that the gap metric induces a topology on a class of LPTV systems with the properties of the graph topology describe above; in Section 4.6, results obtained in preceding sections are applied to the special case of a sampled-data-controlled, closed-loop systems; finally, a brief summary is given in Section 4.7.

4.2 Representations of LPTV Systems

In this section, various representations of LPTV systems are developed. Fundamental to the results presented, is the equivalence, via the time-lifting isomorphism, of LPTV, continuous-time systems to LSI, discrete-time systems (cf. Proposition 2.4 and [BPFT91, BP92]).

4.2.1 LPTV Systems and LSI Equivalents

Let 𝒰 and 𝒴 be (possibly infinite-dimensional) Hilbert spaces and for notational convenience let 𝒱:=𝒰 ⊕ 𝒴. Recall that the graph of an operator P : 𝒟_{P} ⊂ 𝒰 → 𝒴 is defined to be the set of all bounded input-output pairs

𝒢_{P} :=[ \begin{matrix}I \\ P \end{matrix} ]𝒟_{P} ⊂ 𝒱,

and that for notational convenience, the inverse graph is defined to be

𝒢P^{′} :=[ \begin{matrix}0 & I \\ I & 0 \end{matrix} ]𝒢_{P} = [ \begin{matrix}P \\ I \end{matrix} ]𝒟_{P} ⊂ 𝒴 ⊕ 𝒰.

Furthermore, recall that a linear subspace 𝒢 ⊂ 𝒱 is the graph of a linear operator if and only if

[ \begin{matrix}0 \\ y \end{matrix} ] ∈ 𝒢 ⇒ y = 0.

Note that the bounded behaviour of an operator is completely characterised by its graph and as such, graphs prove to be a useful representation of systems, particularly when studying the stability of closed-loops (cf. Sub-Section 2.3.3).

In this chapter, attention is restricted to LPTV systems. In particular, the class of systems considered is defined in Definition 2.3 of Sub-Section 2.3.2, which for convenience is re-stated below:

Definition 4.1 Let 𝒰 and 𝒴 be (possibly infinite-dimensional) Hilbert spaces. Then 𝒫_{𝒰→𝒴} is defined to be the set of causal, LPTV (with period h), continuous-time systems

P : 𝒟_{P} ⊂ L_{ℝ^{+}}^{2}(𝒰) → L_{ℝ^{+}}^{2}(𝒴),

with closed graphs. Furthermore, 𝒫_{𝒰→𝒴}^{e} ⊂ 𝒫_{𝒰→𝒴} is defined to be the subset of causally-extendible systems with locally-Lipschitz-continuous extension and 𝒫_{𝒰→𝒴}^{e,sc} the subset of strongly-causal systems in 𝒫_{𝒰→𝒴}^{e} . A system in 𝒫_{𝒰→𝒴} is said to be \underline{stable} if and only if 𝒟_{P} = L_{ℝ^{+}}^{2}(𝒰) and ∥P ∥ < ∞. Note that any stable system P ∈ 𝒫_{𝒰→𝒴} is an element of 𝒫_{𝒰→𝒴}^{e} . ♠

Recall the important property of LPTV systems, by which they are equivalent via the time-lifting isomorphism to LSI systems (cf. Proposition 2.4 and [BP92, BPFT91]). More precisely, each P ∈ 𝒫_{𝒰→𝒴} is equivalent (via W) to a system in the set 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} defined in Definition 2.5 of Sub-Section 2.3.2. This equivalence is used extensively throughout this chapter.

Remark 4.2 Recall that a system P ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} is equivalent to a system P ∈ 𝒫_{𝒰→𝒴} (via W) if and only if P_{[0]} is a causal map from L_{ℍ}^{2}(𝒰) to L_{ℍ}^{2}(𝒴), where P_{[0]} is the first element of the sequence uniquely identifiable with the Toeplitz representation of P. A system P ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} not satisfying this condition is equivalent (via W) to a continuous-time, LPTV system that is not locally causal in each [kh, (k + 1)h) interval of time (k ∈ ℤ^{+}). ♣

As previously mentioned, the graph is a useful representation of a system. Since an LPTV system is equivalent (via W) to an LSI system, its graph is isomorphic to an LSI subspace. Motivated by this, the following sub-section is concerned with the characterisation of LSI subspaces. To this end, the Beurling-Lax-Halmos Theorem [Lax59, RR85, FF90] is used, which briefly, states that any LSI subspace of H_{𝔻}^{2} can be expressed as the range of a multiplication operator with inner symbol. This result is extended to show that if the LSI subspace is LSI co-ordinatisable, then it can be expressed as the kernel of a multiplication operator with symbol in H_{𝔻}^{∞}. Furthermore, these range and kernel representations are respectively right and left invertible by multiplication operators with symbols in H_{𝔻}^{∞}. Using this, range and kernel representations (with (causal) stable, left and right inverses respectively) of the graphs of stabilisable systems in 𝒫_{𝒰→𝒴}^{e} can be constructed.

4.2.2 LSI Subspaces

Let 𝒢 ⊂ H_{𝔻}^{2}(ℋ) be closed and LSI, where ℋ is an arbitrary Hilbert space. Then 𝒢 is said to be LSI co-ordinatisable if there exists a closed, LSI subspace ℱ ⊂ H_{𝔻}^{2}(ℋ) such that {𝒢, ℱ} is a co-ordinatisation of H_{𝔻}^{2}(ℋ). That is, if 𝒢 ∩ ℱ = {0} and 𝒢 + ℱ = H_{𝔻}^{2}(ℋ). Recall that if {𝒢, ℱ} is a co-ordinatisation of H_{𝔻}^{2}(ℋ), then any η ∈ H_{𝔻}^{2}(ℋ) can be uniquely decomposed as the sum η = γ+φ, where γ ∈ 𝒢 and φ ∈ ℱ (cf. [DGS93]). Furthermore, recall that the operator Π_{𝒢∥ℱ} := η ↦ γ is called the parallel projection onto 𝒢 along ℱ and similarly, that Π_{ℱ∥𝒢} := η ↦ φ is called the parallel projection onto ℱ along 𝒢. Since 𝒢 and ℱ are closed, linear subspaces it follows that the graphs of the parallel projection operators are linear and closed. Hence, by the Closed Graph Theorem [Bol90, pg. 80], the parallel projection operators are bounded and clearly linear. That they are also shift-invariant, is immediate because S𝒢 ⊂ 𝒢 and Sℱ ⊂ ℱ.

Proposition 4.3 (Beurling-Lax-Halmos Theorem) [Lax59, RR85, FF90] Given an LSI subspace 𝒮 of H_{𝔻}^{2}(ℋ) there exists an inner function Σ_{r} ∈ H_{𝔻}^{∞}(B_{ℋ_{1} →ℋ} ) such that 𝒮 = Σ_{r}H_{𝔻}^{2}(ℋ_{1}), where ℋ_{1} can be any Hilbert space isomorphic to 𝒮 ⊖ S𝒮.

So corresponding to the LSI subspaces 𝒢 and ℱ discussed above, there exist inner functions Γ_{r} ∈ H_{𝔻}^{∞}(B_{ℋ_{1} →ℋ} ) and Φ_{r} ∈ H_{𝔻}^{∞}(B_{ℋ_{2} →ℋ} ) such that 𝒢 = Γ_{r}H_{𝔻}^{2}(ℋ_{1}) and ℱ = Φ_{r}H_{𝔻}^{2}(ℋ_{2}), where ℋ_{1} is some Hilbert space isomorphic to 𝒢 ⊖ S𝒢 and ℋ_{2} any Hilbert space isomorphic to ℱ ⊖ Sℱ. Note that since M_{Γ_{r}} and M_{Φ_{r}} are isometries (cf. Proposition 2.2), the orthogonal projections Π_{𝒢} onto 𝒢 and Π_{ℱ} onto ℱ can be expressed as M_{Γ_{r}}(M_{Γ_{r}})^{∗} and M_{Φ_{r}}(M_{Φ_{r}})^{∗} respectively. This fact is used in the proof of the following technical result, which in turn, is fundamental to the main Theorem of this sub-section.

Lemma 4.4 Given two closed, LSI subspaces 𝒢 and ℱ that induce a co-ordinatisation of H_{𝔻}^{2}(ℋ) and inner functions Γ_{r} ∈ H_{𝔻}^{∞}(B_{ℋ_{1} →ℋ} ) and Φ_{r} ∈ H_{𝔻}^{∞}(B_{ℋ_{2} →ℋ} ) such that 𝒢 = Γ_{r}H_{𝔻}^{2}(ℋ_{1}) and ℱ = Φ_{r}H_{𝔻}^{2}(ℋ_{2}), the operators (M_{Γ_{r}})^{∗}Π_{𝒢∥ℱ} and (M_{Φ_{r}})^{∗}Π_{ℱ∥𝒢} are bounded and LSI.

Proof : It is shown that (M_{Γ_{r}})^{∗}Π_{𝒢∥ℱ} is bounded and LSI, the proof that (M_{Φ_{r}})^{∗}Π_{ℱ∥𝒢} is bounded and LSI follows similarly. (M_{Γ_{r}})^{∗}Π_{𝒢∥ℱ} is clearly linear and bounded on H_{𝔻}^{2}(ℋ) since it is the product of two bounded, linear operators. To see that it is also shift invariant, note that any η ∈ H_{𝔻}^{2}(ℋ) is uniquely expressible as η = γ + φ, where γ = Π_{𝒢∥ℱ} η ∈ 𝒢 and φ = Π_{ℱ∥𝒢} η ∈ ℱ. Let ζ be the unique element in H_{𝔻}^{2}(ℋ_{1}) corresponding to γ = M_{Γ_{r}}ζ. Then since M_{Γ_{r}} is an isometry it follows that (M_{Γ_{r}})^{∗}Π_{𝒢∥ℱ} η = ζ. Furthermore, since Sη = Sγ + Sφ, Sγ = M_{Γ_{r}}Sζ ∈ 𝒢 and Sφ ∈ ℱ it is clear that

(M_{Γ_{r}})^{∗}Π_{𝒢∥ℱ} Sη = (M_{Γ_{r}})^{∗}Sγ = Sζ = S(M_{Γ_{r}})^{∗}Π_{𝒢∥ℱ} η.

That is, (M_{Γ_{r}})^{∗}Π_{𝒢∥ℱ} is also shift invariant.

Theorem 4.5 Given a closed, LSI subspace 𝒢 ⊂ H_{𝔻}^{2}(ℋ) that is LSI co-ordinatisable, there exist functions Γ_{r} ∈ H_{𝔻}^{∞}(B_{ℋ_{1} →ℋ} ), Γ_{l} ∈ H_{𝔻}^{∞}(B_{ℋ→ℋ_{2}} ), Φ_{r} ∈ H_{𝔻}^{∞}(B_{ℋ_{2} →ℋ} ) and Φ_{l} ∈ H_{𝔻}^{∞}(B_{ℋ→ℋ_{1}} ) such that

𝒢 = ℛ_{M_{Γr}} = 𝒦_{M_{Γ_{l}}} ,

[ M_{Γ_{r}} M_{Φ_{r}} ][ \begin{matrix}M & Φ_{l} \\ M & Γ_{l} \end{matrix} ] = I and [ \begin{matrix}M & Φ_{l} \\ M & Γ_{l} \end{matrix} ][ M_{Γ_{r}} M_{Φ_{r}} ] = [ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ],

where ℋ_{1} can be any Hilbert space isomorphic to 𝒢⊖S𝒢, ℋ_{2} any Hilbert space isomorphic to ℱ⊖Sℱ and ℱ is any closed, LSI subspace such that {𝒢, ℱ} is a co-ordinatisation of H_{𝔻}^{2}(ℋ).

Proof : Let {𝒢, ℱ} be a co-ordinatisation of H_{𝔻}^{2}(ℋ) and the operators Π_{𝒢∥ℱ} , Π_{ℱ∥𝒢} , M_{Γ_{r}} and M_{Φ_{r}} be defined as above, recalling that 𝒢 = ℛ_{M_{Γr}} and ℱ = ℛ_{M_{Φr}}. Furthermore, note that

Π_{𝒢∥ℱ} = Π_{𝒢}Π_{𝒢∥ℱ} = M_{Γ_{r}}(M_{Γ_{r}})^{∗}Π_{𝒢∥ℱ}  (4.1)

and

Π_{ℱ∥𝒢} = Π_{ℱ} Π_{ℱ∥𝒢} = M_{Φ_{r}}(M_{Φ_{r}})^{∗}Π_{ℱ∥𝒢} .  (4.2)

Now by Lemma 4.4, (M_{Γ_{r}})^{∗}Π_{𝒢∥ℱ} and (M_{Φ_{r}})^{∗}Π_{ℱ∥𝒢} are bounded, LSI operators on H_{𝔻}^{2}(ℋ) and hence, can be expressed as multiplication operators with symbols Φ_{l} ∈ H_{𝔻}^{∞}(B_{ℋ→ℋ_{1}} ) and Γ_{l} ∈ H_{𝔻}^{∞}(B_{ℋ→ℋ_{2}} ) respectively (cf. Proposition 2.2). That is,

M_{Φ_{l}} = (M_{Γ_{r}})^{∗}Π_{𝒢∥ℱ} and M_{Γ_{l}} = (M_{Φ_{r}})^{∗}Π_{ℱ∥𝒢} .  (4.3)

Then since ℱ = ℛ_{M_{Φr}}, it follows that ℛ_{Π_{ℱ∥𝒢}} is orthogonal to 𝒦_{(M_{Φr})^{∗}} and consequently, that 𝒦_{M_{Γ_{l}}} = 𝒦_{Π_{ℱ∥𝒢}} = 𝒢. Similarly, ℱ = 𝒦_{M_{Φ_{l}}}. So it remains to show that

[ M_{Γ_{r}} M_{Φ_{r}} ][ \begin{matrix}M & Φ_{l} \\ M & Γ_{l} \end{matrix} ] = [ \begin{matrix}M & Φ_{l} \\ M & Γ_{l} \end{matrix} ][ M_{Γ_{r}} M_{Φ_{r}} ] = [ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ].

It follows from the relationship Π_{𝒢∥ℱ} + Π_{ℱ∥𝒢} = I and equations (4.1-4.3), that

[ M_{Γ_{r}} M_{Φ_{r}} ][ \begin{matrix}M & Φ_{l} \\ M & Γ_{l} \end{matrix} ] = I.  (4.4)

Now since 𝒢 = ℛ_{M_{Γr}} = 𝒦_{M_{Γ_{l}}}, ℱ = ℛ_{M_{Φr}} = 𝒦_{M_{Φ_{l}}} and {𝒢, ℱ} is a co-ordinatisation of H_{𝔻}^{2}(ℋ), both [ M_{Γ_{r}} M_{Φ_{r}} ] and [ \begin{matrix}M & Φ_{l} \\ M & Γ_{l} \end{matrix} ] have zero kernel. Thus, they are the inverse of each other (cf. equation (4.4)) and

[ \begin{matrix}M & Φ_{l} \\ M & Γ_{l} \end{matrix} ][ M_{Γ_{r}} M_{Φ_{r}} ] = [ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ],

which completes the proof.

4.2.3 Strong-Right/Left Representations and Closed-Loop Stability

Results derived in the previous sub-section are now used to develop range and kernel representations of the graphs of stabilisable systems in 𝒫_{𝒰→𝒴}^{e} , that are themselves (causal) stable, LPTV systems with (causal) stable, LPTV, left and right inverses respectively. All notions of well-posedness and closed-loop stability used in this sub-section are defined in Sub-Section 2.3.2. Recall the standard, closed-loop configuration, denoted [P, C], shown in Figure 4.2. The following result characterises the graph of a stabilisable, LPTV, continuous-time system in terms the range and kernel of stable, LPTV systems that are respectively, left and right invertible by stable, LPTV systems. These representations resemble the coprime-factor representations known to exist for LTI systems (cf. [Vid84, Smi89]) and linear, time-varying, discrete-time systems with finite-dimensional input-output spaces (cf. [DS93]).

[figure: C; P; d_{1}; u_{1}; y_{1}; d_{2}; u_{2}; y_{2}; −; +; +; −]

Figure 4.2 Standard Feedback Configuration

Theorem 4.6 If P ∈ 𝒫_{𝒰→𝒴}^{e} can be stabilised by some C ∈ 𝒫_{𝒴→𝒰}^{e} , then there exist \underline{stable} systems G_{r} ∈ 𝒫_{𝒰→𝒱}^{e} ,^{1} G_{l} ∈ 𝒫_{𝒱→𝒴}^{e} , K_{l} ∈ 𝒫_{𝒱→𝒰}^{e} and K_{r} ∈ 𝒫_{𝒴→𝒱}^{e} such that

𝒢_{P} = ℛ_{G_{r}} = 𝒦_{G_{l}},

and

[ G_{r} K_{r} ][ \begin{matrix}K_{l} \\ G_{l} \end{matrix} ] = [ \begin{matrix}K_{l} \\ G_{l} \end{matrix} ][ G_{r} K_{r} ] = [ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ].

Proof : By assumption, P is stabilised by some C ∈ 𝒫_{𝒰→𝒴}^{e} . So by Proposition 2.12 and Remark 2.13, it follows that 𝒢_{P} and 𝒢_{C} induce a co-ordinatisation of L_{ℝ^{+}}^{2}(𝒱). Let P:=WP W^{−1} and C:=WCW^{−1}, noting that that 𝒢_{P} and 𝒢C^{′} are LSI subspaces of ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒱)). These are respectively, isomorphic (via Z) to the LSI subspaces 𝒢̂_{P}:=Z𝒢_{P} and 𝒢̂C^{′} :=Z𝒢C^{′} of H_{𝔻}^{2}(L_{ℍ}^{2}(𝒱)), which since 𝒢_{P} and 𝒢_{C} induce a co-ordinatisation of L_{ℝ^{+}}^{2}(𝒱), induce a co-ordinatisation of H_{𝔻}^{2}(L_{ℍ}^{2}(𝒱)). So by Theorem 4.5, there exist functions Ĝ_{r} ∈ H_{𝔻}^{∞}(B_{ℋ_{1} →L_{ℍ}^{2}(𝒱)} ), Ĝ_{l} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒱) →ℋ_{2}} ), K̂_{l} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒱) →ℋ_{1}} ) and K̂_{r} ∈ H_{𝔻}^{∞}(B_{ℋ_{2} →L_{ℍ}^{2}(𝒱)} ), such that

𝒢̂_{P} = ℛ_{MĜr} = 𝒦_{MĜ_{l}}

1Recall that 𝒱:=𝒰 ⊕ 𝒴.

and

[ MĜ_{r} MK̂_{r} ][ \begin{matrix}M & K̂_{l} \\ M & Ĝ_{l} \end{matrix} ] = [ \begin{matrix}M & K̂_{l} \\ M & Ĝ_{l} \end{matrix} ][ MĜ_{r} MK̂_{r} ] = [ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ],

where ℋ_{1} is a Hilbert space isomorphic to 𝒢̂_{P} ⊖ S𝒢̂_{P} and ℋ_{2} is a Hilbert space isomorphic to 𝒢̂C^{′} ⊖ S𝒢̂C^{′}. In fact, it is possible to take ℋ_{1} to be L_{ℍ}^{2}(𝒰) and ℋ_{2} to be L_{ℍ}^{2}(𝒴). To see this, note that since P ∈ 𝒫_{𝒰→𝒴} is causally extendible,

P_{0}𝒢_{P} = G_{r[0]}ℋ_{1} = [ \begin{matrix}I \\ P_{[0]} \end{matrix} ]L_{ℍ}^{2}(𝒰),  (4.5)

where G_{r[0]} = Ĝ_{r}(0) and P_{[0]} denotes the first element of the sequence uniquely identifiable with the Toeplitz representation of P. Also, since MK̂_{l}MĜ_{r} = I, it follows that K_{l[0]} = K̂_{l}(0) is a bounded left-inverse of G_{r[0]}.^{2} Define

Q:=K_{l[0]}[ \begin{matrix}I \\ P_{[0]} \end{matrix} ] : L_{ℍ}^{2}(𝒰) → ℋ_{1}.

Then since [ \begin{matrix}I \\ P_{[0]} \end{matrix} ] has zero kernel and K_{l[0]}G_{r[0]} = I, it follows from equation (4.5) that Q is a bijective mapping. Hence, Q(Q^{∗}Q)^{−1} is an isomorphism between L_{ℍ}^{2}(𝒰) and ℋ_{1}, and correspondingly, L_{ℍ}^{2}(𝒰) is isomorphic to 𝒢̂_{P} ⊖ S𝒢̂_{P} ∼ ℋ_{1}. It can be shown in a similar manner that L_{ℍ}^{2}(𝒴) is isomorphic to 𝒢̂C^{′} ⊖ S𝒢̂C^{′}.

Now the multiplication operators MĜ_{r}, MĜ_{l}, MK̂_{l} and MK̂_{r} are equivalent (via Z) to stable systems G_{r} ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒱)} , G_{l} ∈ 𝒟_{L_{ℍ}^{2}(𝒱) →L_{ℍ}^{2}(𝒴)} , K_{l} ∈ 𝒟_{L_{ℍ}^{2}(𝒱) →L_{ℍ}^{2}(𝒰)} and K_{r} ∈ 𝒟_{L_{ℍ}^{2}(𝒴)→L_{ℍ}^{2}(𝒱)} respectively, with 𝒢_{P} = ℛ_{G_{r}} = 𝒦_{G_{l}} and

[ G_{r} K_{r} ][ \begin{matrix}K_{l} \\ G_{l} \end{matrix} ] = [ \begin{matrix}K_{l} \\ G_{l} \end{matrix} ][ G_{r} K_{r} ] = [ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ].

Furthermore, observe that given any stable system Q_{0} ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} and stable systems Q_{1} ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒰)} and Q_{2} ∈ 𝒟_{L_{ℍ}^{2}(𝒴)→L_{ℍ}^{2}(𝒴)} with stable inverses,

𝒢_{P} = ℛ_{G_{r}Q_{1}} = 𝒦_{Q_{2}^{−1}G_{l}}

2That G_{r[0]} has a bounded left-inverse also follows from the fact that Ĝ_{r} may be taken to be inner (see the proof of Theorem 4.5).

and

[ G_{r⋆} K_{r⋆} ][ \begin{matrix}K_{l⋆} \\ G_{l⋆} \end{matrix} ] := [ G_{r}Q_{1} (K_{r} − G_{r}Q_{0})Q_{2} ][ \begin{matrix}Q_{1}^{−1}(K_{l} + Q_{0}G_{l}) \\ Q_{2}^{−1}G_{l} \end{matrix} ]
= [ \begin{matrix}Q_{1}^{−1}(K_{l} + Q_{0}G_{l}) \\ Q_{2}^{−1}G_{l} \end{matrix} ][ G_{r}Q_{1} (K_{r} − G_{r}Q_{0})Q_{2} ]
= [ \begin{matrix}Q_{1}^{−1} & Q_{1}^{−1}Q_{0} \\ 0 & Q_{2}^{−1} \end{matrix} ][ \begin{matrix}K_{l} \\ G_{l} \end{matrix} ][ G_{r} K_{r} ][ \begin{matrix}Q_{1} & −Q_{0}Q_{2} \\ 0 & Q_{2} \end{matrix} ]
= [ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ].  (4.6)

So if Q_{0}, Q_{1} and Q_{2} can be constructed to make (G_{r}Q_{1})_{[0]}, (Q_{2}^{−1}G_{l})_{[0]}, (Q_{1}^{−1}(K_{l} + Q_{0}G_{l}))_{[0]} and ((K_{r} + G_{r}Q_{0})Q_{2})_{[0]} causal mappings on the finite horizon ℍ, it would follow by Remark 4.2, that the stable systems G_{r⋆} ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒱)} , G_{l⋆} ∈ 𝒟_{L_{ℍ}^{2}(𝒱) →L_{ℍ}^{2}(𝒴)} , K_{l⋆} ∈ 𝒟_{L_{ℍ}^{2}(𝒱) →L_{ℍ}^{2}(𝒰)} and K_{r⋆} ∈ 𝒟_{L_{ℍ}^{2}(𝒴)→L_{ℍ}^{2}(𝒱)} are equivalent (via W) to stable systems G_{r⋆} ∈ 𝒫_{𝒰→𝒱}^{e} , G_{l⋆} ∈ 𝒫_{𝒱→𝒴}^{e} , K_{l⋆} ∈ 𝒫_{𝒱→𝒰}^{e} and K_{r⋆} ∈ 𝒫_{𝒴→𝒱}^{e} respectively,^{2} that satisfy

𝒢_{P} = ℛ_{G_{r⋆}} = 𝒦_{G_{l⋆}}

and

[ G_{r⋆} K_{r⋆} ][ \begin{matrix}K_{l⋆} \\ G_{l⋆} \end{matrix} ] = [ \begin{matrix}K_{l⋆} \\ G_{l⋆} \end{matrix} ][ G_{r⋆} K_{r⋆} ] = [ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ].

Accordingly, the rest of the proof is dedicated to showing that this is possible.

Let

[ \begin{matrix}M_{r[0]} \\ N_{r[0]} \end{matrix} ]:=G_{r[0]} and [ −N_{l[0]} M_{l[0]} ]:=G_{l[0]},

where the partitioning is conformal with that of 𝒢_{P}. Then since P ∈ 𝒫_{𝒰→𝒴}^{e} is causally extendible with locally-Lipschitz-continuous extension, it follows that both M_{r[0]} ∈ B_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒰)} and M_{l[0]} ∈ B_{L_{ℍ}^{2}(𝒴)→L_{ℍ}^{2}(𝒴)} are bijective and therefore, boundedly invertible by the Open Mapping Theorem [Bol90, pg. 79]. To see this, first note that since P ∈ 𝒫_{𝒰→𝒴}^{e} ,

ℛ_{M_{r[0]}} = P_{0}𝒟_{P} = P_{h}𝒟_{P} = L_{ℍ}^{2}(𝒰)

2Since these systems are all stable, they have locally-Lipschitz-continuous, causal extensions.

and hence, that M_{r[0]} is surjective.^{3} Suppose now, that there exists some non-zero q^{⊸} ∈ L_{ℍ}^{2}(𝒰) such that M_{r[0]} q^{⊸} = 0. Then by causality of P, it follows that N_{r[0]} q^{⊸} must also be zero. Since K_{l[0]}G_{r[0]} = I, this implies that

0 = K_{l[0]}[ \begin{matrix}M_{r[0]} \\ N_{r[0]} \end{matrix} ] q^{⊸} = q^{⊸},

which is a contradiction. Therefore, the linear operator M_{r[0]} has zero kernel and is by definition, injective. This combined with the fact that it is also surjective, implies that M_{r[0]} is bijective as claimed. Now consider M_{l[0]} and suppose there exists a non-zero q^{⊸} ∈ L_{ℍ}^{2}(𝒴) such that M_{l[0]} q^{⊸} = 0. Then since

𝒦_{[ _{−N_{l[0]} M_{l[0]}} ]} = P_{0}𝒢_{P} = P_{h}𝒢_{P} ,

it follows that [ 0 ; q^{⊸} ]∈ P_{h}𝒢_{P}. But this contradicts the causality of P and hence, q^{⊸} must be 0. Consequently, M_{l[0]} has zero kernel and is thereby injective. Moreover, since G_{l[0]}K_{r[0]} = I,

ℛ_{[ _{−N_{l[0]} M_{l[0]}} ]} = L_{ℍ}^{2}(𝒴).

So for all e^{⊸} ∈ L_{ℍ}^{2}(𝒴), there exists a g^{⊸} =[ \begin{matrix}g^{⊸} & _{1} \\ g^{⊸} & _{2} \end{matrix} ]∈ L_{ℍ}^{2}(𝒱) such that

e^{⊸} = [ −N_{l[0]} M_{l[0]} ][ \begin{matrix}g^{⊸} & 1 \\ g^{⊸} & 2 \end{matrix} ],

In fact,

e^{⊸} = [ −N_{l[0]} M_{l[0]} ][ \begin{matrix}g^{⊸}_{1} + u^{⊸} \\ g^{⊸}_{2} + P_{[0]} u^{⊸} \end{matrix} ]

for all u^{⊸} ∈ P_{h}𝒟_{P}. Since P ∈ 𝒫_{𝒰→𝒴}^{e} , P_{h}𝒟_{P} = L_{ℍ}^{2}(𝒰) and consequently, u^{⊸} can be set to − g^{⊸}_{1}. Doing this, gives e^{⊸} = M_{l[0]}( g^{⊸}_{2} − P_{[0]} g^{⊸}_{1}), from which it is evident that ℛ_{M_{l[0]}} = L_{ℍ}^{2}(𝒴). Therefore, M_{l[0]} is bijective as claimed.

Having shown that M_{r[0]} and M_{l[0]} are boundedly invertible, it is possible to construct the required Q_{0}, Q_{1} and Q_{2}. Let Q_{1} satisfy Q_{1[0]} = M_{r[0]}^{−1} and Q_{1[i]} = 0 for i ≠ 0, where {Q_{1[i]}}_{i=0}^{∞} denotes the sequence uniquely identifiable with the Toeplitz representation of Q_{1}. Then

(G_{r}Q_{1})_{[0]} = [ \begin{matrix}I \\ N_{r[0]}M_{r[0]}^{−1} \end{matrix} ],

3Note that in the equation above P_{0} corresponds to a discrete-time truncation and P_{h} to truncation in a continuous-time signal-space.

which by causality of P is causal on L_{ℍ}^{2}(𝒰). Now let [ Y_{l[0]} X_{l[0]} ]:=K_{l[0]}, where the partition is conformal with 𝒢_{P} and construct Q_{0} so that Q_{0[0]} = −X_{l[0]}M_{l[0]}^{−1} and Q_{0[i]} = 0 for i ≠ 0. Then since N_{r[0]}M_{r[0]}^{−1} = M_{l[0]}^{−1}N_{l[0]} = P_{[0]} and Y_{l[0]}M_{r[0]} + X_{l[0]}N_{r[0]} = I, it follows that

(Q_{1}^{−1}(K_{l} + Q_{0}G_{l}))_{[0]} = (Q_{1}^{−1})_{[0]} (K_{l[0]} + Q_{0[0]}[ −N_{l[0]} M_{l[0]} ])
= M_{r[0]}[ Y_{l[0]} + X_{l[0]}M_{l[0]}^{−1}N_{l[0]} 0 ]
= M_{r[0]}[ M_{r[0]}^{−1} 0 ] = [ I 0 ],

which is obviously causal on L_{ℍ}^{2}(𝒱) as required. Finally, let [ \begin{matrix}−X_{r[0]} \\ Y_{r[0]} \end{matrix} ]:=K_{r[0]}. Then from equation (4.6), it follows that M_{r[0]}X_{l[0]} − X_{r[0]}M_{l[0]} = 0 and hence, that Q_{0[0]} = −X_{l[0]}M_{l[0]}^{−1} = M_{r[0]}^{−1}X_{r[0]}. Defining Q_{2} to satisfy Q_{2[0]} = M_{l[0]} and Q_{2[i]} = 0 for i ≠ 0, gives (Q_{2}^{−1}G_{l})_{[0]} = [ −M_{l[0]}^{−1}N_{l[0]} I ]

and ((K_{r} + G_{r}Q_{0})Q_{2})_{[0]} =[ \begin{matrix}0 \\ I \end{matrix} ]. As required, both are causal on L_{ℍ}^{2}(𝒱) and L_{ℍ}^{2}(𝒴) respectively. This completes the proof.

Remark 4.7 Since G_{r} and G_{l} in Theorem 4.6 are respectively, left and right invertible by stable, LPTV systems, they are called \underline{strong-right} and respectively \underline{strong-left} representations of 𝒢_{P}. Note that they are only unique up to invertible factors. That is, for any stable systems Q_{1}, Q_{1}^{−1} ∈ 𝒫_{𝒰→𝒰}^{e} and Q_{2}, Q_{2}^{−1} ∈ 𝒫_{𝒴→𝒴}^{e} , the systems G_{r}Q_{1} and Q_{2}^{−1}G_{l} are also strong-right and strong-left representations of 𝒢_{P} respectively. ♣

Given a well-posed, plant/controller pair [P, C], the following lemma constitutes a useful characterisation of closed-loop stability in terms of range and kernel representations of 𝒢_{P} and 𝒢C^{′}.

Lemma 4.8 Given a plant P ∈ 𝒫_{𝒰→𝒴}^{e} and controller C ∈ 𝒫_{𝒴→𝒰}^{e} such that [P, C] is well-posed, let G_{r} ∈ 𝒫_{𝒰→𝒱}^{e} be any stable system with zero kernel such that 𝒢_{P} = ℛ_{G_{r}} and K_{l} ∈ 𝒫_{𝒱→𝒰}^{e} be any stable system such that ℛ_{K_{l}} = L_{ℝ^{+}}^{2}(𝒰) and 𝒢C^{′} = 𝒦_{K_{l}}. Then the following are equivalent:

(i) The closed-loop system [P, C] is stable;

(ii) The stable, LPTV system K_{l}G_{r} is bijective (and hence boundedly invertible, although the inverse may not be causal).

Furthermore, if [P, C] is stable and G_{r} and K_{l} are \underline{strong-right} and \underline{strong-left} representations of 𝒢_{P} and 𝒢C^{′} respectively, then (K_{l}G_{r})^{−1} is a stable system in 𝒫_{𝒰→𝒰}^{e} . That is the inverse is also causal.

Proof : Since the closed-loop is well-posed by assumption, Proposition 2.12 holds (cf. Remark 2.13), and using the identities 𝒢_{P} = ℛ_{G_{r}} and 𝒢C^{′} = 𝒦_{K_{l}}, it follows that

[P, C] is stable ⇔ ℛ_{G_{r}} + 𝒦_{K_{l}} = L_{ℝ^{+}}^{2}(𝒱) and ℛ_{G_{r}} ∩ 𝒦_{K_{l}} = {0}.

Then since ℛ_{K_{l}} = L_{ℝ^{+}}^{2}(𝒰) and 𝒦_{G_{r}} = {0}, it is clear that

ℛ_{G_{r}} + 𝒦_{K_{l}} = L_{ℝ^{+}}^{2}(𝒱) and ℛ_{G_{r}} ∩ 𝒦_{K_{l}} = {0}
⇓
ℛ_{K_{l}G_{r}} = L_{ℝ^{+}}^{2}(𝒰) and 𝒦_{K_{l}G_{r}} = {0}.

To see that the converse is also true, let ℛ_{K_{l}G_{r}} = L_{ℝ^{+}}^{2}(𝒰) and 𝒦_{K_{l}G_{r}} = {0}. That ℛ_{G_{r}} ∩𝒦_{K_{l}} = {0} is immediate. Now suppose there exists a v ∈ L_{ℝ^{+}}^{2}(𝒱) such that v ∉ ℛ_{G_{r}} + 𝒦_{K_{l}} and define e:=K_{l}v. Then since ℛ_{K_{l}G_{r}} = L_{ℝ^{+}}^{2}(𝒰) and 𝒦_{K_{l}G_{r}} = {0}, there exist a unique q ∈ L_{ℝ^{+}}^{2}(𝒰) such that e = K_{l}G_{r}q = K_{l}g, where g :=G_{r}q. Consequently, K_{l}(g − v) = 0, which implies that either v = g or (g − v) ∈ 𝒦_{K_{l}}. Since g ∈ ℛ_{G_{r}}, this contradicts v ∉ ℛ_{G_{r}} + 𝒦_{K_{l}} and it follows that [P, C] is stable if and only if the bounded linear operator K_{l}G_{r} is bijective (and hence, boundedly invertible by the Open Mapping Theorem [Bol90, pg. 79]). Note that (K_{l}G_{r})^{−1} is periodically time-varying, but may not be locally causal in L_{ℍ}^{2}(𝒰).

It is now shown that if [P, C] is stable and G_{r} and K_{l} are strong-right and strong-left representations of 𝒢_{P} and 𝒢C^{′} respectively, then (K_{l}G_{r})^{−1} ∈ 𝒫_{𝒰→𝒰}^{e} . First note that for any K_{l} and G_{r} that satisfy the original conditions of the lemma,

Π_{𝒢_{P} ∥𝒢′_{C}} = G_{r}(K_{l}G_{r})^{−1}K_{l}.  (4.7)

To see this, note that since [P, C] is stable, {𝒢_{P} , 𝒢C^{′}} induces a co-ordinatisation of L_{ℝ^{+}}^{2}(𝒱). Therefore, any g ∈ L_{ℝ^{+}}^{2}(𝒱) can be written uniquely as the sum g = p + c, where p ∈ 𝒢_{P} and c ∈ 𝒢C^{′}. Furthermore, since 𝒦_{G_{r}} = {0} and ℛ_{G_{r}} = 𝒢_{P}, there exists a unique q ∈ L_{ℝ^{+}}^{2}(𝒰) such that p = G_{r}q. Consequently,

(Π_{𝒢_{P} ∥𝒢′_{C}} − G_{r}(K_{l}G_{r})^{−1}K_{l})g = p − G_{r}(K_{l}G_{r})^{−1}K_{l}p
= p − G_{r}(K_{l}G_{r})^{−1}K_{l}G_{r}q
= p − G_{r}q = 0,

for all g ∈ L_{ℝ^{+}}^{2}(𝒰), which in turn, implies that Π_{𝒢_{P} ∥𝒢′_{C}} = G_{r}(K_{l}G_{r})^{−1}K_{l}. Now, take G_{r} and K_{l} to be strong-right and strong-left representations. Then by definition, G_{r} and K_{l} satisfy the original conditions of the lemma and there exist stable systems K_{l⋆} ∈ 𝒫_{𝒱→𝒰}^{e} and G_{r⋆} ∈ 𝒫_{𝒰→𝒱}^{e} such that K_{l⋆}G_{r} = I and K_{l}G_{r⋆} = I. Furthermore, from equation (4.7),

(K_{l}G_{r})^{−1} = K_{l⋆}Π_{𝒢_{P} ∥𝒢′_{C}} G_{r⋆}.

Then recalling that Π_{𝒢_{P} ∥𝒢′_{C}} ∈ 𝒫_{𝒱→𝒱}^{e} (cf. Remark 2.14), it follows that (K_{l}G_{r})^{−1} ∈ 𝒫_{𝒰→𝒰}^{e} .

Remark 4.9 Note that by interchanging the roles of P and C in Lemma 4.8, given any stable system G_{l} ∈ 𝒫_{𝒱→𝒴}^{e} such that ℛ_{G_{l}} = L_{ℝ^{+}}^{2}(𝒰) and 𝒢_{P} = 𝒦_{G_{l}} and any stable system K_{r} ∈ 𝒫_{𝒴→𝒱}^{e} with zero kernel such that 𝒢C^{′} = ℛ_{K_{r}}, the following are equivalent :

(i) The closed-loop system [P, C] is stable;

(ii) The stable, LPTV system G_{l}K_{r} is bijective (and hence boundedly invertible, although the inverse may not be causal).

Furthermore, if G_{l} and K_{r} are \underline{strong-left} and \underline{strong-right} representations of 𝒢_{P} and 𝒢C^{′} respectively and [P, C] is stable, then (G_{l}K_{r})^{−1} is a stable system 𝒫_{𝒴→𝒴}^{e} . ♣

Remark 4.10 Given a well-posed, plant/controller pair [P, C] where P ∈ 𝒫_{𝒰→𝒴}^{e} and C ∈ 𝒫_{𝒴→𝒰}^{e} , let P:=WP W^{−1} and C:=WCW^{−1}. Furthermore, let G_{r} ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} and K_{l} ∈ 𝒟_{L_{ℍ}^{2}(𝒴)→L_{ℍ}^{2}(𝒰)} be stable systems that satisfy 𝒦_{G_{r}} = {0}, 𝒢_{P} = ℛ_{G_{r}}, ℛ_{K_{l}} = ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰)) and 𝒢C^{′} = 𝒦_{K_{l}}. Then the closed-loop [P, C] is stable if and only if the bounded linear operator K_{l}G_{r} is boundedly invertible. In this case (K_{l}G_{r})^{−1} is a stable system in 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒰)} and hence, equivalent (via Z) to a multiplication operator with symbol in H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} ) (cf. Proposition 2.2). Moreover, Π_{𝒢_{P}∥𝒢′_{C}} = G_{r}(K_{l}G_{r})^{−1}K_{l}. ♣

It is interesting to note that given a system P ∈ 𝒫_{𝒰→𝒴}^{e} and a stabilising C ∈ 𝒫_{𝒴→𝒰}^{e} , strongright and strong-left representations of 𝒢C^{′} can be constructed from strong-right and strong-left representations of 𝒢_{P} and the respective stable, left and right inverses. This is analogous to necessity of the well-known Youla parameterisation of stabilising controllers for certain classes of systems [YJB76, DLJS80, DG84, DS93] and is summarised in the following lemma.

Lemma 4.11 Given a stabilisable system P ∈ 𝒫_{𝒰→𝒴}^{e} and stable systems G_{r} ∈ 𝒫_{𝒰→𝒱}^{e} , G_{l} ∈ 𝒫_{𝒱→𝒴}^{e} , K_{l} ∈ 𝒫_{𝒱→𝒰}^{e} and K_{r} ∈ 𝒫_{𝒴→𝒱}^{e} such that

𝒢_{P} = ℛ_{G_{r}} = 𝒦_{G_{l}}

and

[ G_{r} K_{r} ][ \begin{matrix}K_{l} \\ G_{l} \end{matrix} ] = [ \begin{matrix}K_{l} \\ G_{l} \end{matrix} ][ G_{r} K_{r} ] = [ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ],

a system C ∈ 𝒫_{𝒴→𝒰}^{e} stabilises P only if there exists a stable Q ∈ 𝒫_{𝒴→𝒰}^{e} such that

K_{r} − G_{r}Q

is a strong-right representation of 𝒢C^{′} and

K_{l} + QG_{l}

is a strong-left representation of 𝒢C^{′}.

Proof : Let C ∈ 𝒫_{𝒴→𝒴}^{e} denote a stabilising controller for P. Then it follows by Theorem 4.6, that there exist stable systems G_{r} ∈ 𝒫_{𝒰→𝒱}^{e} , G_{l} ∈ 𝒫_{𝒱→𝒴}^{e} , K_{l} ∈ 𝒫_{𝒱→𝒰}^{e} and K_{r} ∈ 𝒫_{𝒴→𝒱}^{e} such that

𝒢_{P} = ℛ_{G_{r}} = 𝒦_{G_{l}}

and

[ G_{r} K_{r} ][ \begin{matrix}K_{l} \\ G_{l} \end{matrix} ] = [ \begin{matrix}K_{l} \\ G_{l} \end{matrix} ][ G_{r} K_{r} ] = [ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ].  (4.8)

Furthermore, for any stable system Q ∈ 𝒫_{𝒴→𝒰}^{e} ,

[ G_{r} (K_{r} − G_{r}Q) ][ \begin{matrix}(K_{l} + QG_{l}) \\ G_{l} \end{matrix} ] = [ \begin{matrix}(K_{l} + QG_{l}) \\ G_{l} \end{matrix} ][ G_{r} (K_{r} − G_{r}Q) ] = [ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ]. (4.9)

Now from equation (4.9),

(K_{l} + QG_{l})(K_{r} − G_{r}Q) = 0,

which implies ℛ_{(K_{r}−G_{r}Q)} ⊂ 𝒦_{(K_{l}+QG_{l})}. To see that the opposite inclusion also holds, note from equation (4.9), that

(K_{l} + QG_{l})[ G_{r} (K_{r} − G_{r}Q) ] = [ I 0 ].  (4.10)

Now suppose there exists a non-zero x ∈ 𝒦_{(K_{l}+QG_{l})} that is not in ℛ_{(K_{r}−G_{r}Q)}. Then since

𝒦_{[ _{G_{r} (K_{r} − G_{r}Q)} ]} = {0} and ℛ_{[ _{G_{r} (K_{r} − G_{r}Q)} ]} = L_{ℝ^{+}}^{2}(𝒱),

there exists a unique u =:[ \begin{matrix}u_{1} \\ u_{2} \end{matrix} ]∈ L_{ℝ^{+}}^{2}(𝒱) such that

x = [ G_{r} (K_{r} − G_{r}Q) ][ \begin{matrix}u_{1} \\ u_{2} \end{matrix} ].

Furthermore, 0 = (K_{l} + QG_{l})x = u_{1}, which using equation (4.10), implies that x ∈ ℛ_{(K_{r}−G_{r}Q)}. This is a contradiction and therefore, 𝒦_{(K_{l}+QG_{l})} ⊂ ℛ_{(K_{r}−G_{r}Q)}.

Since [P, C] is stable (or equivalently C is stabilised P), it follows by Theorem 4.6, that there exist strong-right and strong-left representations of 𝒢_{C}. Noting that 𝒢C^{′} :=[ \begin{matrix}0 & I \\ I & 0 \end{matrix} ]𝒢_{C}, these give rise to strong-right and strong-left representations of 𝒢C^{′}, denoted by K_{r⋆} ∈ 𝒫_{𝒴→𝒱}^{e} and K_{l⋆} ∈ 𝒫_{𝒱→𝒰}^{e} respectively. It follows by Lemma 4.8, that K_{l⋆}G_{r} is boundedly invertible in 𝒫_{𝒰→𝒰}^{e} . Now define

Q:= − (K_{l⋆}G_{r})^{−1}K_{l⋆}K_{r}

and note that Q is a stable system in 𝒫_{𝒴→𝒰}^{e} . Then using equation (4.8),

K_{l} − QG_{l} = K_{l} + (K_{l⋆}G_{r})^{−1}K_{l⋆}K_{r}G_{l}
= K_{l} + (K_{l⋆}G_{r})^{−1}K_{l⋆}(I − G_{r}K_{l})
= K_{l} − K_{l} + (K_{l⋆}G_{r})^{−1}K_{l⋆} = (K_{l⋆}G_{r})^{−1}K_{l⋆},

which implies 𝒦_{(K_{l}−QG_{l})} = 𝒦_{K_{l⋆}}. Since ℛ_{(K_{r}+G_{r}Q)} = 𝒦_{(K_{l}−QG_{l})} and ℛ_{K_{r⋆}} = 𝒦_{K_{l⋆}}, it also follows that ℛ_{(K_{r}+G_{r}Q)} = ℛ_{K_{r⋆}}. That is, the strong-right and strong-left representations of 𝒢C^{′} are in the form required.

Remark 4.12 If in Lemma 4.11, the choice of Q ∈ 𝒫_{𝒰→𝒴}^{e} is restricted to those for which ℛ_{(K_{r}−G_{r}Q)} = 𝒦_{(K_{l}+QG_{l})} correspond to the inverse graph of a system C ∈ 𝒫_{𝒰→𝒴}^{e} such that the closed-loop [P, C] is well-posed, then C stabilises P. That is, the Lemma would also be sufficient. To see this, note from equation (4.9) that for such a Q, the system (K_{l} + QG_{l}) is a strong-left representation of 𝒢C^{′}. So by arguments similar to those in the proof of Lemma 4.8 and the fact that (K_{l} + QG_{l})G_{r} = I is obviously invertible, it follows that [P, C] is stable. Again from equation (4.9) and since ℛ_{(K_{r}−G_{r}Q)} = 𝒦_{(K_{l}+QG_{l})} = 𝒢C^{′}, it is also clear that (K_{r} − G_{r}Q) is a strong-right representation of the inverse graph of a stabilising controller. ♣

4.3 The Gap Metric

In this section, the gap metric is formally introduced. The main result of the section is a formula for the directed gap between two LPTV systems. This formula is essentially a generalisation of Georgiou’s [Geo88] for LTI systems. It is fundamental to the main robustness result of this chapter and hence, to the proof that the topology induced by the gap metric on a class of LPTV systems, is the weakest with respect to which closed-loop performance varies continuously and closed-loop stability is a robust property. For additional information on the gap metric the reader is referred to [FGS93, SO93, GS90, Geo88, ZES80, KVZ^{+}72, Kat66].

The gap (or aperture) between two (closed) subspaces ℋ_{0} and ℋ_{1} of a Hilbert space ℋ, is defined to be

δ(ℋ_{0}, ℋ_{1}) :=∥Π_{ℋ_{0}} − Π_{ℋ_{1}} ∥

and it can be shown (cf. [KVZ^{+}72, §15 Chap. 4]) that

δ(ℋ_{0}, ℋ_{1}) = max{δ⃗(ℋ_{0}, ℋ_{1}), δ⃗(ℋ_{1}, ℋ_{0})},

where

δ⃗(ℋ_{i}, ℋ_{j}) = ∥Π_{ℋ_{j}^{⊥}} Π_{ℋ_{i}} ∥ i, j = 0, 1,

is the directed gap. Note that 0 ≤ δ(ℋ_{0}, ℋ_{1}) ≤ 1 and that it defines a metric on ℋ.

Proposition 4.13 [KVZ^{+}72]. For i = 0, 1, let ℋ_{i} be a closed subspace of a Hilbert space ℋ. Then Π_{ℋ_{1}} |_{ℋ_{0}} is a bijective mapping from ℋ_{0} to ℋ_{1} if and only if δ(ℋ_{0}, ℋ_{1}) < 1. Moreover, if δ(ℋ_{0}, ℋ_{1}) < 1 then

δ(ℋ_{0}, ℋ_{1}) = δ⃗(ℋ_{0}, ℋ_{1}) = δ⃗(ℋ_{1}, ℋ_{0}).

The gap between two (closed) operators P_{0} and P_{1} is defined to be the gap between their respective graphs,

δ_{g}(P_{0}, P_{1}) :=δ(𝒢_{P_{0}}, 𝒢_{P_{1}}),

with the directed gap being defined similarly [Kat66]. This is a metric and can be thought of as a measure of the distance between two systems. In the next section the gap metric is used to quantify the robust stability properties of LPTV, closed-loop systems.

Consider a system P_{0} ∈ 𝒫_{𝒰→𝒴} and recall that 𝒢_{P_{0}} is isomorphic to both 𝒢_{P_{0}} and 𝒢̂_{P_{0}} :=Z𝒢_{P_{0}}, where P_{0} :=WP_{0}W^{−1}. Given another system P_{1} ∈ 𝒫_{𝒰→𝒴} , the following hold :

δ_{g}(P_{0}, P_{1}) = δ(𝒢_{P_{0}}, 𝒢_{P_{1}}) = δ(𝒢̂_{P_{0}}, 𝒢̂_{P_{1}}),

for the gap and

δ⃗_{g}(P_{i}, P_{j}) = δ⃗(𝒢_{P_{i}}, 𝒢_{P_{j}}) = δ⃗(𝒢̂_{P_{i}}, 𝒢̂_{P_{j}}), (i, j = 0, 1)

for the directed gap. The following corollary of Proposition 4.13 constitutes a useful characterisation of when the gap between two systems is strictly less than one. It is used in the proof of the main robustness result in the next section. A similar result appeared in [SO93] and [Vin93b] for LTI systems.

Corollary 4.14 Given two systems P_{0} and P_{1} in 𝒫_{𝒰→𝒴} , the following are equivalent :

(i) δ_{g}(P_{0}, P_{1}) = δ(𝒢̂_{P_{0}}, 𝒢̂_{P_{1}}) < 1;

(ii) The bounded linear operator (MĜ_{r1})^{∗}MĜ_{r0} is bijective (and hence boundedly invertible),

where Ĝ_{ri} is any inner function in H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒱)} ) such that 𝒢̂_{P_{i}} :=ZW𝒢_{P_{i}} = Ĝ_{ri}H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰)) for i = 0, 1.

Proof : Since Π𝒢̂_{P_{i}} = MĜ_{ri}(MĜ_{ri})^{∗}, 𝒢̂_{P_{i}} = ℛ_{MĜ_{ri}} and 𝒦_{MĜ_{ri}} = {0}, it follows by Proposition 4.13 (cf. a similar result in Nikol’skii [Nik86, pg. 201] concerning closed subspaces), that δ(𝒢̂_{P_{0}}, 𝒢̂_{P_{1}}) < 1 if and only if MĜ_{r1}(MĜ_{r1})^{∗}MĜ_{r0} is a bijective mapping from H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰)) to 𝒢̂_{P_{1}}. Again using the fact that 𝒢̂_{P_{i}} = ℛ_{MĜ_{ri}} and 𝒦_{MĜ_{ri}} = {0}, this is equivalent to

ℛ_{(MĜ_{r1} )^{∗}MĜ_{r0}} = H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰)) and 𝒦_{(MĜ_{r1} )^{∗}MĜ_{r0}} = {0}.

That is, δ(𝒢̂_{P_{0}}, 𝒢̂_{P_{1}}) < 1 if and only if (MĜ_{r1})^{∗}MĜ_{r0} is bijective (and hence, boundedly invertible by the open mapping theorem [Bol90]).

An important theorem, characterising the directed gap between LPTV systems in terms of an optimisation problem over H_{𝔻}^{∞}, is now presented. It is essentially a generalisation of Georgiou’s result for LTI systems [Geo88] (cf. also [You86, Theorem 1] and [SO93, Corollary 4.4]). The relevant formula plays a crucial role in the proof of the main robustness result in the following section.

Theorem 4.15 For i, j = 0, 1,

δ⃗_{g}(P_{i}, P_{j}) = δ⃗(𝒢̂_{P_{i}}, 𝒢̂_{P_{j}}) = inf_{Q̂∈H𝔻^{∞}(B_{L^{2}_{ℍ}(𝒰)→L^{2}_{ℍ}(𝒰)} )} ∥Ĝ_{ri} − Ĝ_{rj}Q̂∥_{∞},

where Ĝ_{ri} is any inner function in H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒱)} ) such that 𝒢̂_{P_{i}} = Ĝ_{ri}H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰)).

Proof : The proof follows that of an H^{∞} optimisation result in [FF90, pg. 248]. Note that (for i = 0, 1) 𝒢̂_{P_{i}} is an LSI subspace of H_{𝔻}^{2}(L_{ℍ}^{2}(𝒱)). As before, it follows by the Beurling-Lax-Halmos Theorem, that there exists an inner function Ĝ_{ri} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒱)} ) such that 𝒢̂_{P_{i}} = Ĝ_{ri}H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰)). Since MĜ_{ri} is an isometry it can be shown that Π𝒢̂_{P_{i}} = MĜ_{ri}(MĜ_{ri})^{∗} and hence, that

δ⃗(𝒢̂_{P_{i}}, 𝒢̂_{P_{j}}) = ∥Π𝒢̂_{P_{j}}^{⊥} MĜ_{ri}(MĜ_{ri})^{∗}∥ = ∥Π𝒢̂_{P_{j}}^{⊥} MĜ_{ri}∥,

where the last equality follows from the fact that MĜ_{ri} is an isometry. Now recall that 𝒢̂_{P_{j}}^{⊥} :=H_{𝔻}^{2}(L_{ℍ}^{2}(𝒱))⊖ 𝒢̂_{P_{j}} and take any η ∈ H_{𝔻}^{2}(L_{ℍ}^{2}(𝒱)), expressing it as η = γ + φ, where γ = Π𝒢̂_{P_{j}} η and φ = Π𝒢̂_{P_{j}}^{⊥} η. Then Sη = Sγ + Sφ and

Π𝒢̂_{P_{j}}^{⊥} Sη = Π𝒢̂_{P_{j}}^{⊥} Sγ + Π𝒢̂_{P_{j}}^{⊥} Sφ
= Π𝒢̂_{P_{j}}^{⊥} Sφ,

where the second equality follows from the fact that S𝒢̂_{P_{j}} ⊂ 𝒢̂_{P_{j}}. That is,

Π𝒢̂_{P_{j}}^{⊥} Sη = Π𝒢̂_{P_{j}}^{⊥} SΠ𝒢̂_{P_{j}}^{⊥} η = S_{⋆}Π𝒢̂_{P_{j}}^{⊥} η  (4.11)

where

S_{⋆} :=Π𝒢̂_{P_{j}}^{⊥} S|𝒢̂_{P_{j}}^{⊥} : 𝒢̂_{P_{j}}^{⊥} → 𝒢̂_{P_{j}}^{⊥}.

The unilateral shift S is an isometric dilation (lifting) of S_{⋆} in that S_{⋆}^{n} = Π𝒢̂_{P_{j}}^{⊥} S^{n}|𝒢̂_{P_{j}}^{⊥} for all n ≥ 0. In fact, by the way 𝒢̂_{P_{j}}^{⊥} is defined and since 𝒢̂_{P_{j}} is shift-invariant, it follows that S is the minimal, isometric dilatation of S_{⋆} in the sense that H_{𝔻}^{2}(L_{ℍ}^{2}(𝒱)) is the smallest shift-invariant space that contains 𝒢̂_{P_{j}}^{⊥}.^{4} Furthermore, from equation (4.11), it follows that

S_{⋆}Π𝒢̂_{P_{j}}^{⊥} = Π𝒢̂_{P_{j}}^{⊥} S

and consequently, with the generalised, Hankel operator HĜ_{ri} defined to be

HĜ_{ri} :=Π𝒢̂_{P_{j}}^{⊥} MĜ_{ri},  (4.12)

that

S_{⋆}HĜ_{ri} = Π𝒢̂_{P_{j}}^{⊥} SMĜ_{ri} = HĜ_{ri}S.

Since S is a minimal, isometric dilation of S_{⋆}, it follows by the Commutant Lifting Theorem (cf. [FF90, Chapter VII] and [SNF70]), that there exists an LSI operator H : H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰)) → H_{𝔻}^{2}(L_{ℍ}^{2}(𝒱)) that satisfies Π𝒢̂_{P_{j}}^{⊥} H = HĜ_{ri} and

∥H∥ = ∥HĜ_{ri}∥.  (4.13)

Furthermore, by Proposition 2.2, the operator H can be expressed as a multiplication operator with symbol Ĥ ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒱)} ) and

∥H∥ = ∥Ĥ∥_{∞}.  (4.14)

Thus,

Π𝒢̂_{P_{j}}^{⊥} MĤ = HĜ_{ri} = Π𝒢̂_{P_{j}}^{⊥} MĜ_{ri},

which implies that Π𝒢̂_{P_{j}}^{⊥} (MĤ − MĜ_{ir}) = 0 or equivalently (since 𝒢̂_{P_{j}} = Ĝ_{rj}H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰))) that

(Ĥ − Ĝ_{ri})H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰)) ⊂ Ĝ_{rj}H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰)).

4cf. [FF90, SNF70] for a precise definition and treatment of minimal dilations.

Therefore, since Ĝ_{rj} is inner and by Corollary IX.2.2 in [FF90, pp. 239-240] (a corollary of the Beurling-Lax-Halmos Theorem), there exists a Q̂_{⋆} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} ) that satisfies

Ĥ − Ĝ_{ri} = −Ĝ_{rj}Q̂_{⋆}.

Hence, Ĥ = Ĝ_{ri} − Ĝ_{rj}Q̂_{⋆} and

∥Ĝ_{ri} − Ĝ_{rj}Q̂_{⋆}∥_{∞} = ∥HĜ_{ri}∥ = ∥H_{Ĝ_{ri}−Ĝ_{rj} Q̂}∥ ≤ inf_{Q̂∈H𝔻^{∞}(B_{L^{2}_{ℍ}(𝒰)→L^{2}_{ℍ}(𝒰)} )} ∥Ĝ_{ri} − Ĝ_{rj}Q̂∥_{∞},

where the first equality follows from equations (4.13) and (4.14), the second by the fact that H_{Ĝ_{rj} Q̂} = Π𝒢̂_{P_{j}}^{⊥} M_{Ĝ_{rj} Q̂} = 0 for all Q̂ and finally, the third inequality by the fact that ∥Π𝒢̂_{P_{j}}^{⊥} ∥ ≤ 1. Clearly, equality holds when Q̂ = Q̂_{⋆}. In conclusion,

∥Π𝒢̂_{P_{j}}^{⊥} Π𝒢̂_{P_{i}} ∥ = ∥Π𝒢̂_{P_{j}}^{⊥} MĜ_{ri}∥ = ∥HĜ_{ri}∥ = inf_{Q̂∈H𝔻^{∞}(B_{L^{2}_{ℍ}(𝒰)→L^{2}_{ℍ}(𝒰)} )} ∥Ĝ_{ri} − Ĝ_{rj}Q̂∥_{∞},

which completes the proof.

Interestingly, the difference between two stable closed-loops with the same controller but different plants can be bounded (above and below) in terms of the gap between the two plants. As will be seen later, these bounds lead to a particularly simple proof that the gap metric induces what is commonly known as the graph topology (the weakest with respect to which closed-loop performance varies continuously and closed-loop stability is a robust property). Since the parallel projection onto the graph of a plant along the inverse graph of a stabilising controller completely characterises a closed-loop (cf. Remark 2.14), in the following lemma the difference between the closed-loops concerned is quantified by the induced norm of the difference between the respective parallel-projection operators. A similar result has appeared in [Vin93b] for LTI systems. As an aside, note that for the closed-loop configuration specified in the H^{∞}-loopshaping, design procedure proposed in Sub-Section 3.3.2, the pertinent closed-loop operator is simply a weighted, parallel projection (compare equations in Remark 2.14 and equation (3.16)). As such, the following result also characterises explicitly the degradation in performance (in the context of the proposed design procedure) that may be observed if the plant deviates from its nominal value.

Lemma 4.16 For any P_{0}, P_{1} ∈ 𝒫_{𝒰→𝒴}^{e} and C ∈ 𝒫_{𝒴→𝒰}^{e} such that [P_{0}, C] and [P_{1}, C] are stable,

δ_{g}(P_{0}, P_{1}) ≤ ∥Π_{𝒢_{P_{0}}∥𝒢′_{C}} − Π_{𝒢_{P_{1}}∥𝒢′_{C}} ∥ ≤ \frac{δ_{g}(P_{0}, P_{1})}{b_{P_{0},C} b_{P_{1},C}} ,

where b_{P_{i},C} :=∥Π_{𝒢_{P_{i}}∥𝒢′_{C}} ∥^{−1}, i = 0, 1.

Proof : Given any stable plant/controller pair [P_{i}, C] with P_{i} ∈ 𝒫_{𝒰→𝒴}^{e} (i = 0, 1) and C ∈ 𝒫_{𝒴→𝒰}^{e} , recall that 𝒢_{P_{i}} and 𝒢C^{′} induce a co-ordinatisation of L_{ℝ^{+}}^{2}(𝒱) ^{5} (cf. Proposition 2.12 and Remark 2.13) and that the parallel projection Π_{𝒢_{P_{i}}∥𝒢′_{C}} is a bounded, LPTV operator. Now (for i, j = 0, 1) define

Δ := Π_{𝒢_{P_{i}}∥𝒢′_{C}} − Π_{𝒢_{P_{j}} ∥𝒢′_{C}}

and note that

(Π_{𝒢_{P_{i}}∥𝒢′_{C}} − Π_{𝒢_{P_{j}} ∥𝒢′_{C}} ) = Π_{𝒢′_{C} ∥𝒢_{P_{i}}} (Π_{𝒢_{P_{i}}∥𝒢′_{C}} − Π_{𝒢_{P_{j}} ∥𝒢′_{C}} ) = −Π_{𝒢′_{C} ∥𝒢_{P_{i}}} Π_{𝒢_{P_{j}} ∥𝒢′_{C}} .  (4.15)

To see this, take any e ∈ L_{ℝ^{+}}^{2}(𝒱), which since 𝒢_{P_{i}} (i = 0, 1) and 𝒢C^{′} induce a co-ordinatisation of L_{ℝ^{+}}^{2}(𝒱) can be uniquely decomposed as

e = p_{i} + c_{i} = p_{j} + c_{j},  (4.16)

where p_{i} ∈ 𝒢_{P_{i}} ⊂ L_{ℝ^{+}}^{2}(𝒱), p_{j} ∈ 𝒢_{P_{j}} ⊂ L_{ℝ^{+}}^{2}(𝒱) and c_{i}, c_{j} ∈ 𝒢C^{′} ⊂ L_{ℝ^{+}}^{2}(𝒱). Then from equation

(4.16), it follows that

(Π_{𝒢_{P_{i}}∥𝒢′_{C}} − Π_{𝒢_{P_{j}} ∥𝒢′_{C}} )e = (p_{i} − p_{j}) = (c_{j} − c_{i}) ∈ 𝒢C^{′},

by which equation (4.15) is immediate.

Using equation (4.15) and the identity

Π_{𝒢_{P_{j}} ∥𝒢′_{C}} = (Π_{𝒢_{P_{i}}} + Π_{𝒢_{P_{i}}^{⊥}} )Π_{𝒢_{P_{j}}} Π_{𝒢_{P_{j}} ∥𝒢′_{C}} ,

it follows that

Δ = −Π_{𝒢′_{C} ∥𝒢_{P_{i}}} Π_{𝒢_{P_{j}} ∥𝒢′_{C}}
= −Π_{𝒢′_{C} ∥𝒢_{P_{i}}} (Π_{𝒢_{P_{i}}} + Π_{𝒢_{P_{i}}^{⊥}} )Π_{𝒢_{P_{j}}} Π_{𝒢_{P_{j}} ∥𝒢′_{C}}
= −Π_{𝒢′_{C} ∥𝒢_{P_{i}}} Π_{𝒢_{P_{i}}^{⊥}} Π_{𝒢_{P_{j}}} Π_{𝒢_{P_{j}} ∥𝒢′_{C}} ,

where the last equality holds since Π_{𝒢′_{C} ∥𝒢_{P_{i}}} Π_{𝒢_{P_{i}}} = 0. Hence,

∥Δ∥ ≤ ∥Π_{𝒢′_{C} ∥𝒢_{P_{i}}} ∥ · ∥Π_{𝒢_{P_{i}}^{⊥}} Π_{𝒢_{P_{j}}} ∥ · ∥Π_{𝒢_{P_{j}} ∥𝒢′_{C}} ∥,

which by definition of b_{P_{i},C}, the directed gap and its relationship to the gap, implies

∥Δ∥ ≤ \frac{δ_{g}(P_{0}, P_{1})}{b_{P_{0},C}b_{P_{1},C}} .

This is precisely the upper bound stated in the lemma.

5Also recall that 𝒱:=𝒰 ⊕ 𝒴.

To establish the lower bound, consider

Π_{𝒢_{P_{j}}^{⊥}} ΔΠ_{𝒢_{P_{i}}} = Π_{𝒢_{P_{j}}^{⊥}} (Π_{𝒢_{P_{i}}∥𝒢′_{C}} − Π_{𝒢_{P_{j}} ∥𝒢′_{C}} )Π_{𝒢_{P_{i}}}
= Π_{𝒢_{P_{j}}^{⊥}} Π_{𝒢_{P_{i}}} ,

where the second equality follows from the identities Π_{𝒢_{P_{j}}^{⊥}} Π_{𝒢_{P_{j}} ∥𝒢′_{C}} = 0 and Π_{𝒢_{P_{i}}∥𝒢′_{C}} Π_{𝒢_{P_{i}}} = Π_{𝒢_{P_{i}}} . Hence, for i, j = 0, 1

∥Π_{𝒢_{P_{j}}^{⊥}} Π_{𝒢_{P_{i}}} ∥ = ∥Π_{𝒢_{P_{j}}^{⊥}} ΔΠ_{𝒢_{P_{i}}} ∥ ≤ ∥Δ∥,

from which it follows that

δ_{g}(P_{0}, P_{1}) ≤ ∥Δ∥,

as claimed.

4.4 Quantitative Robustness Analysis

A stable, closed-loop system [P, C] is said to be robustly stable if closed-loop stability is maintained when P and C undergo reasonable perturbations. An important property of closed-loop systems is that stability can be desensitised to such perturbations and in the sub-sections to follow, this property is quantitatively analysed in terms of the gap metric for LPTV, closed-loop systems.

4.4.1 Plant Perturbations

In this sub-section, the robustness of closed-loop stability is studied within the context of plant perturbations quantified using the gap metric. Before going on, it is instructive to recall the following definition (cf. Lemma 4.16):

Definition 4.17 Given a stable closed-loop [P, C] with P ∈ 𝒫_{𝒰→𝒴}^{e} and C ∈ 𝒫_{𝒴→𝒰}^{e} , define

b_{P,C}^{−1} :=∥Π_{𝒢_{P} ∥𝒢′_{C}} ∥ = ∥Π_{𝒢_{P}∥𝒢′_{C}} ∥ = ∥G_{r}(K_{l}G_{r})^{−1}K_{l}∥,  (4.17)

where P:=WP W^{−1}, C:=WCW^{−1} and G_{r} ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} and K_{l} ∈ 𝒟_{L_{ℍ}^{2}(𝒴)→L_{ℍ}^{2}(𝒰)} are any stable systems that satisfy 𝒦_{G_{r}} = {0}, 𝒢_{P} = ℛ_{G_{r}}, ℛ_{K_{l}} = ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰)) and 𝒢C^{′} = 𝒦_{K_{l}} (cf. Remark 4.10). For reasons that will soon become apparent, b_{P,C} is called the \underline{robust-stability margin}. ♠ Using results obtained in the previous sections, it is now possible to state and prove the main robust stability result, which identifies the largest gap-ball of LPTV plants, centred at a nominal plant, that a nominal, stabilising, LPTV controller is guaranteed to stabilise. To a certain extent, the result is analogous to the LTI result of [GS90] and the general LTV result of [FGS93]. Note however, that it does not follow directly from either of these. In particular, considerable effort is required to prove necessity, where the approach taken is to construct a causal plant P_{1} of appropriate gap distance from the nominal plant so that [P_{1}, C] is well-posed but not stable.

Theorem 4.18 Let [P_{0}, C] be a stable, closed-loop system, where P_{0} ∈ 𝒫_{𝒰→𝒴}^{e} and C ∈ 𝒫_{𝒴→𝒰}^{e,sc} . Then, the following are equivalent :

(i) b_{P_{0},C} ≥ β;
(ii) [P_{1}, C] is stable for all P_{1} ∈ 𝒫_{𝒰→𝒴}^{e} that satisfy δ_{g}(P_{0}, P_{1}) < β.

Proof : ((i) ⇒ (ii)) By assumption, δ_{g}(P_{0}, P_{1}) = δ(𝒢_{P_{0}}, 𝒢_{P_{1}}) < β and b_{P_{0},C}^{−1} ≤ β^{−1}. For i = 0, 1, let G_{ri} ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒱)} be an isometry such that 𝒢_{P_{i}} = ℛ_{G_{ri}}, where P_{i} :=WP_{i}W^{−1}.^{6} Then Π_{𝒢_{P_{i}}} = G_{ri}(G_{ri})^{∗} and

δ(𝒢_{P_{0}}, 𝒢_{P_{1}}) = ∥Π_{𝒢_{P_{0}}} − Π_{𝒢_{P_{1}}} ∥ = ∥G_{r0}G_{r0}^{∗} − G_{r1}G_{r1}^{∗}∥ < β.  (4.18)

Moreover, since G_{r0} is an isometry,

b_{P_{0},C}^{−1} = ∥(K_{l}G_{r0})^{−1}K_{l}∥ ≤ β^{−1},  (4.19)

where K_{l} ∈ 𝒟_{L_{ℍ}^{2}(𝒱) →L_{ℍ}^{2}(𝒰)} is any stable system that satisfies 𝒦_{K_{l}} = 𝒢C^{′} and ℛ_{K_{l}} = ℓ_{ℤ^{+}}^{2}(𝒰).^{7} Combining equations (4.18) and (4.19) yields

∥(K_{l}G_{r0})^{−1}K_{l}(G_{r0}G_{r0}^{∗} − G_{r1}G_{r1}^{∗})∥ < 1,

which implies

∥I − (K_{l}G_{r0})^{−1}K_{l}G_{r1}G_{r1}^{∗}G_{r0}∥ ≤ ∥G_{r0}^{∗} − (K_{l}G_{r0})^{−1}K_{l}G_{r1}G_{r1}^{∗}∥∥G_{r0}∥ < 1.

Hence, it follows that (K_{l}G_{r0})^{−1}K_{l}G_{r1}(G_{r1}^{∗}G_{r0}) has bounded inverse. Since δ_{g}(P_{0}, P_{1}) < 1, note that the operator G_{r1}^{∗}G_{r0} also has bounded inverse (cf. Corollary 4.14) and therefore, that K_{l}G_{r1} is boundedly-invertible, which by Lemma 4.8, implies stability of the closed-loop [P_{1}, C].

6𝒢_{P_{0}} and 𝒢_{P_{1}} are LSI subspaces and hence, the required G_{r0} and G_{r1} exist by the Beurling-Lax-Halmos Theorem.

7Since [P_{0}, C] is stable, such a K_{l} exists by Theorem 4.6.

((i) ⇐ (ii)) Suppose that (ii) holds for some 1 ≥ β > b_{P_{0},C}.^{8} Under this hypothesis it is sufficient to construct a system P_{1} ∈ 𝒫_{𝒰→𝒴}^{e} such that δ_{g}(P_{0}, P_{1}) < β and [P_{1}, C] is well-posed, but not stable. Let G_{r0} and K_{l} be defined as in the proof of (i) ⇒ (ii). Then by Proposition 2.2, G_{r0} and K_{l} are equivalent (via Z) to multiplication operators with symbols Ĝ_{r0} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒱)} ) and K̂_{l} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒱)→L_{ℍ}^{2}(𝒰)} ) respectively. Furthermore, it follows by Lemma 4.8 and Remark 4.10, that (K̂_{l}Ĝ_{r0})^{−1} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} ). Hence,

K̂_{l⋆}:=(Ĝ_{r0}K̂_{l})^{−1}K̂_{l} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒱)→L_{ℍ}^{2}(𝒱)} )

and K_{l⋆} :=Z^{−1}MK̂_{l⋆}Z is a stable system in 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒱)} that satisfies 𝒦_{K_{l⋆}} = 𝒢C^{′} and ℛ_{K_{l⋆}} = ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰)). In fact, by definition,

b_{P_{0},C}^{−1} :=∥G_{r0}K_{l⋆}∥ = ∥K_{l⋆}∥ = ∥K̂_{l⋆}∥_{∞} = sup_{λ∈𝔻}∥K̂_{l⋆}(λ)∥,  (4.20)

where the second equality holds because G_{r0} is an isometry. Since the function K̂_{l⋆}(λ) is analytic in 𝔻, ∥K̂_{l⋆}(λ)∥ satisfies a maximum principle on any connected, open subset of 𝔻 (cf. Sub-Section 2.2.3). Using this property and equation (4.20), it follows that for any ϵ_{0} > 0 and ϵ_{1} > 0, there exists a λ_{0,ϵ_{0}} ∈ 𝔸_{ϵ_{0}} :={λ : (1 − ϵ_{0}) ≤ |λ| < 1} such that

(b_{P_{0},C} + ϵ_{1})^{−1} ≤ ∥K̂_{l⋆}(λ_{0,ϵ_{0}})∥ ≤ b_{P_{0},C}^{−1}.

Now for any ϵ_{2} > ϵ_{1} it is possible to construct an operator Δ_{0,ϵ_{2}} ∈ B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒱)} with

∥Δ_{0,ϵ_{2}}∥ ≤ (b_{P_{0},C} + ϵ_{2}),  (4.21)

such that (I + K̂_{l⋆}(λ_{0,ϵ_{0}})Δ_{0,ϵ_{2}}) is not invertible in B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} .^{9} Defining

Δ̂_{ϵ_{2}}(λ) :=\frac{λ}{λ_{0,ϵ_{0}}} Δ_{0,ϵ_{2}} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒱)} ),  (4.22)

∥Δ∥ ≤ \frac{1}{∥A∥ − ϵ} .

8δ_{g}(P_{0}, P_{1}) ≤ 1 for all P_{1} ∈ 𝒫_{𝒰→𝒴}^{e} . So it is without loss of generality that β is taken to be less than or equal to 1, otherwise there is an obvious contradiction.

9To see this, let A denote a bounded, linear operator in B_{𝒰→𝒴} . Then for any ϵ > 0, it follows by definition of the induced norm that there exists a u ∈ 𝒰 (which is taken to have unit norm) so that with y :=Au,

∥A∥ ≥ ∥y∥_{𝒴} ≥ ∥A∥ − ϵ.

Defining Δ : 𝒴 → 𝒰 to be the operator that maps αy to −αu for all α ∈ ℂ and every other direction orthogonal to this in 𝒴 to 0, it follows that Clearly then y ∈ 𝒦_{(I+AΔ)} and hence, (I + AΔ) is not invertible.

it follows that

∥(I + K̂_{l⋆}(λ)Δ̂_{ϵ_{2}}(λ))^{−1}∥ → ∞

as λ → λ_{0,ϵ_{0}}, since (I + K̂_{l⋆}Δ̂) is continuous on 𝔻 and (I + K̂_{l⋆}(λ_{0,ϵ_{0}})Δ_{0,ϵ_{2}}) is not invertible.^{10} That is, (I + K̂_{l⋆}Δ̂_{ϵ_{2}}) in not invertible in H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} ).

Let

Ĝ_{r1} :=(Ĝ_{r0} + Δ̂_{ϵ_{2}})Q̂_{1},

for some Q̂_{1}, Q̂_{1}^{−1} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} ) and note that

K̂_{l⋆}Ĝ_{r1} = (I + K̂_{l⋆}Δ̂_{ϵ_{2}})Q_{1}.

Clearly, K̂_{l⋆}Ĝ_{r1} is not invertible in H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} ). Consequently, with G_{r1} :=Z^{−1}MĜ_{r1}Z, the stable system K_{l⋆}G_{r1} ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒰)} is not boundedly invertible. So provided

(a) 𝒦_{G_{r1}} = {0};

(b) ℛ_{G_{r1}} is isomorphic (via W) to the graph of a system P_{1} ∈ 𝒫_{𝒰→𝒴}^{e} ;

(c) and δ_{g}(P_{0}, P_{1}) < β,

a contradiction to the initial hypothesis that (ii) holds for some 1 ≥ β > b_{P_{0},C} can be established using Lemma 4.8 and Remark 4.10. Accordingly, the remainder of the proof is devoted to showing that by appropriate choice of ϵ_{0}, ϵ_{2} and Q̂_{1}, conditions (a), (b) and (c) can be satisfied.

First note that G_{r1} = (G_{r0} + Δ_{ϵ_{2}})Q_{1}, where

G_{r0} :=Z^{−1}MĜ_{r0}Z, Δ_{ϵ_{2}} :=Z^{−1}MΔ̂_{ϵ_{2}} Z, and Q_{1} :=Z^{−1}MQ̂_{1}Z.

Furthermore, note that Δ̂_{ϵ_{2}}(λ) has continuous extension to the closed, unit disc and hence, that ∥Δ̂_{ϵ_{2}}∥_{∞} = max_{ω∈[0,2π)}∥Δ̂_{ϵ_{2}}(e^{jω})∥. It then follows by Proposition 2.2 and from equations (4.21) and (4.22), that

∥Δ_{ϵ_{2}}∥ = ∥Δ̂_{ϵ_{2}}∥_{∞} = \frac{1}{|λ_{0,ϵ_{0}}|}∥Δ_{0,ϵ_{2}}∥ ≤ \frac{b_{P_{0},C} + ϵ_{2}}{1 − ϵ_{0}} .

By hypothesis (β − b_{P_{0},C}) > 0 and setting ϵ_{2} = (β − b_{P_{0},C})/2 > 0, ϵ_{1} < ϵ_{2} and ϵ_{0} < ϵ_{2}/2β, gives

∥Δ_{ϵ_{2}}∥ < β ≤ 1.  (4.23)

It is now immediate that G_{r1} has zero kernel, since G_{r0} is an isometry. That is, condition (a) is satisfied.

10See [Dul94, Lemma 4.3 (only if part)] for a similar result.

Q̂_{1} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} ) is now constructed so that ℛ_{G_{r1}} is isomorphic (via W) to the graph of a system P_{1} ∈ 𝒫_{𝒰→𝒴}^{e} . First let

[ \begin{matrix}M_{r1} \\ N_{r1} \end{matrix} ]:=G_{r1} and [ \begin{matrix}M_{r0} \\ N_{r0} \end{matrix} ]:=G_{r0},

where the partitioning is conformal with 𝒢_{P_{0}}. By construction Δ_{[0]} = 0, where the subscript [0] denotes the first term in the sequence uniquely identifiable with the Toeplitz representation. Thus, G_{r1[0]} = (G_{r0}Q_{1})_{[0]} and

[ \begin{matrix}M_{r1[0]} \\ N_{r1[0]} \end{matrix} ] = ([ \begin{matrix}M_{r0} \\ N_{r0} \end{matrix} ]Q_{1})_{[0]} = [ \begin{matrix}M_{r0[0]} \\ N_{r0[0]} \end{matrix} ]Q_{1[0]}.  (4.24)

Since ℛ_{G_{r0}} is isomorphic to 𝒢_{P_{0}} and P_{0} ∈ 𝒫_{𝒰→𝒴}^{e} , it follows (as is detailed in the proof of Theorem 4.6) that M_{r0[0]} ∈ B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} is boundedly invertible. So by the assumed invertibility of Q_{1} and equation (4.24), M_{r1[0]} is also invertible in B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} . Consequently, M_{r1} has zero kernel, which confirms that ℛ_{G_{r1}} is the graph of a linear operator. In fact, it is isomorphic to the graph of an LPTV system P_{1} defined on a subspace of L_{ℝ^{+}}^{2}(𝒰). It is sufficient therefore, to select Q̂_{1} to ensure that P_{1} is causal and has locally-Lipschitz-continuous extension. To this end, set Q̂_{1} = M_{r0[0]}^{−1}, which is clearly an invertible element in H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} ), so that Q_{1[0]} = M_{r0[0]}^{−1} and

G_{r1[0]} = [ \begin{matrix}M_{r1[0]} \\ N_{r1[0]} \end{matrix} ] = (G_{r0}Q_{1})_{[0]} = [ \begin{matrix}I \\ N_{r0[0]}M_{r0[0]}^{−1} \end{matrix} ].

Since P_{0[0]} = N_{r0[0]}M_{r0[0]}^{−1} is causal on L_{ℍ}^{2}(𝒰), this is also a causal map on L_{ℍ}^{2}(𝒰). It then follows by

Remark 2.6, that [ \begin{matrix}M_{r1} \\ N_{r1} \end{matrix} ]:=G_{r1} = W^{−1}G_{r1}W is a stable system in 𝒫_{𝒰→𝒱}^{e} , which in turn implies that P_{1} is causal. To see this, suppose that P_{1} is not causal. Then by definition, there exists a point [ \begin{matrix}u \\ y \end{matrix} ]∈ 𝒢_{P_{1}} = ℛ_{G_{r1}} and a τ_{1} ∈ ℝ^{+} such that

P_{τ_{1}}[ \begin{matrix}u \\ y \end{matrix} ] = [ \begin{matrix}0 \\ ỹ ≠ 0 \end{matrix} ].

That is, there exists a q ∈ L_{ℝ^{+}}^{2}(𝒰) such that ỹ = P_{τ_{1}}N_{r1}q and 0 = P_{τ_{1}}M_{r1}q. Since M_{r1} is causal,

P_{τ_{1}}M_{r1}q = P_{τ_{1}}W^{−1}P_{n}M_{r1}Wq

= P_{τ_{1}}W^{−1} [ \underbrace{[ \begin{matrix}I & 0 & ⋯ & 0 \\ M_{r1[1]} & ... & ... & ⋮ \\ ⋮ & ... & ... & 0 \\ M_{r1[n]} & ⋯ & M_{r1[1]} & I \end{matrix} ]}_{:=M̃_{n}} [ q^{⊸}_{0} ; q^{⊸}_{1} ; ⋮ ; ⋮ ; q^{⊸}_{n} ] ; 0 ; ⋮ ] ,  (4.25)

where q(=: k ↦ q^{⊸}_{k}) :=Wq, n = ]\frac{τ_{1}}{h} [ + 1 and ] · [ denotes the integer part of a real number. Now note that M̃_{i} is boundedly invertible for all finite i ∈ ℤ^{+} and therefore, has zero kernel and full range.^{11} So if 0 = P_{τ_{1}}M_{r1}q, it follows from equation (4.25), that q^{⊸}_{0} = q^{⊸}_{1} = … = q^{⊸}_{n−1} = 0 and P_{(nh+1−τ_{1})} q^{⊸}_{n} = 0 or equivalently, P_{τ_{1}}q = 0. Then since N_{r1} is causal, this implies that ỹ = P_{τ_{1}}N_{r1}q = 0, which is a contradiction and consequently, P_{1} is causal. This line of reasoning also leads to the conclusion that P_{1} is causally extendible to a locally-Lipschitz-continuous operator on L_{ℝ^{+}}^{2,e}(𝒰). More specifically, since M̃_{i} has full range for all finite i ∈ ℤ^{+},

P_{τ} 𝒟_{P_{1}} = P_{τ} ℛ_{M_{r1}} = P_{τ} W^{−1} [ \begin{matrix}ℛM̃_{n} \\ 0 \end{matrix} ] = P_{τ} L_{[0,nh]}^{2}(𝒰) = L_{[0,τ]}^{2}(𝒰)

for all τ, where n = ]\frac{τ}{h}[ + 1. That is, P_{1} is causally extendible (by definition). Now denote the extension of P_{1} by P_{1e}. Then,

sup_{\begin{matrix}u_{1},u_{2}∈L_{ℝ^{+}}^{2,e}(𝒰) \\ P_{τ} u_{1}≠P_{τ} u_{2} \end{matrix}} \frac{∥P_{τ} (P_{1e}u_{1} − P_{1e}u_{2})∥_{L_{ℝ^{+}}^{2}(𝒴)}}{∥P_{τ} (u_{1} − u_{2})∥_{L_{ℝ^{+}}^{2}(𝒰)}} ≤ sup_{\begin{matrix}u∈P_{n}ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰) ) \\ u≠0 \end{matrix}} \frac{∥Ñ_{n} M̃_{n}^{−1}u∥_{P_{n}}_{ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒴))}}{∥u∥_{P_{n}}_{ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰) )}} ,

where

Ñ_{n} := [ \begin{matrix}N_{r1[0]} & 0 & ⋯ & 0 \\ N_{r1[1]} & ... & ... & ⋮ \\ ⋮ & ... & ... & 0 \\ N_{r1[n]} & ⋯ & N_{r1[1]} & N_{r1[0]} \end{matrix} ] .

Since Ñ_{i} and M̃_{i}^{−1} are both bounded for all finite i ∈ ℤ^{+}, it is then clear (by definition) that P_{1e} is locally Lipschitz-continuous as claimed.

11In fact M̃_{i}^{−1} is block lower-triangular with identities down the diagonal.

It remains to show that δ_{g}(P_{0}, P_{1}) < β. First it is shown that δ⃗_{g}(P_{0}, P_{1}) < β and then, that δ_{g}(P_{0}, P_{1}) = δ⃗_{g}(P_{0}, P_{1}) < β. Note that 𝒢̂_{P_{1}} :=ZW𝒢_{P_{1}} is an LSI subspace of H_{𝔻}^{2}(L_{ℍ}^{2}(𝒱)). Therefore, by the Beurling-Lax-Halmos Theorem, there exists an inner function Ĝ_{r⋆} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒱)} ) such that 𝒢̂_{P_{1}} = Ĝ_{r1⋆}H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰)). Moreover, since ℛ_{MĜ_{r1}} = ℛ_{MĜ_{r1⋆}} and 𝒦_{MĜ_{r1}} = {0}, there exists a Q̂_{⋆}, Q̂_{⋆}^{−1} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} ) such that Ĝ_{r1⋆}Q̂_{⋆} = Ĝ_{r1} = (Ĝ_{r0} + Δ̂)Q̂_{1} (cf. Corollary IX.2.2 of the Beurling-Lax-Halmos Theorem in [FF90]). Then by Theorem 4.15,

δ⃗_{g}(P_{0}, P_{1}) = δ⃗(𝒢̂_{P_{0}}, 𝒢̂_{P_{1}}) = inf_{Q̂∈H𝔻^{∞}(B_{L^{2}_{ℍ}(𝒰)→L^{2}_{ℍ}(𝒰)} )} ∥Ĝ_{r0} − Ĝ_{r1⋆}Q̂∥_{∞}
= inf_{Q̂∈H𝔻^{∞}(B_{L^{2}_{ℍ}(𝒰)→L^{2}_{ℍ}(𝒰)} )} ∥Ĝ_{r0} − (Ĝ_{r0} + Δ̂_{ϵ_{2}})Q̂_{1} Q̂_{⋆}^{−1} Q̂∥_{∞}
≤ ∥Δ̂_{ϵ_{2}}∥_{∞} < β,

where the third inequality follows from the fact that Q̂_{⋆} Q̂_{1}^{−1} ∈ H_{𝔻}^{∞}(B_{L_{ℍ}^{2}(𝒰)→L_{ℍ}^{2}(𝒰)} ) and the last inequality from equation (4.23). Now for any q̂ ∈ H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰)) consider the projection of MĜ_{r1} q̂ ∈ 𝒢̂_{P_{1}} onto 𝒢̂_{P_{0}}. Since MĜ_{r0} is an isometry, Π𝒢̂_{P_{0}} = MĜ_{r0}(MĜ_{r0})^{∗} and the projection described above can be expressed as

MĜ_{r0}(MĜ_{r0})^{∗}(MĜ_{r0} + MΔ̂_{ϵ_{2}} )MQ̂_{1} q̂ = MĜ_{r0} ĝ,

where

ĝ:=(I + (MĜ_{r0})^{∗}MΔ̂_{ϵ_{2}} )MQ̂_{1} q̂.

Furthermore,

∥(MĜ_{r0})^{∗}MΔ̂_{ϵ_{2}} ∥ ≤ ∥MĜ_{r0}∥ · ∥MΔ̂_{ϵ_{2}} ∥ < β ≤ 1.

Thus, (I + (MĜ_{r0})^{∗}MΔ̂_{ϵ}) is a one-to-one mapping onto H_{𝔻}^{2}(L_{ℍ}^{2}(𝒰)) and since MQ̂_{1} is bijective, it follows that the projection described above is also bijective. So by Proposition 4.13,

δ⃗(𝒢̂_{P_{0}}, 𝒢̂_{P_{1}}) = δ⃗(𝒢̂_{P_{1}}, 𝒢̂_{P_{0}}) = δ(𝒢̂_{P_{0}}, 𝒢̂_{P_{1}})

and hence, δ_{g}(P_{0}, P_{1}) = δ(𝒢̂_{P_{0}}, 𝒢̂_{P_{1}}) = δ⃗(𝒢̂_{P_{0}}, 𝒢̂_{P_{1}}) < β as required. This completes the proof.

Consider a closed-loop [P_{0}, C], with P_{0} ∈ 𝒫_{𝒰→𝒴}^{e,sc} and C ∈ 𝒫_{𝒴→𝒰}^{e} . Then by Proposition 2.9, [P_{0}, C] is well-posed and if all permissible perturbations are restricted to 𝒫_{𝒰→𝒴}^{e,sc} , the closed-loop is always well-posed. Furthermore, by the way the destabilising plant P_{1} is constructed in the proof above, P_{0[0]} = P_{1[0]}. That is, the instantaneous behaviour of the nominal plant and the plant constructed to destabilise the closed-loop is the same. As such, if P_{0} ∈ 𝒫_{𝒰→𝒴}^{e,sc} , then P_{1} ∈ 𝒫_{𝒰→𝒴}^{e,sc} . Consequently, (ii) ⇒ (i) would still hold if permissible perturbations were restricted to only strongly causal ones.

Corollary 4.19 Given a stable closed-loop [P_{0}, C], where P_{0} ∈ 𝒫_{𝒰→𝒴}^{e,sc} and C ∈ 𝒫_{𝒴→𝒰}^{e} , the following are equivalent:

(i) b_{P_{0},C} ≥ β;
(ii) [P_{1}, C] is stable for all P_{1} ∈ 𝒫_{𝒰→𝒴}^{e,sc} such that δ_{g}(P_{0}, P_{1}) < β.

Similarly, although it is very difficult to characterise analytically, restricting the set of permissible perturbations to those which do not cause the nominal closed-loop to become ill-posed,^{12} leads to the following restatement of Theorem 4.18.

Corollary 4.20 Given a stable closed-loop [P_{0}, C], where P_{0} ∈ 𝒫_{𝒰→𝒴}^{e} and C ∈ 𝒫_{𝒴→𝒰}^{e} , the following are equivalent:

(i) b_{P_{0},C} ≥ β;

(ii) [P_{1}, C] is stable for all P_{1} ∈ 𝒫_{𝒰→𝒴}^{e} such that δ_{g}(P_{0}, P_{1}) < β and [P_{1}, C] is well-posed.

4.4.2 Simultaneous Plant and Controller Perturbations

In this sub-section, the maximum combined perturbation to both P and C that does not cause instability is quantified in terms of the gap metric. As a corollary of Theorem 4.18, it is shown that a given C stabilises a gap-ball centred at P if and only if P is stabilised by all controllers in a gap-ball of the same radius and centred at C. Then, the maximum combined perturbation (as a sum) to P and C that the closed-loop can tolerate is considered. The results presented are analogous to the LTI results of [GS90, §VI] and the LTV results of [FGS93, §4]. First a technical result.

Lemma 4.21 [DGS93] Let {ℱ, 𝒢} be a co-ordinatisation of 𝒱, where ℱ, 𝒢 and 𝒱 are (closed) Hilbert spaces. If ℱ is a linear subspace then

∥Π_{ℱ∥𝒢} ∥ = ∥Π_{𝒢∥ℱ} ∥.

Recall the definition of b_{P,C} (cf. Definition 4.17). Then since 𝒢C^{′} is a linear subspace, the following corollary of Lemma 4.21 is immediate.

Corollary 4.22

b_{P,C} = b_{C,P} .

12This is necessary for (i) ⇒ (ii) to hold.

Proof : Note that

b_{C,P} := ∥Π_{𝒢_{C}∥𝒢′_{P}} ∥ = ∥[ \begin{matrix}0 & I \\ I & 0 \end{matrix} ]Π_{𝒢′_{C} ∥𝒢_{P}} [ \begin{matrix}0 & I \\ I & 0 \end{matrix} ]∥ = ∥Π_{𝒢′_{C} ∥𝒢_{P}} ∥.

Since [P, C] is stable, {𝒢_{P} , 𝒢C^{′}} induces a co-ordinatisation of L_{ℝ^{+}}^{2}(𝒱) (cf. Proposition 2.12) and so by Lemma 4.21, b_{C,P} = ∥Π_{𝒢′_{C} ∥𝒢_{P}} ∥ = ∥Π_{𝒢_{P} ∥𝒢′_{C}} ∥ = b_{P,C}.

Similarly, the following corollary, which characterises robust stability to perturbations of the controller, is an immediate consequence of Corollary 4.22 and Theorem 4.18.

Corollary 4.23 Let [P, C] be a stable closed-loop system, where P ∈ 𝒫_{𝒰→𝒴}^{e,sc} and C ∈ 𝒫_{𝒴→𝒰}^{e} . Then the following are equivalent :

(i) [P_{1}, C] is stable for all P_{1} ∈ 𝒫_{𝒰→𝒴}^{e,sc} such that δ_{g}(P, P_{1}) < β;
(ii) [P, C_{1}] is stable for all C_{1} ∈ 𝒫_{𝒴→𝒰}^{e} such that δ_{g}(C, C_{1}) < β.

Remark 4.24 As with Corollaries 4.19 and 4.20, Corollary 4.23 can be restated with various different constraints on the strength of causality of the plants and the controllers. ♣

Now, with this established, it is possible to quantitatively characterise robustness to perturbations of both P and its stabilising controller C.

Theorem 4.25 Consider a stable closed-loop system [P, C], where P ∈ 𝒫_{𝒰→𝒴}^{e,sc} and C ∈ 𝒫_{𝒴→𝒰}^{e} . The following are equivalent:

(i) β ≤ b_{P,C};
(ii) [P_{1}, C_{1}] is stable for all P_{1} ∈ 𝒫_{𝒰→𝒴}^{e,sc} and C_{1} ∈ 𝒫_{𝒴→𝒰}^{e} such that
δ_{g}(P, P_{1}) + δ_{g}(C, C_{1}) < β.

Proof : The proof follows directly from Corollary 4.19 of Theorem 4.18 and Corollary 4.23, in exactly the same way as the corresponding result in [GS90, Theorem 7]. Assume that (i) holds so that by Corollaries 4.19 and 4.23, [P, C_{1}] is stable for all C_{1} ∈ ℬ_{δ_{g}}^{e}(C, β), where

ℬ_{δ_{g}}^{e}(F, ϵ) :={F̀ ∈ 𝒫_{𝒴→𝒰}^{e} : δ_{g}(F, F̀) < ϵ}.

Consider a controller C_{1} ∈ 𝒫_{𝒴→𝒰}^{e} , such that τ :=δ_{g}(C, C_{1}) < β. Now by the metric property of the gap, ℬ_{δ_{g}}^{e}(C_{1}, β − τ) ⊂ ℬ_{δ_{g}}^{e}(C, β) and hence, [P, C_{2}] is stable for all C_{2} ∈ ℬ_{δ_{g}}^{e}(C_{1}, β − τ). Using Corollary 4.23, it is then possible to conclude that [P_{1}, C_{1}] is stable for all P_{1} ∈ ℬ_{δ_{g}}^{e,sc}(P, β − τ), where

ℬ_{δ_{g}}^{e,sc}(F, ϵ) :={F̀ ∈ 𝒫_{𝒴→𝒰}^{e,sc} : δ_{g}(F, F̀) < β}.

Since this holds for any C_{1} ∈ 𝒫_{𝒰→𝒴}^{e} such that δ_{g}(C, C_{1}) < β, it is clear that (ii) holds. That (ii) ⇒ (i) is immediate, since (ii) here implies (i) in Corollary 4.19, which is β ≤ b_{P,C}.

4.5 The Graph (Gap) Topology

Combining the quantitative robustness results presented in Section 4.4 with the upper and lower bounds of Lemma 4.16, leads to simple proofs of the following results, which concern the topology induced by the gap metric on the set of stabilisable systems in 𝒫_{𝒰→𝒴}^{e,sc} . This topology will be referred to as the \underline{graph topology} because its properties are the same as those of the graph topology defined on the set of LTI systems (cf. [Vid85] and recall that the graph topology on LTI systems is induced by a number of metrics, of which the gap metric is one [Geo88, Vin93b]).

The following identity (cf. Remark 2.14) is useful in the proof of the next result, which characterises convergent sequences in the topology induced by the gap metric:

H(P, C) = [ \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ]Π_{𝒢_{P} ∥𝒢′_{C}} + [ \begin{matrix}0 & 0 \\ 0 & I \end{matrix} ].

Lemma 4.26 Given a stabilisable P ∈ 𝒫_{𝒰→𝒴}^{e,sc} and an infinite sequence {P_{i}} in 𝒫_{𝒰→𝒴}^{e,sc} , the following are equivalent:

(i) δ_{g}(P, P_{i}) → 0;

(ii) There exists a C ∈ 𝒫_{𝒰→𝒴}^{e} such that [P, C] is stable and H(P_{i}, C) → H(P, C) in the induced-norm topology.

(iii) For any C ∈ 𝒫_{𝒰→𝒴}^{e} such that [P, C] is stable, H(P_{i}, C) → H(P, C) in the induced-norm topology.

Proof : The proof is essentially identical to that of Corollary IV.6 in [Vin93b]. Recall that

ℬ_{δ_{g}}^{e,sc}(P, ϵ) :={P̀ ∈ 𝒫_{𝒰→𝒴}^{e,sc} : δ_{g}(P, P̀) < ϵ}.

Now given any C ∈ 𝒫_{𝒴→𝒰}^{e} such that [P, C] is stable, it follows by Corollary 4.19 of Theorem 4.18, that there is a corresponding open ball ℬ_{δ_{g}}^{e,sc}(P, b_{P,C}) such that [P̀, C] is stable for all P̀ ∈ ℬ_{δ_{g}}^{e,sc}(P, b_{P,C}). Now assume that (i) holds, so that all but a finite number of the terms in the sequence {P_{i}} lie in ℬ_{δ_{g}}^{e,sc}(P, b_{P,C}) and are therefore stabilised by C. By the upper bound in Lemma 4.16 it is then immediate that

Π_{𝒢_{P_{i}}∥𝒢′_{C}} → Π_{𝒢_{P} ∥𝒢′_{C}}

in the induced-norm topology. Consequently, since

∥H(P, C) − H(P_{i}, C)∥ = ∥Π_{𝒢_{P} ∥𝒢′_{C}} − Π_{𝒢_{P_{i}}∥𝒢′_{C}} ∥,

H(P_{i}, C) → H(P, C) in the induced-norm topology. That is, (i) ⇒ (iii).

Now suppose that (ii) holds. Then for all but a finite number of terms in the sequence {P_{i}}, the closed-loop [P_{i}, C] is stable and

∥Π_{𝒢_{P} ∥𝒢′_{C}} − Π_{𝒢_{P_{i}}∥𝒢′_{C}} ∥ → 0.

It is then immediate from the lower bound in Lemma 4.16 that δ_{g}(P, P_{i}) → 0, which is (i). That (iii) ⇒ (ii) is obvious.

The next corollary also appears in [Vin93b] and [Vid85] for LTI systems. Define

𝒮_{C} :={P ∈ 𝒫_{𝒰→𝒴}^{e,sc} : [P, C] is stable}

and

B_{𝒱→𝒱}^{e} :={H ∈ 𝒫_{𝒱→𝒱}^{e} : H is stable}.

Corollary 4.27 For any C ∈ 𝒫_{𝒴→𝒰}^{e} , the map P ↦ H(P, C) from 𝒫_{𝒰→𝒴}^{e,sc} to B_{𝒱→𝒱}^{e} is continuous at all points P ∈ 𝒮_{C} with respect to the graph topology (induced by the gap metric) on 𝒫_{𝒰→𝒴}^{e,sc} and the induced norm topology on B_{𝒱→𝒱}^{e} . Furthermore, the graph topology is the weakest topology on 𝒫_{𝒰→𝒴}^{e,sc} for which this holds.

Proof : The proof is identical to that of Corollary IV.9 in [Vin93b], but is included for completeness. To show that the map P ↦ H(P, C) is continuous at all points in 𝒮_{C}, it is necessary to show that for any P ∈ 𝒮_{C} and any ϵ > 0 there exists a δ > 0 such that

δ_{g}(P, P_{i}) < δ with P_{i} ∈ 𝒫_{𝒰→𝒴}^{e,sc} ⇒ H(P_{i}, C) ∈ B_{𝒱→𝒱}^{e} and ∥H(P, C) − H(P_{i}, C)∥ < ϵ.

Since ∥H(P, C) − H(P_{i}, C)∥ = ∥Π_{𝒢_{P} ∥𝒢′_{C}} − Π_{𝒢_{P_{i}}∥𝒢′_{C}} ∥, the existence of such a δ > 0 is guaranteed by the lower bound in Lemma 4.16.

To see that the graph topology is the weakest on 𝒮_{C} for which the result holds, note that for any other topology τ on 𝒮_{C} to have this property it is necessary for there to exist a neighbourhood 𝒩_{ϵ}(P)^{13} for any given P ∈ 𝒮_{C} and ϵ > 0, such that

H(P_{i}, C) ∈ B_{𝒱→𝒱}^{e} and ∥H(P, C) − H(P_{i}, C)∥ < ϵ for all P_{i} ∈ 𝒩_{ϵ}(P).

If this holds, it follows by the lower bound in Lemma 4.16, that

δ_{g}(P, P_{i}) < ϵ for all P_{i} ∈ 𝒩_{ϵ}(P).

That is, for any P ∈ 𝒮_{C} and ϵ > 0 there exists an open set in τ that is a strict subset of ℬ_{δ_{g}}^{e,sc}(P, ϵ). For this to be possible, the graph topology (induced by the gap metric) must be a sub-collection of τ.

4.6 Robustness of SD Closed-Loop Systems

In this section, results of the last two sections are specialised to SD, closed-loop systems. Set 𝒰 = ℝ^{m} and 𝒴 = ℝ^{p} and consider the feedback configuration shown in Figure 4.3, where P ∈ 𝒫_{𝒰→𝒴}^{e} , C ∈ 𝒟_{𝒴→𝒰} is an LSI, discrete-time controller, A_{D} is an ideal, periodic (period h) sampling-device, D_{A} is a zeroth-order hold-device synchronised with A_{D} and F is a stable, low-pass, LTI filter required to ensure boundedness of the sampling device. Note that F ∈ 𝒫_{𝒴→𝒴}^{e,sc} and hence, that C^{sd} :=D_{A}CA_{D}F ∈ 𝒫_{𝒴→𝒰}^{e,sc} . As such, Theorem 4.18 applies as summarised below.

[figure: C; P; D_{A}; A_{D}; F; d_{1}; u_{1}; y_{1}; d_{2}; u_{2}; y_{2}; −; +; +; −]

Figure 4.3 Sampled-Data Control System

Corollary 4.28 Consider a stable, closed-loop system [P_{0}, C^{sd}]. The following are equivalent : (i) b_{P,C^{sd}} ≥ β.

(ii) [P_{1}, C^{sd}] is stable for all P_{1} ∈ 𝒫_{𝒰→𝒴}^{e} such that δ_{g}(P_{0}, P_{1}) < β.

Remark 4.29 The results presented in Sub-Section 4.4.2 concerning simultaneous perturbations to the plant and controller also apply. One possible application of such a result is in the area of approximating LTI, continuous-time controllers by SD controllers. ♣

13Note the ϵ-dependence of the neighbourhood.

4.7 Summary

The primary motivation of this Chapter stems from a desire to characterise the robustness properties of SD control-systems developed using the design framework proposed in Chapter 3 (cf. Section 4.6). To this end, a more general result concerning the robustness of closed-loop stability to perturbations measured in the gap metric is obtained for (potentially infinite-dimensional) LPTV, closed-loop systems. Specifically, the largest gap-ball of LPTV plants, centred at a nominal plant, that a nominal, stabilising controller is guaranteed to stabilise, is identified. A similar result is obtained for simultaneous plant and controller perturbations and it is shown that the gap metric induces the graph topology on a quite general class of LPTV systems.

Along the way, various results of independent interest are derived. Specifically, it is shown that the graph of a stabilisable LPTV system can be expressed as the range and kernel of stable, LPTV systems, which are respectively, left and right invertible by stable, LPTV systems (cf. Theorem 4.6). The existence of these so-called strong-right and strong-left representations plays a crucial role throughout the chapter. In particular, given a well-posed plant/controller pair, it is shown that closed-loop stability can be readily characterised in terms of strong-right and strong-left representations of the plant and controller graphs. In turn, this leads to a Youla-style parameterisation of stabilising controllers.

In Section 4.3, a formula is derived for the directed gap between two LPTV systems. The formula obtained is essentially a generalisation of the formula derived in [Geo88] for LTI systems. It is fundamental to the framework used to establish robustness results and can be used to compute the gap metric as is detailed in Chapter 5.

The results presented in this chapter may find application in the important area of approximating LTI, continuous-time controllers with SD controllers and may be seen as a step towards characterising the quality of such approximations.

Computing the Gap

5.1 Introduction

In this chapter a procedure is developed for computing (to any desired accuracy) the gap between LPTV systems that admit stabilisable and detectable, state-space realisations. Using a formula derived in Section 4.3, it follows that bounding the directed gap between two LPTV systems is equivalent to solving a LSI, full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem.^{1} More specifically, it is shown that the directed gap between two LPTV, continuous-time systems P_{a} and P_{b} say, is less than some real number γ > 0 if and only if there exists a stabilising, full-information control-law that achieves a closed-loop, ℓ_{ℤ^{+}}^{2}(·)-induced norm of less than γ, when applied to a particular generalised plant. The particular generalised plant is LSI and can be expressed in terms of normalised, rightcoprime factorisations of the equivalent, LSI systems P_{a} :=WP_{a}W^{−1} and P_{b} :=WP_{b}W^{−1}. The existence of a solution to the equivalent, LSI, full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem can be determined using the state-space, necessary and sufficient condition of Proposition 2.22. Given a computationally-tractable characterisation of this necessary and sufficient condition and a method for constructing state-space realisations of the required normalised, right-coprime factors, it follows that a bisection-search can be used to compute (to any desired accuracy) the directed gap (and hence gap) between the two LPTV systems concerned.

Much of this chapter is devoted to constructing state-space realisations of normalised, rightcoprime factors in the time-lifted domain and obtaining computationally-tractable conditions that characterise the existence of solutions to the equivalent, full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem described above. First, given a stabilisable and detectable, state-space realisation of an LPTV system P_{a} say, it is shown how to construct a state-space realisation of a normalised, right-coprime factorisation of P_{a} :=WP_{a}W^{−1}. This is done explicitly in terms of the original state-space description of P_{a} and involves solving a finite-dimensional and hence, computationally-tractable, algebraic, Riccati equation. Using this, a computationally-tractable, necessary and sufficient condition which characterises the existence of a solution to the LSI, full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem associated with bounding the directed gap, is obtained. This characterisation is expressible explicitly in terms of the original stabilisable and detectable, state-space realisations of the two LPTV systems concerned and involves computing the state-transition operator of a periodic, matrix-valued function of time at only one point. The latter step can be achieved by integrating the relevant homogeneous system of differential equations forward in time.

1cf. Sub-Section 2.4.3 for the definition of the full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem

5.2 The Gap Metric and Full-Information Synthesis

Recall (cf. Section 4.3) that the gap between two operators P_{a} and P_{b} is defined to be the gap between their respective graphs as follows:

δ_{g}(P_{a}, P_{b}) := δ(𝒢_{P_{a}}, 𝒢_{P_{b}}) :=∥Π_{𝒢_{Pa}} − Π_{𝒢_{P_{b}}} ∥
= max{δ⃗_{g}(P_{a}, P_{b}), δ⃗_{g}(P_{b}, P_{a})},

where

δ⃗_{g}(P_{a}, P_{b}) :=∥Π_{𝒢_{P_{b}}^{⊥}} Π_{𝒢_{Pa}} ∥ and δ⃗_{g}(P_{b}, P_{a}) :=∥Π_{𝒢_{Pa}^{⊥}} Π_{𝒢_{P_{b}}} ∥

are the directed gaps. Furthermore, recall the formula derived in Section 4.3 for the directed gap between two LPTV systems P_{a} and P_{b} in 𝒫_{𝒰→𝒴} :

δ⃗_{g}(P_{a}, P_{b}) = inf_{stable Q∈𝒟_{L^{2}_{ℍ}(𝒰) →L^{2}_{ℍ}(𝒴)}} ∥G_{a} − G_{b}Q∥,  (5.1)

where 𝒢_{P_{•}} :=W𝒢_{P_{•}} and G_{•} is any isometry in 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒰⊕𝒴)} such that 𝒢_{P_{•}} = ℛ_{G_{•}} (the subscript “•” here denotes a or b). Existence of the required G_{•} is discussed in Section 4.2 (cf. Theorem 4.6 ). An important result used in the sequel is that given a state-space realisation of P_{•} (which must satisfy some mild stabilisability and detectability conditions), it is possible to construct an isometric, strong-right representation G_{•} of 𝒢_{P_{•}}. This is done by first constructing a normalised, right-coprime pair (M_{•}, N_{•}), such that

P_{•} :=WP_{•}W^{−1} = N_{•}M_{•}^{−1}

and then taking

G_{•} = [ \begin{matrix}M_{•} \\ N_{•} \end{matrix} ].

This is described in more detail in Section 5.3.

Remark 5.1 Note that although any system P ∈ 𝒫_{𝒰→𝒴} is equivalent to a system P:=WP W^{−1} in 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} , the converse is not true in general (cf. Remark 2.6). As such, the formula for the directed gap (cf. equation (5.1)) is only valid in the time-lifted domain. ♣

Interestingly, the right-hand side of equation (5.1) can be expressed in the following lower, linearfractional form:

G_{a} − G_{b}Q = F_{ℓ}(H, Q),  (5.2)

where

H:=[ \begin{matrix}G_{a} & G_{b} \\ I & 0 \end{matrix} ] : ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰)) ⊕ ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰)) → ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰 ⊕ 𝒴)) ⊕ ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰)).  (5.3)

The stable system H can be thought of as generalised plant mapping an exogenous disturbance and control signal [ \begin{matrix}w \\ u \end{matrix} ], to a controlled and measured output [ \begin{matrix}z \\ y \end{matrix} ]. Now given state-space realisations of isometric, strong-right representations G_{•} of 𝒢_{P_{•}}, the following equivalence holds.

Lemma 5.2 Given a state-space realisation of the generalised plant H defined in equation (5.3), let x denote the state. Then δ⃗_{g}(P_{a}, P_{b}) < γ if and only if there exists a causal, LSI, stabilising, disturbance-feedforward control-law u = K_{df}w, such that ∥F_{ℓ}(H, K_{df})∥ < γ. Furthermore, this is equivalent to the existence of a causal, LSI, stabilising, full-information control-law u = K_{fi}[ \begin{matrix}x \\ w \end{matrix} ], such that with it in place, the ℓ_{ℤ^{+}}^{2}-induced norm of the mapping from w to z is strictly less than γ.

Proof : Since G_{b} has stable left-inverse in 𝒟_{L_{ℍ}^{2}(𝒰⊕𝒴) →L_{ℍ}^{2}(𝒰)} , it follows from equation (5.2), that F_{ℓ}(H, Q) is stable if and only if Q is stable. That is, the existence of a stable Q ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒰)} such that ∥G_{a} −G_{b}Q∥ < γ, is equivalent to the existence of a causal, LSI, stabilising control-law K_{df} acting on w alone, such that ∥F_{ℓ}(H, K)∥ < γ. Furthermore, since the generalised plant H is a stable system, given a state-space realisation of H and the disturbance w, it is possible to re-construct the state. Thus, without loss of generality, it can be assumed that the control-law has access to the state x in addition to the exogenous disturbance w. That is, the directed gap is less than some number γ if and only if there exists a causal, LSI, stabilising, full-information control-law u = K_{fi}[ \begin{matrix}x \\ w \end{matrix} ] that achieves a closed-loop, ℓ_{ℤ^{+}}^{2}-induced norm strictly less than γ.

5.3 Normalised Right-Coprime Factorisations

A pair of stable systems (M, N), with M ∈ 𝒟_{𝒰→𝒰} and N ∈ 𝒟_{𝒰→𝒴} (𝒰 and 𝒴 possibly infinitedimensional, Hilbert spaces), is said to be a right-coprime pair if there exist stable systems X ∈ 𝒟_{𝒰→𝒰} and Y ∈ 𝒟_{𝒴→𝒰} such that XM + YN = I. Further to this, the pair is said to be \underline{normalised} if

[ M^{∗} N^{∗} ][ \begin{matrix}M \\ N \end{matrix} ] = I.

For a system P ∈ 𝒟_{𝒰→𝒴} a (normalised) right-coprime pair (M, N) is said to constitute a (normalised) right-coprime factorisation of P if P = NM^{−1}. Importantly, the graph of such a P can be characterised in terms of any right-coprime factorisation of it.

Lemma 5.3 If P = NM^{−1} ∈ 𝒟_{𝒰→𝒴} with (M, N) a right-coprime pair, then

𝒢_{P}:=[ \begin{matrix}I \\ P \end{matrix} ]𝒟_{P} = ℛ_{G},

where

G:=[ \begin{matrix}M \\ N \end{matrix} ] ∈ 𝒟_{𝒰→𝒰⊕𝒴} .

In fact, since such a G is stably left-invertible it is a strong-right representation of 𝒢_{P}.

Proof : Suppose that [ \begin{matrix}u \\ y \end{matrix} ]∈ ℛ_{G} ⊂ ℓ_{ℤ^{+}}^{2}(𝒰 ⊕ 𝒴). That is, for some q ∈ ℓ_{ℤ^{+}}^{2}(𝒰), u = Mq and

y = Nq. Then, y = NM^{−1}u and it follows that [ \begin{matrix}u \\ y \end{matrix} ]∈ 𝒢_{P} (or equivalently ℛ_{G} ⊂ 𝒢_{P}). Now to

see the opposite inclusion, suppose that [ \begin{matrix}u \\ y \end{matrix} ]∈ 𝒢_{P}. Then y = NM^{−1}u and defining q:=M^{−1}u, it

follows that [ \begin{matrix}u \\ y \end{matrix} ]= Gq. Since (M, N) is right coprime, there exists a stable pair (Y, X) such that

[ X Y ]G = I. Hence, q = [ X Y ][ \begin{matrix}u \\ y \end{matrix} ]∈ ℓ_{ℤ^{+}}^{2}(𝒰), which implies 𝒢_{P} ⊂ ℛ_{G}.

Given an LPTV system P ∈ 𝒫_{𝒰→𝒴} described by a state-space system of differential equations satisfying some mild stabilisability and detectability conditions, it is shown in the next sub-section how to construct a state-space realisation of a normalised, right-coprime factorisation of the equivalent system P:=WP W^{−1} ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} . It then follows by Lemma 5.3, that this is a state-space realisation of an isometric, strong-right representation of 𝒢_{P}, which can in turn be used for directed gap computation (cf. Lemma 5.2). First though, it is shown how to construct a state-space realisation of a normalised, right-coprime factorisation of any LSI, state-space system that satisfies a stabilisability and detectability condition.

5.3.1 LSI State-Space Systems

Consider the system P ∈ 𝒟_{𝒰→𝒴} (with 𝒰 and 𝒴 possibly infinite-dimensional, Hilbert spaces) described by the LSI, state-space system of difference equations

x_{k+1} = Ax_{k} + Bu_{k}; x_{0} = 0,  (5.4a)
y_{k} = Cx_{k} + Du_{k},  (5.4b)

where it is assumed that the state x_{k} evolves in a finite-dimensional, Hilbert space 𝒳. In this subsection, a state-space realisation of a normalised, right-coprime factorisation (M, N) of P = NM^{−1} is constructed in terms of the state-space operators A, B, C and D. Throughout, it is assumed that the pair (A, B) is stabilisable and (C, A) is detectable. That is, there exists an operator F : 𝒳 → 𝒰 such that spec(A + BF) ⊂ 𝔻 and an operator L : 𝒴 → 𝒳 such that spec(A + LC) ⊂ 𝔻. Define G:=[ \begin{matrix}M \\ N \end{matrix} ] to be the stable system in 𝒟_{𝒰→𝒰⊕𝒴} described by the LSI system of difference equations

s_{k+1} = Ǎs_{k} + B̌q_{k}
:= (A + BF)s_{k} + BVq_{k}; s_{0} = 0,  (5.5a)
[ \begin{matrix}u_{k} \\ y_{k} \end{matrix} ] = Čs_{k} + Ďq_{k}
:= [ \begin{matrix}F \\ C + DF \end{matrix} ]s_{k} + [ \begin{matrix}V \\ DV \end{matrix} ]q_{k},  (5.5b)

where V is any operator in B_{𝒰→𝒰} with bounded inverse. Then it is straight-forward to verify that the pair (M, N) is a right-coprime factorisation of P = NM^{−1} [NJB84]. In fact, a stable left-inverse of G is characterised by the LSI system of difference equations

v_{k+1} = (A + LC)v_{k} − (B + LD)u_{k} + Ly_{k}; v_{0} = 0,
q_{k} = V^{−1}Fv_{k} + V^{−1}u_{k}.

The first step towards characterising normalised, right-coprime factorisations of P is to determine conditions under which the state-space realisation of G given in equations (5.5), is isometric. It follows by causality and shift-invariance, that G can be expressed in terms of the block, lowertriangular, Toeplitz operator

[ \begin{matrix}Ď & 0 & 0 & 0 & ⋯ \\ ČB̌ & Ď & 0 & 0 & ⋯ \\ ČǍB̌ & ČB̌ & Ď & 0 & ⋯ \\ ČǍ^{2}B̌ & ČǍB̌ & ČB̌ & Ď & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Since spec(Ǎ) ⊂ 𝔻, the system of difference equations describes an ℓ_{ℤ^{+}}^{2}(·)-bounded operator and hence, its adjoint is well-defined. In fact, the adjoint can be expressed in terms of the block, upper-triangular, Toeplitz operator

[ \begin{matrix}Ď^{∗} & B̌^{∗}Č^{∗} & B̌^{∗}Ǎ^{∗}Č^{∗} & B̌^{∗}Ǎ^{2∗} Č^{∗} & ⋯ \\ 0 & Ď^{∗} & B̌^{∗}Č^{∗} & B̌^{∗}Ǎ^{∗}Č^{∗} & ⋯ \\ 0 & 0 & Ď^{∗} & B̌^{∗}Č^{∗} & ⋯ \\ 0 & 0 & 0 & Ď^{∗} & ⋯ \\ ⋮ & ⋮ & ⋮ & ⋮ & ... \end{matrix} ] .

Now let X be a self-adjoint, positive-semidefinite operator satisfying the Stein equation X−Ǎ^{∗}XǍ = Č^{∗}Č. Then since lim_{k→∞} Ǎ^{k} = 0, the blocks of the Toeplitz representation of G^{∗}G satisfy

(G^{∗}G)_{[i,j]} = \{ \begin{matrix}Ď^{∗} Ď + B̌^{∗}XB̌ & i = j \\ B̌(Ǎ^{∗})^{i−j}(Č^{∗}Ď + Ǎ^{∗}XB̌) & i > j \\ (Ď^{∗} Č + B̌^{∗}XǍ)Ǎ^{j−i}B̌ & i < j \end{matrix} .

Hence, if there exists a positive-semidefinite, self-adjoint operator X such that

(i)  X − Ǎ^{∗}XǍ = Č^{∗}Č,
(ii)  Ď^{∗} Č + B̌^{∗}XǍ = 0

and

(iii)  Ď^{∗} Ď + B̌^{∗}XB̌ = I,

it follows that G is an isometry. In fact, these conditions are also necessary if F and V are such that (Ǎ, B̌) is completely reachable. Using these conditions, it is possible to select F and V in equation (5.5), to ensure that G corresponds to a normalised (isometric), right-coprime factorisation.

Lemma 5.4 Consider the system P defined by the system of difference equations (5.4). Suppose that (A, B) is stabilisable and (C, A) is detectable. Now define

R:=I + D^{∗}D > 0, R̃:=I + DD^{∗} > 0

and [ \begin{matrix}M \\ N \end{matrix} ] by the system of difference equations

s_{k+1} = (A + BF)s_{k} + BVq_{k}; s_{0} = 0,  (5.6a)
[ \begin{matrix}u_{k} \\ y_{k} \end{matrix} ] = [ \begin{matrix}F \\ C + DF \end{matrix} ]s_{k} + [ \begin{matrix}V \\ DV \end{matrix} ]q_{k},  (5.6b)

where

V := (R + B^{∗}XB)^{− \frac{1}{2}} ,  (5.7)
F := −(R + B^{∗}XB)^{−1}(B^{∗}XA + D^{∗}C)  (5.8)

and X is a self-adjoint, positive-semidefinite solution to the algebraic, Riccati equation

X = A^{∗}XA + C^{∗}C− (D^{∗}C + B^{∗}XA)(R + B^{∗}XB)^{−1}(C^{∗}D + B^{∗}XA),  (5.9)

that satisfies spec(A + BF) ⊂ 𝔻. Then the pair (M, N) is a \underline{normalised, right-coprime factorisation} of P = NM^{−1}.

Proof : Clearly the pair (M, N) is a right-coprime factorisation of P. So it sufficient to show that it is also normalised. That is, [ \begin{matrix}M \\ N \end{matrix} ] defined by the system of difference equation (5.6) is an isometry. First note that under the assumption that (A, B) is stabilisable and (C, A) is detectable, there exists a self-adjoint, positive-semidefinite solution X to the algebraic, Riccati equation (5.9), that satisfies spec(A + BF) ⊂ 𝔻. To see this, note that λ ∈ ℂ is an unobservable mode of (C, A) if and only if it is an unobservable mode of (C^{∗}C− C^{∗}DR^{−1}D^{∗}C, A − BR^{−1}D^{∗}C) = (C^{∗}R̃^{−1}C, A − BR^{−1}D^{∗}C). Then since

[ \begin{matrix}C^{∗}C & C^{∗}D \\ D^{∗}C & R \end{matrix} ] = [ \begin{matrix}I & C^{∗}DR^{−1} \\ 0 & I \end{matrix} ][ \begin{matrix}C^{∗}R̃^{−1}C & 0 \\ 0 & R \end{matrix} ][ \begin{matrix}I & 0 \\ R^{−1}D^{∗}C & I \end{matrix} ] ≥ 0

and (C, A) has no unobservable modes on 𝕋, it follows by Proposition 2.20, that the algebraic, Riccati equation (5.9), admits a self-adjoint, positive-semidefinite solution X so that spec(A + BF) ⊂ 𝔻.

Now, it follows from equation (5.8) that

[ V^{∗} V^{∗}D^{∗} ][ \begin{matrix}F \\ C + DF \end{matrix} ] + V^{∗}B^{∗}X(A + BF)= 0.  (5.10)

Moreover, from equation (5.7),

[ V^{∗} V^{∗}D^{∗} ][ \begin{matrix}V \\ DV \end{matrix} ] + V^{∗}B^{∗}XBV= I.  (5.11)

Note also that the algebraic, Riccati equation (5.9), can be re-written as the Stein equation

X − (A + BF)^{∗}X(A + BF)= [ F^{∗} (C^{∗} + F^{∗}D^{∗}) ][ \begin{matrix}F \\ C + DF \end{matrix} ].  (5.12)

So from the discussion above concerning conditions for a state-space system to be isometric, it follows from equations (5.10), (5.11) and (5.12), that [ \begin{matrix}M \\ N \end{matrix} ] is an isometry as claimed.

5.3.2 LPTV State-Space Systems

Consider the system of differential equations

\frac{dx_{a}}{dt} = A_{a}(t)x_{a}(t) + B_{a}(t)u(t); x_{a}(0) = 0,  (5.13a)
y(t) = C_{a}(t)x_{a}(t) + D_{a}(t)u(t),  (5.13b)

where for fixed t ∈ ℝ^{+} the input u(t) ∈ 𝒰, the output y(t) ∈ 𝒴, the state x_{a}(t) ∈ 𝒳_{a}, and 𝒰, 𝒴 and 𝒳_{a} are all finite-dimensional, Hilbert spaces. It is assumed that A_{a}(t), B_{a}(t), C_{a}(t) and D_{a}(t) are all piecewise-continuous functions of t, so that with the state-transition operator Φ_{A_{a}}(t, τ) : ℝ × ℝ → B_{𝒳_{a} →𝒳_{a}} defined to be the unique, continuous solution to the homogeneous, differential equation

\frac{d}{dt}Φ_{A_{a}}(t, τ) = A_{a}(t)Φ_{A_{a}}(t, τ); Φ_{A_{a}}(τ, τ) = I,

the state-space system of differential equations (5.13), characterises the causal, linear, continuoustime system P_{a}^{e} : L_{ℝ^{+}}^{2,e}(𝒴) → L_{ℝ^{+}}^{2,e}(𝒴) defined by

y(t) :=(P_{a}^{e}u)(t) := ∫_{0}^{t} C_{a}(t)Φ_{A_{a}}(t, τ)B_{a}(τ) dτ + D_{a}(t)u(t)  (5.14)

[Che84, CD91, GL95]. If for some real number h > 0 and all k ∈ ℤ^{+}, A_{a}(t + kh) = A_{a}(t), B_{a}(t+kh) = B_{a}(t), C_{a}(t+kh) = C_{a}(t) and D_{a}(t+kh) = D_{a}(t), then Φ_{A_{a}}(t+kh, τ +kh) = Φ_{A_{a}}(t, τ) and it follows from equation (5.14), that P_{a}^{e} is periodically time-varying in the sense described in Sub-Section 2.3.2. As such, P_{a}^{e} induces a system P_{a} ∈ 𝒫_{𝒰→𝒴}^{e} , which in turn, is equivalent to an LSI system P_{a} :=WP_{a}W^{−1} ∈ 𝒟_{L_{ℍ}^{2}(𝒰) →L_{ℍ}^{2}(𝒴)} . Equations (5.13) are called a state-space realisation of P_{a} and such a systems is said to admit a state-space realisation. A state-space realisation of the equivalent, LSI system P_{a} can be obtained as follows, by characterising the evolution of the state in equation (5.13) in terms of its values at the discrete points in time {kh}_{k∈ℤ^{+}}:

x_{a,k+1} = A_{a}x_{a,k} + B_{a} u^{⊸}_{k}; x_{a,0} = 0,  (5.15a)
y^{⊸}_{k} = C_{a}x_{a,k} + D_{a} u^{⊸}_{k},  (5.15b)

where x_{a,k} = x_{a}(kh), u^{⊸}_{k}(θ) = u(kh + θ), y^{⊸}_{k}(θ) = y(kh + θ) and for all x ∈ 𝒳_{a} and u^{⊸} ∈ L_{ℍ}^{2}(𝒰)

A_{a} : 𝒳_{a} → 𝒳_{a},  A_{a}x:=Φ_{A_{a}}(h, 0)x,  (5.16a)
B_{a} : L_{ℍ}^{2}(𝒰) → 𝒳_{a},  B_{a} u^{⊸}:= ∫_{0}^{h} Φ_{A_{a}}(h, τ)B_{a}(τ) u^{⊸}(τ)dτ,  (5.16b)
C_{a} : 𝒳_{a} → L_{ℍ}^{2}(𝒴),  (C_{a}x)(θ) :=C_{a}(θ)Φ_{A_{a}}(θ, 0)x,  (5.16c)
D_{a} : L_{ℍ}^{2}(𝒰) → L_{ℍ}^{2}(𝒴),  (D_{a} u^{⊸})(θ) :=D_{a} u^{⊸}(θ) + ∫_{0}^{θ} C_{a}(θ)Φ_{A_{a}}(θ, τ)B_{a}(τ) u^{⊸}(τ)dτ.  (5.16d)

If the state-space realisation of P_{a} given in equations (5.13) is stabilisable and detectable (in a sense to be defined shortly), then the state-space realisation of P_{a} given in (5.15) is also stabilisable and detectable in the sense defined in Sub-Section 2.4.1, for LSI, state-space systems. The h-periodic pair (A_{a}(t), B_{a}(t)) is said to be stabilisable if there exists a piecewise-continuous, h-periodic function F_{a}(t) : ℝ^{+} → B_{𝒳_{a} →𝒰} such that

∥Φ_{A_{a}+B_{a}F_{a}}(t_{1}, t_{2})∥ ≤ c_{1}e^{−c_{2}(t_{1}−t_{2})},  (5.17)

for some positive constants c_{1} and c_{2} and all t_{1}, t_{2} ∈ ℝ [RPK92]. Now consider the state-space system

\frac{dx}{dt} = A_{a}(t)x(t) + B_{a}(t)u(t); x(0) = x̃,  (5.18a)
y(t) = x(t),  (5.18b)

with state-feedback u(t) :=F_{a}(t)y(t) = F_{a}(t)x(t), where the (h-periodic) operator F_{a}(t) is chosen so that equation (5.17) holds for some positive constants c_{1} and c_{2}. The system described by equations (5.18) is equivalent (in an input-output sense via W) to

x_{k+1} = A_{a}x_{k} + B_{a} u^{⊸}_{k}; x_{0} = x̃,
y^{⊸}_{k}(θ) = Φ_{A_{a}}(θ, 0)x_{k} + ∫_{0}^{θ} Φ_{A_{a}}(θ, τ)B_{a}(τ)u^{⊸}_{k}(τ) dτ,

where u^{⊸}_{k}(θ) :=(Wu)_{k} = u(kh + θ), y^{⊸}_{k}(θ) :=(Wy)_{k} = y(kh + θ) and x_{k} :=x(kh). However, with the state-feedback u(t) = F_{a}(t)y(t) = F_{a}(t)x(t) in place, y^{⊸}_{k}(θ) = Φ_{A_{a}+B_{a}F_{a}}(θ, 0)x_{k} and hence,

u^{⊸}_{k}(θ) :=F (θ) y^{⊸}_{k}(θ) = F_{a}(θ)Φ_{A_{a}+B_{a}F_{a}}(θ, 0)x_{k}.

So with the bounded, linear operator F_{a} : 𝒳_{a} → L_{ℍ}^{2}(𝒰) defined by

(F_{a}x)(θ) :=F_{a}(θ)Φ_{A_{a}+B_{a}F_{a}}(θ, 0)x

for all x ∈ 𝒳_{a}, it follows that A_{a} + B_{a}F_{a} = Φ_{A_{a}+B_{a}F_{a}}(h, 0) and since Φ_{A_{a}+B_{a}F_{a}}((k + 1)h, kh) = Φ_{A_{a}+B_{a}F_{a}}(h, 0) for all k ∈ ℤ^{+}, that (A_{a} + B_{a}F_{a})^{k} = Φ_{A_{a}+B_{a}F_{a}}(kh, 0). Thus, from equation (5.17), ∥(A_{a} + B_{a}F_{a})^{k}∥ ≤ c_{1}e^{−c_{2}kh} → 0 as k → ∞. This implies that spec(A_{a} + B_{a}F_{a}) ⊂ 𝔻 and therefore, that (A_{a}, B_{a}) is stabilisable. In conclusion, if the pair (A_{a}(t), B_{a}(t)) is stabilisable then (A_{a}, B_{a}) is stabilisable. Similarly, by considering the estimation-error dynamics that result from estimating the state of

\frac{dx}{dt} = A_{a}(t)x(t); x(0) = x̃,
y(t) = C_{a}(t)x(t),

with a simple Luenberger observer

\frac{dx_{e}}{dt} = A_{a}(t)x_{e}(t) + L_{a}(t)(y_{e}(t) − y(t)); x_{e}(0) = x̃_{e},
y_{e}(t) = C_{a}(t)x_{e}(t),

it can be shown that if there exist a h-periodic function L_{a}(t) : ℝ → B_{𝒴→𝒳_{a}} such that

∥Φ_{A_{a}+L_{a}C_{a}}(t_{1}, t_{2})∥ ≤ c_{1}e^{−c_{2}(t_{1}−t_{2})},  (5.19)

for some positive constants c_{1} and c_{2} and all t_{1}, t_{2} ∈ ℝ,^{2} then there exists a bounded, linear operator L_{a} : L_{ℍ}^{2}(𝒴) → 𝒳_{a} such that spec(A_{a} + LC_{a}) ⊂ 𝔻. That is, if the pair (C_{a}(t), A_{a}(t)) is detectable in the sense just described, then the pair (C_{a}, A_{a}) is detectable in the sense defined in Sub-Section 2.4.1, for LSI, state-space systems. So under the assumption that (A_{a}(t), B_{a}(t)) is stabilisable and (C_{a}(t), A_{a}(t)) is detectable, the following corollary follows directly by Lemma 5.4 and Proposition 2.18:

Corollary 5.5 Let G_{a} :=[ \begin{matrix}M_{a} \\ N_{a} \end{matrix} ] be defined by the state-space system of difference equations

v_{a,k+1} = (A_{a} + B_{a}F_{a})v_{a,k} + B_{a}V_{a} q^{⊸}_{a,k}; v_{a,0} = 0,  (5.20a)
[ \begin{matrix}u^{⊸} & k \\ y^{⊸} & k \end{matrix} ] = [ \begin{matrix}F_{a} \\ C_{a} + D_{a}F_{a} \end{matrix} ]v_{a,k} + [ \begin{matrix}V_{a} \\ D_{a}V_{a} \end{matrix} ] q^{⊸}_{a,k},  (5.20b)

where

V_{a} := (R_{a} + B_{a}^{∗}X_{a}B_{a})^{− \frac{1}{2}} ,  (5.21a)
F_{a} := −(R_{a} + B_{a}^{∗}X_{a}B_{a})^{−1}(B_{a}^{∗}X_{a}A_{a} + D_{a}^{∗}C_{a}),  (5.21b)
R_{a} := I + D_{a}^{∗}D_{a},  (5.21c)
R̃_{a} := I + D_{a}D_{a}^{∗},  (5.21d)
À_{a} := A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a}  (5.21e)

and X_{a} is a self-adjoint, positive-semidefinite, stabilising solution to the (discrete-time) algebraic, Riccati equation

X_{a} = À_{a}^{∗}X_{a}(I + B_{a}R_{a}^{−1}B_{a}^{∗}X_{a})^{−1}À_{a} + C_{a}^{∗}R̃_{a}^{−1}C_{a}.  (5.22)

Then the pair (M_{a}, N_{a}) is a normalised, right-coprime factorisation of P_{a} :=WP_{a}W^{−1}, where P_{a} ∈ 𝒫_{𝒰→𝒴}^{e} is characterised by the state-space system of differential equations (5.13).

2This is the definition of (C_{a}(t), A_{a}(t)) detectable [RPK92].

5.4 Computing the Directed Gap

Given a real number γ > 0 and two systems P_{a} and P_{b} in 𝒫_{𝒰→𝒴} that admit stabilisable and detectable, state-space realisations, a necessary and sufficient condition for the directed gap between the two systems to be strictly less than γ is obtained in this section. The necessary and sufficient condition ultimately involves only finite-dimensional operators and is therefore, computationally tractable. Using this result, a bisection-search can be employed to compute the directed gap to any desired accuracy.

5.4.1 A Necessary and Sufficient Condition for Bounding the Directed Gap Consider two LPTV systems P_{a} and P_{b} in 𝒫_{𝒰→𝒴} (with 𝒰 and 𝒴 finite-dimensional, Hilbert spaces) that admit state-space realisations

\frac{dx_{a}}{dt} = A_{a}(t)x_{a}(t) + B_{a}(t)u(t); x_{a}(0) = 0,  (5.23a)
y(t) = C_{a}(t)x_{a}(t) + D_{a}(t)u(t)  (5.23b)

and

\frac{dx_{b}}{dt} = A_{b}(t)x_{b}(t) + B_{b}(t)u(t); x_{b}(0) = 0,  (5.24a)
y(t) = C_{b}(t)x_{b}(t) + D_{b}(t)u(t)  (5.24b)

respectively, where the states x_{a}(t) and x_{b}(t) evolve in the finite-dimensional, Hilbert spaces 𝒳_{a} and 𝒳_{b} respectively, and the state-space operators A_{•}(t), B_{•}(t), C_{•}(t) and D_{•}(t) are all piecewisecontinuous, h-periodic functions of t (the subscript “•” denotes a or b). It is assumed that (A_{•}(t), B_{•}(t)) is stabilisable and (C_{•}(t), A_{•}(t)) is detectable, so that by Corollary 5.5 and Lemma

5.3, the system G_{a} :=[ \begin{matrix}M_{a} \\ N_{a} \end{matrix} ] (respectively G_{b} :=[ \begin{matrix}M_{b} \\ N_{b} \end{matrix} ]) described by the state-space system of

difference equations

s_{a,k+1} = (A_{a} + B_{a}F_{a})s_{a,k} + B_{a}V_{a} q^{⊸}_{a,k}; s_{a,0} = 0,  (5.25a)
[ \begin{matrix}u^{⊸} & a,k \\ y^{⊸} & a,k \end{matrix} ] = [ \begin{matrix}F_{a} \\ C_{a} + D_{a}F_{a} \end{matrix} ]s_{a,k} + [ \begin{matrix}V_{a} \\ D_{a}V_{a} \end{matrix} ] q^{⊸}_{a,k},  (5.25b)

where A_{a}, B_{a}, C_{a} and D_{a} are defined in equations (5.16), V_{a} and F_{a} are defined in equations (5.21) and X_{a} is a self-adjoint, positive-semidefinite, stabilising solution to the algebraic, Riccati equation (5.22) (respectively described by equations (5.25) and the related equations (5.16), (5.21) and (5.22) with the subscript a replaced by b), constitutes a strong-right representation of the graph of the equivalent, LSI system P_{a} :=WP_{a}W^{−1} (respectively P_{b} :=WP_{b}W^{−1}).

Let s_{k} :=[ \begin{matrix}s_{a,k} \\ s_{b,k} \end{matrix} ] and rename q^{⊸}_{a,k} by w^{⊸}_{k} and q^{⊸}_{b,k} by u^{⊸}_{k}. Then the system of difference equations

s_{k+1} = [ \begin{matrix}(A_{a} + B_{a}F_{a}) & 0 \\ 0 & (A_{b} + B_{b}F_{b}) \end{matrix} ]s_{k} + [ \begin{matrix}B_{a}V_{a} \\ 0 \end{matrix} ]w^{⊸}_{k} + [ \begin{matrix}0 \\ B_{b}V_{b} \end{matrix} ]u^{⊸}_{k}; s_{0} = 0, (5.26a)
z^{⊸}_{k} = [ \begin{matrix}F_{a} & F_{b} \\ (C_{a} + D_{a}F_{a}) & (C_{b} + D_{b}F_{b}) \end{matrix} ]s_{k} + [ \begin{matrix}V_{a} \\ D_{a}V_{a} \end{matrix} ]w^{⊸}_{k} + [ \begin{matrix}V_{b} \\ D_{b}V_{b} \end{matrix} ]u^{⊸}_{k},  (5.26b)

is a state-space realisation of the system [ G_{a} G_{b} ] : ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰 ⊕ 𝒰)) → ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒱)), where 𝒱:=𝒰 ⊕ 𝒴. It now follows by Lemma 5.2, that given a real number γ > 0, the directed gap δ⃗_{g}(P_{a}, P_{b}) < γ if and only if there exists a causal, LSI, stabilising control-law u^{⊸} = K[ s ; w^{⊸} ] such that

∥ z^{⊸}∥_{ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒱) )}^{2} − γ^{2}∥w^{⊸}∥_{ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰) )} ≤ −ϵ∥w^{⊸}∥_{ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰) )}

for some ϵ > 0 and all w^{⊸} ∈ ℓ_{ℤ^{+}}^{2}(L_{ℍ}^{2}(𝒰)), where w^{⊸} :=k ↦ w^{⊸}_{k}, u^{⊸} :=k ↦ u^{⊸}_{k}, z^{⊸} :=k ↦ z^{⊸}_{k} and s:=k ↦ s_{k} denote the signals associated with state-space system (5.26). That is, if and only if there exist a solution to the full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem (cf. Proposition 2.22).

Now note that since both spec(A_{a} + B_{a}F_{a}) and spec(A_{b} + B_{b}F_{b}) are subsets of 𝔻, the pair

([ \begin{matrix}(A_{a} + B_{a}F_{a}) & 0 \\ 0 & (A_{b} + B_{b}F_{b}) \end{matrix} ], [ \begin{matrix}0 \\ B_{b}V_{b} \end{matrix} ])

is stabilisable. Furthermore, since V_{b} = V_{b}^{∗} is boundedly invertible, it follows that

[ V_{b}^{∗} V_{b}^{∗}D_{b}^{∗} ][ \begin{matrix}V_{b} \\ D_{b}V_{b} \end{matrix} ] > 0.

Moreover, since (C_{b}, A_{b}) is detectable by assumption, it can be shown that

𝒦[ \begin{matrix}[ \begin{matrix}(A_{a} + B_{a}F_{a}) & 0 \\ 0 & (A_{b} + B_{b}F_{b}) \end{matrix} ] − e^{jθ}I & [ \begin{matrix}0 \\ B_{b}V_{b} \end{matrix} ] \\ [ \begin{matrix}F_{a} & F_{b} \\ (C_{a} + D_{a}F_{a}) & (C_{b} + D_{b}F_{b}) \end{matrix} ] & [ \begin{matrix}V_{b} \\ D_{b}V_{b} \end{matrix} ] \end{matrix} ] = \{0\}

for all θ ∈ [0, 2π). That is, all of the assumption required for Proposition 2.22 to hold are satisfied. The following theorem then follows directly by Lemma 5.2 and Proposition 2.22.

Theorem 5.6 Consider two LPTV systems P_{a} and P_{b} that admit the state-space realisations given in (5.23) and (5.24) respectively, where (A_{a}(t), B_{a}(t)) and (A_{b}(t), B_{b}(t)) are stabilisable and (C_{a}(t), A_{a}(t)) and (C_{b}(t), A_{b}(t)) are detectable. Then given a real number γ > 0, the directed gap δ⃗_{g}(P_{a}, P_{b}) < γ if and only if there exists a self-adjoint, positive-semidefinite operator X_{∞} : 𝒳_{a} ⊕ 𝒳_{b} → 𝒳_{a} ⊕ 𝒳_{b} that satisfies :

(i) the finite-dimensional, (discrete-time) algebraic, Riccati equation

X_{∞} = À^{∗}X_{∞}(I + R̀X_{∞})^{−1}À + Q̀;  (5.27)

(ii) spec(A − BZ^{−1}L) = spec((I + R̀X_{∞})^{−1}À) ⊂ 𝔻;

(iii) Z_{1} − Z_{2}^{∗}Z_{3}^{−1}Z_{2} < 0, where

J:=[ \begin{matrix}I & 0 \\ 0 & −γ^{2}I \end{matrix} ],  À:=A − B(D̃^{∗}JD̃)^{−1}D̃^{∗}JC̃,
R̀:=B(D̃^{∗}JD̃)^{−1}B^{∗},  Q̀ :=C̃^{∗}(I − D̃(D̃^{∗}JD̃)^{−1}D̃^{∗})C̃,
A:=[ \begin{matrix}(A_{a} + B_{a}F_{a}) & 0 \\ 0 & (A_{b} + B_{b}F_{b}) \end{matrix} ],  B:=[ \begin{matrix}B_{a}V_{a} & 0 \\ 0 & B_{b}V_{b} \end{matrix} ],

C̃:= [ \begin{matrix}[ \begin{matrix}F_{a} \\ (C_{a} + D_{a}F_{a}) \end{matrix} & \begin{matrix}F_{b} \\ (C_{b} + D_{b}F_{b}) \end{matrix} ] \\ 0 & 0 \end{matrix} ] ,  D̃ :=[ \begin{matrix}[ \begin{matrix}V_{a} \\ D_{a}V_{a} \end{matrix} ] & [ \begin{matrix}V_{b} \\ D_{b}V_{b} \end{matrix} ] \\ I & 0 \end{matrix} ],

Z :=D̃^{∗}JD̃ + B^{∗}X_{∞}B =: [ \begin{matrix}Z_{1} & Z_{2}^{∗} \\ Z_{2} & Z_{3} \end{matrix} ],  L:=D̃^{∗}JD̃ + B^{∗}X_{∞}A

and A_{•}, B_{•}, C_{•}, D_{•}, F_{•} and V_{•} (“•” denotes a or b) are defined in equations (5.16) and (5.21).

Remark 5.7 Given finite-dimensional expressions (matrix representations) for À, R̀ and Q̀ and a computationally-tractable procedure for testing condition (iii), a bisection-search based on the conditions given in Theorem 5.6 can be used to compute to any desired accuracy the directed gap between any two systems in 𝒫_{𝒰→𝒴} that admit stabilisable and detectable, state-space realisations. Then since the gap is simply the maximum of two directed gaps, this constitutes a procedure for computing the gap between two LPTV systems. ♣

In the sequel the necessary finite-dimensional expressions for À, R̀ and Q̀ and a computationallytractable test for condition (iii) are obtained. To this end, the following assumption is required to hold throughout. Assumption 5.8 It is assumed that D_{a}(t) = D_{b}(t) for all t. In other words, the two systems P_{a} and P_{b} have the same instantaneous response.

5.4.2 Computing À, R̀ and Q̀ in Theorem 5.6

In this sub-section, finite-dimensional expressions are derived for the operators À, R̀and Q̀ appearing in Theorem 5.6. The formulae involve computing the value of a state-transition operator at only one point. The relevant state-transition operator is associated with a time-varying TPBVP that is expressible explicitly in terms of the periodically-time-varying, state-space realisations of the two LPTV systems P_{a} and P_{b}. The TPBVP characterisation is derived in a manner similar to the TPBVP analysis presented in [CG97] (cf. Chapter 6), where finite-dimensional expressions are derived for certain operators involved in the so-called SD-discretisation approach to H^{∞} synthesis of SD control-systems. The derivation here is a long and tedious series of state-space manipulations and as such, the proof is deferred to Appendix C.1.

Lemma 5.9 Given γ > 0, construct the following periodic, matrix function of time :

[E(t)]_{i,j} :=E_{ij}(t),

where the E_{ij}’s (i, j = 1, …, 7) are defined in Appendix C.2. Let

E:=Φ_{E}(h, 0)

and partition it conformably with E, where Φ_{E}(t, τ) denotes the state-transition operator associated with E(t). Then under Assumption 5.8,

Q̀ = − [ 0 I 0 0 ] Ẽ^{−1} [ \begin{matrix}(E_{43} − [ X_{a} 0 ]E_{23}) \\ (E_{53} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{13}) \\ E_{63} \\ (E_{73} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{33}) \end{matrix} ] ,  (5.29)

R̀ = [ E_{24} E_{26} ][ \begin{matrix}−E_{64} & −E_{66} \\ (E_{44} − [ X_{a} 0 ]E_{24}) & (E_{46} − [ X_{a} 0 ]E_{26}) \end{matrix} ]^{−1}[ \begin{matrix}I \\ 0 \end{matrix} ] (5.30)

and

À = (E_{33} − E_{23}) − [ (E_{34} − E_{24}) (E_{35} − E_{25}) (E_{36} − E_{26}) (E_{37} − E_{27}) ] ×
Ẽ^{−1}[ \begin{matrix}(E_{43} − [ X_{a} 0 ]E_{23}) \\ (E_{53} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{13}) \\ E_{63} \\ (E_{73} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{33}) \end{matrix} ],  (5.31)

where

Ẽ:=[ \begin{matrix}(E_{44} − [ X_{a} 0 ]E_{24}) & (E_{45} − [ X_{a} 0 ]E_{25}) & (E_{46} − [ X_{a} 0 ]E_{26}) & (E_{47} − [ X_{a} 0 ]E_{27}) \\ (E_{54} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{14}) & (E_{55} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{15}) & (E_{56} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{16}) & (E_{57} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{17}) \\ E_{64} & E_{65} & E_{66} & E_{67} \\ (E_{74} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{34}) & (E_{75} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{35}) & (E_{76} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{36}) & (E_{77} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{37}) \end{matrix} ],

X_{a} is a self-adjoint, positive-semidefinite, stabilising solution to the algebraic, Riccati equation

X_{a} = À_{a}^{∗}X_{a}(I + R̀_{a}X_{a})^{−1}À_{a} + Q̀_{a},

and X_{b} is a self-adjoint, positive-semidefinite, stabilising solution to the algebraic, Riccati equation

X_{b} = À_{b}^{∗}X_{b}(I + R̀_{b}X_{b})^{−1}À_{b} + Q̀_{b},

with

À_{a} = [ I 0 ] (E_{33} − E_{37}E_{77}^{−1}E_{73})[ \begin{matrix}I \\ 0 \end{matrix} ],  À_{b} = [ 0 I ] (E_{33} − E_{37}E_{77}^{−1}E_{73})[ \begin{matrix}0 \\ I \end{matrix} ],
R̀_{a}= −[ I 0 ]E_{33}E_{77}^{−1}[ \begin{matrix}I \\ 0 \end{matrix} ],  R̀_{b}= −[ 0 I ]E_{33}E_{77}^{−1}[ \begin{matrix}0 \\ I \end{matrix} ],
Q̀_{a} = −[ I 0 ]E_{77}^{−1}E_{73}[ \begin{matrix}I \\ 0 \end{matrix} ]  and Q̀_{b} = −[ 0 I ]E_{77}^{−1}E_{73}[ \begin{matrix}0 \\ I \end{matrix} ].

Proof : The proof, rather long and tedious, is deferred to Appendix C.1.

5.4.3 Testing Condition (iii) in Theorem 5.6

Recall that condition (iii) in Theorem 5.6 involves testing

Z_{1} − Z_{2}^{∗}Z_{3}^{−1}Z_{2} < 0,

where

Z := D̃^{∗}JD̃ + B^{∗}X_{∞}B =: [ \begin{matrix}Z_{1} & Z_{2}^{∗} \\ Z_{2} & Z_{3} \end{matrix} ],

D̃, J and B are defined in Theorem 5.6 and X_{∞} is a self-adjoint, positive-semidefinite, stabilising solution to the finite-dimensional, algebraic, Riccati equation (5.27). It is straightforward to show that

Z_{1} − Z_{2}^{∗}Z_{3}^{−1}Z_{2} = −γ^{2}I + V_{a}^{∗}W̃^{∗}(I − Ỹ(Ỹ^{∗}Ỹ)^{−1}Ỹ^{∗})W̃V_{a},  (5.32)

where

W̃ := [ \begin{matrix}I \\ W \end{matrix} ], W :=[ \begin{matrix}D_{a} \\ X∞^{\frac{1}{2}}[ \begin{matrix}B_{a} \\ 0 \end{matrix} ] \end{matrix} ],  (5.33a)

Ỹ := [ \begin{matrix}I \\ Y \end{matrix} ], Y :=[ \begin{matrix}D_{b} \\ X∞^{\frac{1}{2}}[ \begin{matrix}0 \\ B_{b} \end{matrix} ] \end{matrix} ],  (5.33b)

B_{•} and D_{•} are defined in equations (5.16) and V_{a} is defined in equation (5.21a). Since (I − Ỹ(Ỹ^{∗}Ỹ)^{−1}Ỹ^{∗}) is idempotent, it follows from equation (5.32) that

Z_{1} − Z_{2}^{∗}Z_{3}^{−1}Z_{2} < 0
⇕
−γ^{2}I + V_{a}^{∗}W̃^{∗}(I − Ỹ(Ỹ^{∗}Ỹ)^{−1}Ỹ^{∗})(I − Ỹ(Ỹ^{∗}Ỹ)^{−1}Ỹ^{∗})W̃V_{a} < 0.

This last condition is tantamount to

∥(I − Ỹ(Ỹ^{∗}Ỹ)^{−1}Ỹ^{∗})W̃V_{a}∥ = ∥(I − [ \begin{matrix}I \\ Y \end{matrix} ](I + Y^{∗}Y)^{−1}[ I Y^{∗} ])[ \begin{matrix}I \\ W \end{matrix} ]V_{a}∥< γ.

Now note that the operator

[ \begin{matrix}(I + Y^{∗}Y)^{− \frac{1}{2}} [ I Y^{∗} ] \\ (I + YY^{∗})^{− \frac{1}{2}} [ −Y I ] \end{matrix}]

is unitary and therefore, that

∥(I − [ \begin{matrix}I \\ Y \end{matrix} ](I + Y^{∗}Y)^{−1}[ I Y^{∗} ])[ \begin{matrix}I \\ W \end{matrix} ]V_{a}∥< γ

⇕

∥[ \begin{matrix}(I + Y^{∗}Y)^{− \frac{1}{2}} [ I Y^{∗} ] \\ (I + YY^{∗})^{− \frac{1}{2}} [ −Y I ] \end{matrix}] (I − [ \begin{matrix}I \\ Y \end{matrix} ](I + Y^{∗}Y)^{−1}[ I Y^{∗} ])[ \begin{matrix}I \\ W \end{matrix} ]V_{a}∥< γ

⇕
∥(I + YY^{∗})^{− \frac{1}{2}} (W − Y)V_{a}∥< γ.

Under Assumption 5.8, D_{a}(t) = D_{b}(t) for all t and hence, the operator (W − Y) is compact.^{3} It follows that (I + YY^{∗})^{− \frac{1}{2}} (W − Y)V_{a} is also compact and therefore, that its norm can be determined by calculating its largest singular value (cf. Sub-Section 2.2.2 or [Kre89b]).

Recall that a number σ ∈ ℝ^{+} is a singular value of (I + YY^{∗})^{− \frac{1}{2}} (W − Y)V_{a} if and only if there exist a non-zero x^{⊸} ∈ L_{ℍ}^{2}(𝒰) such that

V_{a}^{∗}(W − Y)^{∗}(I + YY^{∗})^{−1}(W − Y)V_{a} x^{⊸} = σ^{2} x^{⊸}.

Since V_{a} = V_{a}^{∗} is invertible, this is equivalent to the existence of a non-zero z^{⊸} (= V_{a}^{−1} x^{⊸}) such that

(W − Y)^{∗}(I + YY^{∗})^{−1}(W − Y)V_{a}^{2} z^{⊸} = σ^{2} z^{⊸}.  (5.34)

For σ > 0, such a z^{⊸} can be constructed (as follows) in terms of the solution to a TPBVP. To see this, define

w^{⊸} :=(W − Y)^{∗}(I + YY^{∗})^{−1}(W − Y)σ^{−1} u^{⊸}  (5.35a)

and

u^{⊸} :=V_{a}^{2} z^{⊸}.  (5.35b)
y^{⊸}(θ) = [ \begin{matrix}C_{a} & −C_{b} \\ 0 & 0 \end{matrix} ] ∫_{0}^{θ} Φ[ \begin{matrix}A_{a} & 0 \\ 0 & A_{b} \end{matrix} ](θ, τ)[ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ] u^{⊸}(τ) dτ,

ỹ = [ \begin{matrix}0 \\ X∞^{\frac{1}{2}}[ \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ] \end{matrix} ] ∫_{0}^{h} Φ[ \begin{matrix}A_{a} & 0 \\ 0 & A_{b} \end{matrix} ](h, τ)[ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ] u^{⊸}(τ) dτ,

3Under Assumption 5.8, the action of [ y^{⊸} ; ỹ ] = (W − Y) u^{⊸} is completely characterised by the integral operators which are both compact [Kre89a], since there is no direct feedthrough.

It then follows from equations (5.16) and (5.33), that the operator mapping u^{⊸} to w^{⊸} in equation (5.35a) can be expressed in terms of the following TPBVP:

w^{⊸} = [ 0 [ B_{a}^{∗} −B_{b}^{∗} ] ][ a^{⊸} ; b^{⊸} ]  (5.36a)

\frac{d}{dθ} ([ a^{⊸} ; b^{⊸} ]) = [ \begin{matrix}\underline{A} & −[ \begin{matrix}0 \\ B_{b} \end{matrix} ]R_{b}^{−1}[ 0 B_{b}^{∗} ] \\ −[ \begin{matrix}C_{a}^{∗} \\ C_{a}^{∗} \end{matrix} ]R̃_{b}^{−1}[ C_{a} C_{b} ] & −\underline{A}^{∗} \end{matrix} ][ a^{⊸} ; b^{⊸} ](θ)

+[ \begin{matrix}[ & \begin{matrix}B_{a} \\ −B_{b} \end{matrix} & ] \\ & 0 \end{matrix} ]σ^{−1} u^{⊸}(θ),  (5.36b)

with boundary conditions

a^{⊸}(0) = 0 and b^{⊸}(h) = X_{∞} a^{⊸}(h),

where

\underline{A} := [ \begin{matrix}A_{a} & 0 \\ 0 & A_{b} \end{matrix} ] − [ \begin{matrix}0 \\ B_{b} \end{matrix} ]R_{b}^{−1}D_{b}^{∗}[ C_{a} C_{b} ],

R_{•} :=(I + D_{•}^{∗}D_{•}) and R̃_{•} :=(I + D_{•}D_{•}^{∗}).^{4} Similarly, the operator mapping z^{⊸} to u^{⊸} in equation (5.35b) is completely characterised by the TPBVP:

u^{⊸} = R_{a}^{−1} z^{⊸} + [ −R_{a}^{−1}D_{a}^{∗}C_{a} R_{a}^{−1}B_{a}^{∗} ][ e^{⊸} ; g^{⊸} ],  (5.37a)
\frac{d}{dθ} ([ e^{⊸} ; g^{⊸} ]) = [ \begin{matrix}(A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a}) & B_{a}R_{a}^{−1}B_{a}^{∗} \\ C_{a}^{∗}R̃_{a}^{−1}C_{a} & −(A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a})^{∗} \end{matrix} ][ e^{⊸} ; g^{⊸} ](θ)
+[ \begin{matrix}B_{a}R_{a}^{−1} \\ C_{a}^{∗}D_{a}R_{a}^{−1} \end{matrix} ] z^{⊸}(θ),  (5.37b)

with boundary conditions

e^{⊸}(0) = 0 and g^{⊸}(h) = −X_{a} e^{⊸}(h),

4Recall that under Assumption 5.8, D_{a} = D_{b}. However, for clarity of the derivations it is convenient to maintain a notational distinction between them.

where X_{a} is a self-adjoint, positive-semidefinite, stabilising solution to the algebraic, Riccati equation (5.22). Moreover, from equations (5.34) and (5.35), it follows that z^{⊸} = σ^{−1} w^{⊸}. With substitution of this into equations (5.36) and (5.37), elementary algebraic manipulation yields the following TPBVP characterisation of the z^{⊸}’s satisfying equation (5.34):

z^{⊸} = σ^{−1}[ 0 [ 0 [ B_{a}^{∗} −B_{b}^{∗} ] ] ][ r^{⊸} ; v^{⊸} ],  (5.38a)
\frac{d}{dθ} ([ r^{⊸} ; v^{⊸} ]) = [ \begin{matrix}Ǎ^{σ} & B̌^{σ}(Ř^{σ})^{−1}(B̌^{σ})^{∗} \\ (Č^{σ})^{∗}(Q̌^{σ})^{−1}Č^{σ} & −(Ǎ^{σ})^{∗} \end{matrix} ][ r^{⊸} ; v^{⊸} ](θ),  (5.38b)

with boundary conditions

r^{⊸}(0) = [ \begin{matrix}0 \\ 0 \end{matrix} ] and v^{⊸}(h) = [ \begin{matrix}−X_{a} & 0 \\ 0 & X_{∞} \end{matrix} ] r^{⊸}(h),  (5.38c)

where

r^{⊸} :=[ e^{⊸} ; a^{⊸} ],  v^{⊸} :=[ g^{⊸} ; b^{⊸} ],

Ř^{σ} :=[ \begin{matrix}σ^{2}R_{a} & 0 \\ 0 & −R_{b} \end{matrix} ],  Q̌^{σ} :=[ \begin{matrix}σ^{2}R̃_{a} & 0 \\ 0 & −R̃_{b} \end{matrix} ],

B̌^{σ} := [ \begin{matrix}σB_{a} & 0 \\ B_{a} & 0 \\ −B_{b} & B_{b} \end{matrix} ] ,  Č^{σ} := [ \begin{matrix}σC_{a} & 0 & 0 \\ 0 & C_{a} & C_{b} \end{matrix} ]

and

Ǎ^{σ} := [ \begin{matrix}A_{a} & 0 & 0 \\ 0 & A_{a} & 0 \\ 0 & 0 & A_{b} \end{matrix} ] − B̌^{σ}(Ř^{σ})^{−1}[ \begin{matrix}D_{a} & 0 \\ 0 & −D_{b} \end{matrix} ]Č^{σ}.

For z^{⊸} in equation (5.38a) to be non-zero, the TPBVP must have non-trivial solution. With

Ě^{σ}(t) :=[ \begin{matrix}Ǎ^{σ} & B̌^{σ}(Ř^{σ})^{−1}(B̌^{σ})^{∗} \\ (Č^{σ})^{∗}(Q̌^{σ})^{−1}Č^{σ} & −(Ǎ^{σ})^{∗} \end{matrix} ](t)

and ^{5}

Ě^{σ}:=ΦĚ^{σ}(h, 0),  (5.40)

5Recall that Φ_{·}(t, τ) denotes the state-transition operator associated with ·.

existence of a non-trivial solution implies

𝒦(Ě_{22}^{σ}− [ \begin{matrix}−X_{a} & 0 \\ 0 & X_{∞} \end{matrix} ]Ě_{12}^{σ}) ≠ \{0\},

where Ě^{σ} =:[ \begin{matrix}Ě_{11}^{σ} & Ě_{12}^{σ} \\ Ě_{21}^{σ} & Ě_{22}^{σ} \end{matrix} ] and the partition is conformal with that of Ě^{σ}(t). To see this, note that

[ \begin{matrix}r^{⊸}(h) \\ v^{⊸}(h) \end{matrix} ] = [ \begin{matrix}Ě_{11}^{σ} & Ě_{12}^{σ} \\ Ě_{21}^{σ} & Ě_{22}^{σ} \end{matrix} ][ \begin{matrix}r^{⊸}(0) \\ v^{⊸}(0) \end{matrix} ],

which combined with the boundary conditions (5.38c), gives

(Ě_{22}^{σ}− [ \begin{matrix}−X_{a} & 0 \\ 0 & X_{∞} \end{matrix} ]Ě_{12}^{σ}) v^{⊸}(0) = 0.

Now suppose that 𝒦(Ě_{22}^{σ}− [ \begin{matrix}−X_{a} & 0 \\ 0 & X_{∞} \end{matrix} ]Ě_{12}^{σ}) = \{0\}. It then follows that v^{⊸}(0) = 0 and hence,

since r^{⊸}(0) = 0, that the TPBVP has only the trivial solution [ r^{⊸} ; v^{⊸} ] ≡ 0. This contradicts the initial hypothesis that

0 ≠ z^{⊸} = σ^{−2}[ 0 [ 0 [ B_{a}^{∗} −B_{b}^{∗} ] ] ][ r^{⊸} ; v^{⊸} ].

Thus, if z^{⊸} is non-zero then necessarily, 𝒦(Ě_{22}^{σ}− [ \begin{matrix}−X_{a} & 0 \\ 0 & X_{∞} \end{matrix} ]Ě_{12}^{σ}) ≠ \{0\}. The latter condition

is also sufficient. That is, if

det (Ě_{22}^{σ}− [ \begin{matrix}−X_{a} & 0 \\ 0 & X_{∞} \end{matrix} ]Ě_{12}^{σ}) = 0,  (5.41)

then there exists a non-zero z^{⊸} satisfying equation (5.34) and hence, σ is a singular value of (I + YY^{∗})^{− \frac{1}{2}} (W − Y)V_{a}. Again, this can be seen by contradiction. Let equation (5.41) hold. Then there exists a v^{⊸}(0) :=[ \begin{matrix}g^{⊸}(0) \\ b^{⊸}(0) \end{matrix} ]≠ 0, such that

(Ě_{22}^{σ}− [ \begin{matrix}−X_{a} & 0 \\ 0 & X_{∞} \end{matrix} ]Ě_{12}^{σ}) [ \begin{matrix}g^{⊸}(0) \\ b^{⊸}(0) \end{matrix} ] = 0

and consequently, a corresponding non-trivial solution to equation (5.38b) with boundary conditions (5.38c). Now suppose that z^{⊸} ≡ 0. Then u^{⊸} in equation (5.35b) is identically zero and equations (5.36a) and (5.37a) (which define the TPBVP (5.38)), become

\frac{d}{dθ} ([ a^{⊸} ; b^{⊸} ]) = [ \begin{matrix}\underline{A} & −[ \begin{matrix}0 \\ B_{b} \end{matrix} ]R_{b}^{−1}[ 0 B_{b}^{∗} ] \\ −[ \begin{matrix}C_{a}^{∗} \\ C_{a}^{∗} \end{matrix} ]R̃_{b}^{−1}[ C_{a} C_{b} ] & −\underline{A}^{∗} \end{matrix} ][ a^{⊸} ; b^{⊸} ](θ), (5.42a)

\frac{d}{dθ} ([ e^{⊸} ; g^{⊸} ]) = [ \begin{matrix}(A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a}) & B_{a}R_{a}^{−1}B_{a}^{∗} \\ C_{a}^{∗}R̃_{a}^{−1}C_{a} & −(A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a})^{∗} \end{matrix} ][ e^{⊸} ; g^{⊸} ](θ),  (5.42b)

with boundary conditions

a^{⊸}(0) = 0, b^{⊸}(h) = X_{∞} a^{⊸}(h), e^{⊸}(0) = 0 and g^{⊸}(h) = −X_{a} e^{⊸}(h).  (5.42c)

Now consider the operator

[ \begin{matrix}S^{∗}S & 0 \\ 0 & B_{a}V_{a}^{2}B_{a}^{∗} \end{matrix} ] : 𝒳_{a} ⊕ 𝒳_{b} ⊕ 𝒳_{a} → 𝒳_{a} ⊕ 𝒳_{b} ⊕ 𝒳_{a},

where

S:=(I + YY^{∗})^{− \frac{1}{2}} [ \begin{matrix}C_{a} & C_{b} \\ X∞^{\frac{1}{2}}[ \begin{matrix}A_{a} & 0 \\ 0 & A_{b} \end{matrix} ] \end{matrix} ]

and A_{•}, B_{•} and C_{•} are defined in equations (5.16). It is readily verifiable that equations (5.42a) and (5.42b) with boundary conditions

a^{⊸}(0) = x_{1}, b^{⊸}(h) = X_{∞} a^{⊸}(h), e^{⊸}(0) = 0 and g^{⊸}(h) = −X_{a} e^{⊸}(h) + x_{2},  (5.43)

characterise

[ \begin{matrix}S^{∗}S & 0 \\ 0 & B_{a}V_{a}^{2}B_{a}^{∗} \end{matrix} ][ \begin{matrix}x_{1} \\ x_{2} \end{matrix} ] = [ \begin{matrix}b^{⊸}(0) \\ e^{⊸}(h) \end{matrix} ].  (5.44)

Then since the boundary conditions (5.42c) are identical to the boundary conditions (5.43) when x_{1} = 0 and x_{2} = 0, it follows by linearity and equation (5.44) that b^{⊸}(0) = 0 and e^{⊸}(h) = 0. The boundary conditions (5.42c) then imply that e^{⊸}(0) = g^{⊸}(h) = 0 and hence, using equation (5.42b), that

[ \begin{matrix}0 \\ 0 \end{matrix} ] = [ \begin{matrix}e^{⊸}(h) \\ g^{⊸}(h) \end{matrix} ] = Φ_{[ \begin{matrix}(A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a}) & B_{a}R_{a}^{−1}B_{a}^{∗} \\ C_{a}^{∗}R̃_{a}^{−1}C_{a} & −(A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a})^{∗} \end{matrix} ]}(h, 0)[ \begin{matrix}0 \\ g^{⊸}(0) \end{matrix} ].

Since the state-transition operator

Φ_{[ \begin{matrix}(A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a}) & B_{a}R_{a}^{−1}B_{a}^{∗} \\ C_{a}^{∗}R̃_{a}^{−1}C_{a} & −(A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a})^{∗} \end{matrix} ]}(t, τ)

is invertible for all t and τ, this gives g^{⊸}(0) = 0. That is,

v^{⊸}(0) = [ \begin{matrix}g^{⊸}(0) \\ b^{⊸}(0) \end{matrix} ] = 0,

which is a contradiction, since by the initial hypothesis v^{⊸}(0) ≠ 0. Hence, there must exist a non-zero z^{⊸} satisfying equation (5.34), which is the desired result. This is all summarised in the following lemma:

Lemma 5.10 A real number σ > 0 is a singular value of (I + YY^{∗})^{− \frac{1}{2}} (W − Y)V_{a} if and only if

det (Ě_{22}^{σ}− [ \begin{matrix}−X_{a} & 0 \\ 0 & X_{∞} \end{matrix} ]Ě_{12}^{σ}) = 0,  (5.45)

where Ě^{σ} =:[ \begin{matrix}Ě_{11}^{σ} & Ě_{12}^{σ} \\ Ě_{21}^{σ} & Ě_{22}^{σ} \end{matrix} ] is defined in equation (5.40).

By Lemma 5.10, it can be seen that testing the norm of (I+YY^{∗})^{− \frac{1}{2}} (W−Y)V_{a} and hence, condition (iii) of Theorem 5.6, is equivalent to testing the largest σ > 0 that satisfies condition (5.45). As a step towards testing the latter, recall that the set of singular values of a compact operator is discrete, with the only possible accumulation point being 0 [Kre89b]. Therefore, by Lemma 5.10, the zeros of

d(μ) :=det (Ě22^{\frac{1}{μ}}− [ \begin{matrix}−X_{a} & 0 \\ 0 & X_{∞} \end{matrix} ]Ě12^{\frac{1}{μ}})

must constitute a discrete set with no finite accumulation points. Furthermore, note that Ě^{\frac{1}{μ}} and hence d(μ), are analytic functions of μ. So with wno_{r}(d) defined to be the number of clockwise encirclements of the origin made by d(μ) as μ traverses the standard Nyquist contour of radius r,^{6} the following corollary of Lemma 5.10 is a simple consequence of the Principle of the Argument [MH87].^{7}

Corollary 5.11 Condition (iii) of Theorem 5.6 is satisfied, or equivalently given γ > 0

∥(I + YY^{∗})^{− \frac{1}{2}} (W − Y)V_{a}∥ < γ,

6A semi-circle (centred at the origin) into the right half plane and of radius r.

7Prof. Bassam Bamieh must be acknowledged for first suggesting the use of winding numbers in this context.

if and only if

wno_{\frac{1}{γ}} (d) = 0.

Remark 5.12 Although it is not necessary to do so when computing the directed gap, it is possible to compute ∥(I + YY^{∗})^{− \frac{1}{2}} (W − Y)V_{a}∥ to any desired accuracy using Corollary 5.11. That is, one could simply perform a bisection-search along r ∈ ℝ^{+}/0 to find the point at which wno_{r}(d) changes from 0 to a value greater than 0. ♣

Remark 5.13 As an aside, note that if X_{a} were equal to the 11-block of X_{∞}, then

∥(I + YY^{∗})^{− \frac{1}{2}} (W − Y)V_{a}∥ < γ

would be equivalent to the gap between the two finite-horizon, linear operators W and Y being less than γ. As such, the techniques developed in this subsection could be used to compute the gap between the compression of (possibly time-varying) state-space systems. ♣

Remark 5.14 Although it is not explicitly shown here, the technique developed in this chapter could be used to compute the gap between an LTI, continuous-time controller and a low-pass filtered, sampled and held (sampled-data) approximation. The formulae would change slightly because it is not possible to express a sampled and held, LSI, discrete-time system directly in terms of a system of differential equations. However, given a state-space realisation of the LSI, discretetime component of the sampled-data approximation (and of the low-pass filter usually used before the sampling device) it is possible to construct a state-space system of difference equations that characterises the evolution of the sampled-data approximation (in the time-lifted domain). Using such a state-space realisation, the results presented in Sub-Section 5.3.1 could be used to construct a normalised, right-coprime factorisation and hence, strong-right representation of the graph of the sampled-data approximation. The rest would then follow in exactly the same way. ♣

5.5 Summary

Exploiting the equivalence of testing a bound on the directed-gap between two LPTV systems and solving a particular LSI, full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem, a procedure for computing the directed gap between LPTV systems that admit stabilisable and detectable, state-space realisations, is developed in this chapter. Much of the chapter is devoted to establishing computationallytractable tests for a necessary and sufficient condition that characterises the existence of a solution to the equivalent synthesis problem, derived from the necessary and sufficient condition given in Proposition 2.22 for the generic LSI, full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem. These tests can be used in a bisection-search procedure for computing the directed gap (and hence gap) to any desired accuracy.

The equivalent full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem associated with bounding the directedgap involves a generalised plant constructed from isometric, strong-right representations of the time-lifted graphs of the two systems concerned. To this end, it is shown how to construct a statespace realisation of a strong-right representation of an LPTV system’s time-lifted graph. This involves obtaining a normalised, right-coprime factorisation of the time-lifted-equivalent system concerned. A state-space realisation of the relevant coprime factors can be expressed explicitly in terms of the system’s original, continuous-time, state-space realisation and the stabilising solution to an algebraic, Riccati equation.

H^{∞} SD Synthesis and Related Numerical Issues

6.1 Introduction

This chapter is basically an extension of Section 2.5. All of the material presented in Section 2.5 (including specific state-space realisations defined there) is used in this chapter without restatement. As such, it is suggested that the reader be familiar with that section before continuing.

Consider the control-system configuration shown in Figure 6.1, where G is an LTI, continuoustime generalised-plant, K is an LSI, discrete-time controller, A_{D} is an ideal, periodic (period h) sampling-device and D_{A} is a zeroth-order hold-device synchronised with A_{D}. The task of synthesis-

[figure: A_{D}; K; D_{A}; G; w(t); u(t); z(t); y(t); y_{k}; u_{k}]

Figure 6.1 SD Control System

ing an internally-stabilising controller K that achieves some bound on the L_{ℝ^{+}}^{2}-induced norm of the closed-loop operator F_{ℓ}(G, D_{A}KA_{D}), mapping w to z, is complicated by the mixture of continuoustime and discrete-time signals and the fact that the closed-loop is periodically time-varying. Recall however, as discussed in Section 2.3.2, that periodically-time-varying, continuous-time systems are equivalent, via the time-lifting isomorphism, to shift-invariant systems that evolve in discrete-time.

It is therefore not surprising, that the LPTV, SD synthesis problem can be restated as an equivalent LSI problem in discrete-time (cf. Section 2.5 for the details of this). Although this equivalent problem involves a mixture of finite-dimensional and infinite-dimensional input and output spaces, there exist various methods for determining a completely finite-dimensional equivalent (cf. Proposition 2.28 for example and [BP92, Toi92, CF95, KH93, Toi93]). The main purpose of this chapter is to present a new, compact derivation of state-space formulae for constructing a state-space realisation of the SD-equivalent generalised-plant defined in Proposition 2.28 (cf. also Remark 2.29). Furthermore, numerical issues are discussed and numerically robust formulae obtained.

Perhaps the most simple explicit formulae for the state-space matrices of the finite-dimensional, equivalent, discrete-time, synthesis problem are given in [Toi93] and involve computation of only a single matrix exponential. In [Toi93] a finite-dimensional, discrete-time, equivalent problem is derived without using time-lifting. Instead classical, linear-quadratic, optimal-control theory is used to give a closed-loop characterisation of the worst-case, inter-sample disturbance. The relationship between the time-lifting and the linear-quadratic, optimal control approaches is examined in [TS97]. There, it is noted that the time-lifting based approach can be interpreted as consisting of a lifting transformation followed by an open-loop characterisation of the worst-case, inter-sample disturbance. In this way the two methods were said to be equivalent, although expressions for the equivalent, finite-dimensional, discrete-time problem were different. In this chapter, using only the time-lifting technique, essentially identical (except for the accommodation of a direct feed-through term D_{11} here) state-space formulae to those obtained by [Toi93] are developed and hence, the equivalence of the two approaches follows directly. The formulation developed here involves solving a set of TPBVPs, which are used to characterise the behaviour of certain state-space operators that describe the infinite-dimensional, time-lifted, discrete-time, equivalent problem (cf. Proposition 2.24). These TPBVPs lead to state-space formulae that involve the computation of only a single, finite-dimensional matrix-exponential.

The required matrix exponentiation is of a Hamiltonian structured matrix expressible explicitly in terms of the state-space matrices of the original generalised-plant. Since the eigenvalues of a Hamiltonian matrix are symmetric about the imaginary axis, direct computation of its exponential can be dominated by large terms if any of the eigenvalues have large real part. These large terms can cancel or become small in the formulae used to determine the state-space matrices of the finite-dimensional, equivalent problem. Hence, without care the formulae initially derived can be numerically sensitive. With this in mind, it is shown here how to re-structure the state-space formulae to give a numerically robust procedure for computation.

128 H^{∞} SD Synthesis and Related Numerical Issues

Briefly, the structure of the chapter is as follows: in Section 6.2 a new, compact derivation of formulae for constructing a state-space realisation of the SD-equivalent generalised-plant defined in Proposition 2.28, is presented; in Section 6.3, it is shown how to re-structure the new formulae for numerical robustness; in Section 6.4, two assumptions required to make this possible are discussed; finally, a brief summary is given in Section 6.5.

6.2 State-Space Formulae for the SD-Equivalent Generalised-Plant

The purpose of the section is to present a new, compact derivation of formulae for computing the state-space matrices of the SD-equivalent generalised-plant Ğ, defined in Proposition 2.28 (cf. Section 2.5 of Chapter 2). The formulae are expressible explicitly in terms of the state-space matrices of the original generalised-plant G (cf. equation (2.17)) and are similar to those presented in [Toi93, TS97]. Finite-dimensional expressions for the state-space matrices of Ğ are constructed by solving a set of TPBVPs, which characterise the combination of integral operator expressions used in the state-space description of Ğ in Proposition 2.28. The formulae derived accommodate direct feedthrough from the disturbance inputs w to the controlled outputs z (that is, D_{11} ≠ 0).

It follows from equations (2.26–2.37), that in order to explicitly determine the state-space matrices of Ğ, finite-dimensional expressions are required for

γ^{−2}B_{1}(I − γ^{−2}D_{11}^{∗}D_{11})^{−1}B_{1}^{∗},  (6.1)
γ^{2}[ \begin{matrix}C_{1}^{∗} \\ D_{12}^{∗} \end{matrix} ](I − γ^{−2}D_{11}D_{11}^{∗})^{−1}[ C_{1} D_{12} ]  (6.2)

and

γ^{−2}B_{1}D_{11}^{∗}(I − γ^{−2}D_{11}D_{11}^{∗})^{−1}[ C_{1} D_{12} ],  (6.3)

which can be obtained as described below. Given γ > ∥D_{11}∥, define

Q̀:=e^{hÈ} := exp \{h[ \begin{matrix}−À^{∗} & −C̀^{∗}C̀ \\ B̀B̀^{∗} & À \end{matrix} ]\} ,  (6.4)

where

À := Ā + γ^{−2}[ \begin{matrix}B_{1} \\ 0 \end{matrix} ]R^{−1}D_{11}^{∗}[ C_{1} D_{12} ],
C̀^{∗}C̀ := [ \begin{matrix}C_{1}^{∗} \\ D_{12}^{∗} \end{matrix} ]S^{−1}[ C_{1} D_{12} ],
B̀B̀^{∗} := γ^{−2}[ \begin{matrix}B_{1} \\ 0 \end{matrix} ]R^{−1}[ B_{1}^{∗} 0 ]

Ā := [ \begin{matrix}A & B_{2} \\ 0 & 0 \end{matrix} ].  (6.5)

S :=(I − γ^{−2}D_{11}D_{11}^{∗}) and R:=(I − γ^{−2}D_{11}^{∗}D_{11}). Partition Q̀ =:[ \begin{matrix}Q̀_{11} & Q̀_{12} \\ Q̀_{21} & Q̀_{22} \end{matrix} ] conformably with È.

Theorem 6.1 The state-space matrices corresponding to the state-space realisation of the SDequivalent generalised-plant Ğ given in equation (2.35), can be characterised as follows:

Ã = [ I_{n} 0 ](Q̀_{11}^{∗})^{−1}[ \begin{matrix}I_{n} \\ 0 \end{matrix} ],  (6.6)

B̃_{2} = [ I_{n} 0 ](Q̀_{11}^{∗})^{−1}[ \begin{matrix}0 \\ I_{m_{2}} \end{matrix} ],  (6.7)

B̆_{1} a matrix satisfying

B̆_{1}B̆_{1}^{∗} = [ I_{n} 0 ]Q̀_{21}Q̀_{11}^{−1}[ \begin{matrix}I_{n} \\ 0 \end{matrix} ]  (6.8)

and C̆_{1} and D̆_{12} matrices satisfying

[ C̆_{1}^{∗} ; D̆_{12}^{∗} ][ C̆_{1} D̆_{12} ] = −γ^{2}Q̀_{11}^{−1}Q̀_{12}.  (6.9)

Proof : By Proposition 2.28, it is required to determine finite-dimensional expressions for the operators given in equations (6.1), (6.2) and (6.3). Consider the first operator

γ^{−2}B_{1}(I − γ^{−2}D_{11}^{∗}D_{11})^{−1}B_{1}^{∗} := z ↦ x,

where z ∈ ℝ^{n}. Define

y^{⊸} := γ^{−1}B_{1}^{∗}z,  (6.10a)
r^{⊸} := γ^{−1}D_{11} u^{⊸}  (6.10b)

and

u^{⊸} := (I − γ^{−2}D_{11}^{∗}D_{11})^{−1} y^{⊸},  (6.10c)
= γ^{−1}D_{11}^{∗} r^{⊸} + γ^{−1}B_{1}^{∗}z,  (6.10d)

where equation (6.10d) follows from equations (6.10b) and (6.10c). Then clearly, x = γ^{−1}B_{1} u^{⊸}. Now from equations (2.24) and (6.10b) it follows that for θ ∈ ℍ

\frac{d a^{⊸}}{dθ} = A a^{⊸}(θ) + γ^{−1}B_{1} u^{⊸}(θ); a^{⊸}(0) = 0,  (6.11a)
r^{⊸}(θ) = C_{1} a^{⊸}(θ) + γ^{−1}D_{11} u^{⊸}(θ).  (6.11b)

Furthermore, using equations (2.21) and (2.24), the final-condition response of

\frac{d e^{⊸}}{dθ} = −A^{∗} e^{⊸}(θ) − C_{1}^{∗} r^{⊸}(θ); e^{⊸}(h) = z,  (6.12a)
u^{⊸}(θ) = γ^{−1}B_{1}^{∗} e^{⊸}(θ) + γ^{−1}D_{11}^{∗} r^{⊸}(θ),  (6.12b)

gives γ^{−1}B_{1}^{∗}z and the zero, final-condition response gives γ^{−1}D_{11}^{∗} r^{⊸}, which is precisely equation (6.10d). Then from equations (2.21) and (6.11),

x = γ^{−1}B_{1} u^{⊸} = a^{⊸}(h).

Combining equations (6.11) and (6.12) yields a TPBVP. It follows from the boundary conditions, that

[ \begin{matrix}z \\ a^{⊸}(h) \end{matrix} ] = [ \begin{matrix}e^{⊸}(h) \\ a^{⊸}(h) \end{matrix} ] = Q(h)[ \begin{matrix}e^{⊸}(0) \\ 0 \end{matrix} ],  (6.13)

where Q(t) is defined in equation (2.34). Hence,^{1}

a^{⊸}(h) = Q_{21}(h)Q_{11}(h)^{−1}z.

That is, Q_{21}(h)Q_{11}(h)^{−1} constitutes a finite-dimensional expression for the first operator. Now by the structure of È it follows that

Q_{21}(h)Q_{11}(h)^{−1} = [ I_{n} 0 ]Q̀_{21}Q̀_{11}^{−1}[ \begin{matrix}I_{n} \\ 0 \end{matrix} ],

which is equation equation (6.8), as claimed.

Before continuing, a well-known result from matrix theory is required [VL78]. Let M_{11} and M_{22} be square matrices, then for t ≥ 0

exp \{t[ \begin{matrix}M_{11} & M_{12} \\ 0 & M_{22} \end{matrix} ]\} = [ \begin{matrix}e^{tM_{11}} & M̀_{12} \\ 0 & e^{tM_{22}} \end{matrix} ],  (6.14)

where M̀_{12} = ∫_{0}^{t} e^{(t−τ)M_{11}}M_{12}e^{τM_{22}} dτ. Now consider the second and third operators

γ^{2}[ \begin{matrix}C_{1}^{∗} \\ D_{12}^{∗} \end{matrix} ](I − γ^{−2}D_{11}D_{11}^{∗})^{−1}[ C_{1} D_{12} ] := z ↦ x_{1}

and

γ^{−2}B_{1}D_{11}^{∗}(I − γ^{−2}D_{11}D_{11}^{∗})^{−1}[ C_{1} D_{12} ] := z ↦ x_{2}

1Recall that by Remark 2.26, Q_{11}(h) is invertible.

respectively. Define v^{⊸} :=[ C_{1} D_{12} ]z, for some z ∈ ℝ^{n+m_{2}}. It then follows from equations (2.22), (2.25) and (6.14), that for θ ∈ ℍ

v^{⊸}(θ) = [ C_{1} D_{12} ]e^{θĀ}z,  (6.15)

where Ā is defined in equation (6.5). Now define

g^{⊸} :=(I − γ^{−2}D_{11}D_{11}^{∗})^{−1} v^{⊸},  (6.16)

so that

x_{1} = γ^{2}[ \begin{matrix}C_{1}^{∗} \\ D_{12}^{∗} \end{matrix} ] g^{⊸}  (6.17)

and

x_{2} = γ^{−2}B_{1}D_{11}^{∗} g^{⊸}.  (6.18)

With

w^{⊸}(θ) :=γ^{−1}D_{11}^{∗} g^{⊸}(θ),

it follows from equation (2.24) that for θ ∈ ℍ

\frac{dm^{⊸}}{dθ} = −Ā^{∗}m^{⊸}(θ) − [ \begin{matrix}C_{1}^{∗} \\ D_{12}^{∗} \end{matrix} ] g^{⊸}(θ); m^{⊸}(h) = 0,  (6.19a)
w^{⊸}(θ) = [ γ^{−1}B_{1}^{∗} 0 ]m^{⊸}(θ) + γ^{−1}D_{11}^{∗} g^{⊸}(θ).  (6.19b)

Furthermore, from equations (6.16) and (6.15),

g^{⊸}(θ) = v^{⊸}(θ) + γ^{−1}(D_{11} w^{⊸})(θ)
= [ C_{1} D_{12} ]e^{θĀ}z + γ^{−1}(D_{11} w^{⊸})(θ).

Again, using equation (2.24) this can be expressed as

\frac{d n^{⊸}}{dθ} = Ān^{⊸}(θ) + [ \begin{matrix}γ^{−1}B_{1} \\ 0 \end{matrix} ]w^{⊸}(θ); n^{⊸}(0) = z,  (6.20a)
g^{⊸}(θ) = [ C_{1} D_{12} ] n^{⊸}(θ) + γ^{−1}D_{11} w^{⊸}(θ),  (6.20b)

where the zero, initial-condition response gives γ^{−1}D_{11} w^{⊸} and the initial-condition response gives [ C_{1} D_{12} ]e^{θĀ}z. The systems (6.20) and (6.19) form a TPBVP, from which it follows that

[ \begin{matrix}0 \\ n^{⊸}(h) \end{matrix} ] = [ \begin{matrix}m^{⊸}(h) \\ n^{⊸}(h) \end{matrix} ] = Q̀[ \begin{matrix}m^{⊸}(0) \\ z \end{matrix} ],  (6.21)

where Q̀ is defined in equation (6.4). Now from equations (6.17), (6.18), (6.19) and (6.20),

x_{1} = γ^{2}[ \begin{matrix}C_{1}^{∗} \\ D_{12}^{∗} \end{matrix} ] g^{⊸} = γ^{2}m^{⊸}(0),  (6.22)

and

x_{2} = γ^{−2}B_{1}D_{11}^{∗} g^{⊸} = γ^{−1}B_{1} w^{⊸} = [ I_{n} 0 ]{ n^{⊸}(h) − e^{hĀ}z}.  (6.23)

Recall the partitioning of Q̀. Then from equation (6.21), m^{⊸}(0) = −Q̀_{11}^{−1}Q̀_{12}z and combining this with equation (6.22) characterises a finite-dimensional expression for the second operator. Similarly,

n^{⊸}(h) = {Q̀_{22} − Q̀_{21}Q̀_{11}^{−1}Q̀_{12}}z.  (6.24)

Define

J = [ \begin{matrix}0 & −I_{n+m_{2}} \\ I_{n+m_{2}} & 0 \end{matrix} ],  (6.25)

noting that −È^{∗} = JÈJ^{−1} (È is Hamiltonian), so that

(Q̀^{∗})^{−1} = e^{−hÈ^{∗}} = JQ̀J^{−1}.

That is, Q̀^{∗}JQ̀J^{−1} = JQ̀J^{−1}Q̀^{∗} = I, which yields the relationship

Q̀_{22} − Q̀_{21}Q̀_{11}^{−1}Q̀_{12} = (Q̀_{11}^{∗})^{−1}.

Combining this with equations (6.23) and (6.24) then gives a finite-dimensional expression for the third operator. Furthermore, from equation (2.27),

Ã = e^{hA} + [ I_{n} 0 ]\{(Q̀_{11}^{∗})^{−1} − e^{hĀ}\}[ \begin{matrix}I_{n} \\ 0 \end{matrix} ]
= [ I_{n} 0 ](Q̀_{11}^{∗})^{−1}[ \begin{matrix}I_{n} \\ 0 \end{matrix} ]

and from equation (2.29),

B̃_{2} = ∫_{0}^{h} e^{(h−t)A}dtB_{2} + [ I_{n} 0 ]\{(Q̀_{11}^{∗})^{−1} − e^{hĀ}\}[ \begin{matrix}0 \\ I_{m_{2}} \end{matrix} ]
= [ I_{n} 0 ](Q̀_{11}^{∗})^{−1}[ \begin{matrix}0 \\ I_{m_{2}} \end{matrix} ].

To see that Q̀_{11} is invertible, note that from equations (6.21) and (6.14),

Q̀_{11} = [ \begin{matrix}Q_{11}(h) & 0 \\ ? & I_{m_{2}} \end{matrix} ],

where Q_{11} is defined as before and “?” denotes an irrelevant sub-matrix. Thus, Q̀_{11} is invertible if and only if Q_{11} is, which is the case by Remark 2.26. This completes the proof.

6.3 Re-Structuring for Numerical Robustness

In this section, the state-space formulae presented in Theorem 6.1 are re-structured to avoid numerical problems that can arise when calculating the matrix exponential defined there.

As detailed in Theorem 6.1, computing a state-space realisation of Ğ involves taking the matrix exponential of the Hamiltonian-structured matrix È. Since È has Hamiltonian structure, its eigenvalues are symmetric about the imaginary axis. Therefore, if È has eigenvalues with large (with respect to the sampling frequency), positive real-part, the computation of Q̀:=e^{hÈ} can be numerically sensitive, with the calculation being dominated by large terms.^{2} These large terms cancel or become very small in equations (6.6-6.9), which characterise a state-space realisation of Ğ, making any procedure based on these equations directly, numerically sensitive on the whole. In what follows, equation (6.6-6.9) are re-structured to avoid direct computation of the full matrix exponential e^{hÈ}. The aim of the re-structuring is to restrict all matrix exponential calculations to involve only matrices with all eigenvalues having real part less than or equal to some number of order 1. By similar argument, the following re-structuring procedure should also be adopted to modify the formulae given in Lemma 2.25, which can be used to compute bounds on ∥D_{11}∥ (cf. Corollary 2.27).

First some fundamental results from matrix theory [GVL89].

Lemma 6.2 (Schur Decomposition) If A ∈ ℂ^{n×n} then there exists a unitary matrix Q ∈ ℂ^{n×n} such that

Q^{∗}AQ = T = D + N,

where D = diag(λ_{1}, …, λ_{n}) and N ∈ ℂ^{n×n} is strictly upper triangular. Furthermore, Q can be chosen so that the eigenvalues λ_{i} appear in any order along the diagonal.

Lemma 6.3 (Block Diagonalisation) Let T ∈ ℂ^{n×n} be partitioned as follows:

T = \begin{matrix}[ \begin{matrix}T_{11} & T_{12} \\ 0 & T_{22} \end{matrix} ] & \begin{matrix}p \\ q \end{matrix} \\ p & q \end{matrix} .  (6.26)

Define the linear transformation ϕ : ℂ^{p×q} → ℂ^{p×q} by

ϕ(X) = T_{11}X − XT_{22},

2For example, this situation could occur if the original generalised-plant G has stable or unstable modes that are fast relative to the sampling period.

where X ∈ ℂ^{p×q}. Then ϕ is non-singular if and only if spec(T_{11}) ∩ spec(T_{22}) = ∅. If ϕ is nonsingular and Y is defined by

Y = [ \begin{matrix}I_{p} & Z \\ 0 & I_{q} \end{matrix} ], ϕ(Z) = −T_{12},

then Y^{−1}T Y = diag(T_{11}, T_{22}).

Remark 6.4 Lemma 6.3 has a natural extension via an inductive argument to block, upper triangular matrices with finer partitioning than shown in the lemma. ♣

Let M ∈ ℂ^{2(n+m_{2})×2(n+m_{2})}, be an invertible matrix such that

È = [ \begin{matrix}M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix} ][ \begin{matrix}T_{11} & 0 \\ 0 & T_{22} \end{matrix} ][ \begin{matrix}M̃_{11} & M̃_{12} \\ M̃_{21} & M̃_{22} \end{matrix} ],  (6.27)

where M̃ :=M^{−1},

spec(T_{11}) ⊂ ℂ_{>α} :={s : Re(s) > α},
spec(T_{22}) ⊂ ℂ̄_{≤α} :={s : Re(s) ≤ α},

α ∈ [0, l/h] (l of order 1) and the matrices M and M̃ are partitioned conformably with È. That such a matrix exists follows by Lemmas 6.2 and 6.3. Decomposing the matrix È into its spectral components involves first obtaining an ordered Schur decomposition, the accuracy of which, in certain circumstances, can be improved by balancing the original matrix [WR71]. The balancing process involves applying similarity transformations to the original matrix so that corresponding rows and columns have approximately the same vector norms. As implemented in MATLAB^{3}, the similarity transformations can often be ill-conditioned, especially when there are both relatively very large and very small, non-zero elements in the original matrix [MAT92]. Since the computation of È is likely to be embedded in an iterative synthesis or norm-analysis procedure, which could give rise to very small entries that are supposed to be zero, the bad effects of using ill-conditioned balancing transformations are readily observable. As such, caution should be exercised when using such transformations. Also, È involves “squared” terms like B̀B̀^{∗} and C̀^{∗}C̀, which can accentuate differences between the size of elements in B̀ and those in C̀. It is therefore recommended that the original system G be scaled prior to forming È, so that corresponding rows and columns of the statespace matrices are of comparable size in norm. Such an algorithm (syscl) is implemented in the μ-tools^{4} tool-box of MATLAB. Having performed the Schur Decomposition the next step required to obtain the decomposition in equation (6.27) is a Block Triangularisation. The choice of the real Schur form obtained in the previous step and its partitioning determines the sensitivity of the Sylvester equations that must be solved in the diagonalisation process. There can be substantial loss of accuracy whenever the spectral sets spec(T_{ii}) are insufficiently separated [GNVL79]. However, algorithms exist which dynamically determine the partitioning to limit the loss of accuracy [BS79]. Before going on to show how this spectral decomposition can be used to restructure the state-space formulae derived in the previous section, the following technical result is required.

3Copyright Maths Works Inc.

4Copyright Musyn, 1993.

Lemma 6.5 M_{11} has full column-rank if (A, C_{1}) has no unobservable modes in ℂ_{>α}. Similarly, M̃_{11} has full row-rank if (A, B_{1}) has no uncontrollable modes in ℂ_{>α}.

Proof : From equation (6.27) it follows that

È[ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ] = [ \begin{matrix}−À^{∗} & −C̀^{∗}C̀ \\ B̀B̀^{∗} & À \end{matrix} ][ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ] = [ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ]T_{11}.

Since T_{11} has full rank and is square with all eigenvalues having real part strictly greater than α, the proof that M_{11} has full column rank if (A, C_{1}) has no unobservable modes in ℂ_{>α}, is roughly the same as the proof that the Algebraic Riccati equation

A′X + XA + XRX + Q = 0,

has stabilising solution if (A, R) is stabilisable (cf. [ZDG95, Theorem 13.6]). See Appendix D.1 for the full proof. The proof that M̃_{11} has full row rank if (A, B_{1}) has no uncontrollable modes in ℂ_{>α}, follows similarly.

In view of Lemma 6.5, the following assumptions are made:

(A1) (A, B_{1}) has no uncontrollable modes in ℂ_{>α};

(A2) (A, C_{1}) has no unobservable modes in ℂ_{>α}.

For the sake of continuity, justification of these assumptions is postponed to the next section.

With the matrix È decomposed into its ℂ_{>α} and ℂ_{≤α} spectral components, it is now possible to restructure the formulae given in Theorem 6.1, to avoid direct computation of the matrix exponential e^{hT_{11}}. Let p denote the dimension of T_{11} and r:=n + m_{2} − p. Since under assumptions (A1) and (A2) the matrix M_{11} has full column rank and M̃_{11} has full row rank, the following invertible matrices are well-defined:

[ \begin{matrix}M_{11}^{L} \\ M_{11}^{⊥} \end{matrix} ] and [ M̃_{11}^{R} M̃_{11}^{⊥} ],

136 H^{∞} SD Synthesis and Related Numerical Issues

such that

[ \begin{matrix}M_{11}^{L} \\ M_{11}^{⊥} \end{matrix} ]M_{11} = [ \begin{matrix}I_{p} \\ 0_{r,p} \end{matrix} ] and M̃_{11}[ M̃_{11}^{R} M̃_{11}^{⊥} ] = [ I_{p} 0_{p,r} ],

where 0_{i,j} denotes an i × j matrix of zeros. Now define the following matrices:

Ω_{L} := ([ \begin{matrix}e^{−hT_{11}}M_{11}^{L} \\ M_{11}^{⊥} \end{matrix} ]M_{12}e^{hT_{22}} M̃_{21} + [ \begin{matrix}I_{p} \\ 0_{r,p} \end{matrix} ]M̃_{11})^{−1} ,
Ω_{R} := (M_{11}[ I_{p} 0_{p,r} ] + M_{12}e^{hT_{22}} M̃_{21}[ M̃_{11}^{R}e^{−hT_{11}} M̃_{11}^{⊥} ])^{−1} ,
Ψ := [ M̃_{11}^{R}e^{−hT_{11}} M̃_{11}^{⊥} ]Ω_{R},  (6.28)
Φ := (M_{22}e^{hT_{22}} M̃_{21}[ M̃_{11}^{R}e^{−hT_{11}} M̃_{11}^{⊥} ] + M_{21}[ I_{p} 0_{p,r} ]) Ω_{R}  (6.29)

and

Υ :=Ω_{L} ([ \begin{matrix}e^{−hT_{11}}M_{11}^{L} \\ M_{11}^{⊥} \end{matrix} ]M_{12}e^{hT_{22}} M̃_{22} + [ \begin{matrix}I_{p} \\ 0_{r,p} \end{matrix} ]M̃_{12}) .  (6.30)

Note that none of the matrices above involve the exponential of a matrix with eigenvalues having real part greater than l, a number of order 1. Theorem 6.6 The state-space matrices of the SD-equivalent generalised-plant Ğ are given by:

Ã = [ I_{n} 0 ]Ψ^{∗}[ \begin{matrix}I_{n} \\ 0 \end{matrix} ],

B̃_{2} = [ I_{n} 0 ]Ψ^{∗}[ \begin{matrix}0 \\ I_{m_{2}} \end{matrix} ],

B̆_{1} a matrix satisfying B̆_{1}B̆_{1}^{∗} = [ I_{n} 0 ]Φ[ \begin{matrix}I_{n} \\ 0 \end{matrix} ] and

C̆_{1} and D̆_{12} matrices satisfying [ C̆_{1}^{∗} ; D̆_{12}^{∗} ][ C̆_{1} D̆_{12} ] = −γ^{2}Υ.

Proof : Note from equation (6.27) that

Q̀_{11}^{−1} (M_{11}e^{hT_{11}} M̃_{11} + M_{12}e^{hT_{22}} M̃_{21}) = I.

Post-multiplying both sides by [ M̃_{11}^{R}e^{−hT_{11}} M̃_{11}^{⊥} ] gives

Q̀_{11}^{−1}Ω_{R}^{−1} = [ M̃_{11}^{R}e^{−hT_{11}} M̃_{11}^{⊥} ],

and hence from equation (6.28), Ψ = Q̀_{11}^{−1}.

Let Φ = Q̀_{21}Q̀_{11}^{−1}, and note from equation (6.27) that

Φ (M_{11}e^{hT_{11}} M̃_{11} + M_{12}e^{hT_{22}} M̃_{21}) = (M_{21}e^{hT_{11}} M̃_{11} + M_{22}e^{hT_{22}} M̃_{21}) .

Post-multiplying both sides by [ M̃_{11}^{R}e^{−hT_{11}} M̃_{11}^{⊥} ] gives

ΦΩ_{R}^{−1} = (M_{22}e^{hT_{22}} M̃_{21}[ M̃_{11}^{R}e^{−hT_{11}} M̃_{11}^{⊥} ] + M_{21}[ I_{p} 0_{p,r} ]) ,

which is equation (6.29).

Similarly, let Υ = Q̀_{11}^{−1}Q̀_{12}, and note from equation (6.27) that

(M_{11}e^{hT_{11}} M̃_{11} + M_{12}e^{hT_{22}} M̃_{21}) Υ = (M_{11}e^{hT_{11}} M̃_{12} + M_{12}e^{hT_{22}} M̃_{22}) .

Pre-multiplying both sides by [ \begin{matrix}e^{−hT_{11}}M_{11}^{L} \\ M_{11}^{⊥} \end{matrix} ] gives

Ω_{L}^{−1}Υ = ([ \begin{matrix}e^{−hT_{11}}M_{11}^{L} \\ M_{11}^{⊥} \end{matrix} ]M_{12}e^{hT_{22}} M̃_{22} + [ \begin{matrix}I_{p} \\ 0_{r,p} \end{matrix} ]M̃_{12}) ,

which is equation (6.30).

6.4 Comments Concerning Assumptions (A1) and (A2)

Under assumptions (A1) and (A2), the matrices M_{11} and M̃_{11} have full column and full row rank respectively. This is required to enable re-structuring of the formulae given in Theorem 6.1. In this section, it is argued that the assumptions (A1) and (A2) are reasonable.

There are two cases in which the assumptions can be violated. The first case is when any unstable mode, which is unobservable (and/or uncontrollable) from the controlled outputs (disturbance inputs), has a time-constant greater than or equal to the sampling period divided by some number l, of order 1. The second case is when any unstable mode, which is unobservable (and/or uncontrollable) from the controlled outputs (disturbance inputs), has time constant less than the sampling period divided by some number l, of order 1.

In the first case, it is sufficient to simply choose α to be larger than the real part of the unobservable (and/or uncontrollable) mode. In this case such an α will be less than or equal to l/h and therefore, lie in the required interval [0, l/h]. Remark 6.7 With this in mind, a suitable choice for α is the smallest number in [0, l/h], such that (A, C_{1}) and (A, B_{1}) have no unobservable and uncontrollable modes respectively, in the set {s : α < Re(s) ≤ l/h}, where l is a number of order 1. If this cannot be determined exactly then setting α = l/h will suffice. ♣ The second case is more interesting and is now illustrated with a numerical example. Consider a problem for which the original generalised-plant has state-space realisation

G = [ \begin{matrix}A & B_{1} & B_{2} \\ C_{1} & 0 & 0 \\ C_{2} & 0 & 0 \end{matrix} ],

with

A = [ \begin{matrix}−2 & 0 \\ 0 & 50 \end{matrix} ], B_{1} = [ \begin{matrix}1 \\ 1 \end{matrix} ], B_{2} = [ \begin{matrix}1 \\ 1 \end{matrix} ], C_{1} = [ 1 0 ] and C_{2} = [ 1 1 ]

and the sampling period is set to h = 1 sec. The generalised plant G has an unstable mode at 50 which is unobservable from the controlled output. Note that the unstable mode is fast (time constant 0.02 sec) relative to the sampling period (1 sec). In this case

È = [ \begin{matrix}2 & 0 & 0 & −1 & 0 & 0 \\ 0 & −50 & 0 & 0 & 0 & 0 \\ −1 & −1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & −2 & 0 & 1 \\ 0 & 0 & 0 & 0 & 50 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ] ,

which has eigenvalues 0, −1.7321, 1.7321, −50, 50, and 0, two of which are faster than the sampling frequency. For simplicity let l = 1. Since in this example the real part of the unobservable mode is greater than the sampling frequency, it is not possible to select α (which must lie in [0, l/h]) to lie to the right of it. Thus (A1) is violated for all possible choices of α in the sensible interval [0, l/h]. However, it is suggested that from a synthesis point of view, in this case it is the structure of the original problem that is not sensible. The unstable mode is excitable by the disturbance, yet this is not observed from the controlled output and hence, is not “observed” in the optimisation problem formulated to bound the norm of the controlled output for worst case disturbances. Since this unstable mode has a time constant of 0.02 seconds, in one sampling period it can increase by a large amount (e^{50} = 5.18E + 21). It is reasonable then to assume that this will require a large control action in the next sampling period, which if not penalised at the controlled output (that is D_{12} = 0) would be admissible. Furthermore, if penalised at the controlled output (that is D_{12} full column rank) this would result in very poor performance. Additionally, one would intuitively expect that it be very difficult to stabilise a system with such a fast unstable pole using such a slow sampling frequency. Similar arguments hold for when (A, B_{1}) has an uncontrollable mode that is faster than the sampling frequency.

In view of the cases considered, it is claimed that the assumptions (A1) and (A2) are reasonable in the context of H^{∞} SD-synthesis.

6.5 Summary

In this chapter a new, compact derivation of formulae for H^{∞}, SD synthesis is presented. It is shown that these formulae can be re-structured to give a numerically-robust procedure for computing a state-space realisation of the SD-equivalent generalised-plant defined in Proposition 2.28.

The formulae derived are essentially identical (except for the accommodation of a direct feedthrough term, D_{11}, here) to those derived in [Toi93] using classical, linear-quadratic, optimal-control theory. This indicates directly that the time-lifting approach and the classical, linear-quadratic, optimal-control-theoretic approach are equivalent, as was also pointed out in [TS97]. The formulae are simple, requiring computation of only a single matrix exponential. It is the simplicity of the formulae obtained that facilitates the re-structuring summarised in Theorem 6.6, for numerical robustness.

Concluding Remarks

The original motivation for this dissertation stems from the wide-spread use of digital hardware to implement control strategies for continuous-time processes. This invariably leads to closed-loops that exhibit time-varying behaviour, which is often periodic in nature. Although the mathematical analysis is more involved, the periodic structure of the time-variation exhibited by systems studied in this dissertation can be exploited in a manner similar to time-invariance, yielding powerful results analogous to many known to hold for LTI systems. The main contributions of this work and directions for future research are given below.

7.1 Contributions

• A new notion of frequency response is developed for LPTV systems. It is defined in terms of the average power of the asymptotic response to sinusoidal inputs of a single frequency and it can be characterised in terms of the singular values of a frequency-dependent, finitedimensional matrix, called the Average-Power Gain Matrix. The performance-indicating properties of this new notion of frequency response are quite intuitive and it can be used to derive bounds on performance-indicating, closed-loop operators, which facilitate the design of parametric weights employed in a new H^{∞}-loopshaping based design procedure proposed for SD, control-system development.

• It is shown that the graph of a stabilisable, LPTV system can be expressed as the range (respectively kernel) of a stable, LPTV system that has a stable, LPTV left-inverse (respectively right-inverse). This representation is only unique up to right (respectively left) multiplication by stably-invertible factors and is called a strong-right (respectively strong-left) representation. It resembles the right (respectively left), coprime-factor representations known to exist for certain classes of LTI systems.

• Given a well-posed, LPTV plant/controller pair, a useful characterisation of closed-loop stability is obtained in terms of strong-right and strong-left representations the plant and controller graphs. In turn, this leads to a Youla-style parameterisation of stabilising controllers.

• Quantitative, robust-stability results are obtained for LPTV, closed-loop systems in terms of the gap metric. To this end, a formula, expressible as an optimisation problem over H^{∞}, is derived for the directed gap between two LPTV systems. Given a nominal, LPTV plant and a stabilising, LPTV controller, the largest gap-ball of systems centred at the nominal plant, which the controller can be guaranteed to stabilise, is identified. Further to this, the maximally-tolerable, simultaneous gap-perturbation to both the nominal plant and controller is characterised. Importantly, all of these results apply to the special case of LTI systems under periodic, SD control.

• Qualitatively, it is shown that the topology induced by the gap metric on a general class of LPTV systems, is the weakest with respect to which closed-loop stability is a robust property and closed-loop performance varies continuously. That is, the gap metric on LPTV systems induces the graph topology.

• A numerical procedure involving only solutions to finite-dimensional, algebraic, Riccati equations and evaluation of state-transition matrices at only single points in time, is developed for computing (to any desired accuracy) the directed gap (and hence gap) between LPTV, continuous-time systems that admit stabilisable and detectable, state-space realisations.

• A new, compact derivation of state-space formulae is presented for H^{∞}, SD synthesis. It is shown that these formulae can be easily restructured for numerical robustness.

7.2 Directions for Future Research

• Several case studies, in which the new procedure proposed for SD, control-system development is to be employed, are required to gain explicit insight into the design issues involved with the development of such control systems.

• Computation of the gap between LPTV, state-space systems involves calculating the statetransition operator associated with an “almost” Hamiltonian-structured, time-periodic matrix at one point. It would be useful to determine if a restructuring for numerical robustness of the relevant formulae is necessary or indeed possible.

• In Section 2.4.3, a state-space solution to the LSI, full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem is given for state-space systems with possibly infinite-dimensional input and output spaces, but importantly, a finite-dimensional state-space. As seen throughout this work, such state-space systems frequently appear in the study of LPTV, continuous-time systems. An interesting direction of research would be to seek a sensible handle on causality for these equivalent, timelifted (discrete-time), state-space representations, by which it would be possible to use the solution to the LSI, full-information, ℓ_{ℤ^{+}}^{2}(·)-synthesis problem given here and the corresponding H_{𝔻}^{∞}-filtering result, to obtain a state-space solution to the LPTV, output-feedback, L_{ℝ^{+}}^{2}(·)- synthesis problem, involving only finite-dimensional, algebraic, Riccati equations. Such a solution would be easier to work with than a solution involving Riccati, differential equalities.

• Another useful direction for future research would be to develop a ν-gap (cf. [Vin93a]) type metric for LPTV systems, with the same qualitative properties as the gap metric, but by which sharper quantitative results known to hold for LTI systems, could perhaps be obtained for LPTV systems.

A

Appendix to Chapter 2

A.1 Proof of Proposition 2.20

Let F:=(R + B^{∗}XB)^{−1}(S + B^{∗}XA), then a useful identity used in what follows is

A − BF = A − B(R + B^{∗}XB)^{−1}(S + B^{∗}XA)= A_{1} − B(R + B^{∗}XB)^{−1}B^{∗}XA_{1}.  (A.1)

Proof : First note that stabilisability of (A, B) is necessary by definition. Now, since

0 ≤ [ \begin{matrix}I & −S^{∗}R^{−1} \\ 0 & I \end{matrix} ][ \begin{matrix}Q & S^{∗} \\ S & R \end{matrix} ][ \begin{matrix}I & 0 \\ −R^{−1}S & I \end{matrix} ] = [ \begin{matrix}Q − S^{∗}R^{−1}S & 0 \\ 0 & R \end{matrix} ],

it follows that Q − S^{∗}R^{−1}S ≥ 0 and hence, that X̃= 0 is a solution to the Riccati inequality

−X̃+ A^{∗}X̃A + Q − (S + B^{∗}X̃A)^{∗}(R + B^{∗}X̃B)^{−1}(S + B^{∗}X̃A) ≥ 0,

such that (R + BX̃B) > 0. It then follows by [LR95, Theorem 13.11], that under the assumption (A, B) is stabilisable, there exists a self-adjoint solution X ≥ 0 to the algebraic, Riccati equation (2.7) such that spec(A − BF) ⊂ \overline{𝔻} (the closed unit disc). So stabilisability of (A, B) is almost sufficient in that it guarantees existence of a self-adjoint, positive-semidefinite solution, but leaves open the possibility spec(A − BF) ∩ 𝕋 ≠ ∅. In what follows, it is shown that given a self-adjoint, positive-semidefinite solution to equation (2.7), spec(A − BF) ⊂ 𝔻 if and only if (Q_{1}, A_{1}) has no unobservable modes on 𝕋, by which the result claimed holds.

First note that equation (2.7) can be re-written as

X − (A − BF)^{∗}X(A − BF) = Q − S^{∗}R^{−1}S + (F − R^{−1}S)^{∗}R(F − R^{−1}S).  (A.2)

Now let there be no unobservable modes of (Q_{1}, A_{1}) on 𝕋 and suppose there exists a λ ∈ 𝕋 and corresponding x ∈ 𝒳, such that (A − BF)x = λx. Then from equation (A.2), it follows that

(1 − |λ|^{2})⟨x, Xx⟩_{𝒳} = ⟨x, (Q − S^{∗}R^{−1}S + (F − R^{−1}S)^{∗}R(F − R^{−1}S))x ⟩_{𝒳} = 0.

Equivalently,

⟨x, (Q − S^{∗}R^{−1}S)x⟩_{𝒳} = −⟨(F − R^{−1}S)x, R(F − R^{−1}S)x⟩_{𝒳} ,

which since Q_{1} = (Q − S^{∗}R^{−1}S) ≥ 0 and R > 0, implies that ⟨(F − R^{−1}S)x, R(F − R^{−1}S)x⟩_{𝒳} = 0 and that

⟨x, Q_{1}x⟩_{𝒳} = 0.  (A.3)

Therefore, (F − R^{−1}S)x = 0 and consequently, Fx = R^{−1}Sx and

(A − BF)x = (A − BR^{−1}S)x =: A_{1}x = λx.  (A.4)

Furthermore, it follows from equation (A.3), that x ∈ (ℛ_{Q_{1}})^{⊥} = 𝒦_{(Q_{1})^{∗}=Q_{1}} . Combined with equation (A.4), this implies that 𝒦_{(λI−A_{1})} ∩ 𝒦_{(Q_{1})} ≠ {0}, which is a contradiction to the initial hypothesis. So if (Q_{1}, A_{1}) has no unobservable modes on 𝕋, then spec(A−BF) ⊂ 𝔻. It now remains to show that the converse also holds.

Let spec(A − BF) ⊂ 𝔻 and suppose there exists an x ∈ 𝒳 and λ ∈ 𝕋 such that A_{1}x = λx and Q_{1}x = 0 (that is, λ corresponds to an unobservable mode of (Q_{1}, A_{1})). Now, recall from Proposition 2.18, that X is also a solution to

X = A_{1}^{∗}XA_{1}^{∗} + Q_{1} − A_{1}^{∗}XB(R + B^{∗}XB)^{−1}B^{∗}XA_{1}

and hence that,

0 = \frac{(1 − |λ|^{2})}{|λ|^{2}}⟨x, Xx⟩_{𝒳} = ⟨B^{∗}Xx, (R + B^{∗}XB)^{−1}B^{∗}Xx⟩_{𝒳} .

Since R + B^{∗}XB> 0 (which follows from R > 0 and X ≥ 0), it follows that B^{∗}Xx = 0. Using this and the identity in equation (A.1), yields (A − BF)x = A_{1}x − B(R + B^{∗}XB)^{−1}B^{∗}XA_{1}x = λx, which contradicts spec(A − BF) ⊂ 𝔻. Hence, if spec(A − BF) ⊂ 𝔻, then the pair (Q_{1}, A_{1}) necessarily has no unobservable modes on 𝕋.

A.2 Proof of Proposition 2.22

Sufficiency: In the ensuing analysis it is convenient to define

ϕ_{k} :=[ \begin{matrix}w_{k} \\ u_{k} \end{matrix} ] and ψ_{k} :=[ \begin{matrix}z_{k} \\ w_{k} \end{matrix} ],  (A.5)

so that

[ \begin{matrix}x_{k+1} \\ ψ_{k} \end{matrix} ] = [ \begin{matrix}A & B \\ C̃ & D̃ \end{matrix} ][ \begin{matrix}x_{k} \\ ϕ_{k} \end{matrix} ].

Now consider the following algebraic identity:

⟨z_{k}, z_{k}⟩_{𝒵} − γ^{2}⟨w_{k}, w_{k}⟩_{𝒲} + ⟨x_{k+1}, X_{∞}x_{k+1}⟩_{𝒳}
= ⟨[ \begin{matrix}x_{k+1} \\ ψ_{k} \end{matrix} ], [ \begin{matrix}X_{∞} & 0 \\ 0 & J \end{matrix} ][ \begin{matrix}x_{k+1} \\ ψ_{k} \end{matrix} ]⟩_{𝒳⊕𝒵⊕𝒲}
= ⟨[ \begin{matrix}x_{k} \\ ϕ_{k} \end{matrix} ], [ \begin{matrix}A & B \\ C̃ & D̃ \end{matrix} ]^{∗}[ \begin{matrix}X_{∞} & 0 \\ 0 & J \end{matrix} ][ \begin{matrix}A & B \\ C̃ & D̃ \end{matrix} ][ \begin{matrix}x_{k} \\ ϕ_{k} \end{matrix} ]⟩_{𝒳⊕𝒲⊕𝒰}
= ⟨[ \begin{matrix}x_{k} \\ ϕ_{k} \end{matrix} ], [ \begin{matrix}A^{∗}X_{∞}A + C̃^{∗}JC̃ & L^{∗} \\ L & Z \end{matrix} ][ \begin{matrix}x_{k} \\ ϕ_{k} \end{matrix} ]⟩_{𝒳⊕𝒲⊕𝒰} ,  (A.6)

where for the moment X_{∞} is an arbitrary, self-adjoint, positive-semidefinite operator, J is defined at the top of Proposition 2.22 and the operators Z =:[ \begin{matrix}Z_{1} & Z_{2}^{∗} \\ Z_{2} & Z_{3} \end{matrix} ] and L =:[ \begin{matrix}L_{1} \\ L_{2} \end{matrix} ] (partitioned conformably with ϕ) are defined in equations (2.14) and (2.15) respectively.

Suppose that X_{∞} ≥ 0 is a solution to the algebraic, Riccati equation (2.13) such that ∇:=Z_{1} − Z_{2}^{∗}Z_{3}^{−1}Z_{2} < 0 and spec(A−BZ^{−1}L) ⊂ 𝔻. Now note that Z_{3} = D_{2}^{∗}D_{2} +B_{2}^{∗}X_{∞}B_{2} and since D_{2}^{∗}D_{2} > 0 and X_{∞} ≥ 0, that Z_{3} > 0. Therefore, (cf. Remark 2.19) Z_{3} is boundedly invertible and it is possible to write

[ \begin{matrix}Z_{1} & Z_{2}^{∗} \\ Z_{2} & Z_{3} \end{matrix} ] = [ \begin{matrix}I & Z_{2}^{∗}Z_{3}^{−1} \\ 0 & I \end{matrix} ][ \begin{matrix}∇ & 0 \\ 0 & Z_{3} \end{matrix} ][ \begin{matrix}I & 0 \\ Z_{3}^{−1}Z_{2} & I \end{matrix} ].  (A.7)

With ∇ < 0 (which follows by the original hypothesis), ∇ is boundedly invertible (cf. Remark 2.19) and it follows from equation (A.7), that Z is too. A Schur decomposition of the operator on the right-hand side of (A.6) is therefore possible, giving

⟨z_{k}, z_{k}⟩_{𝒵} − γ^{2}⟨w_{k}, w_{k}⟩_{𝒲} + ⟨x_{k+1}, X_{∞}x_{k+1}⟩_{𝒳}

= ⟨[ \begin{matrix}I & 0 \\ Z^{−1}L & I \end{matrix} ][ \begin{matrix}x_{k} \\ ϕ_{k} \end{matrix} ], [ \begin{matrix}A^{∗}X_{∞}A + C̃^{∗}JC̃ − L^{∗}Z^{−1}L & 0 \\ 0 & Z \end{matrix} ][ \begin{matrix}I & 0 \\ Z^{−1}L & I \end{matrix} ][ \begin{matrix}x_{k} \\ ϕ_{k} \end{matrix} ]⟩_{𝒳⊕𝒲⊕𝒰} .

Defining

[ \begin{matrix}w_{k}^{⋆} \\ u_{k}^{⋆} \end{matrix} ]:= − [ \begin{matrix}∇^{−1}L_{∇} & 0 \\ Z_{3}^{−1}L_{2} & Z_{3}^{−1}Z_{2} \end{matrix} ][ \begin{matrix}x_{k} \\ w_{k} \end{matrix} ],

where L_{∇} = L_{1} − Z_{2}^{∗}Z_{3}^{−1}Z_{2}, it follows that since X_{∞} ≥ 0 is a solution to the algebraic, Riccati equation (2.13),

⟨z_{k}, z_{k}⟩_{𝒵} − γ^{2}⟨w_{k}, w_{k}⟩_{𝒲} + ⟨x_{k+1}, X_{∞}x_{k+1}⟩_{𝒳}
= ⟨x_{k}, X_{∞}x_{k}⟩_{𝒳} + ⟨(w_{k} − w_{k}^{⋆}), ∇(w_{k} − w_{k}^{⋆})⟩_{𝒲} + ⟨(u_{k} − u_{k}^{⋆}), Z_{3}(u_{k} − u_{k}^{⋆}) ⟩_{𝒰} .  (A.8)

Note, that u^{⋆} has the form of a causal, LSI, full-information control-law. Moreover, if u = u^{⋆} is a stabilising control-law, then by definition lim_{k→∞} x_{k} = 0 and consequently,

∑_{k=0}^{∞} {⟨x_{k}, X_{∞}x_{k}⟩_{𝒳} − ⟨x_{k+1}, X_{∞}x_{k+1}⟩_{𝒳} } = ⟨x̃, X_{∞}x̃⟩_{𝒳} .

Combining this with equation (A.8), leads to

∥z∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}^{2} − γ^{2}∥w∥_{ℓ_{ℤ^{+}}^{2}(𝒲)}^{2} ≤ ⟨x̃, X_{∞}x̃⟩_{𝒳} − α∥w − w^{⋆}∥_{ℓ_{ℤ^{+}}^{2}(𝒲)}^{2}
( = −α∥w − w^{⋆}∥_{ℓ_{ℤ^{+}}^{2}(𝒲)}^{2} when x̃ = 0),  (A.9)

where α > 0 is any real number such that ∇ ≤ αI. So it remains to show that u = u^{⋆} is indeed stabilising and that there exist an ϵ > 0 such that α∥w − w^{⋆}∥_{ℓ_{ℤ^{+}}^{2}(𝒲)}^{2} ≥ ϵ∥w∥_{ℓ_{ℤ^{+}}^{2}(𝒲)}^{2} . That this is true, follows by the initial hypothesis that spec(A − BZ^{−1}L) ⊂ 𝔻. To see this, first note that with u = u^{⋆},

[ \begin{matrix}x_{k+1} \\ z_{k} \end{matrix} ] = [ \begin{matrix}A − B_{2}Z_{3}^{−1}L_{2} & B_{1} − B_{2}Z_{3}^{−1}Z_{2} \\ C − D_{2}Z_{3}^{−1}L_{2} & D_{1} − D_{2}Z_{3}^{−1}Z_{2} \end{matrix} ][ \begin{matrix}x_{k} \\ w_{k} \end{matrix} ]

and hence, that u = u^{⋆} is stabilising if spec(A − B_{2}Z_{3}^{−1}L_{2}) ⊂ 𝔻. Now note that

[ \begin{matrix}A & B_{2} \\ C & D_{2} \end{matrix} ]^{∗}[ \begin{matrix}X_{∞} & 0 \\ 0 & I \end{matrix} ][ \begin{matrix}A & B_{2} \\ C_{1} & D_{2} \end{matrix} ] = [ \begin{matrix}X_{∞} + L_{∇}^{∗}∇^{−1}L_{∇} & 0 \\ 0 & 0 \end{matrix} ] + [ \begin{matrix}L_{2}^{∗} \\ Z_{3} \end{matrix} ]Z_{3}^{−1}[ L_{2} Z_{3} ].

Multiplying this on the right by [ \begin{matrix}I \\ −Z_{3}^{−1}L_{2} \end{matrix} ] and on the left by [ I −L_{2}^{∗}Z_{3}^{−1} ], gives

X_{∞} = (A − B_{2}Z_{3}^{−1}L_{2})^{∗}X_{∞}(A − B_{2}Z_{3}^{−1}L_{2}) + (C − D_{2}Z_{3}^{−1}L_{2})^{∗}(C − D_{2}Z_{3}^{−1}L_{2}) − L_{∇}^{∗}∇^{−1}L_{∇}.

Recalling that ∇ < 0 (by the initial hypothesis), it follows that ((C − D_{2}Z_{3}^{−1}L_{2})^{∗}(C − D_{2}Z_{3}^{−1}L_{2}) − L_{∇}^{∗}∇^{−1}L_{∇}) > 0 and hence, that any eigenvalue λ of (A − B_{2}Z_{3}^{−1}L_{2}) and corresponding eigenvector x ∈ 𝒳, satisfy

(1 − |λ|^{2})⟨x, X_{∞}x⟩_{𝒳} = ⟨x, ((C − D_{2}Z_{3}^{−1}L_{2})^{∗}(C − D_{2}Z_{3}^{−1}L_{2}) − L_{∇}^{∗}∇^{−1}L_{∇})x⟩_{𝒳} ≥ 0. (A.10)

Since X_{∞} ≥ 0 this implies that spec(A − B_{2}Z_{3}^{−1}L_{2}) ⊂ \overline{𝔻}. In fact, this can be strengthened to spec(A−B_{2}Z_{3}^{−1}L_{2}) ⊂ 𝔻 if (L_{∇}, A−B_{2}Z_{3}^{−1}L_{2}) has no unobservable modes on 𝕋. To see this, suppose that under the additional hypothesis that (L_{∇}, A−B_{2}Z_{3}^{−1}L_{2}) has no unobservable modes on 𝕋, there exist an eigenvalue of (A−B_{2}Z_{3}^{−1}L_{2}) on the unit circle and denote the corresponding eigenvector by x ∈ 𝒳. Then from equation (A.10), ⟨L_{∇}x, ∇^{−1}L_{∇}x⟩_{𝒳} = ⟨(C − D_{2}Z_{3}^{−1}L_{2})x, (C − D_{2}Z_{3}^{−1}L_{2})x⟩_{𝒵} ≥ 0 and hence, since ∇^{−1} < 0, it follows that L_{∇}x = 0, contradicting the additional hypothesis. So in terms of showing u = u^{⋆} is stabilising, it is sufficient to show that spec(A − BZ^{−1}L) ⊂ 𝔻 implies (L_{∇}, A − B_{2}Z_{3}^{−1}L_{2}) has no unobservable modes on 𝕋. This is indeed the case, as can be seen by noting that

A − BZ^{−1}L = (A − B_{2}Z_{3}^{−1}L_{2}) − (B_{1} − B_{2}Z_{3}^{−1}Z_{2})∇^{−1}L_{∇}  (A.11)

and supposing there exists a |λ| = 1 and x ∈ 𝒳 such that (A − B_{2}Z_{3}^{−1}L_{2})x = λx and L_{∇}x = 0. Then using equation (A.11), it follows that (A−BZ^{−1}L)x = λx−0 = λx, contradicting the original hypothesis spec(A − BZ^{−1}L) ⊂ 𝔻.

To complete the sufficiency proof it remains to show that there exist an ϵ > 0 such that α∥w − w^{⋆}∥_{ℓ_{ℤ^{+}}^{2}(𝒲)}^{2} ≥ ϵ∥w∥_{ℓ_{ℤ^{+}}^{2}(𝒲)}^{2} . Let W : 𝒲 → 𝒲 denote the operator mapping w to (w − w^{⋆}) and note that it has state-space realisation

[ \begin{matrix}x_{k+1} \\ w_{k} − w_{k}^{⋆} \end{matrix} ] = [ \begin{matrix}A − B_{2}Z_{3}^{−1}L_{2} & B_{1} − B_{2}Z_{3}^{−1}Z_{2} \\ ∇^{−1}L_{∇} & I \end{matrix} ][ \begin{matrix}x_{k} \\ w_{k} \end{matrix} ].

Since spec(A − B_{2}Z_{3}^{−1}L_{2}) ⊂ 𝔻, W is a bounded operator. In fact it has bounded inverse, since spec(A − BZ^{−1}L) ⊂ 𝔻 and using equation (A.11), a state-space realisation for W^{−1} is

[ \begin{matrix}s_{k+1} \\ w_{k} \end{matrix} ] = [ \begin{matrix}A − BZ^{−1}L & B_{1} − B_{2}Z_{3}^{−1}Z_{2} \\ −∇^{−1}L_{∇} & I \end{matrix} ][ \begin{matrix}s_{k} \\ w_{k} − w_{k}^{⋆} \end{matrix} ].

It follows that \frac{α}{∥W^{−1}∥^{2}}∥w∥_{ℓ_{ℤ^{+}}^{2}(𝒲)} ≤ α∥w−w^{⋆}∥_{ℓ_{ℤ^{+}}^{2}(𝒲)} and therefore, that a suitable ϵ is ϵ =\frac{α}{∥W^{−1}∥^{2}} > 0 (cf. equation (A.9)). This completes the proof of sufficiency.

Necessity: The approach taken by Green and Limebeer [GL95, Appendix B] to prove necessity is based on the solution of the finite-horizon problem with appropriate terminal-state penalty. That is, to define a causal, LSI control-law u = K[ \begin{matrix}x \\ w \end{matrix} ] over the finite-horizon [0, N], such that given γ > 0 and X_{tsp} ≥ 0,

∥z∥_{ℓ_{[0,N]}^{2}(𝒵)}^{2} − γ^{2}∥w∥_{ℓ_{[0,N]}^{2}(𝒲)}^{2} + ⟨x_{N+1}, X_{tsp}x_{N+1}⟩_{𝒳} ≤ −ϵ∥w∥_{ℓ_{[0,N]}^{2}(𝒲)}^{2}  (A.12)

for all w ∈ ℓ_{[0,N]}^{2}(𝒲) and some ϵ > 0, when x̃ = 0. It can be shown [GL95, Theorem B.2.1], that such a control-law exists if and only if

Zk,3^{{N+1,X_{tsp}}} > 0, k = 0, …, N
∇k^{{N+1,X_{tsp}}} ≤ −αI, k = 0, …, N

for some α > 0, where

Zk^{\{N+1,X_{tsp}\}} := D̃^{∗}JD̃ + B^{∗}Xk+1^{\{N+1,X_{tsp}\}} B =: [ \begin{matrix}Zk,1^{\{N+1,X_{tsp}\}} & (Zk,2^{\{N+1,X_{tsp}\}})^{∗} \\ Zk,2^{\{N+1,X_{tsp}\}} & Zk,3^{\{N+1,X_{tsp}\}} \end{matrix} ],
Lk^{\{N+1,X_{tsp}\}} := D̃^{∗}JC̃ + B^{∗}Xk+1^{\{N+1,X_{tsp}\}} A =: [ \begin{matrix}Lk,1^{\{N+1,X_{tsp}\}} \\ L2,k^{\{N+1,X_{tsp}\}} \end{matrix} ],
∇k^{{N+1,X_{tsp}}} := Zk,1^{{N+1,X_{tsp}}} − (Zk,2^{{N+1,X_{tsp}}})^{∗}(Zk,3^{{N+1,X_{tsp}}})^{−1}Zk,2^{{N+1,X_{tsp}}}

and Xk^{{N+1,X_{tsp}}} satisfies the Riccati difference equation

Xk^{{N+1,X_{tsp}}} = A^{∗}Xk+1^{{N+1,X_{tsp}}} A + C̃^{∗}JC̃ − (Lk^{{N+1,X_{tsp}}})^{∗}(Zk^{{N+1,X_{tsp}}})^{−1}Lk^{{N+1,X_{tsp}}} , (A.13)
XN+1^{{N+1,X_{tsp}}} = X_{tsp}.

In this case, Xk^{{N+1,X_{tsp}}} ≥ 0 for k = 0, …, N + 1.

Now assume that (A, B_{2}) is stabilisable and

𝒦[ \begin{matrix}A − e^{jθ}I & B_{2} \\ C & D_{2} \end{matrix} ] = \{0\}

for all θ ∈ [0, 2π) and suppose that a causal, LSI, stabilising, full-information control-law exists such that the infinite-horizon objective (2.12) is achieved. Then for all w ∈ ℓ_{[0,N]}^{2}(𝒲) (that is, w_{k} = 0 for k ≥ N + 1),

−ϵ∥w∥_{ℓ_{[0,N]}^{2}(𝒲)}^{2} ≥ ∥z∥_{ℓ_{ℤ^{+}}^{2}(𝒵)}^{2} − γ^{2}∥w∥_{ℓ_{ℤ^{+}}^{2}(𝒲)}^{2}
≥ ∥z∥_{ℓ_{[0,N]}^{2}(𝒵)}^{2} − γ^{2}∥w∥_{ℓ_{[0,N]}^{2}(𝒲)}^{2} + min_{stab. ctrl−laws} ∑_{k=N+1}^{∞} ⟨z_{k}, z_{k}⟩_{𝒵}
= ∥z∥_{ℓ_{[0,N]}^{2}(𝒵)}^{2} − γ^{2}∥w∥_{ℓ_{[0,N]}^{2}(𝒲)}^{2} + ⟨x_{N+1}, X_{2}x_{N+1}⟩_{𝒳} ,

where X_{2} ≥ 0 is the stabilising solution to the algebraic, Riccati equation (2.10) and the last equality follows by Proposition 2.21. By the finite-horizon results, Xk^{{N+1,X_{2}}} ≥ 0 and

∇k^{{N+1,X_{2}}} ≤ −αI  (A.14)

for all k ≤ N + 1 and some α ≥ ϵ (without loss of generality it is possible to take α = ϵ since this is independent of the problem horizon.) Furthermore, it can be shown that Xk^{{N+1,X_{2}}} is uniformly^{1} bounded above and monotonically increasing in N. Therefore, it follows that

X_{∞} = lim_{N→∞} Xk^{{N+1,X_{2}}} ≥ 0

exists, is independent of k and a is solution to the algebraic, Riccati equation (2.13). Moreover, since α ≥ ϵ is independent of the horizon length, ∇ = lim_{N→∞} ∇k^{{N+1,X_{2}}} < 0, and by monoticity (X_{∞} − X_{2}) ≥ 0. So it remains to show that spec(A − BZ^{−1}L) ⊂ 𝔻. This is done in two steps: first it is shown that it is necessary for spec(A − B_{2}Z_{3}^{−1}L_{2}) to lie inside 𝔻, which is then used this to show that spec(A − BZ^{−1}L) ⊂ 𝔻.

First note that Ω :=X_{∞} − X_{2} ≥ 0 satisfies

Ω = (A − B_{2}Z_{3}^{−1}L_{2})^{∗}(Ω + ΩB_{2}(Z_{3} − B_{2}^{∗}ΩB_{2})^{−1}B_{2}^{∗}Ω)(A − B_{2}Z_{3}^{−1}L_{2}) − L_{∇}^{∗}∇^{−1}L_{∇}.

Now since Z_{3} − B_{2}^{∗}ΩB_{2} = D_{2}^{∗}D_{2} + B_{2}^{∗}X_{2}B_{2}, it is positive-definite, invertible (cf. Remark 2.19) and therefore, has positive-definite square-root [Kre89b, Theorem 9.4-2]. Furthermore, since ∇ < 0, ((A−B_{2}Z_{3}^{−1}L_{2})^{∗}ΩB_{2}(Z_{3} −B_{2}^{∗}ΩB_{2})^{−1}B_{2}^{∗}Ω(A−B_{2}Z_{3}^{−1}L_{2})−L_{∇}^{∗}∇^{−1}L_{∇}) ≥ 0 and hence, any eigenvalue λ of (A − B_{2}Z_{3}^{−1}L_{2}) and corresponding eigenvector x ∈ 𝒳, satisfy

(1 − |λ|^{2})⟨x, Ωx⟩_{𝒳}
= ⟨x, ((A − B_{2}Z_{3}^{−1}L_{2})^{∗}ΩB_{2}(Z_{3} − B_{2}^{∗}ΩB_{2})^{−1}B_{2}^{∗}Ω(A − B_{2}Z_{3}^{−1}L_{2}) − L_{∇}^{∗}∇^{−1}L_{∇})x⟩_{𝒳}
≥ 0.  (A.15)

Since Ω ≥ 0, this implies spec(A − B_{2}Z_{3}^{−1}L_{2}) ⊂ \overline{𝔻}. In fact, this can be strengthened to spec(A − B_{2}Z_{3}^{−1}L_{2}) ⊂ 𝔻 if (B_{2}^{∗}Ω(A − B_{2}Z_{3}^{−1}L_{2}), A − B_{2}Z_{3}^{−1}L_{2}) has no unobservable modes on the unit circle of the complex plane. To see this, suppose that under the hypothesis that (B_{2}^{∗}Ω(A − B_{2}Z_{3}^{−1}L_{2}), A − B_{2}Z_{3}^{−1}L_{2}) has no unobservable modes on 𝕋, there exists an eigenvalue of (A − B_{2}Z_{3}^{−1}L_{2}) on 𝕋 and denote the corresponding eigen-vector by x ∈ 𝒳. Since ∇^{−1} < 0, it follows from equation (A.15), that

∥(Z_{3} − B_{2}^{∗}ΩB_{2})^{− \frac{1}{2}} B_{2}^{∗}Ω(A − B_{2}Z_{3}^{−1}L_{2})x∥_{𝒳}^{2} − ⟨L_{∇}x, ∇^{−1}L_{∇}x⟩_{𝒳} ≤ 0,

This implies that (Z_{3}−B_{2}^{∗}ΩB_{2})^{− \frac{1}{2}} B_{2}^{∗}Ω(A−B_{2}Z_{3}^{−1}L_{2})x = 0, which since 𝒦_{(Z_{3}−B_{2}^{∗}ΩB_{2})^{− \frac{1}{2}}} = \{0\} implies that B_{2}^{∗}Ω(A − B_{2}Z_{3}^{−1}L_{2})x = 0, contradicting the hypothesis. So in terms of showing that spec(A − B_{2}Z_{3}^{−1}L_{2}) ⊂ 𝔻, it remains to show that (B_{2}^{∗}Ω(A − B_{2}Z_{3}^{−1}L_{2}), A − B_{2}Z_{3}^{−1}L_{2}) has no unobservable modes on 𝕋. This is indeed the case, as can be seen by noting that the spectrum of

1Independently of N (and k by time-invariance).

A − B_{2}Z_{3}^{−1}L_{2} + B_{2}(Z_{3} − B_{2}^{∗}ΩB_{2})^{−1}B_{2}^{∗}Ω(A − B_{2}Z_{3}^{−1}L_{2})
= A − B_{2}(D_{2}^{∗}D_{2} + B_{2}^{∗}X_{2}B_{2})^{−1}(D_{2}C + B_{2}^{∗}X_{2}A)  (A.16)

lies inside 𝔻, since it corresponds to the optimally-controlled dynamics of the quadratic-regulator problem. Now, suppose that there exists a |λ| = 1 and x ∈ 𝒳 such that (A − B_{2}Z_{3}^{−1}L_{2})x = λx and B_{2}^{∗}Ω(A − B_{2}Z_{3}^{−1}L_{2})x = 0. Then using equation (A.16), it follows that

(A − B_{2}(D_{2}^{∗}D_{2} + B_{2}^{∗}X_{2}B_{2})^{−1}(D_{2}C + B_{2}^{∗}X_{2}A))x = λx − 0,

which is a contradiction. Hence, (B_{2}^{∗}Ω(A − B_{2}Z_{3}^{−1}L_{2}), A − B_{2}Z_{3}^{−1}L_{2}) has no unobservable modes on 𝕋 and spec(A − B_{2}Z_{3}^{−1}L_{2}) ⊂ 𝔻.

Now, in so far as the proof of necessity goes, it remains to show that spec(A − BZ^{−1}L) ⊂ 𝔻. To this end, recall that

A − BZ^{−1}L = (A − B_{2}Z_{3}^{−1}L_{2}) − (B_{1} − B_{2}Z_{3}^{−1}Z_{2})∇^{−1}L_{∇}  (A.17)

and consider any x ∈ 𝒦_{Ω}. Then from equation (A.15), it follows that

0 ≤ ⟨x, (A − B_{2}Z_{3}^{−1}L_{2})^{∗}ΩB_{2}(Z_{3} − B_{2}^{∗}ΩB_{2})^{−1}B_{2}^{∗}Ω(A − B_{2}Z_{3}^{−1}L_{2})x⟩_{𝒳} = ⟨L_{∇}x, ∇^{−1}L_{∇}x⟩_{𝒳} ,

which since ∇^{−1} < 0 implies that L_{∇}x = 0. Hence, from equation (A.17),

x ∈ 𝒦_{Ω} ⇒ (A − BZ^{−1}L)x = (A − B_{2}Z_{3}^{−1}L_{2})x.  (A.18)

Now, define Ω_{k} :=(X_{∞} − Xk^{{N+1,X_{2}}}) and note that Ω_{k} satisfies the difference equation

Ω_{k} = (A − BZ^{−1}L)^{∗}Ω_{k+1}(I − BZ^{−1}B^{∗}Ω_{k+1})^{−1}(A − BZ^{−1}L),

with boundary condition Ω_{N+1} = Ω :=(X_{∞} − X_{2}). Furthermore, lim_{N→∞} Ω_{k} = 0 (k ≤ N). It follows that for each λ ∈ spec(A − BZ^{−1}L) and corresponding eigen-vector x ∈ 𝒳,

⟨x, Ω_{N}x⟩_{𝒳} = |λ|^{2}⟨x, Ω(I − BZ^{−1}B^{∗}Ω)^{−1}x⟩_{𝒳} .  (A.19)

If λ = 0 then it is immediate that x ∈ 𝒦_{Ω_{N}}. In fact, taking the limit as N → ∞ in equation (A.19), gives ⟨x, Ω(I − BZ^{−1}B^{∗}Ω)^{−1}x⟩_{𝒳} = 0 and hence, x ∈ 𝒦_{Ω_{N}} for λ ≠ 0 as well. If it can

A.2 Proof of Proposition 2.22 151

now be shown that 𝒦_{Ω_{N}} ⊂ 𝒦_{Ω}, then it would follow from equation (A.18), that λ ⊂ 𝔻, since spec(A − B_{2}Z_{3}^{−1}L_{2}) ⊂ 𝔻. To see that this is true, note that

∥z∥_{ℓ_{[k,N]}^{2}(𝒵)}^{2} − γ^{2}∥w∥_{ℓ_{[k,N]}^{2}(𝒲)}^{2} + ⟨x_{N+1}, X_{2}x_{N+1}⟩_{𝒳} =  (A.20)
⟨x_{k}, Xk^{{N+1,X_{2}}} x_{k}⟩_{𝒳} + ∑_{i=k}^{N} ⟨(w_{i} − w_{i}^{⋆}), ∇(w_{i} − w_{i}^{⋆})⟩_{𝒲}  (A.21)

and

∥z∥_{ℓ_{[k,∞)}^{2}(𝒵)}^{2} − γ^{2}∥w∥_{ℓ_{[k,∞)}^{2}(𝒲)}^{2} + ⟨x_{k}, X_{∞}x_{k}⟩_{𝒳} = ∑_{i=k}^{∞} ⟨(w_{i} − w_{i}^{⋆}), ∇(w_{i} − w_{i}^{⋆})⟩_{𝒲} .  (A.22)

Consider the signal

w_{i} = \{ \begin{matrix}0 & i < k \\ w̃_{i} & k ≤ i ≤ N \\ 0 & k > N \end{matrix}

for arbitrary w̃_{i} ∈ 𝒲, k ≤ i ≤ N. Then

∥z∥_{ℓ_{[N+1,∞)}^{2}(𝒵)}^{2} − ⟨x_{N+1}, X_{∞}x_{N+1}⟩_{𝒳} = ∑_{i=N+1}^{∞} ⟨(w_{i} − w_{i}^{⋆}), ∇(w_{i} − w_{i}^{⋆})⟩_{𝒲} .

Subtracting equations (A.20) and (A.22), it follows that

⟨x_{N+1}, (X_{∞} − X_{2})x_{N+1}⟩_{𝒳} = ⟨x_{k}, (X_{∞} − Xk^{{N+1,X_{2}}})x_{k}⟩_{𝒳} .

Since (X_{∞} − X_{2}) ≥ 0, it is possible to conclude from this that 𝒦_{(X_{∞}−X_{k}^{{N+1,X_{2}}} )} ⊂ 𝒦_{(X_{∞}−X_{2})} and hence, that spec(A − BZ^{−1}L) ⊂ 𝔻. This completes the proof of necessity.

B

Appendix to Chapter 3

B.1 Proof of Lemma 3.9

Before going on to prove Lemma 3.9, a few remarks concerning LTI systems and the signals spaces on which they are defined is required (for further details cf. [Fra86]). First note that the Hilbert space of continuous-time signals L_{ℝ^{+}}^{2}(𝒰) is isomorphic to the Hardy space H_{ℂ^{+}}^{2}(𝒰) of functions f̂(s) : ℂ^{+} → 𝒰 that are analytic in ℂ^{+} and for which

\frac{1}{2π} ∫_{μ−j∞}^{μ+j∞} ⟨f̂(jω), f̂(jω)⟩_{𝒰} dω

is uniformly bounded for all μ > 0. The isomorphism between L_{ℝ^{+}}^{2}(𝒰) and H_{ℂ^{+}}^{2}(𝒰) is the Laplace Transform, which for f(t) ∈ L_{ℝ^{+}}^{2}(𝒰) is defined by

f̂(s) :=(Lf)(s) := ∫_{0}^{∞} f(t)e^{−st} dt.

By identifying functions in H_{ℂ^{+}}^{2}(𝒰) with their boundary values on the imaginary axis, H_{ℂ^{+}}^{2}(𝒰) can be embedded as a closed subspace in the Hilbert space L_{jℝ}^{2}(𝒰) of functions f̂(jω) : ℝ → 𝒰 that satisfy ⟨f̂, f̂⟩_{L_{jℝ}^{2}(𝒰)} < ∞, where for f̂, ĝ ∈ L_{jℝ}^{2}(𝒰)

⟨f̂, ĝ⟩_{L_{jℝ}^{2}(𝒰)} :=\frac{1}{2π} ∫_{−∞}^{∞} ⟨f̂(jω), ĝ(jω)⟩_{𝒰} dω.

Furthermore, L_{jℝ}^{2}(𝒰) is isomorphic to the doubly-infinite, continuous-resolution, Hilbert space L_{ℝ}^{2}(𝒰) of functions f : ℝ → 𝒰 that satisfy

∫_{−∞}^{∞} ⟨f(t), f(t)⟩_{𝒰} dt < ∞.

The isomorphism between L_{jℝ}^{2}(𝒰) and L_{ℝ}^{2}(𝒰) is the Fourier transform defined by

f̂(jω) :=(Ff̂)(jω) := ∫_{−∞}^{∞} f(t)e^{−jωt} dt

for all f(t) ∈ L_{ℝ}^{2}(𝒰). By analytic continuation, the Fourier-Transform isomorphism takes functions in L_{ℝ^{+}}^{2}(𝒳) ⊂ L_{ℝ}^{2}(𝒰) to H_{ℂ^{+}}^{2}(𝒰) and functions in L_{ℝ}^{2} (𝒳) :=L_{ℝ}^{2}(𝒰) ⊖ L_{ℝ^{+}}^{2}(𝒳) to H_{ℂ^{+}}^{2⊥}(𝒰) :=L_{jℝ}^{2}(𝒰) ⊖ H_{ℂ^{+}}^{2}(𝒰).

Now as for discrete-time LSI systems, continuous-time LTI systems are equivalent via the Laplace-Transform isomorphism to multiplication operators defined on subspaces of H_{ℂ^{+}}^{2}(𝒰). Furthermore, the multiplication operator equivalent of an LTI system is bounded (that is, its domain is the whole of H_{ℂ^{+}}^{2}(𝒰) and it has finite induced-norm) if and only if its frequency domain symbol is an element of H_{ℂ^{+}}^{∞}(B_{𝒰→𝒴} ), which is the Hardy space of B_{𝒰→𝒴} -valued functions F̂(s) : ℂ^{+} → B_{𝒰→𝒴} that are analytic and bounded in the right-half plane ℂ^{+}. That is, an LTI system F : 𝒟_{F} ⊂ L_{ℝ^{+}}^{2}(𝒰) → L_{ℝ^{+}}^{2}(𝒴) is stable if and only if it is equivalent via the Laplace-Transform isomorphism to a multiplication operator MF̂ with F̂(s) ∈ H_{ℂ^{+}}^{∞}(B_{𝒰→𝒴} ).^{1} Note also, that for any function F̂(jω) : jℝ → B_{𝒰→𝒴} the multiplication operator defined by (MF̂f̂)(jω) :=F̂(jω)f̂(jω) for all f̂ ∈ L_{jℝ}^{2}(𝒰) is bounded and equivalent via the Fourier-Transform isomorphism to an LTI system on the doubly-infinite, continuous-time signal-space L_{ℝ}^{2}(𝒰).

In the following proof, the notion of inner and co-inner functions is required. Let 𝒰 and 𝒴 be finite-dimensional Hilbert-spaces. Then a function F̂(s) ∈ H_{ℂ^{+}}^{∞}(B_{𝒰→𝒴} ) is said to be inner if F̂^{∼}(s)F̂(s) = I and co-inner if F̂(s)F̂^{∼}(s) = I, where F̂^{∼}(s) :=F (−s)^{T} and the superscript T denotes matrix transpose (without conjugation). It can be shown that (MF̂)^{∗} = Π_{H_{ℂ^{+}}^{2}(𝒰)} MF̂^{∼}, where recall that the superscript ∗ denotes the Hilbert space adjoint, which in this case is the adjoint on H_{ℂ^{+}}^{2}(𝒰). To see this, note that MF̂^{∼} is the L_{jℝ}^{2}(𝒰) adjoint of MF̂ and hence, that for any f̂ ∈ H_{ℂ^{+}}^{2}(𝒰) and ĝ ∈ H_{ℂ^{+}}^{2}(𝒴),

⟨ĝ, MF̂f̂⟩_{H_{ℂ^{+}}^{2}(𝒴)} = ⟨ĝ, MF̂f̂⟩_{L_{jℝ}^{2}(𝒴)} = ⟨MF̂^{∼} ĝ, f̂⟩_{L_{jℝ}^{2}(𝒰)} = ⟨Π_{H_{ℂ^{+}}^{2}(𝒰)} MF̂^{∼} ĝ, f̂⟩_{L_{jℝ}^{2}(𝒰)}
= ⟨Π_{H_{ℂ^{+}}^{2}(𝒰)} MF̂^{∼} ĝ, f̂⟩_{H_{ℂ^{+}}^{2}(𝒰)} ,

where all of the equalities follow readily from the fact that H_{ℂ^{+}}^{2}(𝒴) ⊂ L_{jℝ}^{2}(𝒰). That is, (MF̂)^{∗} = Π_{H_{ℂ^{+}}^{2}(𝒰)} MF̂^{∼} as required. So, it follows immediately that if F̂ is inner, then F :=L^{−1}MF̂L is an isometry on L_{ℝ^{+}}^{2}(𝒰). With this established it is now possible to prove the lemma:

Proof : (of Lemma 3.9) That ∥GX∥ = ∥X∥ if Ĝ is inner is obvious, since in this case G is an isometry on L_{ℝ^{+}}^{2}(𝒰). The rest of the proof to show that ∥XG∥ = ∥X∥ when Ĝ is co-inner, follows an idea suggested in [Vin96]. Since all operators concerned are linear

∥XG∥ = ∥G^{∗}X^{∗}∥ = ∥Π_{L_{ℝ^{+}}^{2}(𝒰)} F^{−1}MĜ^{∼}FX^{∗}∥,

1Recall the similar result for discrete-time LSI systems (cf. Proposition 2.2)

where all adjoint above are taken on L_{ℝ^{+}}^{2}(𝒰). Furthermore,

∥Π_{L_{ℝ^{+}}^{2}(𝒰)} F^{−1}MĜ^{∼}FX^{∗}∥ ≤ sup_{0≠y∈L_{ℝ^{+}}^{2}(𝒴)} \begin{matrix}⟨MĜ^{∼}FX^{∗}y, MĜ^{∼}FX^{∗}y⟩ & L_{jℝ}^{2}(𝒰) \\ ⟨y, y⟩ & L{2 \atop ℝ^{+}}(𝒴) \end{matrix}
= ∥X^{∗}∥ = ∥X∥,

where the second equality follows from the fact that ĜĜ^{∼} = I and the fact that MĜ is the L_{jℝ}^{2}(𝒰) adjoint of MĜ^{∼}. Consequently, it is now sufficient to show that

∥X∥ ≤ ∥XG∥.

Without loss of generality suppose ∥X∥ = 1, which implies that for any δ > 0 there exists a w ∈ L_{ℝ^{+}}^{2}(𝒲) with ∥w∥_{L_{ℝ^{+}}^{2}(𝒲)} < 1 such that

∥Xw∥_{L_{ℝ^{+}}^{2}(𝒴)} > 1 − δ.

Furthermore, since X is a bounded, linear operator it is Lipschitz-continuous and hence, there exists an ϵ > 0 such that

∥Xw̃∥_{L_{ℝ^{+}}^{2}(𝒴)} > 1 − δ  (B.1)

for all w̃ ∈ L_{ℝ^{+}}^{2}(𝒲) that satisfy ∥w−w̃∥_{L_{ℝ^{+}}^{2}(𝒲)} < ϵ. Now for notational convenience let w_{[k]} :=U_{kh}w (k ∈ ℤ^{+}), where U_{kh} is the unilateral shift. Then since X is LTPV (with period h)

∥Xw_{[k]}∥_{L_{ℝ^{+}}^{2}(𝒴)} = ∥Xw∥_{L_{ℝ^{+}}^{2}(𝒴)} > 1 − δ,  (B.2)

for all k ∈ ℤ^{+}. Also, since MĜ is a co-isometry on L_{jℝ}^{2}(𝒲), it follows that G is a co-isometry on L_{ℝ}^{2}(𝒲) and thus, that there exists a u ∈ L_{ℝ}^{2}(𝒰) with ∥u∥_{L_{ℝ}^{2}(𝒰)} = ∥w∥_{L_{ℝ^{+}}^{2}(𝒲)} such that

w = Gu

and w_{[k]} = Gu_{[k]} where u_{[k]} :=U_{kh}u. Then since u ∈ L_{ℝ}^{2}(𝒰) there exists a n_{ϵ} ∈ ℤ^{+} such that ∥Π_{L_{ℝ}^{2} (𝒰)} u_{[n_{ϵ}]}∥_{L_{ℝ}^{2}(𝒰)} < ϵ.^{2} Hence, with u_{[n_{ϵ}]}^{+} :=Π_{L_{ℝ^{+}}^{2}(𝒰)} u_{[n_{ϵ}]} ∈ L_{ℝ^{+}}^{2}(𝒰) and w̃_{[n_{ϵ}]} :=Gu_{[n_{ϵ}]}^{+} it follows that

∥w_{[n_{ϵ}]} − w̃_{[n_{ϵ}]}∥_{L_{ℝ^{+}}^{2}(𝒲)} = ∥G(u_{[n_{ϵ}]} − u_{[n_{ϵ}]}^{+})∥_{L_{ℝ}^{2}(𝒲)} = ∥GΠ_{L_{ℝ}^{2} (𝒰)} u_{[n_{ϵ}]}∥_{L_{ℝ}^{2}(𝒲)} < ϵ.

2To see this note that for any v ∈ L_{ℝ}^{2}(𝒰) , lim_{τ→−∞} ∫_{−∞}^{τ} ⟨v(t), v(t)⟩_{𝒰} dt = 0. Define v_{[k]}^{−} :=Π_{L_{ℝ}^{2} (𝒰)} U_{kh}v. Then

∥v_{[k]}^{−}∥_{L_{ℝ}^{2}(𝒰)} = ∫_{−∞}^{−kh} ⟨v(t), v(t)⟩_{𝒰} dt and it follows that lim_{k→∞} ∥v_{[k]}^{−}∥_{L_{ℝ}^{2}(𝒰)} = 0.

So from equations (B.1) and (B.2),

∥Xw̃_{[n_{ϵ}]}∥_{L_{ℝ^{+}}^{2}(𝒴)} = ∥XGu_{[n_{ϵ}]}^{+}∥_{L_{ℝ^{+}}^{2}(𝒴)} > 1 − δ.

Then since ∥u_{[n_{ϵ}]}^{+}∥_{L_{ℝ^{+}}^{2}(𝒰)} ≤ ∥u∥_{L_{ℝ}^{2}(𝒰)} = ∥w∥_{L_{ℝ^{+}}^{2}(𝒲)} < 1 and it is possible to construct such a u_{[n_{ϵ}]}^{+} ∈ L_{ℝ^{+}}^{2}(𝒰) for any δ > 0, it follows that

∥XG∥ := sup_{∥u∥_{L{2 \atop ℝ^{+}}(𝒰)} ≤1} ∥XGu∥_{L_{ℝ^{+}}^{2}(𝒰)}
≥ 1 = ∥X∥,

as required.

B.2 Computing the Average-Power Gain Matrix for SD Systems

In this section state-space formulae are derived for computing the average-power gain matrix of an SD control-system. The formulae derived are expressible explicitly in terms of the state-space matrices of the continuous-time plant and discrete-time controller. Specifically, the stable, closed-

[figure: A_{D}; K; D_{A}; G; w; u; z; y]

Figure B.1 SD Closed-Loop

loop system shown in Figure B.1 is considered, where A_{D} is an ideal, h-periodic sampling device synchronised with the zero-order hold D_{A} and for fixed t ∈ ℝ the signals w(t) ∈ ℝ^{m_{1}}, u(t) ∈ ℝ^{m_{2}}, z(t) ∈ ℝ^{p_{1}} and y(t) ∈ ℝ^{p_{2}}. Let the LTI, generalised plant G and the LSI controller K have state-space realisations

G = [ \begin{matrix}A & B_{1} & B_{2} \\ C_{1} & D_{11} & D_{12} \\ C_{2} & 0 & 0 \end{matrix} ] and K = [ \begin{matrix}A_{d} & B_{d} \\ C_{d} & D_{d} \end{matrix} ],

where the realisation for G has n states and that for K has l states. In the sequel, state-space formulae are derived for computing the average-power gain matrix of the stable, closed-loop system P :=F_{ℓ}(G, D_{A}KA_{D}) ∈ 𝒫_{ℝ^{m_{1}}→ℝ^{p_{1}}} . The formulae derived are expressible explicitly in terms of the state-space realisations of G and K.

Recall that the average-power gain matrix is defined in terms of the transfer operator. To characterise this for the system shown in Figure B.1, it is first necessary to obtain a state-space realisation of

P:=WP W^{−1} :=WF_{ℓ}(G, D_{A}KA_{D})W^{−1} ∈ 𝒟_{L_{ℍ}^{2}(ℝ^{m_{1}})→L_{ℍ}^{2}(ℝ^{p_{1}})} .

This can be achieved using the technique used in section 2.5. Absorbing the ideal sample and hold devices into G and the expressing the evolution of the closed-loop in terms of the state at the sampling instances kh, yields the following discrete-time, LSI state-space realisation of P:

[ \begin{matrix}x_{k+1} \\ s_{k+1} \end{matrix} ] = A_{cl}[ \begin{matrix}x_{k} \\ s_{k} \end{matrix} ] + B_{cl} w^{⊸}_{k}(θ); [ \begin{matrix}x_{0} \\ s_{0} \end{matrix} ] = [ \begin{matrix}0 \\ 0 \end{matrix} ],  (B.3a)
z^{⊸}_{k}(θ) = C_{cl}[ \begin{matrix}x_{k} \\ s_{k} \end{matrix} ] + D_{cl} w^{⊸}_{k}(θ),  (B.3b)

where

A_{cl} := [ \begin{matrix}A + B_{2}D_{d}C_{2} & B_{2}C_{d} \\ B_{d}C_{2} & A_{d} \end{matrix} ] : ℝ^{n+l} → ℝ^{n+l},
B_{cl} := [ \begin{matrix}B_{1} \\ 0 \end{matrix} ] : L_{ℍ}^{2}(ℝ^{m_{1}}) → ℝ^{n+l},
C_{cl} := [ C_{1} + D_{12}D_{d}C_{2} D_{12}C_{d} ] : ℝ^{n+l} → L_{ℍ}^{2}(ℝ^{p_{1}}),
D_{cl} := D_{11} : L_{ℍ}^{2}(ℝ^{m_{1}}) → L_{ℍ}^{2}(ℝ^{p_{1}}),

and for any x ∈ ℝ^{n+l}, u ∈ ℝ^{m_{2}} and w^{⊸} ∈ L_{ℍ}^{2}(ℝ^{m_{1}})

A : ℝ^{n} → ℝ^{n},  Ax:=e^{hA}x,  (B.4a)
B_{2} : ℝ^{m_{2}} → ℝ^{n},  B_{2}u:= ∫_{0}^{h} e^{(h−τ)A} dτ B_{2}u,  (B.4b)
B_{1} : L_{ℍ}^{2}(ℝ^{m_{1}}) → ℝ^{n},  B_{1} w^{⊸}:= ∫_{0}^{h} e^{(h−τ)A}B_{1} w^{⊸}(τ) dτ,  (B.4c)
C_{1} : ℝ^{n} → L_{ℍ}^{2}(ℝ^{p_{1}}),  (C_{1}x)(θ) :=C_{1}e^{θA}x,  (B.4d)
D_{11} : L_{ℍ}^{2}(ℝ^{m_{1}}) → L_{ℍ}^{2}(ℝ^{p_{1}}),  (D_{11} w^{⊸})(θ) :=D_{11} w^{⊸}(θ) + ∫_{0}^{θ} C_{1}e^{(θ−τ)A}B_{1} w^{⊸}(τ) dτ  (B.4e)

and

D_{12} : ℝ^{m_{2}} → L_{ℍ}^{2}(ℝ^{p_{1}}),  (D_{12}u)(θ) :=D_{12}u + ∫_{0}^{θ} C_{1}e^{(θ−τ)A} dτB_{2}u.  (B.4f)

The transfer operator P̂(λ) is defined to be the symbol of the multiplication operator equivalent of P, where the equivalence is via the Z-Transform isomorphism. Using the state-space realisation given in equation (B.3), it is possible to characterise P̂(λ) in terms of A_{cl}, B_{cl}, C_{cl} and D_{cl} as follows: Let z^{⊸} = Pw^{⊸}, then using equation (B.3) it is straightforward to show that

ˆz^{⊸}(λ) :=(Z z^{⊸})(λ) := ∑_{k=0}^{∞} z^{⊸}_{k}λ^{k} = ∑_{k=0}^{∞} λ^{k} (∑_{n=0}^{k−1} C_{cl}A_{cl}^{k−n−1}B_{cl}w^{⊸}_{n} + D_{cl}w^{⊸}_{k}) + D_{cl}w^{⊸}_{0},

which since spec(A_{cl}) ⊂ 𝔻 (because P is stable), converges (with respect to ∥ · ∥_{L_{ℍ}^{2}(•)}) for all λ ∈ 𝔻. Simple rearrangement of terms yields

ˆz^{⊸}(λ) = (∑_{l=0}^{∞}(λ^{l+1}C_{cl}A_{cl}^{l}B_{cl}) + D_{cl}) ∑_{q=0}^{∞} w^{⊸}_{q}λ^{q}
= (C_{cl}λ(I − λA_{cl})^{−1}B_{cl} + D_{cl}) ˆw^{⊸}(λ) =: P̂(λ) ˆw^{⊸}(λ).

It then follows that (cf. equations (3.2–3.5))

Θ_{P} (jω)Θ_{P} (jω) := \frac{1}{h}(Γ_{P} (jω))^{∗}Γ_{P} (jω) :=(Ω(jω))^{∗}(P̂(e^{jω_{0}h}))^{∗}P̂(e^{jω_{0}h})Ω(jω),
= \frac{1}{h}(Ω(jω))^{∗}(B_{cl}^{∗}Â(e^{jω_{0}h})^{∗}C_{cl}^{∗} + D_{cl}^{∗})(C_{cl}Â_{cl}(e^{jω_{0}})B_{cl} + D_{cl})Ω(jω), (B.5)

where ω_{0} :=ω − ]\frac{ωh}{2π}[ \frac{2π}{h} and Â(e^{jω_{0}})^{∗} :=(e^{jω_{0}h}I − A_{cl})^{−1}. By expanding the expression in equation (B.5), it follows that it is necessary to determine finite-dimensional expressions for the following operators:

A, B_{2}, B_{1}Ω(jω), [ \begin{matrix}C_{1}^{∗} \\ D_{12}^{∗} \end{matrix} ][ C_{1} D_{12} ],
Ω(jω)^{∗}D_{11}^{∗}D_{11}Ω(jω), Ω(jω)^{∗}D_{11}^{∗}[ C_{1} D_{12} ].

This can be achieved using equations (B.4) and a technique used in [CG97] (cf. also Chapters 5 and 6), to characterise finite-dimensional combinations of these types of integral operator expressions in terms of TPBVPs. Subsequent computation of Θ_{P} (jω) is then a simple matter of standard finitedimensional matrix arithmetic. State-Space formulae for the required finite-dimensional expressions are given in Theorem B.1 below. Note that the formulae are expressed explicitly in terms of the state-space matrices of the realisations of G and K and involve a single matrix exponential of a Hamiltonian-structured matrix. Since the resultant formula are similar to those obtained for the SD, H^{∞}-synthesis problem (cf. [CG97] and Theorem 6.1), it is possible to re-structure the formulae presented here in a similar manner to that described in Section 6.3, to render the formulae numerically robust.

The following matrices are required in Theorem B.1. Let Ǎ:=[ \begin{matrix}A & B_{2} \\ 0 & 0 \end{matrix} ],

Ã := [ \begin{matrix}Ǎ & [ \begin{matrix}B_{1} \\ 0 \end{matrix} ] \\ 0 & (jωI_{m}) \end{matrix} ],  C̃ :=[ C_{1} D_{12} D_{11} ],

E := [ \begin{matrix}Ã & 0 \\ −C̃′ C̃ & −Ã′ \end{matrix} ]  (B.6)

and

Q := [ \begin{matrix}Q_{11} & Q_{12} \\ Q_{21} & Q_{22} \end{matrix} ] = e^{hE},  (B.7)

where the partitioning of Q conforms with that of E. Theorem B.1 Recall that m_{1} (respectively n) is the number of inputs to (respectively the number of states of the realisation of) 11-block of G and that m_{2} is the number of outputs from K_{d}. For notational convenience, define i:=n + m_{2}, k :=n + m_{1} and m:=m_{1} + m_{2}. Then,

Ω(jω)^{∗}D_{11}^{∗}D_{11}Ω(jω) = −[ 0_{m_{1},i} I_{m} ]Q_{22}^{−1}Q_{21}[ \begin{matrix}0_{i,m_{1}} \\ I_{m_{1}} \end{matrix} ],
Ω(jω)^{∗}D_{11}^{∗}[ C_{1} D_{12} ] = −[ 0_{m_{1},i} I_{m} ]Q_{22}^{−1}Q_{21}[ \begin{matrix}I_{k} \\ 0_{m_{2},k} \end{matrix} ],
[ \begin{matrix}C_{1}^{∗} \\ D_{12}^{∗} \end{matrix} ][ C_{1} D_{12} ] = −[ I_{k} 0_{k,m_{2}} ]Q_{22}^{−1}Q_{21}[ \begin{matrix}I_{k} \\ 0_{m_{2},k} \end{matrix} ],
B_{1}Ω(jω) = [ I_{n} 0_{n,m} ]Q_{11}[ \begin{matrix}0_{i,m_{1}} \\ I_{m_{1}} \end{matrix} ],
A = [ I_{n} 0_{n,m} ]Q_{11}[ \begin{matrix}I_{n} \\ 0_{m,n} \end{matrix} ],

B_{2} = [ I_{n} 0_{n,m} ]Q_{11}[ \begin{matrix}0_{n,m_{2}} \\ I_{m_{2}} \\ 0_{m_{1},m_{2}} \end{matrix} ],

where 0_{p,r} denotes an p × p matrix of zeros and I_{p} the p × p identity. Proof : Consider z̃ = Ω(jω)^{∗}D_{11}^{∗}D_{11}Ω(jω)z for some z ∈ ℝ^{m_{1}} and let u^{⊸} :=Ω(jω)z, y^{⊸} :=D_{11}u and v^{⊸} :=D_{11}^{∗} y^{⊸}. By the definition of Ω(jω) (cf. equation (3.3)), u^{⊸} :=Ω(jω)z can be characterised as

\frac{d a^{⊸}}{dθ} = (jωI_{m_{1}}) a^{⊸}(θ); a^{⊸}(0) = z,  (B.8a)
u(θ) = a^{⊸}(θ).  (B.8b)

B.2 Computing the Average-Power Gain Matrix for SD Systems 159

It follows then, that z̃ = Ω(jω)^{∗} v^{⊸} can be expressed as

\frac{d b^{⊸}}{dθ} = −(jωI_{m_{1}})^{∗} b^{⊸}(θ) − v^{⊸}(θ); b^{⊸}(h) = 0,  (B.9a)
z̃ = b^{⊸}(0).  (B.9b)

From equation (B.4e), it follows that y^{⊸} = D_{11} u^{⊸} and respectively v^{⊸} = D_{11}^{∗} y^{⊸}, can be characterised by the system of equations

\frac{d g^{⊸}}{dθ} = Ǎg^{⊸}(θ) + [ \begin{matrix}B_{1} \\ 0 \end{matrix} ] u^{⊸}(θ); g^{⊸}(0) = 0,  (B.10a)
y^{⊸}(θ) = [ C_{1} D_{12} ] g^{⊸}(θ) + D_{11} u^{⊸}(θ),  (B.10b)

respectively

\frac{d e^{⊸}}{dθ} = −Ǎ^{∗} e^{⊸}(θ) − [ \begin{matrix}C_{1}^{∗} \\ D_{12}^{∗} \end{matrix} ] y^{⊸}(θ); e^{⊸}(h) = 0,  (B.11a)
v^{⊸}(θ) = [ B_{1}^{∗} 0 ] e^{⊸}(θ) + D_{11}^{∗} y^{⊸}(θ).  (B.11b)

Combining equations (B.8–B.11) gives the TPBVP

\frac{d}{dθ} ([ g^{⊸} ; a^{⊸} ; e^{⊸} ; b^{⊸} ]) = E [ g^{⊸} ; a^{⊸} ; e^{⊸} ; b^{⊸} ] (θ), [ \begin{matrix}g^{⊸}(0) \\ a^{⊸}(0) \\ e^{⊸}(h) \\ b^{⊸}(h) \end{matrix} ] = [ \begin{matrix}0 \\ z \\ 0 \\ 0 \end{matrix} ]

where E is defined in equation (B.6). The solution to this TPBVP satisfies

[ \begin{matrix}g^{⊸}(h) \\ a^{⊸}(h) \\ e^{⊸}(h) \\ b^{⊸}(h) \end{matrix} ] = e^{tE} [ \begin{matrix}g^{⊸}(0) \\ a^{⊸}(0) \\ e^{⊸}(0) \\ b^{⊸}(0) \end{matrix} ] .

Using the terminal condition [ \begin{matrix}e^{⊸}(h) \\ b^{⊸}(h) \end{matrix} ] = [ \begin{matrix}0 \\ 0 \end{matrix} ] it follows that

[ \begin{matrix}e^{⊸}(0) \\ b^{⊸}(0) \end{matrix} ] = −Q_{22}^{−1}Q_{21}[ \begin{matrix}g^{⊸}(0) \\ a^{⊸}0 \end{matrix} ] = −Q_{22}^{−1}Q_{21}[ \begin{matrix}0 \\ z \end{matrix} ].

where Q is defined in equation (B.7). Hence, noting that z̃ = b^{⊸}(0) = Ω(jω)^{∗}D_{11}^{∗}D_{11}Ω(jω)z gives

Ω(jω)^{∗}D_{11}^{∗}D_{11}Ω(jω) = −[ 0_{m,l} I_{m} ]Q_{22}^{−1}Q_{21}[ \begin{matrix}0_{l,m} \\ I_{m} \end{matrix} ].

The proof follows in a similar manner for the other operators and is therefore omitted.

Appendix to Chapter 5

C.1 Proof of Lemma 5.9

Although Assumption 5.8 holds throughout, in what follows it is convenient to maintain the notational distinction between D_{a} and D_{b}. First consider the operator

Q̀ :=C̃^{∗} (I − D̃ (D̃^{∗}JD̃)D̃^{∗}) C̃ =
[ \begin{matrix}F_{a}^{∗} & (C_{a}^{∗} + F_{a}^{∗}D_{a}^{∗}) \\ F_{b}^{∗} & (C_{b}^{∗} + F_{b}^{∗}D_{b}^{∗}) \end{matrix} ] ([ \begin{matrix}I & 0 \\ 0 & I \end{matrix} ]

−[ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ] ([ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ][ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ] − [ \begin{matrix}γ^{2}V_{a}^{−2} & 0 \\ 0 & 0 \end{matrix} ])^{−1} [ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]) ×

[ \begin{matrix}F_{a} & F_{b} \\ (C_{a} + D_{a}F_{a}) & (C_{b} + D_{b}F_{b}) \end{matrix} ]:=z ↦ x,  (C.1)

where C_{a}, C_{b}, D_{a}, D_{b}, F_{a}, F_{b} and V_{a} are defined in equations (5.16) and (5.21) and z ∈ 𝒳_{a} ⊕ 𝒳_{b}. Let

y^{⊸}_{1} := [ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ] y^{⊸},  (C.2)
u^{⊸}_{1} := ([ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ][ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ] − [ \begin{matrix}γ^{2}V_{a}^{−2} & 0 \\ 0 & 0 \end{matrix} ])^{−1} y^{⊸}_{1},  (C.3)
u^{⊸} := [ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ] u^{⊸}_{1}  (C.4)

and

v^{⊸}_{1} := [ \begin{matrix}γ^{2}V_{a}^{−2} & 0 \\ 0 & 0 \end{matrix} ] u^{⊸}_{1} = [ \begin{matrix}γ^{2}(I + D_{a}^{∗}D_{a} + B_{a}^{∗}X_{a}B_{a}) & 0 \\ 0 & 0 \end{matrix} ] u^{⊸}_{1}  (C.5a)
= [ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ] u^{⊸} − y^{⊸}_{1} = [ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]( u^{⊸} − y^{⊸}),  (C.5b)

where y^{⊸} will be defined later and equation (C.5b) follows from equation (C.3). It follows from equations (5.16) and (C.4) that for 0 ≤ θ ≤ h, ^{1}

\frac{d a^{⊸}}{dθ} = [ \begin{matrix}A_{a} & 0 \\ 0 & A_{b} \end{matrix} ] a^{⊸}(θ) + [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ] u^{⊸}_{1}(θ); a^{⊸}(0) = 0,  (C.6a)
u^{⊸}(θ) = [ \begin{matrix}0 & 0 \\ C_{a} & C_{b} \end{matrix} ] a^{⊸}(θ) + [ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ] u^{⊸}_{1}(θ).  (C.6b)

Furthermore (cf. equation (C.5b)),

\frac{d b^{⊸}}{dθ} = −[ \begin{matrix}A_{a}^{∗} & 0 \\ 0 & A_{b}^{∗} \end{matrix} ] b^{⊸}(θ) − [ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ]( u^{⊸} − y^{⊸})(θ); b^{⊸}(h) = 0,  (C.7a)
v^{⊸}_{1}(θ) = [ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ] b^{⊸}(θ) + [ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]( u^{⊸} − y^{⊸})(θ)  (C.7b)

and similarly (cf. equation (C.5a)),

\frac{d c^{⊸}}{dθ} = −A_{a}^{∗} c^{⊸}(θ) − [ C_{a}^{∗}C_{a} 0 ] a^{⊸}(θ) − [ C_{a}^{∗}D_{a} 0 ] u^{⊸}_{1}(θ); c^{⊸}(h) = [ X_{a} 0 ] a^{⊸}(h), (C.8a)
v^{⊸}_{1}(θ) = [ \begin{matrix}γ^{2}B_{a}^{∗} \\ 0 \end{matrix} ] c^{⊸}(θ) + [ \begin{matrix}γ^{2}D_{a}^{∗}C_{a} & 0 \\ 0 & 0 \end{matrix} ] a^{⊸}(θ) + [ \begin{matrix}γ^{2}(I + D_{a}^{∗}D_{a}) & 0 \\ 0 & 0 \end{matrix} ] u^{⊸}_{1}(θ),  (C.8b)

where X_{a} is a self-adjoint, positive-semidefinite, stabilising solution to the algebraic Riccati equation (5.22). Using the two expressions for v^{⊸}_{1} in equations (C.7b) and (C.8b), u^{⊸}_{1} can be expressed in terms of a^{⊸}, b^{⊸}, c^{⊸} and y^{⊸} only. The following TPBVP characterisation of the operator mapping y^{⊸} to u^{⊸} can then be obtained via elementary state-space manipulation of equations (C.6-C.8):

\frac{d a^{⊸}}{dθ} = ([ \begin{matrix}A_{a} & 0 \\ 0 & A_{b} \end{matrix} ] − [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ]R^{−1}[ \begin{matrix}(1 − γ^{2})D_{a}^{∗}C_{a} & D_{a}^{∗}C_{b} \\ D_{b}^{∗}C_{a} & D_{b}^{∗}C_{b} \end{matrix} ]) a^{⊸}(θ)

−[ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ]R^{−1}[ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ] b^{⊸}(θ)

+[ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ]R^{−1}[ \begin{matrix}γ^{2}B_{a}^{∗} \\ 0 \end{matrix} ] c^{⊸}(θ) + [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ]R^{−1}[ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ] y^{⊸}(θ), (C.9a)

1For notational convenience time (θ) dependence of the state-space matrices is suppressed.

\frac{d b^{⊸}}{dθ} = −[ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ] ([ \begin{matrix}0 & 0 \\ C_{a} & C_{b} \end{matrix} ] − [ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}(1 − γ^{2})D_{a}^{∗}C_{a} & D_{a}^{∗}C_{b} \\ D_{b}^{∗}C_{a} & D_{b}^{∗}C_{b} \end{matrix} ]) a^{⊸}(θ)

+ ([ \begin{matrix}−A_{a}^{∗} & 0 \\ 0 & −A_{b}^{∗} \end{matrix} ] + [ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ][ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ]) b^{⊸}(θ)

−[ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ][ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}γ^{2}B_{a}^{∗} \\ 0 \end{matrix} ] c^{⊸}(θ)

+[ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ] (I − [ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]) y^{⊸}(θ),  (C.9b)

\frac{d c^{⊸}}{dθ} = ([ C_{a}^{∗}D_{a} 0 ]R^{−1}[ \begin{matrix}(1 − γ^{2})D_{a}^{∗}C_{a} & D_{a}^{∗}C_{b} \\ D_{b}^{∗}C_{a} & D_{b}^{∗}C_{b} \end{matrix} ] − [ C_{a}^{∗}C_{a} 0 ]) a^{⊸}(θ)

[ C_{a}^{∗}D_{a} 0 ]R^{−1}[ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ] b^{⊸}(θ) − (A_{a}^{∗} + [ C_{a}^{∗}D_{a} 0 ]R^{−1}[ \begin{matrix}γ^{2}B_{a}^{∗} \\ 0 \end{matrix} ]) c^{⊸}(θ)

−[ C_{A}^{∗}D_{a} 0 ]R^{−}1[ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ] y^{⊸}(θ)  (C.9c)

with boundary conditions

a^{⊸}(0) = 0, b^{⊸}(h) = 0, c^{⊸}(h) = [ X_{a} 0 ] a^{⊸}(h),  (C.9d)

and

u^{⊸}(θ) = ([ \begin{matrix}0 & 0 \\ C_{a} & C_{b} \end{matrix} ] − [ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}(1 − γ^{2})D_{a}^{∗}C_{a} & D_{a}^{∗}C_{b} \\ D_{b}^{∗}C_{a} & D_{b}^{∗}C_{b} \end{matrix} ]) a^{⊸}(θ)

−[ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ] b^{⊸}(θ) + [ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}γ^{2}B_{a}^{∗} \\ 0 \end{matrix} ] c^{⊸}(θ)

+[ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ] y^{⊸}(θ),  (C.9e)

where

R:=[ \begin{matrix}(1 − γ^{2})(I + D_{a}^{∗}D_{a}) & (I + D_{a}^{∗}D_{b}) \\ (I + D_{b}^{∗}D_{a}) & (I + D_{b}^{∗}D_{b}) \end{matrix} ].

To see that R is invertible, recall that by Assumption 5.8 D_{a}(t) = D_{b}(t) for all t and hence, note that

R = [ \begin{matrix}(1 − γ^{2})I & I \\ I & I \end{matrix} ][ \begin{matrix}(I + D_{a}^{∗}D_{a}) & 0 \\ 0 & (I + D_{a}^{∗}D_{a}) \end{matrix} ].

Given γ > 0, both terms on the right-hand side are clearly invertible and the invertibility of R follows.

Now, define

y^{⊸} := [ \begin{matrix}F_{a} & F_{b} \\ (C_{a} + D_{a}F_{a}) & (C_{b} + D_{b}F_{b}) \end{matrix} ]z  (C.10)
= ([ \begin{matrix}0 & 0 \\ C_{a} & C_{b} \end{matrix} ] − [ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ][ \begin{matrix}V_{a}^{2} & 0 \\ 0 & V_{b}^{2} \end{matrix} ][ \begin{matrix}(B_{a}^{∗}X_{a}A_{a} + D_{a}^{∗}C_{a}) & 0 \\ 0 & (B_{b}^{∗}X_{b}A_{b} + D_{b}^{∗}C_{b}) \end{matrix} ]) z,

for some some arbitrary z ∈ 𝒳_{a} ⊕ 𝒳_{b}, where A_{a} and A_{b} are defined in equation (5.16) and X_{b} is a self-adjoint, positive-semidefinite, stabilising solution to the algebraic, Riccati equation (5.22) with the subscript a replaced by b. Moreover, let

v^{⊸} := [ \begin{matrix}V_{a}^{2} & 0 \\ 0 & V_{b}^{2} \end{matrix} ][ \begin{matrix}(B_{a}^{∗}X_{a}A_{a} + D_{a}^{∗}C_{a}) & 0 \\ 0 & (B_{b}^{∗}X_{b}A_{b} + D_{b}^{∗}C_{b}) \end{matrix} ]z.  (C.11)

Then, using equation (5.21a), it follows that

0 = [ \begin{matrix}(B_{a}^{∗}X_{a}A_{a} + D_{a}^{∗}C_{a}) & 0 \\ 0 & (B_{b}^{∗}X_{b}A_{b} + D_{b}^{∗}C_{b}) \end{matrix} ]z
+[ \begin{matrix}(I + D_{a}^{∗}D_{a} + B_{a}^{∗}X_{a}B_{a}) & 0 \\ 0 & (I + D_{b}^{∗}D_{b} + B_{b}^{∗}X_{b}B_{b}) \end{matrix} ](− v^{⊸}).  (C.12)

Again, this can all be expressed in terms of a TPBVP. From equation (5.16), it follows that the initial-condition response of

\frac{d e^{⊸}}{dθ} = [ \begin{matrix}A_{a} & 0 \\ 0 & A_{a} \end{matrix} ] e^{⊸}(θ) + [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ](− v^{⊸})(θ); e^{⊸}(0) = z,  (C.13a)
w^{⊸}(θ) = [ \begin{matrix}C_{a} & 0 \\ 0 & C_{b} \end{matrix} ] e^{⊸}(θ) + [ \begin{matrix}D_{a} & 0 \\ 0 & D_{b} \end{matrix} ](− v^{⊸})(θ),  (C.13b)

gives [ \begin{matrix}C_{a} & 0 \\ 0 & C_{b} \end{matrix} ]z+[ \begin{matrix}A_{a} & 0 \\ 0 & A_{b} \end{matrix} ]z, and that the zero, initial-condition response gives [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ](− v^{⊸})+

[ \begin{matrix}D_{a} & 0 \\ 0 & D_{b} \end{matrix} ](− v^{⊸}). Similarly, from equation (C.12),

\frac{d g^{⊸}}{dθ} = −[ \begin{matrix}A_{a} & 0 \\ 0 & A_{b} \end{matrix} ] g^{⊸}(θ) − [ \begin{matrix}C_{a} & 0 \\ 0 & C_{b} \end{matrix} ]w^{⊸}(θ); g^{⊸}(h) = [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ] e^{⊸}(h), (C.14a)
0 = [ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ] g^{⊸}(θ) + [ \begin{matrix}D_{a}^{∗} & 0 \\ 0 & D_{b}^{∗} \end{matrix} ]w^{⊸}(θ) + (− v^{⊸})(θ).  (C.14b)

164 Appendix to Chapter 5

Then, elementary manipulation of equations (C.13) and (C.14) yields the following TPBVP characterisation of the operator mapping z to y^{⊸}:

\frac{d e^{⊸}}{dθ} = [ \begin{matrix}(A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a}) & 0 \\ 0 & (A_{b} − B_{b}R_{b}^{−1}D_{b}^{∗}C_{b}) \end{matrix} ] e^{⊸}(θ)
−[ \begin{matrix}B_{a}R_{a}^{−1}B_{a}^{∗} & 0 \\ 0 & B_{b}R_{b}^{−1}B_{b}^{∗} \end{matrix} ] g^{⊸}(θ),  (C.15a)
\frac{d g^{⊸}}{dθ} = −[ \begin{matrix}C_{a}^{∗}R̃_{a}^{−1}C_{a} & 0 \\ 0 & C_{b}^{∗}R̃_{b}^{−1}C_{b} \end{matrix} ] e^{⊸}(θ)
−[ \begin{matrix}(A_{a}^{∗} − C_{a}^{∗}D_{a}R_{a}^{−1}B_{a}^{∗}) & 0 \\ 0 & (A_{b}^{∗} − C_{b}^{∗}D_{b}R_{b}^{−1}B_{b}^{∗}) \end{matrix} ] g^{⊸}(θ),  (C.15b)

with boundary conditions

e^{⊸}(0) = z, g^{⊸}(h) = [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ] e^{⊸}(h)  (C.15c)

and

y^{⊸}(θ) = ([ \begin{matrix}0 & 0 \\ C_{A} & C_{b} \end{matrix} ] − [ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ][ \begin{matrix}R_{a}^{−1} & 0 \\ 0 & R_{b}^{−1} \end{matrix} ][ \begin{matrix}D_{a} & 0 \\ 0 & D_{b} \end{matrix} ][ \begin{matrix}C_{a} & 0 \\ 0 & C_{b} \end{matrix} ]) e^{⊸}(θ)
−[ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ][ \begin{matrix}R_{a}^{−1} & 0 \\ 0 & R_{b}^{−1} \end{matrix} ][ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ] g^{⊸}(θ),  (C.15d)

where R_{a} :=I + D_{a}^{∗}D_{a}, R_{b} :=I + D_{b}^{∗}D_{b}, R̃_{a} :=I + D_{a}D_{a}^{∗} and R̃_{b} :=I + D_{b}D_{b}^{∗}. Note that from equation (C.1), (C.4) and (C.10),

x := C̃^{∗} (I − D̃ (D̃^{∗}JD̃)D̃^{∗}) C̃z  (C.16)
= [ \begin{matrix}F_{a}^{∗} & (C_{a}^{∗} + F_{a}^{∗}D_{a}^{∗}) \\ F_{b}^{∗} & (C_{b}^{∗} + F_{b}^{∗}D_{b}^{∗}) \end{matrix} ]( y^{⊸} − u^{⊸})
= ([ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ]

−[ \begin{matrix}(A_{a}^{∗}X_{a}B_{a} + C_{a}^{∗}D_{a}) & 0 \\ 0 & (A_{b}^{∗}X_{b}B_{b} + C_{b}^{∗}D_{b}) \end{matrix} ][ \begin{matrix}V_{a}^{2} & 0 \\ 0 & V_{b}^{2} \end{matrix} ][ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]) ( y^{⊸} − u^{⊸}).

Now define

r^{⊸} :=[ \begin{matrix}V_{a}^{2} & 0 \\ 0 & V_{b}^{2} \end{matrix} ][ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b} \end{matrix} ]( y^{⊸} − u^{⊸}),  (C.17)

C.1 Proof of Lemma 5.9 165

so that

x = [ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ]( y^{⊸} − u^{⊸}) + [ \begin{matrix}(A_{a}^{∗}X_{a}B_{a} + C_{a}^{∗}D_{a}) & 0 \\ 0 & (A_{b}^{∗}X_{b}B_{b} + C_{b}^{∗}D_{b}) \end{matrix} ](− r^{⊸}).

As in the preceding paragraphs it is possible to express all of this in terms of a TPBVP.

\frac{dm^{⊸}}{dθ} = [ \begin{matrix}A_{a} & 0 \\ 0 & A_{b} \end{matrix} ]m^{⊸}(θ) + [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ](− r^{⊸})(θ); m^{⊸}(0) = 0,  (C.18a)
p^{⊸} = [ \begin{matrix}C_{a} & 0 \\ 0 & C_{b} \end{matrix} ]m^{⊸}(θ) + [ \begin{matrix}D_{a} & 0 \\ 0 & D_{b} \end{matrix} ](− r^{⊸})(θ),  (C.18b)
\frac{d n^{⊸}}{dθ} = −[ \begin{matrix}A_{a}^{∗} & 0 \\ 0 & A_{b}^{∗} \end{matrix} ] n^{⊸}(θ) − [ \begin{matrix}C_{a}^{∗} & 0 \\ 0 & C_{b}^{∗} \end{matrix} ] p^{⊸}(θ)
−[ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ]( y^{⊸} − u^{⊸})(θ); n^{⊸}(h) = [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]m^{⊸}(h),  (C.19a)
x = n^{⊸}(0).  (C.19b)

Note that in equation (C.19), the final-condition response plus the response to the forcing term p^{⊸} gives [ \begin{matrix}(A_{a}^{∗}X_{a}B_{a} + C_{a}^{∗}D_{a}) & 0 \\ 0 & (A_{b}^{∗}X_{b}B_{b} + C_{b}^{∗}D_{b}) \end{matrix} ](− r^{⊸}) and that the response to the forcing term ( y^{⊸} − u^{⊸}) gives [ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ]( y^{⊸} − u^{⊸}). Using equation (C.17), r^{⊸} can be expressed in terms of m^{⊸}, n^{⊸} and ( y^{⊸} − u^{⊸}) only:

r^{⊸}(θ) = [ \begin{matrix}R_{a}^{−1} & 0 \\ 0 & R_{b}^{−1} \end{matrix} ] ([ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ] n^{⊸}(θ) + [ \begin{matrix}D_{a}^{∗}C_{a} & 0 \\ 0 & D_{b}^{∗}C_{b} \end{matrix} ]m^{⊸}(θ) + [ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]( y^{⊸} − u^{⊸})) .

Substituting this into, and further manipulating equations (C.18) and (C.19), yields the following TPBVP characterisation of the operator mapping ( y^{⊸} − u^{⊸}) to x:

\frac{dm^{⊸}}{dθ} = [ \begin{matrix}(A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a}) & 0 \\ 0 & (A_{b} − B_{b}R_{b}^{−1}D_{b}^{∗}C_{b}) \end{matrix} ]m^{⊸}(θ) − [ \begin{matrix}B_{a}R_{a}^{−1}B_{a}^{∗} & 0 \\ 0 & B_{b}R_{b}^{−1}B_{b}^{∗} \end{matrix} ] n^{⊸}(θ)
−[ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ][ \begin{matrix}R_{a}^{−1} & 0 \\ 0 & R_{b}^{−1} \end{matrix} ][ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]( y^{⊸} − u^{⊸})(θ),  (C.20a)
\frac{d n^{⊸}}{dθ} = −[ \begin{matrix}C_{a}^{∗}R̃_{a}^{−1}C_{a} & 0 \\ 0 & C_{b}^{∗}R̃_{b}^{−1}C_{b} \end{matrix} ]m^{⊸}(θ) − [ \begin{matrix}(A_{a}^{∗} − C_{a}^{∗}D_{a}R_{a}^{−1}B_{a}^{∗}) & 0 \\ 0 & (A_{b}^{∗} − C_{b}^{∗}D_{b}R_{b}^{−1}B_{b}^{∗}) \end{matrix} ] n^{⊸}(θ)
− ([ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ] − [ \begin{matrix}C_{a}^{∗} & 0 \\ 0 & C_{b}^{∗} \end{matrix} ][ \begin{matrix}D_{a} & 0 \\ 0 & D_{b} \end{matrix} ][ \begin{matrix}R_{a}^{−1} & 0 \\ 0 & R_{b}^{−1} \end{matrix} ][ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]) ( y^{⊸} − u^{⊸})(θ), (C.20b)

166 Appendix to Chapter 5

with boundary conditions

m^{⊸}(0) = 0, n^{⊸}(h) = [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]m^{⊸}(h)  (C.20c)

and

x = n^{⊸}(0).  (C.20d)

Finally, combining equations (C.9-C.20) yields the TPBVP

\frac{d}{dθ} ([ m^{⊸} ; a^{⊸} ; e^{⊸} ; c^{⊸} ; n^{⊸} ; b^{⊸} ; g^{⊸} ]) = [ \begin{matrix}E_{11} & E_{12} & E_{13} & E_{14} & E_{15} & E_{16} & E_{17} \\ E_{21} & E_{22} & E_{23} & E_{24} & E_{25} & E_{26} & E_{27} \\ E_{31} & E_{32} & E_{33} & E_{34} & E_{35} & E_{36} & E_{37} \\ E_{41} & E_{42} & E_{43} & E_{44} & E_{45} & E_{46} & E_{47} \\ E_{51} & E_{52} & E_{53} & E_{54} & E_{55} & E_{56} & E_{57} \\ E_{61} & E_{62} & E_{63} & E_{64} & E_{65} & E_{66} & E_{67} \\ E_{71} & E_{72} & E_{73} & E_{74} & E_{75} & E_{76} & E_{77} \end{matrix} ] [ m^{⊸} ; a^{⊸} ; e^{⊸} ; c^{⊸} ; n^{⊸} ; b^{⊸} ; g^{⊸} ] (θ),

with boundary conditions

[ \begin{matrix}m^{⊸}(0) \\ a^{⊸}(0) \\ e^{⊸}(0) \\ c^{⊸}(h) \\ n^{⊸}(h) \\ b^{⊸}(h) \\ g^{⊸}(h) \end{matrix} ] = [ \begin{matrix}0 \\ 0 \\ z \\ [ X_{a} 0 ] a^{⊸}(h) \\ [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]m^{⊸}(h) \\ 0 \\ [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ] e^{⊸}(h) \end{matrix} ] ,

and

x = n^{⊸}(0),

where the E_{ij}’s are defined in Appendix C.2. This completely characterises the operator C̃^{∗} (I − D̃ (D̃^{∗}JD̃)D̃^{∗}) C̃:=z ↦ x.

Let E:=Φ_{E(t)}(h, 0), where Φ_{E(t)}(t, τ) is the state-transition operator associated with E(t). It follows readily that

−Ẽ[ \begin{matrix}c^{⊸}(0) \\ n^{⊸}(0) \\ b^{⊸}(0) \\ g^{⊸}(0) \end{matrix} ] = [ \begin{matrix}(E_{43} − [ X_{a} 0 ]E_{23}) \\ (E_{53} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{13}) \\ E_{63} \\ (E_{73} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{33}) \end{matrix} ] z,  (C.21)

where

Ẽ = [ \begin{matrix}(E_{44} − [ X_{a} 0 ]E_{24}) & (E_{45} − [ X_{a} 0 ]E_{25}) & (E_{46} − [ X_{a} 0 ]E_{26}) & (E_{47} − [ X_{a} 0 ]E_{27}) \\ (E_{54} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{14}) & (E_{55} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{15}) & (E_{56} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{16}) & (E_{57} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{17}) \\ E_{64} & E_{65} & E_{66} & E_{67} \\ (E_{74} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{34}) & (E_{75} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{35}) & (E_{76} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{36}) & (E_{77} − [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]E_{37}) \end{matrix} ].

To see that Ẽ is invertible, take z = 0 and suppose that there exists a non-zero

[ \begin{matrix}c^{⊸} & 0 \\ n^{⊸} & 0 \\ b^{⊸} & 0 \\ g^{⊸} & 0 \end{matrix} ] ∈ 𝒦Ẽ.  (C.22)

This clearly satisfies equation (C.21) and since z = 0, the unique solution to equations (C.13) and (C.14) is y^{⊸} ≡ e^{⊸} ≡ g^{⊸} ≡ 0. Consequently, the unique solution to equations (C.6-C.8) is a^{⊸} ≡ b^{⊸} ≡ c^{⊸} ≡ 0 and thus, the only possible non-zero element of (C.22) is n^{⊸}_{0}. This contradicts linearity of the operator mapping z to x = n^{⊸}_{0} and hence, 𝒦Ẽ = {0}. Equation (5.29) (which constitutes a finite-dimensional characterisation of Q̀) now follows directly from equation (C.21).

Consider the operator À. Let z be an arbitrary element in 𝒳_{a} ⊕ 𝒳_{b} and recall that

Àz = (A − B (D̃^{∗}JD̃)^{−1} D̃^{∗}JC̃)z
= [ \begin{matrix}(A_{a} + B_{a}F_{a}) & 0 \\ 0 & (A_{b} + B_{b}F_{b}) \end{matrix} ]z + [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ] u^{⊸}_{1}
= [ \begin{matrix}A_{a} & 0 \\ 0 & A_{b} \end{matrix} ]z + [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ](− v^{⊸}) − [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ] u^{⊸}_{1}
= e^{⊸}(h) − a^{⊸}(h),

where u^{⊸}_{1}, a^{⊸}, v^{⊸} and e^{⊸} are defined in equations (C.3), (C.6a), (C.11) and (C.13a) respectively. Since z is arbitrary, the finite-dimensional characterisation of À given in equation (5.31) follows directly from equation (C.21).

Similarly, consider the bounded, linear operator

R̀ := B (D̃^{∗}JD̃)B^{∗}
= [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ] ([ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ][ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ] − [ \begin{matrix}γ^{2}V_{a}^{−2} & 0 \\ 0 & 0 \end{matrix} ])^{−1} [ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ]:=z ↦ x,

for z ∈ 𝒳_{a}⊕𝒳_{b}. If the initial condition in equation (C.13a) is set to 0 then e^{⊸}(θ) ≡ g^{⊸}(θ) ≡ y^{⊸}(θ) ≡ 0. Now replacing equation (C.2) by

y^{⊸}_{1} = [ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ]z,

equation (C.5) becomes

v^{⊸}_{1} := [ \begin{matrix}γ^{2}V_{a}^{−2} & 0 \\ 0 & 0 \end{matrix} ] u^{⊸}_{1} = [ \begin{matrix}γ^{2}(I + D_{a}^{∗}D_{a} + B_{a}^{∗}X_{a}B_{a}) & 0 \\ 0 & 0 \end{matrix} ] u^{⊸}_{1}  (C.23a)
= [ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ] u^{⊸} − y^{⊸}_{1} = [ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ] u^{⊸} + [ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ](−z).  (C.23b)

It then follows from equation (5.16) and (C.23b) that

\frac{d b^{⊸}}{dθ} = −[ \begin{matrix}A_{a}^{∗} & 0 \\ 0 & A_{b}^{∗} \end{matrix} ] b^{⊸}(θ) − [ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ] u^{⊸}(θ); b^{⊸}(h) = −z,  (C.24a)
v^{⊸}_{1}(θ) = [ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ] b^{⊸}(θ) + [ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ] u^{⊸}(θ).  (C.24b)

The only differences between equation (C.24) and equation (C.7) are the final condition and since it is identically 0, y^{⊸} has been omitted in (C.24). Similarly, using equation (C.23a), v^{⊸}_{1} can be expressed in terms of a^{⊸} and c^{⊸} defined in equations (C.6) and (C.8). Furthermore, it can be shown that

x = a^{⊸}(h).

Since m^{⊸} and n^{⊸} do not come into the equations defining a^{⊸}, b^{⊸} and c^{⊸}, they can be ignored. Hence, it follows that the action of the operator R̀ can be expressed in terms of a TPBVP with the same system of differential equations a the TPBVP characterisation of Q̀ but different boundary conditions, given by

[ \begin{matrix}m^{⊸}(0) \\ a^{⊸}(0) \\ e^{⊸}(0) \\ c^{⊸}(h) \\ θ(h) \\ b^{⊸}(h) \\ g^{⊸}(h) \end{matrix} ] = [ \begin{matrix}0 \\ 0 \\ 0 \\ [ X_{a} 0 ] a^{⊸}(h) \\ [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ]m^{⊸}(h) \\ −z \\ [ \begin{matrix}X_{a} & 0 \\ 0 & X_{b} \end{matrix} ] e^{⊸}(h) \end{matrix} ] .

Consequently, it is possible to write

R̀= [ E_{24} E_{26} ][ \begin{matrix}−E_{64} & −E_{66} \\ (E_{44} − [ X_{a} 0 ]E_{24}) & (E_{46} − [ X_{a} 0 ]E_{26}) \end{matrix} ]^{−1}[ \begin{matrix}I \\ 0 \end{matrix} ].

It remains to establish finite-dimensional expressions for the operators that appear in the two algebraic, Riccati equations associated with the normalised, coprime-factorisation of the two, original, LPTV systems. Consider

x:=[ \begin{matrix}R̀_{a} & 0 \\ 0 & R̀_{b} \end{matrix} ]z :=[ \begin{matrix}B_{a}(I + D_{a}^{∗}D_{a})^{−1}B_{a}^{∗} & 0 \\ 0 & B_{b}(I + D_{b}^{∗}D_{b})^{−1}B_{b}^{∗} \end{matrix} ]z,

for some arbitrary z ∈ 𝒳_{a} ⊕ 𝒳_{b}. Define

w^{⊸} := [ \begin{matrix}(I + D_{a}^{∗}D_{a})^{−1} & 0 \\ 0 & (I + D_{b}^{∗}D_{b})^{−1} \end{matrix} ][ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ]z,  (C.25a)

and

v^{⊸} := [ \begin{matrix}D_{a} & 0 \\ 0 & D_{b} \end{matrix} ]w^{⊸}.  (C.25b)

Then, from equation (C.25a),

w^{⊸} = [ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ]z − [ \begin{matrix}D_{a}^{∗} & 0 \\ 0 & D_{b}^{∗} \end{matrix} ] v^{⊸}.  (C.25c)

Using equations (5.16), it follows from equation (C.25b), that

\frac{d e^{⊸}}{dθ} = [ \begin{matrix}A_{a} & 0 \\ 0 & A_{a} \end{matrix} ] e^{⊸}(θ) + [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ]w^{⊸}(θ); e^{⊸}(0) = 0,  (C.26a)
w^{⊸}(θ) = [ \begin{matrix}C_{a} & 0 \\ 0 & C_{b} \end{matrix} ] e^{⊸}(θ) + [ \begin{matrix}D_{a} & 0 \\ 0 & D_{b} \end{matrix} ]w^{⊸}(θ),  (C.26b)

and from equation (C.25c), that

\frac{d g^{⊸}}{dθ} = −[ \begin{matrix}A_{a}^{∗} & 0 \\ 0 & A_{b}^{∗} \end{matrix} ] g^{⊸}(θ) − [ \begin{matrix}C_{a}^{∗} & 0 \\ 0 & C_{b}^{∗} \end{matrix} ] v^{⊸}(θ); g^{⊸}(h) = −z,  (C.27a)
w^{⊸} = −[ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ] g^{⊸}(θ) − [ \begin{matrix}D_{a}^{∗} & 0 \\ 0 & D_{b}^{∗} \end{matrix} ] v^{⊸}(θ),  (C.27b)

where the final-condition response gives [ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ]z and the zero, final-condition response gives

−[ \begin{matrix}D_{a}^{∗} & 0 \\ 0 & D_{b}^{∗} \end{matrix} ] v^{⊸}. Furthermore,

x:=[ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ]w^{⊸} = e^{⊸}(h).

Elementary manipulation of equations (C.27) and (C.26) leads to a TPBVP with dynamics described by equations (C.15a) and (C.15b) subject to the boundary conditions

e^{⊸}(0) = 0 and g^{⊸}(h) = −z.

By the structure of E and consequently, that of E:=Φ_{E(t)}(h, 0), it follows that

[ \begin{matrix}e^{⊸}(h) \\ g^{⊸}(h) \end{matrix} ] = [ \begin{matrix}E_{33} & E_{37} \\ E_{73} & E_{77} \end{matrix} ][ \begin{matrix}e^{⊸}(0) \\ g^{⊸}(0) \end{matrix} ]  (C.28)

and hence, since z is arbitrary, that

[ \begin{matrix}R̀_{a} & 0 \\ 0 & R̀_{b} \end{matrix} ] = −E_{37}E_{77}^{−1}.

Note that [ \begin{matrix}E_{33} & E_{37} \\ E_{73} & E_{77} \end{matrix} ] is invertible since it is simply the state-transmission operator associated with the differential system described by equations (C.15a) and (C.15b). To see that E_{77} is also invertible, suppose there exist a non-zero ˜ g^{⊸} ∈ 𝒦_{E_{77}} and take z = g^{⊸}(h) = 0. Then by linearity, x = e^{⊸}(h) = 0 and hence from equation (C.28) that

[ \begin{matrix}E_{37} \\ E_{77} \end{matrix} ] ˜ g^{⊸} = 0,

which contradicts the invertibility of [ \begin{matrix}E_{33} & E_{37} \\ E_{73} & E_{77} \end{matrix} ]. Hence, 𝒦_{E_{77}} = \{0\} and E_{77} is invertible.

The expression for Q̀_{a}, Q̀_{b}, À_{a} and À_{b} presented in the lemma statement follow similarly. This completes the proof of Lemma 5.9.

C.2 Definition of Matrices Used in Lemma 5.9 171

C.2 Definition of Matrices Used in Lemma 5.9

For notational convenience time-dependence is not expressed explicitly.

R_{a} := I + D_{a}^{∗}D_{a}, R̃_{a} :=I + D_{a}D_{a}^{∗},
R_{b} := I + D_{b}^{∗}D_{b}, R̃_{b} :=I + D_{b}D_{b}^{∗},
R := [ \begin{matrix}(1 − γ^{2})R_{a} & I + D_{a}^{∗}D_{b} \\ I + D_{b}^{∗}D_{a} & R_{b} \end{matrix} ],

S_{1} := [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ][ \begin{matrix}R_{a}^{−1} & 0 \\ 0 & R_{b}^{−1} \end{matrix} ][ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ],
S_{2} := (I − [ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]) ,
S_{3} := ([ \begin{matrix}0 & 0 \\ C_{a} & C_{b} \end{matrix} ] − [ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ][ \begin{matrix}R_{a}^{−1} & 0 \\ 0 & R_{b}^{−1} \end{matrix} ][ \begin{matrix}D_{a}^{∗} & 0 \\ 0 & D_{b}^{∗} \end{matrix} ][ \begin{matrix}C_{a} & 0 \\ 0 & C_{b} \end{matrix} ]) ,
S_{4} := ([ \begin{matrix}0 & 0 \\ C_{a} & C_{b} \end{matrix} ] − [ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}(1 − γ^{2})D_{a}^{∗}C_{a} & D_{a}^{∗}C_{b} \\ D_{b}^{∗}C_{a} & D_{b}^{∗}C_{b} \end{matrix} ]) ,

E_{11} := [ \begin{matrix}A_{a} − B_{a}R_{a}^{−1}D_{a}^{∗}C_{a} & 0 \\ 0 & A_{b} − B_{b}R_{b}^{−1}D_{b}^{∗}C_{b} \end{matrix} ],
E_{12} := S_{1}S_{2}S_{4},
E_{13} := −S_{1}S_{2}S_{3},
E_{14} := S_{1}[ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}γ^{2}B_{a}^{∗} \\ 0 \end{matrix} ],
E_{15} := −[ \begin{matrix}B_{a}R_{a}^{−1}B_{a}^{∗} & 0 \\ 0 & B_{b}R_{b}^{−1}B_{b}^{∗} \end{matrix} ],
E_{16} := −S_{1}[ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ],
E_{17} := S_{1}S_{2}S_{1}^{∗},

E_{21} := 0,^{E_{22} := ([ \begin{matrix}A_{a} & 0 \\ 0 & A_{b} \end{matrix} ] − [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ]R^{−1}[ \begin{matrix}(1 − γ^{2})D_{a}^{∗}C_{a} & D_{a}^{∗}C_{b} \\ D_{b}^{∗}C_{a} & D_{b}^{∗}C_{b} \end{matrix} ]) ,}^{E_{23} := [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ]R^{−1}[ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]S_{3},}^{E_{24} := [ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ]R^{−1}[ \begin{matrix}γ^{−2}B_{a}^{∗} \\ 0 \end{matrix} ],}^{E_{25} := 0,}^{E_{26} := −[ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ]R^{−1}[ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ],}^{E_{27} := −[ \begin{matrix}B_{a} & 0 \\ 0 & B_{b} \end{matrix} ]R^{−1}[ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]S_{1}^{∗},}^{E_{31} := 0,}^{E_{32} := 0,}^{E_{33} := E_{11},}^{E_{34} := 0,}^{E_{35} := 0,}^{E_{36} := 0,}^{E_{37} := E_{15},}^{E_{41} := 0,}^{E_{42} := ([ C_{a}^{∗}D_{a} 0 ]R^{−1}[ \begin{matrix}(1 − γ^{2})D_{a}^{∗}C_{a} & D_{a}^{∗}C_{b} \\ D_{b}^{∗}C_{a} & D_{b}^{∗}C_{b} \end{matrix} ] − [ C_{a}^{∗}C_{a} 0 ]) ,}^{E_{43} := −[ C_{a}^{∗}D_{a} 0 ]R^{−1}[ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]S_{3},}^{E_{44} := − (A_{a}^{∗} + [ C_{a}^{∗}D_{a} 0 ]R^{−1}[ \begin{matrix}γ^{2}B_{a}^{∗} \\ 0 \end{matrix} ]) ,}^{E_{45} := 0,}^{E_{46} := [ C_{a}^{∗}D_{a} 0 ]R^{−1}[ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ],} E_{47} := [ C_{a}^{∗}D_{a} 0 ]R^{−1}[ \begin{matrix}I & D_{a}^{∗} \\ I & D_{b}^{∗} \end{matrix} ]S_{1}^{∗},

E_{51} := −[ \begin{matrix}C_{a}^{∗}R̃_{a}^{−1}C_{a} & 0 \\ 0 & C_{b}^{∗}R̃_{b}^{−1}C_{b} \end{matrix} ],^{E_{52} := S_{3}^{∗}S_{4},}^{E_{53} := −S_{3}^{∗}S_{2}S_{3},}^{E_{54} := S_{3}^{∗}[ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}γ^{2}B_{a} \\ 0 \end{matrix} ],}^{E_{55} := −E_{11}^{∗},}^{E_{56} := −S_{3}^{∗}[ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ],}^{E_{57} := S_{3}^{∗}S_{2}S_{1}^{∗},}^{E_{61} := 0,}^{E_{62} := −[ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ]S_{4},}^{E_{63} := [ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ]S_{2}S_{3},}^{E_{64} := −[ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ][ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}γ^{2}B_{a}^{∗} \\ 0 \end{matrix} ],}^{E_{65} := 0,}^{E_{66} := − ([ \begin{matrix}A_{a}^{∗} & 0 \\ 0 & A_{b}^{∗} \end{matrix} ] − [ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ][ \begin{matrix}I & I \\ D_{a} & D_{b} \end{matrix} ]R^{−1}[ \begin{matrix}B_{a}^{∗} & 0 \\ 0 & B_{b}^{∗} \end{matrix} ]) ,}^{E_{67} := −[ \begin{matrix}0 & C_{a}^{∗} \\ 0 & C_{b}^{∗} \end{matrix} ]S_{2}S_{1}^{∗}}^{E_{71} := 0,}^{E_{72} := 0,}^{E_{73} := E_{51},}^{E_{74} := 0,}^{E_{75} := 0,}^{E_{76} := 0,}^{E_{77} := −E_{11}^{∗}.}

Appendix to Chapter 6

D.1 Proof of Lemma 6.5

Before proving Lemma 6.5, the following technical result is required. Lemma D.1 The matrix M_{11}^{∗}M_{21} = M_{21}^{∗}M_{11} and M̃_{11}M̃_{12}^{∗} = M̃_{12}M̃_{11}^{∗}. Proof : This proof is similar to that of Theorem 13.3 in [ZDG95]. From equation (6.27), it follows that

È[ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ] = [ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ]T_{11}

and therefore, that

[ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ]^{∗}JÈ[ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ] = [ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ]^{∗}J[ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ]T_{11},  (D.1)

where J is defined in equation (6.25). Since È is Hamiltonian, JÈ is symmetric and hence, so is the right-hand side of the equation (D.1). That is,

[ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ]^{∗}J[ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ]T_{11} = −T_{11}^{∗}[ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ]^{∗}J[ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ],

which leads to the Lyapunov equation

(−M_{11}^{∗}M_{21} + M_{21}^{∗}M_{11})T_{11} − (−T_{11}^{∗})(−M_{11}^{∗}M_{21} + M_{21}^{∗}M_{11}) = 0.

Since spec(T_{11}) ∩ spec(−T_{11}^{∗}) = ∅, this has unique solution −M_{11}^{∗}M_{21} + M_{21}^{∗}M_{11} = 0, and therefore, M_{21}^{∗}M_{11} is symmetric. Similarly, by noting from equation (6.27) that

[ M̃_{11} M̃_{12} ]È = T_{11}[ M̃_{11} M̃_{12} ],

it follows that M̃_{11}M̃_{12}^{∗} is symmetric.

Proof of Lemma 6.5: This proof is similar to that of Theorem 13.6 in [ZDG95]. From equation (6.27), it follows that

È[ \begin{matrix}M_{11} \\ M_{12} \end{matrix} ] = [ \begin{matrix}−À^{∗} & −C̀^{∗}C̀ \\ B̀B̀^{∗} & À \end{matrix} ][ \begin{matrix}M_{11} \\ M_{12} \end{matrix} ] = [ \begin{matrix}M_{11} \\ M_{12} \end{matrix} ]T_{11},  (D.2)

and therefore, that

−À^{∗}M_{11} − C̀^{∗}C̀M_{21} = M_{11}T_{11}.  (D.3)

Let (A, C_{1}) have no unobservable modes in ℂ_{>α} and suppose that M_{11} is column-rank deficient. That is, 𝒦_{M_{11}} ≠ {0}. For any x ∈ 𝒦_{M_{11}}, post-multiply (D.3) by x and pre-multiply by x^{∗}M_{21}^{∗}. Then

−x^{∗}M_{21}^{∗}À^{∗}M_{11}x − x^{∗}M_{21}^{∗}C̀^{∗}C̀M_{21}x = x^{∗}M_{21}^{∗}M_{11}T_{11}x,

which by the symmetry of M_{21}^{∗}M_{11} (cf. Lemma D.1) implies that

−x^{∗}M_{21}^{∗}C̀^{∗}C̀M_{21}x = 0.

Since M_{21}^{∗}C̀^{∗}C̀M_{21} is positive semi-definite, this implies C̀M_{21}x = 0. So simply post-multiplying (D.3) by x gives M_{11}T_{11}x = 0. Since T_{11} has full rank, it follows that 𝒦_{M_{11}} is T_{11}-invariant. That is there exists a λ ∈ ℂ and a corresponding eigenvector v ∈ 𝒦_{M_{11}}, such that

T_{11}v = λv.

Note that since spec(T_{11}) ⊂ ℂ_{>α}, λ > α ≥ 0. Now, from equation (D.2) it follows that

B̀B̀^{∗}M_{11} + ÀM_{21} = M_{21}T_{11},

which when post-multiplied by v gives,

(À − λI)M_{21}v = 0.

Recall that for v ∈ 𝒦_{M_{11}}, C̀M_{21}v = 0 and hence, note that

[ À − λI ; C̀ ]M_{21}v = 0.  (D.4)

176 Appendix to Chapter 6

Furthermore,

[ À − λI ; C̀ ] = [ \begin{matrix}I & γ^{−2}B_{1}R^{−1}D_{11}^{∗} & 0 \\ 0 & 0 & −I \\ 0 & S^{− \frac{1}{2}} & 0 \end{matrix} ] [ \begin{matrix}A − λI & B_{2} \\ C_{1} & D_{12} \\ 0 & λI \end{matrix} ] ,  (D.5)

where importantly the first matrix on the right-hand side has full rank. Since λ > α ≥ 0 and (A, C_{1}) has no unobservable modes in ℂ_{>α}, λI has full rank and [ \begin{matrix}A − λI \\ C_{1} \end{matrix} ] has full column rank. Thus, it follows from equation (D.5), that [ À − λI ; C̀ ] also has full column rank. From equation (D.4), this then implies that M_{21}v = 0, which since M_{11}v = 0 contradicts the fact that [ \begin{matrix}M_{11} \\ M_{21} \end{matrix} ] has full column rank. Therefore, if (A, C_{1}) has no unobservable modes in ℂ_{>α}, M_{11} has full column rank. Similarly, noting that

[ M̃_{11} M̃_{12} ]È = T_{11}[ M̃_{11} M̃_{12} ],

it can be shown that if (A, B_{1}) has no uncontrollable modes in ℂ_{>α}, then M̃_{11} has full row rank.

Bibliography

[AHI93] M. Araki, T. Hagiwara, and Y. Ito. Frequency-response of sampled-data systems II: Closed-loop consideration. In Proceedings of the 12th IFAC World Congress, pages 293–296, 1993.

[Ahl79] L. V. Ahlfors. Complex Analysis. Mathematics Series. McGraw-Hill International Editions, Singapore, 1979.

[AI93] M. Araki and Y. Ito. Frequency-response of sampled-data systems I: Open-loop consideration. In Proceedings of the 12th IFAC World Congress, pages 289–292, 1993.

[Bel62] R. E. Bellman. Applied Dynamic Programming. Princeton University Press, New Jersey, 1962.

[BGPK85] R. R. Bitmead, M. R. Gevers, I. R. Petersen, and R. J. Kaye. Monoticity and stabilisability properties of solutions of the Riccati difference equation: Propositions, lemmas, theorems, fallacious conjectures and counterexamples. Systems and Control Letters, 5:309–315, 1985.

[BLW91] S. Bittanti, A. J. Laub, and J. C. Willems, editors. The Riccati Equation, Berlin, 1991. Springer Verlag.

[Bol90] B. Bollobás. Linear Analysis : An Introductory Course. Cambridge Mathematical Textbooks. Cambridge University Press, 1990.

[BP92] B. A. Bamieh and J. B. Pearson. A general framework for linear periodic systems with applications to H_{∞} sampled-data control. IEEE Transactions of Automatic Control, 37:418–435, 1992.

[BPFT91] B. A. Bamieh, J. B. Pearson, B. A. Francis, and A. R. Tannenbaum. A lifting technique for linear periodic systems with applications to sampled-data control. Systems and Control Letters, 17:79–88, 1991.

[Bro70] R. W. Brockett. Finite-Dimensional Linear Systems. Wiley, New York, 1970.

[BS79] C. Bavely and G. W. Stewart. An algorithm for computing reducing subspaces by block diagonalisation. SIAM Journal of Numerical Analysis, 16:359–367, 1979.

[CD91] F. M. Callier and C. A. Desoer. Linear Systems Theory. Springer Verlag, New York, 1991.

[CF90] T. Chen and B. A. Francis. On the L_{2}-induced norm of a sampled-data system. Systems and Control Letters, 15:211–219, 1990.

[CF91] T. Chen and B. A. Francis. Input-output stability of sampled-data systems. IEEE Transactions on Automatic Control, 36:50–58, 1991.

[CF95] T. Chen and B. A. Francis. Optimal Sampled-Data Control Systems. Communications and Control Engineering Series. Springer-Verlag, London, 1995.

[CG96] M. W. Cantoni and K. Glover. A design-framework for continuous-time systems under sampled-data control. In Proceedings of the 35th IEEE CDC, pages 458–463, 1996.

[CG97] M. W. Cantoni and K. Glover. H_{∞} sampled-data synthesis and related numerical issues. Automatica, 33:2233–2241, 1997.

[Che84] C. T. Chen. Linear Systems Theory and Design. Holt, Rinehart and Winston, 1984.

[DFT92] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum. Feedback Control Theory. Maxwell Macmillan International, Oxford, 1992.

[DG84] C. A. Desoer and C. L. Gustafson. Algebraic theory of linear multivariable feedback systems. IEEE Transactions on Automatic Control, 29:909–917, 1984.

[DGG86] C. E. DeSouza, M. R. Gevers, and G. C. Goodwin. Riccati equations in optimal filtering of nonstabilisable systems having singular state transition matrices. IEEE Transaction on Automatic Control, 31:831–838, 1986.

[DGKF89] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis. State space solutions to standard H_{2} and H_{∞} control problems. IEEE Transaction on Automatic Control, 34:831–847, 1989.

[DGS93] J. C. Doyle, T. T. Georgiou, and M. C. Smith. The parallel projection operators of a nonlinear feedback system. Systems and Control Letters, 20:79–85, 1993.

[DL70] P. Dorato and A. H. Levis. Optimal linear regulators: The discrete-time case. IEEE Transactions on Automatic Control, 16:613–620, 1970.

[DLJS80] C. A. Desoer, Murray Liu, R-W., J., and R. Saeks. Feedback system design: The fractional representation approach to analysis and synthesis. IEEE Transactions on Automatic Control, 25:399–412, 1980.

[DS88] N. Dunford and J. T. Schwartz. Linear Operators, Part I : General Theory and Part II : Spectral Theory–Self Adjoint Operators in Hilbert Space. Wiley Classics Library. Wiley, New York, 1988.

[DS93] W. N Dale and M. C. Smith. Stabilizability and existence of system representations for discrete-time time-varying systems. SIAM Journal of Control and Optimization, 31:1538–1557, 1993.

[Dul94] G. E. Dullerud. Control of Uncertain Sampled-Data Systems. PhD thesis, University of Cambridge, England, 1994.

[DV75] C. A. Desoer and M. Vidyasagar. Feedback Systems: Input–Output Properties. Academic Press, New York, 1975.

[ES85] A. K. El-Sakkary. The gap metric: Robustness of stabilisation of feedback systems. IEEE Transactions on Automatic Control, 30:240–247, 1985.

[FF90] C. Foiaş and A. Frazho. The Commutant Lifting Approach to Interpolation Problems, volume 44 of Operator Theory : Advances and Applications. Birkhäuser Verlag, Berlin, 1990.

[FG88] B. A. Francis and T. T. Georgiou. Stability theory for linear time-invariant plants with periodic digital control. IEEE Transactions on Automatic Control, 33:820–83, 1988.

[FGS93] C. Foiaş, T. T. Georgiou, and M. C. Smith. Robust stability of feedback systems: A geometric approach using the gap metric. SIAM Journal of Control and Optimization, 31:1518–1537, 1993.

[Fra86] B. A. Francis. A Course in H^{∞} Control Theory. Springer-Verlag, Berlin, 1986.

[FS82] A. Feintuch and R. Saeks. System Theory : a Hilbert Space Approach. Pure and Applied Mathematics,. Academic Press, 1982.

[GA94] P. Gahinet and P. Apkarian. A linear matrix inequality approach to H_{∞} control. International Journal of Robust and Nonlinear Control, 4:421–448, 1994.

[Geo88] T. T. Georgiou. On the computation of the gap metric. Systems and Control Letters, 11:253–257, 1988.

[GL95] M. Green and D. J. N. Limebeer. Linear Robust Control. Information and Systems Science Series. Prentice Hall, New Jersey, 1995.

[GNVL79] G. H. Golub, S. Nash, and C. F. Van Loan. A hessenberg-schur method for the matrix problem AX + XB = C. IEEE Transactions on Automatic Control, 24:909–913, 1979.

[GS90] T. T. Georgiou and M. C. Smith. Optimal robustness in the gap metric. IEEE Transactions on Automatic Control, 35:673–686, 1990.

[GS93] T. T. Georgiou and M. C. Smith. Graphs, causality and stabilizability: Linear, shiftinvariant systems on L^{2}([0, ∞)). Mathematics of Control, Signals and Systems, 6:195– 223, 1993.

[GS94] G. C. Goodwin and M. Salgado. Frequency domain sensitivity functions for continuoustime systems under sampled-data control. Automatica, 30:1263–1270, 1994.

[GVL89] G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins Series in the Mathematical Sciences. The Johns Hopkins University Press, Baltimore, second edition, 1989.

[Hal82] P. R. Halmos. A Hilbert Space Problem Book. Springer Verlag, New York, 1982.

[Hof62] K. Hoffman. Banach Spaces of Analytic Functions. Prentice-Hall, New Jersey, 1962.

[Hyd95] R. A. Hyde. H^{∞} Aerospace Control Design - A VSTOL Flight Application. Advances in Industrial Control Series. Springer-Verlag, 1995.

[IG91] P. A. Iglesias and K. Glover. State-space approach to discrete-time H_{∞} control. International Journal of Control, 54:1031–1073, 1991.

[IOW97] W. Ionescu, C. Oară, and M. Weiss. General matrix pencil techniques for the solution of algebraic Riccati equations. IEEE Transactions on Automatic Control, 42:1085–1098, 1997.

[Kai80] T. Kailath. Linear Systems. Information and System Science Series. Prentice Hall, New Jersey, 1980.

Bibliography 181

[Kat66] T. Kato. Perturbation Theory for Linear Operators. Springer Verlag, New York, 1966.

[KF75] A. N. Kolmogorov and S. V. Fomin. Introductory Functional Analysis. Dover, New York, 1975.

[KH93] P. T. Kabamba and S. Hara. Worst-case analysis and design of sampled-data control systems. IEEE Transactions on Automatic Control, 38:1337–1357, 1993.

[KHI90] R. Kondo, S. Hara, and T. Itou. Characterisation of discrete-time H_{∞} controllers via bilinear transformation. In Proceedings of the 29th IEEE CDC, pages 1763–1768, 1990.

[Kre89a] R. Kress. Linear Integral Equations. Number 82 in Applied Mathematical Sciences. Springer-Verlag, 1989.

[Kre89b] E. Kreyszig. Introductory Functional Analysis With Applications. Wiley Classics Library. Wiley, New York, 1989.

[Kuč72] V. Kučera. A contribution to matrix quadratic equations. IEEE Transaction on Auto-

matic Control, 17:344–347, 1972.

[KVZ^{+}72] M. A. Krasnosel’skii, G. M. Vainikko, P. P. Zabreiko, Rutitskii, Y. B., and V. Y. Stetsenko. Approximate Solution of Operator Equations. Wolters-Noordhoff Publishing, Groningen, The Netherlands, 1972.

[Lax59] P. D. Lax. Translation invariant subspaces. Acta Mathematica, 101:163–178, 1959.

[LR95] P. Lancaster and L. Rodman. Algebraic Riccati Equations. Oxford Science Publications, Oxford, 1995.

[MAT92] The Maths Works Inc., Natick, Massachusetts. High-Performance Numeric Computation and Visualisation Software: Reference Guide, 1992.

[MG90] D. C. McFarlane and K. Glover. Robust Controller Design Using Normalised Coprime Factorisation Plant Descriptions. Lecture Notes in Control and Information Sciences. Springer-Verlag, Berlin, 1990.

[MH87] J. E. Marsden and M. J. Hoffman. Basic Complex Analysis. Freeman, New York, second edition, 1987.

[Nik86] N. K. Nikol’skii. Treatise on the Shift Operator. Springer-Verlag, 1986.

[NJB84] C. N. Nett, C. A. Jacobson, and N. J. Balas. A connection between state-spaces and doubly coprime fractional representations. IEEE Transactions on Automatic Control, 29:831–832, 1984.

[OS91] R. J. Ober and J. A. Sefton. Stability of control systems and graphs of linear systems. Systems and Control Letters, 17:265–280, 1991.

[Pap98] G. Papageorgiou. Robust Control System Design: H^{∞} Loop shaping and Aerospace Applications. PhD thesis, University of Cambridge, Cambridge, England, 1998.

[PLS80] T. Pappas, A. J. Laub, and N. R. Sandell. On the numerical solution of the discretetime algebraic Riccati equation. IEEE Transaction on Automatic Control, 25:631–641, 1980.

[RPK92] R. Ravi, A. M. Pascoal, and P. P Khargonekar. Normalized coprime factorizations for linear time-varying systems. Systems and Control Letter, 18:455–465, 1992.

[RR85] M. Rosenblum and J Rovnyak. Hardy Classes and Operator Theory. Oxford University Press, 1985.

[Rud86] W. Rudin. Real and Complex Analysis. Mathematics Series. McGraw-Hill International Editions, Singapore, 1986.

[Rud91] W. Rudin. Functional Analysis. Mathematics Series. McGraw-Hill International Editions, Singapore, 1991.

[SLC90] M. G. Safonov, D. N. J. Limebeer, and R. Y. Chiang. Simplifying the H^{∞} theory via loop-shifting, matrix pencil and descriptor concepts. International Journal of Control, 50:2467–2488, 1990.

[Smi89] M. C. Smith. On stabilisation and the existence of coprime factorisations. IEEE Transactions on Automatic Control, 34:1005–1007, 1989.

[SNF70] B. Sz-Nagy and C. Foiaş. Harmonic Analysis of Operators on Hilbert Space. North-Holland Publishing Company, Amsterdam, 1970.

[SO93] J. A. Sefton and R. J. Ober. On the gap metric and coprime factor perturbations. Automatica, 29:723–734, 1993.

[Sut93] W. A. Sutherland. Introduction to Metric and Topological Spaces. Oxford Science Publications, Oxford, 1993.

[Toi92] H. T. Toivonen. Sampled-data control of continuous-time systems with an H_{∞} optimality criterion. Automatica, 28:45–54, 1992.

[Toi93] H. T. Toivonen. Worst-case sampling for sampled-data H_{∞} design. In Proceedings of the 32nd IEEE CDC, San Antonio, Texas, pages 337–342, 1993.

[TS97] H. T. Toivonen and M. F. Sagfors. The sampled-data H_{∞} problem: A unified framework for discretization based methods and Riccati equation solution. International Journal of Control, 66:289–309, 1997.

[Vid78] M. Vidyasagar. Nonlinear Systems Analysis. Networks Series. Prentice Hall, New Jersey, 1978.

[Vid84] M. Vidyasagar. The graph metric for unstable plants and robustness estimates for feedback stability. IEEE Transactions on Automatic Control, 29:403–417, 1984.

[Vid85] M. Vidyasagar. Control System Synthesis : A Factorization Approach. The MIT Press, Cambridge, Massachusetts, 1985.

[Vin93a] G. Vinnicombe. Frequency domain uncertainty and the graph topology. IEEE Transactions on Automatic Control, 38:1371–1383, 1993.

[Vin93b] G. Vinnicombe. Measuring the Robustness of Feedback Systems. PhD thesis, University of Cambridge, England, 1993.

[Vin96] G. Vinnicombe. Personal communication, 1996.

[Vin98] G. Vinnicombe. Uncertainty and Feedback. Imperial College Press, London, In Press 1998.

[VK86] M. Vidyasagar and H. Kimura. Robust controllers for uncertain linear multivariable systems. Automatica, 22:85–94, 1986.

[VL78] C. F. Van Loan. Computing integrals involving the matrix exponential. IEEE Transactions on Automatic Control, 23:395–404, 1978.

[VSF83] M. Vidyasagar, H. Schneider, and B. A. Francis. Algebraic and topological aspects of feedback stabilisation. IEEE Transactions on Automatic Control, 27:880–894, 1983.

[Wil71] J. C. Willems. The Analysis of Feedback Systems, volume 62 of Research Monographs. The MIT Press, Cambridge, Massachusetts, 1971.

[Won85] W. M. Wonham. Linear Multivariable Control. Springer Verlag, Berlin, 1985.

[WR71] J. H. Wilkinson and C. Reinsch. Handbook for Automatic Computation, volume II Linear Algebra. Springer-Verlag, Berlin, 1971.

[YA94] Y. Yamamoto and M. Araki. Frequency responses for sampled-data systems - their equivalence and relationships. Linear Algebra and its Applications, 205-206:1319–1339, 1994.

[Yam94] Y. Yamamoto. A function space approach to sampled-data control systems and tracking problems. IEEE Transactions on Automatic Control, 39:703–713, 1994.

[YJB76] D. C. Youla, H. A. Jabr, and J. J. Bongiorno. Modern Wiener-Hopf design of optimal controllers - part II: The multivariable case. IEEE Transactions on Automatic Control, 21:319–338, 1976.

[YK96] Y. Yamamoto and P. P. Khargonekar. Frequency response of sampled-data systems. IEEE Transactions on Automatic Control, 41:166–176, 1996.

[You86] N. Young. The nevanlinna-pick problem for matrix-valued functions. Journal of Operator Theory, 15:239–265, 1986.

[Zam66] G. Zames. On the input-output stability for time-varying nonlinear feedback systems – Part I conditions using concepts of loop gain, conicity and positivity. IEEE Transactions on Automatic Control, 2:228–238, 1966.

[Zam81] G. Zames. Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms and approximate inverses. IEEE Transactions on Automatic Control, 26:301–320, 1981.

[ZDG95] K. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall, New Jersey, 1995.

[ZES80] G. Zames and A. K. El-Sakkary. Unstable systems and feedback: The gap metric. In Proceedings of the Allerton Conference, pages 380–385, 1980.