Robust Control System Design: H_{∞} Loop Shaping and Aerospace

Applications

George Papageorgiou

Darwin College Cambridge

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A dissertation submitted for the degree of Doctor of Philosophy July, 1998

“En oida oti ouden oida” (All I know is that I do not know anything)

Socrates

Sthn oikogeneia mou me agaph (to my family with love)

Abstract

In this thesis, a control system design procedure is developed that is particularly suited for designing flight controllers. The design procedure is based on H_{∞} loop shaping and recent theoretical developments that enable the synthesis of a robust gain scheduled controller. It is shown that this procedure can systematically deal with multi-input multi-output plants, plants with nonlinear and parameter-dependent dynamics, and uncertainty in the mathematical model of the plant used for the controller design. The designer can also build many of the closed loop specifications into the cost function used for the controller synthesis. To this end, H_{∞} loop shaping is discussed in some detail, a procedure for computing non-diagonal pre- and/or post-compensators is developed, and the H_{∞} loop shaping cost function is modified to include specifications on the transfer matrix from references to outputs. To be able to use the robust gain scheduling synthesis results, a “quasi-LPV” model of the longitudinal dynamics of an aircraft is developed. This model is tested by designing a robust gain scheduled flight controller for the High Incidence Research Model used in the GARTEUR design challenge. In an effort to increase the designer’s faith in the proposed procedure, a wind tunnel model was designed and built. This wind tunnel model is shown to be a challenging rig for active control experiments. A number of flight controllers were designed for the model and subsequently flight tested successfully.

Keywords: H_{∞} loop shaping, Weight selection, Model matching, Gain scheduling, LPV modelling, Flight control, Experimental rigs.

Preface

I am very grateful to my supervisor Professor Keith Glover. He has not only inspired my work, but has given me many opportunities, made sure my stay in Cambridge has been very comfortable, and showed understanding when I was not calm. I would also like to thank Dr. Glenn Vinnicombe and Dr. Jan Maciejowski for helping me out on control related topics, and Dr. Malcolm Smith for introducing me to robust and optimal control during my first two terms in Cambridge. My previous supervisor, Dr. Cowling, also deserves mention for urging me to pursue a PhD in control engineering.

There have been many stimulating discussions and entertaining moments up in the penthouse where I have a desk. This has been due to Gavin, Haig, Johannes, Mihai, Mike, Nelson, Richard F., Richard L., Richard W., Sinichi, Teddy and Tim. Mihai’s quick fixes and Richard Ford’s tea break timing have been quite outstanding! Thanks to Mihai and Richard F. who proof read this dissertation, the number of typos has been significantly reduced.

The wind tunnel model, described in Chapter 6, would not have been built without the support of the CUED workshops and design offices. Many thanks to Dave Green, Roger Hume, Samuel Lesley, Dr. Peter Long, Dave Nightingale, Alistair Ross, Richard Taylor and Tim Wilmshurst. I would also like to thank Dr. David Maull and Dr. Holger Babinsky for their advice, and Jason Waters and David Nunn for helping me during wind tunnel tests. A lot of work on the wind tunnel model was done by my fourth year students Kelvin Halsey and Chris Pedersen; it has been a great help. I am grateful to Dr. Rick Hyde from whom I learnt a lot about flight control.

This work has been funded by DERA Bedford and EPSRC. Special thanks to Dr. Yoge Patel, our contact at DERA Bedford, for making sure that this project ran as smoothly as possible.

Life in Cambridge would not have been as enjoyable without the Greek Society (a memorable occasion being the 1995 annual dinner), the Darwin Bball Club and its European tours, and the nights out with Ioannis, Kenneth, Mihai, Mike, Parham and Takis. I would like to thank my very good friends Aleko, Lefteri, Leonida, Saki and Vassili for the unpredictable times while on holiday in Greece! Hopefully, there are more to come!

I am extremely grateful to my parents for tolerating me for the last twenty six years (this has not been so difficult because I have been ever so pleasant!), my brothers Edward and Charles, and my grand-parents.

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.

This thesis is less than sixty two thousand words long.

George Papageorgiou

Darwin College Cambridge July 14, 1998

Acronyms

AMI  Affine Matrix Inequality
BIBO  Bounded-Input Bounded-Output
CoG  Centre of Gravity
CUED  Cambridge University Engineering Department
DERA  Defence Evaluation and Research Agency
DoF  Degree of Freedom
EoM  Equation of Motion
GARTEUR Group for Aeronautical Research and Technology in EURope
GHIRM  GARTEUR HIRM
HIRM  High Incidence Research Model
lcf  left coprime factorisation
LFT  Linear Fractional Transformation
LHP  Left-Half Plane
LMI  Linear Matrix Inequality
LTI  Linear Time-Invariant
LTV  Linear Time-Varying
LPV  Linear Parameter-Varying
LUT  Look-Up Table
MIMO  Multi-Input Multi-Output
MISO  Multi-Input Single-Output
MoI  Moment of Inertia
NLTV  NonLinear Time-Varying
rcf  right coprime factorisation
PI  Proportional and Integral filter
PIO  Pilot-Induced Oscillations
RCAM  Research Civil Aircraft Model
RHP  Right-Half Plane
SISO  Single-Input Single-Output
SVD  Singular Value Decomposition
VAAC  Vectored thrust Aircraft Advanced flight Control

Contents

1 Introduction 1

2 H_{∞} loop shaping 5 2.1 Motivation for H_{∞} control 5 2.2 Control system design 12 2.3 Overview of the design procedure 20 2.4 Justification of the set-up 20 2.5 Choice of weights 21 2.6 Designing controllers for coupled, ill-conditioned plants 28 2.7 The significance of the stability margin ϵ 30 2.8 The design cycle and design philosophy 34 2.9 Remarks on the architecture of flight controllers 37 2.10 A design example 38

2.10.1 Presentation of Model Used 38 2.10.2 Design Specifications 38 2.10.3 Scaling 39 2.10.4 Choice of Weights 40 2.10.5 Time Responses 42

2.11 Designing controllers for non-square plants 42 2.12 Limitations and ideal plant 44 2.13 H_{∞} loop shaping and other robust control design methods 45 2.14 Industrial viewpoint 46

3 Non-diagonal weights 48 3.1 Motivation for non-diagonal weights 48 3.2 The proposed design procedure 50 3.3 The design procedure requirements 51 3.4 Computing W_{1} and W_{2} 52 3.5 How well does the proposed design procedure work? 54 3.6 How would the designer use the proposed design procedure? 57 3.7 Some comments on the proposed design procedure 57 3.8 A design example 59

4 Robust model matching 66 4.1 Motivation for robust model matching 66 4.2 The guarantees required 67

4.2.1 The stability margin γ_{RS}^{−1} 68 4.2.2 The ideal reference model T_{0} 68 4.2.3 The performance margin γ_{RP}^{−1} 68 4.2.4 The nominal and robust performance levels δ_{NP} and δ_{RP} 69

4.3 The proposed setup 72 4.4 A note on nominal performance 78 4.5 A note on the history of the new setup 79 4.6 The proposed design methods 80

4.6.1 The first method 80 4.6.2 The second method 84 4.6.3 The third method 89 4.6.4 The fourth method 91

4.7 Some remarks on the new setup 93 4.8 H_{∞} control with preview 95 4.9 Design example 96

4.9.1 The first controller 97 4.9.2 The second controller 99 4.9.3 The third controller 100 4.9.4 A preview example 103

5 Robust gain scheduling 108 5.1 Motivation for robust gain scheduling 108 5.2 Single and parameter-dependent Lyapunov function approaches 111

5.2.1 Notes on the single Lyapunov function approach 116 5.2.2 Comments on the parameter-dependent Lyapunov function approach 120 5.2.3 Recent developments in LPV gain scheduling 123

5.3 LFT gain scheduling 124

5.3.1 Comments on the LFT method 126 5.4 Obtaining a reliable LPV model 127 5.5 A “quasi-LPV” model of the longitudinal aircraft dynamics 132 5.6 Synthesis of a robust gain scheduled controller 142 5.7 A design example 145

6 Design, development and testing of a wind tunnel model 169 6.1 Introduction 169 6.2 The wind tunnel model 170 6.3 Open loop analysis 176 6.4 Open loop identification 177 6.5 Control law design 179 6.6 Experimental results 185

7 Conclusions 196 7.1 The main contributions 196 7.2 Further work 198

Bibliography 200

Introduction

Fifty years ago, aircraft were not equipped with flight controllers. The stick, pedals and throttle level were all connected with mechanical links to the aircraft’s control surfaces and engines. In the last fifty years though, there has been a strong drive to make civil and military aircraft safer and easier to fly, less expensive to run, and more agile. A flight controller can address all the above issues, and consequently has become one of the most vital systems of an aircraft. What follows are a number of examples where automatic pilots can help improve the performance of an aircraft. By moving the centre of gravity of an aircraft further aft, it is possible to reduce its weight and make it more maneuverable. This makes the aircraft more fuel efficient and agile, but less stable and hence more difficult to fly, e.g. a pilot cannot fly an F-16 fighter aircraft without a flight controller as it is open loop unstable. Some aircraft impose a high workload on the pilot, e.g. the Harrier aircraft, and others are not built only to fly, e.g. stealthy aircraft. It is possible to enhance the performance (agility, safety and service life) of an aircraft by adding extra features to its flight controller, e.g. direct lift and sideforce capabilities, flutter suppression, and gust load alleviation.

The next generation fighter aircraft will be very maneuverable, because they will be able to pitch up to high angles of attack [Her80]. Flight at high angles of attack requires extra control power; part of the reason being that the rudder is in the wake of the fuselage. Consequently, the next generation fighter aircraft will be equipped with canards, vectored thrust, and possibly an actuator that will control the vortex shedding off the nose of the aircraft, e.g. nose suction [RJE90, RJE91]. Thus the flight envelope and configuration of fighter aircraft is changing, and designing a flight controller for such aircraft will be a challenging problem; their equations of motion are nonlinear and parameter-dependent, the aerodynamics at high angles of attack are very unsteady and uncertain, and it is not entirely clear how to best use the new actuators.

Classical control system design techniques have been successfully used to design flight controllers, but in our opinion have just about reached the limit of their capability. This is because they cannot deal systematically with multi-input and multi-output systems, and systems whose dynamics are nonlinear and parameter-dependent. Consequently, designing the flight controller of a future fighter aircraft with a classical control system design technique may require a large number of iterations, that could take a long time and cost a lot of money.

The performance of a flight controller is initially evaluated by performing exhaustive computer simulations based on a mathematical model of the aircraft. Then follow piloted simulations based on the same mathematical model used for the computer simulations, and if all is satisfactory the flight controller is subsequently flight tested. As flight testing is an expensive exercise, it is desired that the flight controller performs well on the first test flight. Unfortunately, this is not always the case as the mathematical model used for controller design is only an approximation of the true dynamics of the aircraft. In the last few years, the aircraft industry has suffered a number of setbacks due to the failure of the flight control system of an aircraft during flight testing, e.g. the YF-22 and SAAB Gripen both crashed during flight testing.

In this thesis, a control system design procedure is developed that is particularly suited for designing flight controllers, as it can systematically deal with multi-input multi-output plants, plants with nonlinear and parameter-dependent dynamics and uncertainty in the mathematical model of the plant used for the controller design, and the designer can build many of the flight controller specifications into the cost function used for the controller synthesis. This design procedure is essentially based on the H_{∞} loop shaping design procedure proposed by McFarlane and Glover [MG90], and recent theoretical developments that enable the synthesis of a gain scheduled controller for a linear parameter-varying (LPV) plant that guarantees robust stability of the resulting LPV closed loop system [BP94, Woo95]. The work described in this thesis can be seen as a continuation of the design work carried out by Hyde [Hyd95].

A flight controller designed using the procedure proposed in this thesis comes with a number of guarantees about the closed loop system, but it is not guaranteed that it will work on the real plant. If the guarantees with which the flight controller comes are “sensible”, then the chances that the controller will work on the real plant are much greater. Apart from demonstrating in this thesis that the proposed control system design procedure can be used to design controllers that come with “sensible” guarantees, a wind tunnel model is designed and built to assess the true performance of the designed flight controllers. The wind tunnel testing is a cheap intermediate step between computer simulations and flight testing, that can help pinpoint possible deficiencies of the controller and increase the designer’s faith in the proposed design procedure.

The thesis consists of six chapters, the contents of which are outlined below:

Chapter 2: H_{∞} loop shaping The aim of this chapter is to motivate the use of H_{∞} loop shaping as a control system design technique. After giving an outline of the design procedure and justifying the cost function minimised, insight is given on how the weights can be used to specify the desired closed loop behaviour. The significance of the stability margin is discussed in some detail, and a design cycle and philosophy are presented. A design example is used to illustrate how well the method works in practice. Finally, H_{∞} loop shaping is compared with other robust control system design techniques and the design approach currently used by the aerospace industry.

Chapter 3: Non-diagonal weights This chapter presents a procedure for computing a pre- and/or post-compensator that enable the designer to achieve the desired loop shape. How well the proposed procedure works in practice is discussed in some detail. At the end of the chapter a design example is presented.

Chapter 4: Robust model matching In this chapter it is shown how the H_{∞} loop shaping cost function can be modified to include specifications on the transfer matrix from references to outputs. Unfortunately, the resulting optimisation problem cannot be solved exactly. Four different ways of approximately solving the resulting optimisation problem are presented. The relative merits of each of the methods are discussed, and then illustrated by means of a design example. If the reference trajectory is known a number of time steps ahead, it is shown how the controller can take advantage of this information to improve certain closed loop properties.

Chapter 5: Robust gain scheduling Two different ways of synthesising robust gain scheduled controllers are presented together with their strengths and weaknesses. Both methods require an LPV model of the real plant. How to obtain an LPV model that accurately captures the dynamics of the real plant is discussed in some detail. A “quasi-LPV” model of the longitudinal dynamics of an aircraft is derived. This model is subsequently used to design a flight controller for the High Incidence Research Model that was used in the GARTEUR design challenge.

Chapter 6: Design, development and testing of a wind tunnel model In this chapter, the design and development of a wind tunnel model for active control experiments is described. The previous four chapters essentially develop a control system design technique that is particularly suited for designing flight controllers. A number of control laws are designed for the wind tunnel model and are subsequently flight tested successfully.

Introduction 4

Chapter 7: Conclusions In this last chapter the main contributions of the thesis are outlined, and some suggestions for future research are made.

H_{∞} loop shaping

The aim of this chapter is to motivate the use of H_{∞} loop shaping as a control system design procedure. It will be shown that an H_{∞} loop shaping controller comes with “sensible” guarantees about the resulting closed loop system. If the guarantees with which a controller comes are “sensible”, then the chances that the controller will work on the real plant are much greater.

2.1 Motivation for H_{∞} control

Generally speaking, the goal of a control system is to make some plant outputs (y) behave in a desired way by manipulating some plant inputs (u). This thesis deals with control system design. The approach taken is to design a control system by minimising the effect of some disturbances (d) on some errors (e). For example, the designer could attempt to design a controller by minimising the effect of the reference command (r) on r − y, i.e. choose r to be the disturbance and r − y to be the error. The energy of a signal will be used as a measure of the size of the signal. It will be shown in the sequel, that given a signal d of known energy, the energy of e can be bounded in terms of the energy of d and the “amplification” of the transfer matrix from d to e (T_{d→e}). Thus the control system will be designed by making the “amplification” of T_{d→e} sufficiently small.

The gain of a matrix A ∈ ℝ^{n×m} can be defined as \frac{∥Ax∥_{2}}{∥x∥_{2}} , where x ∈ ℝ^{m} is an input vector and ∥ · ∥_{2} denotes the Euclidean 2-norm. It can easily be deduced, after a few calculations, that the gain of A will depend on the direction of the input vector x. To see this we define the singular value decomposition (SVD) of a matrix (see [ZDG96, pp. 32-35]). For example the SVD of a matrix A ∈ ℝ^{2×2} is

A = [u_{1} u_{2}] [\begin{matrix}\overline{σ}(A) & 0 \\ 0 & \underline{σ}(A) \end{matrix}] [\begin{matrix}v_{1} & ∗ \\ v_{2} & ∗ \end{matrix}] ,

2.1 Motivation for H_{∞} control 6

where [u_{1} u_{2}] , [v_{1} v_{2}] ∈ ℝ^{2×2} are unitary matrices (see [ZDG96, p. 19]), \overline{σ}(A) denotes the maximum singular value of A, and \underline{σ}(A) the minimum singular value of A. Therefore as

[\begin{matrix}v_{1} & ∗ \\ v_{2} & ∗ \end{matrix}] [v_{1} v_{2}] = [\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}] ,

if x = v_{1} then the gain of A will be \overline{σ}(A) with Ax in the direction of u_{1}. Similarly, for x = v_{2} the gain of A will be \underline{σ}(A) with Ax in the direction of u_{2}.

Define the condition number of A as

κ(A) = \frac{\overline{σ}(A)}{\underline{σ}(A)}.

Hence an ill-conditioned matrix, i.e. a matrix with a large condition number, has significantly different gains in different directions. A “round matrix” is a matrix with a condition number near unity.

L_{∞} is a Banach space of matrix-valued functions that are essentially bounded on jℝ, with norm

∥G∥_{∞} := ess sup_{ω∈ℝ} \overline{σ}[G(jω)].

The rational subspace of L_{∞}, denoted by RL_{∞}, consists of all proper and real transfer matrices with no poles on the imaginary axis.

H_{∞} is a closed subspace of L_{∞} with functions that are analytic and bounded in the open right-half plane (RHP). The H_{∞} norm is defined as

∥G∥_{∞} := sup_{Re(s)>0} \overline{σ}[G(s)] = ess sup_{ω∈ℝ} \overline{σ}[G(jω)].

The real rational subspace of H_{∞} is denoted by RH_{∞}, and consists of all proper and real rational stable transfer matrices.

The Hilbert space L_{2}(−∞, ∞) consists of all square integrable and Lebesgue measurable vector-valued signals defined on ℝ with a 2-norm defined as

∥u∥_{2} := (∫_{−∞}^{+∞} u(t)^{∗}u(t)dt)^{\frac{1}{2}} < ∞.

Assume that we are given a stable linear time-invariant (LTI) system G with m inputs and p outputs, i.e. G(jω) ∈ ℂ^{p×m}. All inputs u(t) into the system are assumed to have finite energy, i.e. u(t) ∈ L_{2}. It is easy to prove (see [DFT92, pp. 18-25]) that if ∥u∥_{2} ≤ 1, the energy of the output y(t) will be bounded by

sup {∥y∥_{2} : ∥u∥_{2} ≤ 1} = ∥G∥_{∞}.  (2.1)

∥G∥_{∞} denotes the maximum gain of G over all frequencies and all input directions. Consequently, if for example we wanted good disturbance rejection at the output of a plant, we would try to make the ∞-norm of the output sensitivity sufficiently small.

L_{2}^{+} denotes the space L_{2}[0, ∞), consisting of Lebesgue measurable vector-valued signals that are square integrable over [0, ∞) and zero elsewhere. The space L_{2}[0, T] consists of Lebesgue measurable vector-valued signals that are square integrable over [0, T] and zero elsewhere. ∥ · ∥_{2,[0,T]} denotes the 2-norm of a signal in L_{2}[0, T]. The extended L_{2}^{+} space denoted by L_{2,e}^{+}, is defined as

L_{2,e}^{+} := {u ∈ L_{2}[0, T] for all T < ∞} .

Equation (2.1) also holds for u(t) ∈ L_{2,e}^{+}.

If P is the transfer matrix of a given plant model, then P = M̃^{−1}Ñ is a normalised left coprime factorisation (lcf ) of P, if M̃ and Ñ are stable rational coprime transfer matrices satisfying the normalisation constraint

ÑÑ^{∼} + M̃M̃^{∼} = I, ∀s,

where Ñ^{∼}(s) = [Ñ(−s̄)]^{∗}. P = NM^{−1} is a normalised right coprime factorisation (rcf ) of P (see [ZDG96, pp. 370-372]).

[figure: M; Δ; w_{1}; e_{1}; e_{2}; w_{2}]

Figure 2.1 The interconnected system

The motivation for using H_{∞} techniques to design robust controllers is provided by the small gain theorem. The notions of well-posedness and internal stability are defined in [ZDG96, pp. 119-125].

Theorem 2.1 [ZDG96] [Small Gain Theorem] Suppose M ∈ RH_{∞} and let γ > 0. Then the interconnected system in Fig. 2.1 is well posed and internally stable for all Δ(s) ∈ RH_{∞} with ∥Δ∥_{∞} ≤ 1/γ if and only if ∥M(s)∥_{∞} < γ.

Proof of sufficiency

Without loss of generality, assume γ = 1. It is clear that M(s)Δ(s) is stable since both M(s) and Δ(s) are stable. Thus, by the Nyquist stability theorem the closed loop system is stable if det(I − MΔ) has no zero in the closed right-half plane for all Δ ∈ RH_{∞} and ∥Δ∥_{∞} ≤ 1. Equivalently, the closed loop system is stable if

inf_{s∈ℂ̄_{+}} \underline{σ}(I − M(s)Δ(s)) ≠ 0

for all Δ ∈ RH_{∞} and ∥Δ∥_{∞} ≤ 1. But this follows from^{1}

inf_{s∈ℂ̄_{+}} \underline{σ}(I − M(s)Δ(s)) ≥ 1 − sup_{s∈ℂ̄_{+}} \overline{σ}(M(s)Δ(s)) =
= 1 − ∥M(s)Δ(s)∥_{∞} ≥ 1 − ∥M(s)∥_{∞} > 0.
□

Remark 2.2 The small gain theorem implies that the smaller the ∞-norm of the stable transfer matrix M, the larger the ∞-norm of the smallest stable perturbation Δ that destabilises the interconnected system [M, Δ].

A typical H_{∞} control problem boils down to minimising the ∞-norm of a closed loop transfer matrix F_{l}(P, K), see Fig. 2.2 where F_{l}(P, K) = P_{11} + P_{12}K(I − P_{22}K)^{−1}P_{21}, over all stabilising controllers K, i.e.

γ_{opt} = min_{stab K} ∥F_{l}(P, K)∥_{∞}.  (2.2)

A controller is said to be optimal if it achieves the optimal cost γ_{opt}. The synthesis of a stabilising controller that satisfies

∥F_{l}(P, K)∥_{∞} < γ,  (2.3)

with γ > γ_{opt}, is based on the solution of two Riccati equations [DGKF89]. Such a controller is said to be sub-optimal. In fact, all stabilising controllers that satisfy (2.3) are parametrised in [DGKF89]. Optimisation problem (2.2) is usually solved using a bisection algorithm, i.e. γ_{opt} is usually approximated by solving (2.3) for successively smaller values of γ (γ-iteration). P is sometimes called the generalised plant. The generalised plant we choose to look at depends on the type of uncertainty thought to be associated with the nominal plant model G, and the performance specifications.

The first step of a typical H_{∞} control system design procedure, is to decide on the uncertainty model to be used, see [ZDG96, Table 9.1]. This is difficult and requires good physical knowledge of the plant. For example, the real plant G_{Δ} in Fig. 2.2 is thought to be in the set Π = {G+Δ : Δ ∈ RH_{∞}}, i.e. we have chosen an additive uncertainty model. It can be shown that the interconnected system [G_{Δ}, K] is internally stable, if [G, K]

1For any A, B ∈ ℂ^{n×n}, \overline{σ}(A) < \underline{σ}(B) ⇒ det(A + B) ≠ 0.

[figure: P; K; y; u; Δ; G; Δ; K; d; T_{d→e} = F_{l}(P, K); d; e; u; y; e; G_{Δ}]

Figure 2.2 Robust stability with respect to additive uncertainty

and [F_{l}(P, K), Δ] are internally stable. [G, K] is internally stable if K is a stabilising controller, and [F_{l}(P, K), Δ] is internally stable for all Δ ∈ RH_{∞} with ∥Δ∥_{∞} ≤ 1/a if ∥F_{l}(P, K)∥_{∞} < a. It is therefore obvious that [G_{Δ}, K] will be internally stable if K is a stabilising controller that satisfies ∥F_{l}(P, K)∥_{∞} < a, i.e. minimisation (2.2) has a solution and γ_{opt} < a. It must be noted that performance specifications can be expressed as robust stability requirements.

The second step of the design procedure, is to choose frequency dependent weights and solve the optimisation problem (2.2), see [ZDG96, pp. 213-245]. The weights are chosen according to robustness and performance specifications. If the achieved γ is not satisfactory, then the weights are redesigned.

The small gain theorem can easily be modified to include weighting functions. Using similar arguments to those employed to prove the sufficiency of Theorem 2.1, and the following property of the determinant of a matrix

det(I − W^{−1}W F_{l}(P, K)Δ) = det(I − W F_{l}(P, K)ΔW^{−1}),

it can easily be shown that [F_{l}(P, K), Δ] is well posed and internally stable for all Δ(s) ∈ RH_{∞} with \overline{σ}(Δ(jω)W (jω)^{−1}) ≤ 1/γ(ω) if \overline{σ}(W (jω)F_{l}(P, K)(jω)) < γ(ω). The weighting function W (s) is assumed to be minimum phase and stable, and γ is now a function of frequency. Moreover, if \overline{σ}(W (jω)F_{l}(P, K)(jω)) < γ(ω) then (the dependence on ω is not shown)

A statement about nominal performance:

\overline{σ}(F_{l}(P, K)) = \overline{σ}(W^{−1}W F(P, K)) ≤ \overline{σ}(W^{−1})\overline{σ}(W F(P, K)) < γ/\underline{σ}(W).

A statement about robust stability: It can be deduced from the small gain theorem and the nominal performance bound above, that [F_{l}(P, K), Δ] remains well posed and internally stable for all Δ(s) ∈ RH_{∞} with \overline{σ}(Δ) ≤ \underline{σ}(W)/γ. This statement is conservative since there exist perturbations that satisfy \overline{σ}(ΔW^{−1}) ≤ 1/γ, but not \overline{σ}(Δ) ≤ \underline{σ}(W)/γ, e.g. if the weight is chosen to be W = diag(\overline{σ}, \underline{σ}) one such perturbation is Δ = diag(\overline{σ}/γ, \underline{σ}/γ).

Assume that the nominal plant model G in Fig. 2.2 has an unstable pole. The interconnected system [G_{Δ}, K] is well posed and internally stable for all G_{Δ} ∈ Π with ∥Δ∥_{∞} ≤ 1/γ, if K is a stabilising controller and ∥F_{l}(P, K)∥_{∞} < γ. This robust stability guarantee has no engineering value since Δ ∈ RH_{∞}, and thus the real plant must have the same unstable poles as the nominal plant model. It is almost certain that the unstable poles of the real plant will be different from those of the nominal plant model. It is shown in [Vin98, Section 2.3] using homotopy arguments, that [G_{Δ}, K] is well posed and internally stable for all Δ ∈ RL_{∞} with ∥Δ∥_{∞} ≤ 1/γ, if K is a stabilising controller, ∥F_{l}(P, K)∥_{∞} < γ, and G_{Δ} has the same number of unstable poles as G. Therefore, the set Π̂ = {G + Δ : Δ ∈ RL_{∞}, η(G + Δ) = η(G)} is much larger than Π (η(·) denotes the number of open RHP poles). In the case of coprime factor uncertainty, the difference between the corresponding sets Π̂_{cf} and Π_{cf} is much more subtle^{2}.

The sufficiency of the small gain theorem was established in [Zam66] for general nonlinear time-varying (NLTV) systems. Let G : L_{2,e}^{+} → L_{2,e}^{+}, and define the incremental gain of G as

g̃(G) := sup \frac{∥Gw − Gw̃∥_{2,[0,T]}}{∥w − w̃∥_{2,[0,T]}} ,

where the supremum is taken over all w, w̃ ∈ L_{2,e}^{+}, for which w ≠ w̃, and finite T > 0.

Theorem 2.3 [Zam66] Let the systems Δ : L_{2,e}^{+} ↦ L_{2,e}^{+} and M : L_{2,e}^{+} ↦ L_{2,e}^{+} in Fig. 2.2 have finite incremental gains such that g̃(Δ)g̃(M) < 1. Then for all w_{1}, w_{2} ∈ L_{2}^{+}, there exist unique solutions e_{1}, e_{2} ∈ L_{2}^{+}.

Remark 2.4 Assume that M = F_{l}(P, K), Fig. 2.2. Bounding the induced norm of M, ∥M∥_{∞} < γ, guarantees, subject to certain additional well-posedness assumptions, that the interconnected system [G_{Δ}, K] will be well posed and internally stable for any linear, possibly time-varying, infinite-dimensional perturbation Δ with an induced norm less than or equal to 1/γ. The same is true for any nonlinear Δ having g̃(Δ) ≤ 1/γ; G and K are assumed to be linear, but not necessarily LTI, and all that has been assumed about Δ is that it maps 0 to 0.

2It is usually the case that any G_{Δ} ∈ Π̂_{cf} is also in Π_{cf}, but the ∞-norm of the Δ ∈ RL_{∞} is smaller, see [Vin98, p. 40]

The small gain theorem for LTI plants subject to LTI perturbations (Theorem 2.1) is also necessary. In a similar context, the necessity of the small gain theorem for robust bounded-input bounded-output (BIBO) stability was established for LTI systems subject to linear time-varying (LTV) perturbations, and extended to LTV systems subject to LTV perturbations. A general result regarding the stable invertibility of a broad class of NLTV operators is established in [Sha91]. It is shown that an appropriately modified version of the small gain theorem, which makes use of the notion of asymptotic gain, is also necessary. This result is established for the class of NLTV systems with fading memory.

If the perturbation Δ has a known structure, then the small gain condition can be conservative. It is easy to modify the small gain theorem so that the structure of Δ is exploited. If M, Δ ∈ RH_{∞} this is done by using stable, minimum phase dynamic scalings D(s) that commute with Δ, i.e. DΔ = ΔD. Thus if D(s) can be chosen so that

\overline{σ}(D(jω)M(jω)D(jω)^{−1}) < \overline{σ}(M(jω)), ∀ω,  (2.4)

the small gain condition will be conservative.

This motivates the structured singular value μ. The matrix function μ(M) is defined with respect to an uncertainty structure

Δ = {diag(δ_{1}I_{r_{1}}, …, δ_{S}I_{r_{S}}, Δ_{1}, …, Δ_{F}) : δ_{i} ∈ ℂ, Δ_{j} ∈ ℂ^{m_{j}×m_{j}}} ⊂ ℂ^{n×n},  (2.5)

which consists of S repeated scalar blocks and F full blocks, where

∑_{i=1}^{S} r_{i} + ∑_{j=1}^{F} m_{j} = n.

The full blocks do not have to be square, but restricting them as such saves a great deal in terms of notation.

Definition 2.5 For M ∈ ℂ^{n×n}, μ_{Δ}(M) is defined as

μ_{Δ}(M) := \frac{1}{min\{\overline{σ}(Δ) : Δ ∈ Δ, det(I − MΔ) = 0\}},

unless no Δ ∈ Δ makes I − MΔ singular, in which case μ_{Δ}(M) := 0.

Let M(Δ) denote the set of all block diagonal and stable rational transfer functions, that have block structures such as Δ.

M(Δ) := {Δ(s) ∈ RH_{∞} : Δ(s_{o}) ∈ Δ ∀s_{o} ∈ ℂ̄_{+}}.

The modified small gain theorem is stated below.

[figure: d; e; M_{p}; Δ(s)]

Figure 2.3 Evaluating robust performance using μ

Theorem 2.6 [ZDG96] [μ Robust Stability Theorem] Let M ∈ RH_{∞} and β > 0. The loop in Fig. 2.1 is well posed and internally stable for all Δ(s) ∈ M(Δ) with ∥Δ(s)∥_{∞} < 1/β if and only if sup_{ω∈ℝ} μ_{Δ}(M(jω)) ≤ β.

The generalised plant M_{p} ∈ RH_{∞} in Fig. 2.3, has q_{1} + q_{2} inputs and p_{1} + p_{2} outputs. Let Δ ⊂ ℂ^{q_{1}×p_{1}} be a block structure defined as in (2.5), and define the augmented block structure

Δ_{p} := {diag(Δ, Δ_{f}) : Δ ∈ Δ, Δ_{f} ∈ ℂ^{q_{2}×p_{2}}}.

The following theorem can be used to quantify how the size of T_{d→e} changes in the face of uncertainty Δ(s). This theorem is of great engineering importance.

Theorem 2.7 [ZDG96] [μ Robust Performance Theorem] Let β > 0. For all Δ(s) ∈ M(Δ) with ∥Δ(s)∥_{∞} < 1/β, the loop shown in Fig. 2.3 is well posed, internally stable, and ∥F_{u}(M_{p}, Δ(s))∥_{∞} ≤ β if and only if sup_{ω∈ℝ} μ_{Δ_{p}}(M_{p}(jω)) ≤ β.

In Theorems 2.6 and 2.7 the perturbations are assumed to be stable. This is not a restriction with parametric uncertainty as constant parameters are clearly stable, but can be a restriction if the uncertainty is dynamic. This was discussed earlier on in this section. It can be shown that Theorems 2.6 and 2.7 are still valid if the perturbations are in RL_{∞}. The structured singular value can be calculated using ideas similar to those described by Equation (2.4). A more rigorous account of μ can be found in [PD93].

2.2 Control system design

There are two reasons for designing feedback systems: 1) disturbance rejection can only be achieved when using feedback, and 2) feedback can reduce the effect of modelling uncertainty on the closed loop system. Consider open loop control of a single-input singleoutput (SISO) plant model G with a dynamic filter Q. |S_{Δ}| can be used as a measure of how the transfer matrix from references to outputs changes (T_{r→y}: nominal, and T_{r→y}^{Δ}: perturbed), when the plant model is perturbed to G_{Δ}.

|S_{Δ}| := |\frac{T_{r→y}^{Δ} − T_{r→y}}{T_{r→y}^{Δ}} | = |\frac{G_{Δ}Q − GQ}{G_{Δ}Q}| = |\frac{G_{Δ} − G}{G_{Δ}} | .

Now consider feedback control of G with an internally stabilising controller, see Fig. 2.4. |S_{Δ}| becomes,

|S_{Δ}| = |\frac{(1 + G_{Δ}K)^{−1}G_{Δ}K − (1 + GK)^{−1}GK}{(1 + G_{Δ}K)^{−1}G_{Δ}K} | = |(1 + GK)^{−1}| |\frac{G_{Δ} − G}{G_{Δ}} | . (2.6)

Thus if K is suitably designed, feedback can be used to shrink the ball of uncertainty associated with T_{r→y}^{Δ}.

The control system design specifications are:

• closed loop stability,

• disturbance rejection at the input, and the output of the plant,

• tracking and output decoupling,

• noise rejection, and

• robustness with respect to loop perturbations,

within the limitations imposed by

• the plant, and

• measurement noise.

If the closed loop system in Fig. 2.4 is internally stable, then the following equations hold:

y = T_{o}(r − n) + S_{o}Gd_{i} + S_{o}d,  (2.7)
r − y = S_{o}(r − d) + T_{o}n − S_{o}Gd_{i},
u = KS_{o}(r − n) − KS_{o}d − T_{i}d_{i},  (2.8)
u_{p} = KS_{o}(r − n) − KS_{o}d + S_{i}d_{i},  (2.9)

where L_{o} = GK is the output loop transfer matrix, L_{i} = KG is the input loop transfer matrix, S_{o} = (I + L_{o})^{−1} is the output sensitivity transfer matrix, and T_{o} = I − S_{o} is the output complementary sensitivity transfer matrix. The input sensitivity transfer matrix (S_{i}) and the input complementary sensitivity transfer matrix (T_{i}) are defined in a similar way. It is simple to show that S_{o}G = GS_{i}, and S_{i}K = KS_{o}^{3}.

Disturbance rejection at the plant output y, Equation (2.7), can be achieved by making \overline{σ}(S_{o}) = 1/\underline{σ}(I + GK) small to reject d, and \overline{σ}(S_{o}G) = \overline{σ}(GS_{i}) small to reject d_{i} (for notational convenience the dependence of the singular values on ω is not shown). Similarly,

3S_{o}G = (I + GK)^{−1}G(I + KG)(I + KG)^{−1} = (I + GK)^{−1}(I + GK)G(I + KG)^{−1} = GS_{i}.

[figure: K; G; u; u_{p}; d_{i}; d; y; n; −; r]

Figure 2.4 A typical closed loop system

disturbance rejection at the plant input u_{p}, Equation (2.9), can be achieved by making \overline{σ}(S_{i}) = 1/\underline{σ}(I + KG) small to reject d_{i}, and \overline{σ}(S_{i}K) = \overline{σ}(KS_{o}) small to reject d. Since (some very important singular value inequalities are stated and proved in [GL95, pp. 31-33])

\begin{matrix}\underline{σ}(KG) − 1 ≤ \underline{σ}(I + KG) ≤ \underline{σ}(KG) + 1 \\ \underline{σ}(GK) − 1 ≤ \underline{σ}(I + GK) ≤ \underline{σ}(GK) + 1, \end{matrix}

we can deduce that if \underline{σ}(KG) > 1 and \underline{σ}(GK) > 1 then

\frac{1}{\underline{σ}(KG) + 1} ≤ \overline{σ}(S_{i}) ≤ \frac{1}{\underline{σ}(KG) − 1}
\frac{1}{\underline{σ}(GK) + 1} ≤ \overline{σ}(S_{o}) ≤ \frac{1}{\underline{σ}(GK) − 1}.

Therefore, for disturbance rejection of d at y, and d_{i} at u_{p}

\begin{matrix}\overline{σ}(S_{o}) ≪ 1 ⟺ \underline{σ}(GK) ≫ 1 \\ \overline{σ}(S_{i}) ≪ 1 ⟺ \underline{σ}(KG) ≫ 1. \end{matrix}

Also, if \underline{σ}(GK) ≫ 1 or \underline{σ}(KG) ≫ 1, and G and K are square and invertible frequency by frequency, then

• \overline{σ}(S_{o}G) ≈ 1/\underline{σ}(K). Therefore, for disturbance rejection of d_{i} at y the singular values of the controller should be large.

• \overline{σ}(KS_{o}) ≈ 1/\underline{σ}(G). This gives the input required to cancel the influence of d and n on u_{p}. This will be small if \underline{σ}(G) ≫ 1, but cannot be set by the designer and constitutes a limitation imposed by the plant.

The shape of the sensitivity function is constrained by RHP zeros (see [FL85]), and the control power available; the control power essentially imposes a constraint on the closed loop bandwidth. It can be deduced from Bode’s sensitivity integral with a loop bandwidth constraint, that good disturbance rejection at certain frequencies will result in amplification of disturbances at other frequencies. Consider a SISO plant model, i.e.

L = GK = KG, and suppose that L has poles p_{i} in the open RHP. If L has a relative degree of at least two (this assumption does not restrict the practical importance of the result), then the sensitivity function must satisfy

∫_{0}^{∞} ln |S(jω)|dω = π ∑_{i} Rep_{i}.  (2.10)

If L has no open RHP poles, then the above integral will be equal to zero. Consider also the following loop bandwidth constraint,

|L| < \frac{1}{ω^{2}} , ∀ ω ≥ ω_{0},

with ω_{0} > 1. It can be shown that the positive area over the interval [ω_{0}, ∞) is limited. Consequently, if |S| is made small over some subinterval of [0, ω_{0}], |S| must necessarily become larger somewhere else in [0, ω_{0}]. Bode’s sensitivity integral can be generalised to multi-input multi-output (MIMO) systems [ZDG96, pp. 145-147]. If L has RHP zeros then a stronger result can be obtained [ZDG96, p. 148].

Tracking and output decoupling^{4} are equivalent to rejecting d at the plant output because T_{o} + S_{o} = I. Therefore forcing S_{o} to zero forces T_{o} to the identity. The shape of the complementary sensitivity function is constrained by RHP poles, and the control power available. Disturbance rejection, tracking, and output decoupling are important at low frequency.

\overline{σ}(T_{o}), and thus \overline{σ}(L_{o}), must be made small in the frequency range where noise rejection is required. Noise rejection at low frequency conflicts with disturbance rejection as T_{o} + S_{o} = I. If good quality sensors are used, noise rejection is only important at high frequency.

\overline{σ}(L) \underline{σ}(L)  log ω ω_{h} ω_{l}

Figure 2.5 Loop gain boundaries

Often good nominal performance boils down to achieving high loop gain at low frequency, and low loop gain at high frequency. Fig. 2.5 illustrates the desired shape of L_{o} and L_{i}. High low-frequency loop gain is usually achieved with a controller that has high gain at low frequency. The controller must not have high gain outside the open loop bandwidth, since large loop gains outside the bandwidth of G can make the control activity quite unacceptable, Equation (2.8).

4By good tracking it is meant that y_{i} follows r_{i}, and by output decoupling it is meant that r_{i} does not significantly affect other outputs apart from y_{i}.

Consider for the moment a SISO plant model. The Nyquist stability criterion can be used to check whether the closed loop system is stable. If G is a stable plant model, then according to the Nyquist stability criterion the loop gain |L| must not encircle or pass through the -1 point. The designer must now consider the phase of L. It is essential that the closed loop system is robust with respect to loop perturbations, see Fig. 2.6. Loop perturbations arise because of modelling errors and controller implementation. Assume that L is perturbed to

L_{Δ} = (1 + δ_{T})L(1 − δ_{S})^{−1},

and consider the allowable sizes of δ_{T} and δ_{S} that ensure stability. |δ_{S}| (inverse multiplicative uncertainty) can be large at frequencies where |S| is small, i.e. in the low frequency range. |δ_{T} | (multiplicative uncertainty) can be large at frequencies where |T | is small, i.e. in the high frequency range. Thus the closed loop system is guaranteed to tolerate large perturbations of the loop transfer function at high and low frequency, if the shape of |L| does not violate the boundaries in Fig. 2.5.

[figure: δ_{T}; δ_{S}; G; K]

Figure 2.6 Perturbed closed loop system

At frequencies around the loop cross-over (ω_{c})

|S(jω)^{−1}| = |1 + L(jω)| = \sqrt{1 + |L(jω)|^{2} − 2|L(jω)| cos(π + ∠L(jω))} := α
|T (jω)^{−1}| = |1 + L(jω)^{−1}| = \frac{α}{|L(jω)|}

For good robustness with respect to loop perturbations around ω_{c}, the designer must keep |S| and |T | small in that region. As |L(jω_{c})| = 1,

|1 + L(jω_{c})| = |1 + L(jω_{c})^{−1}| = 2 |sin \frac{π + ∠L(jω_{c})}{2}| ,

where π + ∠L(jω_{c}) is the phase margin. Bode’s gain and phase relation for a minimum phase and stable loop transfer function,

∠L(jω_{0}) = \frac{1}{π} ∫_{−∞}^{∞} \frac{d ln |L|}{dν}f(ν)dν,

where

ν = ln(\frac{ω}{ω_{0}} ), f(ν) = ln coth \frac{|ν|}{2} = ln |\frac{ω + ω_{0}}{ω − ω_{0}} |,

relates ∠L(jω_{0}) to the slope of |L|. The slope of |L| is weighted by f(ν). This function takes large values around ω_{0}. Therefore, if the slope of |L| is big near the loop cross-over frequency, π +∠L(jω_{c}) will be small and |S|, |T | will be large resulting in poor robustness. In addition, the step response of y will overshoot. The smaller the slope of |L| around ω_{c}, the more robust the closed loop system is likely to be^{5}. Bode’s gain and phase relation can be modified if L has RHP poles and/or zeros [DFT92, pp. 130-132].

Thus, for good robustness and nominal performance the slope of |L| around ω_{c} should be less than -40dB/decade^{6}. Although Bode’s gain and phase relation does not carry through to multi-input multi-output systems, the generalisation of Equation (2.10) to MIMO systems with a loop bandwidth constraint indicates, that the same guideline applies to the slope of the singular values of L_{o} and L_{i}.

Remark 2.8 Control systems are sensitive to uncertainty around the loop gain bandwidth, as |S| and |T | can simultaneously take large values in that region.

Consider setting up an H_{∞} control problem. The first step of an H_{∞} control system design procedure is to choose the generalised plant. The above discussion motivates using inverse multiplicative and multiplicative uncertainty models. The inverse multiplicative uncertainty model will be used to guarantee robust stability with respect to loop perturbations at low frequency, and the multiplicative uncertainty model to guarantee robust stability with respect to loop perturbations at high frequency. Around the loop cross-over, both uncertainty models have similar structures as (1 − δ_{S})^{−1} ≃ 1 + δ_{S} if |δ_{S}| ≪ 1. For MIMO systems the location of the uncertainty is also important^{7}. The block diagram corresponding to the generalised plant chosen is depicted in Fig. 2.7.

This corresponds to solving the following optimisation problem:

γ_{opt} = min_{stab K} ∥[\begin{matrix}W_{S}S \\ W_{T} T \end{matrix}]∥_{∞} .  (2.11)

The second step of the design procedure is to choose frequency dependent weights. The control system requirements dictate that W_{T}^{−1} is a low-pass filter and W_{S}^{−1} a high-pass filter. Both weights are usually stable and minimum phase. The use of unstable and

5One can find examples where the slope of |L| around ω_{c} is not too big but the system is not robust, e.g. systems with a lightly damped pole above the bandwidth are not very robust to gain perturbations.

6If d ln |L|/dν = −c then ∠L(jω_{0}) = −cπ/2, [DFT92, p. 113].

7Dynamic uncertainty models are chosen according to whether robust stability is desired with respect to loop perturbations at high or low frequency, and at the plant input or output.

[figure: G; K; W_{T}; W_{S}; δÑ; δM̃]

Figure 2.7 Suggested setup

nonproper weights in H_{∞} control is discussed in [Mei95]. The closed loop system [G_{Δ}, K] will remain internally stable for all δ_{T} , δ_{S} ∈ RH_{∞} with ∥[δ_{T} W_{T}^{−1} δ_{S}W_{S}^{−1}]∥_{∞} ≤ 1/γ, where γ is the achieved norm bound, G_{Δ} = (1 − δ_{S})^{−1}G(1 + δ_{T}), and δ_{S}, δ_{T} are defined in Fig. 2.6. The following level of nominal performance is guaranteed

|S| ≤ \frac{γ}{|W_{S}|}, |SG| ≤ \frac{γ|G|}{|W_{S}|}, |KS| ≤ \frac{γ}{|G| |W_{T} |}, |T | ≤ \frac{γ}{|W_{T} |}.

Some level of robust performance is also guaranteed,

|S_{Δ}| = |\frac{(1 − L_{Δ})^{−1}L_{Δ} − (1 − L)^{−1}L}{(1 − L_{Δ})^{−1}L_{Δ}} | = |(1 − L)^{−1}| |\frac{L_{Δ} − L}{L_{Δ}} | .

Note that

|\frac{L_{Δ} − L}{L_{Δ}} | = |\frac{δ_{T} + δ_{S}}{1 + δ_{T}} | .

The first part of this section discusses how the designer can satisfy the control system requirements by shaping the loop gain. In the case of this H_{∞} control problem the designer shapes the loop gain indirectly using the weighting functions W_{T} and W_{S}. As S and T are related by the equation S + T = 1 (note that a SISO plant model is considered), if W_{T} and W_{S} are chosen separately, then the resulting value of γ could be large; γ can be considered as an indicator of whether the desired closed loop properties can be achieved (note that γ_{opt} is flat across frequency). At low and high frequency it is relatively easy to make sure that the weights do not conflict with each other; the difficulty arises around the desired loop bandwidth.

We are now going to consider an alternative weighting scheme that leads to the celebrated H_{∞} loop shaping design procedure. The uncertainty structure is the same as before and the new setup is given in Fig. 2.8. The designer now chooses weights W_{1} and W_{2} with the only constraint that there are no pole/zero cancellations in the RHP between the weights and G. It is obvious that by shaping the loop gain the designer cannot shape

[figure: K_{∞}; W_{T}; W_{S}; δÑ; δM̃; G; W_{1}; W_{2}]

Figure 2.8 Alternative weighting scheme

S and T separately. Note that we have not as yet chosen the weights W_{S} and W_{T}.

L = (1 − W_{S}δM̃)^{−1}W_{2}GW_{1}(1 + W_{T} δÑ)K_{∞}
= (1 − W_{S}δM̃)^{−1}M̃^{−1}Ñ(1 + W_{T} δÑ)K_{∞}
= (M̃ − M̃W_{S}δM̃)^{−1}(Ñ + ÑW_{T} δÑ)K_{∞},

where {M̃, Ñ} is a lcf of G_{s} = W_{2}GW_{1}. If the factorisation is normalised, then at every frequency

|M̃| = \frac{1}{\sqrt{1 + |G_{s}|^{2}}} , |Ñ| = \frac{|G_{s}|}{\sqrt{1 + |G_{s}|^{2}}} .

Hence if the weights W_{1} and W_{2} are chosen so that |G_{s}| follows the trends illustrated in Fig. 2.5, M̃ is a high-pass filter and Ñ is a low-pass filter. If W_{T} = Ñ^{−1} and W_{S} = M̃^{−1}, then the weights W_{T} and W_{S} will not conflict with each other around the loop bandwidth (this will become apparent later on in this chapter). The block diagram in Fig. 2.8 can now be redrawn, see Fig. 2.9. Hence the choice of W_{T} = Ñ^{−1} and W_{S} = M̃^{−1} leads

[figure: K_{∞}; Ñ; M̃^{−1}; δÑ; δM̃]

Figure 2.9 Coprime factor uncertainty

to a robust stabilisation problem with respect to uncertainty in the coprime factors of G_{s} (coprime factor uncertainty). It will be shown in the rest of this chapter that this is the most sensible H_{∞} control problem to consider; nominal and robust performance of a certain level can be guaranteed by solving a robust stability problem. Furthermore, it extends naturally to MIMO systems which is not the case for the first problem considered.

2.3 Overview of the design procedure

The H_{∞} loop shaping design procedure, which was proposed by McFarlane and Glover [MG90], is described below:

1. Shape G open loop, with frequency dependent weights W_{1} and W_{2}, according to closed loop objectives. The weighted plant, G_{s} = W_{2}GW_{1}, is depicted in Fig. 2.10.

[figure: K_{∞}; W_{1}; G; W_{2}; z_{2}; z_{1}; w_{2}; w_{1}]

Figure 2.10 The H_{∞} loop shaping standard block diagram

2. Minimise the ∞-norm of the transfer matrix from disturbances w_{1} and w_{2} to outputs z_{1} and z_{2} over all stabilising controllers K_{∞}.

min_{stab K_{∞}} ∥[\begin{matrix}w_{1} \\ w_{2} \end{matrix}] → [\begin{matrix}z_{1} \\ z_{2} \end{matrix}]∥_{∞} = min_{stab K_{∞}} ∥[\begin{matrix}K_{∞} \\ I \end{matrix}] (I − G_{s}K_{∞})^{−1} [I G_{s}]∥_{∞} = γ.

(2.12)

Check the achieved γ; γ is a measure of how robust the desired loop shape is.

3. Choose the position of K_{∞} in the loop. Model reduce the controller, and design the command pre-filters.

4. Check the time simulations and frequency responses of the resulting closed loop system, to verify robust performance. Reiterations may be required.

2.4 Justification of the set-up

Nominal performance is specified by shaping the singular values of the plant G with weights W_{1} and W_{2}, see Fig. 2.10. The theoretical basis for H_{∞} loop shaping is that K_{∞} does not modify the desired loop shape significantly at low and high frequencies, if γ is not too large [ZDG96, Theorems 18.9 and 18.10]. Thus shaping G, corresponds to shaping the loop gains GK and KG, where K = W_{1}K_{∞}W_{2}. All closed loop transfer functions can be bounded in terms of G_{s}, W_{1}, W_{2} and γ [ZDG96, pp. 493-494]. The fact that K_{∞} is well-conditioned becomes apparent by examining the cost function minimised, and noting that this transfer matrix can be written in two ways, see Equation (2.20).

When setting up a robust control problem, a decision has to be made about the type of uncertainty to be used. This can be difficult as it requires good knowledge of the plant. γ is a measure of how robust the desired loop shape is to perturbations in its coprime factors. Coprime factor uncertainty is a general type of uncertainty, in the same way that SISO gain and phase margins are (robustness with respect to loop gain perturbations). It is of engineering importance that the real plant and nominal plant model, do not have to have the same number of RHP poles and zeros. When there is little detailed knowledge about the uncertainty present in a plant, H_{∞} loop shaping is a good method for designing robust controllers.

The cost function minimised provides robust stability. It will be shown in Section 2.6 that robust performance of a certain level is also guaranteed. Calculating the best γ requires no γ-iteration [MG90]. Application of the method to real plants has shown that a value of ϵ = 1/γ between 0.2 − 0.3 is satisfactory, in the same way that a gain margin of ±6dB, and a phase margin of 45^{o} are for a SISO design. For a suitably scaled weighted plant, ϵ is a nondimensional quantity that characterises the behaviour of the closed loop system. Hereafter, ϵ will be referred to as the “stability margin”.

H_{∞} loop shaping is a very intuitive method for designing robust controllers, as the notions of classical loop shaping readily carry through. The designer can specify closed loop requirements, such as disturbance and noise rejection, by simply shaping G open loop. An important feature of H_{∞} loop shaping is that it enables the designer to push for the best achievable closed loop performance subject to a required level of robustness. This is because the designer has control over the cross-over frequencies of the singular values of G_{s}. In contrast with classical loop shaping, the designer does not need to explicitly shape the phase of G.

In [SG90] it is shown that K_{∞} can be written as an exact plant observer with state feedback. Hence H_{∞} loop shaping controllers can be gain scheduled; this was done successfully in [Hyd95]. The observer structure also adds visibility, and enables the handling of actuator limitations. The poles of K_{∞} are usually clustered around the bandwidth of G_{s}. Therefore the designer has some control, over how fast the poles of the controller are.

2.5 Choice of weights

An H_{∞} loop shaping controller can be implemented in three ways: in the forward path, in the feedback path, and in the observer form.

The controller K_{∞} = Ṽ^{−1}Ũ in Fig. 2.11 has been implemented in the observer form. Formulae for the observer and state feedback gains can be derived by solving the dual of the problem solved in [SG90].

[figure: W_{2}; N_{dc}; Ṽ^{−1}; W_{1}; W_{2}; Ũ; r; y_{i}; d_{i}; u; y_{o}; d_{o}; y; G]

Figure 2.11 The implemented closed loop

Theorem 2.9 [ZDG96] Let {M, N} be a right, and {M̃, Ñ} a left coprime factorisation of G_{s} over RH_{∞}. Then the set of all proper controllers achieving internal stability is parametrised either by,

K_{∞} = (U_{0} + MQ_{r})(V_{0} + NQ_{r})^{−1}, det(I + V_{0}^{−1}NQ_{r})(∞) ≠ 0

for Q_{r} ∈ RH_{∞} or by

K_{∞} = (Ṽ_{0} + Q_{l}Ñ)^{−1}(Ũ_{0} + Q_{l}M̃), det(I + Q_{l}ÑṼ_{0}^{−1})(∞) ≠ 0

for Q_{l} ∈ RH_{∞} where U_{0}, V_{0}, Ũ_{0}, Ṽ_{0} ∈ RH_{∞} satisfy the Bezout identities:

Ṽ_{0}M − Ũ_{0}N = I, M̃V_{0} − ÑU_{0} = I.

Moreover, if U_{0}V_{0}^{−1} = Ṽ_{0}^{−1}Ũ_{0} and define U = U_{0} + MQ, V = V_{0} + NQ, Ũ = Ũ_{0} + QM̃, and Ṽ = Ṽ_{0} + QÑ, then

[\begin{matrix}Ṽ & −Ũ \\ −Ñ & M̃ \end{matrix}] [\begin{matrix}M & U \\ N & V \end{matrix}] = [\begin{matrix}I & 0 \\ 0 & I \end{matrix}] .

Note that Q = Q_{l} = Q_{r}.

Using the results stated in Theorem 2.9 it can be deduced after some algebraic manipulations that

T_{d_{o}→y} = W_{2}^{−1}V M̃W_{2} = W_{2}^{−1}NŨW_{2} + I,  (2.13)
T_{d_{o}→y_{o}} = W_{2}^{−1}NŨW_{2} = W_{2}^{−1}V M̃W_{2} − I,  (2.14)
T_{d_{i}→u} = W_{1}MṼ W_{1}^{−1} = W_{1}UÑW_{1}^{−1} + I,  (2.15)
T_{d_{i}→y_{i}} = W_{1}UÑW_{1}^{−1} = W_{1}MṼ W_{1}^{−1} − I,  (2.16)
T_{d_{o}→y_{i}} = W_{1}MŨW_{2} = W_{1}UM̃W_{2},  (2.17)
T_{d_{i}→y_{o}} = W_{2}^{−1}V ÑW_{1}^{−1} = W_{2}^{−1}NṼ W_{1}^{−1},  (2.18)
T_{r→y} = W_{2}^{−1}NN_{dc}W_{2},  (2.19)

where N_{dc} is a constant matrix that satisfies N(0)N_{dc} = I. If {M, N} and {M̃, Ñ} are normalised coprime factorisations, it can be proved that (see [ZDG96, p. 485])

∥[\begin{matrix}I \\ K_{∞} \end{matrix}] (I − G_{s}K_{∞})^{−1}M̃^{−1}∥_{∞} = ∥M^{−1}(I − K_{∞}G_{s})^{−1} [I K_{∞}]∥_{∞} .  (2.20)

It can be shown, using Theorem 2.9 and (2.12), that (2.20) is equivalent to

∥[\begin{matrix}V \\ U \end{matrix}]∥_{∞} = ∥[Ṽ Ũ]∥_{∞} = γ.  (2.21)

It is straightforward to show that at each frequency ω, the following inequalities hold

\overline{σ}(T_{d_{o}→y}) ≤ min\{γ\overline{σ}(M̃)κ(W_{2}), 1 + γ\overline{σ}(N)κ(W_{2})\},
\overline{σ}(T_{d_{o}→y_{o}}) ≤ min\{1 + γ\overline{σ}(M̃)κ(W_{2}), γ\overline{σ}(N)κ(W_{2})\},
\overline{σ}(T_{d_{i}→u}) ≤ min\{γ\overline{σ}(M)κ(W_{1}), 1 + γ\overline{σ}(Ñ)κ(W_{1})\},
\overline{σ}(T_{d_{i}→y_{i}}) ≤ min\{1 + γ\overline{σ}(M)κ(W_{1}), γ\overline{σ}(Ñ)κ(W_{1})\},
\overline{σ}(T_{d_{o}→y_{i}}) ≤ γ\overline{σ}(M)\overline{σ}(W_{1})\overline{σ}(W_{2}),
\overline{σ}(T_{d_{i}→y_{o}}) ≤ \frac{γ\overline{σ}(Ñ)}{\underline{σ}(W_{1})\underline{σ}(W_{2})}.  (2.22)

Let G_{s} be a square transfer matrix^{8}, i.e. G_{s}(jω) ∈ ℂ^{n×n}. Because {M, N} is a right, and {M̃, Ñ} a left normalised coprime factorisation of G_{s}, it can be shown that at each frequency ω and for every i = 1, …, n

σ_{i}(Ñ) = σ_{i}(N) = \frac{σ_{i}(G_{s})}{\sqrt{1 + σ_{i}^{2}(G_{s})}} < 1,
σ_{i}(M̃) = σ_{i}(M) = \frac{1}{\sqrt{1 + σ_{n+1−i}^{2}(G_{s})}} < 1.  (2.23)

Thus, the maximum singular value of every closed loop transfer function at each frequency is bounded from above. The upper bounds are a function of γ, the weighting functions W_{1} and W_{2}, and the desired loop shape G_{s}.

If the weights contain integrators and are chosen so that the singular values of G_{s} follow the trends in Fig. 2.5, it is easy to show that for every i = 1, …, n

σ_{i}(N(jω)) ⟶^{ω→0} 1, σ_{i}(N(jω)) ⟶^{ω→∞} 0,
σ_{i}(M(jω)) ⟶^{ω→0} 0, σ_{i}(M(jω)) ⟶^{ω→∞} 1.

Note that G_{s} is assumed to be strictly proper. The same trends hold for Ñ and M̃. Typical plots of σ_{i}(N) and σ_{i}(M) are depicted in Fig. 2.12. Using the Bezout identities

8Non-square plants will be considered in Section 2.11.

Figure 2.12 Singular values of N and M

defined in Theorem 2.9, the above limits and (2.21), it is straightforward to show that

\overline{σ}(Ũ(jω)) ⟶^{ω→0} 1, \overline{σ}(Ũ(jω)) ⟶^{ω→∞} \sqrt{γ^{2} − 1},
\overline{σ}(Ṽ (jω)) ⟶^{ω→0} \sqrt{γ^{2} − 1},  \overline{σ}(Ṽ (jω)) ⟶^{ω→∞} 1.

The above trends hold if K_{∞} achieves the infimum of the cost function minimised in (2.12), i.e. γ is flat across frequency (see Fig. 2.13). If K_{∞} is a sub-optimal controller,

Figure 2.13 Optimal and sub-optimal costs

then \overline{σ}(Ũ(jω)) ≤ \sqrt{γ^{2} − 1} at high ω. Experience gained from designing H_{∞} loop shaping controllers indicates that the plots of \overline{σ}(Ũ) and \overline{σ}(Ṽ) across frequency follow the trends illustrated in Fig. 2.14. \overline{σ}(U) and \overline{σ}(V) exhibit similar behaviour.

Consider the transfer matrix T_{d_{i}→y_{i}} which is associated with input multiplicative uncertainty, Equation (2.16). At low frequency the singular values of M are small, thus

Figure 2.14 Singular values of Ṽ and Ũ for a sub-optimal value of γ

σ_{i}(T_{d_{i}→y_{i}}) ≃ 1.

1 − γ\overline{σ}(M)κ(W_{1}) ≤ \underline{σ}(T_{d_{i}→y_{i}}) ≤ \overline{σ}(T_{d_{i}→y_{i}}) ≤ 1 + γ\overline{σ}(M)κ(W_{1}).

As ω → 0, it is evident that T_{d_{i}→y_{i}} → −I. At high frequency T_{d_{i}→y_{i}} rolls off, because the singular values of Ñ roll off and \overline{σ}(U) is bounded from above.

\frac{\underline{σ}(U)\underline{σ}(Ñ)}{κ(W_{1})} ≤ \underline{σ}(T_{d_{i}→y_{i}}) ≤ \overline{σ}(T_{d_{i}→y_{i}}) ≤ κ(W_{1})\overline{σ}(U)\overline{σ}(Ñ).

Around the bandwidth

\overline{σ}(T_{d_{i}→y_{i}}) ≤ \overline{σ}(U)\overline{σ}(Ñ)κ(W_{1}) ≤ γκ(W_{1}).

Engineering experience suggests that this upper bound could be conservative. This is deduced by examining the trends illustrated in Figs. 2.12 and 2.14. Even so, the bounds on the shape of T_{d_{i}→y_{i}} are usefully tight^{9}. The same is true for the other closed loop transfer functions.

An upper bound of the maximum singular value of S_{o} is derived in [Vin98, Section 3.4.4]. If the desired loop shape G_{s} is well-conditioned around its bandwidth, then this upper bound is tighter than \overline{σ}(S_{o}) ≤ γκ(W_{2}). If for example γ = 3, W_{2} = I, and κ(G_{s}) ≤ 2 at all frequencies, \overline{σ}(S_{o}) ≤ 2.15 as opposed to \overline{σ}(S_{o}) ≤ 3 (see [Vin98, Fig. 3.8]). A similar bound can be derived for S_{i}. This will be of significance when discussing the guaranteed gain and phase margins in Section 2.7.

Some control system specifications are given in the frequency domain and others in the time domain. H_{∞} control system design techniques are frequency domain methods.

9For good robust stability the weights should not be ill-conditioned around the desired loop bandwidth.

Figs. 2.12, 2.13 and 2.14 have been generated using the controller designed in Section 2.10.

What has not been discussed is translating the time response requirements into frequency response requirements. Time response requirements are usually specified in terms of overshoot (M_{p}), settling time (t_{s}) and rise time (t_{r}) of the step response of a plant output, see [FPEN94, Fig. 3.16].

These requirements are set by shaping the loop gain near cross-over and choosing the bandwidth of the closed loop. What we must first understand is what kind of information we can extract from the frequency response of a stable system. For example, when looking at a singular value plot of the output sensitivity of a system (stable transfer matrix) one can easily see, at a particular frequency, what the maximum gain is. So requirements of the type that \overline{σ}[S_{o}(jω)] ≤ 0.1 for ω < 0.1rad/s can easily be included in the design procedure by forcing \underline{σ}(L_{o}) ≥ 11 at frequencies smaller than 0.1rad/s.

Assume that the controller is implemented in the forward path. The Fourier series of a square wave with period 2π/ω_{0} is given by:

r(t) = \frac{4}{π} ∑_{n=1}^{N} \frac{1}{2n − 1} sin(2n − 1)ω_{0}t.

For a reasonably accurate representation of a square wave it is sufficient to take N = 6. Hence a square wave of frequency ω_{0} = 1rad/s can be considered to contain frequencies ranging from 1 to 11rad/s. In reality we cannot achieve a perfect square wave as the high frequency components will be filtered out. If the signal r(t) is inserted at the input of the closed loop then the output y(t) will be

y(t) = \frac{4}{π} ∑_{n=1}^{N} \frac{1}{2n − 1} |T_{o}[j(2n − 1)ω_{0}]| sin\{(2n − 1)ω_{0}t + ∠T_{o}[j(2n − 1)ω_{0}]\}.

To follow^{10} a square wave of frequency ω_{0} (this frequency is related to t_{s}) we must make the output sensitivity sufficiently small over frequencies ω_{0} → 11ω_{0}, hence control the gain of T_{o} and get it as close to unity as possible. This can be done by increasing the loop gain in this frequency range. To achieve this it might be necessary to increase the bandwidth of the system (the bandwidth is related to t_{r}). We must also make sure that the system is sufficiently well damped, and therefore control the phase of L_{o} (the phase of L_{o} is related to M_{p} and the phase of T_{o}). This may mean decreasing the bandwidth due to actuator and sensor limitations. It could also mean decreasing the phase lags introduced by the weights at cross-over.

Even though it is not straightforward to translate time response requirements into the frequency domain there are general trends that can be followed. The rise time and overshoot are related to the damping of the system. The less damped the system the smaller the rise time and the greater the overshoot. For a desired damping ratio < 1, which is usually the case for aircraft, rise time depends very little on damping. A well damped closed loop is achieved by making sure that the roll-off of the weighted plant singular values at cross-over is typically smaller than −40dB/decade, see Section 2.2. Rise time is controlled by setting the bandwidth of the system. A fast system corresponds to a system with a small rise time and in most cases a small settling time as well (see pp. 126-131 in [FPEN94]). The above discussion assumes that the value of γ is small and hence K_{∞} does not modify the desired loop shape too much (γ is always greater than 1). It is evident from Section 2.2 that γ depends on the behaviour of loop gains around cross-over.

10By follow it is meant that M_{p} is small and t_{r} < t_{s} < \frac{π}{ω_{0}} .

Now assume that the controller has been implemented in the observer form. It can be deduced from (2.13) and (2.19), that the disturbance rejection problem is decoupled from the nominal tracking problem. Thus, nominal tracking depends only on W_{2} and G_{s}. As \overline{σ}(N) < 1 in most cases good damping is guaranteed. The rise time is set by the bandwidth of G_{s} and the settling time by the low frequency gain of G_{s}.

Assume that one of the control system specifications is ramp following, i.e. G_{s} must contain a double integrator. If K_{∞} is implemented in the forward path, then in steady state the plant output y will follow the reference r exactly. If however K_{∞} is implemented in the observer form, it is easy to show using the final value theorem that in steady state

r_{ss} − y_{ss} = lim_{s→0} sW_{2}^{−1}(I − NN_{dc})W_{2} \frac{1}{s^{2}} = lim_{s→0} −W_{2}^{−1}N(Ũ + N_{dc})W_{2} \frac{1}{s} = α.

α is a positive constant that depends on the settling time, i.e. on the low frequency gain of G_{s}. Thus y_{ss} will lag behind r_{ss} by an amount α. This problem may be rectified by the use of a command pre-filter.

It becomes obvious that the notions of classical loop shaping readily carry through. Hence all control system requirements boil down to shaping G. The general trends to be followed are high low frequency gain (disturbance rejection at both the input and the output of the plant, output decoupling and tracking, settling time), low high frequency gain (noise rejection) and a smooth transition in between (robust stability, gain and phase margins, overshoot and damping). The rise time is set by the loop cross-over frequency (the cross-over frequency is the first thing the designer should decide on). High low frequency gain can be achieved with a proportional and integral (PI) filter. Low high frequency gain can be achieved with a low-pass filter. Smooth transition can be achieved with a lead-lag filter.

There are however features of H_{∞} loop shaping that do not exist in classical loop shaping. The designer can use the conditioning of the weights to attempt to trade off properties at the input and output of the plant. How effectively the designer can trade off the input and output properties will depend on how tight the upper bounds (2.22) are. This will depend on how coupled G_{s} is, and in which frequency range the weights are ill-conditioned. Consider for example designing a controller for the model HIRM presented in Section 6.3. If κ(W_{1}) is chosen to be large so that each loop has similar performance, the robustness to input multiplicative uncertainty could be poor. To improve robustness to input multiplicative uncertainty the designer must reduce κ(W_{1}). To maintain a similar loop shape and hence similar nominal performance, κ(W_{2}) must be made larger. If the plant is multi-input single-output (MISO), the task of translating time response requirements into the frequency domain is relatively straightforward, as shaping the desired loop gain corresponds to setting the rise time, overshoot and settling time of that one output. For a MIMO plant though, the task is not always straightforward as the singular values of a transfer matrix do not necessarily correspond to the gain of a particular transfer function of the transfer matrix (unless the plant is diagonal). It can therefore be difficult to translate the controller specifications into weights. Engineering intuition sometimes helps.

The procedure for designing the weights can be systematic, e.g. Hyde’s procedure for designing diagonal weights [Hyd95]. It is obviously easier to design controllers for plants with dominant diagonal terms. In some cases it is difficult to shape the loop gain accurately with diagonal weights. In Chapter 3 a procedure for designing weights that pre- and post-compensate the open loop plant to give a weighted plant with the desired loop shape is presented.

2.6 Designing controllers for coupled, ill-conditioned plants

For tight control of ill-conditioned plants, the controller should compensate for the strong directionality by applying large input signals in the directions where the plant gain is low. However, because of uncertainty, the direction of the large input may not correspond exactly to the low plant gain direction, and amplification of those large input signals may be much larger than expected from the model. This will result in large values of the controlled variables, leading to poor performance or even instability [SMD88].

The design difficulties due to coupling and directionality manifest themselves in several ways^{11}. Feedback properties must be evaluated at both the plant input and output. It can happen that nominal properties at the plant input are quite unacceptable, even though at the plant output they are quite good. Inequalities (2.22) clearly illustrate how such tradeoffs can be achieved when designing controllers using H_{∞} loop shaping. Assume that G is ill-conditioned around the bandwidth of G_{s}. If W_{1} is chosen so that κ(G_{s}) ≃ 1 and W_{2} = I, the closed loop system will have poor properties at the plant

11Freudenberg has studied the control of coupled, ill-conditioned plants [Fre90].

[figure: W_{d_{2}}; W_{d_{1}}; W_{e_{2}}; W_{e_{1}}; G; K; e_{2}; d_{2}; d_{1}; e_{1}]

Figure 2.15 A robust performance problem

input. Moreover, it can happen that the properties at one point are extremely sensitive to uncertainty elsewhere. Let us investigate how the size of the output sensitivity changes due to input multiplicative uncertainty. Consider the block diagram in Fig. 2.15. Let Δ = diag(Δ_{1}, Δ_{2}), and

T_{d→e} = [\begin{matrix}W_{e_{1}}S_{o}W_{d_{1}} & W_{e_{1}}S_{o}GW_{d_{2}} \\ W_{e_{2}}KS_{o}W_{d_{1}} & W_{e_{2}}T_{i}W_{d_{2}} \end{matrix}] .

It can be shown that at each frequency (see [ZDG96, p. 293])

μ_{Δ}(T_{d→e}) ≤ \sqrt{κ(W_{d_{1}}^{−1}GW_{d_{2}})}[\overline{σ}(W_{e_{2}}T_{i}W_{d_{2}}) + \overline{σ}(W_{e_{1}}S_{o}W_{d_{1}})].

Thus from Theorem 2.7

\overline{σ}(W_{e_{1}}S_{o}^{Δ}W_{d_{1}}) = \overline{σ}[F_{l}(T_{d→e}, Δ_{2})] ≤ μ_{Δ}(T_{d→e}), ∀ \overline{σ}(Δ_{2}) < \frac{1}{μ_{Δ}(T_{d→e})},

where S_{o}^{Δ} is the perturbed output sensitivity transfer matrix. This suggests that the robust performance margin is inversely proportional to \sqrt{κ(G)}, if W_{d_{1}} = I and W_{d_{2}} = I.

If the controller has been designed using H_{∞} loop shaping, and W_{e_{1}} = W_{2}, W_{d_{1}} = W_{2}^{−1}M̃^{−1}, W_{d_{2}} = W_{1}Ñ^{−1}, and W_{e_{2}} = W_{1}^{−1}

μ_{Δ}(T_{d→e}) ≤ \sqrt{2}γ ⟹ \overline{σ}(S_{o}^{Δ}) ≤ \sqrt{2}γκ(W_{2})\overline{σ}(M̃), ∀ \overline{σ}(Δ_{2}) < \frac{1}{\sqrt{2}γ} .

Consequently, if κ(W_{1}) is large, S_{o}^{Δ} will not change too much from its nominal value if the input multiplicative uncertainty Δ_{im} = W_{d_{2}}Δ_{2}W_{e_{2}} is small^{12}. Frequency by frequency,

\overline{σ}(Δ_{2}) = \overline{σ}(W_{d_{2}}^{−1}Δ_{im}W_{e_{2}}^{−1}) < \frac{1}{\sqrt{2}γ} .

2.7 The significance of the stability margin ϵ 30

Thus, reliable control of an ill-conditioned plant requires that the uncertainty is modelled as precisely as possible. It is also evident that H_{∞} loop shaping guarantees some level of robust performance (it is possible to obtain guarantees for other robust performance tests as well).

12The upper bound is larger by a factor of \sqrt{2} from the nominal, as \overline{σ}(S_{o}) ≤ γκ(W_{2})\overline{σ}(M̃).

Low condition number controllers are less sensitive to uncertainty but the response is strongly dependent on the disturbance direction. Changing the plant may make even an ill-conditioned plant insensitive to uncertainty. This is only possible if there is a large number of inputs and outputs to choose from.

2.7 The significance of the stability margin ϵ

The stability margin ϵ of a weighted plant G_{s} with a controller K_{∞} is primarily a measure of robust stability; ϵ is a measure of how robust the closed loop is to perturbations in the coprime factors of G_{s}. Robust stability with respect to other types of uncertainty is also guaranteed, see (2.22). A poor stability margin will signify that the desired loop shape is not compatible with robust stability. In such a case K_{∞} will probably modify G_{s} significantly so that the closed loop system is nominally stable and as robust as possible; thus nominal performance specifications will not be met. The only nominal transfer function that can be independent of ϵ is T_{r→y}. It was demonstrated in the previous section that the stability margin also guarantees some level of robust performance.

It is desirable to derive expressions that relate the guaranteed gain and phase margins of each loop, while leaving the other loops closed, with the achieved stability margin ϵ. In Fig. 2.16 the first loop at the plant input is considered. The analysis that follows, is valid for any loop at the input or output of the plant. Such expressions are of engineering importance, as they relate a relatively new robustness measure with quantities the engineer is accustomed to.

[figure: K; G; K; G; e_{1}; u_{1}; d_{1}; n; n]

Figure 2.16 Single loop stability margins

Consider a SISO control system^{13}. It is easy to show that the guaranteed gain and phase margins, depend on the closed loop transfer function considered [GL95, p. 43]. The results are listed in Table 2.1.

If all the norm bounds are similar in size, then the last column gives the least conservative guarantees. The largest possible maximum gain margin guaranteed based on

13Positive feedback is used.

|S| ≤ γ_{S}  |T | ≤ γ_{T} |S + T | ≤ γ_{ST}
\overline{GM} ≥  \frac{γ_{S}}{γ_{S}−1}  \frac{γ_{T} +1}{γ_{T}}  \frac{γ_{ST} +1}{γ_{ST} −1}
\underline{GM} ≤  \frac{γ_{S}}{γ_{S}+1}  \frac{γ_{T} −1}{γ_{T}}  \frac{γ_{ST} −1}{γ_{ST} +1}
PM ≥ 2 arcsin \frac{1}{2γ_{S}} 2 arcsin \frac{1}{2γ_{T}} 2 arctan \frac{1}{γ_{ST}}

Table 2.1 Guaranteed gain and phase margins

|S + T | is ∞ (\overline{GM} ∈ [1, ∞)), the smallest possible minimum gain margin guaranteed is 0 (\underline{GM} ∈ [0, 1]), and the maximum possible phase margin guaranteed is 90^{∘} (PM ∈ [0^{∘}, 90^{∘}]). Note that all norm bounds range in [1, ∞).

Consider the first loop at the plant input, see Fig. 2.16. L_{i} is partitioned as follows,

L_{i} = W_{1}K_{∞}W_{2}G = [\begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}] .

As the plant model has n + 1 inputs, A_{11}(jω) ∈ ℂ, A_{12}^{∗}(jω) ∈ ℂ^{n}, A_{21}(jω) ∈ ℂ^{n}, and A_{22}(jω) ∈ ℂ^{n×n}. The first loop transfer function is

L_{i}^{1} = A_{11} + A_{12}(I_{n} − A_{22})^{−1}A_{21}.

It is easy to show that

T_{d_{1}→u_{1}} = S_{i}^{1} = (1 − L_{i}^{1})^{−1}, T_{d_{1}→e_{1}} = T_{i}^{1} = (1 − L_{i}^{1})^{−1}L_{i}^{1}.

It can be deduced from the matrix inversion lemma, and det(I − L_{i}) ≠ 0 that

S_{i}^{1} = [1 … 0] S_{i} [1 ⋮ 0] , T_{i}^{1} = [1 … 0] T_{i} [1 ⋮ 0] .

From the inequalities (2.22)

|T_{i}^{1}| =  |[1 … 0] W_{1}Ṽ MW_{1}^{−1} [1 … 0]^{T} |  ≤ γκ(W_{1}),
|S_{i}^{1}| =  |[1 … 0] W_{1}UÑW_{1}^{−1} [1 … 0]^{T} |  ≤ γκ(W_{1}),
|S_{i}^{1} + T_{i}^{1}| = |[1 … 0] W_{1}(UÑ + Ṽ M)W_{1}^{−1} [1 … 0]^{T} | ≤ 2γκ(W_{1}).

If W_{1} is diagonal, i.e. W_{1} = diag(w_{1}^{1}, …, w_{1}^{n+1}), then the above bounds become tighter as the condition number of W_{1} drops out. If the open loop plant is not too ill-conditioned around the desired loop bandwidth, then the second bound can be tightened using results from [Vin98, Section 3.4.4], and the third bound can be tightened using (2.23). It becomes obvious, by examination of the trends illustrated in Fig. 2.14, that the bounds are prone to conservatism. Using the above norm bounds and the results in Table 2.1 the designer can get a feel for the guaranteed single loop gain and phase margins. If γ = 3 and W_{1} is diagonal, then \overline{GM} ≥ 3.5dB, \underline{GM} ≤ −3.5dB, and PM ≥ 19.2^{∘}.

Furthermore if W_{1} is diagonal

L_{i}^{1} = w_{1}^{1}[Â_{11} + Â_{12}(I_{n} − A_{22})^{−1}A_{21}],

where A_{11} = w_{1}^{1}Â_{11} and A_{12} = w_{1}^{1}Â_{12}. Thus assuming that the achieved γ does not change too much from one design to another, the designer can use w_{1}^{1} to shape the first loop.

[figure: K; G; \begin{aligned}δ_{1} \\ & Δ \end{aligned}; e_{1}; u_{1}]

Figure 2.17 Multi-loop stability margins

It is desirable that the closed loop system withstands the application of simultaneous and independent gain and phase offsets at the input of each one of the actuators without becoming unstable. This amounts to evaluating the gain and phase margins of L̂_{i}^{1}, which in the case of input multiplicative uncertainty is equal to

L̂_{i}^{1} = F_{l}(L_{i}, I_{n} + Δ) = A_{11} + A_{12}(I_{n} + Δ)(1 − A_{22}(I_{n} + Δ))^{−1}A_{21}.

Let Δ̂ = diag(δ_{1}, Δ) with Δ = diag(δ_{2}, …, δ_{n+1}), see Fig. 2.17. Bounds on the perturbed transfer functions |T̂_{i}^{1}| and |Ŝ_{i}^{1}| can now be derived by invoking the μ robust performance theorem. For example

|T̂_{i}^{1}| ≤ μΔ̂(T_{i}) ≤ \overline{σ}(T_{i}) ≤ γκ(W_{1}), ∀ ∥Δ∥_{∞} ≤ \frac{1}{γκ(W_{1})}.

Hence the relations in Table 2.1 can be used to check the guaranteed gain and phase margins. Note that at low and high frequency the system can tolerate large amounts of uncertainty, see Section 2.2.

In [Vin93, Theorem V.6] the phase and gain margins of a stable SISO closed loop system, PM and GM respectively, are related to the obtained ϵ.

GM ≥ (1 + ϵ)/(1 − ϵ), PM ≥ 2 arcsin(ϵ).

The utility of the bounds is that a “good” value of γ should guarantee that the required gain and phase margins are achieved. Typical flight controller specifications are given in terms of exclusion regions in the Nichols plots of L_{i}^{k}, L̂_{i}^{k}, for k = 1, …, n+1, see [MBT97]. The norm bounds, e.g. |S| ≤ γ_{S}, do not only provide guarantees about the achieved PM and GM of a closed loop system, but also define regions in the complex plane which L_{i}^{k}, L̂_{i}^{k} are guaranteed not to enter [GL95, p. 43].

Consider for example the control system designed for the HIRM wind tunnel model in Section 3.8. Scalar weights^{14} are used and the achieved γ = 3.22. Consequently, κ(W_{1}) drops out of the formulae used to compute the guaranteed gain and phase margins which are given in Table 2.2. Fig. 2.18 illustrates the Nichols plot of the yaw loop L_{i}^{2} (solid line);

|S| ≤ 3.22  |T | ≤ 3.22  |S + T | ≤ 6.44
\overline{GM} ≥ 1.45 (3.23dB)  1.31  1.37
\underline{GM} ≤  0.76  0.69 (-3.22dB)  0.73
PM ≥  17.87^{∘}  17.87^{∘}  17.65^{∘}

Table 2.2 Guaranteed gain and phase margins

the closed loop has been broken at the controller output, i.e. before the actuators. The actual gain and phase margins are -6dB and 40^{∘} respectively. If the dynamic part of w_{1}^{2} is changed from \frac{s+1/3}{s} to \frac{s+3}{s}, the Nichols plot of L_{i}^{2} becomes the dashed line in Fig. 2.18 (achieved γ = 3.62, PM ≃ 35^{∘}, and GM ≃ −5.5dB). The modified Nichols plot of L_{i}^{2} has the expected shape as the achieved γ has not changed too much; higher low frequency gain and more high frequency phase lag. The trapezium is the exclusion region which has been plotted using the specifications given in [MBT97]. The dash-dot ellipses are the guaranteed exclusion regions plotted using |T | ≤ γ_{T}. The smaller one has been plotted using γ_{T} = 3.22, and the larger one using γ_{T} = 3.22/1.5 (the bound on |T | is known to be conservative).

In Fig. 2.19 the multi-loop specifications and guarantees are plotted. It can be seen that the specifications are less stringent and hence the guarantees less conservative.

The stability margin ϵ is related to the ν-gap metric (δ_{ν}) defined in [Vin93, Section V.2].

14By scalar weights it is meant a transfer function multiplied by the identity matrix.

Figure 2.18 Single loop example

Figure 2.19 Multi-loop guarantees and specifications

It can be concluded that the stability margin summarises a lot of information about the behaviour of a closed loop system.

2.8 The design cycle and design philosophy

The typical design cycle given a plant G is as follows:

Scale G: The open loop plant must be scaled according to the desired output decoupling and actuator usage. This is because the open loop singular values cannot be associated with any one input or output (see [SP96, Chapter 1] and [Hyd95, p. 42]). A badly scaled plant is equivalent to a badly formulated problem. Consider designing an autopilot for the longitudinal dynamics of the RCAM, see Section 2.10. Assume that the cost function minimised is ∥W_{S}S_{o}∥_{∞}, where W_{S}^{−1} is a high-pass filter. If the output scaling is such that ż (climb rate) is measured in km/s and V (forward speed) in m/s, then a step disturbance on V will have a large cross coupling into ż. This is because the size of S_{o} is minimised. Thus all the elements of S_{o} are of the same magnitude, but incompatible units.

Choose the weights W_{1} and W_{2}: With non-diagonal weights the designer can specify the desired loop shape more accurately (the designer exploits more degrees of freedom, see Chapter 3). Diagonal weights are appealing as the designer can weight each input and output directly, e.g. the designer can reduce the settling time of an output by weighting that output. Consider for the moment diagonal weights. Integrators (or near integrators in the case of rate following) are placed in W_{1} to boost the low frequency gain. This ensures a zero steady state error if we are tracking an attitude, disturbance rejection and output decoupling/tracking. To increase robust stability, hence decrease phase lag at cross-over (i.e. the slope of the weighted plant singular values), a proportional gain is added to the pre-compensator. The value of the gain (position of the resulting zero) is a trade-off between speed of response (moves the integrator open loop pole away from the origin in closed loop) and robustness. The bandwidth is made as high as possible within the actuator and sensor capabilities, i.e. robust stability considerations. If the open loop is unstable care must be taken not to make the closed loop too fast so that disturbance rejection leads to input saturation and hence loss of controllability. W_{2} is chosen for noise rejection, hence it contains low-pass filters. W_{1} is made to contain PI filters on its diagonal. Each PI filter affects the usage of the actuator it is weighting, and the single loop phase and gain margins of the loop that contains the actuator it is weighting. As a whole W_{1} affects robust stability, the settling times, disturbance and noise rejection, output decoupling and tracking. There are three reasons for which W_{1} is made to contain the integrators. Firstly, noise at frequencies above the bandwidth is not integrated, as the signal that goes into W_{1} has been filtered through W_{2} and K_{∞}. Secondly, the designer can get away without having a trim map. This leads to a more robust design. Note that a nonlinear model can be converted to a “quasi-LPV” model without trim loops only if the nonlinear model has integrators on its inputs, see Section 5.4. Finally, if W_{2} contains the integrators then the signals going into the summation block are ever increasing even if the plant is at a trim point, see Fig. 2.11. W_{2} is made to contain low-pass filters for noise rejection. W_{2} can also contain lead-lag filters to reduce the phase lag at cross-over.

As W_{1} can contain integrators, prolonged input saturation could lead to integrator wind-up. Integrator wind-up degrades the performance; the plant output overshoots its reference. In the worst case it can lead to instability. The Hanus anti-windup scheme, detailed in [Hyd95], has been found to perform satisfactorily. A systematic procedure for designing the gain of the pre-compensator is given in Section 5.7. The weight designed will include the input scaling. This procedure does not always work well. If the designer wishes to align the singular values of the weighted plant at some frequency, then W_{1} may contain a non-diagonal term. The singular values are usually aligned at the bandwidth and this is done when all channels have the same rise time specification. Aligning singular values is not advisable for ill-conditioned plants, as this could affect the achievable stability margin and degrade properties at the plant input.

Choose the position of the controller: An H_{∞} loop shaping controller can be implemented in three ways. Placed in the forward path the step response of T_{r→y} has a smaller rise time, but overshoots as high frequency controller dynamics are excited by the references. Also any RHP zeros of the controller, are RHP zeros of T_{r→y}. Placing the controller in the feedback path, leads to a slower more damped response, but any RHP poles of the controller are RHP zeros of T_{r→y}. The controller could also be implemented in the observer form as proposed in [Vin93, pp. 72-78]. This constitutes the optimal way of introducing the controller into the loop.

Design the command pre-filters: The pre-filters are designed to meet the handling quality requirements. Let T_{0} be an ideal reference model that reflects the desired handling qualities. The obvious way to design a command pre-filter Q is by solving the model matching problem, min_{Q∈RH_{∞}} ∥T_{0} − NQ∥_{∞}. This procedure does not necessarily lead to robust performance. This will be discussed further in Chapter 4. Robust performance is closely linked to pilot-induced oscillations (PIOs). Command pre-filters can be designed to improve the response from the references to the plant input and output. Command pre-filters are not a substitute for badly designed weights. If the weights are badly designed, the closed loop properties may be poor, and the command pre-filters will not work well on plants other than the nominal (poor robust performance). It may be possible that no command pre-filter is required if the weights are well designed. Tracking is limited by closed loop non-minimum phase zeros (RHP zeros).

Perform time simulations and analysis to prove robust performance: H_{∞} loop shaping readily provides robust stability. We can achieve nominal performance but

must test for robust performance.

Within this general framework one design strategy is to make the loop dynamics as fast as possible, within the limitations imposed by the plant. Designing command prefilters that satisfy the handling quality requirements should then be a straightforward task. The price that one pays for a high closed loop bandwidth is high actuator usage. The benefits though are good disturbance rejection and good robust performance. The lower the bandwidth of the command pre-filter the higher the level of the robust performance (see Chapter 4). Good robust performance implies that gain scheduling may not be necessary.

It has been shown that feedback systems are sensitive to uncertainty around the bandwidth. It is therefore sensible to position the bandwidth in a frequency range where little uncertainty is expected. For example, the closed loop bandwidth of a longitudinal autopilot should be higher than the short period dynamics.

2.9 Remarks on the architecture of flight controllers

The designer should always feed back any available rate signal that is reliable, i.e. not too noisy. Using a rate signal can reduce the actuator usage during disturbance rejection. If the rate signal is not used and only the attitude is fed back, closed loop robustness is improved by including lead-lag filters in W_{2}, thus the actuator usage is increased (it is advisable to remove the lead term from the W_{2} in the reference command path as this reduces actuator usage during a reference command, see Fig. 2.11).

A classical autopilot has an inner and outer loop structure. The inner loop is for stabilisation (rate feedback) and the outer loop is for tracking (attitude feedback). The reference to the autopilot could be either an attitude or rate command. What is proposed is signal blending. One loop is designed for both functions. The feedback variable used is the weighted sum of the attitude and the rate signals. The weights are chosen so that the cross-over dynamics are dominated by the rate signal. The robustness of such a structure is characterised by one robustness measure instead of two. The number of states associated with the controller are less and it is clear how to implement different flight modes. Controllers designed using H_{∞} loop shaping can tolerate large amounts of uncertainty at high and low frequency. The attitude signal affects the low frequency dynamics. Thus a pitch attitude hold can easily be replaced by a flight path angle hold. This fact also enables switching between a rate and attitude commanded closed loop.

A double integrator is required if the designer wishes to reject the effect of a rate disturbance on the attitude. The nominal performance of the real system can be expressed in terms of the nominal performance of the blended system (system for which the controller has been designed).

2.10 A design example

We are going to present a design example to illustrate the above points, and how the designer could produce a good or bad design.

2.10.1 Presentation of Model Used

We are going to design a longitudinal controller for the RCAM [GAR96]. The inputs into the linear model G are tailplane (δ_{T}), thrust (δ_{T H}), the outputs climb rate (ż), forward speed (V) and the states pitch rate (q), pitch attitude (θ), longitudinal velocity (u_{B}), normal velocity (w_{B}). The linearised plant model G can be found in [GAR96]. All angles are in rad and velocities in m/s (note that δ_{T H} is measured in rads).

G has poles at −0.011 ± j0.126, −0.830 ± j − 1.107 and zeros at 4.338, −4.390. We can easily distinguish between the phugoid and short period modes. The phugoid is slower and lightly damped. The plant is open loop stable with a non-minimum phase zero. The RHP zero exists because the vertical accelerometer is located behind the centre of rotation of the aircraft. The physical meaning of a non-minimum phase zero is that the plant goes initially in the opposite direction to that desired, so when the aircraft pitches up ż is initially going to be positive. Non-minimum phase zeros within the bandwidth of the plant limit the achievable performance.

The actuator dynamics and loop delays are given below. The loop delay transfer function is based on a first order Padé approximation (MATLAB μ-tools commands are used).

>> tailplane = nd2sys(1,[0.15 1]);
>> engine = nd2sys(1,[1.5 1]);
>> delay = nd2sys([-0.06 1],[0.06 1]); % delay of 0.12s

2.10.2 Design Specifications

The tracking requirements are as follows [GAR96]:

V_{air}  γ_{f}
t_{r} < 12sec t_{r} < 5sec
t_{s} < 45sec t_{s} < 20sec
M_{p} < 5% for h > 305m
M_{p} < 30% for h < 305m

Table 2.3 Time response requirements

Note that ż = V sin γ_{f}, hence for a small flight path angle γ_{f} the climb rate becomes ż ≈ V γ_{f}. For good disturbance rejection a 13m/s wind step should not induce a deviation in airspeed greater than 2.6m/s, for more than 15s. There are no cross-coupling requirements defined between V and ż.

2.10.3 Scaling

The open loop plant is scaled according to output decoupling and actuator usage. For convenience the units of the actuators are converted to degrees (d2r: degrees to radians). It was thought that 1 degree of tailplane is analogous to 0.5 of a degree of thrust. This is compatible with the physical limits of the actuators [GAR96]. Increasing for example 0.5 to 1 will increase the usage of the throttle by increasing the bandwidth of the system.

Bsc = diag([d2r 0.5*d2r]); % input scaling
Csc = diag([3 1]); % output scaling

Similarly, increasing the second entry in Csc from 1 to 2, will increase the speed of response of the airspeed and the decoupling with ż. Fig. 2.20 shows the open loop scaled RCAM

Csc*G*Bsc.

Figure 2.20 Csc*G*Bsc

Weighting the throttle with a big number would result in increasing the closed loop bandwidth even though the throttle actuator is slow resulting in poor stability.

Figure 2.21 G_{s}, G_{s}K_{∞} and K_{∞}G_{s}

2.10.4 Choice of Weights

To ensure that a diagonal weight W_{1} augments each of the singular values of the scaled plant independently a weight Ṽ was designed as described in [PG97a]. The total precompensator becomes Ṽ W_{1}. Ṽ used in this design has no dynamics. In the MATLAB code Ṽ is the variable preW V.

preW_V =
   9.8313e-01  -1.0565e-01
  -1.0565e-01  -9.8313e-01

Integrators are added in W_{1} to boost the low frequency gain. Having just an integrator reduces the robustness as 90^{∘} of phase is added at cross-over. Hence a proportional matrix gain is added to W_{1}. The position of the zeros is a trade-off between speed of response and robustness. The post-compensator W_{2} is designed for noise rejection. Again the closer to cross-over that the singular values are rolled-off at, the bigger the reduction of the achieved robustness. The weights chosen are:

w1_1 = nd2sys(0.25*[3 1],[3 0]);
w1_2 = nd2sys([3 1],[3 0]);
preW = daug(w1_1,w1_2);
w2 = nd2sys(1,[0.2 1]);
postW = daug(w2,w2);

The bandwidth in each loop is pushed up as high as possible subject to a desired level of robustness. The weighted plant G_{s} is shown in Fig. 2.21 (solid line). Fig. 2.21 also illustrates that K_{∞} does not alter the desired loop shape too much. Hence shaping G open loop is equivalent to shaping both G_{s}K_{∞} (dotted) and K_{∞}G_{s} (dash-dot). So for example, L_{o} = GW_{1}K_{∞}W_{2} ⇒ \underline{σ}(L_{o}) ≥ \underline{σ}(G_{s}K_{∞})/κ(W_{2}) = \underline{σ}(G_{s}K_{∞}). Note that the singular values of G_{s}K_{∞} and K_{∞}G_{s} are virtually the same.

The augmented plant has 12 states. The controller was synthesised using

>> [sysK,emax] = ncfsyn(sys,1.1);
>> emax
emax =
   3.3307e-01
>> emargin(sys,sysK)
ans =
   3.0560e-01

Hence the achieved ϵ = 0.31. The controller sysK was model reduced to 7 states. Therefore the overall controller Ṽ W_{1}K_{∞}W_{2} has 11 states. Most of the poles in K_{∞} are located around the bandwidth. The singular values of the model reduced controller sysK2 (dashed) and the original controller sysK (solid) controller are shown in Fig. 2.22. The difference can hardly be seen. sysK2 gives an ϵ = 0.30. The equations in Section 2.7 give a feel for the magnitude of the achieved gain and phase margins.

The controller was implemented in the forward path. The pre-filters were chosen to be first order lags. The singular values of S_{o} (solid) and T_{o} (dotted) are shown in Fig. 2.23. A μ-analysis could be carried out, as in [BDG^{+}93, pp. 3-36], to check robust performance. Note that the large positive area under the S_{o} curve is due to the RHP zero (see the waterbed effect in [DFT92, pp. 97-103]).

Figure 2.22 sysK and sysK2

Figure 2.23 T_{o} and S_{o}

2.10.5 Time Responses

The SIMULINK block diagram of the linear model is shown in Fig. 2.24.

Figure 2.24 SIMULINK block diagram of the linear model

Fig. 2.25 shows the response to a command on ż (solid) at t = 1s. Note the initial undershoot due to the non-minimum phase zero. Fig. 2.26 illustrates the response to an airspeed command (dash-dot). Fig. 2.27 illustrates the rejection of a wind-shear of 13m/s.

It can be deduced that all the requirements are met. The interested reader is encouraged to go through the example and change the scaling and weighting functions to obtain a feel of how the different parameters influence the design. It is also suggested that the controller is implemented in the observer form. Other examples, tutorials, of designing H_{∞} loop shaping controllers can be found in [Hyd95, SP96, BDG^{+}93, Vin93, BO95].

2.11 Designing controllers for non-square plants

Consider a non-square plant with m inputs and p outputs, i.e. G_{s}(jω) ∈ ℝ^{p×m}. Assume that m > p; a fat nominal plant model is of greater interest to the flight control engineer. Let the SVD of G_{s} be frequency by frequency G_{s} = UΣV^{∗}, where U ∈ ℂ^{p×p}, V ∈

Figure 2.25 Step on ż command

Figure 2.26 Step on V_{air} command

ℂ^{m×m}, Σ ∈ ℝ^{p×m}. Let {M, N} be a right normalised coprime factorisation of G_{s}. It is easy to show that at each frequency,

NN^{∗} = U(I + ΣΣ^{∗})^{−1}ΣΣ^{∗}U^{∗}
MM^{∗} = V (I + Σ^{∗}Σ)^{−1}V^{∗}

Thus \overline{σ}(M) = 1. Similarly it can be shown that \underline{σ}(Ñ) = 0, i.e. the closed loop system cannot reject input disturbances in all directions. This is a well known fact.

There are two ways of designing a controller for a non-square plant. The first way is to design a non-square controller. The controller that results from solving the H_{∞} problem will blend the inputs in a way that maximises closed loop robustness. The designer can use the input scaling matrix to set the usage of each actuator. Alternatively, the designer

Figure 2.27 Wind-shear rejection

can make the open loop plant square, i.e. enhance the performance of the closed loop by building structure into the controller. This can be done by input blending (complementary filters or slaving) or daisy chaining. Not using all the available inputs necessarily compromises the achievable closed loop performance.

Daisy chaining is outlined below. Assume that (A, B, C) is a state-space realisation of G_{s}. Partition the inputs, ẋ = Ax + B_{1}u_{1} + B_{2}u_{2}. Design the controller using inputs u_{1}. The closed loop bandwidth should not be faster than any of the actuators corresponding to inputs u_{2}. If the inputs u_{1} saturate, the excess demand is passed to u_{2} via the matrix gain B_{2}^{#}B_{1}, where B_{2}^{#} is the pseudo-inverse of B_{2}. This was proposed in [EBHS94].

2.12 Limitations and ideal plant

Some plants have features that restrict the achievable performance (see [ZDG96, pp. 143-153] and [SP96, Chapters 5 and 6]). Such limitations are for example RHP poles outside the bandwidth, RHP zeros within the bandwidth and ill-conditioning. These limitations are design method independent.

H_{∞} controller design techniques are frequency domain based methods. This is because robustness issues are more easily addressed in the frequency domain. The potential weakness of H_{∞} loop shaping is that there is sometimes difficulty in translating time response requirements to the desired loop shape. This difficulty can be overcome by pushing for the highest possible closed loop bandwidth. Hence the designer aims for a bandwidth higher than that required to satisfy the handling quality requirements, subject to obtaining reasonable robustness.

The design problem becomes harder when the plant is open loop unstable in which

2.13 H_{∞} loop shaping and other robust control design methods 45

case a high bandwidth could lead to input saturation and loss of controllability during disturbance rejection. In such circumstances the bandwidth must be lowered. Decoupling becomes a more difficult task, particularly if the specifications for each loop vary significantly.

The ideal plant for controlling with an H_{∞} loop shaping controller would be a plant that has similar properties in all loops. By similar properties we mean equally fast and powerful actuators and sensors, and not too ill-conditioned. Hence the cross-over frequency for all singular values can be made the same.

When using classical control the designer designs the controller directly. This is not the case when using H_{∞} design techniques as the controller is the product of an optimisation, and hence the designer has to set-up the cost function to be minimised. There is an evident transfer of tasks. When designing using H_{∞} loop shaping the designer has only got to worry about translating the performance specifications to the desired loop shape.

2.13 H_{∞} loop shaping and other robust control design methods

The philosophy of H_{∞} loop shaping is different to that of other robust control system design techniques. For example when designing a controller using μ-synthesis, the designer weights transfer functions that are indicative of performance and of robustness. If the achieved norm bound is greater than 1 (assuming that the cost function has been normalised), it might be that the specifications cannot be met or that the weights are conflicting with each other. Redesigning the weights is not an easy problem. When designing using H_{∞} loop shaping the designer specifies performance and then checks whether the desired performance is compatible with robustness requirements.

The value of a general H_{∞} minimisation is in analysis, not synthesis. For example, assume that we have designed a control system using H_{∞} loop shaping and are not happy with a certain single loop PM, but cannot see how to modify the weights to increase it. A natural question that arises, is whether the poor PM is due to a limitation imposed by the plant or because of badly designed weights. The designer can solve an H_{∞} minimisation using different uncertainty structures to check for the best achievable PM.

Theorems 2.6 and 2.7 motivate the use of the structured singular value as an analysis tool. Robust stability with respect to structured uncertainty, and robust performance of some closed loop transfer matrix with respect to some uncertainty structure can be evaluated.

There are certain cost functions ideally suited for μ-synthesis; the cost function considered in Chapter 4 and the one outlined in Fig. 2.28, where Δ is parametric real uncertainty. It can be shown that the controller K will have the properties of the H_{∞} loop shaping controller.

[figure: K; G; W_{2}; W_{1}; Δ]

Figure 2.28 Proposed μ-synthesis cost function

2.14 Industrial viewpoint

For new types of controllers to be used by the aerospace industry, there are two major pre-requisites. Firstly benefits need to be quantified so as to justify a change in approach. Secondly, once the benefits have been established, the flight clearance aspects need to be addressed.

H_{∞} loop shaping has much in common with the design approach currently used by industry, sometimes referred to as “classical control”. The loop shaping part of the design procedure is carried out in exactly the same way that classical design is carried out: inputs and outputs are matched up, and single loop shaping is carried out to ensure that low frequency gain is large enough, roll-off at cross-over is not excessive, and that sufficient high frequency roll-off is provided. Once this is done, then the optimal H_{∞} loop shaping controller is synthesised for this so-called “weighted plant”, see Section 2.3. Hence, it is perfectly possible to take a classical design, and to augment it with a corresponding H_{∞} controller, which will then modify the feedback structure so as to allow for the multivariable nature of the system.

When designing using classical control where the system has inherent cross-coupling (as for example with most yaw - roll augmentation systems), corresponding cross-terms are put into the controller. Design of these terms is not always straightforward in that their effect on the feedback loops is not addressed directly, and some iteration may be required. With H_{∞} loop shaping, these cross-terms can be left to the H_{∞} synthesis part of the design. One of the prime motivations for H_{∞} loop shaping is the potential for design time reduction, particularly when dealing with multi-input multi-output systems with strong cross-coupling. As such better performance and robustness may be obtained.

Some common control system requirements are step response specifications, gain and phase margins, guidelines for avoiding pilot induced oscillations, and structural coupling specifications. H_{∞} loop shaping addresses all of the above requirements, some directly and others indirectly.

H_{∞} loop shaping has a number of key attributes which make it particularly suitable for designing flight controllers. An important attribute is that the resulting controller can be written as an exact plant observer plus state feedback. This structure allows gain scheduling, as designs at different operating points have the same state-space structure, and hence the observer and state feedback gains can be interpolated. The observer also allows the handling of input limitations (e.g. authority and rate limits) by driving the observer with achieved plant inputs rather than demanded ones. A third very important attribute of H_{∞} loop shaping controllers is that the observer structure adds visibility in that physical units and interpretation can be applied to the controller coefficients and states. This may be advantageous with respect to flight clearance. However, having state-space elements within the control law as opposed to SISO transfer functions, is still a new approach in the United Kingdom. This will probably not be an insurmountable problem, particularly given the numerical and efficiency advantages of using state-space implementations.

Non-diagonal weights

The weighting functions W_{1} and W_{2} are usually designed in two stages. In the first stage the desired loop shape^{1} is determined. This involves translating time response requirements into the frequency domain. To do this, engineers rely largely on intuition provided by the loop shaping concepts described in Section 2.5 and past experience. In the second stage the designer has to select weights W_{1} and W_{2}, so that the shape of the singular value plot of G_{s} = W_{2}GW_{1} is close to the desired loop shape. Choosing diagonal weights is often adequate [Hyd95]. Experience gained with designing H_{∞} loop shaping controllers indicates, that diagonal weights do not work well with plants that have strong cross-coupling between channels. The task of choosing W_{1} and W_{2} for such plants can be difficult and time consuming if done in an ad-hoc manner. Furthermore, there is no guarantee that the desired loop shape will be achieved.

This chapter develops a systematic procedure for designing a pre- and/or post- compensator, that enable the designer to shape each singular value of the nominal plant separately, and therefore get close the desired loop shape. This procedure does not restrict the achievable performance and/or robustness of the designed control system.

F is said to be a unit in RH_{∞} if F, F^{−1} ∈ RH_{∞}, i.e. F is stable and minimum phase. A transfer matrix F ∈ RH_{∞} is inner, if F^{∼}F = I. F^{∼} denotes the conjugate system of F, and F^{∼}(s) = [F (−s̄)]^{∗}. Therefore, if F ∈ RH_{∞} then F^{∼} ∈ RH_{∞}^{−}, i.e. F^{∼} is analytic in the LHP (anti-stable).

3.1 Motivation for non-diagonal weights

It was concluded in Section 2.5 that the designer can specify all closed loop objectives by shaping G (the open loop plant) open loop. If the designer can shape G accurately, then:

1The desired shape of the plot of the singular values of the weighted plant (G_{s}) against frequency will be referred to as the desired loop shape.

1. The designer is more likely to achieve the nominal tracking requirements. If the controller has been implemented in the observer form, the transfer function from references (r) to outputs (y) is

T_{r→y} = W_{2}^{−1}NN_{dc}W_{2}.

The implemented closed loop system is depicted in Fig. 2.11. As the singular values of N depend on the singular values of the weighted plant (see Equation (2.23)), and the bandwidth of W_{2} is usually larger than the cross-over frequency of G_{s} and consequently the bandwidth of N, shaping the singular values of G corresponds to shaping the singular values of T_{r→y}, i.e. specifying nominal tracking requirements. N_{dc} has been chosen so that T_{r→y} = I at ω = 0. Thus N_{dc} is a unitary matrix and σ_{i}(NN_{dc}) = σ_{i}(N). It must be noted that shaping G accurately, does not in general enable the designer to specify in detail the response of each output y_{i} to a reference command r_{i}. This is because the designer is shaping singular values not transfer functions.

2. The designer can specify requirements given in the frequency domain more accurately, e.g. avoidance of structural coupling [Mui97]. It was shown in Section 2.5, that the upper bound of the maximum singular value of every closed loop transfer matrix depends on γ, the weights and the weighted plant. Consider for example the output sensitivity transfer matrix (S_{o}). At every frequency,

\overline{σ}(S_{o}) ≤ γ\overline{σ}(M̃)κ(W_{2}).

As the singular values of M̃ depend on the singular values of the weighted plant (see Equation (2.23)), and typically the designer is aiming for γ ≃ 3, requirements of the type \overline{σ}(S_{o}) ≤ α over some frequency range can be met by shaping G, i.e. it is sufficient that at every frequency of interest

γκ(W_{2})\overline{σ}(M̃) ≤ α ⟺ \underline{σ}(G_{s}) ≥ \sqrt{(\frac{γκ(W_{2})}{α})^{2} − 1} ≃ \sqrt{(\frac{3κ(W_{2})}{α})^{2} − 1}.

3. The designer can systematically trade off properties at the input and output of the plant. If for example W_{1} is ill-conditioned around the bandwidth of G_{s}, then the designer can attempt to improve robustness with respect to input multiplicative uncertainty (Δ) by reducing κ(W_{1}) at frequencies around the bandwidth.

μ_{Δ}(T_{i}) ≤ \overline{σ}(T_{i}) ≤ γ\overline{σ}(Ñ)κ(W_{1}).  (3.1)

Note that Δ ∈ Δ, where Δ is the structure of the input multiplicative uncertainty. Robustness with respect to uncertainty at the plant input is related to the input single and multi-loop gain and phase margins defined in Section 2.7. To maintain similar nominal tracking performance the designer must increase κ(W_{2}), potentially degrading robustness at the plant output. Such trade-offs can only be made in a systematic manner if the designer can shape the singular values of the open loop plant accurately. Reducing the condition number of W_{1} will improve the robustness with respect to input multiplicative uncertainty, if the upper bound in (3.1) is tight. How tight the upper bound is, will depend on the structure of the uncertainty.

3.2 The proposed design procedure

Consider a linear time-invariant nominal plant model G with m inputs and p outputs. To keep the notation simple, it shall be assumed in the sequel that G has the same number of inputs and outputs equal to m. It must be noted that G is the scaled open loop plant.

The first step of the proposed design procedure for selecting W_{1} and W_{2}, is to determine the desired loop shape. Let σ_{i} (i = 1, …, m) denote the i^{th} singular value of G. A typical singular value plot of an open loop plant is depicted in Fig. 3.1. Note that every σ_{i} varies smoothly with frequency. Hence σ_{1} is not necessarily the maximum singular value at all frequencies. The ordering of the singular values of G

Σ_{G} = diag(σ_{1}, …, σ_{m}),

is according to the order at ω = 0. Every σ_{i} is associated with a transfer function f_{i}. The gain of f_{i}, denoted by |f_{i}|, is chosen using engineering intuition and experience. Hence it is desired that the i^{th} singular value of the weighted plant is σ_{i}|f_{i}|. f_{i} is restricted to be a unit in RH_{∞}^{2}. The designer effectively chooses a diagonal transfer matrix W = diag(f_{1}, …, f_{m}), such that the desired singular values of the weighted plant are

Σ_{G_{s}}^{des} = Σ_{G} W = diag(σ_{1}, …, σ_{m}) × diag(|f_{1}|, …, |f_{m}|).

W is split into two diagonal transfer matrices Π, Γ such that W = Π Γ. Γ reflects the desired conditioning and singular values of W_{1}, and Π the desired conditioning and singular values of W_{2}. Note that the designer could choose Γ or Π equal to the identity. There should be no pole/zero cancellations between Γ and Π. If there are pole/zero cancellations, then the order of Γ and Π is unnecessarily inflated.

The second step is to find stable and minimum phase weights, W_{1} and W_{2}, that shape the open loop plant such that the singular values of the weighted plant are close to

Σ_{G_{s}}^{des} = Π Σ_{G} Γ = diag(σ_{1}|f_{1}|, …, σ_{m}|f_{m}|).

2If f_{i} is a unit in RH_{∞}, it cannot contain integrators. This will be discussed further in Section 3.7.

Figure 3.1 A typical singular value plot

If the inputs and outputs of the open loop plant are re-ordered so that the open loop plant is as diagonal as possible, it may be the case that the desired loop shape can be achieved using diagonal weights. This was pointed out in [Hyd95, Chapter 4]. According to Hyde, W_{1} should in general contain the low frequency dynamics of W, and W_{2} the high frequency dynamics of W. If the designer wishes to align the singular values of the weighted plant at some frequency, W_{1} may contain a non-diagonal term. The singular values are usually aligned at the loop bandwidth. This is done when all channels have the same rise time specification, and the open loop plant is not too ill-conditioned around the loop bandwidth.

The procedure described in [Hyd95], was developed from the perspective of designing longitudinal controllers for the Harrier aircraft. The open loop plant model of the longitudinal dynamics of a typical fixed wing aircraft, is close to being diagonal around the desired loop bandwidth. It has been found that selecting weights using the method proposed by Hyde for open loop plants with coupled dynamics (e.g. the lateral dynamics of an aircraft), is a time consuming task that involves trial and error. Furthermore, it is not guaranteed that the desired loop shape will be achieved.

3.3 The design procedure requirements

3.4 Computing W_{1} and W_{2} 52

W_{1} and W_{2} must be chosen so that the weighted plant G_{s} contains no unstable hidden modes [MG90]. This is required to be able to prove internal stability of the implemented closed loop system depicted in Fig. 3.2. This leaves open the possibility of unstable and/or non-minimum phase weights. However, it is well known that open loop RHP poles outside the loop bandwidth, and open loop RHP zeros within the loop bandwidth restrict the achievable closed loop performance [FL85]. Thus any procedure for computing W_{1} and W_{2} should not produce weights that have RHP poles outside the loop bandwidth, and RHP zeros within the loop bandwidth.

[figure: G; K_{∞}; W_{1}; W_{2}]

Figure 3.2 The implemented closed loop system

Pole/zero cancellations between the open loop plant and the weights in the LHP do not result in an unstable closed loop system, but could lead to poor robust performance if the weights cancel any lightly damped poles of the open loop plant [SG90]. Thus the weights must not cancel any lightly damped plant poles.

The design procedure must not remove insight from the designer about possible closed loop trade-offs and limitations. Furthermore, the design procedure must not be cumbersome, i.e. difficult to use and too computationally intensive to allow for design iterations.

3.4 Computing W_{1} and W_{2}

In the first step of the proposed design procedure described in Section 3.2, two stable and minimum phase diagonal transfer matrices Π, Γ were chosen, that reflect the desired conditioning and singular values of W_{2} and W_{1} respectively. What is described in this section is a procedure that enables the designer to compute stable and minimum phase non-diagonal weights W_{1} and W_{2}, so that the singular values of the weighted plant are close to Σ_{G_{s}}^{des}, i.e.

|σ_{i}(W_{2}GW_{1}) − σ_{i}|f_{i}|| ≤ β, ∀ i = 1, …, m,

where β is a function of frequency and β > 0, ∀ ω. The significance of β will be discussed in Section 3.5. The procedure is described below:

1. Choose a frequency range [ω_{L}, ω_{H}] that contains the dynamics of G around the target closed loop bandwidth. Grid the range. At each frequency of the grid perform a singular value decomposition of G(jω)

G(jω) = U_{G}(jω)Σ_{G}(jω)V (jω)^{∗},

such that the ordering of the singular values in Σ_{G} is uniform across the frequency range considered. By uniform it is meant that each singular value as a function of frequency varies smoothly with frequency, see for example Fig. 3.1. This ensures “continuous variation” of the elements of V across [ω_{L}, ω_{H}] (see Fig. 3.3), when the all-pass factors up to which V is unique are carefully selected (see Equation (3.2)). By “continuous variation” it is meant continuous variation when the grid is infinitely dense.

Figure 3.3 Magnitude and phase of an element of V

It is assumed that all the dynamics of U_{G} and V are contained in the frequency range [ω_{L}, ω_{H}]; this can be easily verified. If not, the designer must increase the frequency range.

2. Fit a transfer function to each element V_{jl} of V (j, k = 1, …, m), without restricting the transfer functions to be stable and minimum phase. This can be done using the μ-tools command fitsys [BDG^{+}93]. Denote the resulting transfer matrix by V̂. The better the fit the smaller the value of δ

\overline{σ}[V̂ (jω) − V (jω)] ≤ δ, ∀ ω.

The above inequality ensures that V̂ is close to V outside the frequency range [ω_{L}, ω_{H}].

3. W = Π Γ, with Π = diag(π_{1}, …, π_{m}) and Γ = diag(γ_{1}, …, γ_{m}). If δ is small σ_{i}(GV̂ Γ) ≃ σ_{i}|γ_{i}|, but V̂ Γ is not necessarily minimum phase and stable. Obtain a co-spectral factorisation of Ψ = V̂ Γ(V̂ Γ)^{∼}

V̂ Γ(V̂ Γ)^{∼} = W_{1}W_{1}^{∼},

where W_{1} denotes the co-spectral factor of Ψ and W_{1}, W_{1}^{−1} ∈ RH_{∞}. The co-spectral factorisation can be obtained using formulae^{3} given in [Fra87, pp. 90-94]. Thus, W_{1} is stable and minimum phase and σ_{i}(GW_{1}) = σ_{i}(GV̂ Γ). W_{1} is the required precompensator.

3These formulae can only be used if V̂ Γ has no imaginary axis singularities and V̂ Γ(∞) > 0.

4. To compute W_{2}, perform an SVD of GW_{1} at each grid point

GW_{1}(jω) = U(jω)Σ_{GW_{1}}(jω)V_{GW_{1}}(jω)^{∗},

and fit transfer functions to the elements of U. Denote the resulting transfer matrix by Û. Then take a spectral factorisation of UΠ^{∼}ΠU^{∼} to obtain the post-compensator W_{2}. Note that step 1 has to be repeated for GW_{1}, as U(jω) ≠ U_{G}(jω) because V̂ is only an approximation of V.

As indicated in step 1, care must be taken when performing the SVD. In particular, the designer must ensure that the elements of V (and U in step 4) vary “continuously” across the frequency range [ω_{L}, ω_{H}], so that the fit obtained in step 2 is sensible^{4}. This can be achieved by ensuring that the ordering of the singular values is uniform across the frequency range considered, in combination with careful selection frequency by frequency of the all-pass factors up to which V is unique. The μ-tools command vsvd arranges the singular values in decreasing order frequency by frequency, and as such cannot be used directly. However, vsvd can be modified in a straightforward manner to guarantee uniform ordering of the singular values across [ω_{L}, ω_{H}]; this is done by looking at the singular vectors frequency by frequency. If G( jω) has distinct singular values then V is determined up to a right diagonal all-pass factor Φ = diag(e^{jθ_{1}}, …, e^{jθ_{m}}) with all θ_{i} ∈ ℝ [HJ93]. If the designer ensures that the ordering of the singular values of G is uniform across the grid, and solves the 2-norm minimisation

α_{k,i}^{opt} = arg min_{α} ∥v_{ω_{k},i} − v_{ω_{k+1},i} e^{jα}∥_{2}  (3.2)

at each pair of grid frequencies (ω_{k}, ω_{k+1}), where v_{ω_{k},i} denotes the i^{th} column of V ( jω_{k}), then post-multiplying V ( jω_{k+1}) by the all-pass factor e^{j diag(α_{k,1}^{opt},...,α_{k,m}^{opt})} guarantees “continuous variation” of the elements of V. It is easy to show that minimisation (3.2) has an analytic solution

α_{k,i}^{opt} = −∠(v_{ω_{k},i}^{∗} · v_{ω_{k+1},i}).

3.5 How well does the proposed design procedure work?

The proposed design procedure is only of use to the engineer if the singular values of the weighted plant σ_{i}(G_{s}) are close to |π_{i}|σ_{i}|γ_{i}|. Thus the success of the proposed design procedure depends on how well V̂ approximates V and Û approximates U. To keep the notation simple, consider that W_{1} is non-diagonal and W_{2} is equal to the identity. It is evident that at each frequency (the dependence on ω is not shown)

4Note that a differentiable matrix function V is guaranteed to exist, see [Woo95, Lemma 7.8.3] and [Kat80].

\overline{σ}(E)^{2} = \overline{λ}(E^{∗}E) ≤ ∑_{j=1}^{m} λ_{j}(E^{∗}E) = Trace(E^{∗}E),  (3.3)

where E = V̂ − V. As

E^{∗}E = [\begin{matrix}E_{11}^{∗} & … & E_{m1}^{∗} \\ ⋮ & ... & ⋮ \\ E_{1m}^{∗} & … & E_{mm}^{∗} \end{matrix}] [\begin{matrix}E_{11} & … & E_{1m} \\ ⋮ & ... & ⋮ \\ E_{m1} & … & E_{mm} \end{matrix}] = [\begin{matrix}∑_{j=1}^{m} E_{i1}^{∗}E_{i1} & … & ⋆ \\ ⋮ & ... & ⋮ \\ ⋆ & … & ∑_{j=1}^{m} E_{im}^{∗}E_{im} \end{matrix}] ,

where ⋆ stands for “do not care”, at each frequency

\overline{σ}(E)^{2} ≤ Trace(E^{∗}E) = ∑_{k=1}^{m} ∑_{j=1}^{m} |E_{jk}|^{2} ≤ ∑_{k=1}^{m} ∑_{j=1}^{m} max_{ω} |E_{jk}|^{2}.

Hence it follows that

\overline{σ}[V̂ (jω) − V (jω)] ≤ δ, ∀ ω,  (3.4)

with

δ = \sqrt{∑_{k=1}^{m} ∑_{j=1}^{m} max_{ω} |V̂_{jk}(jω) − V_{jk}(jω)|^{2}}.

As σ_{i}(V) = 1, it is easy to show using (3.4) that^{5}

1 − δ ≤ \underline{σ}(V̂) ≤ \overline{σ}(V̂) ≤ 1 + δ.  (3.5)

Using the minimax characterisation of singular values (see [GL95, p. 30]) and that σ_{i}(GW_{1}) = σ_{i}[G(V + E)Γ], it can be easily proved that

|σ_{i}(GW_{1}) − σ_{i}|γ_{i}|| ≤ β, with β = δ \overline{σ}(G)\overline{σ}(Γ), ∀ i = 1, …, m.  (3.6)

Thus the smaller the value of δ, the closer the singular values of the weighted plant will be to the specified desired loop shape σ_{i}|γ_{i}|. To make δ small it may be necessary to approximate each element of V with a high order transfer function^{6}. It can be deduced using (3.5), that the condition number of the weighting function W_{1} is bounded above by

κ(W_{1}) ≤ \frac{1 + δ}{1 − δ} κ(Γ).  (3.7)

5The inequalities are proved using the singular value inequalities stated in [GL95, pp. 31-33].

6A continuous bounded irrational function can be approximated arbitrarily well on the imaginary axis by a rational function of sufficiently high order.

The bounds given in (3.6) can be conservative at frequencies where Γ = I. It is easy to show that if Γ = I the following bounds are tighter:

(1 − δ)σ_{i}(G) ≤ σ_{i}(GW_{1}) ≤ (1 + δ)σ_{i}(G), ∀ i = 1, …, m.  (3.8)

It may be necessary to reduce the order of W_{1} for implementation. Assume that V̂ has been reduced to V̂_{red} using some model reduction technique which guarantees that

\overline{σ}[V̂_{red}(jω) − V̂ (jω)] ≤ δ_{red}, ∀ ω.

The constant δ_{red} usually depends on the Hankel singular values of V̂. The reduced W_{1} can be computed as described in step 3 of the proposed design procedure but using V̂_{red} instead of V̂. Equations (3.5), (3.6), (3.7) and (3.8) can easily be modified to include the model reduction error. For example (3.6) becomes

|σ_{i}(GW_{1}) − σ_{i}|γ_{i}|| ≤ (δ + δ_{red})\overline{σ}(G)\overline{σ}(Γ), ∀ i = 1, …, m.

Alternatively the designer can model reduce W_{1} directly, as was done in the design example presented in Section 3.8.

The design procedure requirements outlined in Section 3.3 are all satisfied. By construction the weights do not contain RHP poles and zeros. Design examples have shown that the weights designed using the proposed method, do not inherently cancel LHP plant poles and/or zeros. The designer does not lose insight into the problem and the method is easy to use.

A sensible question to ask is whether the proposed design method inherently leads to poor robustness, since the designer is essentially inverting the singular vectors of the plant. Consider for example robustness to input multiplicative uncertainty:

\overline{σ}(T_{d_{i}→y_{i}}) ≤ γ \overline{σ}(Ñ) κ(W_{1}) ≤ γ \frac{1 + δ}{1 − δ} κ(Γ).

The condition number κ(Γ) is set by the designer. If the slope of each singular value of the weighted plant is not too large around cross-over, then the closed loop system will probably be robust, i.e. a good value of γ will probably be achieved. This is suggested by the generalisation of Bode’s sensitivity integral to MIMO systems, with a loop bandwidth constraint [ZDG96, pp. 148-151]. Therefore, if σ_{i}|γ_{i}| are chosen sensibly and δ is small, γ will be good. As the designer has direct control over κ(Γ) and δ, and indirect control over γ, the proposed design procedure does not inherently lead to control systems with poor robustness.

3.6 How would the designer use the proposed design procedure?

The proposed design procedure can be used in two different ways:

1. The designer can design non-diagonal weights that shape the singular values of the scaled open loop plant. As the weights are fully populated, the designer cannot for example decrease the usage of one of the actuators in a simple and intuitive manner. If the weights were diagonal the designer would modify the filter that compensates the actuator with the large usage. In this case, the engineer may have to change the input scaling or change the shape of one or more of the singular values of the weighted plant. Non-diagonal weights used in this manner can remove some of the designer’s insight.

2. A more intuitive approach is to add the proposed design procedure to the end of the procedure proposed in Section 2.8. The designer chooses diagonal weights W_{1} and W_{2} and checks whether the control system design specifications are satisfied. If not, then non-diagonal weights are designed for the weighted plant W_{2}GW_{1}. If for example robustness to input multiplicative uncertainty is poor and W_{1} is illconditioned, then the designer must decrease the condition number of W_{1} (which is a diagonal weight), and design a non-diagonal post-compensator to maintain similar nominal tracking performance.

3.7 Some comments on the proposed design procedure

The spectral factorisations in steps 3 and 4 can be obtained using formulae given in [Fra87]. These standard formulae however, do not accommodate imaginary axis poles and zeros. Hence there is a problem if W has poles and/or zeros on the imaginary axis. It is common for W to contain integrators. If all channels have integrators then they can be factored out of W and absorbed into W_{1}, once W_{1} and W_{2} have been computed. If integrators are not placed in all channels, then it is suggested that each integrator is approximated by a constant at frequencies lower than the dynamics of the nominal plant, before doing the spectral factorisation^{7}. Alternatively, the designer can use more elaborate spectral factorisation methods, that are numerically intensive, but allow for imaginary axis poles and zeros [CG89].

When using this method, it becomes trivial to design H_{∞} loop shaping controllers for the class of plants considered in [HBS96]. In [HBS96], the optimality of SVD controllers is considered for plants with scalar dynamics multiplied by a constant matrix (e.g. the well celebrated distillation column example used in [SMD88, Hyd95]), and circulant symmetric plants. These plants have the property that the singular vector matrices are real and constant across frequency.

7This is not a restriction, as good engineering practice suggests that integrators are approximated by a constant at low frequency before implementing the controller.

If G is ill-conditioned and the designer wishes to achieve good disturbance rejection in all directions at the output (input) of the plant, robust stability with respect to perturbations at the input (output) of the plant will be poor, see Section 2.6 (the same trend applies for robust performance). H_{∞} loop shaping allows the designer to design satisfactory controllers for ill-conditioned plants, as properties at the plant input and output are traded off using the condition number of the weights. The proposed design procedure for computing W_{1} and W_{2} does not change this fact, as the conditioning of the weights is essentially set by the designer, see for example (3.7).

If the controller has been implemented in the observer form, the transfer function from references to outputs depends only on the post-compensator and N. If W_{2} is scalar, or the dynamics of W_{2} are faster than the dynamics of the weighted plant, the designer can set σ_{i}(T_{r→y}) accurately. In the general case that W_{2} is full block, the designer may have difficulty in specifying nominal tracking performance accurately. In most of the design examples that were used to test the method, W_{2} was not required to be non-diagonal, as output scaling and weighting the outputs with high-pass filters was adequate.

Care must be taken when approximating the singular vectors (see step 1 of the proposed procedure), so as not to inflate the order of the weights unnecessarily. δ is a measure of how close the singular values of the weighted plant are to the desired. The higher the order of the approximation the smaller the value of δ. It was found that a δ ≃ 0.1 is adequate.

Gu et al. [GPGC97] have developed an expert system for selecting a scalar precompensator. The designer specifies time domain requirements for the transfer matrix from references to outputs. The expert system then translates the time domain specifications into the frequency domain (assuming a second order model for each channel), and performs an “educated search”, where heuristics select W_{1} with a structure and parameters that satisfy the time response requirements without a large amount of searching. The authors have effectively coded the design procedure suggested by Hyde. By automating the weight selection process and concentrating only on time domain specifications, the designer loses insight into what performance is achievable for a specified level of robustness. The design process also becomes less flexible. Furthermore, the expert system does not include performance specifications on input disturbance rejection, and does not utilise the extra degrees of freedom provided by W_{2} and a full block W_{1}.

Vinnicombe (see [Vin93, pp. 75-76]) proposes a method for designing a full block precompensator W_{1}, to make the transfer matrix from references to outputs equal to some ideal reference model T_{0}^{8}. The designer initially chooses a weight W_{first} using the loop shaping principles discussed in Section 2.5. Then, a minimum phase and stable weight W_{full} is computed so that N, where {M, N} is a right normalised coprime factorisation of G_{s} = G W_{first}W_{full}, is equal to T_{0}. Hence the pre-compensator is W_{1} = W_{first}W_{full}. As the controller is implemented in the observer form, the closed loop transfer matrix from references to outputs will be equal to T_{0}. It can shown that W_{full} cancels open loop plant poles in the LHP (consider Equation V.13 in [Vin93] with a SISO plant model). This can lead to poor robust performance (see Section 3.3). Furthermore, this method only helps the designer achieve nominal tracking performance.

3.8 A design example

To illustrate the proposed design procedure, two H_{∞} loop shaping controllers are designed for the wind tunnel model described in Chapter 6. The first controller is designed using diagonal weights, and the second controller using a non-diagonal pre-compensator. The linear time-invariant model G of the wind tunnel model, has 2 inputs, roll and yaw thrusters, and four states, roll (ϕ) and yaw (ψ) angle and their corresponding rates. The moment produced by the thrusters is measured in Nm, and the angles in rad. The LTI model has been obtained by identifying the wind tunnel model about steady level flight (see Section 6.4). The open loop model includes actuator dynamics. The valves have been

Figure 3.4 Open loop singular values

modelled as a first order transfer function with a corner frequency of 100rad/s, in series with a first order Padé approximation of a 50ms time delay. The delay is due to a length of pipe between the valve and the thruster outlet. Hence, the open loop plant model has 8 states. The singular value plot of the open loop plant is given in Fig. 3.4.

8It is assumed that ∥T_{0}∥_{∞} < 1.

The model G is unstable; the unstable mode has a natural frequency of about 4rad/s. The RHP zero of the Padé approximation has a natural frequency of 40rad/s. The performance specifications described in Section 6.4, dictate that the desired closed loop bandwidth of both channels is about 5rad/s.

W^{2}: low-pass and lead-lag filters G  W_{1}: PI filters

Figure 3.5 The weighting scheme

Figure 3.6 Weighted plant singular values - diagonal weights

The open loop plant requires no scaling, as the units of the inputs and outputs reflect the desired actuator usage and output decoupling respectively. The feedback variables are ϕ and ψ. The first controller is designed using diagonal weights. The weighting scheme is shown in Fig. 3.5. W_{2} consists of low-pass filters for noise rejection, and lead-lag filters to improve robustness by decreasing the roll-off of the open loop singular values around the loop cross-over. W_{1} contains PI filters. The integrators are used to boost the low frequency gain and thus improve performance. By performance it is meant a small steady state error, output decoupling and tracking, and disturbance rejection at both the plant input and output. The proportional matrix gain is used to reduce the phase lag introduced by the integrators around cross-over, and to set the actuator usage. The overall gain of W_{1} and W_{2} is used to specify the loop bandwidth. The weights chosen are

W_{1} = 4 \frac{s + \frac{1}{3}}{s} I_{2}, and W_{2} = \frac{200}{s + 200} \frac{s + 5}{0.02s + 1} I_{2}.

The singular value plot of the weighted plant is illustrated in Fig. 3.6.

Figure 3.7 Closed loop time response - diagonal weights

The achieved γ = 3.22. Fig. 3.7 depicts the time history of the controlled outputs to a step command of ϕ = 1^{∘}. Note that the controller has been implemented in the observer form (see Fig. 2.11, where the W_{2} in the forward path is equal to \frac{5}{0.02s+1} I_{2}). The settling time^{9} of the roll angle is 4.9s. The large settling time is due to the low gain of the weighted plant in some input directions at frequencies between 0.2 and 2rad/s (see the singular value plot illustrated in Fig. 3.6).

It was not possible to find diagonal weights that increased the minimum singular value of the weighted plant without changing the loop shape considerably, i.e. changing the nominal tracking performance. Thus a second controller is designed using a non-diagonal W_{1}. W_{1} consists of a diagonal part, 4 \frac{s+1/3}{s} I_{2}, and a non-diagonal part which was designed using the design procedure outlined in Section 3.2. The post-compensator chosen, is the same as the one used in the first design. The non-diagonal part of the pre-compensator was designed using the open loop plant G, i.e. the model G is used in the procedure described in Section 3.4, as the rest of pre-compensator and W_{2} are diagonal. Γ is

Γ = [\begin{matrix}1 & 0 \\ 0 & 2.5 \frac{0.4s+1}{s+1} \end{matrix}] .

The resulting V̂ has 45 states with an associated δ = 0.09. The non-diagonal part of W_{1} is model reduced to 2 states using optimal Hankel norm approximation. One of the two

9The settling time is the time it takes the system transients to decay within ±1% of the setpoint.

Figure 3.8 Weighted plant singular values - non-diagonal pre-compensator

poles (both poles are real) is replaced by a constant, as it has a corner frequency much higher than the loop bandwidth. The singular values of the weighted plant (solid) and desired loop shape (dashed) are shown in Fig. 3.8. As can be seen, there is very little difference between the two.

The singular value plots of the implemented non-diagonal part of W_{1} and Γ are depicted in Fig. 3.9. The implemented non-diagonal part of W_{1} has a pole at -0.98rad/s and a zero at -2.46rad/s. It can be deduced by examining the poles and zeros of G, that there are

Figure 3.9 Singular values of W_{1} (solid) and Γ (dashed)

no pole/zero cancellations between the weight and the open loop plant. The achieved γ = 3.34 is slightly higher than that achieved in the first design. This is probably because of the higher roll-off rate of minimum singular value around cross-over. The time history of the controlled outputs to a step command of ϕ = 1^{∘} is shown in Fig. 3.10. The settling time of ϕ is now 1.5s.

Figure 3.10 Closed loop time response - non-diagonal pre-compensator

Remark 3.1 If round weights have been selected, perturbations of the singular vectors of the open loop plant will not affect the singular values of the weighted plant, i.e. nominal tracking performance. This is not the case if the weights are ill-conditioned. The use of ill-conditioned weights, requires knowledge of the size and location of the uncertainty present in the open loop plant. Thus ill-conditioned non-diagonal weights must be used with care.

Figure 3.11 Singular values of E

Fig. 3.11 depicts the singular value plot of E = V̂ − V. In the frequency range shown, the maximum value of \overline{σ}(E) is 0.07. Hence the estimated upper bound of \overline{σ}(E) (see

Figure 3.12 Plot of \sqrt{Trace(E^{∗}E)} against frequency

Equation (3.4)), which is computed to be δ = 0.09, is not overly conservative. However, as \overline{σ}(E) ≃ 0.07 at high frequency, at low frequency the value of δ can lead to conservative estimates of how close the singular values of G_{s} are to σ_{i}|f_{i}|. This motivates a weighted approximation of each element of V. The μ-tools command fitsys supports such a feature. The plot of \sqrt{Trace(E^{∗}E)} against frequency is depicted in Fig. 3.12. Note that \overline{σ}(E) ≤ \sqrt{Trace(E^{∗}E)} ≤ δ, see (3.3). The singular value plot of V̂ is illustrated in Fig. 3.13. It is predicted by (3.5) that 0.91 ≤ \underline{σ}(V̂) ≤ \overline{σ}(V̂) ≤ 1.09.

Figure 3.13 Singular values of V̂

It can be deduced from (3.6) that

[1 − δ κ(Γ)] w_{2} \overline{σ}(G) γ_{2} w_{1} ≤ \overline{σ}(G_{s}) ≤ [1 + δ κ(Γ)] w_{2} \overline{σ}(G) γ_{2} w_{1},

where w_{1} = 4 \frac{s+1/3}{s} and w_{2} = \frac{200}{s+200} \frac{s+5}{0.02s+1}. Similarly,

[1 − δ κ(G)] w_{2} \underline{σ}(G) γ_{1} w_{1} ≤ \underline{σ}(G_{s}) ≤ [1 + δ κ(G)] w_{2} \underline{σ}(G) γ_{1} w_{1}.

For frequencies ω ≤ 2.5rad/s, δ κ(Γ) ≃ 0.23 and δ κ(G) ≃ 0.23, therefore the bounds are tight (see Fig. 3.8). Note that w_{2} \overline{σ}(G) w_{1} and w_{2} \underline{σ}(G) w_{1} are illustrated in Fig. 3.6. For ω ≥ 2.5rad/s, Γ = I and hence tighter bounds can be derived (see Equation (3.8))

(1 − δ) w_{2} σ_{i}(G) w_{1} ≤ σ_{i}(G_{s}) ≤ (1 + δ) w_{2} σ_{i}(G) w_{1}.

Both controllers have been flight tested successfully; see Chapter 6. A second design example, that illustrates the use of the proposed design procedure, can be found in [PG97b].

Robust model matching

This chapter describes a design procedure, that enables the designer to meet the time domain specifications defined with respect to the time response of the controlled outputs to a reference command (or a subset of the controlled outputs), accurately and robustly. In the case of designing flight controllers this type of specification is very common; for example see [Mui97, Section 27.3.5].

4.1 Motivation for robust model matching

The designer cannot always specify T_{r→y} accurately (T_{r→y} is the transfer matrix from references (r) to outputs (y), see Fig. 2.11). Assume that the controller has been implemented in the forward path, i.e. T_{r→y} = (I − L_{o})^{−1}L_{o}. It can be difficult to translate the time response requirements defined with respect to T_{r→y} into frequency domain specifications of L_{o} (output loop transfer matrix). This difficulty arises, because it is not always clear how the individual transfer functions of L_{o} affect T_{r→y}. Even if it is, it is not straightforward to shape the individual loop transfer functions. To add to this, the weighted plant will be modified to some extent by the synthesised controller K_{∞}, though if the value of γ is small and κ(W_{2}) is not too large, L_{o} will be close to the weighted plant (see [ZDG96, Section 18.3]).

If the controller has been implemented in the observer form, T_{r→y} = W_{2}^{−1}NN_{dc}W_{2}. Non-diagonal weights enable the designer to shape the singular values of T_{r→y} accurately (if W_{2} is chosen sensibly), but in general shaping the singular values of T_{r→y} does not correspond to shaping its individual transfer functions. If for example the rise time of one of the controlled outputs is poor, the designer can remedy the problem by increasing the bandwidth of the minimum singular value of N; but this may also result in reducing the rise times of other controlled outputs.

In was shown in Section 2.2 that feedback can reduce the effect of modelling uncertainty on the closed loop. Robust performance is one of the two reasons for using feedback control. Robust performance can be expressed in many ways, e.g. it may be desired to maintain good disturbance rejection at the plant output in the presence of input multiplicative uncertainty (see Section 2.6). A very natural and useful robust performance requirement is that of maintaining good time response characteristics of the controlled outputs in the presence of loop perturbations. In the case of flight controllers such a guarantee could help avoid pilot-induced oscillations.

In this chapter a design procedure is developed, that enables the designer to systematically design controllers that meet the time response requirements defined with respect to T_{r→y}, and maintain these requirements to a certain extent in the presence of loop perturbations. The design procedure also guarantees some level of robust stability with respect to perturbations ([ΔM̃ ΔÑ], where ΔM̃, ΔÑ ∈ RL_{∞}) in the normalised coprime factors of the weighted plant (G_{s} = M̃^{−1}Ñ, where {M̃, Ñ} is a normalised lcf of G_{s}).

4.2 The guarantees required

The following guarantees about the closed loop system capture, in a robust control framework, the requirements outlined in the previous section.

1. Robust stability with respect to normalised coprime factor uncertainty

∥[ΔM̃ ΔÑ]∥_{∞} ≤ γ_{RS}^{−1},

where γ_{RS}^{−1} is the stability margin.

2. Nominal performance with respect to an ideal reference model T_{0}

∥T_{r→y} − T_{0}∥_{∞} < δ_{NP} ,

where T_{r→y} is the nominal transfer matrix from references to outputs, and δ_{NP} the nominal performance level.

3. Robust performance with respect to an ideal reference model T_{0}

∥T̂_{r→y} − T_{0}∥_{∞} < δ_{RP} , ∀ ∥[ΔM̃ ΔÑ]∥_{∞} ≤ γ_{RP}^{−1} ,

where T̂_{r→y} is the perturbed transfer matrix from references to outputs (perturbed due to perturbations in the normalised coprime factors of the weighted plant), δ_{RP} is the robust performance level, and γ_{RP}^{−1} the performance margin.

A design procedure that enables the designer to synthesise a controller that comes with the above guarantees, is only of use if the designer can choose γ_{RS}, T_{0}, δ_{NP} , δ_{RP} and γ_{RP} in a meaningful manner. In the rest of this section, a number of observations will be made about γ_{RS}, T_{0}, δ_{NP} , δ_{RP} and γ_{RP}, that will demonstrate that they can be chosen in a meaningful manner.

4.2.1 The stability margin γ_{RS}^{−1}

The significance of the stability margin was discussed in Section 2.7; γ_{RS}^{−1} summarises a lot of information about the closed loop system. Nominal performance, single loop and multi-loop gain and phase margins and some types of robust performance can all be related to γ_{RS}^{−1}. Engineering experience indicates that if the value of γ_{RS}^{−1} is between 0.2–0.3, then the closed loop system will be robust and behave as specified.

4.2.2 The ideal reference model T_{0}

It is easy to translate the time response specifications of each channel into the frequency domain, i.e. into a transfer function T_{0}^{i} (i = 1, …, p, where p is the number of controlled outputs). This results in a diagonal transfer matrix, diag(T_{0}^{1}, …, T_{0}^{p}), which is called the ideal reference model and is denoted by T_{0}. The fact that T_{0} is diagonal emphasises output decoupling. In the case of military aircraft the time response specifications of each channel are provided by the MIL standard. An alternative approach for designing T_{0} would be to design a feedback controller with a performance oriented method, such as nonlinear dynamic inversion or eigenstructure assignment, and use the resulting transfer function from references to outputs as the ideal reference model. This approach has the advantage that T_{0} can actually be achieved, i.e. T_{0} contains any limitations inherited from the open loop system (e.g. RHP zeros) and the cross-coupling is realistic. Using an ideal reference model that has been designed in such a way may result in a better level of robust performance (δ_{RP}).

4.2.3 The performance margin γ_{RP}^{−1}

The designer usually has a feel for type and size of the uncertainty associated with the nominal plant model. For example, the nominal HIRM models used in the GARTEUR design challenge were supplied with a set of plausible perturbed models. A ν-gap analysis can provide a sensible measure of the desired performance margin (γ_{RP}^{−1} ). Thus

γ_{RP}^{−1} = max_{i} δ_{ν}(W_{2}G_{i}W_{1}, W_{2}GW_{1}),

where G_{i} are the perturbed plant models.

4.2.4 The nominal and robust performance levels δ_{NP} and δ_{RP} The guarantee

∥T̂_{r→y} − T_{0}∥_{∞} < δ_{RP} ,

can be used to provide estimates of the time response properties of T̂_{r→y} (rise time, settling time and overshoot), if the robust performance level is smaller than 1, i.e. δ_{RP} < 1. It can be shown that at each frequency

σ_{i}(T_{0}) − δ_{RP} < σ_{i}(T̂_{r→y}) < σ_{i}(T_{0}) + δ_{RP} ,

with σ_{i}(T_{0}) usually not much greater than 1. Similarly, for each transfer function of T̂_{r→y}

|T_{0}^{ij}| − δ_{RP} < |T̂_{r→y}^{ij}| < |T_{0}^{ij}| + δ_{RP} .  (4.1)

Thus boundaries can be placed around every singular value and transfer function of T̂_{r→y}. Fig. 4.1 illustrates boundaries (dotted) placed around the gain of an ideal transfer function |T_{0}^{ij}| (solid), which has a natural frequency of ω_{0} = 1rad/s and a damping ratio of ζ_{0} = 0.5, for a robust performance level of δ_{RP} = 0.7. Fig. 4.1 also illustrates the time and frequency

Figure 4.1 Ideal and perturbed transfer functions with the same damping ratio

responses of two perturbed plants (dashed and dash-dot), which have the same damping ratio as the ideal model but different natural frequencies, and fit within the boundaries. If T̂_{r→y}^{ij} is assumed to be a second order system^{1}, then the plot illustrated in Fig. 4.2 can be used to deduce how the natural frequency of T̂_{r→y}^{ij} (ω_{n}) can vary for every value of δ_{RP}. Thus a robust performance level of 0.7 guarantees a frequency ratio of 1.93, i.e.

1For a good controller design, T̂_{r→y}^{ij} can be closely approximated by a second order system. In some cases, to capture variations in the settling time, a third order system must be considered.

Figure 4.2 Plot of the frequency ratio against the level of robust stability for ζ_{0} = 0.5

ω_{n}^{min} = \frac{ω_{0}}{1.93} = 0.52 and ω_{n}^{max} = 1.93 × ω_{0} = 1.93, if the damping ratio of the ideal and perturbed models is 0.5. The designer can use the approximations

t_{r} ≃ \frac{1.8}{ω_{n}}, and t_{s} ≃ \frac{4.6}{ζω_{n}}

found in [FPEN94, p. 128], to estimate the rise time (t_{r}) and settling time (t_{s}) of the perturbed plant models.

Fig. 4.3 illustrates the upper boundary (dash-dot) placed around the gain of an ideal transfer function |T_{0}^{ij}| (solid), which has a natural frequency of ω_{0} = 1rad/s and a damping ratio of ζ_{0} = 0.5, for a robust performance level of δ_{RP} = 0.7. The dashed line represents a perturbed plant, which has the same natural frequency as the ideal model but a different damping ratio. If T̂_{r→y}^{ij} is assumed to be a second order system, then the plot illustrated in Fig. 4.4 can be used to deduce the worst case damping ratio for every value of δ_{RP}, if the damping ratio of T_{0}^{ij} is ζ_{0} = 0.5. Thus a robust performance level of 0.7 guarantees a damping ratio ≥ 0.31, if the damping ratio of the ideal model is 0.5.

A worst case scenario is illustrated in Fig. 4.5. The ideal model (solid) is the same as that considered in the previous two examples as is the robust performance level. The perturbed plant model (dashed) has a natural frequency of 0.52rad/s and a damping ratio of 0.31. It can be calculated that

∥T̂_{r→y}^{ij} − T_{0}^{ij}∥_{∞} = 0.59 < δ_{RP} = 0.7.

Hence plots as those illustrated in Figs. 4.2 and 4.4 can be used to calibrate δ_{RP}. There is some conservatism associated with Equation (4.1) and hence Figs. 4.2 and 4.4,

4.2 The guarantees required 71

Figure 4.3 Ideal and perturbed transfer functions with the same natural frequency Figure 4.4 Plot of the damping ratio against the level of robust stability for ζ_{0} = 0.5

Figure 4.5 A possible worst case scenario

particularly at low frequency. This stems from the fact that δ_{RP} summarises information about every transfer function of T̂_{r→y}, and that ∥T̂_{r→y} − T_{0}∥_{∞} < δ_{RP} is only valid for

∥[ΔM̃ ΔÑ]∥_{∞} ≤ γ_{RP}^{−1} .

Furthermore, if it is assumed that W_{1} contains integrators in all channels, and that the nominal plant is uncertain not the weights, then in most cases T̂_{r→y} = T_{r→y} at ω = 0.

Remark 4.1 The data used in Fig. 4.2 has been evaluated using the constraint imposed by the boundary (dotted in Fig. 4.1) at ω = ω_{n}. The data used in Fig. 4.4 has been evaluated assuming that the peak of the frequency response of the perturbed transfer function in Fig. 4.3 is achieved at ω = \sqrt{1 − 2ζ_{0}^{2}} ω_{n}. Both assumptions result in fairly accurate plots if ζ_{0} is small (say smaller than 0.6). For more accurate plots more complex equations have to be solved.

Finally, δ_{NP} must be small so that T_{r→y} is close to ideal reference model T_{0}.

4.3 The proposed setup

In this section a new setup will be motivated. This setup will enable the designer to synthesise controllers that come with the guarantees discussed in Section 4.2. Consider the block diagram illustrated in Fig. 4.6. Let {M̃, Ñ} be a normalised lcf of the weighted plant G_{s}, T_{0} be the ideal reference model, and ρ_{i}, ρ_{o} be positive constants. A normalised left coprime factor uncertainty model is chosen (T_{ϕ→[y u]^{T}}). The transfer function from r to z will be used to deduce information about the nominal and robust performance. The

[figure: K; r; β; Ñ; M̃^{−1}; ρ_{o}I; ρ_{i}I; T_{0}; u; ϕ; y; z; −]

Figure 4.6 A new configuration

controller K can be partitioned as K = [K_{1} K_{2}], where K_{1} is the command pre-filter and K_{2} the feedback controller.

The weighted plant G_{s} is assumed to have m inputs and p outputs, i.e. u ∈ ℝ^{m} and r, y, z ∈ ℝ^{p}. It is assumed that performance requirements are placed on all the feedback signals; this does not have to be the case, it is done for notational convenience. Define

Δ := {diag(Δ_{1}, Δ_{2}) : Δ_{1} ∈ ℂ^{p×p}, Δ_{2} ∈ ℂ^{p×p+m}},

and

M(s) := \{Δ(s) = diag(Δ_{perf}, [ΔM̃ ΔÑ]) ∈ RL_{∞} : Δ(s_{0}) ∈ Δ, ∀ s_{0} ∈ \overline{ℂ}_{+}\}.

Let the transfer matrix from [r ϕ]^{T} → [z y u]^{T} be denoted by M, where

M = [\begin{matrix}ρ_{o}I \\ & I \end{matrix}] [ \begin{matrix}(I − G_{s}K_{2})^{−1}G_{s}K_{1} − T_{0} & (I − G_{s}K_{2})^{−1}M̃^{−1} \\ G_{s}(I − K_{2}G_{s})^{−1}K_{1} & (I − G_{s}K_{2})^{−1}M̃^{−1} \\ (I − K_{2}G_{s})^{−1}K_{1} & K_{2}(I − G_{s}K_{2})^{−1}M̃^{−1} \end{matrix} ] [\begin{matrix}ρ_{i}I \\ & I \end{matrix}] .

Partition M

M = [ \begin{matrix}M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix} ] , where M_{11} = ρ_{o}[(I − G_{s}K_{2})^{−1}G_{s}K_{1} − T_{0}]ρ_{i}.

Consider the following minimisation

min_{stab K} sup_{ω∈ℝ} μ_{Δ}(M(jω)), with ∥M_{22}∥_{∞} < γ_{RS}.  (4.2)

Let

Π = {Ĝ_{s} = (M̃ − ΔM̃)^{−1}(Ñ + ΔÑ) : [ΔM̃ ΔÑ] ∈ RL_{∞}} ,

denote the set of perturbed models of the weighted plant. Assume that minimisation (4.2) is solved and sup_{ω∈ℝ} μ_{Δ}(M(jω)) < γ_{RP}. It is obvious that the closed loop system will be stable for all Ĝ_{s} with

∥[ΔM̃ ΔÑ]∥_{∞} ≤ γ_{RS}^{−1}.

The μ robust performance theorem (Theorem 2.7) guarantees that

∥(I − Ĝ_{s}K_{2})^{−1}Ĝ_{s}K_{1} − T_{0}∥_{∞} < δ_{RP} , where δ_{RP} = \frac{γ_{RP}}{ρ_{i} ρ_{o}} ,  (4.3)

for all Ĝ_{s} with

∥[ΔM̃ ΔÑ]∥_{∞} ≤ γ_{RP}^{−1} .

Equation (4.3) also guarantees a nominal performance level of δ_{NP} = δ_{RP}, i.e. the guaranteed robust and nominal performance levels are equal. Including a constraint of the type

∥M_{11}∥_{∞} < γ_{NP} , with δ_{NP} = \frac{γ_{NP}}{ρ_{i} ρ_{o}}

in minimisation (4.2) could result in a better level of nominal performance if δ_{NP} < δ_{RP}. This will be discussed in Section 4.4.

Therefore optimisation (4.2) can be used to synthesise controllers that come with the desired guarantees.

The designer uses a pre-compensator W_{1}, and the constants ρ_{o} and ρ_{i} to specify the control system requirements. If a post-compensator W_{2} is used, then (4.3) is not meaningful as such. Denote the open loop perturbed plant by Ĝ. If it is assumed that no uncertainty is associated with W_{2}, then at each frequency

\begin{matrix}\overline{σ}[(I − ĜW_{1}K_{2}W_{2})^{−1}ĜW_{1}K_{1}W_{2} − T_{0}] ≤ \\ \overline{σ}(W_{2}^{−1})\overline{σ}[(I − W_{2}ĜW_{1}K_{2})^{−1}W_{2}ĜW_{1}K_{1} − W_{2}T_{0}W_{2}^{−1}] \overline{σ}(W_{2}). \end{matrix}

If W_{2} and T_{0} are assumed to be diagonal, then from (4.3) it is deduced that

\overline{σ}[(I − ĜW_{1}K_{2}W_{2})^{−1}ĜW_{1}K_{1}W_{2} − T_{0}] < κ(W_{2}) δ_{RP} .

The weight W_{2} is usually used for noise rejection, thus it is common that W_{2} is a diagonal transfer matrix of low-pass filters. If T_{0} is also chosen to be a diagonal transfer matrix, emphasising good output decoupling, then κ(W_{2}) δ_{RP} is the guaranteed level of robust performance. As κ(W_{2}) is commonly 1 at frequencies of interest, the guaranteed robust performance level will be equal to δ_{RP}.

Define the set

D_{l} := {diag(dI_{p}, I_{p+m}) : d ∈ ℝ, d > 0}.

Every D_{l} ∈ D_{l} is associated with a D_{r} = diag(d^{−1}I_{p}, I_{p}). It can be shown that for the Δ under consideration, i.e. block diagonal with two blocks, at each frequency ω

μ_{Δ}(M(jω)) = inf_{D_{l}∈D} \overline{σ}(D_{l}M(jω)D_{r}).  (4.4)

Hence sup_{ω∈ℝ} μ_{Δ}(M(jω)) can be calculated exactly. In general (4.4) is an inequality (≤), see [PD93, Table 1]. Minimisation (4.2) without the robust stability constraint is a standard μ synthesis problem. This synthesis problem has not yet been fully solved. Equation (4.4) motivates approximating the μ synthesis problem with

min_{stab K} inf_{D_{l}(s)∈D_{l}(s)} ∥D_{l}(s)M(s)D_{r}(s)∥_{∞} ,  (4.5)

where the set D_{l}(s) is defined as

D_{l}(s) := {D_{l}(s) = diag(d(s)I_{p}, I_{p+m}) : d(s), d^{−1}(s) ∈ RH_{∞}}.

Every D_{l}(s) ∈ D_{l}(s) is associated with a D_{r}(s) = diag(d^{−1}(s)I_{p}, I_{p}). D_{l}(s) and D_{r}(s) are chosen so that D_{r}^{−1}(s)Δ(s) = Δ(s)D_{l}(s). The optimisation problem stated in Equation (4.5) is “solved” by iteratively solving for K and D_{l}(s). This is the so-called D-K iteration^{2}. For a fixed D_{l}(s), min_{K} ∥D_{l}(s)M(s)D_{r}(s)∥_{∞} is a standard H_{∞} optimisation problem. For a given K, inf_{D_{l}(s)∈D_{l}(s)} ∥D_{l}(s)M(s)D_{r}(s)∥_{∞} is a standard convex optimisation problem. As given the minimising D_{l} across frequency, there is always a rational function D_{l}(s) ∈ D_{l}(s) approximating each D_{l}, this convex optimisation problem can be solved point-wise in the frequency domain

sup_{ω∈ℝ} inf_{D_{l}∈D} \overline{σ}(D_{l}M(jω)D_{r}).

A μ synthesis tutorial can be found in [BDG^{+}93]. It is evident from (4.4) that for the Δ under consideration, the convex optimisation problem corresponds to evaluating sup_{ω∈ℝ} μ_{Δ}(M(jω)), and hence conservatism from the D step of the D-K iteration arises only from approximating the minimising D_{l} across frequency.

As minimisation (4.2) is hard to solve, throughout this chapter the following optimisation problem will be considered

min_{stab K} inf_{D_{l}(s)∈D_{l}(s)} ∥D_{l}(s)M(s)D_{r}(s)∥_{∞} , with ∥M_{22}∥_{∞} < γ_{RS},  (4.6)

and will be “solved” using the D-K iteration. A possible design procedure is outlined below:

1. Choose a possible range for ρ^{2} = ρ_{o} ρ_{i} and grid the range.

2. At each point of the grid solve minimisation (4.6); how to solve the minimisation will be discussed in Section 4.6. Because of the structure of D_{l}(s), two different pairs of (ρ_{o}, ρ_{i}) which have the same product lead to the same solution (this actually depends on how well D_{l}(s) approximates the individual D_{l}).

2The cost of (4.5) can be shown to decrease monotonically if D_{l}(s) interpolates the individual D_{l} exactly, but there is no guarantee that a global minimum will be achieved.

3. Plot δ_{RP} and γ_{RP} against ρ^{2}. The plot of δ_{RP} versus ρ^{2} is monotonically decreasing, and the plot of γ_{RP} versus ρ^{2} monotonically increasing (see for example Fig. 4.10). Synthesise the controller for the value of ρ^{2} which corresponds to the desired γ_{RP}. If the corresponding value of δ_{RP} is too large, then reconsider the desired γ_{RS} and γ_{RP}.

All stabilising controllers K in (4.6) can be parametrised in terms of a free parameter Q ∈ RH_{∞}. Let M̂(s) = D_{l}(s)M(s)D_{r}(s) and P be the generalised plant that corresponds to M̂

P = [ \begin{matrix}−ρ^{2} T_{0} & ρ_{o} d M̃^{−1} & ρ_{o} d G_{s} \\ 0 & M̃^{−1} & G_{s} \\ 0 & 0 & I \\ ρ_{i} d^{−1} I & 0 & 0 \\ 0 & M̃^{−1} & G_{s} \end{matrix} ] , with P_{22} = [\begin{matrix}0 \\ G_{s} \end{matrix}] .

Note that in the sequel the dependence on s will be made clear from the context. If the realisation of P_{22} is stabilisable and detectable, then the interconnected system [P, K] is internally stable if and only if [P, K] is internally stable [ZDG96, Lemma 12.2]. It will be shown in Subsection 4.6.2 that the realisation of P_{22} will be stabilisable and detectable if the realisation of G_{s} is stabilisable and detectable and T_{0}, d, d^{−1} ∈ RH_{∞}. By definition T_{0}, d, d^{−1} are always going to be stable transfer matrices. The normalised lcf and rcf of P_{22} are

P_{22} = [\begin{matrix}0 \\ G_{s} \end{matrix}] = [\begin{matrix}I & 0 \\ 0 & M̃^{−1} \end{matrix}] [ 0 ; Ñ] = [\begin{matrix}0 \\ N \end{matrix}] M^{−1},

where {M, N} is a normalised rcf of the weighted plant. If Q = [Q_{1} Q_{2}], then Theorem 2.9 can be used to parametrise all stabilising controllers. Hence

[K_{1} K_{2}] = [MQ_{1} − (U_{0} + MQ_{2})(V_{0} + NQ_{2})^{−1}NQ_{1} (U_{0} + MQ_{2})(V_{0} + NQ_{2})^{−1}] ,

or

[K_{1} K_{2}] = [(Ṽ_{0} + Q_{2}Ñ)^{−1}Q_{1} (Ṽ_{0} + Q_{2}Ñ)^{−1}(Ũ_{0} + Q_{2}M̃)] ,

where Ṽ_{0}, Ũ_{0}, V_{0}, U_{0} ∈ RH_{∞} and

[ \begin{matrix}Ṽ_{0} & 0 & −Ũ_{0} \\ 0 & I & 0 \\ −Ñ & 0 & M̃ \end{matrix} ] [ \begin{matrix}M & 0 & U_{0} \\ 0 & I & 0 \\ N & 0 & V_{0} \end{matrix} ] = [ \begin{matrix}I & 0 & 0 \\ 0 & I & 0 \\ 0 & 0 & I \end{matrix} ] .

Thus M̂ becomes

M̂ = [\begin{matrix}ρ_{o} d I_{p} \\ & I_{p+m} \end{matrix}] ([ \begin{matrix}−T_{0} & V_{0} \\ 0 & V_{0} \\ 0 & U_{0} \end{matrix} ] + [ \begin{matrix}N \\ N \\ M \end{matrix} ] [ Q_{1} Q_{2} ]) [\begin{matrix}ρ_{i} d^{−1} I_{p} \\ & I_{p} \end{matrix}] . (4.7)

All stabilising K_{2} and K_{∞} (K_{∞} is the H_{∞} loop shaping controller presented in Chapter 2) have the same parametrisation. Consequently, Equations (2.13) to (2.18) hold for this new setup and the maximum singular value of every closed loop transfer matrix can be bounded from above using the technique outlined in Section 2.5.

Because of the robust stability constraint, optimisation problem (4.6) is an infinitedimensional problem, and hence is difficult to solve^{3}. The fact that (4.6) is not a finitedimensional problem will become evident in Subsection 4.6.4. Using Equation (4.7), the cost function in (4.6) can be written in terms of the free parameter Q. Three different approaches to solving (4.6) will be considered in detail in Section 4.6:

1. Optimise over Q_{1} and Q_{2} separately.

2. Remove the robust stability constraint and optimise over Q_{1} and Q_{2} simultaneously.

3. Make the problem finite-dimensional and optimise over Q_{1} and Q_{2} simultaneously.

A fourth approach which enables the designer to include meaningful nominal performance constraints, will also be presented. Each approach offers valuable insight to the designer.

The following bounds of the structured singular value will be used in the sequel. Assume that K is a stabilising controller, i.e. M ∈ RH_{∞}. Equation (4.4) implies that at each frequency ω

μ_{Δ}(M(jω)) = inf_{d∈ℝ_{+}} \overline{σ} ([\begin{matrix}M_{11}(jω) & d M_{12}(jω) \\ d^{−1} M_{21}(jω) & M_{22}(jω) \end{matrix}]) .  (4.8)

The equality is because Δ has two blocks. An approximation to d can be obtained by approximating the right hand side of Equation (4.8), see [ZDG96, pp. 289-291]. This leads to the following upper bound

μ_{Δ}(M(jω)) ≤ \sqrt{∥M_{11}(jω)∥^{2} + ∥M_{22}(jω)∥^{2} + 2∥M_{12}(jω)∥∥M_{21}(jω)∥}.  (4.9)

M_{11} characterises nominal performance and M_{22} robust stability. All blocks affect robust performance. From the definition of μ

μ_{Δ}(M(jω)) ≥ max_{i}(∥M_{ii}(jω)∥).  (4.10)

3As the D-K-iteration will be used it is a convex problem.

If the off-diagonal blocks ∥M_{12}(jω)∥ and ∥M_{21}(jω)∥ are large, then

μ_{Δ}(M(jω)) ≥ \sqrt{∥M_{12}(jω)∥ ∥M_{21}(jω)∥} − max(∥M_{11}(jω)∥, ∥M_{22}(jω)∥)  (4.11)

derived in [Fre89] may be tighter than (4.10). Freudenberg also derives an alternative upper bound to (4.9)

μ_{Δ}(M(jω)) ≤ \sqrt{∥M_{12}(jω)∥ ∥M_{21}(jω)∥} + max(∥M_{11}(jω)∥, ∥M_{22}(jω)∥).  (4.12)

Equations (4.11) and (4.12) are only valid if Δ is block diagonal with two blocks.

4.4 A note on nominal performance

The control system design specifications dictate that T_{r→y}(0) = I. This is not guaranteed by solving minimisation (4.6); all that is guaranteed is that

\overline{σ}(T_{r→y}(0) − T_{0}) ≤ δ_{RP} .

Thus a constant pre-filter T_{dc} must be added to the control system so that

T_{r→y} T_{dc} = (I − G_{s}K_{2})^{−1}G_{s}K_{1}T_{dc} = I, at ω = 0.

Unfortunately T_{dc} will change the robust performance level. The modified transfer matrix from [r ϕ]^{T} → [z y u]^{T} is

M_{mod} = [\begin{matrix}ρ_{o} I_{p} \\ & I_{p+m} \end{matrix}] [ \begin{matrix}NQ_{1}T_{dc} − T_{0} & V_{0} + NQ_{2} \\ NQ_{1}T_{dc} & V_{0} + NQ_{2} \\ MQ_{1}T_{dc} & U_{0} + MQ_{2} \end{matrix} ] [\begin{matrix}ρ_{i} I_{p} \\ & I_{p} \end{matrix}] .  (4.13)

Design experience indicates that ρ^{2} \overline{σ}(NQ_{1}T_{dc} − T_{0}) ≤ γ_{RS}. Hence it can be deduced from Equation (4.12) and \overline{σ}(M̂) < γ_{RP}, that

μ_{Δ}(M_{mod}(jω)) < \sqrt{\overline{σ}(T_{dc})} γ_{RP} + γ_{RS}.

As it can be shown that

\frac{1}{1 + δ_{RP}} < \overline{σ}(T_{dc}) < \frac{1}{1 − δ_{RP}} ⟹ μ_{Δ}(M_{mod}(jω)) < \sqrt{\frac{1}{1 − δ_{RP}}} γ_{RP} + γ_{RS}.

This upper bound has been found to be quite conservative. Nevertheless it indicates that the smaller the value of δ_{RP} the smaller μ_{Δ}(M_{mod}(jω)) will be. It has been observed that T_{dc} produces better model matching at all frequencies because the optimisation process tends to give T_{r→y} the same frequency response shape as T_{0}.

A constraint on nominal performance could be included in minimisation (4.6). Intuition suggests that this could be achieved by weighting r or z in Fig. 4.6 with a transfer matrix W, with W^{−1} a high-pass filter. Unfortunately this leads to high values of γ_{RP} as W also weights M_{12} or M_{21}. This scheme essentially optimises nominal performance by optimising the robust performance at low frequency which is not desired. Including a constraint of the type ∥T_{r→y} − T_{0}∥_{∞} < δ_{NP} was found to be too rigid around the desired closed loop bandwidth. The solution is to include constraints at discrete frequency points, e.g. at ω = 0, or to include a constraint of the type

∥W (NQ_{1} − T_{0})∥_{∞} < δ_{NP} .

Such a constraint can only be included in the first and fourth proposed way of solving minimisation (4.6). The constraints at discrete frequency points can only be imposed in the fourth proposed method.

4.5 A note on the history of the new setup

The setup illustrated in Fig. 4.6 was first considered in [LKP93]. The following optimisation problem was proposed

min_{stab K} ∥[ \begin{matrix}ρ^{2} [(I − G_{s}K_{2})^{−1}G_{s}K_{1} − T_{0}] & ρ (I − G_{s}K_{2})^{−1}M̃^{−1} \\ ρ G_{s}(I − K_{2}G_{s})^{−1}K_{1} & (I − G_{s}K_{2})^{−1}M̃^{−1} \\ ρ (I − K_{2}G_{s})^{−1}K_{1} & K_{2}(I − G_{s}K_{2})^{−1}M̃^{−1} \end{matrix} ]∥_{∞} .  (4.14)

The authors in [LKP93] suggest that the designer uses only a pre-compensator W_{1} and the parameter ρ to specify the control system requirements.

Minimisation (4.14) does not always result in a control system with a satisfactory stability margin. As ρ increases δ_{RP} decreases, but γ_{RP} increases reducing the guaranteed stability margin and the size of the set of plants for which the robust performance guarantee is valid. These trends are the result of observations made while designing controllers using this method. Thus ρ is not an effective tuning parameter.

The weighting function W_{2} is not used to shape G. As discussed in Section 2.5, W_{2} is used for noise rejection; filtering the controlled outputs becomes very important when controlling the real plant.

As μ_{Δ}(M(jω)) is bounded above by \overline{σ}(M(jω)), minimisation (4.14) will result in values of γ_{RP} greater than or equal to the values of γ_{RP} resulting from minimisation (4.6) without the robust stability constraint.

4.6 The proposed design methods

This section deals with solving minimisation (4.6). As the optimisation problem is infinitedimensional because of the robust stability constraint, it will approximated. Four different solutions will be presented. Each solution gives the designer additional insight about how to best achieve the design objectives.

4.6.1 The first method

The controller K is designed in two steps. In the first step, a feedback controller K_{2} is designed using H_{∞} loop shaping to achieve the desired value of γ_{RS}. In the second step a command pre-filter K_{1} is designed, so that the control system responds in the desired manner to a reference command. Essentially, the designer optimises over Q_{1} and Q_{2} separately, see Equation (4.7) but without the D-scales (the D-K iteration is not used).

K_{1} is usually designed after having designed the feedback controller by solving

min_{Q_{1}∈RH_{∞}} ∥W (NQ_{1} − T_{0})∥_{∞},  (4.15)

where W is minimum phase and stable, and hence K_{1} = (Ṽ_{0} + Q_{2}Ñ)^{−1}Q_{1}. Designing K_{1} in this way could lead to poor robust performance. In the case of flight controllers poor robust performance could lead to PIOs. If the actuators rate limit, additional phase lag is added to the loop. This could degrade closed loop performance to such an extent, that the closed loop system of the pilot and the augmented aircraft is unstable. Pilot-induced oscillations can lead to the loss of aircraft. The flight control system software of the JAS 39 Gripen includes a phase-correction “filter” designed to keep a pilot from loosing control of an advanced computer-controlled fighter aircraft, such as the Gripen, by eliminating the lag between a pilot’s stick movements and the signals from the flight control system to the aircraft’s control surfaces [Shi95]. If the feedback controller K_{2} has been implemented in the observer form, see Fig. 2.11, then Q_{1} will be the command pre-filter (phase-correction “filter”).

The cost function used for designing the command pre-filter is motivated by the upper bound given in Equation (4.9). Consider the optimisation problem

min_{Q_{1}∈RH_{∞}} ∥[\begin{matrix}W_{e_{1}} \\ & W_{e_{2}} \end{matrix}] ([\begin{matrix}−T_{0} \\ 0 \end{matrix}] + [\begin{matrix}N \\ I \end{matrix}] Q_{1})∥_{∞} < γ_{1},  (4.16)

where the weighting functions W_{e_{1}} and W_{e_{2}} are units in RH_{∞}. The corresponding setup is illustrated in Fig. 4.7. It is easy to show that if (4.16) has a solution, then

\overline{σ}(NQ_{1} − T_{0}) < \frac{γ_{1}}{\underline{σ}(W_{e_{1}})} and \overline{σ}(Q_{1}) < \frac{γ_{1}}{\underline{σ}(W_{e_{2}})}.  (4.17)

[figure: Q_{1}; T_{0}; N; W_{e_{2}}; W_{e_{1}}; −; d; e_{1}; e_{2}]

Figure 4.7 Setup used for designing Q_{1}

The achievable norm bound is restricted by any RHP zeros of the weighted plant. If G_{s} is SISO and has a RHP zero (z), then γ_{1} ≥ |W_{e_{1}}(z)| |T_{0}(z)|. It can be deduced from (4.9) that at each frequency ω

μ_{Δ}(M(jω)) < \sqrt{(ρ^{2} \frac{γ_{1}}{\underline{σ}(W_{e_{1}}(jω))})^{2} + γ_{RS}^{2} + 2 ρ^{2} \frac{γ_{1}}{\underline{σ}(W_{e_{2}}(jω))} γ_{RS}}.  (4.18)

To prove that Equation (4.18) holds, note that \overline{σ}([NQ_{1} MQ_{1}]^{T}) = \overline{σ}(Q_{1}). Equation (4.18) provides guidelines on how to choose the weights W_{e_{1}} and W_{e_{2}} and the stability margin γ_{RS}^{−1} to achieve a good level of robust performance.

For good nominal and robust performance W_{e_{1}}^{−1} must be a high-pass filter and W_{e_{2}}^{−1} a low-pass filter. Thus, \overline{σ}(NQ_{1} − T_{0}) is forced to be small at low frequencies and \overline{σ}(Q_{1}) is forced to be small at high frequencies, see Equation (4.17). γ_{1} will be close to 1 if the bandwidth of T_{0} is smaller than the bandwidth of N, and σ_{i}(W_{e_{1}}) and σ_{i}(W_{e_{2}}) do not take large values in the same frequency region^{4}. The bandwidth of N is set by shaping the singular values of the weighted plant G_{s}. Fig. 4.8, that has been plotted from data used to design controller LTI3 (see Section 6.5), depicts the singular value plots of N (dashed), T_{0} (solid), W_{e_{1}} (dash-dot) and W_{e_{2}} (dotted).

It is obvious from Equation (4.18), that if γ_{RS} is small then γ_{RP} will also be small^{5}. It also becomes apparent that γ_{RP} can be reduced by degrading the stability margin γ_{RS}^{−1}. The term 2ρ^{2}γ_{1}γ_{RS}/\underline{σ}(W_{e_{2}}) on the right hand side of Equation (4.18) is small at high frequency but approximately equal to 2ρ^{2} \overline{σ}(V_{0} + NQ_{2}) at low frequency (note that \overline{σ}(V_{0} + NQ_{2}) < γ_{RS}). Thus if the designer can reduce \overline{σ}(V_{0} + NQ_{2}) at low frequency without increasing γ_{RS} too much, the achieved γ_{RP} could decrease. One possible way of achieving this is to pre-compensate the H_{∞} loop shaping cost function with a weight W_{RP} which is minimum phase and stable, and has a singular value plot similar in shape

4It is assumed that the cost function in Equation (4.16) has been normalised, i.e. W_{e_{2}} is equal to the identity in the low frequency range. Thus for a good design, i.e. γ_{1} close to 1, the maximum singular value of Q_{1} will be approximately equal to 1 in the low frequency range.

5Note that γ_{RP} = max_{ω} μ_{Δ}(M(jω)).

Figure 4.8 Singular value plots of N(dashed), T_{0}(solid), W_{e_{1}}(dash-dot) and W_{e_{2}}(dotted)

to the singular value plot of V = V_{0} + NQ_{2}, see Fig. 2.14. If the frequency range over which σ_{i}(W_{RP}) > 1 is not too large, then γ_{RS} will not increase too much. The H_{∞} loop shaping cost function cannot be made too small over a large frequency range as it is related to the input and output sensitivities. The cost functions minimised in Subsections 4.6.2 and 4.6.3, optimise γ_{RP} by making use of this trade-off. Fig. 4.9, that has been plotted from data used to design controller LTI3, illustrates the singular value plots of the weighted (solid) and unweighted (dashed) H_{∞} loop shaping costs together with the W_{RP} used (dash-dot).

Figure 4.9 Reducing the value of γ_{RP} by allowing γ_{RS} to increase

Equation (4.16) is equivalent to

min_{stab Q_{1}} ∥ F_{l} ([ \begin{matrix}−W_{e_{1}}T_{0} & W_{e_{1}}N \\ 0 & W_{e_{2}} \\ I & 0 \end{matrix} ] , Q_{1}) ∥_{∞} .  (4.19)

This is shown in [Fra87]. Thus optimisation (4.16) can easily be solved using the well documented command hinfsyn from [BDG^{+}93]. Q_{1} is guaranteed to be stable, as by solving (4.19)

[\begin{matrix}−W_{e_{1}}T_{0} \\ 0 \end{matrix}] + [\begin{matrix}W_{e_{1}}N \\ W_{e_{2}} \end{matrix}] Q_{1} ∈ RH_{∞},

and W_{e_{2}} is chosen to be minimum phase and stable.

Q_{1} has as many states as the weighted plant G_{s} plus the ideal reference model T_{0} plus the weighting functions W_{e_{1}} and W_{e_{2}}, assuming that W_{RP} is not used. When implementing a controller it is the number of states of its feedback portion that is important. The feedback controller K_{2} has the same number of states as the weighted plant.

Figure 4.10 The predicted (solid) and actual (dashed) values of γ_{RP}^{−1} and δ_{RP}

Design experience indicates that the bound (4.18) is usefully tight, e.g. Fig. 4.10. It is evident that the bound is less tight for large values of ρ. Fig. 4.10 has been plotted from data used to design controller LTI3; the weight W_{RP} was not used, γ_{RS} = 3.76, γ_{1} = 1.02, W_{e_{1}} = 5 and W_{e_{2}} = 1. It can be shown that

lim_{ρ→∞} δ_{RP} = \frac{γ_{1}}{min_{ω} \underline{σ}(W_{e_{1}})}, and lim_{ρ→∞} γ_{RP} = ∞.

This is proved by taking the limits of the upper and lower bounds of μ_{Δ}(M(jω)), given in Equations (4.9) and (4.10) respectively, as ρ → ∞.

If the bandwidth of T_{0} is much smaller than the bandwidth of N, then solving optimisation (4.15) could result in a Q_{1} that is compatible with robust performance. It can be shown that if

∥W (NQ_{1} − T_{0})∥_{∞} < γ̂_{1} ⟹ \overline{σ}(Q_{1}) < \frac{γ̂_{1} + \overline{σ}(W) \overline{σ}(T_{0})}{\underline{σ}(N) \underline{σ}(W)}.

Thus \overline{σ}(Q_{1}) ≃ \overline{σ}(T_{0}) at low frequency if W is round. At high frequency \overline{σ}(Q_{1}) will roll off if T_{0} rolls off at a faster rate than N. Note that the above trends hold if γ̂_{1} is sufficiently small. It must also be noted that if the bandwidth of T_{0} is much smaller than the bandwidth of N, γ_{RP} will be small as the frequency range over which \overline{σ}(V_{0} + NQ_{2}) has to be pushed down is small.

This design method allows the designer to directly control the stability margin and the nominal performance level, but the controller is not optimal with respect to γ_{RP}. Nevertheless, insight is given on how the choice of T_{0} affects γ_{RP}, and how the designer can trade off γ_{RS} to achieve good robust performance. Because K_{2} is designed using H_{∞} loop shaping, the feedback controller can be written as a plant observer with state feedback.

Remark 4.2 A good value of γ_{RP} can be achieved at the expense of degrading the stability margin γ_{RS}^{−1}. This trade off is very evident when the bandwidth of T_{0} is similar to the bandwidth of N. The design method presented above works well if the bandwidth of T_{0} is smaller than the bandwidth of N.

4.6.2 The second method

In this subsection optimisation (4.6) is considered without the robust stability constraint

min_{stab K} inf_{D_{l}(s)∈D_{l}(s)} ∥D_{l}(s)M(s)D_{r}(s)∥_{∞} .  (4.20)

This optimisation is “solved” using the D-K iteration. The designer essentially optimises over Q_{1} and Q_{2} without taking the robust stability constraint into account.

It will be shown in the sequel, that any controller K that results from solving optimisation (4.20), has a structure that enables “classical” gain scheduling and can handle input nonlinearities (the controller is driven with achieved rather than demanded plant inputs). Consider the setup illustrated in Fig. 4.11. The gain matrix S_{o} ∈ ℝ^{p̂×p} is used, if the designer does not wish to impose robust performance specifications on all the controlled outputs (y ∈ ℝ^{p}); note that z, ẑ ∈ ℝp̂. Thus S_{o} is a fat matrix, i.e. has more inputs than outputs. It is assumed that ρ = ρ_{i} = ρ_{o}, as it was shown in Section 4.3 that this does not restrict the achievable norm bound.

[figure: D^{−1}; ρ; K; Ñ; M̃^{−1}; ϕ; u; y; r̂; r; S_{o}; ρ; D; T_{0}; z; ẑ; −]

Figure 4.11 Setup considered in Subsection 4.6.2

Let G_{s}, T_{0} and D^{−1} have the following state-space realisations

G_{s} = [ \begin{matrix}A & B \\ C & 0 \end{matrix} ] , T_{0} = [ \begin{matrix}A_{0} & B_{0} \\ C_{0} & 0 \end{matrix} ] , D^{−1} = [ \begin{matrix}A_{d} & B_{d} \\ C_{d} & D_{d} \end{matrix} ] .

Note that D(s) = d(s)Ip̂ and D_{l}(s) = diag(d(s)Ip̂, I_{p+m}). Thus D^{−1} is a unit in RH_{∞} and is invertible. Because S_{o} has been added to the setup, the definition of D_{l}(s) used in this subsection has different dimensions to that used in Section 4.3.

[Ñ M̃] = [ \begin{matrix}A + HC & B & H \\ C & 0 & I \end{matrix} ] ,

where H = −Y C^{T} and Y = Y^{T} ≥ 0 is the stabilising solution of the following algebraic Riccati equation

AY + Y A^{T} − Y C^{T} CY + BB^{T} = 0.

It is assumed that G_{s} is strictly proper. This is not a restrictive assumption, because even if the nominal plant has a D-term the singular values of the weighted plant are usually rolled off at high frequency for noise rejection and good robustness with respect to high frequency uncertainty.

Let P denote the generalised plant corresponding the cost function minimised in (4.20). Thus, D_{l}(s)M(s)D_{r}(s) = F_{l}(P, K). It can be shown that P has the following state-space realisation

[ ẋ ; z ; y ; u ; r̂ ; y ] = [ \begin{matrix}Â & B̂_{1} & B̂_{2} \\ Ĉ_{1} & D̂_{11} & D̂_{12} \\ Ĉ_{2} & D̂_{21} & D̂_{22} \end{matrix} ] [ \begin{matrix}x \\ r \\ ϕ \\ u \end{matrix} ] = P_{ss} [ \begin{matrix}x \\ r \\ ϕ \\ u \end{matrix} ] ,

where x is the state vector of P, and P_{ss}

[ \begin{matrix}A & 0 & 0 & 0 & 0 & −H & B \\ 0 & A_{0} & ρB_{0}C_{d} & 0 & ρB_{0}D_{d} & 0 & 0 \\ 0 & 0 & A_{d} & 0 & B_{d} & 0 & 0 \\ −ρB_{d}D_{d}^{−1}S_{o}C & ρB_{d}D_{d}^{−1}C_{0} & 0 & A_{d} − B_{d}D_{d}^{−1}C_{d} & 0 & −ρB_{d}D_{d}^{−1}S_{o} & 0 \\ ρD_{d}^{−1}S_{o}C & −ρD_{d}^{−1}C_{0} & 0 & D_{d}^{−1}C_{d} & 0 & ρD_{d}^{−1}S_{o} & 0 \\ C & 0 & 0 & 0 & 0 & I & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & I \\ 0 & 0 & ρC_{d} & 0 & ρD_{d} & 0 & 0 \\ C & 0 & 0 & 0 & 0 & I & 0 \end{matrix} ] .

In [SG90] it is proved that an H_{∞} loop shaping controller can be written as a plant observer with state feedback. The same proof can be used to show that any controller K that results from solving optimisation (4.20), has the structure illustrated in Fig. 4.12. The outline of the proof is given below:

1. Normalise the realisation of P by normalising D̂_{21}, see [ZDG96, p. 454]. Denote the normalised generalised plant by P̂.

2. As the (2,1) block of the D-matrix of P̂ is equal to I, γ_{RP} > \overline{σ}([ρD_{d}^{−1}S_{o} I]^{T}), see [ZDG96, p. 453].

3. P̂ has a disturbance feedforward structure. Thus from [SG90, Proposition A.1], the controller K̂ = F_{l}(K̂_{DF} , K_{F I}) internally stabilises P̂ (the structure of K̂_{DF} is given in [SG90]).

4. To obtain K_{F I} solve the control algebraic Riccati equation (CARE) corresponding to P̂, see [ZDG96, p. 471], and denote its stabilising solution by X_{∞}. Again γ_{RP} > \overline{σ}([ρD_{d}^{−1}S_{o} I]^{T}) and

K_{F I} = −B̂_{2}^{T} X_{∞} = −B^{T} [X_{∞11} X_{∞12} X_{∞13} X_{∞14}] .

5. Calculate K̂ = F_{l}(K̂_{DF} , K_{F I}) and undo the normalisation to get K. The structure illustrated in Fig. 4.12 then follows.

In [Wal96] it is shown that any controller that results from (4.14) has a structure similar to that given in Fig. 4.12.

[figure: G_{s}; H; ∫; A; C; B; −; B^{T} X_{∞11}; ∫; ρB_{d}D_{d}^{−1}; A_{d} − B_{d}D_{d}^{−1}C_{d}; B^{T} X_{∞14}; −; [I 0]; B^{T} [X_{∞12} X_{∞13}]; C_{0}; ∫; [\begin{matrix}A_{0} & 0 \\ 0 & A_{d} − B_{d}D_{d}^{−1}C_{d} \end{matrix}]; [ \begin{matrix}B_{0} \\ \frac{1}{ρ} B_{d}D_{d}^{−1} \end{matrix}]; r̂; y; y_{0}; −; −; −; S_{o}]

Figure 4.12 The structure of K

X_{∞} is the solution of the CARE corresponding to P̂. Partition X_{∞} conformably with the realisation of P

X_{∞} = [\begin{matrix}X_{∞11} & X_{∞12} & X_{∞13} & X_{∞14} \\ X_{∞21} & X_{∞22} & X_{∞23} & X_{∞24} \\ X_{∞31} & X_{∞32} & X_{∞33} & X_{∞34} \\ X_{∞41} & X_{∞42} & X_{∞43} & X_{∞44} \end{matrix}] .

It can be shown by simple algebraic manipulation that Y_{∞} = 0 is the unique stabilising solution of the filtering algebraic Riccati equation (FARE) and thus to obtain the best γ_{RP} only the CARE is considered. It also becomes apparent that the realisation of P_{22} is stabilisable and detectable if the realisation of G_{s} is stabilisable and detectable, a claim made in Section 4.3. The realisation of P_{22} is detectable because Â − B̂_{1}Ĉ_{2} is stable, and stabilisable because

Â − B̂_{2} B̂_{2}^{T} [\begin{matrix}X & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]

is stable, where X is the stabilising solution of the CARE of G_{s}.

The structure of K adds visibility. If K is a flight controller, this visibility is very important for its certification. The feedback controller has more states than the controller presented in Subsection 4.6.1. The number of extra states is equal to the number of states of the ideal reference model.

Even though the designer cannot control γ_{RS} directly, it has been found that the method suggested in this subsection can lead to a satisfactory design. The design procedure is similar to that suggested in [LKP93]. As discussed in Section 4.5, the constant parameter ρ can only be used to crudely trade off robust stability and robust performance. The reason why the suggested method can result in a good design, is that sup_{ω} μ_{Δ}(M(jω)) is minimised as opposed to sup_{ω} \overline{σ}(M(jω)), and

μ_{Δ}(M(jω)) ≤ \overline{σ}(M(jω)).

Design examples have shown, that using the structured singular value can improve γ_{RP} by up to 25%.

Remark 4.3 If ρ^{2} is chosen too small then the optimisation will result in a controller with Q_{1} = 0. This is easy to see from Equation (4.7). Note that if M is upper or lower triangular then μ_{Δ}(M(jω)) = max[\overline{σ}(M_{11}(jω)), \overline{σ}(M_{22}(jω))].

A scaling is required to make the gain of the transfer function from references to outputs equal to the identity at ω = 0. This will degrade the robust performance margin to γ̂_{RP}^{−1} , see Section 4.4. It was attempted to perform the scaling by choosing a different controller than the central controller, i.e. take advantage of the parametrisation of all stabilising controllers. Unfortunately this was not successful.

Minimising \overline{σ}  Minimising μ
ρ γ_{RP} γ_{RS} γ̂_{RP} γ_{RP} γ_{RS} γ̂_{RP}
1.5 4.23 3.95 4.20 3.04 2.99 4.10
2 5.00 4.53 5.00 3.58 3.10 5.00

Table 4.1 A comparison of controllers resulting from (4.14) and (4.20)

Table 4.1 compares two controllers designed for the longitudinal dynamics of the RCAM (the RCAM was presented in Section 2.10).

4.6.3 The third method

In this subsection, the constrained cost function (4.6) is “solved” by iteratively solving for K and D_{l}(s). To obtain K for a fixed D_{l}(s), optimisation problem

min_{stab K} ∥D_{l}(s)M(s)D_{r}(s)∥_{∞} with ∥M_{22}∥_{∞} < γ_{RS}  (4.21)

will be approximated with a finite-dimensional optimisation problem that will be solved using efficient numerical methods. Solving for D_{l}(s) for a given K is done as discussed in Section 4.3. Hence the designer optimises over a subset of Q = [Q_{1} Q_{2}].

The finite-dimensional problem is derived using the method proposed in [SGC97], which deals with multi-objective output-feedback synthesis for MIMO LTI systems. The linear time-invariant plant under consideration is described by P_{ss}. As D̂_{22} = 0 the method proposed in [SGC97] can be used. It is desired to compute a dynamical output feedback controller K(s)

K(s) := [ \begin{matrix}A_{K} & B_{K} \\ C_{K} & D_{K} \end{matrix} ] ,

that internally stabilises the closed loop, and meets the closed loop specifications on robust stability and robust performance. Each specification can be formulated with respect to some close loop transfer function of the form:

T_{j}(s) = L_{j} M̂(s) R_{j}, j = 1, 2.

The matrices L_{j} and R_{j} select the appropriate input and output channels. For the minimisation under consideration, constraints will be imposed upon ∥M̂∥_{∞} and ∥M̂_{22}∥_{∞} = ∥M_{22}∥_{∞}. Let T_{1} denote M̂ and T_{2} denote M̂_{22}. The matrices L_{2} and R_{2} corresponding to M̂_{22} are

L_{2} = [\begin{matrix}0 & I_{p} & 0 \\ 0 & 0 & I_{m} \end{matrix}] , and R_{2} = [\begin{matrix}0 \\ I_{p} \end{matrix}] .

A state-space realisation of T_{j}(s) is given by

[ \begin{matrix}A & B_{j} \\ C_{j} & D_{j} \end{matrix} ] := [ \begin{matrix}Â + B̂_{2}D_{K} Ĉ_{2} & B̂_{2}C_{K} & B̂_{j} + B̂_{2}D_{K} F̂_{j} \\ B_{K} Ĉ_{2} & A_{K} & B_{K} F̂_{j} \\ Ĉ_{j} + Ê_{j}D_{K} Ĉ_{2} & Ê_{j}C_{K} & D̂_{j} + Ê_{j}D_{K} F̂_{j} \end{matrix} ] ,

with B̂_{j} := B̂_{1}R_{j}, Ĉ_{j} := L_{j}Ĉ_{1}, D̂_{j} := L_{j}D̂_{11}R_{j}, Ê_{j} := L_{j}D̂_{12}, F̂_{j} := D̂_{21}R_{j}.

Let A_{cl} and x_{cl} denote the closed loop state matrix and state vector respectively. Since the controller has to be internally stabilising, the closed loop system must admit a quadratic Lyapunov function

V (x_{cl}) = x_{cl}^{T} P x_{cl}, where P > 0 : A^{T} P + PA < 0.  (4.22)

The approach proposed in [SGC97] consists of expressing each control specification as an additional constraint on the admissible closed loop Lyapunov functions satisfying (4.22). For each ∞-norm constraint, the bounded real lemma provides an analysis result by which A is stable and ∥T_{j}∥_{∞} < γ if and only if there exists a P_{j} = P_{j}^{T} > 0 such that

[\begin{matrix}A^{T} P_{j} + P_{j}A & P_{j}B_{j} & C_{j}^{T} \\ B_{j}^{T} P_{j} & −γI & D_{j}^{T} \\ C_{j} & D_{j} & −γI \end{matrix}] < 0.

Thus, optimisation (4.21) can be posed in terms of a set of inequalities whose variables are the controller matrices A_{K}, B_{K}, C_{K}, D_{K} and two Lyapunov matrices P_{1}, P_{2}. Since expressions like A^{T} P_{j} + P_{j}A involve products of P_{j} and the controller variables, the resulting feasibility problem is nonlinear. To recover convexity, it is required that both specifications are enforced by a single closed loop Lyapunov function P > 0. This amounts to imposing the constraint P_{1} = P_{2} = P. With this constraint in force, all inequalities can be further reduced to LMIs without additional conservatism.

The approach described is sub-optimal as all closed loop specifications are expressed by means of a single Lyapunov function. Nevertheless, it has been found that the resulting synthesis technique works well in practise. It is numerically tractable since it leads to an LMI problem which is readily solvable using commercial software [GNLC95], and exploits all degrees of freedom in P. This can be seen as a systematic way of exploiting the parametrisation, albeit a subset of, of all stabilising controllers.

To solve (4.21) the designer must solve a set of LMIs. This set consists of one LMI for each ∞-norm constraint (the structure of this LMI is given below)

[\begin{matrix}ÂX + XÂ^{T} + B̂_{2}C̃ + (B̂_{2}C̃)^{T} & ∗ & ∗ & ∗ \\ Ã + (Â + B̂_{2}D̃Ĉ_{2})^{T} & Â^{T} Y + Y Â + B̃Ĉ_{2} + (B̃Ĉ_{2})^{T} & ∗ & ∗ \\ (B̂_{j} + B̂_{2}D̃F̂_{j})^{T} & (Y B̂_{j} + B̃F̂_{j})^{T} & −γI & ∗ \\ Ĉ_{j}X + Ê_{j}C̃ & Ĉ_{j} + Ê_{j}D̃Ĉ_{2} & D̂_{j} + Ê_{j}D̃F̂_{j} & −γI \end{matrix}] < 0.

and the coupling constraint

[\begin{matrix}X & I \\ I & Y \end{matrix} ] < 0.

To recover the controller matrices, equation MN^{T} = I − XY is solved for M, N, e.g. by taking an SVD of I − XY. In the full order case, M and N are square and invertible.

Ã := NA_{K}M^{T} + NB_{K} Ĉ_{2}X + Y B̂_{2}C_{K}M^{T} + Y (Â + B̂_{2}D_{K} Ĉ_{2})X,
B̃ := NB_{K} + Y B̂_{2}D_{K},
C̃ := C_{K}M^{T} + D_{K} Ĉ_{2}X,
D̃ := D_{K}.

Consequently, the variables A_{K}, B_{K}, C_{K}, D_{K} can be replaced by Ã, B̃, C̃, D̃ without loss of generality, since for given Ã, B̃, C̃, D̃, X, Y the designer can always solve for the controller matrices.

Ignoring that D_{l}(s) is the result of fitting to frequency response data, it is clear that the this procedure is guaranteed to be monotonically decreasing.

A constant pre-filter T_{dc} is required. To avoid the use of a pre-filter a nominal performance constraint of the type ∥T_{r→y} − T_{0}∥_{∞} < δ_{NP} was included in the cost function. Unfortunately, this constraint was found to be too tight around the target closed loop bandwidth, and thus did not result in a controller with satisfactory closed loop behaviour.

4.6.4 The fourth method

The solution presented in this subsection enables the designer to build nominal performance constraints into cost function (4.21). The resulting cost can then be minimised using efficient optimisation tools. These nominal performance constraints are either placed at discrete frequency points, e.g. at frequency ω_{1}

\overline{σ} [N(jω_{1})Q_{1}(jω_{1}) − T_{0}(jω_{1})] < δ,

where δ is a small positive constant, or are of the type

∥W (NQ_{1} − T_{0})∥_{∞} < γ_{NP} ,

where W is minimum phase and stable. The overall optimisation is still solved iteratively, D_{l}(s) being obtained as suggested in Section 4.3.

Let the constant matrices F and L be chosen so that Â + B̂_{2}F and Â + LĈ_{2} are stable. As D̂_{22} = 0, all proper controllers K achieving internal stability have the structure depicted in Fig. 4.13, where Q(s) is a stable transfer matrix with a state-space realisation given by

Q(s) := [ \begin{matrix}A_{Q} & B_{Q} \\ C_{Q} & D_{Q} \end{matrix} ] .

A proof of this statement can be found in [ZDG96, Chapter 12]. It can be shown that the upper part of the controller depicted in Fig. 4.13, i.e. the controller K with Q = 0, can be redrawn as in Fig. 4.12 (to see this normalise P, choose F = −B̂_{2}X_{∞} and L = −B̂_{1} and undo the normalisation; see Subsection 4.6.2).

[figure: ∫; Â; Ĉ_{2}; B̂_{2}; L; F; −; ∫; A_{Q}; B_{Q}; D_{Q}; C_{Q}; P; [r ϕ]^{T}; [z y u]^{T}; [r̂ y]^{T}; u]

All stable closed loops M̂ can also be parametrised by Q, see [ZDG96, pp. 322-323]

M̂ = T_{1} + T_{2}QT_{3}, where [\begin{matrix}z \\ y \\ u \end{matrix}] = M̂ [\begin{matrix}r \\ ϕ \end{matrix}]

Figure 4.13 Structure of all stabilising controllers

and T_{1}, T_{2}, T_{3} are stable transfer matrices that depend on the realisation of P and the constant matrices F, L. If the control system requirements are satisfied, then at each frequency ω the following inequalities must hold

\overline{σ} [T_{1}(jω) + T_{2}(jω)Q(jω)T_{3}(jω)] < γ_{RP}
\overline{σ} \{[\begin{matrix}0 & I_{p} & 0 \\ 0 & 0 & I_{m} \end{matrix}] [T_{1}(jω) + T_{2}(jω)Q(jω)T_{3}(jω)] [\begin{matrix}0 \\ I_{p} \end{matrix}]\} < γ_{RS}
\overline{σ} \{W (jω) \{[I_{p} 0 0] [T_{1}(jω) + T_{2}(jω)Q(jω)T_{3}(jω)] [\begin{matrix}I_{p} \\ 0 \end{matrix}]\}\} < γ_{NP} ,

where Q(jω) = C_{Q}(jω−A_{Q})^{−1}B_{Q}+D_{Q}. Thus the optimisation problem can be written as a set of LMIs which are affine in (B_{Q}, D_{Q}) if (A_{Q}, C_{Q}) are fixed, or (C_{Q}, D_{Q}) if (A_{Q}, B_{Q}) are fixed^{6}. The designer must choose a stable A_{Q}, the dimension of which will determine how many extra states the resulting controller will have over the controllers designed in the previous two subsections. It is also evident that the designer can include constraints at discrete frequency points. Internal stability is not affected by gridding the frequency range as the designer chooses A_{Q} to be stable.

The controller has a particular structure and scaling is now not required. The drawback associated with this method is that it is not clear how to choose A_{Q}. More work needs to be done in this direction.

4.7 Some remarks on the new setup

The pre-compensator W_{1} is chosen using the guidelines given in Chapters 2 and 3. If a post-compensator W_{2} is used, for the robust performance guarantee to hold, W_{2} and T_{0} must have a specific structure (see Section 4.3). If the designer wishes to use a rate signal for feedback, but does not desire to guarantee robust performance with respect to the time history of that signal, then the designer can use the output selector S_{o}. In general, the ideal reference model is designed using a performance oriented control system design method. The first design method will give the designer a feel for whether T_{0} is compatible with the robust performance requirements. It is possible that a controller synthesised using the second design procedure suffices. This will depend on whether the level of robust stability is satisfactory. If it is not, then the designer can use the third design method.

A robust performance measure for the single DoF H_{∞} loop shaping controller is derived in [Vin93, p. 76]. Let G_{s} be the weighted plant and {N, M} a normalised rcf of G_{s}. If the H_{∞} loop shaping controller has been implemented in the observer form, then the nominal transfer matrix from references to outputs is T_{r→y} = N (it is assumed that W_{2} = I). If the weighted plant is perturbed to Ĝ_{s}, then the transfer matrix from references to outputs will be perturbed to T̂_{r→y}. The robust performance measure is given by

6Since ∥Z∥ < γ is equivalent to [\begin{matrix}γI & Z \\ Z^{T} & γI \end{matrix}] > 0, ∥Z∥ < γ can be converted into an LMI.

∥T̂_{r→y} − N∥_{∞} ≤ \frac{α}{sin(arcsin ϵ − arcsin α)}, ∀ Ĝ_{s} : δ_{ν}(G_{s}, Ĝ_{s}) ≤ α,

where ϵ is the achieved stability margin. Design experience has shown that this robust performance measure is larger than the values of δ_{RP} achieved when designing controllers using the method described in Subsection 4.6.3 for δ_{ν}(G_{s}, Ĝ_{s}) = γ_{RP}^{−1} . Furthermore, N may not be close to the desired transfer matrix from references to outputs. The power of Vinnicombe’s bound is that it is a function of δ_{ν}(G_{s}, Ĝ_{s}). The design method described in Subsection 4.6.3 does not come with such a guarantee (Equation (4.9) can be used to give a weaker guarantee).

[figure: r; ρ_{o}I; −; ρ_{i}I; T_{0}; z; K; W_{1}; G; u; d_{i}]

Figure 4.14 Robust performance with respect to input multiplicative uncertainty

The performance margin (γ_{RP}^{−1} ) guarantees robust performance of the closed loop with respect to other types of uncertainty, e.g. input multiplicative uncertainty and output inverse multiplicative uncertainty (robust performance always with respect to an ideal reference model). Consider input multiplicative uncertainty. From Fig. 4.14

[\begin{matrix}z \\ u \end{matrix}] = [\begin{matrix}ρ_{o}I \\ & I \end{matrix}] [\begin{matrix}NQ_{1} − T_{0} & V ÑW_{1}^{−1} \\ W_{1}MQ_{1} & W_{1}UÑW_{1}^{−1} \end{matrix}] [\begin{matrix}ρ_{i}I \\ & I \end{matrix}] [\begin{matrix}r \\ d_{i} \end{matrix}] ,

where V = V_{0} + NQ_{2} and U = U_{0} + MQ_{2}. Hence

[\begin{matrix}z \\ u \end{matrix}] = [\begin{matrix}I \\ & W_{1} \end{matrix}] [\begin{matrix}ρ_{o}I \\ & I \end{matrix}] [\begin{matrix}NQ_{1} − T_{0} & V \\ MQ_{1} & U \end{matrix}] [\begin{matrix}ρ_{i}I \\ & I \end{matrix}] [\begin{matrix}I \\ & ÑW_{1}^{−1} \end{matrix}] [\begin{matrix}r \\ d_{i} \end{matrix}] .

If the weight on u is chosen to be W_{1}^{−1} and the weight on d_{i} to be W_{1}Ñ^{−1}, and the D-scales are added, then the achieved norm bound is γ_{RP}

sup_{ω} μ_{Δ_{i}} ([\begin{matrix}r \\ d_{i} \end{matrix}] → [\begin{matrix}z \\ u \end{matrix}]) < γ_{RP} ,

4.8 H_{∞} control with preview 95

[figure: r; ρ_{0}I; −; ρ_{i}I; T_{0}; z; K; M^{−1}; N; d_{i}; d_{o}; e; −]

Figure 4.15 Robust performance with respect to right coprime factor uncertainty

for Δ_{i} with a suitable structure.

Consider robust performance with respect to right coprime factor uncertainty, see Fig. 4.15. Let

[\begin{matrix}z \\ e \end{matrix}] = M_{r} [\begin{matrix}r \\ d_{i} \\ d_{o} \end{matrix}] = [\begin{matrix}ρ_{o}I \\ & I \end{matrix}] [\begin{matrix}NQ_{1} − T_{0} & V Ñ & V M̃ \\ Q_{1} & Ṽ & Ũ \end{matrix}] [\begin{matrix}ρ_{i}I \\ & I \end{matrix}] [\begin{matrix}r \\ d_{i} \\ d_{o} \end{matrix}] ,

where Ṽ = Ṽ_{0} +NQ_{2} and Ũ = Ũ_{0} +NQ_{2}. All stabilising controllers K are parametrised as in Section 4.3, as the generalised plants corresponding to M and M_{r} have the same (2, 2) block. Unfortunately, it cannot be shown that μ_{Δ}(M) = μ_{Δ_{r}}(M_{r}) with Δ_{r} appropriately defined, even though the upper bound of μ_{Δ_{r}}(M_{r}) given by Equation (4.9) is the same as that of μ_{Δ}(M). Thus robust performance with respect to right coprime factor uncertainty is not the same as robust performance with respect to left coprime factor uncertainty. Nevertheless, examples have shown that the guaranteed performance levels are similar.

4.8 H_{∞} control with preview

If the reference trajectory is known a priori, then the setup in Fig. 4.6 can be modified so that the controller uses this information about the reference to optimise certain closed loop objectives. This problem is solved in discrete time.

The modified setup is depicted in Fig. 4.16. Assume that the reference (r) is known T seconds ahead and that the sampling time is t_{s}, i.e. the reference is known S = T/t_{s} timesteps ahead. Hence the only signals in Fig. 4.16 that are known in the future are r and β. The output of the plant (y(z)) is desired to follow the setpoint

s(z) = T_{0}(z) (∑_{i=1}^{N} a_{i} z^{−n_{i}}) z^{S}r(z),

[figure: r; β; ρ_{o}I; ρ_{i}I; z; K; −; ∑_{i} a_{i}z^{−n_{i}}; T_{0}; s; α; G; W_{1}^{−1}; W_{1}; W_{2}; W_{2}^{−1}; u; y]

Figure 4.16 Setup for H_{∞} control with preview

where ∑_{i=1}^{N} a_{i} = 1 with a_{i} ∈ ℝ_{+}, and n_{i} ∈ ℕ with n_{i} ∈ [0, S]. Thus, y(k) is made to follow s(k) which is a filtered version of

∑_{i=1}^{N} a_{i} r(k + τ_{i}), where τ_{i} = S − n_{i}.

The constants a_{i} of the FIR filter ∑_{i=1}^{N} a_{i} z^{−n_{i}} are used to specify how the a priori information about the reference should be used. Thus a step on r(k) can be broken up into smaller steps using the FIR filter. The designer chooses a_{i} according to closed loop objectives, e.g. passenger comfort. The filter T_{0} is used to specify the time response characteristics of y with respect to a reference command. The overall aim is to optimise the robust performance of the closed loop system.

The price paid for the preview is the potentially large number of states of the controller. More work needs to be done on this setup.

4.9 Design example

The design methods detailed in Subsections 4.6.1, 4.6.2 and 4.6.3 are used to design three controllers for the wind tunnel model presented in Chapter 6. The designs are carried out using the same nominal plant model and actuator models as those used in Section 3.8, and the pre-compensator

4 \frac{s + 3}{s} \frac{s + 5}{0.02s + 1} W_{full},

where W_{full} is the non-diagonal weight designed in Section 3.8. No post-compensator is used, and the ideal reference model T_{0} is chosen to be

T_{0} = \frac{7^{2}}{s^{2} + 2 × 0.7 × 7s + 7^{2}} I_{2}.

By making T_{0} diagonal, good output decoupling is emphasised. It can be seen in Fig. 4.17, that the bandwidth of N is around 10rad/s. Hence, an ideal reference model with a bandwidth higher than 10rad/s will not be compatible with robust performance. Time simulations will be carried out using perturbed plant models that result from changing the aerodynamic coefficients \frac{∂ϕ̈}{∂ϕ} and \frac{∂ψ̈}{∂ψ} of the nominal plant model.

Figure 4.17 Singular value plots of N(dashed), T_{0}(solid), W_{e_{1}}(dash-dot) and W_{e_{2}}(dotted)

4.9.1 The first controller

In this subsection a controller is designed using the design method presented in Subsection 4.6.1. γ_{1} = 1.06 with

W_{e_{1}} = 5 \frac{s + 2}{s + 0.01} and W_{e_{2}} = \frac{0.05s + 1}{0.001s + 1}.

The singular value plots of N, T_{0}, W_{e_{1}} and W_{e_{2}} are illustrated in Fig. 4.17. γ_{RS} = 3.76 with γ_{RS}^{opt} = 3.59. The predicted and actual values of γ_{RP} and δ_{RP} are depicted in Fig. 4.18; the predicted values have been plotted using (4.18). The synthesised feedback controller has 13 states, i.e. the same number of states as the weighted plant. Q_{1} can be model reduced to 8 states from 17 with no cost. Anti-windup designed using the Hanus scheme [HKH87] has been placed on W_{1}, and the feedback controller has been implemented in the observer form. The plant observer is driven with achieved plant inputs rather than demanded ones.

The nominal response to step commands is illustrated in Fig. 4.19, where ϕ (solid), the demanded roll angle (dash-dot), ψ (dashed) and demanded yaw angle (dotted). Note how the demanded and actual responses overlap.

A perturbed plant model is created by changing the aerodynamic coefficients mentioned above to zero. The ν-gap between the perturbed and nominal weighted plant models is 0.15. The perturbed response to step commands is illustrated in Fig. 4.20.

4.9 Design example 98

Figure 4.18 The predicted (solid) and actual (dashed) values of γ_{RP}^{−1} and δ_{RP}

Figure 4.19 Nominal response for the first design method Figure 4.20 Perturbed response for the first design method

4.9.2 The second controller

In this subsection a controller is designed using the method outlined in Subsection 4.6.2. Minimisation (4.20) is solved for various values of ρ, and the results are illustrated in Table 4.2. Note that γ_{RP}^{−1} is the performance margin that is achieved by minimising (4.20), γ_{RS}^{−1} the stability margin and γ̂_{RP}^{−1} the corrected performance margin due to the inclusion of the scaling T_{dc}.

\begin{matrix}ρ & γ_{RP} & γ_{RS} & γ̂_{RP} & γ_{RP} using (4.14) \\ 1.5 & 3.71 & 3.70 & 4.84 & 5.34 \\ 2 & 3.98 & 3.91 & 5.26 & 6.25 \\ 2.5 & 4.30 & 4.17 & 5.55 & 7.25 \\ 3 & 4.75 & 4.46 & 5.94 & 8.24 \\ 3.5 & 5.42 & 5.12 & 6.61 & 9.27 \end{matrix}

Table 4.2 Results for various values of ρ

As the ν-gap between the perturbed and nominal weighted plants is equal to 0.15, a value of ρ = 3 is chosen. γ_{RP} = 4.75 was reached after 3 iterations using 2^{nd} order Dscales. The singular values of T_{dc} are 1.43 and 1.42. The synthesised controller which has 27 states has been implemented as illustrated in Fig. 4.12. Anti-windup has been placed on W_{1} and the plant observer is driven with achieved rather than demanded plant inputs.

The nominal response to step commands is illustrated in Fig. 4.21, and the perturbed response is depicted in Fig. 4.22. By comparing Figs. 4.20 and 4.22 it can be deduced that the perturbed time response has improved and the actuator usage has reduced.

Figure 4.21 Nominal response for the second design method

4.9.3 The third controller

In this subsection a controller is designed using the method outlined in Subsection 4.6.3. The constraint γ_{RS}^{−1} > 0.25 is imposed on the stability margin^{7}. The results of the D-K iteration for ρ = 3 are given in Table 4.3 (note that the singular values of T_{dc} are 1.60 and 1.45). The synthesised controller is model reduced from 27 to 15 states with no cost.

The time response of the nominal model is illustrated in Fig. 4.23 and of the perturbed model in Fig. 4.24.

Fig. 4.25 illustrates the Nichols plot of the transfer function that results from breaking the yaw loop at the plant input for both the constraint and unconstraint case. By imposing a constraint on the stability margin, the gain and phase margins have improved while

7Constraints are also imposed on the size of X and Y (the trace penalty in Table 4.3).

4.9 Design example 101

Figure 4.22 Perturbed response for the second design method

iteration γ_{RP} γ_{RS} γ̂_{RP} order of D-scales trace penalty
1  15.90 3.67 7.62  0  10^{−4}
2  6.53 3.69 7.61  1  10^{−4}
3  6.48 3.69 7.62  1  10^{−4}
4  5.94 3.81 7.40  1  10^{−6}
5  5.55 3.95 6.62  2  10^{−6}
6  6.05 3.98 6.61  2  10^{−8}
7  5.71 3.93 6.44  2  10^{−6}

Table 4.3 D-K iteration for the LMI controller

4.9 Design example 102

Figure 4.23 Nominal response for the third design method Figure 4.24 Perturbed response for the third design method

maintaining good step response properties (gain margin: -4.9dB to -5.4dB, and phase margin: 29.7^{∘} to 32.0^{∘}).

Figure 4.25 Nichols plots for constraint (solid) and unconstraint (dashed) case

Figs. 4.26, 4.27 and 4.28 illustrate responses to step commands using a set of perturbed plant models for each of the three controllers designed. The set of perturbed plant models is created by taking combinations that result from varying the aerodynamic coefficients \frac{∂ϕ̈}{∂ϕ} and \frac{∂ψ̈}{∂ψ} by up to ±100%. The ν-gap between the perturbed weighted plants and the nominal weighted plant is ≤ 0.15. Again the benefit of the controllers designed for robust performance is evident.

4.9.4 A preview example

The plant used in this example is the RCAM model presented in Section 2.10. Having designed a weighting function W_{1} that provides good disturbance rejection, the design specifications in Table 2.3 can be included directly in the design procedure using a two DOF approach. The following ideal reference model T_{0} was used

T_{0} = [\begin{matrix}\frac{0.5^{2}}{s^{2}+2×0.7×0.5s+0.5^{2}} & 0 \\ 0 & \frac{0.3^{2}}{s^{2}+2×0.7×0.3s+0.3^{2}} \end{matrix} ] .

A sampling time of t_{s} = 1s, and a preview time of T = 10s is chosen. N = 2 with a_{1} = 0.5, n_{1} = 5 and a_{2} = 0.5, n_{2} = 10. The integrators of the pre-compensator are approximated at low frequency with constants so that W_{1} is stable. No W_{2} is used. The setup in Fig. 4.16 is used for to synthesise the controller with ρ_{i} = 4.5 and ρ_{o} = 1.5.

First a continuous time controller is designed without preview. The resulting γ_{RP} = 3.08. The step responses are shown in Figs. 4.29 and 4.30.

4.9 Design example 104

Figure 4.26 Perturbed time responses - first design method Figure 4.27 Perturbed time responses - second design method

4.9 Design example 105

Figure 4.28 Perturbed time responses - third design method Figure 4.29 Step on ż without preview

Figure 4.30 Step on V_{air} without preview

The preview controller is designed using discrete time H_{∞} techniques (the command dhfsyn of the μ-analysis and synthesis toolbox). The achieved norm bound is γ_{RP} = 3.65. The time responses are shown in Figs. 4.31 and 4.32. Note the reduced actuator usage and anticipation of the reference.

4.9 Design example 107 Figure 4.31 Step on ż with preview Figure 4.32 Step on V_{air} with preview

Robust gain scheduling

This chapter deals with the design of controllers for plants with widely ranging nonlinear and/or parameter-dependent dynamics. In some sections the discussion will be carried out for a specific type of plant, namely the longitudinal dynamics of a rigid body fixed wing aircraft.

5.1 Motivation for robust gain scheduling

Gain scheduling is a method used to design controllers for systems with widely varying nonlinear and/or parameter-dependent dynamics. Gain scheduling is essentially an attempt to use linear controller design techniques to design a parameter-dependent controller for a plant with widely varying nonlinear and/or parameter-dependent dynamics. The idea is to select several operating points which cover the range of the plant’s dynamics. At each of these points, the designer makes an LTI approximation to the plant and designs a linear compensator for each linearised plant. In between operating points, the “gains” of the compensators are then interpolated resulting in a global feedback compensator.

Despite the lack of sound theoretical analysis, gain scheduling is a design methodology which is known to work in practice, e.g. the gain scheduled flight controller designed for the VAAC Harrier in [Hyd95]. However, in the absence of such an analysis, these designs come with no guarantees; even though the local point designs may have excellent feedback properties, the global gain scheduled design need not have any of these properties. That is, the designer typically cannot assess a priori the guaranteed stability, robustness and performance properties of gain scheduled designs. Any such properties are inferred from extensive computer simulations.

In addition to simulations, gain scheduled designs are guided by heuristic rules of thumb. The most fundamental guidelines are:

1. the scheduling variables should vary slowly, and

2. the scheduling variables should capture the plant’s nonlinearities.

These guidelines are simply reminders that the designs were based on LTI approximations to the actual plant. Thus, these approximations must be sufficiently accurate if one expects the local feedback properties to carry over to the overall scheduled system. In what follows, it will be attempted to give the reader a feel for why the rules of thumb mentioned above are intuitively correct.

The class of nonlinear systems considered in this chapter are linear parameter-varying (LPV) systems, i.e. systems whose dynamics can be written as

ẋ = A(ρ)x + B(ρ)u,
y = C(ρ)x + D(ρ)u,

where A, B, C, D are continuous matrix valued functions of ρ, and ρ is the vector of scheduling variables. It is usually the case that A, B, C, D are nonlinear functions of the scheduling variables. Linear parameter-varying systems were first studied in [SA91, SA92]. Furthermore, it will be assumed that the scheduling variables are all states of the open loop system. Such an LPV system will be referred to as a “quasi-LPV” system. Many real systems of interest, e.g. a rigid body model of the longitudinal dynamics of an aircraft, can be written as “quasi-LPV” systems; this statement will be qualified in Section 5.4. Consider a “quasi-LPV” plant model

ẋ = f(x, u) = A(x)x + B(x)u.  (5.1)

It is assumed for notational convenience that ρ = x ∈ ℝ^{n}, i.e. the scheduling variables capture all of the plant’s nonlinearities. The function f(x, u) is linearised about the equilibrium point (x_{0}, u_{0})

δẋ = [A(x) + \frac{dA(x)}{dx}x + \frac{dB(x)}{dx}u]|_{(x_{0},u_{0})} δx + B(x_{0})δu,  (5.2)

where

\frac{dA(x)}{dx}x = [\frac{∂A(x)}{∂x_{1}}x … \frac{∂A(x)}{∂x_{n}}x] , \frac{dB(x)}{dx}u = [\frac{∂B(x)}{∂x_{1}}u … \frac{∂B(x)}{∂x_{n}}u] ,

δx = x − x_{0}, and δu = u − u_{0}. An equilibrium point (x_{0}, u_{0}) is defined by f(x_{0}, u_{0}) = 0. The function f(x, u) is also linearised about a non-equilibrium point (x_{0}, u_{d})

δẋ = A(x_{0})x_{0} + B(x_{0})u_{d} + [A(x) + \frac{dA(x)}{dx}x + \frac{dB(x)}{dx}u]|_{(x_{0},u_{d})} δx + B(x_{0})(u − u_{d}).

As A(x_{0})x_{0} = −B(x_{0})u_{0},

δẋ = [A(x) + \frac{dA(x)}{dx}x + \frac{dB(x)}{dx}u]|_{(x_{0},u_{d})} δx + B(x_{0})δu.  (5.3)

Note that (5.2) and (5.3) are only first order approximations of f(x, u). The systems (5.2) and (5.3) have different A-matrices (these are denoted by A_{eq} and A_{tr} respectively), the difference being

A_{eq} − A_{tr} = \frac{dB(x)}{dx}u|_{(x_{0},u_{0})} − \frac{dB(x)}{dx}u|_{(x_{0},u_{d})} .  (5.4)

If the scheduling variables are fast variables, i.e. in this case the closed loop dynamics are fast, then a large reference command will result in a large input demand u_{d}. Thus from (5.4) it becomes evident why the LTI approximations of (5.1) cannot capture the transient behaviour of the “quasi-LPV” plant model. However, if the scheduling variables vary slowly, then the LTI approximations will be close to the transient models (5.3).

Now consider a “quasi-LPV” plant model

[ż ; ẇ] = [\begin{matrix}A_{11}(z) & A_{12}(z) \\ A_{21}(z) & A_{22}(z) \end{matrix}] [\begin{matrix}z \\ w \end{matrix}] + [\begin{matrix}B_{1}(z) \\ B_{2}(z) \end{matrix}] u,  (5.5)

where z is the vector of scheduling variables. An LTI approximation to (5.5) about the equilibrium point (z_{0}, w_{0}, u_{0}) is

\begin{matrix}[\begin{matrix}δż \\ δẇ \end{matrix}] = [\begin{matrix}A_{11}(z) + \frac{dA_{11}(z)}{dz}z + \frac{dA_{12}(z)}{dz}w + \frac{dB_{1}(z)}{dz}u & A_{12}(z) \\ A_{21}(z) + \frac{dA_{21}(z)}{dz}z + \frac{dA_{22}(z)}{dz}w + \frac{dB_{2}(z)}{dz}u & A_{22}(z) \end{matrix}]|_{(z_{0},w_{0},u_{0})} [\begin{matrix}δz \\ δw \end{matrix}] + \\ + [\begin{matrix}B_{1}(z_{0}) \\ B_{2}(z_{0}) \end{matrix}] δu. \end{matrix} (5.6)

Let A_{11} be a function of x = [z, w]^{T}. The first row of (5.6) then becomes

\begin{matrix}δż = [A_{11}(z, w) + \frac{∂A_{11}(z, w)}{∂z}z + \frac{dA_{12}(z)}{dz}w + \frac{dB_{1}(z)}{dz}u]|_{(z_{0},w_{0},u_{0})} δz + \\ + [\frac{∂A_{11}(z, w)}{∂w}z + A_{12}(z)]|_{(z_{0},w_{0})} δw + B_{1}(z_{0})δu. \end{matrix}

If w is not part of ρ, then the LTI approximations of (5.5) might not be representative of the dynamics of the “quasi-LPV” plant model. Thus the scheduling variables must capture the plant’s nonlinearities.

Two very useful properties of gain scheduled systems are formalised in [SA90]. Firstly, the stability margin of the gain scheduled system is proportional to the level of frozen operating point stability. Secondly, the stability margin is inversely proportional to the rate of change of the scheduling variables.

To be able to schedule the “gains” of the LTI controllers, all the controllers must have the same structure. Controllers designed using H_{∞} control design techniques do not in general have a structure that is suitable for gain scheduling^{1}.

What has been described is “classical” gain scheduling. This has been the most popular method for designing a parameter-dependent controller for a plant with widely varying nonlinear and/or parameter-dependent dynamics. An alternative method for designing a controller for such plants is nonlinear dynamic inversion (NDI). The main advantage of this method is that the designer has not got to decide on how to interpolate the controller “gains”; this is done automatically. Again, the controller comes with no robustness guarantees^{2}.

A lot of work has been done in the last ten years in trying to develop a technique for synthesising what will be referred to as robust gain scheduled controllers, i.e. controllers that depend on the scheduling variables and guarantee some level of robust stability across the whole envelope of operation. Recent theoretical developments have produced two generically different methods for synthesising robust gain scheduled controllers for LPV systems. Both methods are detailed in the following sections.

To be able to use these robust gain scheduling techniques, the designer must have an LPV model of the nonlinear plant. Considerable effort has been made in trying to find ways of transforming the EoM of a nonlinear plant into LPV form, without having to make approximations that result in the LPV model not capturing the transient behaviour of the plant. How to develop LPV models that capture the transient behaviour of a nonlinear system will also be discussed in this chapter.

5.2 Single and parameter-dependent Lyapunov function approaches

Consider a finite-dimensional linear system whose state-space entries depend continuously on a time-varying parameter vector ρ(t) ∈ ℝ^{s}. It is assumed that the trajectory of ρ(t) is not known in advance, although its value is known in real-time. This measurement of ρ(t) gives real time information on variations in the plant’s characteristics, so it is desirable to design a controller which uses this information. The synthesis problem is to find a linear finite-dimensional controller, whose state-space entries can also depend continuously on ρ, such that the closed loop system is exponentially stable and achieves good performance with respect to variations in ρ. This synthesis problem will be referred to as the quadratic LPV γ-performance problem.

1It was shown in [SG90] that a controller designed using H_{∞} loop shaping can be written as a plant observer with state feedback. Hyde exploited this structure to design a gain scheduled controller for the VAAC Harrier, see [Hyd95]. This controller was flight tested successfully.

2A novel methodology for designing flight controllers is presented in [RBG95]. The methodology is a combination of NDI and μ-synthesis. An inner loop controller, designed by NDI, is used to linearise the aircraft dynamics. This inner-loop controller lacks guaranteed robustness; thus a robust, linear outer-loop controller is designed using μ-synthesis. Good results are reported.

Given a compact set P ⊂ ℝ^{s}, the parameter variation set F_{P} denotes the set of all piecewise continuous functions ρ mapping ℝ (time) into P with a finite number of discontinuities in any interval. Assume that the following are given: a compact set P ⊂ ℝ^{s}, and continuous functions A : ℝ^{s} → ℝ^{n×n}, B : ℝ^{s} → ℝ^{n×n_{d}}, C : ℝ^{s} → ℝ^{n_{e}×n} and D : ℝ^{s} → ℝ^{n_{e}×n_{d}}. These represent an n^{th} order linear parameter-varying (LPV) system, whose dynamics evolve as

[\begin{matrix}ẋ(t) \\ e(t) \end{matrix}] = [\begin{matrix}A(ρ(t)) & B(ρ(t)) \\ C(ρ(t)) & D(ρ(t)) \end{matrix}] [\begin{matrix}x(t) \\ d(t) \end{matrix}] , where ρ ∈ F_{P}.  (5.7)

Denote the above LPV system as Σ(P, A, B, C, D) or Σ_{P} for short. Given a quadratically stable LPV system Σ_{P}, for zero initial conditions, define

∥G_{F_{P}} ∥ := sup_{ρ∈F_{P}} sup_{{∥d∥_{2}≠0 \atop d∈L_{2}}} \frac{∥e∥_{2}}{∥d∥_{2}} .  (5.8)

Note that for zero initial conditions ∥G_{F_{P}} ∥ is always finite for a quadratically stable LPV system; this is shown in [BP94, Lemma 2.6].

What follows, is the main analysis lemma that is used to solve the quadratic LPV γ-performance problem.

Lemma 5.1 [BP94, Lemma 2.10] Given the LPV system Σ(P, A, B, C, D), and a scalar γ > 0. If there exists an X ∈ ℝ^{n×n}, X = X^{T} > 0 such that for all ρ ∈ P,

[\begin{matrix}A^{T} (ρ)X + XA(ρ) & XB(ρ) & γ^{−1}C^{T} (ρ) \\ B^{T} (ρ)X & −I & γ^{−1}D^{T} (ρ) \\ γ^{−1}C(ρ) & γ^{−1}D(ρ) & −I \end{matrix}] < 0,  (5.9)

then (1) the function A is quadratically stable over P, and (2) there exists a β < γ such that ∥G_{F_{P}} ∥ ≤ β.

A controller K_{ρ} that solves the quadratic LPV γ-performance problem will be derived for the setup illustrated in Fig. 5.1, which corresponds to the H_{∞} loop shaping setup. The general synthesis result in derived in [Bec93]. The dynamics of the weighted LPV plant G_{ρ} are governed by

[\begin{matrix}ẋ(t) \\ e_{1}(t) \end{matrix}] = [ \begin{matrix}A(ρ(t)) & B(ρ(t)) \\ C(ρ(t)) & 0 \end{matrix} ] [\begin{matrix}x(t) \\ e_{2}(t) \end{matrix}] .  (5.10)

[figure: e_{2}; e_{1}; u; y; d_{1}; d_{2}; G_{ρ}; K_{ρ}]

Figure 5.1 The H_{∞} loop shaping setup

The weighted LPV plant is assumed to be strictly proper to keep the equations simple. The LPV generalised plant corresponding to the setup depicted in Fig. 5.1 is

[\begin{matrix}ẋ(t) \\ e_{1}(t) \\ e_{2}(t) \\ y(t) \end{matrix}] = [ \begin{matrix}A(ρ(t)) & B(ρ(t)) & 0 & B(ρ(t)) \\ C(ρ(t)) & 0 & 0 & 0 \\ 0 & I & 0 & I \\ C(ρ(t)) & 0 & I & 0 \end{matrix} ] [\begin{matrix}x(t) \\ d_{1}(t) \\ d_{2}(t) \\ u(t) \end{matrix}] .  (5.11)

Given a nonnegative integer m, and continuous functions A_{K} : ℝ^{s} → ℝ^{m×m}, B_{K} : ℝ^{s} → ℝ^{m×n_{y}}, C_{K} : ℝ^{s} → ℝ^{n_{u}×m}, and D_{K} : ℝ^{s} → ℝ^{n_{u}×n_{y}} the parameter-dependent linear feedback controller K_{ρ} can be written as

[\begin{matrix}u(t) \\ ẋ_{c}(t) \end{matrix}] = [ \begin{matrix}D_{K}(ρ(t)) & C_{K}(ρ(t)) \\ B_{K}(ρ(t)) & A_{K}(ρ(t)) \end{matrix} ] [\begin{matrix}y(t) \\ x_{c}(t) \end{matrix}] .  (5.12)

Let x_{clp}^{T} := [x^{T} , x_{c}^{T}], e^{T} (t) := [e_{1}^{T} (t), e_{2}^{T} (t)], and d^{T} (t) := [d_{1}^{T} (t), d_{2}^{T} (t)]. Then using the above controller the closed loop system becomes

[\begin{matrix}ẋ_{clp}(t) \\ e(t) \end{matrix}] = [ \begin{matrix}A_{clp}(ρ(t)) & B_{clp}(ρ(t)) \\ C_{clp}(ρ(t)) & D_{clp}(ρ(t)) \end{matrix} ] [\begin{matrix}x_{clp}(t) \\ d(t) \end{matrix}] ,  (5.13)

where

\begin{matrix}A_{clp}(ρ) = [\begin{matrix}A(ρ) + B(ρ)D_{K}(ρ)C(ρ) & B(ρ)C_{K}(ρ) \\ B_{K}(ρ)C(ρ) & A_{K}(ρ) \end{matrix}] , & B_{clp}(ρ) = [\begin{matrix}B(ρ) & B(ρ)D_{K}(ρ) \\ 0 & B_{K}(ρ) \end{matrix}] , \\ C_{clp}(ρ) = [\begin{matrix}C(ρ) & 0 \\ D_{K}(ρ)C(ρ) & C_{K}(ρ) \end{matrix}] , & D_{clp}(ρ) = [\begin{matrix}0 & 0 \\ I & D_{K}(ρ) \end{matrix}] . \end{matrix}

Using the analysis result given in Lemma 5.1, the synthesis problem referred to as the quadratic LPV γ-performance problem can be stated as follows:

Definition 5.2 [BP94, Definition 3.1] Given the LPV generalised plant (5.11), and γ > 0. The quadratic LPV γ-performance problem is solvable if there exists an m ≥ 0, a finitedimensional, m-state controller (5.12), and an X ∈ ℝ^{(n+m)×(n+m)}, X = X^{T} > 0 such that for all ρ ∈ P,

[\begin{matrix}A_{clp}^{T}(ρ)X + XA_{clp}(ρ) & XB_{clp}(ρ) & γ^{−1}C_{clp}^{T}(ρ) \\ B_{clp}^{T}(ρ)X & −I & γ^{−1}D_{clp}^{T}(ρ) \\ γ^{−1}C_{clp}(ρ) & γ^{−1}D_{clp}(ρ) & −I \end{matrix}] < 0.  (5.14)

As inequality (5.14) is not linear in the matrices A_{K}, B_{K}, C_{K}, D_{K} and X, it cannot easily be used to synthesise a controller K_{ρ} that solves the quadratic LPV γ-performance problem. However, it can be shown that the feasibility problem consisting of (5.14) and X = X^{T} > 0 is equivalent to a feasibility problem that consists of a set of five linear matrix inequalities (LMIs) in two matrix variables (see Theorem 5.3 below). For a given γ, a controller K_{ρ} that solves the quadratic LPV γ-performance problem can then be computed from every pair of matrices that satisfy those LMIs. The synthesis result is stated below and can be derived as a specialisation of [Bec93, Theorem 5.3.7].

Theorem 5.3 Given P, the generalised plant (5.11), and γ > 0, the quadratic LPV γperformance problem is solvable if and only if there exist matrices X_{11} ∈ ℝ^{n×n}, X_{11} = X_{11}^{T} > 0, and Y_{11} ∈ ℝ^{n×n}, Y_{11} = Y_{11}^{T} > 0, such that for all ρ ∈ P

[\begin{matrix}Y_{11}A^{T} (ρ) + A(ρ)Y_{11} − B(ρ)B^{T} (ρ) & Y_{11}C^{T} (ρ) \\ C(ρ)Y_{11} & −I \end{matrix}] < 0,

[\begin{matrix}A^{T} (ρ)X_{11} + X_{11}A(ρ) − C^{T} (ρ)C(ρ) & C^{T} (ρ) & 0 & X_{11}B(ρ) \\ C(ρ) & −γ^{2}I & 0 & 0 \\ 0 & 0 & −γ^{2}I & I \\ B^{T} (ρ)X_{11} & 0 & I & −I \end{matrix}] < 0,

[\begin{matrix}X_{11} & γ^{−1}I \\ γ^{−1}I & Y_{11} \end{matrix}] ≥ 0.  (5.15)

We explicitly state the formulas for one n-dimensional, strictly proper controller that solves the quadratic LPV γ-performance problem. Let Z = (X_{11} − γ^{−2}Y_{11}^{−1})^{−1} and define

H(ρ) := −[Y_{11}^{−1}A(ρ) + A^{T} (ρ)Y_{11}^{−1} − Y_{11}^{−1}B(ρ)B^{T} (ρ)Y_{11}^{−1} + C^{T} (ρ)C(ρ)],
L(ρ) := −ZC^{T} (ρ),
F (ρ) := −B^{T} (ρ)Y_{11}^{−1}.

Then we define state-space data for the parametrically-dependent controller (5.12) as

K_{ρ} := [ \begin{matrix}A(ρ) + B(ρ)F (ρ) + L(ρ)C(ρ) − γ^{−2}ZH(ρ) & −L(ρ) \\ F (ρ) & 0 \end{matrix} ] .

Note that the three LMIs (5.15) represent convex constraints on the matrices X_{11} and Y_{11}, and that X_{11} = X_{11}^{T} > 0, and Y_{11} = Y_{11}^{T} > 0 trivially represent convex constraints. Since the first two LMIs depend continuously on ρ ∈ P, the solution to the quadratic LPV γ-performance problem can be formulated as a convex feasibility problem with an infinite number of constraints. Typically, one must resort to gridding the set P and solving approximations to the actual constraints. If no solution can be found for a finite subset of P, then clearly no solution exists for P. On the other hand if a solution is found satisfying the LMIs on a finite subset of P, then in general the best one can do is to check that the LMIs hold on a very dense grid of P.

A controller K_{ρ} that solves the quadratic LPV γ-performance problem, will guarantee exponential stability for arbitrarily fast variations of ρ. In many applications though \underline{ν} ≤ ρ̇ ≤ \overline{ν}, where \underline{ν}, \overline{ν} ∈ ℝ^{s}, e.g. if ρ consists of one or more of the states of the weighted plant (5.10). By taking \underline{ν}, \overline{ν} into account during synthesis it may be possible to reduce the achieved value of γ. This can be done by exploiting the notion of extended quadratic stability; for a definition of extended quadratic stability see [Woo95, Definition 2.4.6]. Although the notion of extended quadratic stability is weaker than the more commonly used notion of quadratic stability, it is still a fairly strong form of stability which can guarantee exponential stability and thus bound performance. Using the notion of extended quadratic stability Definition 5.2 can be modified as follows:

Definition 5.4 Given the LPV generalised plant defined by (5.11), and γ > 0. The extended quadratic LPV γ-performance problem is solvable if there exists an m ≥ 0, a finitedimensional, m-state controller (5.12), and a continuously differentiable X(ρ) : ℝ^{s} → ℝ^{(n+m)×(n+m)}, X(ρ) = X^{T} (ρ) > 0 such that for all ρ ∈ P with \underline{ν} ≤ ρ̇ ≤ \overline{ν},

[\begin{matrix}Ẋ(ρ) + A_{clp}^{T}(ρ)X(ρ) + X(ρ)A_{clp}(ρ) & X(ρ)B_{clp}(ρ) & γ^{−1}C_{clp}^{T}(ρ) \\ B_{clp}^{T}(ρ)X(ρ) & −I & γ^{−1}D_{clp}^{T}(ρ) \\ γ^{−1}C_{clp}(ρ) & γ^{−1}D_{clp}(ρ) & −I \end{matrix}] < 0.  (5.16)

Similarly, Theorem 5.3 can be modified, see [Woo95, Theorem 8.2.2] and [Wu95, Theorem 5.3.2], and the feasibility problem now consists of five LMIs in X_{11}(ρ), Y_{11}(ρ) : ℝ^{s} → ℝ^{n×n}.

The five LMIs are X_{11}(ρ) = X_{11}^{T}(ρ) > 0, Y_{11}(ρ) = Y_{11}^{T}(ρ) > 0,

[\begin{matrix}−Ẏ_{11}(ρ) + Y_{11}(ρ)A^{T} (ρ) + A(ρ)Y_{11}(ρ) − B(ρ)B^{T} (ρ) & Y_{11}(ρ)C^{T} (ρ) \\ C(ρ)Y_{11}(ρ) & −I \end{matrix}] < 0,

[\begin{matrix}Ẋ_{11}(ρ) + A^{T} (ρ)X_{11}(ρ) + X_{11}(ρ)A(ρ) − C^{T} (ρ)C(ρ) & C^{T} (ρ) & 0 & X_{11}(ρ)B(ρ) \\ C(ρ) & −γ^{2}I & 0 & 0 \\ 0 & 0 & −γ^{2}I & I \\ B^{T} (ρ)X_{11}(ρ) & 0 & I & −I \end{matrix}] < 0,

[\begin{matrix}X_{11}(ρ) & γ^{−1}I \\ γ^{−1}I & Y_{11}(ρ) \end{matrix}] ≥ 0. (5.17)

A controller that solves the extended quadratic LPV γ-performance problem, has the same structure as the controller that is given in Theorem 5.3, but with X_{11} and Y_{11} now parameter-dependent and

\begin{matrix}H(ρ) = −[−Y_{11}^{−1}(ρ)Ẏ_{11}(ρ)Y_{11}^{−1}(ρ) + Y_{11}^{−1}(ρ)A(ρ) + A^{T} (ρ)Y_{11}^{−1}(ρ) − \\ −Y_{11}^{−1}(ρ)B(ρ)B^{T} (ρ)Y_{11}^{−1}(ρ) + C^{T} (ρ)C(ρ)]. \end{matrix}  (5.18)

Without loss of generality,

X_{11}(ρ) = ∑_{i} f_{i}(ρ)X_{i}, and Y_{11}(ρ) = ∑_{j} g_{j}(ρ)Y_{j},  (5.19)

where X_{i} = X_{i}^{T} , Y_{j} = Y_{j}^{T} with X_{i}, Y_{j} ∈ ℝ^{n×n}, and f_{i}, g_{j} are continuously differentiable functions of ρ. Thus,

Ẋ_{11}(ρ) = ρ̇ ∑_{i} \frac{df_{i}(ρ)}{dρ}X_{i}, and Ẏ_{11} = ρ̇ ∑_{j} \frac{dg_{j}(ρ)}{dρ}Y_{j}.  (5.20)

If the designer chooses f_{i}, g_{j} then LMIs (5.17) represent convex constraints on X_{i}, Y_{j} and ρ̇. As the rate of change of ρ is bounded by \underline{ν} and \overline{ν}, the LMIs depend continuously on ρ and the solution of the synthesis problem can be formulated as a convex feasibility problem with an infinite number of constraints. In the sequel, the first method will be referred to as the single Lyapunov function approach and the second method as the parameterdependent Lyapunov function approach.

5.2.1 Notes on the single Lyapunov function approach

The main analysis lemma (Lemma 5.1) is only a sufficient condition. Hence the designer cannot deduce that there does not exist a controller that solves the quadratic LPV γperformance problem if the feasibility problem stated in Theorem 5.3 has no solution.

The solution of the quadratic LPV γ-performance problem is a natural extension of the solution of the H_{∞} problem as presented in [GA94]. This is because if the state-space data (5.11) does not depend on ρ, then the main analysis lemma becomes the bounded real lemma^{3}.

Lemma 5.1 is derived by using a quadratic Lyapunov function of the form

V (x_{cl}) = x_{cl}^{T} X x_{cl},  (5.21)

where X is a constant matrix. The use of such Lyapunov functions can lead to conservative values of γ, e.g. see [Hel95, p. 31]. The motivation for using quadratic Lyapunov functions of the form (5.21), is that they can be used to derive simple stability and performance tests. In [PH93, pp. 57-71] four alternative ways of constructing a Lyapunov function for stability investigations are presented.

In many cases the resulting value of γ is conservative, because even though in practice ρ cannot vary arbitrarily fast, the synthesised controller K_{ρ} guarantees ∥G_{F_{P}} ∥ < γ for arbitrarily large ρ̇.

The synthesis problem is typically solved by gridding the set P. If the parameter ρ consists of some of the states of the open loop plant, then there might be some grid points that correspond to values of ρ that cannot be achieved in practice; in steady state or during a transient maneuver. Excluding these grid points during synthesis could lead to a smaller value of γ. Consider for example designing a flight controller that is scheduled on angle of attack (α) and forward speed (V). As high α maneuvers at a high speed can lead to a structural failure of the airframe, it is assumed that the pilot will avoid such maneuvers and hence grid points that correspond to high values of α and V can be excluded during synthesis. Excluding such grid points could result in a smaller performance bound, as the dynamics of an aircraft vary considerably with angle of attack and forward speed.

The synthesised controller K_{ρ} comes with the following guarantee about the LPV closed loop

sup_{ρ∈F_{P}} sup_{{∥d∥_{2}≠0 \atop d∈L_{2}}} \frac{∥e∥_{2}}{∥d∥_{2}} < γ < ∞.  (5.22)

As the weighted plant (5.10) and K_{ρ} are linear, the small-gain theorem (Theorem 2.3) and (5.22) guarantee that the interconnected system of the perturbed weighted plant and K_{ρ} will be well-posed and internally stable for any nonlinear, possibly time-varying, infinitedimensional perturbation Δ : e ↦ d, with an incremental gain smaller than or equal to 1/γ. The same is true for any linear perturbation Δ having \overline{σ}(Δ) ≤ 1/γ. It will be shown in Section 5.4 (if the nonlinear plant model satisfies certain assumptions and the weighted LPV plant model is derived in a special way), that the weighted LTI plant model that results from evaluating (5.10) at some permissible value of ρ, has the same state-space properties with the weighted Jacobi linearisation of the nonlinear plant model about the same value of ρ. A value of ρ is said to be permissible if it is contained within the grid on which the synthesis problem is solved. Hence it becomes an easier task to calibrate γ. It can be deduced from the bounded real lemma^{4} and (5.22), that the optimal LTI stability margin at every ρ is an upper bound of γ^{−1}. Thus, it is expected that the values of γ will be larger than those achieved for LTI designs.

3The bounded real lemma is also a necessary condition.

The synthesis problem (see Theorem 5.3) is typically solved by gridding the set P and solving approximations to the actual constraints (5.15). It becomes apparent that the number of LMI constraints depends on the number of grid points. If for example ρ = [α, V]^{T} and the range of α, V is spanned by n_{α}, n_{V} points respectively, then the number of LMI constraints is

N_{SLF} = 2 n_{α} n_{V} + 3.

Solving the feasibility problem becomes a more computationally intensive task as N_{SLF} increases. In Subsection 5.2.2 it is discussed how the time required to solve the feasibility problem depends on the number of LMIs and decision variables^{5}.

It is of engineering importance to know how dense the grid must be, so that satisfying the constraints on the grid points ensures that the actual constraints (5.15) are also satisfied. Some work in this direction has been done in [Wu95, Lemma 5.4.2]. Consider for example the problem of finding a matrix X = X^{T} > 0 such that

A^{T} (ρ)X + XA(ρ) < 0.  (5.23)

The parameter ρ is assumed to be scalar and the grid points are chosen to be ρ_{i}. If

A^{T} (ρ_{i})X + XA(ρ_{i}) ≤ −δI ∀i, and \overline{σ}(X) ≤ T ⟺ X ≤ T I,

where δ > 0, then it can be proved (see for example [Wu95, Lemma 5.4.2]) that (5.23) will be satisfied if

2 T |ρ_{i+1} − ρ_{i}| max_{ξ∈[ρ_{i},ρ_{i+1}]} \overline{σ}[\frac{dA^{T} (ξ)}{dρ}] < δ.  (5.24)

Having decided on the values of δ and T, the designer can choose the grid points ρ_{i} so that (5.23) is satisfied, by plotting \overline{σ}[dA^{T} (ξ)/dρ] against ξ for ξ in the parameter range, and choosing ρ_{i} so that

|ρ_{i+1} − ρ_{i}| max_{ξ∈[ρ_{i},ρ_{i+1}]} \overline{σ}[\frac{dA^{T} (ξ)}{dρ}] < \frac{δ}{2 T} .

4The key point being that the bounded real lemma is a necessary and sufficient condition.

5Each independent entry of X is a decision variable.

The grid points do not need to be evenly spaced. It has been found that this approach results in a larger number of LMIs than is actually necessary, i.e. a feasibility problem that takes longer to solve. In practice, after solving the synthesis problem on a grid with N grid points, Inequalities (5.15) are evaluated on a very dense grid. If this analysis shows that some constraint is violated, then the synthesis problem is solved on a denser grid, i.e. a grid with more than N grid points.

The quadratic LPV γ-performance problem can be reduced to a convex problem with a finite number of constraints by imposing additional restrictions on P, and the dependence of the state-space data (5.10) on ρ. It is required that the set P is a convex polytope whose finite set of extreme points is denoted by E, that B, C are constant matrices, and that the dependence of the matrix valued function A(ρ) on ρ ∈ P is multi-affine. Under these assumptions, to solve the quadratic LPV γ-performance problem, it is sufficient that the LMI constraints hold on the finite set E of extreme points of P (the proof of this statement can be found in [BP94, Theorem 4.3]). The assumption that A(ρ) must be a multi-affine function of ρ, is a restricting assumption as most physical systems of interest are nonlinear in the scheduling variables. The second assumption is not as restricting, as pre- and post-multiplying G_{ρ} with low-pass filters ensures that B, C are constant matrices, e.g. see [AGB95]. The number of LMI constraints now becomes N_{SLF} = 2 × 2^{s} + 3, where s is the number of scheduling variables.

It is assumed throughout this section, that the value of ρ is known exactly. This is not true because of sensor noise. It is difficult to analyse the effect of a noisy ρ measurement on the robustness of the closed loop system, as this type of sensor noise cannot be included in the framework considered. If the plant is very nonlinear and the control system not too robust, then it is intuitively obvious that a large sensor bias could lead to closed loop instability. It is also assumed throughout this section that ρ is known in real-time. Time delays in sensing the scheduling variables can result in an unstable closed loop system. Again it is difficult to analyse the effect of such time delays on the robustness of the closed loop. Another natural question that arises, is what happens when ρ is not in F_{P} (this may occur if ρ consists of some of the states of the open loop plant). The view taken, is that the controller will be robust enough to cope with all three types of uncertainty.

The synthesised controller K_{ρ} can written as an observer with state feedback, see the state-space realisation of the controller given in Theorem 5.3. Hence K_{ρ} can be implemented in the same way as a “classical” gain scheduled controller, e.g. see [Hyd95]. Note that the observer is not a plant observer as in the LTI case, but an observer of

[ \begin{matrix}A(ρ) − γ^{−2}ZH(ρ) & B(ρ) \\ C(ρ) & 0 \end{matrix} ] .

To reduce the on-line computation during implementation, the matrix valued functions A(ρ), B(ρ), C(ρ) can be evaluated on a grid and stored in look-up tables (LUTs). The robustness of the approximated controller can be evaluated using the feasibility problem stated in Definition 5.2. Note that (5.14) can be used to assess the robustness of any gain scheduled controller with ρ̇ arbitrarily large.

Quadratic stability is not a system property; rather it depends on the state-space realisation of the system. Given the system ẋ = A(ρ) x, the matrix valued function A(ρ) is exponentially stable if there exists a matrix X = X^{T} > 0 such that

A^{T} (ρ)X + XA(ρ) < 0, ∀ ρ ∈ P.

Consider the state transformation x(t) = T (ρ)z(t) (it is assumed that T^{−1}(ρ) exists for all allowable values of ρ). Thus ẋ = A(ρ) x becomes

ż = Â(ρ) z(t) = T^{−1}(ρ) [−Ṫ (ρ) + A(ρ)T (ρ)] z(t).

It can be deduced that X does not necessarily guarantee exponential stability of Â(ρ). It is not clear how to exploit this property to design controllers which achieve a better stability margin.

5.2.2 Comments on the parameter-dependent Lyapunov function approach The main analysis lemma used to solve the extended quadratic LPV γ-performance problem is still only a sufficient condition. The motivation for using the parameter-dependent Lyapunov function approach, is that it may be possible to design a more robust closed loop than if the single Lyapunov function approach was used, because the bounds of ρ̇ are taken into account.

As was discussed in the previous subsection, if the scheduling variables are also states of the open loop plant, then there might be some values of ρ that cannot be achieved in practice, and if they are not excluded during synthesis then the closed loop system may have poor robustness. In addition, solving LMIs (5.17) at ρ̇ = \underline{ν} and ρ̇ = \overline{ν} for every value of ρ could lead to a poor stability margin. Consider for example the dynamical system ẋ = A(ρ) x. The matrix valued function A(ρ) will be exponentially stable if there exists a continuously differentiable matrix function X(ρ) = X^{T} (ρ) > 0 such that

Ẋ(ρ) + A^{T} (ρ)X(ρ) + X(ρ)A(ρ) < 0,

that is

∑_{i=1}^{s} \frac{∂X(ρ)}{∂ρ_{i}} ρ̇_{i} + A^{T} (ρ)X(ρ) + X(ρ)A(ρ) < 0.  (5.25)

As ρ_{i} = T_{i} x, where T_{i} is a constant row vector, Inequality (5.25) becomes

∑_{i=1}^{s} \frac{∂X(ρ)}{∂ρ_{i}}T_{i} A(ρ) x + A^{T} (ρ)X(ρ) + X(ρ)A(ρ) < 0.  (5.26)

To solve LMI (5.26) it is necessary to grid over the whole state-space. In the case of the extended quadratic LPV γ-performance problem, solving LMIs (5.17) at ρ̇ = \underline{ν} and ρ̇ = \overline{ν} for every value of ρ may not have such a large effect on γ, as ẋ_{cl} also depends on the disturbance signal d. Nevertheless, it is not clear how to modify (5.16) to take into account that some of the states are also scheduling variables.

As in the case of designing the closed loop system using the single Lyapunov function approach, K_{ρ} comes with guarantee (5.22). In this case though, (5.22) is valid for all ρ ∈ F_{P} with \underline{ν} ≤ ρ̇ ≤ \overline{ν}.

The feasibility problem (5.17) is usually solved by gridding the set P, because X_{11} and Y_{11} are functions of ρ. Making the problem convex in the way discussed in the previous subsection, could lead to poor values of γ. The number of LMIs is greater than that in the single Lyapunov function approach for the same number of grid points. Each grid point contributes 2^{s} × 2 + 3 LMIs, thus for the example considered in the previous subsection the total number of LMIs becomes

N_{P DLF} = 11 n_{α} n_{V} .

The number of decision variables will depend on how X_{11} and Y_{11} depend on ρ, e.g. if the dependence of X_{11} and Y_{11} on ρ = [α, V]^{T} is affine, then the number of decision variables is 4n(n + 1), where n is the number of states of G_{ρ}. Thus if n = 7, the number of decision variables is 224, i.e. 168 more than in the single Lyapunov function approach^{6}. Numerical experiments show that it takes O(m^{2.1}L^{1.2}) arithmetic operations to solve a problem with m decision variables and L constraints using the primal-dual interior-point method (see [BGEB94, Section 2.4.4]). Thus the solution time depends heavily on the number of decision variables.

The density of the grid that is required to guarantee that the actual constraints are satisfied can be calculated using [Wu95, Lemma 5.4.2]. In practice it has been found that this approach results in a larger number of grid points than is actually necessary. The approach taken during design is to uniformly grid P with a finite number of points and to solve the LMIs on this finite set. Then, to see if the solution holds over the entire set, the inequalities are evaluated on a much denser grid of P.

Before solving LMIs (5.17), the designer must choose how the matrix valued functions X_{11}, Y_{11} depend on the parameter ρ. The minimum value of γ that can be achieved will depend on this choice. The most general situation would be to allow X (and similarly Y; the subscript has been dropped) to be

6The design example considered in Section 5.7 has 7 states.

X(ρ) = ∑_{i=1}^{n} ϕ_{i}(ρ)X_{i} > 0, X_{i} = X_{i}^{T} , i = 1, …, n.

There is a lot of freedom in the choice of basis functions ϕ_{i}(ρ) from which the set of trial functions X(ρ), Y (ρ) can be constructed. A convenient choice of basis functions is the set of cubic splines. Partition the interval [ρ_{l}, ρ_{h}] into m equal intervals and define the associated knot points (ρ is assumed to be scalar)

ρ_{j+1} − ρ_{j} = h = \frac{ρ_{h} − ρ_{l}}{m} , j = 0, 1, …, m − 1.

It is well known that any continuous function can be approximated arbitrarily closely on a closed interval by a cubic spline which interpolates that function at a set of sufficiently close knot points. The linear space of cubic splines is defined as follows

\begin{matrix}ϕ_{i}(ρ) = \frac{1}{4h^{3}} [(ρ − ρ_{i−2})_{+}^{3} − 4(ρ − ρ_{i−1})_{+}^{3} + 6(ρ − ρ_{i})_{+}^{3} − 4(ρ − ρ_{i+1})_{+}^{3} + (ρ − ρ_{i+2})_{+}^{3}], \\ i = −1, 0, …, m + 1 where (x)_{+} = \{ \begin{matrix}x, x ≥ 0 \\ 0, x < 0 \end{matrix} \end{matrix}

Each basis function ϕ_{i}(ρ) has compact support on the interval [ρ_{i−2}, ρ_{i+2}] and is twice continuous and differentiable. At a knot point ρ_{i}

X(ρ_{i}) = \frac{1}{4}X_{i−1} + X_{i} + \frac{1}{4}X_{i+1}, and \frac{∂X}{∂ρ} (ρ_{i}) = −\frac{3}{4h}X_{i−1} + \frac{3}{4h}X_{i+1}.

The drawback of this method is that as the number of knot points increases so does the number of decision variables. Note that it is not necessary for all grid points to be knot points. The above basis is discussed in [RTJ95], and was first used for solving (5.17) in [Woo95]. If ρ is a vector, then product splines must be used; product splines are discussed in [RTJ95]. If the computational cost of using cubic splines is too high, then the designer can select the trial functions in an ad-hoc manner. It was reported in [Bec95] that good results were achieved by making the trial functions depend on ρ in the same way that G_{ρ} does.

It is assumed that the value of ρ is known exactly and in real-time. The validity of these assumptions was discussed in the previous subsection.

The synthesised controller K_{ρ} can be implemented like a “classical” gain scheduled controller because it can be written as an observer with state feedback. Unfortunately, K_{ρ} depends on ρ̇, see Equation (5.18). It is not known without performing time simulations how sensitive the closed loop system is to time delays and inaccuracies in the measurement of ρ̇. Controllers that do not depend on ρ̇ can be synthesised by choosing a constant Y_{11} or a constant X_{11} if the dual synthesis problem is solved. For example, if the parameter-dependent Lyapunov function approach is used to design a flight controller that is scheduled on V, α, the flight controller will also depend on V̇ , α̇. As there are no sensors that give a reliable α̇ measurement at high α, a flight controller that is not a function of α̇ can be synthesised by solving (5.17) and choosing a trial function Y_{11} that depends only on V. The price paid is that the achieved value of γ will be sub-optimal. If ρ is scalar, then the rate dependence of the controller can be removed by employing a Lyapunov transformation as suggested in [Woo95, p. 144].

Unlike quadratic stability, extended quadratic stability is a system property and thus the achievable stability margin does not depend on the realisation of the G_{ρ}.

5.2.3 Recent developments in LPV gain scheduling

A controller synthesised using the single or parameter-dependent Lyapunov function approach comes with the guarantee (5.22). In (5.22) γ is constant signifying performance for not only worst-case disturbances, but also worst-case parameter variation. In [LS97] the extended quadratic LPV γ-performance problem is modified so that γ can be a function of ρ. Allowing γ to be parameter-varying offers some advantages. It gives the designer an idea of the robustness and consequently design difficulty associated with each value of ρ. Also, performance is not degraded because of a few troublesome operating points.

In this chapter it has been assumed that the uncertainty Δ associated with the closed loop system (5.13) has no structure. In many applications Δ will have a known structure and by taking this structure into account it may be possible to reduce the achieved value of γ. The synthesis problem of finding a K_{ρ} that takes the structure of Δ into account has been approached in a number of different ways in the literature.

In [BFP^{+}97] dynamic D-scalings are used to improve the performance of the LPV controller, i.e. a D-K iteration is carried out. In the first step K_{ρ} is obtained by solving the quadratic LPV γ-performance problem. In the second step dynamic D-scalings are calculated at each grid point by performing a μ-analysis of the corresponding frozen closed loop system. These scalings are then wrapped into each generalised plant and the first step in repeated. The iteration stops when the achieved value of γ is satisfactory or does not decrease any further. This procedure guarantees robust stability and performance of each LTI closed loop with respect to linear time-invariant uncertainty.

In [LS96] a similar procedure is proposed but using parameter-dependent D-scalings which are not dynamic. This procedure guarantees robust stability and performance of the LPV closed loop with respect to linear time-varying uncertainty. At each grid point the constant D-scalings are evaluated by performing a μ-analysis.

In [BWS96] the so-called scaled differential bounded real lemma provides an analysis result that given the structure and nature of the uncertainty can be used to evaluate γ. The scaled differential bounded real lemma is proved in [Hel95, Section 2.9] using [Hel95, Theorem 2.2]. The analysis result can be turned into a synthesis problem which is solved iteratively. This is a more rigorous approach than that taken in [LS96], and can result in smaller values of γ in the case of time-varying parametric uncertainty. This method guarantees robust stability and performance of the LPV closed loop with respect to linear time-varying uncertainty, non-expansive dynamic uncertainty, and combinations of the two. A similar approach to [BWS96] has been taken in [AA98], the difference being the LMIs solved to obtain K_{ρ}.

5.3 LFT gain scheduling

In this section an alternative gain scheduling technique will be presented that was introduced in [Pac94] (discrete time) and [AG95] (continuous and discrete time). Consider a parameter-dependent plant modelled as an linear fractional transformation (LFT) of a time-varying, block-structured matrix Δ(t) and an LTI system P, see Fig. 5.2. In [Hel95,

[figure: P; e; y; α; β; d; u; Δ(t)]

Figure 5.2 Parameter-dependent plant

Chapter 4] it is explained in detail how to convert an LPV model into LFT form. The parameter dependence of the plant is due to the time-varying matrix

Δ(t) := diag[δ_{1}(t)I_{n_{1}}, …, δ_{f}(t)I_{n_{f}}] ∈ ℂ^{n×n}.

It is assumed that the value of Δ(t) is guaranteed a priori to lie in a known set, i.e.

sup_{t} |δ_{i}(t)| ≤ 1, ∀ i 1 ≤ i ≤ f,

and that the exact value of Δ(t) is known only in real-time. The controller for this parameter-dependent LFT plant will also be parameter-dependent, and will be restricted to have a similar structure, i.e. a linear fractional interconnection of a linear, time-invariant system with copies of the same δ_{i} which affect the plant, see Fig. 5.3. It can be shown (e.g. see [Pac94]) that for the synthesis problem considered (a sub-optimal problem that is

[figure: K; Δ(t); u; y; α̃; β̃]

Figure 5.3 Parameter-dependent controller

given by the conditions in Lemma 5.5), each linear fractional parameter must be repeated in the controller no more than the number of times it appears in the plant. The synthesis problem can be written as a set of LMIs only if each δ_{i} is repeated in the controller as many times as it appears in the plant, and this is why the controller is made to depend on Δ(t).

The parameter-dependent closed-loop system is depicted in Fig. 5.4. The time-varying

[figure: {Δ \atop Δ}; P_{gen}; K; e; d; y; u; β̃; α̃; β̃; α̃; α; β]

Figure 5.4 Parameter-dependent closed loop

parameters have been collected together, Δ_{tot} = diag(Δ, Δ). The LTI system P_{gen} is completely known, and K is the LTI portion of the LFT controller. The control objective is to design an LTI system K such that for all allowable parameter trajectories δ_{i}(t), the closed loop system is internally exponentially stable and has a small induced L_{2} norm from d to e. The synthesis problem appears as a robust control problem with the generalised plant P_{gen} and the uncertainty Δ_{tot} having a very special structure. The induced norm of the d → e channel under time-varying Δ(t), can be bounded (albeit conservatively) using the optimally scaled small gain theorem. The scaling matrices J are restricted to be constant, block diagonal, invertible matrices that commute with Δ_{tot}, i.e. J ∈ ℝ^{2n×2n} and JΔ_{tot} = Δ_{tot}J. Note that the rows of Δ_{tot} have been reordered so that the copies of δ_{i} from the plant and the controller are adjacent. The synthesis problem can be cast as a convex feasibility problem which can be solved using the LMI-lab, see [GNLC95].

The main analysis lemma is stated below.

Lemma 5.5 If there is a J and a stabilising, finite-dimensional, linear time-invariant K such that

∥[\begin{matrix}J & 0 \\ 0 & I_{n_{e}} \end{matrix}] F_{l}(P_{gen}, K) [\begin{matrix}J^{−1} & 0 \\ 0 & I_{n_{d}} \end{matrix}]∥_{∞} < 1,  (5.27)

then there is a 0 ≤ γ < 1, such that for all parameter trajectories δ_{i}(t) with sup_{t} |δ_{i}(t)| ≤ 1, the system in Fig. 5.4 is well-posed (for a given disturbance signal d, all signals α, β, α̃, β̃, e, y and u are unique, and each depends causally on δ and d) and internally exponentially stable, moreover for zero initial conditions, if d ∈ L_{2}, then ∥e∥_{2} ≤ γ∥d∥_{2}.

The LFT control synthesis problem is based on the previous lemma. The solution of this synthesis problem yields only one controller that satisfies (5.27).

5.3.1 Comments on the LFT method

To be able to use the LFT gain scheduling method, the parameter-dependent plant must be modelled as a linear fractional transformation of a time-varying matrix containing the scheduling variables and an LTI system. The LFT model is produced in two steps. First an LPV model is produced and then the LPV model is transformed into an LFT model. To produce an LFT model that accurately captures the dynamics of the LPV plant model, it may be necessary to fit high order polynomials to each state-space entry

ẋ = A(ρ)x + B(ρ)u, y = C(ρ)x.

Consequently, Δ(t) may be a large matrix with a large number of repeated entries. This is typical of what may happen when designing a flight controller for a supermaneuverable aircraft. The dynamics of such an aircraft vary considerably over the flight envelope and hence an accurate LFT model may necessitate a big Δ with many repeated entries. This will have two undesirable effects.

The first effect is due to a parameter δ_{i}(t) being repeated many times in Δ(t). A controller K_{ρ} that has been designed using the LFT gain scheduling method comes with the guarantee

sup_{{∥Δ(t)∥_{2}≤1 \atop ∥d∥_{2}≠0,d∈L_{2}}} \frac{∥e∥_{2}}{∥d∥_{2}} < γ.  (5.28)

Consider that the parameter δ_{i}(t) is repeated twice. It is easy to see that (5.28) guarantees robust stability for any pair [δ_{i}^{1}(t), δ_{i}^{2}(t)] in the unit circle even though δ_{i}^{1}(t) cannot vary independently from δ_{i}^{2}(t). Thus the synthesis problem that has been formulated only partly takes into account the structure of Δ(t). The larger the number of times that δ_{i}(t) is repeated in Δ(t) the more likely it is that the LFT gain scheduling approach will not yield a satisfactory controller.

The second effect is because of the large dimension of Δ(t). The controller is given by

K_{ρ} = F_{l}(K, Δ(t)) = K_{11} + K_{12}Δ(t)(I − K_{22}Δ(t))^{−1}K_{21},

where

K = [ \begin{matrix}K_{11} & K_{12} \\ K_{21} & K_{22} \end{matrix} ] = [ \begin{matrix}A & B_{1} & B_{2} \\ C_{1} & D_{11} & D_{22} \\ C_{2} & D_{21} & D_{22} \end{matrix} ] .

Thus to implement the controller, at each sampling time the matrix I − D_{22}Δ(t) must be inverted. If Δ(t) has a large dimension, this inversion could incur a large computational cost that prohibits a small sampling time. Alternatively, instead of inverting I − D_{22}Δ(t) the entries of (I − D_{22}Δ(t))^{−1} could be stored in LUTs. Again it is not clear that the number of computations will be small; furthermore, this approach may result in memory problems.

It can easily be deduced from the small-gain theorem, that (5.28) guarantees robustness of the LPV closed loop system with respect nonlinear, possibly time-varying, infinitedimensional perturbations of the open loop plant model.

In the previous subsection it was assumed that Δ(t) can be a complex matrix. Scheduling variables can obviously only be real, and thus treating Δ as a complex matrix can result in conservative values of γ. In [Hel95] the synthesis problem is modified to take the realness of Δ(t) into account. This is done by using real-μ ideas.

It was mentioned in Section 5.2, that by taking the rate of change of the scheduling variables into account during synthesis, it may be possible to achieve a smaller value of γ. Unfortunately, the LFT control synthesis problem cannot be modified to take Δ̇(t) into account and thus can lead to large values of γ. Once the controller has been synthesised the designer can evaluate its robustness with respect to M_{l} ≤ Δ̇ ≤ M_{h} by using Definition 5.4.

The existence of a controller K_{ρ} that satisfies the scaled small-gain condition (5.27) can be reformulated into a finite-dimensional convex feasibility problem, which is attractive from a computational viewpoint.

As was discussed in Subsection 5.2.3, in many applications the uncertainty associated with the transfer matrix from d → e has some structure that if exploited can reduce the achieved value of γ. Lemma 5.5 can be modified using the scaled bounded real lemma to take the structure of the uncertainty into account, e.g. see [BAG97, SNB97].

5.4 Obtaining a reliable LPV model

To be able to apply a robust gain scheduling technique, the designer must have a “good” LPV model of the nonlinear plant. It is essential that the LPV plant model captures the transient behaviour of the real plant. If this is not the case then there is less motivation for applying a robust gain scheduling technique. Assume the designer is using a family of Jacobi linearisations. The guaranteed γ will be valid, as long as the parameter rate remains within the pre-specified bounds (assume that the parameter-dependent Lyapunov function approach was used to design the LPV controller). But a good value of γ could lead to false conclusions about the stability of the real closed loop system, as the dynamics of an LTI model and a possible transient model about the same operating point could be very different (this was pointed out in Section 5.1). Hence although the design procedure is more systematic than that of “classical” gain scheduling, and the controller K_{ρ} comes with guarantees about the LPV closed loop system, K_{ρ} will not necessarily perform well on the real plant. In this case the designer must still follow the rule of thumb that “the scheduling variables should vary slowly”. This rule of thumb is a reminder that LTI models are used to design an LPV controller. It was assumed in this thought experiment that the scheduling variables capture the plant’s nonlinearities.

The discussion will be restricted to the class of nonlinear systems considered in Section 5.1, i.e. “quasi-LPV” systems. Note that the two synthesis methods discussed in Sections 5.2 and 5.3 can be used to synthesise robust gain scheduled controllers for “quasi-LPV” systems, albeit conservatively. To our knowledge there are two possible ways of systematically creating an LPV model of a nonlinear plant^{7}:

1. A “quasi-LPV” model can be generated by interpolating the state-space data of a family of Jacobi linearisations.

2. In [SC93] it is shown how to rewrite the equations of motion of a certain class of nonlinear systems in “quasi-LPV” form.

In general, a family of Jacobi linearisations cannot capture the transient behaviour of the real plant. However, if the nonlinear plant has a special structure, e.g. if the dynamics of the plant are described by (5.1) and B is a constant matrix, then a family of LTI models can capture the essential features of the transient response. It is not straightforward to deduce, without prior experience, which parameters and/or states capture the plant’s nonlinearities. For example, the state vector of the longitudinal HIRM model used in the GARTEUR design challenge is [V, α, γ, q, h]^{T}. It is not clear whether the designer should schedule on α and V, or α, V and h. It is assumed that γ and q remain small over most of the flight. If the designer chooses to schedule on α and V, it is not clear how to choose the values of h about which the nonlinear plant model should be linearised. Thus the family of Jacobi linearisations does not give the designer insight into how the state-space data depends on the states.

7Both these methods are based on the assumption that there exists an analytic model of the plant’s dynamics; which is true for most mechanical systems.

The class of nonlinear systems considered is described by

[ż ; ẇ] = [\begin{matrix}k_{1}(z) \\ k_{2}(z) \end{matrix}] + [\begin{matrix}A_{11}(z) & A_{12}(z) \\ A_{21}(z) & A_{22}(z) \end{matrix}] [\begin{matrix}z \\ w \end{matrix}] + [\begin{matrix}B_{1}(z) \\ B_{2}(z) \end{matrix}] u,  (5.29)

where z ∈ ℝ^{n_{z}}, w ∈ ℝ^{n_{w}}, and u ∈ ℝ^{n_{u}}. It can be shown that if n_{u} + n_{w} = n where n = n_{z} + n_{w}, then (5.29) can be transformed into a “quasi-LPV” model whose state-space data is a function of z. Thus it must be possible to measure the states z in real-time. The nonlinearity k(z) will be referred to as an output nonlinearity as z can be measured.

Assume that there exist differentiable functions w_{eq}, u_{eq} such that for every z

[\begin{matrix}0 \\ 0 \end{matrix}] = [\begin{matrix}k_{1}(z) \\ k_{2}(z) \end{matrix}] + [\begin{matrix}A_{11}(z) & A_{12}(z) \\ A_{21}(z) & A_{22}(z) \end{matrix}] [\begin{matrix}z \\ w_{eq} \end{matrix}] + [\begin{matrix}B_{1}(z) \\ B_{2}(z) \end{matrix}] u_{eq}.  (5.30)

The dependence of w_{eq}, u_{eq} on z is not shown. Subtracting (5.30) from (5.29) results in

[ż ; ẇ] = [\begin{matrix}0 & A_{12}(z) \\ 0 & A_{22}(z) \end{matrix}] [\begin{matrix}z \\ w − w_{eq} \end{matrix}] + [\begin{matrix}B_{1}(z) \\ B_{2}(z) \end{matrix}] (u − u_{eq}).  (5.31)

As

ẇ_{eq} = \frac{dw_{eq}}{dt} = \frac{dw_{eq}}{dz} ż,

it can be deduced from (5.31) that

[ ż ; ẇ − ẇ_{eq}] = [\begin{matrix}0 & A_{12}(z) \\ 0 & A_{22}(z) − \frac{dw_{eq}}{dz} A_{12}(z) \end{matrix}] [\begin{matrix}z \\ w − w_{eq} \end{matrix}] + [\begin{matrix}B_{1}(z) \\ B_{2}(z) − \frac{dw_{eq}}{dz} B_{1}(z) \end{matrix}] (u − u_{eq}). (5.32)

Thus (5.29) has been transformed into a “quasi-LPV” model. The state-space model (5.32) can be seen as the product of a linearising feedback loop, see Fig. 5.5. If A, B, k are also functions of an exogenous variable ρ, e.g. in the case of an aircraft ρ could be the mass of the aircraft, then this method can be used to create an LPV system that has inputs [u − u_{eq}, ρ̇]^{T}. The cost function minimised will depend on whether a measurement of ρ̇ is available.

[figure: real plant; u_{eq}(·); z; a; u; b; ”quasi-LPV”; [\begin{matrix}z \\ w − w_{eq} \end{matrix}]] Figure 5.5 The “quasi-LPV” model

The linearising feedback loop could lead to a closed loop system with poor robustness properties. The synthesised controller K_{ρ} will guarantee robustness with respect to uncertainty at the input of the “quasi-LPV” model (point b in Fig. 5.5), but the uncertainty actually occurs at the plant input (point a). A good robustness margin at point b does not imply a good robustness margin at point a. This is because the feedback loop essentially cancels part of the plant’s dynamics; Equation (5.31) is the result of subtracting (5.30) from (5.29). The feedback loop can be eliminated by pre-compensating the inputs of the nonlinear plant with integrators before transforming into “quasi-LPV” form. If

u = ∫ v,

Equation (5.29) becomes

[ż ; ẇ ; u̇] = [\begin{matrix}k_{1}(z) \\ k_{2}(z) \\ 0 \end{matrix}] + [\begin{matrix}A_{11}(z) & A_{12}(z) & B_{1}(z) \\ A_{21}(z) & A_{22}(z) & B_{2}(z) \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix}z \\ w \\ u \end{matrix}] + [\begin{matrix}0 \\ 0 \\ I \end{matrix}] v,  (5.33)

and (5.32)

[ ż ; ẇ − ẇ_{eq} ; u̇ − u̇_{eq}] = [\begin{matrix}0 & A_{12}(z) & B_{1}(z) \\ 0 & A_{22}(z) − \frac{dw_{eq}}{dz} A_{12}(z) & B_{2}(z) − \frac{dw_{eq}}{dz} B_{1}(z) \\ 0 & −\frac{du_{eq}}{dz} A_{12}(z) & −\frac{du_{eq}}{dz} B_{1}(z) \end{matrix}] [\begin{matrix}z \\ w − w_{eq} \\ u − u_{eq} \end{matrix}] + [\begin{matrix}0 \\ 0 \\ I \end{matrix}] v. (5.34)

Note that the functions w_{eq}, u_{eq} are defined by (5.30). Thus if K_{ρ} is designed using (5.34) and without using the states u − u_{eq}(z) for feedback, the robustness margin at point b can be used to predict the robustness margin at point a. If the controller is designed using the H_{∞} loop shaping cost function, W_{1} can contain the integrators necessary for eliminating the feedback loop.

Assume that the nonlinear plant (5.29) has been pre-compensated with integrators and that only scheduling variables are used for feedback, i.e.

y = [C 0 0] [\begin{matrix}z \\ w − w_{eq} \\ u − u_{eq} \end{matrix}] ,

where y ∈ ℝ^{n_{y}} and C ∈ ℝ^{n_{y}×n_{z}}. It can be shown that the state-space model obtained by evaluating the “quasi-LPV” model (5.34) at z_{0}

G_{LP V} = [ \begin{matrix}0 & A_{12} & B_{1} & 0 \\ 0 & A_{22}(z) − \frac{dw_{eq}}{dz} A_{12} & B_{2} − \frac{dw_{eq}}{dz} B_{1} & 0 \\ 0 & −\frac{du_{eq}}{dz} A_{12} & −\frac{du_{eq}}{dz} B_{1} & I \\ C & 0 & 0 & 0 \end{matrix} ]|_{z_{0}} ,  (5.35)

and the Jacobi linearisation of (5.33) about z_{0}, w_{eq}(z_{0}) and u_{eq}(z_{0})

G_{LT I} = [ \begin{matrix}−A_{12} \frac{dw_{eq}}{dz} − B_{1} \frac{du_{eq}}{dz} & A_{12} & B_{1} & 0 \\ −A_{22} \frac{dw_{eq}}{dz} − B_{2} \frac{du_{eq}}{dz} & A_{22} & B_{2} & 0 \\ 0 & 0 & 0 & I \\ C & 0 & 0 & 0 \end{matrix} ]|_{z_{0}} ,  (5.36)

have the same input/output properties, i.e. poles, zeros and singular values. This is true because

G_{LP V} = [\begin{matrix}T^{−1} & 0 \\ 0 & I \end{matrix}] G_{LT I} [\begin{matrix}T & 0 \\ 0 & I \end{matrix}] ,

where

T = [\begin{matrix}I & 0 & 0 \\ \frac{dw_{eq}}{dz} & I & 0 \\ \frac{du_{eq}}{dz} & 0 & I \end{matrix}]|_{z_{0}} .  (5.37)

The above also holds if states w with zero equilibrium values (w_{eq}(z) = 0) are used as feedback variables. This property of the “quasi-LPV” model (5.34) is used to link the stability margin of the LPV closed loop system to the stability margin of a family of LTI closed loop systems. This property will also lead to a natural choice of parameterdependent weights for the “quasi-LPV” model. As the frozen point stability margin is an upper bound of the LPV stability margin, at each z_{0} the weights are chosen to maximise the frozen point stability margin. As the “quasi-LPV” model evaluated at z_{0} has the same input/output properties with a Jacobi linearisation about z_{0}, w_{eq}(z_{0}) and u_{eq}(z_{0}), the weights at each z_{0} are chosen as in the LTI case. Interpolating the “gains” of the LTI weights, assuming a similar structure, gives rise to parameter-dependent weights.

Any plant model that is described by (5.29), can be written as a “quasi-LPV” model without having to make any approximations, e.g. to linearise. Thus the “quasi-LPV” model is valid during transient maneuvers. The “quasi-LPV” model makes it easier for the designer to work out which states capture the plant’s nonlinearities. This will be demonstrated in the next section. Systems of practical interest do not always fall into the class of systems considered. Hence the designer must make assumptions, based on engineering intuition, that turn the nonlinear differential equations into the desired form. A very common problem is that the dynamics are not linear in the plant input.

5.5 A “quasi-LPV” model of the longitudinal aircraft dynamics

In this section, the aerodynamic model of the High Incidence Research Model (HIRM) will be used to help develop a “quasi-LPV” model of the longitudinal dynamics of an aircraft. The HIRM is described in some detail in [Mui97].

The differential equations that describe the behaviour of a rigid body aircraft are well known, e.g. see [ER96]. As the lift, drag and pitching moment coefficients of the HIRM are nonlinear functions of angle of attack (α), the equations of motion (EoM) will be written in wind axes; this way α can be a scheduling variable. Assuming wings level flight, the EoM of the longitudinal dynamics of a rigid body aircraft written in wind axes are:

m V̇ = T cos α − D − mg sin γ,
α = θ − γ,
m V γ̇ = L + T sin α − mg cos γ,
I_{y} q̇ = M + z_{atp} T,
ḣ = V sin γ,  (5.38)

where L, D, M are the aerodynamic lift, drag and pitching moment, T is the thrust (the thrust line is parallel to the body x-axis), V, γ, q, h are the forward speed, flight path angle, pitch rate and altitude, m, I_{y}, g are the mass of the aircraft, pitch moment of inertia and acceleration due to gravity, θ is the Euler pitch angle and z_{atp} is the vertical distance of the engines from the centre of gravity (CoG). As wings level flight is assumed, q = θ̇. The lift, drag and pitching moment can be expressed as

L = q̄S C_{L},
D = q̄S C_{D},
M = q̄S c̄C_{M},  (5.39)

where C_{L}, C_{D}, C_{M} are the lift, drag and pitching moment coefficients, q̄ = \frac{1}{2} ρ V^{2} is the dynamic pressure, S is the reference surface area of the wing, c̄ is the mean aerodynamic chord and ρ is the air density which varies as a function of h. The lift coefficient is the sum of the following components:

C_{L} = C_{L_{δ_{T S}}} (α, δ_{T S}) + C_{L_{δ_{CS}}} (α, δ_{T S}) δ_{CS} + C_{L_{q}}(α, δ_{CS}) \frac{q c̄}{2 V} ,  (5.40)

where δ_{T S} is the symmetrical tailplane deflection and δ_{CS} is the symmetrical canard deflection. The aerodynamic coefficients C_{L_{δ_{T S}}}, C_{L_{δ_{CS}}}, C_{L_{q}} have been determined experimentally, and their values are stored in LUTs. Similarly, the drag and pitching moment coefficients depend on C_{D_{δ_{T S}}}, C_{D_{δ_{CS}}}, C_{D_{q}} and C_{M_{δ_{T S}}}, C_{M_{δ_{CS}}}, C_{M_{q}} respectively.

The structure of the HIRM’s aerodynamic model^{8} is representative of any other fixed wing aircraft. There are however two differences. Firstly, the canard efficiency (the lift canard efficiency for example is denoted by C_{L_{δ_{CS}}}) is a function of symmetrical tailplane deflection, which is not the case in aerodynamic models of other aircraft. If the canard efficiency does not depend on δ_{T S}, then the “quasi-LPV” model will predict the transient behaviour of the aircraft more accurately; this fact will become evident in the sequel. Secondly and most importantly, the aerodynamic coefficients of the HIRM do not depend on Mach number. As most fighter aircraft can break the sound barrier, compressibility effects must be taken into account. It will be shown how to include variations due to Mach number in the “quasi-LPV” model. Hence it is claimed, that the “quasi-LPV” model that will be developed is valid for any other aircraft.

The objective is to derive a “quasi-LPV” model that captures the aircraft’s dynamics as accurately as possible, and its outputs are V, q, θ, γ and h so that any flight mode can be designed for. The EoM (5.38) can be rewritten using (5.39) and (5.40) as

[V̇ ; α̇ ; γ̇ ; q̇ ; ḣ] = [\begin{matrix}−\frac{q̄S}{m} C_{D_{δ_{T S}}} (α, δ_{T S}) − g sin γ \\ −\frac{q̄S}{m V} C_{L_{δ_{T S}}} (α, δ_{T S}) + \frac{g cos γ}{V} \\ \frac{q̄S}{m V} C_{L_{δ_{T S}}} (α, δ_{T S}) − \frac{g cos γ}{V} \\ \frac{q̄S c̄}{I_{y}}C_{M_{δ_{T S}}} (α, δ_{T S}) \\ 0 \end{matrix}] +

+ [\begin{matrix}0 & 0 & 0 & −\frac{q̄S}{m} C_{D_{q}}(α, δ_{CS})\frac{c̄}{2 V} & 0 \\ 0 & 0 & 0 & 1 − \frac{q̄S}{m V} C_{L_{q}}(α, δ_{CS})\frac{c̄}{2 V} & 0 \\ 0 & 0 & 0 & \frac{q̄S}{m V} C_{L_{q}}(α, δ_{CS})\frac{c̄}{2 V} & 0 \\ 0 & 0 & 0 & \frac{q̄S c̄}{I_{y}}C_{M_{q}}(α, δ_{CS})\frac{c̄}{2 V} & 0 \\ sin γ & 0 & 0 & 0 & 0 \end{matrix}] [\begin{matrix}V \\ α \\ γ \\ q \\ h \end{matrix}] + [\begin{matrix}−\frac{q̄S}{m} C_{D_{δ_{CS}}} (α, δ_{T S}) & \frac{cos α}{m} \\ −\frac{q̄S}{m V} C_{L_{δ_{CS}}} (α, δ_{T S}) & −\frac{sin α}{m V} \\ \frac{q̄S}{m V} C_{L_{δ_{CS}}} (α, δ_{T S}) & \frac{sin α}{m V} \\ \frac{q̄S c̄}{I_{y}}C_{M_{δ_{CS}}} (α, δ_{T S}) & \frac{z_{atp}}{I_{y}} \\ 0 & 0 \end{matrix}] [\begin{matrix}δ_{CS} \\ T \end{matrix}]

(5.41)

These differential equations are not in the desired form, see Equation (5.29). As (5.41) is not linear in δ_{T S}, the EoM are linearised about a tailplane position δ_{T S_{eq}}, e.g. in the case of the aerodynamic coefficient C_{D_{δ_{T S}}} (α, δ_{T S})

C_{D_{δ_{T S}}} (α, δ_{T S}) ≃ C_{D_{δ_{T S}}} (α, δ_{T S_{eq}}) + DC_{D_{δ_{T S}}} |_{eq}(δ_{T S} − δ_{T S_{eq}}),

where D(·) = \frac{∂(·)}{∂δ_{T S}}. A plot of how the density of the standard atmosphere varies with altitude is illustrated in Fig. 5.6. The standard atmosphere is defined in [Mui97]. Thus ρ

8The structure of the aerodynamic model is described by (5.40).

Figure 5.6 Plot of density against altitude

varies approximately linearly with altitude, i.e.

ρ ≃ b h + a.

The smaller the largest altitude of flight, the better this linear approximation will be. In the case of q̄C_{D_{δ_{T S}}} (α, δ_{T S})

\begin{matrix}q̄C_{D_{δ_{T S}}} (α, δ_{T S}) ≃ q̄DC_{D_{δ_{T S}}} |_{eq} (δ_{T S} − δ_{T S_{eq}}) + q̄_{eq} C_{D_{δ_{T S}}} (α, δ_{T S_{eq}}) + \\ +\frac{1}{2} V^{2} C_{D_{δ_{T S}}} (α, δ_{T S_{eq}}) b (h − h_{eq}), \end{matrix}

where h_{eq} is some altitude, q̄_{eq} = \frac{1}{2}ρ_{eq}V^{2} and ρ_{eq} = b h_{eq} + a. The scheduling variables are chosen to be z = [V, α]^{T}, and hence w = [γ, q, h]^{T}; the justification for this choice of scheduling variables will become evident later on in this section. As (5.41) is not linear in γ, the EoM are linearised about a flight path angle γ_{eq}

sin γ ≃ sin γ_{eq} + cos γ_{eq} (γ − γ_{eq}), cos γ ≃ cos γ_{eq} − sin γ_{eq} (γ − γ_{eq}).

The EoM are now in the desired form. By setting the left-hand side of (5.41) equal to zero and assuming that δ_{CS_{eq}} = 0, it is clear that q_{eq} = 0 and

0 = [\begin{matrix}−\frac{q̄_{eq} S}{m} C_{D_{δ_{T S}}} (α, δ_{T S_{eq}}) − g sin γ_{eq} \\ \frac{q̄_{eq} S}{m V} C_{L_{δ_{T S}}} (α, δ_{T S_{eq}}) − \frac{g cos γ_{eq}}{V} \\ \frac{q̄_{eq} S c̄}{I_{y}}C_{M_{δ_{T S}}} (α, δ_{T S_{eq}}) \\ 0 \end{matrix}] + [\begin{matrix}0 \\ 0 \\ 0 \\ sin γ_{eq} \end{matrix}] V + [\frac{cos α}{m} ; \frac{sin α}{m V} ; \frac{z_{atp}}{I_{y}} ; 0] T_{eq}.  (5.42)

It is not necessary to choose δ_{CS_{eq}} = 0, this choice is made to keep the notation simple. Any other trim map that results in n_{u} + n_{w} = n (this equation was introduced in Section 5.4), is also feasible. Note that the number of controlled outputs must be greater or equal to n_{u}. As each input is weighted with an integrator (this is done to eliminate the feedback loops, see Section 5.4), if the number of controlled outputs is smaller than n_{u}, then the resulting “quasi-LPV” model will not be detectable. Thus, it will not be possible to solve the synthesis problem.

From the last row of (5.42) it can be deduced that γ_{eq} = 0, and consequently

sin γ ≃ γ, cos γ ≃ 1.

Hence the EoM become

[V̇ ; α̇ ; γ̇ ; q̇ ; ḣ] = [\begin{matrix}0 & 0 & −g & −\frac{q̄S}{m} C_{D_{q}}(α, δ_{CS})\frac{c̄}{2 V} & −\frac{1}{2}V^{2} \frac{S}{m}C_{D_{δ_{T S}}} (α, δ_{T S_{eq}})b \\ 0 & 0 & 0 & 1 − \frac{q̄S}{m V} C_{L_{q}}(α, δ_{CS})\frac{c̄}{2 V} & −\frac{1}{2}V^{2} \frac{S}{m V} C_{L_{δ_{T S}}} (α, δ_{T S_{eq}})b \\ 0 & 0 & 0 & \frac{q̄S}{m V} C_{L_{q}}(α, δ_{CS})\frac{c̄}{2 V} & \frac{1}{2}V^{2} \frac{S}{m V} C_{L_{δ_{T S}}} (α, δ_{T S_{eq}})b \\ 0 & 0 & 0 & \frac{q̄S c̄}{I_{y}}C_{M_{q}}(α, δ_{CS})\frac{c̄}{2 V} & \frac{1}{2}V^{2} \frac{S c̄}{I_{y}}C_{M_{δ_{T S}}} (α, δ_{T S_{eq}})b \\ 0 & 0 & V & 0 & 0 \end{matrix}] [\begin{matrix}V \\ α \\ γ \\ q \\ h − h_{eq} \end{matrix}] +

+ [\begin{matrix}\frac{q̄S}{m} DC_{D_{δ_{T S}}} (α, δ_{T S})|_{δ_{T Seq}} & −\frac{q̄S}{m} C_{D_{δ_{CS}}} (α, δ_{T S}) & \frac{cos α}{m} \\ −\frac{q̄S}{m V} DC_{L_{δ_{T S}}} (α, δ_{T S})|_{δ_{T Seq}} & −\frac{q̄S}{m V} C_{L_{δ_{CS}}} (α, δ_{T S}) & −\frac{sin α}{m V} \\ \frac{q̄S}{m V} DC_{L_{δ_{T S}}} (α, δ_{T S})|_{δ_{T Seq}} & \frac{q̄S}{m V} C_{L_{δ_{CS}}} (α, δ_{T S}) & \frac{sin α}{m V} \\ \frac{q̄S c̄}{I_{y}}DC_{M_{δ_{T S}}} (α, δ_{T S})|_{δ_{T Seq}} & \frac{q̄S c̄}{I_{y}}C_{M_{δ_{CS}}} (α, δ_{T S}) & \frac{z_{atp}}{I_{y}} \\ 0 & 0 & 0 \end{matrix}] [\begin{matrix}δ_{T S} − δ_{T S_{eq}} \\ δ_{CS} \\ T − T_{eq} \end{matrix} ] .

(5.43)

Before pre-compensating (5.43) with integrators, the designer must decide on how to blend the canard with the tailplane. The blending scheme must satisfy the constraint δ_{CS_{eq}} = 0. Once this has been done, a “quasi-LPV” model of the longitudinal dynamics of an aircraft can be derived using the procedure outlined in the previous section. If for example the canard is not used at all, then the “quasi-LPV” model is given by (5.34) with z = [V, α]^{T}, w = [γ, q, h]^{T}, u = [δ_{T S}, T]^{T},

u = ∫ v, A_{12} = [\begin{matrix}−g & −\frac{q̄S}{m} C_{D_{q}}(α, δ_{CS})\frac{c̄}{2 V} & −\frac{1}{2}V^{2} \frac{S}{m}C_{D_{δ_{T S}}} (α, δ_{T S_{eq}})b \\ 0 & 1 − \frac{q̄S}{m V} C_{L_{q}}(α, δ_{CS})\frac{c̄}{2 V} & −\frac{1}{2}V^{2} \frac{S}{m V} C_{L_{δ_{T S}}} (α, δ_{T S_{eq}})b \end{matrix}] ,

and

B_{1} = [\begin{matrix}\frac{q̄S}{m} DC_{D_{δ_{T S}}} (α, δ_{T S})|_{δ_{T Seq}} & \frac{cos α}{m} \\ −\frac{q̄S}{m V} DC_{L_{δ_{T S}}} (α, δ_{T S})|_{δ_{T Seq}} & −\frac{sin α}{m V} \end{matrix}] .

Note that the “quasi-LPV” model only depends on V, α.

The state-space data of the “quasi-LPV” model depends on the functions h_{eq}, δ_{T S_{eq}}, T_{eq} and their derivatives with respect to the scheduling variables V, α. Before solving (5.42) to obtain h_{eq}, δ_{T S_{eq}}, T_{eq}, the designer must fit 2-D functions to the look-up tables of C_{L_{δ_{T S}}} (α, δ_{T S}), C_{D_{δ_{T S}}} (α, δ_{T S}), C_{M_{δ_{T S}}} (α, δ_{T S}). There exist several fitting procedures that the designer can use; the most common choice being a least-squares fit. The fitting procedure used is based on the following idea.

Consider a 1-D look-up table with data points (x_{k}, y_{k}), where k = 0, …, n + 1. Given an ϵ > 0, the function that interpolates the data on the interval x ∈ [x_{i−1} + ϵ, x_{i+1} − ϵ] is chosen to be

y = f_{i}(x) = \{ \begin{matrix}x ∈ [x_{i−1} + ϵ, x_{i} − ϵ], & y = \frac{y_{i}−y_{i−1}}{x_{i}−x_{i−1}} (x − x_{i−1}) + y_{i−1} \\ x ∈ [x_{i} − ϵ, x_{i} + ϵ], & y = a_{i} x^{2} + b_{i} x + c_{i} \\ x ∈ [x_{i} + ϵ, x_{i+1} − ϵ], & y = \frac{y_{i+1}−y_{i}}{x_{i+1}−x_{i}} (x − x_{i}) + y_{i} \end{matrix} ,  (5.44)

where the coefficients a_{i}, b_{i}, c_{i} are given by

[\begin{matrix}(x_{i} − ϵ)^{2} & x_{i} − ϵ & 1 \\ (x_{i} + ϵ)^{2} & x_{i} + ϵ & 1 \\ 2(x_{i} − ϵ) & 1 & 0 \end{matrix}] [\begin{matrix}a_{i} \\ b_{i} \\ c_{i} \end{matrix}] = [f_{i}(x_{i} − ϵ) ; f_{i}(x_{i} + ϵ) ; \frac{y_{i}−y_{i−1}}{x_{i}−x_{i−1}}] .

Fig. 5.7 illustrates three successive data points of the 1-D look-up table, and the interpolating function f_{i}. Note that the function f_{i} does not pass through (x_{i}, y_{i}); f_{i}(x_{i}) − y_{i} can be made arbitrarily small by choosing a suitable value of ϵ. This fitting procedure can easily be extended to two dimensions.

y  x x_{i+1} x_{i} x_{i−1} x_{i} − ϵ  x_{i} + ϵ y_{i+1} y_{i} y_{i−1}

Figure 5.7 The interpolating function

This fitting procedure is motivated by the fact that the designer has control over the derivative of f_{i}. If this fitting procedure is used to interpolate the LUTs of all the aerodynamic coefficients, then the “quasi-LPV” model evaluated at some operating point (frozen “quasi-LPV” model) will have the same input/output properties as the Jacobi linearisation of the nonlinear model (5.41) about the same operating point pre-compensated with integrators^{9}. This would not be the case if the LUTs were interpolated using a standard least-squares fitting procedure, e.g. the MATLAB command polyfit. Note that even though the state-space data of the “quasi-LPV” model of the longitudinal dynamics of an aircraft does not only depend on z = [V, α]^{T}, because the dynamics are linearised about δ_{T S_{eq}}, q_{eq} = δ_{CS_{eq}} = 0, and for the specified flight envelope ρ = b h+a, the above statement holds.

The nonlinear equations (5.42) can be solved at each operating point using the MAT-LAB command fsolve, see [Gra92]. Unfortunately, there is no guarantee that a solution exists at every operating point, or that the solution varies smoothly with operating point. The existence of differentiable functions h_{eq}, δ_{T S_{eq}}, T_{eq} is only guaranteed between data points^{10}. For the HIRM and a flight envelope defined by V ∈ [70, 200] m/s and α ∈ [5^{∘}, 40^{∘}], it turns out that δ_{T S_{eq}} ∈ [−9.5^{∘}, −3.5^{∘}]. As the LUTs of C_{L_{δ_{T S}}} (α, δ_{T S}), C_{D_{δ_{T S}}} (α, δ_{T S}), C_{M_{δ_{T S}}} (α, δ_{T S}) have data points at δ_{T S} = −40^{∘}, −20^{∘}, −10^{∘}, 0^{∘}, 10^{∘}, Equation (5.42) is linear in δ_{T S_{eq}} and thus differentiable functions h_{eq}, δ_{T S_{eq}}, T_{eq} always exist. It is easy to show that h_{eq}(V, α), δ_{T S_{eq}}(α), T_{eq}(α), i.e. only h_{eq} is a function of V.

The state vector of the “quasi-LPV” model is [V, α, γ, q, h − h_{eq}]^{T}. If h − h_{eq} is used as a feedback variable to design an altitude hold, then the zeros and singular values of the “quasi-LPV” model evaluated at some operating point will not be the same with the zeros and singular values of the Jacobi linearisation of (5.41) about the same operating point pre-compensated with integrators. Hence, it will not be possible to draw any conclusions about the robustness of the LTI closed loops. Furthermore, the accuracy of the hold function will depend on the uncertainty associated with h_{eq}. Thus, it may be difficult to design a reliable altitude hold.

To synthesise the gain scheduled controller K_{ρ}, the “quasi-LPV” model has to be evaluated at each operating point of the flight envelope. This will become evident in the next section. The differences between the nonlinear model and the model used for synthesis are the following:

1. The “quasi-LPV” model is not valid for maneuvers during which δ_{T S} ≫ δ_{T S_{eq}}(V, α), e.g. a fast pitch up. This is because the aerodynamic coefficients C_{L_{δ_{T S}}} (α, δ_{T S}), C_{D_{δ_{T S}}} (α, δ_{T S}), C_{M_{δ_{T S}}} (α, δ_{T S}) which are nonlinear functions of δ_{T S}, have been linearised about δ_{T S_{eq}}(V, α).

2. The efficiency of the canard is computed at δ_{T S_{eq}}, and the tailplane and canard contributions at h_{eq}, e.g. the canard contribution to the forward speed EoM is

9Note that the Jacobi linearisation referred to, results from linearising the nonlinear plant model with the MATLAB commands trim and linmod. At operating points around the data points the input/output properties are arbitrarily close depending on ϵ.

10In reality for an aircraft with a conventional flight envelope, given a sensible V, α it is always possible to find a h, δ_{T S}, T so that the aircraft is in trim with γ = 0.

−\frac{q̄S}{m} C_{D_{δ_{CS}}} (α, δ_{T S}) δ_{CS} ≃ −\frac{q̄_{eq} S}{m} C_{D_{δ_{CS}}} (α, δ_{T S_{eq}}) δ_{CS}.

3. The model is only valid for small γ.

4. Density is assumed to be a linear function of height, ρ = b h + a.

5. The pitch rate aerodynamic coefficients are evaluated at δ_{cs_{eq}}. In addition, the pitch rate contribution is computed at h_{eq}. For example, the pitch rate contribution to the pitch EoM is

\begin{matrix}\frac{q̄S}{m} C_{D_{q}}(α, δ_{CS})\frac{c̄}{2 V} q ≃ \frac{q̄_{eq} S}{m} C_{D_{q}}(α, δ_{CS_{eq}})\frac{c̄}{2 V} q + \\ +\frac{1}{2} V^{2} \frac{S}{m} C_{D_{q}}(α, δ_{CS_{eq}}) b \frac{c̄}{2 V} q_{eq} (h − h_{eq}). \end{matrix}

The second term on the right-hand side is ignored during synthesis as q_{eq} = 0.

The gain scheduled controller K_{ρ} has been designed using the “quasi-LPV” plant model, but must obviously work well with the nonlinear plant. By adding some nonlinear elements to the controller K_{ρ}, it is possible to make the nonlinear plant behave more like the “quasi-LPV” plant model. Consequently, the performance of the real closed loop will be closer to the performance of the designed closed loop. The differences between the nonlinear and “quasi-LPV” plant models have been listed above. In what follows engineering intuition will be used to show how K_{ρ} can be modified, so that the effect of each difference is minimised. The differences that cannot be taken into account will be taken care by the robust stability and performance of the controller^{11}.

Consider the aerodynamic coefficient C_{M_{δ_{T S}}} (α, δ_{T S}). Assume that there is data for this coefficient at tailplane positions δ_{T S_{i−1}}, δ_{T S_{i}} and δ_{T S_{i+1}}. At some point during flight δ_{T S_{eq}} ∈ (δ_{T S_{i−1}}, δ_{T S_{i}}) and δ_{T S} ∈ (δ_{T S_{i}}, δ_{T S_{i+1}}), thus

C_{M_{δ_{T S}}} (α, δ_{T S}) = C_{M_{δ_{T S}}} (α, δ_{T S_{eq}}) + DC_{M_{δ_{T S}}} (α, δ_{T S})|_{δ_{T Seq}} (δ_{T S_{i}} − δ_{T S_{eq}}) +
+ DC_{M_{δ_{T S}}} (α, δ_{T S})|_{δ_{T S_{int}}} (δ_{T S} − δ_{T S_{i}})
= C_{M_{δ_{T S}}} (α, δ_{T S_{eq}}) + DC_{M_{δ_{T S}}} (α, δ_{T S})|_{δ_{T Seq}} g_{δ_{T S}} (δ_{T S} − δ_{T S_{eq}}),

where δ_{T S_{int}} ∈ (δ_{T S_{i}}, δ_{T S_{i+1}}) and

g_{δ_{T S}} = \frac{δ_{T S_{i}} − δ_{T S_{eq}}}{δ_{T S} − δ_{T S_{eq}}} + \frac{DC_{M_{δ_{T S}}} (α, δ_{T S})|_{δ_{T S_{int}}}}{DC_{M_{δ_{T S}}} (α, δ_{T S})|_{δ_{T Seq}}} \frac{δ_{T S} − δ_{T S_{i}}}{δ_{T S} − δ_{T S_{eq}}} .

11A μ-synthesis controller could be designed as the uncertainty associated with each state-space entry can be accurately quantified.

Thus if the tailplane position is known, the tailplane can be pre-compensated with the nonlinear gain 1/g_{δ_{T S}}. This gain will only correct the contribution of the tailplane to the pitch EoM. This EoM was chosen, as the tailplane primarily effects pitch. The same can be done for the canard efficiency, the nonlinear gain being

g_{δ_{CS}} = \frac{C_{M_{δ_{CS}}} (α, δ_{T S_{int}})}{C_{M_{δ_{CS}}} (α, δ_{T S_{eq}})}.

As mentioned at the beginning of this section, for most aircraft the canard efficiency does not depend on δ_{T S} and hence this modification is not required.

Remark 5.6 It is clear that if the tailplane position does not vary too quickly, then the above nonlinear gains are not necessary. A fast closed loop could still be achieved by allocating the high frequency demand to the canard.

The tailplane and canard are pre-compensated by ρ_{eq}/ρ. This is standard practice in the aerospace industry and is extremely effective. Thus, even though K_{ρ} depends on (V, α), the implemented controller also depends on ρ.

[figure: NL HIRM; f(·); [\begin{matrix}V \\ α \\ γ \\ ρ \\ δ_{T S} \end{matrix} ]; [δ̂_{CS} ; T̂]; [\begin{matrix}δ_{CS} \\ T \end{matrix}]]

Figure 5.8 Inner loop compensation

To account for large variations of flight path angle an inner loop is proposed, see Fig. 5.8. The input vector [δ̂_{CS}, T̂]^{T} is calculated by solving the minimisation

min_{δ̂_{CS}, T̂} ∥W (B + A [δ̂_{CS} ; T̂])∥_{2} ,  (5.45)

where W ∈ ℝ^{3×3},

B = [\begin{matrix}−mg (sin γ − γ) \\ −mg (cos γ − 1) \\ 0 \end{matrix}] , A = [\begin{matrix}−q̄S C_{D_{δ_{CS}}} (α, δ_{T S}) & cos α \\ q̄S C_{L_{δ_{CS}}} (α, δ_{T S}) & sin α \\ q̄S c̄C_{M_{δ_{CS}}} (α, δ_{T S}) & z_{atp} \end{matrix}] .

The solution to this minimisation is

[δ̂_{CS} ; T̂] = −(A^{T} W^{T} W A)^{−1}A^{T} W^{T} W B.

The weight W is used by the designer to control the error associated which each EoM. If vectored thrust is used in addition to δ_{T S} and T, then the cost of the corresponding minimisation can be made zero. Around wings level flight where the approximations of sin γ and cos γ are accurate, the inner loop has no effect. Unfortunately, the equation ḣ = V sin γ cannot be corrected for large flight path angles^{12}.

The assumption that density varies linearly with altitude is a very realistic one; a plot of density against altitude is illustrated in Fig. 5.6.

The different pitch rate contributions cannot be accounted for directly. If the closed loop dynamics of q are not too fast and the canard position does not change too rapidly, then the error will be smaller. Note that in some cases it may be beneficial to compute the pitch rate contribution at an h chosen by the designer, because h_{eq} can take unrealistic values.

The aerodynamic coefficients of a supersonic aircraft depend on Mach number (M) and angle of attack. Mach number is defined as M = \frac{V}{a}, where a is the speed of sound. For h ∈ [0, 11000] m, a ∈ [295, 340] m/s. The simplest way of dealing with such aircraft is to assume a constant speed of sound and rewrite the LUTs of the aerodynamic coefficients in terms of V and α. More work has to be done in this direction.

A trim map f(V, α) can be easily included in the “quasi-LPV” model, see Fig. 5.9. The function f(V, α) can be chosen in a number of different ways, e.g. such as to minimise the aerodynamic drag, D = C_{D_{δ_{T S}}} (α, δ_{T S}) + C_{D_{δ_{CS}}} (α, δ_{T S}) δ_{CS}, in trim. Note that the presence of the trim map will change Equation (5.42).

[figure: nonlinear logic; ω_{f}; s; \frac{1}{s+w_{f}}; f(V, α); k; δ_{T S}; δ_{CS}; −]

Figure 5.9 Blending, control allocation and trim map

The tailplane and the canard can be blended in a number of different ways. The scheme illustrated in Fig. 5.9 is based on a complementary filter, i.e. it is a frequency based blending scheme. An alternative strategy would be to drive both surfaces in tandem.

12The approximation sin γ ≃ γ is accurate within 11% for flight path angles up to 45^{∘}.

Both strategies and the considerations that must be taken into account before choosing a blending strategy are detailed in [PGH97].

The gain k is chosen so that daisy chaining can be implemented. Daisy chaining was introduced in [EBHS94]. The idea is to pass any excess demand from a rate limiting and/or saturating actuator to another actuator. The “nonlinear logic” block in Fig. 5.9 is a rate limiter in series with a saturation block. The gain k is chosen to be

k = \frac{DC_{M_{δ_{T S}}} (α, δ_{T S})|_{eq}}{C_{M_{δ_{CS}}} (α, δ_{T S_{eq}})} .

When passing any excess demand from the tailplane to the canard, it is implicitly assumed that the effect of using the canard instead of the tailplane on the forward speed and angle of attack EoM is not significant. This is because the tailplane and canard primarily affect the pitching motion of the aircraft. If vectored thrust is available, then more sophisticated daisy chaining schemes can be implemented, e.g. the effect of using the canard instead of the tailplane on the angle of attack EoM can be eliminated.

A number of unsuccessful attempts were made at trying to produce a “quasi-LPV” model that approximates the longitudinal dynamics of an aircraft accurately, before arriving at the solution presented above. What follows are the lessons learned from the various attempts:

1. Fitting polynomial functions that are nonlinear in α and linear in δ_{T S} and δ_{CS} to the LUTs of the aerodynamic coefficients, does not result in a “quasi-LPV” model that approximates the nonlinear plant model accurately^{13}. In one example the frozen “quasi-LPV” model had a RHP pole which its corresponding Jacobi linearisation did not have. This motivated the fitting procedure that was presented previously.

2. A “quasi-LPV” model was derived using the first 4 EoM in (5.38), 3 scheduling variables (V, α, γ), 3 inputs (δ_{CS}, T_{x}, T_{z}) and consequently 3 integrators and 3 outputs (V, q + k_{θ} θ, γ). Note that k_{θ} ∈ ℝ, T_{x} is the thrust along the aircraft’s body x-axis, and T_{z} is the thrust along the body z-axis^{14}. The position of the tailplane at trim (δ_{T S_{eq}}) was set with an inner loop^{15}. Thus, differentiable functions δ_{CS_{eq}}, T_{x_{eq}}, T_{z_{eq}} are guaranteed to exist as the EoM are linear in δ_{CS}, T_{x}, T_{z}. Designing a pilot friendly command pre-filter that does not saturate the aircraft’s inputs could be difficult, e.g. as γ is a feedback variable, a pitch rate demand corresponds to a change of angle of attack; how to vary γ_{dem} during a pitch up so that the aircraft climbs is not straightforward. Density variations are not taken into account and controller synthesis with three scheduling variables is not feasible from a computational viewpoint.

13By accurate approximation it is meant, that the “quasi-LPV” model evaluated at some operating point and the Jacobi linearisation of the nonlinear model about the same operating point pre-compensated with integrators, have similar input/output properties.

14Most modern fighters are equipped with vectored thrust.

15The tailplane can also be used for closed loop control.

3. To reduce the computation during controller synthesis, the number of scheduling variables was reduced to 2. Thus a “quasi-LPV” model was derived using the first 4 EoM in (5.38), 2 scheduling variables (V, α), 2 inputs (δ_{CS}, T) and consequently 2 integrators and 2 outputs (V, q + k_{θ} θ). The position of the tailplane at trim was set with an inner loop, so that γ_{eq} = 0 and hence θ can be used as a feedback variable. The inner loop was found not to be robust to modelling errors. Again density variations are not accounted for.

4. An attempt was then made to eliminate the inner loop. Thus a “quasi-LPV” model was derived using the first 4 EoM in (5.38), 2 scheduling variables (V, α), 2 inputs (δ_{T S}, T) and consequently 2 integrators and 2 outputs (V, q + k_{θ} θ). The position of the canard at trim (δ_{CS_{eq}}) was set to zero with the use of a complementary filter. To put the EoM in “quasi-LPV” form it was necessary to linearise the EoM about γ_{eq}. It was found that γ_{eq} takes very large values in some parts of the flight envelope; thus the “quasi-LPV” model is not valid as for most of the flight γ ≃ 0. Furthermore, density variations are not taken into account, the equilibrium functions do not exist at all operating points, and θ cannot be used as a feedback variable.

5. By including the height EoM density variations can be accounted for, and γ_{eq} = 0 so θ can be used as a feedback variable.

To design a “classical” gain scheduled controller, engineers use a family of Jacobi linearisations about some operating points (V, α, h), with h chosen in an ad-hoc manner. The “quasi-LPV” model developed enables the designer to choose h systematically. The “quasi-LPV” model presented would approximate the longitudinal dynamics of a civil aircraft extremely well. This is because the longitudinal dynamics of a civil aircraft are linear in δ_{T S} and γ is never large, see [LBLM97].

5.6 Synthesis of a robust gain scheduled controller

There are two different methods for designing a robust gain scheduled controller. The gain scheduled controller will be designed using the parameter-dependent Lyapunov function approach. To design a controller using the LFT approach, the designer must produce an LFT model of the “quasi-LPV” model. To do this, each state-space entry of the “quasi-LPV” model is approximated with a polynomial function of V, α. As the longitudinal dynamics of a modern fighter aircraft are nonlinear in V, α, the order of each polynomial function could be high^{16}. Hence, the matrix Δ associated with the LFT plant model could have many repeated entries. Consequently, the LFT approach is likely to result in a closed loop system with a poor stability margin for the reason given in Subsection 5.3.1. In the case of the HIRM, the “quasi-LPV” model has 7 states, 2 inputs and 2 outputs. Assume that the polynomial functions are of order n_{V} in V and n_{α} in α. It is always possible to produce an LFT model with a Δ of size n_{Δ} = 9 (n_{V} + n_{α}). The LFT realisation of a 2-D power series is given in [Bra96, Appendix E]. So if for example n_{V} = 3 and n_{α} = 4, then n_{Δ} = 63 with V appearing 27 times in Δ and α 36 times. Reducing the size of Δ by either taking into account the structure of the “quasi-LPV” model, e.g. the C matrix is constant and D = 0, or by using LFT reduction techniques has not been investigated.

Remark 5.7 The Lyapunov function approach can handle nonlinear plants, but the designer must grid to obtain a gain scheduled controller. With the LFT approach the designer approximates the nonlinear plant and then solves a finite-dimensional problem to obtain the controller. The price paid for a finite-dimensional problem, is a potentially poor stability margin.

The H_{∞} loop shaping setup is used, see Fig. 5.1. The synthesis of a controller K_{ρ} that achieves a stability margin γ^{−1}, boils down to solving the LMIs (5.17). The synthesised controller K_{ρ} guarantees robust stability with respect to nonlinear, possibly time-varying, infinite-dimensional perturbations with an incremental gain smaller than or equal to 1/γ. It has been shown that the guaranteed stability margin of any LTI closed loop system that is formed from a Jacobi linearisation of the nonlinear plant model about any operating point V, α, is at least γ^{−1}. The maximum singular value of any frozen closed loop transfer matrix can be bounded from above. This can be proved using techniques employed in [ZDG96, Chapter 18].

As the dynamics of a modern fighter are significantly different from one operating point to another, parameter-dependent weights are selected. At each operating point the weights are designed using the guidelines given in Chapter 2. When choosing the weights, the designer must remember that the LTI stability margin is an upper bound of the LPV stability margin. The parameter-dependent weights arise by interpolating the “gains” of the LTI weights assuming that all LTI weights have the same structure.

The LPV controller has a structure similar to the LTI H_{∞} loop shaping controller, see Fig. 5.10. Note that Â = A − γ^{−2}ZH, where Z, H are defined in Section 5.2. The transfer

16The dynamics are very nonlinear in α because a modern fighter has post-stall capabilities.

[figure: ∫; Â(z); C(z); B(z); L(z); F (z); −; y; u; G; y_{0}; f(z); r; r̂]

Figure 5.10 Controller structure

matrix from r̂ to y is

[ẋ_{cl} ; y] = [ \begin{matrix}A(z) + B(z)F (z) & B(z)F (z) & B(z) \\ Â(z) − A(z) & Â(z) + H(z)C(z) & 0 \\ C(z) & 0 & 0 \end{matrix} ] [x_{cl} ; r̂] .  (5.46)

As ρ̇ → 0 and at each operating point Y_{11}^{−1} tends to the solution of the Riccati equation

Y_{11}^{−1}(z)A(z) + A(z)Y_{11}^{−1}(z) − Y_{11}^{−1}(z)B(z)B^{T} (z)Y_{11}^{−1}(z) + C^{T} (z)C(z) = 0,

then

T_{r̂→y} → N(z) = [ \begin{matrix}A(z) + B(z)F (z) & B(z) \\ C(z) & 0 \end{matrix} ] ,

where {N(z), M(z)} is a normalised rcf of G(z). This is the reason for choosing the setup illustrated in Fig. 5.1 and not the setup discussed in [SG90]. For the design described in Section 6.5, it was found that T_{r̂→y} is close to N. More work is required to establish how ρ̇ and Y_{11}^{−1} affect T_{r̂→y}.

It is straightforward to implement K_{ρ} in the forward path. However, such an implementation could lead to poor time response characteristics. If the controller is implemented in the observer form (see Fig. 5.10), then a command pre-filter f(z) is required to ensure that at ω = 0, T_{r→y} = I. The command pre-filter is a function of the scheduling variables and can be calculated using (5.46). To avoid an extra feedback loop, the command prefilter is scheduled on commanded z and not measured z. Unfortunately, z is not always a controlled output. In such cases, an idea would be to schedule the command pre-filter on a signal that results from passing the measured z through a low-pass filter. More work needs to be done in this direction.

What follows are a number of points related to the solution time of the feasibility problem (5.17). It was found that the solution time reduced by scaling the states of the “quasi-LPV” model. This was done by randomly choosing an operating point and using the μ-tools command syscl to calculate a diagonal similarity transformation matrix. This transformation matrix was then used at all other operating points. The use of a parameterdependent state transformation matrix was not investigated. As discussed in Subsection 5.2.2, the time it takes to solve the LMIs depends on the basis functions of the trial functions X_{11}, Y_{11}, i.e. the number of decision variables. Choosing the set of cubic splines as the basis of the trial functions can lead to computationally intensive problems. The number of LMI constraints increases as the number of scheduling variables increases. It was found that in order to obtain a solution in any reasonable amount of time, the number of scheduling variables should not be greater than two. As the LMIs are not affine in γ, the best value of γ is calculated using a search, i.e. a γ-iteration. An alternative strategy that reduces the amount of computation involved, but leads to a sub-optimal value of γ, is proposed in [Woo95]. Good results are reported.

5.7 A design example

A gain scheduled flight controller is designed for the longitudinal dynamics of the HIRM. The HIRM model and design specifications^{17} used, are the same as those used for the GARTEUR design challenge, see [Mui97]. The flight controller is designed using H_{∞} loop shaping, and the structure of the weighted plant is depicted in Fig. 5.11.

[figure: q; V; α; γ; k_{θ}; NL HIRM; P+I; P+I; HPF; LPF; k; q + k_{θ}θ; δ_{T S}; δ_{CS}; T]

Figure 5.11 The structure of the weighted plant

The inputs are δ_{T S}, δ_{CS}, T, and the outputs are q+k_{θ} θ, V, where k_{θ} ∈ ℝ. The tailplane and canard are blended using a complementary filter. The complementary filter, which is implemented using the LPF and HPF blocks, feeds the high frequency portion of the demand to the canard, and the low frequency portion of the demand to the tailplane. The constant k ∈ ℝ is added to enable daisy chaining; how to choose k was discussed in Section 5.5. The P+I filters form a diagonal pre-compensator (W_{1}) that is used to shape the singular values of the open loop plant. This structure was used to successfully design an LTI controller for the HIRM as part of the GARTEUR design challenge. Even though this controller was LTI, it performed reasonably well over the whole specified flight envelope. The design of the LTI controller and the reasons for choosing the structure illustrated in Fig. 5.11 are detailed in [PGH97].

17By design specifications it is meant: flight envelope, time response specifications, gain and phase margins, etc.

The “quasi-LPV” model developed in Section 5.5 will be used to design the gain scheduled controller. The outputs were scaled with the matrix gain ScO = diag(1, \frac{1}{38}) which reflects the coupling requirements. Because the HIRM’s dynamics vary from one operating point to another, W_{1} is designed to be parameter-dependent, i.e.

W_{1} = [\begin{matrix}w_{1}^{1}(V, α) \frac{s+τ_{1}}{s} \\ & w_{1}^{2}(V, α) \frac{s+τ_{2}}{s} \end{matrix} ] ,

where w_{1}^{1}, w_{1}^{2} are continuous functions of V, α, and τ_{1}, τ_{2} are chosen to be constant in this design example (τ_{1} = 2, τ_{2} = 1). No post-compensator was used. The corner frequency of the complementary filter was chosen to be ω_{f} = 2rad/s and k = −2.2, i.e. the same as in [PGH97]. The value of k_{θ} is set to 0.5. Note that k_{θ} is used to specify the performance of the attitude hold, see Section 2.9. The weighted plant can now be formed by precompensating the scaled “quasi-LPV” model, written as in Equation (5.43), with the complementary filter, the gain k, and the dynamic part of the weight W_{1}. The functions w_{1}^{1}(V, α), w_{1}^{2}(V, α) are chosen as follows:

1. At each operating point plot the gain of the (1,1) and (2,2) elements of the weighted plant against frequency, e.g. see Fig. 5.12 for V = 100m/s and α = 12^{∘} (i.e. operating point ini3005 in [Mui97]).

2. Choose the desired cross-over frequencies for the pitch and forward speed loops, denoted by q_{ω_{c}} and V_{ω_{c}} respectively, according to closed loop specifications. Note that q_{ω_{c}} and V_{ω_{c}} must be continuous functions of the scheduling variables.

3. Calculate the gains w_{1}^{1}, w_{1}^{2} so that the desired cross-over frequencies are achieved.

It was mentioned in Section 5.2, that the solution time of the feasibility problems (5.15) and (5.17) depend on the number of decision variables. To keep the number of decision variables small, the sensor and actuator models are not included in the weighted plant when solving for K_{ρ}; this is also the reason for not using a dynamic post-compensator. Thus the weighted plant has 7 states; the complementary filter state does not come into the picture as ω_{f} = τ_{1}. The desired cross-over frequencies for the pitch and forward speed loops were chosen to be constant across the flight envelope, q_{ω_{c}} = 10rad/s and V_{ω_{c}} = 3m/s. These

Figure 5.12 Plot of (1,1) and (2,2) elements of the weighted plant

are the highest the cross-over frequencies can go before the LTI robustness margins are necessarily reduced^{18}. The singular value plots of the weighted plant at all five operating points given by GARTEUR are illustrated in Fig. 5.13.

Figure 5.13 Singular value plots at all operating points given by GARTEUR

The flight envelope considered is illustrated in Fig. 5.14, and includes all operating points (V, α) that have h_{eq}(V, α) ≤ 6000m. It must be noted that this flight envelope is smaller than that specified in the GARTEUR design challenge; the reason being, that as the envelope becomes larger so does the number of LMI constraints and hence the solution time of feasibility problems (5.15) and (5.17).

18The LTI robustness margins have been evaluated with a weighted plant that includes the actuator and sensor models.

Figure 5.14 The flight envelope

The synthesis problem, for bounded and unbounded parameter rate variation, can be formulated as a convex feasibility problem with an infinite number of constraints. The approach taken was to grid the parameter space, solve the synthesis problem, and then check that the resulting solution is valid on a much denser grid.

The synthesis problem was first solved for unbounded rate of variation of the scheduling variables. The results are listed in Table 5.1; the number of decision variables being 56 and the LTI cost γ_{LT I} ≃ 1.6. All computations were carried out on a SUNW,Ultra-1, which

design range α achieved γ # of LMIs time (hr) iterations
U1  [7^{∘}, 12^{∘}]  4.44  201  5.63  5
U2 [12^{∘}, 17^{∘}]  2.57  157  7.23  8
U3 [17^{∘}, 22^{∘}]  1.97  131  1.18  7
U4 [22^{∘}, 27^{∘}]  1.81  117  1.13  9
U5 [27^{∘}, 30^{∘}]  1.85  71  1.10  8
U6  [7^{∘}, 17^{∘}]  > 10  327  -  -
U7 [12^{∘}, 22^{∘}]  > 10  261  -  -
U8 [22^{∘}, 30^{∘}]  > 10  167  -  -

Table 5.1 Unbounded rate of variation

has a 143MHz cpu and 131Mb of RAM, using the LMI-lab [GNLC95]. For this particular example, i.e. designing a longitudinal controller for the HIRM, the limiting factor is the processor power as no job was greater than 131Mb. It must be noted that the machine is networked, but most of the time was not used by anyone else.

After simulating a number of maneuvers, it was established that α̇ ∈ [−10, 10]^{∘}/s and V̇ ∈ [−10, 10]m/s^{2}. The synthesis problem was then solved for bounded parameter rate variation using trial functions X_{11}(ρ), Y_{11}(ρ) that are affine in ρ = [V, α]^{T}. The results are listed in Table 5.2; the number of decision variables being 224. For zero parameter rate variation and α ∈ [7^{∘}, 12^{∘}], γ = 1.67.

design range α achieved γ # of LMIs time (hr) iterations
B1  [7^{∘}, 12^{∘}]  2.54  297  13.54  6
B2 [12^{∘}, 17^{∘}]  1.97  231  11.32  7
B3  [7^{∘}, 17^{∘}]  > 10  484  -  -
Table 5.2 Bounded rate variation, X_{11}, Y_{11} = f(V, α)

In an effort to improve design B3, trial functions that are first order in V and second order in α were used. Note that the synthesis problem was solved for the same parameter rate variation α̇ ∈ [−10, 10]^{∘}/s and V̇ ∈ [−10, 10]m/s^{2}. The results are listed in Table 5.3; the number of decision variables now being 336. For zero parameter rate variation and α ∈ [7^{∘}, 12^{∘}], γ = 1.63. Unfortunately, it was not possible to improve on the achieved

design range α achieved γ # of LMIs time (hr) iterations
B4 [7^{∘}, 12^{∘}]  2.32  297  32.95  6
B5 [7^{∘}, 17^{∘}]  > 10  484  -  -
Table 5.3 Bounded rate variation, X_{11}, Y_{11} = f(V, α^{2})

stability margin of design B3. At this stage it is not known whether this is due to bad designing, the fact that the feasibility problem is only a sufficient condition for the existence of a gain scheduled controller, or just a limitation imposed by the available computing power, as using more sophisticated trial functions, e.g. using the set of cubic splines as a basis to construct the trial functions, was prohibited by the large amount of time it took to solve the resulting feasibility problem. What was done instead, was to gain schedule between controllers B1 and B2 as done in [Hyd95]. Thus, what is proposed as a computationally cheaper solution, is to divide the flight envelope into a number of regions, solve the feasibility problem associated with each region, and gain schedule between the resulting controllers^{19}. Theoretically speaking, the achieved stability margin of the overall gain scheduled controller can be evaluated using Definition 5.4. In practice though, this is a computationally intensive exercise. A faster processor and/or more efficient LMI solvers could improve the designer’s insight into the problem.

19The matrix functions X_{11}^{i}(ρ) and Y_{11}^{i}(ρ), for i = 1, …, n where n is the number of regions, are interpolated using a generalisation of the fitting procedure presented in Section 5.5.

The implemented closed loop system is illustrated in Fig. 5.15. In the spirit of Section

[figure: \frac{ω_{f}}{s+ω_{f}}; \frac{s}{s+ω_{f}}; \frac{s+τ_{1}}{s}; \frac{s+τ_{2}}{s}; w_{1}^{1}; w_{1}^{2} \frac{ρ_{0}}{ρ}; δ_{T S}; δ_{CS}; T; {NL \atop HIRM}; W_{1}; r; K_{ρ}; [V α V̇ α̇]; [V α ρ]; CP; 1; \frac{1}{38}; W_{2}; V; k; LUT^{−1}; q + 0.5θ]

Figure 5.15 The implemented closed loop system

5.5, the closed loop performance can be improved by adding a number of nonlinear elements to the synthesised controller K_{ρ}. The object of these elements is to make the nonlinear plant behave closer to the “quasi-LPV” model that was used for design and synthesis. In this design example only two elements have been added: 1) the weight W_{1} is made a function of density (ρ_{0} is the density of air at sea level), and 2) the thrust nonlinearity (the LUT block in Fig. 5.15 is part of the engine model which is described in [Mui97]) has been inverted out. The controller K_{ρ} has been implemented in the forward path, and the command pre-filter (CP) is

CP = ScO [\frac{1.344s+1}{(0.576s+1)(0.4s+1)} ; \frac{1}{2.5s+1}] .

The pitch rate command pre-filter has been designed to meet the drop-back requirement specified in the GARTEUR design challenge, see [PGH97, Subsection 29.3.4]. The forward speed command pre-filter has been designed to make maximum use of the engine performance for a 20m/s speed demand.

In remainder of this section a number of time simulations will be presented. The simulations detailed in Table 5.4 have been generated using a nonlinear aircraft model^{20},

20This nonlinear model has been produced by assuming that air density varies as a linear function of altitude. This assumption is valid for the flight envelope considered, 0 ≤ h ≤ 6000m.

controller V_{0} (m/s) α_{0} (deg) h_{0} (m)  demand  figure #
U1  131.4  8.7  3213 q_{dem} = ±5^{∘}/s  5.16
U1  131.4  8.7  3213 V_{dem} = 20m/s  5.17
U1  161.1  7.1  4569 q_{dem} = ±5^{∘}/s  5.18
U1  161.1  7.1  4569 V_{dem} = 20m/s  5.19
B1+B2  120  8.7  1299 q_{dem} = ±5^{∘}/s  5.20
B1+B2  91  13  379 V_{dem} = 20m/s  5.21
B2  90  15  2024 q_{dem} = ±5^{∘}/s  5.22
B2  80  17  915 V_{dem} = 20m/s  5.23

Table 5.4 Response details for simulations with no actuator and sensor models

but no actuator or sensor models. The tailplane and canard deflection limits have been included in the nonlinear aircraft model^{21}. Note that the demanded values rather than the achieved values of δ_{T S}, δ_{CS} are plotted in Figs. 5.16 to 5.23. As there are no sensors to measure V̇ and α̇, these rates have been calculated using the measured values of V and α and applying a backward difference. Note that q, V have been used as feedback variables, not q + 0.5 θ, V. The robustness properties of the closed loop are not altered significantly by feeding back q. This is because the θ portion of the signal q + 0.5 θ does not modify the loop gain at cross-over too much.

It is evident that very good responses have been achieved at all operating points.

controller V_{0} (m/s) α_{0} (deg) h_{0} (m)  demand  figure #
B1+B2  131.4  8.7  3048  q_{dem} = ±4^{∘}/s  5.24
B1+B2  131.4  8.7  3048 q_{dem} = ±3^{∘}/s twice  5.25
B1+B2  131.4  8.7  3048  V_{dem} = 20m/s  5.26
B1+B2  100.3  12  1524  q_{dem} = ±4^{∘}/s  5.27
B1+B2  100.3  12  1524 q_{dem} = ±3^{∘}/s twice  5.28
B1+B2  100.3  12  1524  V_{dem} = 20m/s  5.29
U4  67.8  21.4  304.8  q_{dem} = ±3^{∘}/s  5.30
U3  67.8  21.4  304.8  V_{dem} = 20m/s  5.31

Table 5.5 Response details for simulations with actuator dynamics

The time simulations listed in Table 5.5 have been generated using the full nonlinear model of the HIRM together with actuator dynamics. The signals q + k_{θ} θ, V are used

21The deflection limits are: δ_{T S} = +10^{∘}, −40^{∘} and δ_{CS} = +10^{∘}, −20^{∘}.

5.7 A design example 152

Figure 5.16 U1, pitch rate doublet, ini4010

5.7 A design example 153 Figure 5.17 U1, step on speed, ini4010

5.7 A design example 154

Figure 5.18 U1, pitch rate doublet, ini5015

5.7 A design example 155 Figure 5.19 U1, step on speed, ini5015 Figure 5.20 B1+B2, pitch rate doublet, V_{0} = 120 m/s and α_{0} = 8.7^{∘} Figure 5.21 B1+B2, step on speed, V_{0} = 91 m/s and α_{0} = 13^{∘}

5.7 A design example 158

Figure 5.22 B2, pitch rate doublet, V_{0} = 90 m/s and α_{0} = 15^{∘} Figure 5.23 B2, step on speed, V_{0} = 80 m/s and α_{0} = 17^{∘}

as feedback variables. The gain k_{θ} is either equal to 0.5 or 0 depending on whether the pilot wishes an attitude hold or not. For the time responses illustrated in Figs. 5.26, 5.29 and 5.31 the attitude hold is engaged^{22}. Anti-windup designed using the Hanus scheme [HKH87] has been placed on the pre-compensator W_{1} (note that W_{1} contains integrators). Any excess demand due to saturation and/or rate limiting of the canard is passed onto the tailplane. This daisy chaining scheme would perform better, if k was made a function of the scheduling variables as suggested in Section 5.5.

It is obvious from the above plots that the engines are used to their maximum capacity; the achieved thrust is rate limited in all responses. This could lead to closed loop instability. A more robust closed loop (and hence a smaller overshoot of V due to a speed demand) can be designed by reducing V_{ω_{c}} and possibly q_{ω_{c}}, and including a linearised engine model in the weighted plant used for synthesising the gain scheduled controller. No sensor models are used for the above simulations. Including the sensor dynamics degrades the closed loop performance slightly; again this can be remedied by including the linearised sensor models in the weighted plant used for synthesis. To reduce the pitch rate crosscoupling due to a forward speed demand, the designer must add more low frequency gain to the (1,1) element of W_{1}. Nevertheless, the achieved performance is very satisfactory.

It can be seen in Fig. 5.30 that the Euler pitch angle θ decreases instead of remaining approximately constant. This is due to an aerodynamic phenomenon that occurs at high angles of attack. At high α, there is asymmetric separation of the nose cone vortices of an aircraft that cause a large yawing moment. So in Fig. 5.30 the HIRM is actually yawing and dipping its nose.

This design example indicates that the proposed design procedure can systematically produce robust gain scheduled controllers that achieve good closed loop performance. This claim is also backed by the fact that one such gain scheduled controller has been successfully flight tested on a wind tunnel model; the results of the wind tunnel tests are detailed in the next chapter. The next step will be to design a gain scheduled controller for the VAAC Harrier. For this design further issues will be investigated, e.g.

1. Whether it is beneficial to implement the gain scheduled controller in the observer form.

2. The effect of noisy and time delayed signals V̇ , α̇ on closed loop stability and performance.

3. The stability margin that can be achieved by controllers that do not depend on the rate of the scheduling variables.

4. The effectiveness of the nonlinear elements of K_{ρ} described in Section 5.5.

22A sophisticated attitude hold scheme is described in [PGH97, Section 29.2.1].

5.7 A design example 161

Figure 5.24 B1+B2, pitch rate doublet, ini4010

5.7 A design example 162

Figure 5.25 B1+B2, two pitch rate doublets, ini4010

5.7 A design example 163

Figure 5.26 B1+B2, step on speed, ini4010

5.7 A design example 164

Figure 5.27 B1+B2, pitch rate doublet, ini3005

5.7 A design example 165

Figure 5.28 B1+B2, two pitch rate doublets, ini3005

5.7 A design example 166

Figure 5.29 B1+B2, step on speed, ini3005

5.7 A design example 167

Figure 5.30 U4, pitch rate doublet, ini2001

5.7 A design example 168 Figure 5.31 U3, step on speed, ini2001

Design, development and testing of a wind tunnel model

6.1 Introduction

A super-maneuverable aircraft has post-stall (PST) capabilities. PST represents the ability of an aircraft to perform controlled tactical maneuvers beyond stall up to at least 70^{∘} of incidence. The tactical advantage of super-maneuverability has been successfully demonstrated in a series of air combat simulations with guns and short range missiles and against multiple targets [Her80]. Assume for example that the pilot of a high thrust to weight ratio fighter aircraft wants to perform a U-turn in the sky and return to some point as fast as possible, at the same altitude and with the same speed. Increasing the angle of incidence to which the aircraft is permitted to pitch up to, enables the aircraft to maneuver in less air space and to complete the maneuver in a shorter amount of time (see [Her80, Fig. 8]).

Controlling aircraft performing PST maneuvers is a challenging problem. Modern fighter aircraft are longitudinally unstable. Consequently, prolonged input saturation could lead to loss of stabilisability. At high angles of attack the pitch and yaw channels are very coupled, and thus the usual assumption that the longitudinal and lateral dynamics are decoupled does not hold. Modern aircraft have a large number of sensors on board that essentially provide state feedback. How to use this extra information is an interesting question. Super-maneuverable aircraft have additional actuators such as thrust vectoring and nose suction. Control allocation schemes must be developed to make use of the extra control power. Flight envelopes are continuously expanding and no single controller performs adequately over the whole flight envelope. Hence there is a need to develop robust scheduling techniques. The aerodynamics of an aircraft operating in the post-stall regime are very unsteady. Therefore, the control law design methods employed should be robust to modelling uncertainty. Nonlinear effects, such as the random nature of the vortex separation off the forebody of an aircraft at high angles of attack, must also be included in the design.

A wind tunnel model was built to help develop a control law design method that successfully addresses the above issues. The rig was set up such that it has the control related features of a real aircraft. Once a controller has been designed it can be implemented within a small time period and at a low cost. Existing nonlinear aircraft models are not accurate if the rate of change of the state of the aircraft is large. Hence the wind tunnel model will help evaluate the true robustness of the designed controllers. It is our desire that the wind tunnel model acts as the intermediate test between computer simulations and real flight. The model will also serve as a test-bed for other control related exercises, e.g. identification and model validation.

The wind tunnel model is a scaled down version of the High Incidence Research Model (HIRM). Detailed drawings of the HIRM were provided by DERA Bedford. DERA Bedford also provided us with an aerodynamic model of the HIRM. The wind tunnel model was built according to the specifications described in Section 6.2. The arm, interior of the wind tunnel model, and instrumentation and actuation associated with the wind tunnel model were designed in collaboration with the CUED design offices. The rig was built in the CUED workshops, and all the wind tunnel tests were carried out in the Markham tunnel at CUED.

6.2 The wind tunnel model

At high incidence, the asymmetric shedding of vortices off the forebody of an aircraft causes a large yawing moment that destabilises the aircraft due to lack of sufficient actuator power. At high angles of attack the rudder losses effectiveness since it is in the wake of the fuselage. Hence aircraft performing post-stall maneuvers require additional control power [Her80]. A suitable solution would be the deflection of engine thrust since the PST maneuvers are flown at high power settings. Alternatively, nose suction can be used to control the asymmetric shedding of the vortices and hence the yawing motion of an aircraft at high angles of attack [RJE91].

The wind tunnel model is geometrically similar to the HIRM originally used to investigate the feasibility of using nose suction as an actuator at high angles of attack [RJE90, RJE91]. The original HIRM is a control configured fighter aircraft with conventional control surfaces; tailplane, rudder and canard, and new concept actuators; thrust vectoring and nose suction. It has a length of 4.335m and a semi-span of 1.3m. The wind tunnel model, shown in Fig. 6.1, is scaled down by a factor of 5. The size of the model is essentially limited by the wind tunnel working section. The longer the nose cone the larger the yawing moment generated by the nose suction.

The rig, illustrated in Fig. 6.2, is placed inside a 5\frac{1}{2}ft×4ft low speed turbulent wind

Figure 6.1 A photograph of the model HIRM

tunnel. The model aircraft has two degrees of freedom (DoF); it can yaw about the vertical rod (the yaw angle is denoted by ψ) and roll about an axis that coincides with its body x-axis when ψ = 0^{∘} (the roll angle is denoted by ϕ). The model is set at a pitch angle θ_{0} that remains fixed throughout each experiment.

Figure 6.2 A photograph of the rig

There are two actuators for each DoF. The differential tailplane and thrusters at the top of the vertical rod provide rolling power, and the nose suction and air jets at the rear of the plane provide yawing power. The required rolling moment could not be achieved by placing the roll jets inside the aircraft fuselage since the moment arm would be too small. The differential tailplane is actuated by a high speed, high torque model aircraft servo. The valves used to regulate the jets and suction have a very high bandwidth. There is feedback on all of the actuators. A potentiometer is used for the servo and flow meters for the valves. The symmetrical tailplane and canards are set before each experiment to alleviate the root bending moment of the arm.

The attitude of the aircraft (ϕ, ψ) is measured by two high resolution optical encoders. The x position (in body axes [ER96]) of the centre of gravity (CoG) of the model is regulated by a lead weight inside the fuselage. The lead weight is driven by a high torque d.c. motor. The pivot point is at 12.5% of the quarter chord. The pivot point is the point at which the vertical rod intercepts the x-axis of the model aircraft.

The arm has been designed so that it is not excited by flow induced vibrations, see [Ble77] and [Pap95, Section 3.3]. The two bearings used are low friction high work-load bearings. The vertical CoG position of the model is regulated by altering the mass of the balance weight. This provides a way of changing the roll dynamics of the model.

The rig was assumed to be two separate rigid bodies. The equations of motion (EoM) of the rig with the CoG of the model HIRM at the pivot point, Equations (6.1) and (6.2), are derived by examining each rigid body separately^{1}. This way it becomes possible to keep track of the internal forces and moments. The analysis employed to each rigid body is similar to that carried out in [ER96] when deriving the EoM of an aircraft.

N_{4} = ψ̈I_{z_{4}z_{4}} − ϕ̈ cos ψI_{z_{4}x_{4}} − (ϕ̇)^{2} cos ψ sin ψĀ,  (6.1)
L_{4} cos ψ − M_{4} sin ψ = ϕ̈B̄ + ϕ̇ ψ̇ sin 2ψĀ − ψ̈ cos ψI_{z_{4}x_{4}} + (ψ̇)^{2} sin ψI_{z_{4}x_{4}}. (6.2)

Note that Ā = I_{y_{4}y_{4}} −I_{x_{4}x_{4}}, B̄ = I_{x_{4}x_{4}} cos^{2} ψ +I_{y_{4}y_{4}} sin^{2} ψ +I_{x_{3}x_{3}(arm)}, and L_{4}, M_{4}, N_{4} are the total rolling, pitching and yawing moments in axis system 4, AS(4). All moments are a function of the state of the model and the actuator settings, e.g. L_{4} = f(α, β, p, q, r, …). The I quantities are the moments of inertia (MoI) of the aircraft model and arm. The origin of all axis systems is the pivot point which is stationary. AS(1) is earth fixed, AS(2) is earth fixed at a pitch angle of θ_{0}, AS(3) is frame fixed rotating at ϕ̇ with respect to AS(2), and AS(4) is aircraft fixed rotating at ψ̇ with respect to AS(3).

In deriving Equations (6.1) and (6.2), the friction and damping caused by the bearings and pipes that connect the valves with the jets have been ignored. It has been assumed that the mass of the arm does not produce moments of significant magnitude and hence does not contribute to L_{4}, M_{4} and N_{4}. The EoM of the rig can easily be modified to include the above effects. The aerodynamic effect of the arm has been ignored. This is because the flow over the arm is stalled. The validity of this assumption is currently being assessed by experiment. It may be necessary to add fairings over the arm to make quantifying the forces and moments it generates an easier task.

1The EoM have been derived in [Pap95, Chapter 4]. In [Hal97, Appendix A] the EoM of the rig have been derived for the case when the CoG of the model HIRM is not at the pivot point.

The angle of attack (α) and sideslip (β) of the model are given in Equations (6.3) and (6.4) as a function of the measured angles ϕ and ψ and the fixed pitch setting θ_{0}.

tan α = \frac{sin θ_{0} cos ϕ}{cos θ_{0} cos ψ + sin θ_{0} sin ϕ sin ψ} ,  (6.3)
sin β = − cos θ_{0} sin ψ + sin θ_{0} sin ϕ cos ψ.  (6.4)

Equation (6.5) gives the rates in AS(4) as a function of the measured rates and angles; note that p, q and r are the body roll, pitch and yaw rates of the model aircraft.

p = ϕ̇ cos ψ,
q = −ϕ̇ sin ψ,
r = ψ̇.  (6.5)

A full aerodynamic model has been developed for the original HIRM [RJE90, RJE91]. This data was used to create the 6 DoF aircraft model used in the GARTEUR design challenge [Mui97]. If this data is going to be used in the simulation model of our rig, the model aircraft must have dynamic similarity with the original HIRM. It can be shown that for dynamic similarity the Mach number (M) and Reynolds number (Re) must be the same in both cases assuming geometrically similar bodies. This is described by the Rayleigh equation [HC82]. It has been found, for most bodies of aerodynamic interest, that the effects of compressibility are negligible for stream Mach numbers less than 0.5 and in cases where this limit is not exceeded, Reynolds number may be used as the only criterion for dynamic similarity [HC82]. Hence as post-stall maneuvers are usually performed at Mach numbers below 0.5, M can be ignored for the purposes of dynamic similarity.

Achieving the same Re for both models is impossible as our model has been scaled down by a factor of 5 from the original and the maximum wind tunnel speed is about 30m/s. Low values of Re may permit a laminar boundary layer to extend into the adverse pressure gradient of the aerofoil. As a laminar boundary layer is much less able than a turbulent boundary layer to overcome an adverse pressure gradient, the flow will separate from the surface at a lower angle of incidence. This is a problem which always exists in model testing. Transition can be fixed artificially on the model by roughening the model surface with carborundum powder. Therefore dynamic similarity can be assumed^{2}. It is also thought that at high angles of attack, when the flow has stalled, Re may not be as important.

The nondimensional version of the yaw EoM, Equation (6.1), is given in Equation (6.6). A brief account of nondimensional analysis can be found in [ER96].

2If the wind tunnel speed is 30m/s, then the Reynolds number of the model is smaller than the Reynolds number of the original HIRM by a factor of 5.

\frac{1}{8}C_{n_{4}} = i_{z} \frac{d^{2}ψ}{dt̂^{2}} − i_{zx} \frac{d^{2}ϕ}{dt̂^{2}} cos ψ − \frac{1}{2}(\frac{dϕ}{dt̂})^{2} sin 2ψ [i_{y}(\frac{c̄}{b})^{3} − i_{x}].  (6.6)

C_{n_{4}} is the total yawing moment coefficient defined as N_{4}/\frac{1}{2}ρV^{2}Sb. The MoI I_{x_{4}x_{4}}, I_{z_{4}z_{4}} and I_{z_{4}x_{4}} have been nondimensionalised using the parameter ρSb^{3}, and ρSc̄^{3} has been used for I_{y_{4}y_{4}}. c̄, b, ρ and V are the mean aerodynamic chord, the wing span, the air density and wind tunnel speed respectively. The nondimensional time t̂is given by t̂= t/t^{∗}, where t^{∗} = b/2V. The roll EoM, Equation (6.2), can be nondimensionalised in a similar way.

As the wind tunnel model was built to help develop a control law design method that can be used to design control laws for a real aircraft, the dynamics of the wind tunnel model must be similar to the dynamics of a real aircraft. Thus both aircraft must have the same nondimensional EoM. Consequently, the following conditions must be satisfied:

1. both aircraft must be geometrically similar, i.e. c̄/b is the same for both aircraft,

2. the parameters i_{x}, i_{y}, i_{z} and i_{zx} must be the same for both aircraft, and

3. there must be dynamic similarity between both aircraft, i.e. the aerodynamic moment coefficients are the same for both aircraft.

The fact that both aircraft must have the same nondimensional EoM does not impose any restrictions on the mass of the model. This is because our model does not translate and hence only the rotational EoM come into play.

The wind tunnel model is chosen to behave like the aircraft used in the GARTEUR design challenge (GHIRM). The GHIRM has the aerodynamic properties and shape of the original HIRM and the mass distribution of an F-18 fighter aircraft. Thus the mass distribution of the wind tunnel model, with its CoG at the pivot point, is chosen to be similar with that of an F-18 fighter aircraft. With the help of a finite element model of the wind tunnel model it was possible to get close to the target moments of inertia^{3}.

It turns out that for a wind tunnel speed of 30m/s and the GHIRM flying at 100m/s (100m/s is a typical speed for a PST maneuver)

t_{GHIRM}^{∗} = 6.75 × t_{model}^{∗}.

Hence the open loop bandwidth of the wind tunnel model is about 7 times larger than that of the GHIRM. Consequently, the closed loop bandwidth of the wind tunnel model must be about 7 times larger than that of the GHIRM, i.e. approximately 14rad/s; note that the roll and yaw specifications of the GHIRM can be satisfied with a closed loop bandwidth of about 2rad/s, see [Mui97]. In real applications the closed loop bandwidth of a system is limited by the sampling frequency of the controller, the actuator bandwidth, delays in the system and the control power available. Input saturation of a plant that is open loop unstable could lead to loss of stabilisability. In our case the sampling frequency of the controller poses no limitations. The bandwidth of the valves is 100rad/s but the servo is substantially slower. Therefore, a faster servo may be needed if comparisons are to be made with the control system of a real fighter.

3The target moments of inertia were calculating assuming that the arm MoI is negligible.

The flight envelope is given in Table 6.1. Having chosen the flight envelope a decision

Wind tunnel speed: 20 to 30m/s
Angle of attack:  0^{∘} to 60^{∘}
Sideslip:  ±10^{∘}
Air density:  1.225Kg/m^{3}

Table 6.1 The flight envelope

was made about the actuator power required. This was done by calculating the control power needed to keep the model in trimmed flight at the extreme points of the design envelope; at this point the simulation model was not available. To allow for uncertainty the thruster power was chosen so that the model can maneuver throughout the whole design envelope without using the differential tailplane and nose suction.

The simulation model of the rig, which was derived using the aerodynamic model of the original HIRM, the EoM (6.1) and (6.2) and the kinematic expressions (6.3), (6.4) and (6.5), was written in C and included as an S-function in the SIMULINK block diagram of the closed loop system. The most critical assumption made in the aerodynamic model is that when the nose cone vortices separate, always at the fixed incidence of 28.5^{∘}, the aircraft yaws to starboard. In reality the direction of yaw is completely random. This may cause problems during implementation if the controller is not robust enough. The states of the S-function representing the model aircraft are roll angle (rad), roll rate (rad/s), yaw angle (rad) and yaw rate (rad/s). The inputs into the model are differential tailplane (deg), rolling moment (Nm), yawing moment (Nm) and nose suction (10^{6}m^{3}/s).

dSPACE GmbH software and hardware are used to control and monitor the model. An incremental counter was built in-house to provide the controller with the angles ϕ and ψ. This is because the optical encoders have no fixed reference.

6.3 Open loop analysis

The nonlinear simulation model has been set-up using Equations (6.1) and (6.2). The model was linearised about 30^{∘} of incidence, a tunnel speed of 30m/s and wings level flight, i.e. ϕ = ψ = 0^{∘}. This linearisation will be used to demonstrate some features of the rig. All analysis is carried out on the transfer matrix (G) from the roll and yaw thrusters to ϕ and ψ. The target MoI have been used in this analysis and the MoI of the arm has been ignored. The model was trimmed using the differential tailplane and nose suction, and the aircraft is flown around the design envelope using the thrusters.

The model is open loop unstable. The unstable mode is lightly damped and has a frequency of about 9rad/s. This imposes a lower limit on the loop bandwidth [FL85]. The transfer matrix G has no transmission zeros. The valves used to regulate the jets have been modelled as a first order transfer function in series with a first order Padé approximation of a 50ms time delay. The time delay occurs because of a length of pipe between the valves and the jets^{4}. The first order Padé approximation introduces a RHP zero at a frequency of 40rad/s. Consequently, for a robust closed loop system the loop bandwidth must be smaller than 40rad/s and greater than 9rad/s. This is compatible with the specification of a closed loop bandwidth of about 14rad/s, see Section 6.2.

Figure 6.3 Open loop singular values

As can be seen from Fig. 6.3, the minimum singular value of the plant up to about 10rad/s is \underline{σ}(G) ≈ 0.18. The transfer function from the measurement noise n to the plant input u is given by K(I − GK)^{−1}, assuming positive feedback, where K is the designed controller. For good tracking and disturbance rejection \underline{σ}(GK) ≫ 1 at low frequency. Assuming that the plant G is invertible, then the maximum singular value of T_{n→u} becomes approximately equal to 1/\underline{σ}(G) ≈ 5.6. As a result measurement noise could saturate the plant inputs; 1^{∘} of measurement noise is equivalent to 0.1Nm of moment demand. As the maximum moment that can be generated is ±1.7Nm, the requirement for high resolution optical encoders is evident. Optical encoders with 3600 pulses per revolution are used.

4The pipe lag is quantified in [Ped97, Chapter 5].

Figure 6.4 Achievable attitudes

At low frequency the plant condition number is about 45, i.e. at low frequency the plant gain varies significantly with input direction. This indicates that the achievable design envelope has the shape of an ellipse. Fig. 6.4 illustrates the attitude (ϕ, ψ) that can be reached from the trim point using only the thrusters. It is difficult to achieve similar closed loop properties at the input and output of an ill-conditioned plant. This is because the gain of GK can be very different from that of KG. The physical reason behind the high condition number is that I_{z_{4}z_{4}} is much greater than I_{x_{4}x_{4}}. The condition number decreases when the arm moment of inertia is included.

The roll and yaw channels are very coupled. This can be seen by plotting the left and right singular vectors of the plant against frequency. Coupling can also be deduced from Fig. 6.5. This figure illustrates the open loop time response of the model to a step of 0.1Nm on the yaw thrusters between 0s and 0.1s. Open loop simulations indicate that the classical aircraft modes do not carry through to the wind tunnel model.

Remark 6.1 The wind tunnel model is a challenging plant to design a good performance controller for, as it is coupled, ill-conditioned, open loop unstable and non-minimum phase.

6.4 Open loop identification

Before designing a controller for the rig, a number of parameters that appear in the EoM of the rig were determined experimentally.

Figure 6.5 Open loop time response

The moments of inertia of the model HIRM were measured using the trifilar suspension technique, and the roll moment of inertia of the arm was measured by swinging the rig in roll, see [Hal97, Chapter 2]. The target and measured MoI are listed in Table 6.2. As the

MoI  target (Kg m^{2}) measured (Kg m^{2}) ratio
I_{x_{4}x_{4}}  7.72 × 10^{−3}  2.90 × 10^{−2}  3.76
I_{y_{4}y_{4}}  5.14 × 10^{−2}  1.94 × 10^{−1}  3.77
I_{z_{4}z_{4}}  5.76 × 10^{−2}  2.12 × 10^{−1}  3.68
I_{z_{4}x_{4}}  −9.83 × 10^{−4}  3.00 × 10^{−3}  -
I_{x_{3}x_{3}(arm)}  0  1.72 × 10^{−1}  -

Table 6.2 Target and measured MoI

measured MoI are about 4 times larger than the target MoI, the open loop bandwidth of the rig will be smaller than that anticipated, see Fig. 6.3. The effect of a nonzero I_{x_{3}x_{3}(arm)} is to reduce the condition number of the transfer matrix G at low frequency^{5}.

The stiffness and damping caused by the bearings and the pipes that connect the valves with the air jets have been measured, and the EoM of the rig modified accordingly, see [Hal97, Chapter 2 and Appendix A].

It was relatively straightforward, with the apparatus available, to measure the aerodynamic coefficients of the model HIRM with respect to ϕ and ψ about wings level flight, see [Hal97, Chapter 2]. The values of the aerodynamic coefficients that have not been identified, have been taken from the aerodynamic model of the original HIRM.

5The transfer matrix G is defined in Section 6.3.

The time delay that arises because of the pipe connecting each valve and its corresponding jet was determined to be about 50ms, see [Ped97, Chapter 5]. The valve dynamics have be modelled as a first order transfer function with a corner frequency of 100rad/s, in series with a first order Padé approximation of the 50ms time delay. As the force of the air jet in steady state is not a linear function of the input voltage that drives the valve, a look-up table has been included in the controller to invert out this nonlinearity; the relationship between the input voltage and the force of the jet has been determined experimentally [Ped97, Chapter 5]. The encoder dynamics have been modelled as a first order transfer function with a corner frequency of 200rad/s.

The flow field around the rig was studied in detail; the observations are reported in [Ped97, Chapters 7, 8 and 9]. Unfortunately, it is not entirely clear how the vertical rod affects the aerodynamics of the model HIRM. There is no sign of the asymmetric shedding of vortices off the nose cone at high angles of attack, and nose suction was found not to produce any measurable moment. More work has to be done in this direction.

The above data was used to produce an LTI model (G_{id}) of the rig about 30^{∘} of incidence, a wind tunnel speed of 30m/s, and wings level flight [Hal97, Chapter 1]. The open loop singular values of G_{id} are illustrated in Fig. 3.4. The identified model, which has a real unstable pole at about 4rad/s, is easier to control than the model G studied in the previous section. By comparing the frequencies of the unstable poles of G and G_{id}, the target closed loop bandwidth is set to about 5rad/s, i.e. a rise time of about 0.35s.

6.5 Control law design

The controllers presented in this section have been designed to provide good attitude tracking. The feedback variables are the two measured angles ϕ and ψ.

Five LTI controllers were designed for the rig using the LTI model G_{id} and the following control system design techniques:

1. H_{∞} loop shaping with diagonal weights (LTI1),

2. H_{∞} loop shaping with a non-diagonal pre-compensator and a diagonal post-compensator (LTI2),

3. robust model matching in two steps (LTI3),

4. robust model matching using μ-synthesis (LTI4), and

5. robust model matching using multi-objective output-feedback control (LTI5).

The design of controllers LTI1 and LTI2 is described in Section 3.8. In this case though, the diagonal part of the pre-compensator is chosen to be

W_{diag} = 4 \frac{s + 3}{s} I_{2}, and W_{2} = \frac{500}{s + 500} \frac{s + 5}{0.02s + 1}.

The diagonal part of the pre-compensator has been modified to increase the mid frequency loop gain and thus improve the attitude tracking. Consequently, the difference between the two controllers is not as marked as in Section 3.8. The stability margin of LTI1 is γ_{RS}^{−1} = 0.28 and that of LTI2 is γ_{RS}^{−1} = 0.27. LTI1 has 20 states and LTI2 has 22 states. Both controllers are implemented in the observer form, anti-windup designed using the Hanus scheme is placed on the pre-compensator (anti-windup is used in all the presented designs), and the plant observer is driven with achieved plant inputs rather than demanded ones. The command pre-filter used is

CP = \frac{5}{0.02s + 1},

i.e. W_{2} without the derivative term and the low-pass filter. It can be deduced from Figs. 6.6 and 6.7, that the closed loop time response specification of Section 6.4 is satisfied^{6}.

Figure 6.6 Time simulation with LTI1

6The demanded plant inputs can be obtained from the time simulations presented by multiplying the rolling and yawing moments with -1.

Figure 6.7 Time simulation with LTI2

The model matching controllers have been designed as described in Section 4.9, but using the ideal reference model

T_{0} = \frac{2^{2}}{s^{2} + 2 × 0.8 × 2s + 2^{2}} \frac{1}{0.05s + 1} I_{2}.

To improve the robustness of the transfer matrix from reference demands to controlled outputs, the ideal reference model has a bandwidth smaller than the specified 5rad/s.

W_{1} = 4 \frac{s + 3}{s} \frac{s + 5}{0.02s + 1} W_{full},

where W_{full} is the non-diagonal part of the pre-compensator which has been designed in Section 3.8. No post-compensator is used. LTI3 achieves a stability margin of γ_{RS}^{−1} = 0.28 and a robust performance level (δ_{RP}) which is plotted in Fig. 6.9 as a function of ρ (γ_{RP} is defined in Section 4.2). The feedback controller is implemented in the same way as LTI1 and has 18 states. The command pre-filter has 8 states. The controller LTI4 achieves γ_{RS} = 4.28 and γ_{RP} = 5.38 for a ρ = 3. Second order D-scales are used, LTI4 is implemented as illustrated in Fig. 4.12 and has 32 states. LTI5 achieves γ_{RS} = 3.93 and γ_{RP} = 5.40 for a ρ = 3. Second order D-scales are used and LTI5 has 19 states.

Time simulations of all five controllers with the linear model G_{id} are depicted in Figs. 6.6, 6.7, 6.8, 6.10 and 6.11. The differences between the various controllers are described in Chapters 3 and 4. The dash-dot and dotted lines in Figs. 6.8, 6.10 and 6.11 are the outputs of the ideal reference model.

Figure 6.8 Time simulation with LTI3

A “quasi-LPV” model of the rig was derived using a family of Jacobi linearisations. As the efficiency of the air jets is approximately constant in the design envelope considered^{7}, it is thought that a family of Jacobi linearisations captures the transient behaviour of the rig, see Section 5.1. The Jacobi linearisations have been generated from a nonlinear model that includes the stiffness and damping effects caused by the bearings and pipes, and uses the aerodynamic model of the original HIRM and the identified MoI. Two LPV controllers were designed for the rig using the following design techniques:

1. the single Lyapunov function approach with the H_{∞} loop shaping cost function and an LTI non-diagonal pre-compensator and LTI diagonal post-compensator (LPV1), and

2. the parameter-dependent Lyapunov function approach with the H_{∞} loop shaping cost function and an LTI non-diagonal pre-compensator and LTI diagonal postcompensator (LPV2).

Both LPV controllers have been designed in a similar way to the flight controller designed for the longitudinal dynamics of the GHIRM in Section 5.7. LPV1 has been

7The design envelope under consideration is α = 30^{∘}, V = 30m/s and ϕ, ψ = ±10^{∘}.

6.5 Control law design 183

Figure 6.9 The robust performance level of LTI3 Figure 6.10 Time simulation with LTI4

Figure 6.11 Time simulation with LTI5

designed using the following weighting functions:

W_{1} = 4 \frac{s + 3}{s} W_{full}, and W_{2} = \frac{s + 5}{0.02s + 1} I_{2}.

No encoder dynamics are included in the weighted plant used for synthesis. The resulting feasibility problem has 243 LMIs and 182 decision variables and took 10.1 hours to solve using the SUNW, Ultra-1 described in Section 5.7. After 6 iterations, γ_{RS} = 3.74. Note that across the whole flight envelope γ_{LT I} ≃ 3.08. LPV1 has been implemented in the observer form (see Section 5.6) and has 18 states.

The controller LPV2 has been designed using:

W_{1} = 4 \frac{s + \frac{1}{3}}{s} W_{full}, and W_{2} = \frac{s + 5}{0.02s + 1} I_{2}.

The encoder dynamics and LPF portion of the valve dynamics are not included in the weighted plant used for synthesis. After a number of time simulations it was determined that ϕ̇, ψ̇ ∈ [−20, 20]^{∘}/s. Trial functions that are affine in ϕ, ψ were chosen. The resulting feasibility problem has 1089 LMI constraints and 528 decision variables and took 108.6 hours to solve. After 4 iterations, γ_{RS} = 2.98. Note that γ_{LT I} ≃ 2.86 across the whole design envelope. LPV2 has been implemented in the observer form and has 16 states. The signals ϕ̇, ψ̇ have been obtained by differentiating the measured values of ϕ, ψ.

The time simulations depicted in Figs. 6.12 and 6.13 have been obtained using the nonlinear model which was used to produce the “quasi-LPV” model of the rig, and actuator and sensor dynamics. The nonlinear model was trimmed using the thrusters. The trim input is not zero, because it is assumed that at high angles of attack the asymmetric shedding of vortices off the forebody of the model causes a yawing moment. It can be seen in Fig. 6.12 that there is a steady-state error. This is due to the way in which LPV1 has been implemented. To enable real time implementation (this is discussed in the next section), LPV1 has been implemented using look-up tables. The steady-state error can be reduced by increasing the data points in the LUTs. Very good performance is achieved.

Figure 6.12 Time simulation with LPV1

None of the above control laws use the nose suction. This is because it was not possible to produce any moments of significant magnitude by nose suction. The differential tailplane is only used in open loop to simulate atmospheric disturbances.

6.6 Experimental results

The first six controllers designed in the previous section were tested on the rig. Some time responses are illustrated in Figs. 6.14 to 6.29. All controllers have been implemented in the observer form with no anti-windup on the pre-compensator. Each controller was only redesigned once; the time response from references to outputs was found to improve

Figure 6.13 Time simulation with LPV2

significantly by increasing the frequency of the zero of W_{1} from 0.3 to 3rad/s. Very satisfactory performance has been achieved. A classical controller (two SISO controllers with no cross-terms) has also been designed for the rig but did not perform as well, see [Hal97, Section 3.3]. The simulations presented in the previous section predict the response of the rig very well, e.g. compare the actuator usage in Figs. 6.6 and 6.14.

The overshoot of the yaw angle in Fig. 6.16 is due to input saturation and subsequent integrator windup; compare with Fig. 6.15. A yaw maneuver uses the actuators more heavily than a roll maneuver of the same type, e.g. see Figs. 6.21 and 6.22. This is because the yaw moment of inertia is larger than the roll moment of inertia, see Table 6.2.

There is no significant difference between LTI1 and LTI2. The feature in G_{id} that motivated the use of a non-diagonal W_{1}, does not seem to be present in the real model. The model matching controllers all perform quite well; LTI4 seems to be the best in terms of time responses, see Figs. 6.21 and 6.22, and LTI5 uses the actuators most heavily, see Figs. 6.23 and 6.24. All model matching controllers use the actuators substantially more than LTI1 and LTI2. This is probably because the weighted plant used to synthesise LTI3, LTI4 and LTI5 did not contain low-pass filters for noise rejection.

Time responses with LPV1 are illustrated in Figs. 6.25 to 6.29. In Fig. 6.29 the differential tailplane is used to simulate an atmospheric disturbance; the tailplane demand being a square wave of 10^{∘} amplitude and 0.5Hz frequency. The most striking difference between the LTI2 and LPV1 is the different actuator usage; the LPV controller uses the actuators very heavily^{8}. This is probably because of two reasons: 1) the mathematical model used to design LPV1 uses the aerodynamic model of the original HIRM and not the identified aerodynamic data, and 2) the post-compensator used to design LPV2 does not contain low-pass filters for noise rejection. Because different mathematical models were used to design LTI2 and LPV1, one cannot deduce which controller performs better. One difference due to the use of different mathematical models that can easily be seen, is the reduction of the cross-coupling into roll due to a yaw demand with LPV1 compared to LTI2, see Figs. 6.18 and 6.26.

It can be seen in all time histories that at high yaw angles, there is a large periodic disturbance on yaw angle. This is probably due to vortex shedding off the forebody of the wind tunnel model. More work has to be done in this direction to see whether it is possible to reject this disturbance.

dSPACE GmbH software and hardware are used to control the wind tunnel model. The processor used runs at 40MHz (DS1003 DSP board). The LTI controllers were all implemented at 100Hz and LPV1 at 91Hz. If LPV1 was not implemented using lookup tables, the execution time would be about 19ms as opposed to about 10ms. The execution time is the time it takes to compute the output of the controller at each sample event. Running at a sampling time of 19ms resulted in an unstable closed loop system. Unfortunately, it was not possible to implement LPV2 with the existing processor. A faster processor that will enable implementation of LPV2 can help assess how sensitive the closed loop system is to inaccuracies of the rate measurements ϕ̇, ψ̇.

Finally, it must be noted that a large actuator usage drained the tank that supplied the compressed air, and as a result closed loop performance degraded, e.g. see Figure 6.24.

8LPV1 is compared with LTI2 because they have been designed using similar weights and the same cost function.

Figure 6.14 ϕ_{dem} = ±10^{∘} with LTI1

Figure 6.15 ψ_{dem} = ±5^{∘} with LTI1 Figure 6.16 ψ_{dem} = ±10^{∘} with LTI1 Figure 6.18 ψ_{dem} = ±5^{∘} with LTI2 Figure 6.20 ψ_{dem} = ±10^{∘} with LTI3 Figure 6.22 ψ_{dem} = ±10^{∘} with LTI4 Figure 6.24 ψ_{dem} = ±10^{∘} with LTI5

Figure 6.25 ϕ_{dem} = ±10^{∘} with LPV1

Figure 6.26 ψ_{dem} = ±10^{∘} with LPV1

Figure 6.27 ψ_{dem} = ±10^{∘} at ϕ = −10^{∘} with LPV1 Figure 6.28 ϕ_{dem} = 10^{∘} sin(0.5 t) at ψ = −10^{∘} with LPV1

Figure 6.29 Disturbance rejection with LPV1

7_{Conclusions}

In this thesis, a control system design procedure has been developed. This control system design procedure is particularly suited for designing flight controllers, as it can systematically deal with multi-input multi-output plants, plants with nonlinear and parameterdependent dynamics and uncertainty in the mathematical model of the plant used for the controller design, and the designer can build many of the closed loop specifications into the cost function used for the controller synthesis. The proposed control system design procedure has been tested by using it to design six controllers for a wind tunnel model, which were all flight tested successfully.

It must be noted that the control system design procedure developed in this thesis, relies heavily on the work done by other researchers [MG90, Hyd95, Vin98, LKP93, PD93, BP94, Woo95].

7.1 The main contributions

A personal view of H_{∞} loop shaping: It has been shown that an H_{∞} loop shaping controller comes with “sensible” guarantees about the resulting closed loop system. If the guarantees with which a controller comes are “sensible”, then the chances that the controller will work on the real plant are much greater.

Computing non-diagonal weights: A procedure for computing a pre- and/or postcompensator that enable the designer to achieve the desired loop shape has been developed. This procedure was motivated by the difficulty that was encountered when trying to shape the singular values of a plant with strong cross-coupling between channels with diagonal weights.

Robust model matching: The H_{∞} loop shaping setup has been modified to include an ideal reference model. This is motivated by the fact that some flight controller specifications are given with respect to the transfer matrix from references to outputs. Although this modified setup was first proposed in [LKP93], the cost function used for the controller synthesis is new. This cost function can be used to synthesise controllers which perform better than those designed in [LKP93, HHL91].

H_{∞} control with preview: If the reference trajectory is known a number of time steps ahead, it is shown how the modified H_{∞} loop shaping setup^{1} can be further modified, so that the controller uses this additional information to optimise certain closed loop properties. This problem is solved in discrete time.

A review of robust gain scheduling: Two robust gain scheduling synthesis techniques have been reviewed; their merits and shortcomings are discussed in some detail. A suggestion is made as to which technique is more suitable for designing flight controllers for future fighter aircraft.

Development of a reliable “quasi-LPV” model: A reliable “quasi-LPV” model of the longitudinal dynamics of an aircraft has been developed. This model is only a function of angle of attack, forward speed and air density. It is shown that the model used for synthesis does not depend on air density, and that trim maps and control allocation schemes can easily be included into the “quasi-LPV” model.

Design of a robust gain scheduled controller: A robust gain scheduled controller has been designed for the High Incidence Research Model used in the GARTEUR design challenge. This is essentially an extension of the work carried out in [PGH97]. To our knowledge this is the first time robust gain scheduling has been systematically applied to a challenging benchmark.

Development of a control system design procedure: The work done in this thesis, together with work previously done in the area, results in a control system design method that is particularly suited for designing flight controllers.

Design, development and testing of a wind tunnel model: A wind tunnel model has been built for active control experiments. Six flight controllers have been designed for the wind tunnel model, and all controllers were flight tested successfully. This success increases the designer’s faith in the proposed control system design procedure. To our knowledge this is the first time a controller designed using the single Lyapunov function approach has been tested on a real rig.

1By modified H_{∞} loop shaping setup it is meant, the H_{∞} loop shaping setup with an ideal reference model.

7.2 Further work

• The next generation fighter aircraft will have additional actuators. As it is not entirely clear how to use the extra actuators effectively, more work has to be done on developing robust and good performance control allocation schemes.

• The setup illustrated in Fig. 2.28 should be further investigated. The synthesised controller may turn out to have some interesting properties.

• An appealing design procedure for robust model matching is: 1) design the preand/or post-compensator, 2) design Q_{1} using the cost function (4.16), and 3) design Q_{2} by solving the following minimisation

min_{Q_{2}∈RH_{∞}} ∥[\begin{matrix}ρ_{o} d I_{p} \\ & I_{p+m} \end{matrix}] ([\begin{matrix}NQ_{1} − T_{0} & V_{0} \\ NQ_{1} & V_{0} \\ MQ_{1} & U_{0} \end{matrix}] + [\begin{matrix}0 & N \\ 0 & M \\ 0 & M \end{matrix}] Q_{2}) [\begin{matrix}ρ_{i} d^{−1} I_{p} \\ & I_{p} \end{matrix}]∥_{∞} .

The designer is effectively asking the question: “How robust is the desired nominal performance?” How to solve the above minimisation is not immediately obvious. More work has to be done in this direction.

• It is not clear how to choose the matrix A_{Q} in Section 4.6.4. Consequently, the designer cannot assess the optimality of the solution proposed in Section 4.6.3.

• A reliable “quasi-LPV” model of the whole aircraft must be developed. This will enable the design of a full axis robust gain scheduled flight controller.

• It would be beneficial to develop a gap type metric for LPV systems. Such a metric would enable the designer to quantify the difference between using a “quasi-LPV” plant model (exact model of nonlinear plant), and an LPV plant model created from a family of Jacobi linearisations or an approximation of the “quasi-LPV” plant model. Ideally, the designer would like to measure the distance in a gap sense between a nonlinear plant and the LPV plant model used for synthesis.

• The effect of noisy and time delayed measurements of ρ, ρ̇ on the robustness of the LPV closed loop should be investigated. A simple analysis test would be of great use to the designer.

• Further work has to be done on whether it is beneficial to implement an LPV controller in the observer form.

• The wind tunnel model would become a very realistic test-bed if it was made to pitch. This is not an easy design task but would be very rewarding.

• It would be of great interest to test generically different flight controllers on the rig, e.g. a controller designed using nonlinear dynamic inversion.

Our future work will include designing a flight controller for the VAAC Harrier using the control system design techniques developed in this thesis; the VAAC Harrier is described in [Hyd95].

Bibliography

[AA98] P. Apkarian and R.J. Adams. Advanced gain-scheduling techniques for uncertain systems. IEEE Transactions on Control Systems Technology, 6(1):21–32, 1998.

[AG95] P. Apkarian and P. Gahinet. A convex characterisation of gain-scheduled H_{∞} controllers. IEEE Transactions on Automatic Control, 40(5):853–864, 1995.

[AGB95] P. Apkarian, P. Gahinet, and G. Becker. Self-scheduled H_{∞} control of linear parameter-varying systems: a design example. Automatica, 31(9):1251–1261, 1995.

[BAG97] J-M. Biannic, P. Apkarian, and W.L. Garrard. Parameter varying control of a high-performance aircraft. Journal of Guidance, Control, and Dynamics, 20(2):225–231, 1997.

[BDG^{+}93] G.J. Balas, J.C. Doyle, K. Glover, A. Packard, and R. Smith. μ analysis and synthesis toolbox. The Math Works, 1993.

[Bec93] G. Becker. Quadratic Stability and Performance of Linear Parameter Dependent Systems. PhD thesis, U.C. Berkeley, 1993.

[Bec95] G. Becker. Parameter-dependent control of an under-actuated mechanical system. In Conference on Decision and Control, 1995.

[BFP^{+}97] G.J. Balas, I. Fialho, A. Packard, J. Renfrow, and C. Mullaney. On the design of LPV controllers for the F-14 aircraft lateral-directional axis during powered approach. In American Control Conference, 1997.

[BGEB94] S. Boyd, L. El Ghaoui, E. Eric, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, 1994.

[Ble77] R.D. Blevins. Flow-Induced Vibration. Van Nostrand Reihold Company, first edition, 1977.

[BO95] D. Baldelli and H. Ohta. Flutter margin augmentation synthesis using normalized coprime factors approach. Journal of Guidance, Control, and Dynamics, 18(4):802–811, 1995.

[BP94] G. Becker and A. Packard. Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback. Systems and Control Letters, 23(3):205–215, 1994.

[Bra96] M. Brandstetter. Robust Air-Fuel Ratio Control for Combustion Engines. PhD thesis, Department of Engineering, University of Cambridge, 1996.

[BWS96] S. Bennani, D.M.C. Willemsen, and C.W. Scherer. Robust LPV control with bounded parameter rates. In AIAA Guidance, Navigation, and Control Conference, 1996.

[CG89] D.J. Clements and K. Glover. Spectral factorization by hermitian pencils. Linear Algebra and its Applications, 122-124:797–846, 1989.

[DFT92] J.C. Doyle, B.A. Francis, and A.R. Tannenbaum. Feedback Control Theory. Macmillan Publishing Company, first edition, 1992.

[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 Transactions on Automatic Control, 17(AC-34):831–847, August 1989.

[EBHS94] D. Enns, D. Bugajski, R. Hendrick, and G. Stein. Dynamic inversion: an evolving methodology for flight control design. International Journal of Control, 59(1):71–91, 1994.

[ER96] B. Etkin and L.D. Reid. Dynamics of Flight. John Wiley & Sons, third edition, 1996.

[FL85] J.S. Freudenberg and D.P. Looze. Right half plane poles and zeros and design tradeoffs in feedback systems. IEEE Transactions on Automatic Control, AC-30(6):555–565, June 1985.

[FPEN94] Gene F. Franklin, J. David Powell, and Abbas Emami-Naeini. Feedback Control of Dynamic Systems. Addison-Wesley, third edition, 1994.

[Fra87] B.A. Francis. A Course in H_{∞} Control Theory. Lecture Notes in Control and Information Sciences. Springer-Verlag, first edition, 1987.

[Fre89] J. Freudenberg. Analysis and design for ill-conditioned plants - part 1. lower bounds on the structured singular value. International Journal of Control, 49(3):851–871, 1989.

[Fre90] J. Freudenberg. Plant directionality, coupling and multivariable loop-shaping. International Journal of Control, 51(2):365–390, 1990.

[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.

[GAR96] GARTEUR/TP-088-3. Robust flight control design challenge problem formulation and manual: the Research Civil Aircraft Model (RCAM), Version 2, February 1996.

[GL95] M. Green and D.J.N. Limebeer. Linear Robust Control. Prentice Hall, first edition, 1995.

[GNLC95] P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali. LMI Control Toolbox. The Math Works, 1995.

[GPGC97] D.W. Gu, I. Postlethwaite, S.J. Goh, and P. Carter. Rx: An expert environment for robust control systems design. Technical Report 97-3, Department of Engineering, Leicester University, January 1997.

[Gra92] A. Grace. Optimization Toolbox. The Math Works, 1992.

[Hal97] K.M. Halsey. High incidence flight control through H_{∞} loopshaping. Master’s thesis, Department of Engineering, University of Cambridge, 1997.

[HBS96] M. Hovd, R. Braatz, and S. Skogestad. Optimality of svd controllers. Technical report, Norwegian University of Science and Technology, April 1996.

[HC82] E.L. Houghton and N.B. Carruthers. Aerodynamics for Engineering Students. Edward Arnold, third edition, 1982.

[Hel95] A. Helmersson. Methods for Robust Gain Scheduling. PhD thesis, Department of Electrical Engineering, Linköping University, 1995.

[Her80] W.B. Herbst. Future fighter technologies. Journal of Aircraft, 17(8):561–566, August 1980.

[HHL91] D. Hoyle, R. Hyde, and D.J.N. Limebeer. An H_{∞} Approach to Two-Degree-Of-Freedom Design. Proceedings of the IEEE CDC, pages 1581–1585, December 1991.

[HJ93] R.A. Horn and C.R. Johnson. Matrix Analysis. Cambridge University Press, first paperback edition, 1993.

[HKH87] R. Hanus, M. Kinnaert, and J.L. Henrotte. Conditioning technique, a general anti-windup and bumpless transfer method. Automatica, 23(6):729–739, 1987.

[Hyd95] R.A. Hyde. H_{∞} Aerospace Control Design - A VSTOL Flight Application. Advances in Industrial Control Series. Springer-Verlag, 1995.

[Kat80] T. Kato. Perturbation Theory for Linear Operators. Springer-Verlag, second edition, 1980.

[LBLM97] P. Lambrechts, S. Bennani, G. Looye, and D. Moormann. Robust Flight Control: A Design Challenge, GARTEUR, volume 224 of Lecture Notes in Control and Information Sciences, chapter 14, pages 149–179. Springer-Verlag, 1997.

[LKP93] D.J.N. Limebeer, E.M. Kasenally, and J.D. Perkins. On the design of robust two degree of freedom controllers. Automatica, 29(1):157–168, 1993.

[LS96] L.H. Lee and M. Spillman. Control of slowly varying LPV systems: an application to flight control. In AIAA Guidance, Navigation, and Control Conference, 1996.

[LS97] L. Lee and M. Spillman. A parameter-dependent performance criterion for linear parameter-varying systems. In Conference on Decision and Control, 1997.

[MBT97] J-F. Magni, S. Bennani, and J. Terlouw, editors. Robust Flight Control: A Design Challenge. Number 224 in Lecture Notes in Control and Information Sciences. Springer-Verlag, first edition, 1997.

[Mei95] G. Meinsma. Unstable and nonproper weights in H_{∞} control. Automatica, 31(11):1655–1658, 1995.

[MG90] D.C. McFarlane and K. Glover. Robust Controller Design Using Normalized Coprime Factor Plant Descriptions. Lecture Notes in Control and Information Sciences. Springer-Verlag, first edition, 1990.

[Mui97] E. Muir. Robust Flight Control: A Design Challenge, GARTEUR, volume 224 of Lecture Notes in Control and Information Sciences, chapter 27, pages 419–443. Springer-Verlag, 1997.

[Pac94] A. Packard. Gain scheduling via linear fractional transformations. Systems and Control Letters, 22:79–92, 1994.

[Pap95] G. Papageorgiou. Application of advanced robust control techniques to aircraft performing post-stall manoeuvres, 1995. First year report; Department of Engineering, University of Cambridge.

[PD93] A. Packard and J. Doyle. The complex structured singular value. Automatica, 29(1):71–109, 1993.

[Ped97] C.B. Pedersen. System identification of HIRM aircraft model. Master’s thesis, Department of Engineering, University of Cambridge, 1997.

[PG97a] G. Papageorgiou and K. Glover. Design and development of a wind tunnel model for high angle of attack active control experiments. In European Control Conference, July 1997.

[PG97b] G. Papageorgiou and K. Glover. A systematic procedure for designing nondiagonal weights to facilitate H_{∞} loop shaping. In Conference on Decision and Control, December 1997.

[PGH97] G. Papageorgiou, K. Glover, and R.A. Hyde. Robust Flight Control: A Design Challenge, GARTEUR, volume 224 of Lecture Notes in Control and Information Sciences, chapter 29, pages 464–483. Springer-Verlag, 1997.

[PH93] P.C. Parks and V. Hahn. Stability Theory. Prentice Hall, 1993.

[RBG95] J. Reiner, G. Balas, and W.L. Garrard. Robust dynamic inversion for control of a highly maneuverable aircraft. Journal of Guidance, Control, and Dynamics, 18(1):18–24, 1995.

[RJE90] A. Jean Ross, E.B. Jefferies, and G.F. Edwards. Control of forebody vortices by suction at the nose of the RAE High Incidence Research Model. AGARD, CP-494(Paper 27), 1990.

[RJE91] A. Jean Ross, E.B. Jefferies, and G.F. Edwards. Dynamic wind tunnel tests on control of forebody vortices with suction. AGARD, CP-497(Paper 17), 1991.

[RTJ95] V. Rehbock, K.L. Teo, and L.S. Jennings. Suboptimal feedback control for a class of nonlinear systems using spline interpolation. preprint, 1995.

[SA90] J.S. Shamma and M. Athans. Analysis of gain scheduled control for nonlinear plants. IEEE Transactions on Automatic Control, 35(8):898–907, 1990.

[SA91] J.S. Shamma and M. Athans. Guaranteed properties of gain scheduled control for linear parameter-varying plants. Automatica, 27(3):559–564, 1991.

[SA92] J.S. Shamma and M. Athans. Gain scheduling: Potential hazards and possible remedies. IEEE Control Systems, 12(3):101–107, June 1992.

[SC93] J.S. Shamma and J.R. Cloutier. Gain-scheduled missile autopilot design using linear parameter varying transformations. Journal of Guidance, Control, and Dynamics, 16(2):256–263, 1993.

[SG90] J. A. Sefton and K. Glover. Pole/zero cancellations in the general H_{∞} problem with reference to a two block design. Systems and Control Letters, 14:295–306, 1990.

[SGC97] C. Scherer, P. Gahinet, and M. Chilali. Mutliobjective output-feedback control via LMI optimization. IEEE Transactions on Automatic Control, 42(7):896– 911, 1997.

[Sha91] J.S. Shamma. The necessity of the small-gain theorem for time-varying and nonlinear systems. IEEE Transactions on Automatic Control, 36(10):1138– 1147, 1991.

[Shi95] C.A. Shifrin. New JAS 39 software clears initial tests. Aviation Week & Space Technology, 12^{th} June 1995.

[SMD88] S. Skogestad, M. Morari, and J. C. Doyle. Robust control of ill-conditioned plants: High-purity distillation. IEEE Transactions on Automatic Control, 33(12):1092–1105, 1988.

[SNB97] C.W. Scherer, R.G.E. Njio, and S. Bennani. Parametrically varying flight control system design with full block scalings. In Conference on Decision and Control, 1997.

[SP96] S. Skogestad and I. Postlethwaite. Multivariable Feedback Control. John Wiley & Sons, first edition, 1996.

[Vin93] G. Vinnicombe. Measuring Robustness of Feedback Systems. PhD thesis, Department of Engineering, University of Cambridge, 1993.

[Vin98] G. Vinnicombe. Uncertainty and Feedback: H_{∞} loop-shaping and the ν-gap metric. Imperial College Press, draft edition, 1998.

[Wal96] D.J. Walker. On the Structure of a 2-Degrees-Of-Freedom Controller. International Journal of Control, vol. 63, No 6, pages 1105–1127, 1996.

[Woo95] G.D. Wood. Control of Parameter-Dependent Mechanical Systems. PhD thesis, Department of Engineering, University of Cambridge, 1995.

[Wu95] F. Wu. Control of Linear Parameter Varying Systems. PhD thesis, U.C. Berkeley, 1995.

[Zam66] G. Zames. On the input-output stability of time-varying nonlinear feedback systems - part i: Conditions derived using concepts of loop gain, conicity, and positivity. IEEE Transactions on Automatic Control, 11(2):228–238, 1966.

[ZDG96] K. Zhou, J.C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall, first edition, 1996.