Stability of constrained MBPC using an internal model control structure

S.A. Heise and J.M. Maciejowski

Abstract

Using a state space formulation, Model-Based Predictive Control (MBPC) is studied in the presence of active state and input constraints. A piecewise linear time invariant control scheme is described corresponding to all possible fixed constraint conditions. Posing the MBPC system in terms of an internal model control framework, tractable stability requirements are then described for stable and unstable plants with perfect models acting under constraints.

1 Introduction

Since its inception, Model-Based Predictive Control (MBPC) has become popular with process plant operators and researchers alike due to its unique ability to systematically deal with constraints. This popularity is evidenced by the numerous branches of MBPC that have emerged in recent years, including Dynamic Matrix Control (Cutler 1983), Model Algorithmic Control (Rouhani & Mehra 1982), and Generalized Predictive Control (Clarke, Mohtadi & Tuffs 1987a), (Clarke, Mohtadi & Tuffs 1987b), (Clarke & Mohtadi 1989), (Tsang & Clarke 1988), to cite a few. All of these methods possess certain common characteristics, most notably the on-line determination of a sequence of optimal (in the sense of some objective function) control moves to minimize deviation of a set of future system outputs from their setpoints, subject to constraints and control penalties. Since only the first of these computed control moves is implemented and the entire set is redetermined at each sample time, MBPC functions over a “receding” or “moving” horizon.

One further feature intrinsic to all MBPC techniques is the incorporation of an internal model in the controller (usually to predict the future outputs based upon future input moves). This characteristic forms the basis for Internal Model Control (IMC), a controller parametrization method introduced by Garcia and Morari in the first of a series of comprehensive papers on the topic (Garcia & Morari 1982), (Garcia & Morari 1985a), (Garcia & Morari 1985b), (Rivera, Morari & Skogestad 1986), (Economou & Morari 1986a), (Economou & Morari 1986b). Although IMC does not address constraints other than actuator saturation (Garcia & Morari 1982), it possesses several valuable characteristics, including the ability to transform a system into an effectively open loop structure in which a perfect and stable plant model and a stable controller guarantee system stability. In the sequel we will seek to take advantage of this structure to examine MBPC systems operating under constraints.

To date, however, little has been established regarding the stability of MBPC systems in the presence of constraints. Notable exceptions include the Quadratic Dynamic Matrix Control work of Zafiriou and colleagues (Zafiriou 1990), (Zafiriou & Marchal 1991), (Gattu & Zafiriou 1992), and the stabilizing receding horizon control methods of Mayne and Michalska (Mayne & Michalska 1990), (Mayne & Michalska 1991). We seek to address the issue of nominal stability of MBPC systems in the presence of active input, rate and state/output constraints.

In our study a state space, rather than convolution, framework is utilized to allow consideration of unstable systems, and the piecewise linear time invariant (LTI) nature of MBPC acting under specific constraints is established for a quadratic cost function. We then take advantage of the inherent properties of IMC type systems to establish sufficient conditions for stability under possible combinations of changing constraints and switching.

2 MBPC - a state space formulation

We begin by assuming a system model is available in a general discrete state space form:

x(k + 1) = Ax(k) + B_{u}u(k) + B_{d}d(k)  (1)
y(k) = Cx(k) + w(k − 1) + ν(k)  (2)

where x(k) ∈ ^{ζ} is the system state vector, u(k) ∈ ^{ξ} is the system manipulated variable (control input), and y(k) ∈ ^{η} is an output vector of controlled variables at time k. d(k) ∈ ^{υ} is a general state disturbance input, including both measured and unmeasured variables, and w(k) ∈ ^{ρ} is an output disturbance vector, possibly representing the effects of plant-model mismatch or unmodelled dynamics. Either of these disturbance vectors may be deterministic or stochastic depending upon the application under consideration. ν(k) is an additional noise vector corrupting the measured outputs.

In order to represent the MBPC problem in terms of the change in the control vector at time k, Δu(k) = u(k) − u(k − 1), the system is expressed in the form:

[ \begin{matrix}Δx(k + 1) \\ y_{x}(k + 1) \end{matrix} ] = Â [ \begin{matrix}Δx(k) \\ y_{x}(k) \end{matrix} ] + B̂_{u}Δu(k) + B̂_{d}Δd(k) + B̂_{w}Δw(k)  (3)
y(k) = Ĉ [ \begin{matrix}Δx(k) \\ y_{x}(k) \end{matrix} ] + ν(k)  (4)

where Â = [ \begin{matrix}A & 0 \\ CA & I \end{matrix} ] , B̂_{u} = [ \begin{matrix}B_{u} \\ CB_{u} \end{matrix} ] , B̂_{d} = [ \begin{matrix}B_{d} \\ CB_{d} \end{matrix} ]  (5)
B̂_{w} = [ \begin{matrix}0 \\ I \end{matrix} ] , Ĉ = [ 0 I ]

Δd(k) = d(k) − d(k − 1) is the change in the state disturbance at time k, Δw(k) = w(k) − w(k − 1) is the change in the output disturbance, and y_{x} is a vector of outputs uncorrupted by measurement noise. The state equation (3) may also be augmented depending upon assumptions made about the disturbances (Gelormino, Lee & Morari 1990). For example, Δd(k) and Δw(k) may be modelled as linear systems driven by white noise, in which case these disturbance variables will be augmented to the state vector. We will still refer to the augmented system by the state space triple (Â, B̂_{u}, Ĉ), where B̂_{u} will address known inputs.

2.1 State prediction

Since in practice certain elements of the state vector may not be available for measurement at time k, we wish to determine a set of state estimates for [ Δx(k)^{T} y(k)^{T} ]^{T}, or this vector augmented by disturbance variables Δd(k) and Δw(k). The state estimates will be denoted as x̂(k) ∈ ^{κ}, and they will rely upon selecting a particular observer. In general, the state estimation will take the following form in terms of a particular point l on a prediction horizon p:

x̂(k + l + 1) = F (l)x̂(k + l) + GΔu(k + l)
+ H(l)y(k) (l = 0, …, p − 1)  (6)

Letting q be an optimization horizon, it is assumed that Δu(k + l) = 0 for l > q. The matrices F (l), G and H(l) depend upon (Â, B̂_{u}, Ĉ) and the specific observer chosen. From (6), and letting

Δũ(k) = [ Δu(k)^{T} … Δu(k + q)^{T} ]^{T} ,  (7)

the future state predictions may be written as

[ x̂(k + 1) ; ⋮ ; x̂(k + p) ] = Fx̂(k) + GΔũ(k) + Hy(k)  (8)

where

F = [ F (0) ; F (1)F (0) ; ⋮ ; ∏_{j=p−1}^{0} F (j) ]} p  (9)

G = \overbrace{[ \begin{matrix}G & 0 & ⋯ & 0 \\ F (1)G & G & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ [∏_{j=q−1}^{1} F (j)] G & [∏_{j=q−1}^{2} F (j)] G & ⋯ & G \\ ⋮ & ⋮ & ⋮ & ⋮ \\ [∏_{j=p−1}^{1} F (j)] G & [∏_{j=p−1}^{2} F (j)] G & ⋯ & [∏_{j=p−1}^{q+1} F (j)] G \end{matrix} ]\} p}^{q+1}  (10)

H = [ H(0) ; F (1)H(0) ; ⋮ ; [∏_{j=p−1}^{1} F (j)] H(0) ]} p  (11)

Here ∏_{j=m}^{n} F (j) denotes F (m)F (m − 1) … F (n), where m > n (Maciejowski & Heise 1993).

2.2 Constraint handling

We apply constraints on u(k) and Δu(k) across the optimization horizon, and on y(k) across the prediction horizon. These constraints are posed in a form allowing upper and lower limits on the actuator “levels,” actuator rates and state estimates (which include output constraints), as well as linear combinations of constraints:

LΔũ(k) ≤ l(k)  (12)
M [ u(k) ; ⋮ ; u(k + q) ] ≤ m(k)  (13)

N [ x̂(k + 1) ; ⋮ ; x̂(k + p) ] ≤ n(k)  (14)

where l(k) ∈ ^{λ}, m(k) ∈ ^{μ}, and n(k) ∈ ^{ν}. Here λ represents the total number of rate constraints across the control horizon. Similarly μ is the number of input constraints over q, and ν is the number of constraints on the state estimates across the prediction horizon. Noting that

u(k + l) = u(k − 1) + ∑_{j=0}^{l} Δu(k + j),  (15)

using equation (8) and performing some algebraic manipulation, these equations may be posed in vector form in terms of Δũ(k) (Maciejowski & Heise 1993):

DΔũ(k) ≤ E [ \begin{matrix}u(k − 1) \\ x̂(k) \\ y(k) \\ l(k) \\ m(k) \\ n(k) \end{matrix} ]  (16)

where

D = [ \begin{matrix}L \\ MΛ \\ NG \end{matrix} ]  (17)

for Λ a block lower triangular matrix of identity matrices I_{ξ} ∈ ^{ξ,ξ}. Also,

E = [ \begin{matrix}0 & 0 & 0 & I_{λ} & 0 & 0 \\ −MI & 0 & 0 & 0 & I_{μ} & 0 \\ 0 & −NF & −NH & 0 & 0 & I_{ν} \end{matrix} ]  (18)

where I_{λ} ∈ ^{λ,λ}, I_{μ} ∈ ^{μ,μ} and I_{ν} ∈ ^{ν,ν} are all identity matrices, and

I = [\overbrace{I_{ξ} I_{ξ} … I_{ξ}}^{q+1}]^{T}  (19)

In the standard case where each input, rate and output variable is simply bounded by a maximum and a minimum value, L, M and N are given by L_{s}, M_{s} and N_{s} and λ = μ = 2(q + 1)ξ, ν = 2pκ:

L_{s} = M_{s} = [ \begin{matrix}−I_{(q+1)ξ} \\ I_{(q+1)ξ} \end{matrix} ]  (20)
N_{s} = [ \begin{matrix}−I_{pκ} \\ I_{pκ} \end{matrix} ]  (21)

2.3 Cost function

Subject to the constraints of equation (16), the objective of the model-based control problem is to minimize the difference between the plant output and a desired reference trajectory, as well as to place a penalty upon control effort through the following performance index:

J(k) = ∑_{l=1}^{p} ∥Ĉx̂(k + l) − s(k + l)∥_{Q(l)}^{2} + ∑_{l=0}^{q} ∥Δu(k + l)∥_{R(l)}^{2}  (22)

where s(k) ∈ ^{η} is a vector of output setpoints at time k, which may include state setpoints, and Q(l) and R(l) are weighting matrices which may vary across the prediction and control horizons, respectively, but are assumed to be independent of k. Define

Q = diag [ Q(1) … Q(p) ]  (23)
R = diag [ R(0) … R(q) ]  (24)
C = diag [ Ĉ … Ĉ ]  (25)

Then, expanding equation (22) in terms of l and using equations (7) and (8), this cost function may be written as

J(k) = Δũ(k)^{T} AΔũ(k) + [ x̂(k)^{T} y(k)^{T} s̃(k)^{T} ] BΔũ(k) + K(k)  (26)

where

A = G^{T} C^{T} QCG + R  (27)
B = 2 [ CF CH −I_{pη} ]^{T} QCG  (28)
s̃(k) = [ s(k + 1)^{T} … s(k + p)^{T} ]^{T}  (29)

and K contains all terms independent of Δũ(k). The control optimization can then be described in terms of a quadratic program which is solved at each time k:

min_{Δũ(k)} Δũ(k)^{T} AΔũ(k) + [ x̂(k)^{T} y(k)^{T} s̃(k)^{T} ] BΔũ(k)  (30)

subject to DΔũ(k) ≤ E [ \begin{matrix}u(k − 1) \\ x̂(k) \\ y(k) \\ c(k) \end{matrix} ]  (31)

for

c(k) = [ l(k)^{T} m(k)^{T} n(k)^{T} ]^{T}  (32)

2.4 Piecewise LTI MBPC

It has been noted by several researchers ((Zafiriou 1990), (Maciejowski & Heise 1993)) that when a fixed set of constraints is active, the control law resulting from solving the quadratic programming problem of equations (30) and (31) is linear time invariant. This fact is evident from examining the optimal solution when the ith set of ϕ possible constraint conditions is active . Let D̄_{i} and Ē_{i} be the arrays corresponding to the active rows of equation (31). This solution is then given by

[ \begin{matrix}2A & D̄_{i}^{T} \\ D̄_{i} & 0 \end{matrix} ] [ \begin{matrix}Δũ_{i}^{∗} \\ λ_{i}^{∗} \end{matrix} ] = [ \begin{matrix}& −B^{T} [ x̂(k)^{T} y(k)^{T} s̃(k)^{T} ]^{T} \\ Ē_{i} & [ u(k − 1)^{T} x̂(k)^{T} y(k)^{T} c(k)^{T} ] & T \end{matrix} ] for i = 1, … ϕ (33)

where Δũ_{i}^{∗} is the optimal control sequence over q, and λ_{i}^{∗} is a set of Lagrange multipliers. It has been shown (Fletcher 1980) that if the solution to the optimization problem is unique and feasible, then the inverse of the matrix on the left hand side of (33) exists and is given by

[ \begin{matrix}2A & D̄_{i}^{T} \\ D̄_{i} & 0 \end{matrix} ]^{−1} = [ \begin{matrix}L_{1i} & L_{2i} \\ L_{2i}^{T} & L_{3i} \end{matrix} ]  (34)

The optimal solution Δũ_{i}^{∗} is then

Δũ_{i}^{∗} = −L_{1i}B^{T} [ x̂(k)^{T} y(k)^{T} s̃(k)^{T} ]^{T}
+ L_{2i}Ē_{i} [ u(k − 1)^{T} x̂(k)^{T} y(k)^{T} c(k)^{T} ]^{T} for i = 1, … ϕ  (35)

Specifically, if L_{2i} is partitioned as [ L_{2li} L_{2mi} L_{2ni} ] where L_{2li} ∈ ^{(q+1)ξ,λ̄_{i}}, L_{2mi} ∈ (q+1)ξ,μ̄_{i} and L_{2ni} ∈ ^{(q+1)ξ,ν̄_{i}}, and λ̄_{i}, μ̄_{i} and ν̄_{i} indicate the number of active rate, input and state constraints, respectively, it is seen that

Δũ_{i}^{∗} = [ (−2L_{1i}G^{T} C^{T} Q^{T} C − L_{2ni}N̄_{i}) F (−2L_{1i}G^{T} C^{T} Q^{T} C − L_{2ni}N̄_{i}) H ] [ \begin{matrix}x̂(k) \\ y(k) \end{matrix} ]

−L_{2mi}M̄_{i}Iu(k − 1) + 2L_{1i}G^{T} C^{T} Q^{T} s̃(k) + L_{2i}c̄_{i}(k) (36)

where c̄_{i}(k) consists of the active rows of c(k) corresponding to the ith constraint set, and similarly for M̄_{i} and N̄_{i}. Now consider only the first increment Δu(k) of Δũ(k)_{i}^{∗} for implementation. Taking the first ξ rows of equation (36), we can write a state space form for the piecewise LTI controller, where x_{u}(k) = u(k − 1) is the controller state:

x_{u}(k + 1) = [I_{ξ} − (L_{2mi}M̄_{i}I)_{ξ}] x_{u}(k) + V_{ξ} [ \begin{matrix}x̂(k) \\ y(k) \\ s̃(k) \\ c(k) \end{matrix} ]  (37)

Δu(k) = −(L_{2mi}M̄_{i}I)_{ξ}x_{u}(k) + V_{ξ} [ \begin{matrix}x̂(k) \\ y(k) \\ s̃(k) \\ c(k) \end{matrix} ]  (38)

where the subscript ξ denotes the first ξ rows and

V = [ (−2L_{1i}G^{T} C^{T} Q^{T} C − L_{2ni}N̄_{i}) F
(−2L_{1i}G^{T} C^{T} Q^{T} C − L_{2ni}N̄_{i}) H 2L_{1i}G^{T} C^{T} Q^{T} L_{2i}J_{i} ]  (39)

Also note that J_{i} ∈ ^{λ̄_{i}+μ̄_{i}+ν̄_{i},λ+μ+ν} must be a matrix consisting of rows of the identity matrix corresponding to active constraints in c(k). As expected, if no constraints are active, then the LTI controller for the unconstrained system is

Δu(k) = [ −VCF −VCH V 0 ]_{ξ} [ \begin{matrix}x̂(k) \\ y(k) \\ s̃(k) \\ c(k) \end{matrix} ]  (40)

where the subscript ξ again denotes the first ξ rows and

V = A^{−1}G^{T} C^{T} Q^{T}  (41)

3 MBPC in the IMC framework

As noted earlier, IMC is an appealing structure for control design since it transforms a feedback system into an essentially feedforward system influenced only by disturbances and modelling error. As such, we seek to restructure the MBPC problem presented in equations (30) and (31) into this feedforward format. The MBPC system is initially modelled as shown in fig. (1):

Figure 1: Model Based Predictive Control System

where G_{p} represents the system plant, E is the state estimator, and Q_{i} is the model predictive controller of equations (37) and (38) corresponding to a specific set of active constraints. s is the set of reference setpoints, c is the set of constraint “levels” of equation (32), d is the state disturbance, w is the output disturbance, y is the system output, and x̂ represents the state vector estimate. In order to take advantage of the IMC structure (Garcia & Morari 1982), we reformulate this system in terms of a predictor/optimizer acting upon the error between the plant and model output. To this end, we select a specific observer (Åström & Wittenmark 1984) to implement equation (6):

x̂(k + 1) = Âx̂(k) + B̂_{u}Δu(k) + H[y(k) − Ĉx̂(k)]  (42)

The predictor state may then formed by reconstructing the state estimate in terms of an internal plant model and a model correction:

x̂(k) = x_{m}(k) + x_{c}(k)  (43)

where the internal plant model state is given by

x_{m}(k + 1) = Âx_{m}(k) + B̂_{u}Δu(k)  (44)

and the model correction is defined as a one-step-ahead corrector:

x_{c}(k + 1) = Âx_{c}(k) + H[y(k) − Ĉx̂(k)]  (45)

The predictor is thus represented in state space form as

[ \begin{matrix}x_{m}(k + 1) \\ x_{c}(k + 1) \end{matrix} ] = [ \begin{matrix}Â & 0 \\ 0 & Â − HĈ \end{matrix} ] [ \begin{matrix}x_{m}(k) \\ x_{c}(k) \end{matrix} ] + [ \begin{matrix}B̂_{u} & 0 \\ 0 & H \end{matrix} ] [ \begin{matrix}Δu(k) \\ e(k) \end{matrix} ]  (46)
[ \begin{matrix}x̂(k) \\ y(k) \end{matrix} ] = [ \begin{matrix}I & I \\ Ĉ & 0 \end{matrix} ] [ \begin{matrix}x_{m}(k) \\ x_{c}(k) \end{matrix} ] + [ \begin{matrix}0 & 0 \\ 0 & I \end{matrix} ] [ \begin{matrix}Δu(k) \\ e(k) \end{matrix} ]  (47)

where the error e(k) = y(k) − Ĉx_{m}(k). The system of figure (1) is now realized by the IMC type structure of figure (2):

Figure 2: IMC Type Predictor-Optimizer System

In this figure, G̃ represents the internal plant model, and we define a new plant which operates on Δu(k) as

G(z) = G_{p}(z)\frac{z}{z − 1}  (48)

where z^{−1} denotes the delay operator. In addition, P is the predictor of equations (46) and (47), and r, v_{1} and v_{2} are possible external system inputs. Q_{pi} will denote the transfer matrix from e to Δu. We wish to examine the internal stability of this system, assuming a perfect model (G = G̃). This will be done initially for systems with a stable G, and then for those with unstable G. A similar system is well documented in the literature (Morari & Zafiriou 1989) wherein the predictor P is absent.

3.1 Stability of MBPC with stable plants

For the case where P is not present (Morari & Zafiriou 1989), achieving internal stability for the IMC system requires both a stable plant G and a stable controller Q_{i}. In the case of figure (2), this requirement equates to needing stability of the closed loop system from e to Δu, which may be achieved by applying the small gain theorem (Maciejowski 1989). Denote P_{1} and P_{2} as the transfer matrices from Δu and e, respectively, to [ x̂^{T} y^{T} ]^{T} such that P = [P_{1} P_{2}]. Then the block Q_{pi} shown in figure (2) is an LTI predictor-optimizer associated with the ith set of active constraints:

Q_{pi} = (I − Q_{i}P_{1})^{−1} Q_{i}P_{2}  (49)

This relation allows us to pose the following theorem for stability of the MBPC system acting under a specific set of active constraints:

Theorem 1 Assume both P and Q_{i} are stable. Then the system of figure (2) is stable under the ith constraint set if

∥Q_{i}∥_{∞} < \frac{1}{∥DP_{1}D^{−1}∥_{∞}} i = 1 … ϕ  (50)
where D = diag[ d_{1}, d_{2}, …, d_{κ+η−1} ], d_{i} > 0  (51)

Furthermore, the following theorem holds for each constraint condition and switching between constraints:

Theorem 2 Assume (Â,Q(l)^{\frac{1}{2}} Ĉ) is observable, l = 1 … p. Then if equation (50) holds for i = 1 … ϕ, the closed loop system is bounded-input bounded-output stable.

Proof Assume equation (50) is true for i = 1 … ϕ such that, from theorem 1, Q_{pi} is stabilizing, i = 1 … ϕ. Also assume that all exogenous inputs of figure (2) are bounded, and that y(k+l), l = 1 … p, is bounded while the ith constraint set is active. Express equation (22) as

J(k) = J_{x}(k) + J_{s}(k) + J_{u}(k)  (52)

where

J_{x}(k) = ∑_{l=1}^{p}[Ĉx̂(k + l)]^{T} Q(l)Ĉx̂(k + l) = ∑_{l=1}^{p} y(k + l)^{T} Q(l)y(k + l)  (53)

J_{s}(k) corresponds to the portion of the cost function containing all setpoint terms, and is assumed to be finite since the setpoint trajectory is bounded. Similarly, J_{u}(k) contains all rate input terms, and is also assumed to be finite since the actuator rates are bounded by constraints. From equation (53), it can be seen that since y(k + l) is bounded for the ith constraint condition, J_{x}(k) and thus J(k) is bounded. Now assuming, without loss of generality, that Δu(k + l) = 0, l = 0 … q, and rewriting (53) as

J_{x}(k) = x̂(k)^{T} Q̄x̂(k)  (54)
where Q̄ = [ Â^{T} Ĉ^{T} Q(1)\frac{1}{2} … Â^{p^{T}} Ĉ^{T} Q(p)\frac{1}{2} ] [ Q(1)^{\frac{1}{2}} ĈÂ ; ⋮ ; Q(p)^{\frac{1}{2}} ĈÂ^{p} ]  (55)

it is evident that, using the assumption of observability such that the matrices in equation (55) are full rank, a bounded performance index implies a finite norm bound on the state estimates while the ith constraint set is active. This norm bound is given by

∥x̂(k)∥ ≤ \frac{|J_{x}(k)^{\frac{1}{2}} |}{σ_{min} \binom{}{Q̄^{\frac{1}{2}}}}  (56)

where σ_{min} denotes the minimum singular value. Now having assumed the external inputs are bounded, i.e.

∥[ \begin{matrix}r(k) \\ v(k) \end{matrix} ]_{i}∥ < b for some b < ∞ and ν(k) = [ ν_{1}(k)^{T} ν_{2}(k)^{T} ]^{T} ,  (57)

and having assumed the output is bounded for the ith constraint set, with the inputs bounded by constraints,

∥[ \begin{matrix}y[x̂(k; x̂_{0}, k − 1), k] \\ u[u(k − 1), x̂(k; x̂_{0}, k − 1), k, s̃(k), c(k)] \end{matrix} ]_{i}∥ < c for some c < ∞,  (58)

since the norm bound on equation (56) provides a bounded initial state estimate x̂_{0} at the time of switching, these inequalities (57) and (58) will also be true for the (i + 1)th constraint set. Using the initial assumption that the system of figure (2) is bounded-input bounded-output stable, the following inequality is clearly true for i = 1:

\begin{matrix}∥[ \begin{matrix}y \\ u \end{matrix} ]_{i}∥ & ∞ \\ ∥[ \begin{matrix}r \\ v \end{matrix} ]_{i}∥ & ∞ \end{matrix} ≤ β_{i} for some β_{i} < ∞  (59)

By induction, the inequality (59) then holds for i = 1 … ϕ. Now generalizing for any fixed ith constraint condition,

∥[ \begin{matrix}y \\ u \end{matrix} ]_{i}∥_{∞} ≤ β_{i} ∥[ \begin{matrix}r \\ v \end{matrix} ]_{i}∥_{∞}  (60)

Taking the maximum of the right hand side of this equation over each i:

∥[ \begin{matrix}y \\ u \end{matrix} ]_{i}∥_{∞} ≤ β_{i} ∥[ \begin{matrix}r \\ v \end{matrix} ]∥_{∞}  (61)

But since ∥[ \begin{matrix}y \\ u \end{matrix} ]∥_{∞} = max_{i} \{∥[ \begin{matrix}y \\ u \end{matrix} ]_{i}∥_{∞}\} ,  (62)
then ∥[ \begin{matrix}y \\ u \end{matrix} ]∥_{∞} ≤ max_{i} β_{i} ∥[ \begin{matrix}r \\ v \end{matrix} ]∥_{∞}  (63)

Thus \begin{matrix}∥[ \begin{matrix}y \\ u \end{matrix} ]∥ & ∞ \\ ∥[ \begin{matrix}r \\ v \end{matrix} ]∥ & ∞ \end{matrix} ≤ max_{i} β_{i} < ∞  (64)

□

This requirement is useful in that the right hand side of equation (50) is independent of the current active constraints, and so the criterion is easily evaluated for ϕ possible constraint sets.

3.2 Stability of MBPC with consideration of unstable plants

For the case of unstable plants, the structure of figure (2) is used for design of the Q_{i} and P, and a new structure which will not require stable G is then implemented as in (Morari & Zafiriou 1989). Let

C_{pi} = Q_{pi}(I − G̃Q_{pi})^{−1}  (65)

or

Q_{pi} = C_{pi}(I + G̃C_{pi})^{−1}  (66)

where Q_{pi} is defined in equation (49). Then the desired equivalent structure is given by figure (3):

Figure 3: Equivalent Predictor-Optimizer System

where r and v are again possible external inputs. From this figure, applying the perfect model assumption and using equation (65), it is evident that for internal stability for a fixed constraint condition, each of the transfer matrices in the system

[ \begin{matrix}y \\ u \end{matrix} ] = [ \begin{matrix}GQ_{pi} & (I − GQ_{pi})G \\ Q_{pi} & −Q_{pi}G \end{matrix} ] [ \begin{matrix}r \\ v \end{matrix} ] = S_{i} [ \begin{matrix}r \\ v \end{matrix} ]  (67)

must be stable. This internal stability is guaranteed for a set S which includes each S_{i} by applying the following theorem by Morari and Zafiriou:

Theorem 3 (Morari & Zafiriou 1989) Assume that G = G̃ and that G̃ has no zeros at the location of its unstable poles. Also assume that S_{i} is stable for some Q_{pi} = Q_{p0}. Then all controllers Q_{pi} which stabilize S_{i} are given by

Q_{pi}(s) = Q_{p0}(s) + Q_{p1}(s)  (68)

for Q_{p1} a stable transfer matrix and GQ_{p1}G stable.

Noting the structure of the control law in equations (37), (38), and (40), we can now state the following lemma:

Lemma 4 Assume that G = G̃ and that G̃ has no zeros at the location of its unstable poles. Furthermore, let each LTI predictor-optimizer pair be described by

Q_{pi} = Q_{p0} + ΔQ_{pi} (i = 1 … ϕ)  (69)

where Q_{p0} corresponds to the predictor-optimizer for the unconstrained system S_{0} (no constraints active) and ΔQ_{pi}, i = 1 … ϕ, is an additional increment corresponding to a particular fixed constraint condition. Also assume that Q_{p0} stabilizes S_{0}. Then the predictoroptimizer Q_{pi} stabilizes S_{i} iff ΔQ_{pi} and GΔQ_{pi}G are stable.

This result leads to the following theorem for each possible constraint and switching condition:

Theorem 5 Assume that G = G̃ and that G̃ has no zeros at the location of its unstable poles. Also assume that Q_{p0} stabilizes S_{0} and that (Â,Q^{\frac{1}{2}} (l)Ĉ) is observable, l = 1 … p. If ΔQ_{pi} and GΔQ_{pi}G are stable for i = 1 … ϕ, then the closed loop system is bounded-input bounded-output stable.

Proof Assume ΔQ_{pi} and GΔQ_{pi}G are stable, i = 1 … ϕ. Then from lemma 2, Q_{pi} is stabilizing, i = 1 … ϕ, and the argument of theorem 2 applies.

□

Thus we have a sufficient condition for BIBO stability of the MBPC system in the presence of all possible constraint conditions, and for switching between constraints, which begins with checking stability of the system with no constraints active (surely a necessary requirement). This condition is easily evaluated, but it may be unnecessarily conservative since it is unlikely that all ϕ constraint combinations and all ϕ(ϕ − 1) switching conditions will be activated during system operation. Furthermore, checking stability for all ϕ constraint conditions may require excessive numerical computation. In that case, one could check to ensure stability under only the most likely constraint conditions and then employ a scheme wherein if other constraints are activated, the predictor-optimizer checks stability and slightly changes the tuning parameters if an instability results. Another approach may be to a priori choose tuning parameters such that certain constraint conditions are unlikely to be activated. This would involve minimizing a specific norm of the closed loop operator from the disturbance vector to the vector of constrained variables, as was suggested in (Maciejowski & Heise 1993) in a heuristic fashion.

4 Conclusions

In this paper we have sought to exploit the piecewise linear time invariant nature of MBPC given that a specific set of constraints is active, as well as to take advantage of specific features of IMC. To this end, a set of linear controllers was formed corresponding to each possible/probable constraint condition. In order to evaluate the stability of the system in each condition, the state space model predictive system was restructured into an internal model control format, wherein the system is essentially open loop for a perfect model assumption. This structure provided a framework for establishing tractable, easily evaluated sufficient conditions for guaranteeing stability under all constraint conditions and switching. To relax the perfect model assumption, extension to robustness in the presence of parameter variations is needed.

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