혼합한정무한시간대선형2차최적제어와 모델예측제어의 미래

최진훈, 이광순

서강대학교 화학공학과

Mixed Constrained Infinite Horizon Linear Quadratic Optimal Control

and Furture of Model Predictive Control

Jinhoon Choi, Kwang Soon Lee

Department of Chemical Engineering

Sogang University

Introduction

Model predictive control (MPC), also called as receding horizon control or moving horizon control, is a technique to implement the finite horizon open loop optimal control sequences in the infinite horizon feedback setting. MPC was not an interesting control strategy in the early 60's when the open loop optimal control problem was vigorously studied, because the infinite horizon linear quadratic optimal control (IHLQOC) problem admits a nice linear feedback type optimal solution. Nevertheless, in the late 70's, the chemical engineers found that MPC has a significant advantage over the other control strategies [7]. Indeed they showed that MPC can handle effectively various constraints that are always present in any practical control problems and cannot be effectively handled by linear control strategies.  Moreover, constrained MPC has proven its usefulness in practice. Since then, the constrained MPC was the core research topic in chemical enigineering control community. As a result of these research efforts, the complete exponential stability properties of constrained model predictive regulation (MPR) are recently established [4].

The solution method for constrained infinite horizon linear quadratic optimal control (CIHLQOC) problem was unknown until Sznaier and Damborg [12] presented a technique in late 80's. Since then, there were few attenpts to the problem [2], [11]. However, these techniques are conputationally more involved than the constrained MPC. Since the major drawback of constrained MPC was computational burden, the success of these techniques in practice is questionable. Very recently, the authors [5] developed very powerful        technique to solve the CIHLQOC problem, in which the computational burden

is compatible with the constrained MPC. In this paper, we first present this new technique. Then the future directions in MPC research are identified.

Exponential Stability

Consider

[eqn: x(k+1)=Ax(k) +Bu(k),]  [eqn: x(0)=x_0]                 (1)

[eqn: y(k)=Cx(k)]

where [eqn: x(k) IN R^n], [eqn: u(k) IN R^q] and [eqn: y(k) IN R^l]. Throughout the paper, the system is assumed to be stabilizable and detectable. We also assume that the control and the state of the system need to satisfy the following inequalities:

[eqn: u^{min} <= u(k) <= u^{max},]   [eqn: Gx(k) <= g,]  [eqn: k=0,1, CDOTS],

where [eqn: G IN R^{p TIMES n}] and [eqn: g IN R^p]. Consider the following mixed constrained IHLQOC problem:

[eqn: J( { x}_{ 0} ) = { min}_{ epsilon ,u,x} SUM from { { i}=1} to inf x(i { )}^{T }Rx(i)+ SUM from { { i}=0} to inf u(i { )}^{ T} Au(i)+ { epsilon }^{T } Q epsilon]

subject to (1),

[eqn: { u}^{ min} <= u(i) <= { u}^{max } ,]  [eqn: Gx(i) <= g+ epsilon ,]   [eqn: i=0,1, CDOTS]

where R>0, S>0, P>0 are symmetric matrices, Q>0 is diagonal matrix, [eqn: G IN { R}^{p TIMES n } .]

Exponentially Stable Plants:

It is shown in [5] that on [eqn: { x}_{k } IN { R}^{n } ,]

[eqn: { lambda }_{min } (P) {LEFT | { x}_{k } RIGHT|}^{2 } <= J( { x}_{ k} )]

[eqn: <= LEFT [ { 1} over {1- {lambda }_{max}(A{)}^2 } {lambda}_{max} (R) + {lambda}_{max}(Q) p {LEFT || G RIGHT ||}^{2} {LEFT || A RIGHT ||}^2 RIGHT ] {LEFT | { x}_{k } RIGHT|}^{2 } ,]

[eqn: DELTA J({x}_{k}) <= - {lambda }_{max}(R) {LEFT | {x}_{k} RIGHT | }^{2}.]

Hence, the closed loop system with CSFIHLQOC is globally exponentially stable.

Marginal Plants:

As shown in [10], a marginal plant is semi-globally exponentially stabilizable using linear state feedback. Namely, for any bounded set W, there exists a state feedback law [eqn: u(k)=- sigma (Kx(k))] such that the closed loop system satisfies [eqn: LEFT | x(k) RIGHT | <= a LEFT | { x}_{ 0} RIGHT | { rho }^{i }] for all[eqn: { x}_{ 0} IN W]. Using this result, it is shown in [5] that for all [eqn: { x}_{k } IN { R}^{n } ,]

[eqn: { lambda }_{min } (P) {LEFT | { x}_{k } RIGHT|}^{2 } <= J( { x}_{ k} )]

[eqn: <= LEFT [ {{ 1} over {1- {rho }^{2}}} LEFT ( 1+ {LEFT || F RIGHT ||}^{2} RIGHT ) {lambda }_{max}(R)+ {lambda}_{max}(Q) p {LEFT || G RIGHT ||}^{2} {rho}^2 RIGHT ] {a}^2{LEFT| x_k RIGHT|}^{2 } ,]

[eqn: DELTA J({x}_{k}) <= - {lambda }_{max}(R) {LEFT | {x}_{k} RIGHT | }^{2}.]

Hence, the closed loop system with CSFIHLQOC is exponentially stable on

any bounded set W.

Unstable Plants:

As shown in [3], unstable plants subject to input constraints are exponentially stabilizable by linear periodic variable structure state feedback control on any bounded set W contained in the asymptotic null controllable set [eqn: { C}_{ INF } = LEFT { x_0 IN R^n | EXIST u(k) IN U, lim from { k -> inf } x(k) =0 RIGHT } .] Using this result, it is shown in [5] that for all [eqn: { x}_{k } IN { R}^{n } ,]

[eqn: { lambda }_{min } (P) {LEFT | { x}_{k } RIGHT|}^{2 } <= J( { x}_{ k} )]

[eqn: <= LEFT [ {{ 1} over {1- {rho }^{2}}} LEFT ( 1+ { max}_{l,k } {LEFT || { K}`_{ l} ^{k} RIGHT ||}^{2} RIGHT ) {lambda }_{max}(R)+ {lambda}_{max}(Q) p {LEFT || G RIGHT ||}^{2} {rho}^2 RIGHT ] {a}^2{LEFT| x_k RIGHT|}^{2 } ,]

[eqn: DELTA J({x}_{k}) <= - {lambda }_{max}(R) {LEFT | {x}_{k} RIGHT | }^{2}.]

Hence, the closed loop system with CSFIHLQOC is exponentially stable on any bounded set W whose closure is contained in [eqn: C_INF].

Feedback Implementation of CIHLQOC

Maximal output admissible set [8] is defined by

[eqn: O_inf = LEFT { x IN R^n | LEFT ( A-BF RIGHT )^k x IN Y , k >= 0 RIGHT }]

[eqn: Y= LEFT { x IN R^n | Gx <= g , u^{min} <= -Fx <= u^{max} RIGHT }] .

Theorem 1 [8]: Suppose the following assumptions hold: i) A-BF is exponentially stable, ii) (G,A-BF) is observable, iii) Y is bounded, iv) [eqn: 0 IN i n t Y] . Then there exists [eqn: t^*] such that

[eqn: O_inf = LEFT { x IN R^n | G(A-BF)^k -g <= 0, u^{min} <= -F(A-BF)^k x <= u^{max} , 0 <= k <= t^* RIGHT }].

If [eqn: x(N) IN O_INF], Problem (P) reduces to

[eqn: J_N ( { x}_{ 0} ) = { min}_{ epsilon ,u_N ,x_N} SUM from { { i}=1} to {N-1} x(i { )}^{T }Rx(i)+ SUM from { { i}=0} to {N-1} u(i { )}^{ T} Au(i)+ x(N)^T Px(N)+{ epsilon }^{T } Q epsilon]

subject to

[eqn: { u}^{ min} <= u(i) <= { u}^{max } ,]  [eqn: Gx(i) <= g+ epsilon ,]  [eqn: i=0,1, CDOTS ,N-1]

where [eqn: P=A^T LEFT ( P-PB LEFT ( B^T PB + S RIGHT )^{-1} B^T P RIGHT ) A+Q]. Hence for feedback implementation of CIHLQOC, we need to find [eqn: N^*] such that [eqn: x(N^* ) IN O_INF]. For this let B be the largest ball contained in [eqn: O_INF]. As shown in [2], the radius r of B can be found solving [eqn: r^2 =min_i LEFT { i n f_{x IN { O}`_{ inf} ^{ i}} x^T x RIGHT }] where [eqn: { O}`_{ inf} ^{i }] is the complement of the set defined by an inequality for [eqn: O_INF] in Theorem 1. Notice that the infinization problems are convex QP's. Since CIHLQOC is exponentially stable, there exist a,b,c>0 such that

[eqn: a LEFT | x_k RIGHT | ^2 <= J(x_k ) <= b LEFT | x_k RIGHT | ^2 ,]         [eqn: DELTA J(x_k ) <= -c LEFT | x_k RIGHT | ^2 .]

Then [eqn: LEFT | x(k) RIGHT | <= {b over a} LEFT | x_0 RIGHT | LEFT ( 1- {c over b} RIGHT ) ^k]. Since [eqn: 1-{c over b) IN (0,1)][eqn: 1 -{c over b} IN (0,1)], there exists [eqn: k^*] such

that [eqn: LEFT | x_0 RIGHT | LEFT ( 1-{c over b} RIGHT ) ^{k^*} < r], then [eqn: x(k^* ) IN O_INF].

Future of Model Predictive Control

Disturbances are always exist in practice. Hence, the disturbance attenuation problem is an important control problem. The unconstrained model predictive disturbance attenuation (MPDA) problem is already solved [9]. However, similar to the MPR case, unconstrained MPDA is not very interesting because there exists a number of powerful infinite horizon disturbance attenuation algorithms. Hence, the incorporation of the constraints into MPDA problem is important and will become one of the future MPC research topics.

The model uncertainty was identified as a key reason why the modern control algorithms failed in practice in the late 70s. Since then, the robustness of various control strategies were one of the hottest issues in control         research. In this context, the robustness of the constrained MPC is also an important issue in constrained MPC. Robust MPC in the presence of the structured uncertainties is already addressed [1]. However, there still exist a number of robustness issues in MPC such as robust MPC in face of the unstructured uncertainties.

An alternate strategy to deal with the structured uncertainty is the adaptation. The MPC combined with adaptation is already proposed [6]. However, the theoretical background for adaptive MPC is not well established and will be one of the core research topics in future MPC research.

For MPC design, a good state space model for the plant is imperative. However, the indentification of such models is not well establshied. Moreover, the indentification of uncertainty is indispensible for MPDA and adpative MPC. Hence, the development of simultaneous plant and disturbance identification techniques such as subspace identification [13] will be necessary for the advanced MPC techniques to be meaningful.

References

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