안정한 공정의 한정 H∞ 모델예측제어

최진훈, 유성엽, 이광순

서강대학교 화학공학과

Constrained H∞ MPC for Stable Plants

Jinhoon Choi,  Sung Yeup Yoo,  Kwang Soon Lee

Department of  Chemical Engineering, Sogang University

1. Introduction

Model predictive control(MPC) is an attractive tool for dynamic optimization and control of multivariable systems with time-varying performance requirement and constraints. MPC has recently received much attention from both theoreticians and practitioners in diverse fields.  Although different fields have adopted different names and favorite model forms, the underlying concept is the same:  Solve a finite or infinite horizon open-loop optimal control problem at every sample time.  Updating the current states using measurement before each optimization results in feedback control.  The great success of this MPC in practice was mainly driven by its ability to handle various constraints.

Although there always exist disturbances, the study of MPC has been focused on the regulation problem.  Recently unconstrained H∞ model predictive control has been addressed by several researchers, [1], [2], [6].  However, as mentioned above, MPC is especially useful when the constraints are handled.  Hence we address the constrained H∞ MPC problem in this paper.

In this paper,  we first formulate the constrained H∞ MPC for stable plants.  Then its solution strategy and stability properties are presented.

2. Preliminaries

In this section, we present the open-loop saddle point solution of unconstrained finite linear quadratic dynamic game as exposed in [4], [5].

Consider the time-invariant, discrete time, linear system

[eqn: rm{x(k+j+1|k)=Ax(k+j|k)+Bu(k+j|k)+Dd(k+j|k)}]     (1)

where [eqn: rm{k in [0,infty]}]; [eqn: rm{x(k|k) in R^n}] is the state vector; [eqn: rmu(k|k) in R^m] is the manipulated input; [eqn: rm{d(k|k) in R^p}] is the unknown disturbance.  Consider the unconstrained finite linear quadratic game problem:

[eqn: rm{J_k = rm{min} atop {u} rm{max} atop{d}{1over 2 } SUM from { { j}=1} to N }x^T (k+j|k) Q x(k+j|k) + }][eqn: rm{{1over 2 } SUM from { { j}=0} to N-1 }u^T (k+j|k) u(k+j|k)}]                    [eqn: rm{-{1over 2 } SUM from { { j}=0} to N-1 }d^T (k+j|k) d(k+j|k) }]                       (2)

subject to (1), where [eqn: rmQ&ge&0].

[eqn: rmQ&ge&0] essentially makes Jk strictly convex in u(k).  In order to formulate a meaningful problem, we also have to require Jk to be strictly concave in  d(k), since  disturbance can be make the value of  unboundedly Jk large otherwise. The following lemma now provides a necessary and sufficient condition for the above objective functional to be strictly concave in d(k).

Lemma 1. For the linear-quadratic two-person zero-sum dynamic game, introduced above the objective functional  Jk  is strictly concave in d(k) if and only if

[eqn: rm I - D^T S_k+1 D& > &0]                       (3)

where [eqn: rm S_k] is given by

[eqn: rmS_k = Q + A^T S_k+1 A + A^T S_k+1 D[I- D^T S_k+1 D]^-1 D^T S_k+1 A],

[eqn: rmS_N = Q].

If the above lemma is satisfied, the unconstrained linear-quadratic dynamic game has unique saddle point solution.  We now summarize the unique saddle point solution of he unconstrained linear-quadratic dynamic game.

Theorem 1 For the linear-quadratic zero-sum game described by (2), let Condition (3) be satisfied and [eqn: rm Lambda_k,& M_k] be appropriate dimensional matrices defined through

[eqn: rm Lambda_k = [I+(BB^T -DD^T ) M_k+1 ]]

[eqn: rm M_k+1 = Q + AM_k+1 LAMBDA^-1 A& ;& M_N = Q],

then

(i)[eqn: rm Lambda_k], thus recursively defined, is  invertible,

(ii) the game admits a unique open-loop saddle-point solution given by

[eqn: rm u^* (k) = -BM_k+1 LAMBDA_k ^-1 A x^* (k)]

[eqn: rm d^* (k) = +BM_k+1 LAMBDA_k ^-1 A x^* (k)]

where [eqn: LEFT{x^* (k+1|k)} RIGHT]is the corresponding state trajectory determined from

[eqn: rm x^* (k+1|k) = Lambda_k ^-1 A x^* (k)].

3. Input Constrained H∞ MPC

Now we consider the input constrained H∞ MPC.  The system is the same as the  above.  However the objective functional is altered and considered is the following input constrained H∞ MPC problem:

[eqn: rm {J_k = rm{min} atop {u} rm{max} atop{d}{1over 2 } SUM from { { j}=1} to infty }x^T (k+j|k) Q x(k+j|k) + }][eqn: rm {{1over 2 } SUM from { { j}=0} to infty}u^T (k+j|k) u(k+j|k) }]

[eqn: rm {-{1over 2 } SUM from { { j}=0} to infty }d^T (k+j|k) d(k+j|k) }]                    (4)

subject to

[eqn: rm u_min& le& u(k+j|k)& le& u_max , (1 le j le N-1),] [eqn: rm u(k+j|k)=0 &(j ge N)].

Suppose A is stable and time-invariant, and then

[eqn: rm{x(k+N+j|k) = A^j x (k+N|k) + SUM from { { i}=0} to j-1 A^j-i-1 Dd(k+N+i|k), }][eqn: rm{ j = 0, 1, cdots, inf}] ,

and Equation (4) becomes

[eqn: rm{ J_k =rm{min} atop {u} rm{max} atop{d}{1over 2 } SUM from { { j}=1} to N-1 }x^T (k+j|k)Q x(k+j|k) + }][eqn: rm {{1over 2 } SUM from { { j}=0} to N-1}u^T (k+j|k) u(k+j|k)}]                  [eqn: rm {-{1over 2 } SUM from { { j}=0} to N-1 }d^T (k+j|k) d(k+j|k) }]

[eqn: rm+{ max atop{d}{1over 2 } SUM from { { j}=N} to INF } [x^T (k+j|k) Q x(k+j|k) -d^T (k+j|k) d(k+j|k) }]]

and the second maximization reduces to

[eqn: rm{ max atop{d}{1over 2 } SUM from { { j}=N} to INF } [x^T (k+j|k) Q x(k+j|k) -d^T (k+j|k) d(k+j|k) }]]

[eqn: = rm-{ min atop{d}{1over 2 } SUM from { { j}=N} to INF } [-x^T (k+j|k)Q x(k+j|k)+d^T (k+j|k) d(k+j|k) }]]

[eqn: rm{= {1 over 2} x^T (k+N |k) tilde{Q} x (k+N|k)}]

where [eqn: rm tilde Q] is the solution of the above optimization problem and it is given by

[eqn: rm {tilde{Q} = S_inf - Q }]

[eqn: rm S_ inf = Q + A^T S_inf A + A^T S_inf D[I-D^T S_inf D]^-1 D^T S_inf A].

Therefore Equation (4) can be written as follows:

[eqn: rm {J_k = rm{min} atop {u} rm{max} atop{d} {1 over 2} x^T (k+N |k) tilde{Q} x (k+N|k)+ {1over 2 } SUM from { { j}=1} to N-1 }x^T (k+j|k) Q x(k+j|k) }]

[eqn: rm {+{1over 2 } SUM from { { j}=0} to N-1}[u^T (k+j|k) u(k+j|k)-d^T (k+j|k) d(k+j|k)]}]   (5)

subject to

[eqn: rm u_min& le& u(k+j|k)& le& u_max , (1 le j le N-1),& u(k+j|k)=0 &(j ge N)].

Here we can solve analytically the maximization problem in the above min-max problem.

Proposition 1. Consider the maximization problem:

[eqn: rm{ max atop {d} 1 over 2 SUM from { {j }=1} to N-1 x^T (k+j |k) Q x(k+j|k) -{1 over 2} SUM from { { j}=0} to N-1 d^T (k+j|k) d(k+j|k)}]

subject to (1).

This maximal problem admits the unique solution

[eqn: rm {d^* (k+j |k) = P_k S_k+1 A x(k+j|k) +P_k S_k+1 B u (k+j|k) - P_k v_k+1 , j in [0, N-1] }]

where

[eqn: rm P_k = [I- DS_k+1 D]^-1 D^T]

[eqn: rm S_k = Q + A S_k+1 [I - D P_k S_k+1 ] A;& S_N = Q]

[eqn: rm {v_k = -A^T [S_k+1 Xi B u +left{S_k+1 XiDD^T -I right} v_k+1 }]

[eqn: rm Xi = I+DP_k S_k+1].

[eqn: rm d^* (k+j|k)] can also be written as

[eqn: rm d^* (k+j|k) =D^T S_k+1 x(k+j+1|k) -D^T v_k+1].

Substituting d(k+j|k) in Equation (5) with this and rearranging, we obtain

[eqn: rm{ J_k = min atop {u}{ 1 over 2} x^T (k+N|k)Q_F x(k+N|k) +{ 1 over 2} SUM from { { j}=1} to N-1 x^T (k+j|k)(Q -K^T K) x(k+j|k) }]

[eqn: rm{+ {1 over 2} SUM from { { j}=0} to N-1 u^T (k+j|k) u(k+j|k) }]

[eqn: rm{+ {1 over 2} SUM from { { j}=0} to N-1 [x^T (k+j|k) K_k^T D^T v_k + v_k ^T D K_k x(k+j|k)+v_k ^T D D^T v_k ] }]

subject to

[eqn: rm{x(k+j+1|k)=Ax(k+j|k)+Bu(k+j|k)+Dd(k+j|k)}]

[eqn: rm u_min& le& u(k+j|k)& le& u_max , (1 le j le N-1),& u(k+j|k)=0 &(j ge N)]

[eqn: rm {v_k = -A^T [S_k+1 Xi B u +left{S_k+1 XiDD^T -I right} v_k+1 },& v_N = 0]

where

[eqn: rm {Q_F = tilde{Q} - K_k ^T K_k}], [eqn: rm {K_k = D^T S_k}].

To this end, the constrained H∞ MPC problem is reduced to a quadratic program.

We now establish the attractivity of the closed loop system.

Theorem 2. Let [eqn: rm{u^* (k)}] and [eqn: rm{d^* (k)}] denote the solution of the optimization (2).  Then the closed-loop system is attractive for all disturbance satisfying (3).  In other words, [eqn: rm{u (k) -> 0}] and [eqn: rm{x (k) -> 0}] as  [eqn: rm{k -> inf}].

Proof. Let[eqn: rm{u^* (k)}] and[eqn: rm{d^* (k)}] be the argument of the optimization in the above.  Then

[eqn: rm {J_k = rm{min} atop {u} rm{max} atop{d}{1over 2 } SUM from { { j}=1} to infty }x^T (k+j|k) Q x(k+j|k) + }][eqn: rm {{1over 2 } SUM from { { j}=0} to infty}u^T (k+j|k) u(k+j|k) }]                   [eqn: rm {-{1over 2 } SUM from { { j}=0} to infty }d^T (k+j|k) d(k+j|k) }]

with[eqn: rm{ u (k) = u^* (k), u_min le u(k+j) le u_max (0 le j le N-1), u(k+j)=0(j le N)}]

[eqn: rm {ge rm{min} atop {u} rm{max} atop{d}{1over 2 } SUM from { { j}=1} to infty }x^T (k+j|k) Q x(k+j|k) + }][eqn: rm {{1over 2 } SUM from { { j}=0} to infty}u^T (k+j|k) u(k+j|k) }]

[eqn: rm {-{1over 2 } SUM from { { j}=0} to infty }d^T (k+j|k) d(k+j|k) }]

with[eqn: rm{x(k+1|k)=Ax^* (k|k)+Bu^* (k)+Dd^* (k)}],[eqn: rm{u(k)= u^* (k)}],[eqn: rm{d(k|k)= d(k) =0}]

[eqn: rm{ u (k) = u^* (k), u_min le u(k+j) le u_max (0 le j le N-1), u(k+j)=0(j le N)}]

[eqn: rm{= J_k+1 + {1 over 2}x^*T (k+1|k) Q x^* (k+1|k)+ {1 over 2}u^*T (k) u(k)}].

This means

[eqn: rm{ J_k+1 - J_k le-left{ {1 over 2}x^*T (k+1|k) Q x^* (k+1|k)+ {1 over 2}u^*T (k) u(k)right}} < 0].

The inequality implies that [eqn: rm {J_k }] is converging sequence, and x(k) and u(k) must vanish as [eqn: rm{k -> inf}], implying attractivity.                                      □

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