한정 H∞ 최적제어

최진훈, 고훈석, 이광순

서강대학교 화학공학과

Constrained H∞ Optimal Control

Jinhoon Choi,  Hoon Seok Ko,  Kwang Soon Lee

Department of  Chemical Engineering, Sogang University

1. Introduction

In 1980, Zames [6] proposed the H∞ disturbance attenuation problem. Since then, H∞ optimal control problem has been the hottest research topic in control community. Using state space approach, a very simple and efficient solution technique based on Riccati equations has been obtained by Doyle, Glover, Khargonekar and Francis [4]. However, constraints are always present in any practical control problems. For instance, the physical restriction of the actuator limits the value the input can assume. Moreover due to safety, environmental regulation and so on, the states of the plant are desired to lie within a designated area in the state space. Under the presence of these constraints, the closed loop system becomes nonlinear and the current analysis is no longer valid. However due to the difficulty caused by the nonlinearity, the constrained H∞ optimal control problem remains unsolved. In this paper,  we show that the constrained H∞ optimal control problem that is a constrained infinite dimensional dynamic game problem can be reduced to a constrained finite dimensional quadratic programming problem. The feedback implementation of proposed technique based on the quadratic programming problems is also discussed.

2. Preliminaries

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}, { } atop { } x_{k|k}=x_k}]     (1)

where [eqn: rm{k in [0,infty]}]; [eqn: rm{x_{k+j|k} in R^n}] is the state vector; [eqn: rmu_{k+j|k} in R^m] is the manipulated input; [eqn: rm{d_{k+j|k} in R^p}] is the unknown disturbance.  Consider the H∞ optimal control problem:

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

subject to (1), where [eqn: rmQ&ge&0]. Using[eqn: {1 over gamma}D]  instead of [eqn: D], we can assume [eqn: gamma=1].

Theorem 1. [1]. [2] Suppose the generalized algebraic Riccati equation:

[eqn: rm M = Q + A^T M LAMBDA^-1 A],                      (3)

[eqn: rm Lambda = I+(BB^T -DD^T ) M]

admits a positive semi-definite solution M' satisfying

[eqn: rm I - D^T M' D& > &0.]

(i) There exists a minimal positive definite solution M.

(ii) The corresponding cost function is [eqn: J_u (x_{k})=x_{k}^T M x_{k}.]

(iii) The controller and the corresponding maximizing disturbance:

[eqn: rm u_{k+j|k}^* = -B^T M LAMBDA ^-1 A x_{k+j|k}^* =-Fx_{k+j|k}^*]

[eqn: rm d_{k+j|k}^* = D^T M LAMBDA ^-1 A x_{k+j|k}^* =Lx_{k+j|k}^*]

results in

[eqn: rm x_{k+j+1|k}^* = Lambda^-1 A x_{k+j|k}^* .].

The maximal output admissible set [5] is defined by

[eqn: O_INF = LEFT { x in R^n : (A-BF+DL)^k x in Y , k geq 0 RIGHT }]

[eqn: Y= LEFT { x in R^n : u^min leq -Fx leq u^max RIGHT } .]

3. Main Results

Consider the following constrained H∞ optimal control problem:

[eqn: rm {J(x_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 (1),

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

Since [eqn: O_INF] contains a neighborhood of the origin, the constrained H∞ optimal control in this neighbor is linear and, thus, is exponentially stable. For stable plants, similar to the stable plant case in [3], the closed loop system with the constrained H∞ optimal control  is globally exponentially stable.

If the solution of the constrained H∞ optimal control problem exists, the optimal control must converge to zero as [eqn: j -> INF] because of [eqn: {1 over 2} sum_{j=0}^{inf}{u}`_{k+j|k}^T{u}_{k+j|k}] in the cost function. Now consider the following truncated constrained H∞ optimal control problem:

[eqn: rm {J_N (x_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) }]

subject to (1),

[eqn: rm u_min& le& u(k+j|k)& le& u_max , ~~~(0 le j le N-1).]     (5)

Since the optimal control of the constrained H∞ optimal control problem converges to zero, there exists [eqn: N] such that the constraints are not active for all [eqn: j ge N] and, thus, [eqn: J] and [eqn: J_N] are equivalent. Let [eqn: u_N] be the vector that consists of the first [eqn: N] elements of u and [eqn: u^N] the sequence of the rest of the elements of u. [eqn: d_N] and [eqn: d^N] are defined similarly. Now notice that [eqn: J_N] can be rewritten as

[eqn: {J}_N ({x}_k ) = {min atop u_N} left{ {min atop u^N} {max atop d_N} {1 over 2} left[ sum_{j=0}^{N-1}{x}`_{k+j|k}^T {Q}{x}_{k+j|k}+ sum_{j=0}^{N-1}{u}`_{k+j|k}^T{u}_{k+j|k}]

[eqn: -sum_{j=0}^{N-1} d`_{k+j|k}^T{d}_{k+j|k} + {bar J}_N ({x}_{k+N|k}) right] right}]

subject to (1) and (5), where

[eqn: {bar J}_N ({x}_{k+N|k}) ={max atop d_N} {1 over 2} left[ sum_{j=N}^inf {x}`_{k+j|k}^T{Q}{x}_{k+j|k}+ sum_{j=N}^{inf}{u}`_{k+j|k}^T{u}_{k+j|k}]

[eqn: -sum_{j=N}^inf d`_{k+j|k}^T{d}_{k+j|k} right]].

Before we proceed, consider the following three auxiliary problems:

[eqn: {tilde J}_N ({x}_k ) ={min atop u_N} left{ {max atop d_N} {min atop u^N} {1 over 2} left[ sum_{j=0}^{N-1}{x}`_{k+j|k}^T{Q}{x}_{k+j|k}+ sum_{j=0}^{N-1}{u}`_{k+j|k}^T{u}_{k+j|k}]

[eqn: -sum_{j=0}^{N-1} d`_{k+j|k}^T{d}_{k+j|k} + {bar J}_N ({x}_{k+N|k} ) right] right}]

subject to (1) and (5),

[eqn: {hat J}_N ({x}_k ) ={min atop u} {max atop d} {1 over 2} left{ sum_{j=0}^inf {x}`_{k+j|k}^T {Q}{x}_{k+j|k}+ sum_{j=0}^{inf}{u}`_{k+j|k}^T{u}_{k+j|k}]

[eqn: -sum_{j=0}^inf d`_{k+j|k}^T{d}_{k+j|k} right}]

subject to (1),

[eqn: {u}_{min} le {u}_{k+j|k} le {u}_{max}~~(0 le j le N-1) ~~~~~ u_{k+j|k}=-Fx_k+j|k ~~(j ge N)],

[eqn: {check J}_N ({x}_k ) ={min atop u_N} {max atop d_N} {1 over 2}left{ {x}`_{k+N|k}^T{M}{x}_{k+N|k}+ sum_{j=0}^{N-1} {x}`_{k+j|k}^T{Q}{x}_{k+j|k}]

[eqn: + sum_{j=0}^{N-1} {u}`_{k+j|k}^T{u}_{k+j|k} -sum_{j=0}^{N-1} d`_{k+j|k}^T{d}_{k+j|k} right}]

subject to (1) and (5).

Lemma 1: [eqn: J_N], [eqn: {tilde J}_N] ,[eqn: {hat J}_N] and [eqn: {check J}_N]  are all equivalent.

Proof: The lemma follows from the following inequalities:

[eqn: J_N ({x}_k ) ge {tilde J}_N ({x}_k )= {check J}_N ({x}_k ) = {hat J}_N ({x}_k )ge J_N ({x}_k ) .]           □

Theorem 2: Let [eqn: {u}`_{k+j|k}^dagger] and [eqn: {d}`_{k+j|k}^dagger] be the olution of [eqn: J_N ({x}_k )].  If

[eqn: x`_{k+N|k}^dagger in O_inf], [eqn: J] and [eqn: J_N] are equivalent.

Proof: Notice that [eqn: J_N ({x}_k ) le J({x}_k ).] If [eqn: x`_{k+N|k}^dagger in O_inf], [eqn: {u}`_{k+j|k}^dagger =-Kx`_{k+j|k}^dagger] for all [eqn: j ge N]. Hence [eqn: {u}`_{k+j|k}^dagger] is the feasible point for [eqn: J({x}_k )] and [eqn: {d}`_{k+j|k}^dagger] is the corresponding maximizing disturbance. Hence the theorem follows.         □

Fact 1: [2] The maximization problem in [eqn: J_{N^*}] admits the unique solution:

[eqn: {d}`_{k+j|k}^* = {P}_{j}{S}_{j+1}{A}{x}_{k+j|k}+{P}_{j}LEFT[{S}_{j+1}{B}{u}_{k+j|k}- {v}_{j+1} RIGHT],~~j in [0,N-1]]

where

[eqn: {P}_j = [{I}-{D}^T{S}_{j+1}{D}]^{-1}{D}^T]

[eqn: {S}_j = {Q}+{A}^T {S}_{j+1} [{I}+{D}{P}_j{S}_{j+1}]{A} ;~~{S}_N=M]

[eqn: {v}_j = {A}^T [{I}+{D} {P}_j {S}_{j+1}]^T [{v}_{j+1} -{S}_{j+1}{B}{u}_{k+j|k}];~~{v}_N =0].

Furthermore, the optimal cost is

[eqn: {1 over 2}{x}`_{k|k}^T S_0 {x}_{k|k} -{x}`_{k|k}^T v_0 -{q}_0]

where

[eqn: {q}_0 = {1 over 2} sum_{j=0}^{N-1} LEFT[-{u}`_{k+j|k}^T{B}^T{S}_{j+1}{B}{u}_{k+j|k}-LEFT({S}_{j+1}{B}{u}_{k+j|k}-{v}_{j+1}RIGHT)^T]

[eqn: {P}`_{j}^T {D}^T LEFT({S}_{j+1}{B}{u}_{k+j|k}-{v}_{j+1} RIGHT) +2{u}`_{k+j|k}^T{B}^T {v}_{j+1} RIGHT ]].

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

[eqn: {J}_N ({x}_k ) = {min atop u_N} LEFT{ {1 over 2} sum_{j=0}^{N-1} u^T`_{k+j|k} {u}_{k+j|k} + LEFT[ {1 over 2}{x}`_{k|k}^T {S}_0 {x}_{k|k} -{x}`_{k|k}^T {v}_0 -{q}_0 RIGHT] RIGHT}]

subject to

[eqn: {v}_j={A}^T [{I}+{D} {P}_j {S}_{j+1}]^T [{v}_{j+1} -{S}_{j+1}{B}{u}_{k+j|k}];~~{v}_N=0,]

[eqn: {u}_{min} <= {u}_{k+j|k}<= {u}_{max}~~(0 <= j <= N-1).]

We now summarize the feedback implementation.

Off-line Computation:

1) Compute [eqn: O_INF] as proposed in [5].

On-line Computation:

1) If [eqn: x_k in O_INF], implement [eqn: u_k = -Kx_k].

2) Otherwise, choose N and solve [eqn: J_n (x_k )].

3) If [eqn: x_k+N|k^dagger in O_INF], implement the first control input. Otherwise, increase N and go to 2).

References

[1] Basar, T. and Brenhard, P. : "H∞-optimal control and related minimax design problem", Birkhauser, Boston(1995)

[2] Basar, T. and Olsder, G. J. : "Dynamic Noncooperative game theory", Academic Press, New York(1982)

[3] Choi, J. and Lee, K. S. : IEEE Transactions on Automatic Control, submitted.

[4] Doyle, J., Glover, K., Khargonekar, P. P. and Francis, B. A. : IEEE Transactions on Automatic Control, 34, 831(1988)

[5] Gilbert, E. G. and Tan, K. T. : IEEE Transactions on Automatic Control, 36, 1008(1991)

[6] Zames, G. : IEEE Transactions on Automatic Control, 26, 301(1981)