H, Control for Singularly Perturbed Sampled Data Systems

Tong V. Vu and Mahmoud E. Sawan Department of Electrical Engineering

The Wichita State University Wichita, Kansas 67208 USA

Abstract - The H, control problem for singularly perturbed discrete-time systems is studied. The suboptimal state feedback controllers for such systems are formulated using the theory of the two-person zero-sum differential game.

INTRODUCTION

Development of discrete-time control systems theory has attracted many researchers due to the availability of low cost digital computers, the advantages of digital control systems in the application, and many physical systems are naturally represented by the models of discrete-time systems. In the recent years, several papers on the discrete-time H, control problem have been presented [5],[9],[8],[ 1],[2]. However, H, control problems for singularly perturbed discrete-time systems has not yet been considered. In this present, the suboptimal state feedback controller for singularly perturbed sampled data system that the transfer function satisfies the Hm norm bound is designed. In order to receive the suboptimal controller, it is required to find the controllers of two lower order subsystems: the continuous-time slow subsystem and the discrete-time fast subsystem. The optimal controller for the continuous-time slow subsystem has been examined [lo]. The optimal controller for the discrete-time fast subsystem is obtained in this paper.

PRELIMINARIES

stabilizable. A is Hurwitz.

Define the performance index:

y is a given attenuation level.

Define R = DTD (4) Q, = CT(I-DR"DT)C ( 5 ) A,, = A - BR-'D'C (6)

Theorem Let K > 0 be a solution of the Riccati quat ion

K = Q, + ATK[ 1 + (BR" B' - y * F F " ) K ] ~ ' ~ (7)

and K satisfies the condition

yzI - FTKF > 0 (8)

Then the unique feedback saddle point strategies of the zero sum dynamic game described by (1),(2),and (3) are

u'(k) = -R1{D'C+BiKII + (BR 1B1-y2FF1)K)"A,,}x(k) (9 )

w'(k) = y2F1K[I +(BR 'B'-y2FF')K]"P$x(k) ( I O ) for k=O,1,2, ...

Consider the linear time invariant system: The control policy u'(k) of (9) assures that

x(k+ 1) = Ax(k)+Bu(k)+Fw(k) ;x(O)=O (1) y(k) = Cx(k)+Du(k) k=0,1,2, ... (2)

where where T,Qw) denotes the transter function trom w(k) to y(k) and the infinity norm of transfer function is defined by x(k)ER" is the state vector.

u(k)ERm is the controlled input.

y(k)eR is the system output. A,B,C,D,E,F are constant matrices of the appropriate

Assume (A,B,C) is stabilizable-detectable, (A,F) is

w(k)ERP is disturbance. IJ TOW) II m = SZP ~ m l T ( ~ ~ ) ) (12)

U- E maximum singular value. dimensions.

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Proof: (see Appendix) PROBLEM STATEMENT

Consider the singularly perturbed system

x (k+ l ) = (I+EA,,)x(k) + cA,,z(k) + cB,u(k) + cF,w(k) ( 1 3 4

(13b) Y(k) = C,x(k) + C,Z&) + DNk) (14)

z(k+l) = AZ,x(k) + A&k) + B,u(k) + Fzw(k)

x(O)=O and z(O)=O, k=0,1, ... where

E is a small positive parameter. x(k)ER"', z(k)eRd are the states. u(k)ERm, w(k)ERP, y(k)eR are the control, disturbance,

Aij,Bi,Ci,Fi,D for i = 1,2 and j = I ,2 are constant matrices

(I-A& is nonsingular.

and the system output, respectively.

of the appropriate dimensions.

The performance index (7 1:

J, = 'h I%( II YAt) II f - Y2 II w,(t) I1 I)dt (26)

We assume that (A,,B,,CJ is stabilizable-detectable and (A,,F,) is stabilizable.

The state feedback control that provides the infinity norm transfer function of the slow subsystem to satisfy a bound y is given by [ I O ]

u:(t) = -(D%W,D,)-'(DfW,C, + B:K, + DrE,V, 'FiK,)x,(t) (27)

Find a stabilizing control policy

u(k) = G,x(k) + G,z(k)

K, 20 is a solution of the algebraic Riccati equation:

&Ao + AZK, + Q, - K,R,K, = 0 (15) (28)

to satisfy the infinity norm bound

II T,W II - 5 Y where y is a given attenuation level.

where A, = 4 - B,(D'~W,D,)~'D~W,C,

+ F,( y21-EfS,E,)"E~S,C, (29 )

Q, = QL = Ct[I + E,(y21-EIS,EJ 'EbS, - D,(DTW,DJ'DfW,]C. (30)

Using the link between the H, optimal control and the linear quadratic difference game theory [3], we denote : R, = RT = B,(DTW,D,)"[B~E'+ D'IE,V IFS]

+ [B,(D'IW,D,)-'DLE,V IFtj' - F,(yZI-ETS,E,)-'FT (31)

R, = D':D, (32) v = 2 (33 ) 5 Y 1 - EIE, S3 = I - D,R>'D! (34) W* = 1 + EaV>IEA (35)

J = '/4 11 y(k) 11 b2 11 w(k) 11 3 ;k=O,1,2,... (17) k=

and The unique control policy, u*(k), forces the system (13)

to satisfy the norm (16).

STATE FEEDBACK CONTROLLERS

Let T,(jw) be transfer function from w,(t) to y,(t). u,(t) The system (13) can be decomposed into slow and fast subsystems using the method introduced by Blankenship [4]. Of (27) leads to After simple work, slow and fast subsystems are found as below: II TAw) II - 5 Y (36)

A. Slow subproblem B. Fasl subproblem

Slow subsystem is a continuous time system: Fast subsystem is a discrete t i m i system

where GO) = z(0)-Z(0) where A, = A, , + A12(I-A2J-'Azl (20)

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The performance index [7] : decomposed into the continuous time slow subsystem and discrete time fast subsystem. The linear quadratic dynamc game theory is used to design the optimal controllers for the

k= slow and fast subsystems. The optimal controller for the slow subsystem is found in [ IO]. The optimal controller for fast subsystem is designed in this paper. Finally, the suboptimal controller is achieved successfully by composing the slow and fast controllers.

J, = lh $( 11 ykk) 11 i-y' 11 wkk) 11 :) ;k=0,1,2,...(39)

we aSSume that (A,,B,,c;) is stabilizable-detectable and

ne state feedback control that minimizes the infinity (AnrFJ is stabilizable.

norm of the transfer function of the fast subsystem is

APPENDIX

Consider the dynamic game described by ( l ) , (2) , and (3). The Hamiltonian hnction is defined 161

K f 2 0 is the solution of the algebraic Riccati equation:

The maximum principle shows that the necessary condition for the cost to be optimized is

DTCx(k) + DTDu(k) + B"p(k+ 1) = 0

Let TXjw) be transfer function from wdk) to ydk), u;(k)

(A21 of (40) gives

C. Composite Control

(45) FTp(k + 1) - y2w(k) = 0

The state and costate equations are

x(k + I ) = Ax(k) + Bu(k) + Fw(k) ;x(O) = 0 (A41 Let

p(k) = CTCx(k) + CTDu(k) + ATp(k + 1 ) G, = -(DTW,D,)-'(DTW,C, +BT&+ DZE,V ;IF:&) G, = Gz = -R-'{DTCZ+BTKdI

(46)

; p(m)=O (AS)

(Ab)

+ (BzR-'BT-~2F~~)Kf]"A,) a k ) (47) G, = (y2I-E~S,E,)'(ETS,CS +FTq-ED,R; 'BTKJ (48) Let p(k) = K(k)x(k)

The composite state feedback control law is formed as Then from (A2),(A3),and (A4), we have

D'Cx(k) + D'Du(k) + B"K(k+ l)[Ax(k)+Bu(k)+Fw(k)) = 0 (A7)

(49)

FTK(k + l)[Ax(k) + Bu(k) + Fw(k)j-y2w(k) = 0 (A8) where Go = [ I + G ~ ( I - A ~ ) " B ~ J G , + G ~ ( I - A ~ 2 ) " [ A , l +F,G,] (50) Therefore, the condition that J(u'(k),w'(k)) IS strictly

concave in w(k) so that the maximizing responsa to u'(k) 15

unique: Let T,(jw) be transfer function from w(k) to y(k), we have

y2I - F'K(k+ 1)F > 0

Substitute (A2),(A3) into (A4), WG get

x(k+ I ) = [I+(BR"B'-y'FF')K(k+ I ) ) ' CONCLUSION

The H, control problem for singularly perturbed discrete time system is considered. The system is

.[ A-BR"D'C Ix(k). (A10)

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where R = DTD REFERENCES

Let 4 = A - BRIDTC Q, = CT(I-DR-'DT)C

121 Using (A5) and (AlO), Equation (A6) becomes

K(k) = Q, +A:K(k+ 1)(I+(BR-'BT-y-'FFT)K(k+ l)]-'% (A1 1)

If condition (A9) is satisfied vk then K(k) 2 K(k + 1) vk and K(k) converges to K as k+m [3], where K is the solution of

K = Q, + A;K[I +(BR-'BT-y-*FF')K]-'A,, (A 12)

and K satiafies yzI - FTKF > 0.

Similarly to the continuous time case [ 101, the optimal controller

u*(k) = -R-'{D'C +BTKII + (BR-'BT-yZFF1)K]-'A,)x(k) (A 13)

forces the infinity norm of the transfer function from w(k) to y(k) of the system (1),(2) to be bounded by the attenuation y.

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I Y I

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