Decentralized Quadratic Stabilization of Interconnected Systems

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

DECENTRALIZED QUADRATIC STABILIZA TION OF INTERCONNECTED SYSTEMS K. Yasuda Depanment of Computer and Systems Engineering, Kobe University, Kobe 657, Japan

Abstract. In this paper, the problem of quadratically stabilizing a linear interconnected system with parameter Wlcertainties by decentralized control is considered. The obtained stabilizability condition is given in terms of quadratic stabilizability with disturbance attenuation for each subsystem and the attenuation values are determined using only upper bOWlds of the strength of interconnections among subsystems. A decentralized control for quadratically stabilization is also given to demonstrate the feasibility of the design. Key Words. Decentralized controlj large-scale systems; quadratic stabilization; robust control; disturbance rejection

quadratic stabilizability with disturbance attenuation for each subsystem and the attenuation values can be determined using only upper bounds of the strength of interconnections among subsystems. In addition, a decentralized state feedback for quadratic stabilization is given to demonstrate the feasibility of the design.

1. INTRODUCTION

A large number of papers concerning decentralized control for interconnected systems which are composed of a large number of subsystems has been reported so far. Especially, the design of decentralized connectively stabilizing controllers for such a interconnected systems has attracted considerable attention in the literature (e.g. Siljak, 1991). Connective stability is a kind of robust stability and the property guarantees that the stability of the overall closed-loop system is preserved under perturbations in the interconnections such that the strength of each coupling between any two subsystem is bounded. In recent year, this kind of decentralized stabilization problem has been considered for uncertain interconnected systems (Chen, 1992; Cheng et al., 1992).

2. PROBLEM FORMULATION Consider an uncertain interconnected system S which is composed of N subsystems Si described by state-space model of the form

Xi = (Ai + ~Ai(t))xi + (Bu + ~Bli(t))Vi +(B2i + ~B2i(t))Ui Zi = (Cu + ~Cli(t))Xi + (Du + ~DU(t))Ui Yi = (C2i + ~C2i(t))Xi + (D2i + ~D2i(t))Ui i=1,2, ... ,N (1)

In the decentralized controller design, it is desirable that the stabilizability condition for an interconnected system is decomposed into conditions for each isolated subsystem, because a local controller can be designed independently of those of the other subsystems. Such stabilizability conditions are appeared in few papers except the special cases such that the interconnection matrix satisfies the matching condition.

where Xi E nn; is the state, Ui E n r ; is the control input, Vi E n k ; is the interconnection input, Zi E n l; is the interconnection output, Yi E n m ; is the measured output of the subsystem, Ai, Bu, B 2i, CH, C2i , DH, D2i are constant matrices of appropriate dimensions and ~Ai(')' ~Bli(')' ~B2i(')' ~Cli(')' ~C2iO, ~Dli(')' ~D2i( ' ) are matrices of uncertain parameters, and interactions among subsystems given by

In this paper, the problem of quadratically stabilizing uncertain interconnected systems by decentralized control is considered. The objective is to show that the decentralized quadratic stabilizability condition can be given in terms of

N

Vi

=L

j=l

499

Gij(t)Zj

(2)

where GijO is unknown matrix function satisfying (3) with the elements of Gij (-) being Lebesgue measurable and gij > 0 a given constant being the uncertainty bound. It is assumed that the pair (Ai, B 2i) is stabilizable and (C2i , Ai) is detectable.

3. PRELIMINARY DEFINITIONS Definition 3 Given a scalar Pi > 0, the uncertain subsystem (6) is said to be quadratically stable with disturbance attenuation Pi if there exists a symmetric positive-definite matrix Pi and a positive constant C:i such that for all admissible uncertainties ~AiO, ~B1iO and ~CliO,

[Ai + ~Ai(t)lT Pi + p;[A i + ~Ai(t)l + P;[B 1i + ~Bli(t)lpi2[B1i + ~Bli(t)f Pi T + [Ch + ~C1i(t)l [Cli + ~C1i(t)l + c:;I = O. (8)

For stabilization of this interconnected system, we consider the decentralized state feedback Uj=J(iXi,

i=I,2, ...

,N

(4)

where J(j is a constant matrix, and the decentralized dynamic output feedback Ui

= J(i(S)Yi, i = 1,2, ... , N

(5)

where J(i(S) is a linear controller. By applying the decentralized control (4) or (5), t.he resulting closed-loop system is described by the state equation of the form

ii = (Ai + ~Ai(t))Xi + (Bli + ~B1i(t))Vi (6) Zi = (C\i + ~Cli(t))Xi and then the overall system with the interconnection (2) is written as

i

= (A + ~A(t))x

where x = [xi xI ... x~

(7)

IT.

Remark 1 Definition 3 is a natural extension from that defined in the case of ~BliO = 0 and ~Cli( ' ) = 0 by Xie et al. (1992b) . Therefore, when the subsystem (6) is quadratic ally stable with disturbance attenuation Pi, with zero-initial condition for Xj(t), IIzdl2 < Pillvjl12 for all admissible uncertainty ~AiO, ~Bli(-)' ~C1i(') and all non zero Vi E L 2 [ 0,(0), where 11 . 112 denotes the usual L 2 [ 0,00 )-norm. Definition 4 Given Pi > 0, the uncertain subsystem (1) is said to be quadratically stabilizable with disturbance attenuation Pi if there exists a decentralized control law of (4) or (5) such that for all admissible uncertainties ~AiO, ~BliO, ~B2i( ' )' ~CliO, ~C2iO, ~Dli( ' )' ~D2i(-)' the resulting closed-loop system (6) is quadratically stable with disturbance attenuation Pi. Definition 5 Let A be a real square matrix with non-positive off-diagonal elements. Then the matrix A is said to be a semi-M-matrix if and only if the principal minors are all non-negative.

Definition 1 The overall closed-loop system (7) is said to be quadratically stable if there exists a symmetric positive-definite matrix P and a positive constant c: such that the following condition holds: given any admissible uncertainty ~A(.), the Lyapunov derivative corresponding to (7) and the Lyapunov function V(x) xT Px satisfies the bound

=

V(x(t))

4. STABILIZABILITY CONDITION To present main results of this paper, let us define an N x N matrix G as

= 2xT (t)P [A + ~A 1x(t) ~ -c: 11 x(t) 112

G for all x(t) and t, where norm.

11·11

= matrix [gij 1

denotes the Euclidean where gij is an uncertainty bound given in (3).

Definition 2 The interconnected system S is said to be quadratically stabilizable via decentralized control if there exists a decentralized control (4) or (5) such that the resulting overall closedloop system (7) is quadratically stable.

Theorem 1 (Appendix A) Let G be irreducible = diag.{ 11,/2, ... , IN} be positive defand inite. If for li > 0 such that G is a

r

r-

semi-M-matrix, every closed-loop subsystem (6) is quadratically stable with disturbance attenuation li- 1 , then the overall closed-loop system composed of (6) and (2) is quadratically stable.

The problem considered in this paper is to find a condition under which the uncertain interconnected system S is quadratically stabilizable via decentralized control.

This theorem readily yields the following result on quadratic stabilizability via decentralized control. 500

Theorem 2 Let G be irreducible. If for li > 0 such that r - G is a semi-M-matrix, every subsystem (1) is quadratically stabi/izable with disturbance attenuation li- 1, then the interconnected system S composed of (1) and (2) is quadratically stabilizable via decentralized control.

LlBli(t) = HBliFB1i(t)EBli LlCu(t)

= HChFCli(t)Ec1i

where H. and E. are known constant real matrices and F.(t) is unknown matrix function with Lebesgue measurable elements and such that F.(tf F.(t) ~ I .

r-

Theorem 3 (Appendix B) If there exists positive numbers t:i, t:Ai, t:B1i, t:B2i, t:Cli and a symmetric positive-definite matrix Pi such that the inequalities

In the theorems, the irreducibility of the matrix G is assumed, however the restriction can be easily removed. If G is reducible, it can be reformed by renumbering the subsystems as

G11 G 21 [

= HB2iFB2i(t)EB2i

LlC2i (t) = H C2 ;Fc2i(t)Ec2i

Remark 2 The disturbance attenuation value li- 1 such that G is a semi-M-matrix is not unique, so that li- 1 can be appropriately chosen to strike a balance of the load imposed to each subsystem for stabilization .

G=

LlB 2i(t)

1- t:BliEtliEB1i

>0

(9)

I - t:ctiHc1iH~li

>0

(10)

and the Riccati equation PiAi+ATpi 1 - t:i Pi B2i (I - Et2iEB2i)Bf;Pi

G~f1

- t:B~iPiB2iEt2iEt~Bf;Pi -1 H HT 2 -1 H HT + P.i [ t:Ai Ai Ai +'it:Bh B1i Bli

where G ii are all irreducible. This decomposition of G means that N subsystems is separated into M groups. A system composed of subsystems belonging to a group and interactions within the group is referred here as a sub-interconnected system.

+,[ Bli(I - t:BhE~liEB1;)-1 B{;

+t:B~iHB2iHJJ2i J Pi + t:AiE~iEAi + t:C1iE~liEcli T )-1 C1i + C 1iT (I - 1 t:C1iH Ch H C1i

Corollary 2.1 If every sub-interconnected system is quadratically stabilizable via decentralized control, then the overall interconnected system is also quadratically stabilizable via decentralized control.

+ ci I = 0 (11)

are satisfied, then the subsystem (1) is quadratically stabilizable with disturbance attenuation li- 1 via state feedback. And a suitable state feedback gain Ki is given by

" hi

5. CONTROLLER SYNTHESIS

1 (I-EB2iEB2i + ) +--EB2iEB2i 1 + +Tj B T P i. = -[-2 2i Ci CB2i

In the above equations, .+ denotes the MoorePenrose inverse.

According to the above results, the synthesis of a decentralized controller for quadratically stabilizing the interconnected system turns out the design of a quadratically stabilizing controller with disturbance attenuation li- 1 for each subsystem. Such design methods are developed by Xie et al. (1992a), Xie et al. (1992b) and Cheng et al. (1992), however they are considerably restrictive . In this section, a controller for the subsystem with a wider class of parameter uncertainties shall be proposed.

Remark 3 The solvability of the Riccati equation (11) is guaranteed for any li- 1 under some conditions on parameter uncertainties. To investigate this, consider the following case: LlB2i

= 0,

EAi

= EC1i = C 1i .

In this case, the Riccati equation (11) has a positive-definite solution Pi for any li if the following conditions hold (Petersen, 1988) .

For only simple description, it is assumed here that the state are available for feedback and

(i) Both B2i and Cu are full rank. (ii) (Ai, B2i ) is stabilizable. (iii) (Cli, Ai ) is detectable.

Let the other parameter uncertainties be of the form

(iv) rank [

501

Ai - sI Ch

B2i]

0

=ni+li,

Re(s»

O.

where

6. CONCLUDING REMARKS In this paper, it is shown that the decentralized quadratic stabilization problem is reduced into that of designing quadratically stabilizing controller with disturbance attenuation 1j-l for each subsystem. The approach adopted here is based upon the Riccati equation and semi-M-matrix, so that it can be extensively applied to discrete-time system case and nonlinear system case.

€i

= 11 [Bli + ~Bli(t) ]T PiXi 11

7Ji = 11

[C\i + ~C\i(t) ]Xj

11·

Let V(x) = L~l WiV;(Xi) be a candidate of the Lyapunov function for the overall system. Its time-derivative is calculated as N

N

V ~ E Wi{ -rd -

1l€l - cixTxi

i=1

= _[ ..eT

7. REFERENCES

T] [ rw r 7J -Fl'W

1-

+ ~i EgijTJj} j=l

WF ] [ € ] 7J

W

N

Araki, M. (1977). M-matrices -IV. Systems and Control, 21, p.214 (in Japanese). Chen, Y.R . (1992) . Decentralized robust control for large-scale uncertain systems: a design based on the bound of uncertainty. J. of

-

~ -T~WiCiXi Xi

(A2)

i=l

= [66

where € Therefore,

Dynamic Systems, Measurement, and Control, 114, 1- 9. Cheng, C.-F., Wang, W.-J., and Lin, Y.-P. (1992). Quadratically decentralized stabilization for uncertain structured interconnected systems. Proc. 31st IEEE Conf. on Decision and Control, Tucson, pp.2846-2847. Petersen, I.R. (1988). Some new results on algebraic Riccati equations arising in linear quadratic differential games and stabilization of uncertain linear systems. Systems fj Control Letters, 10, 341-348. Siljak, D.D. (1991). Decentralized Control of Complex Systems. Academic Press, San Diego. Xie, 1., and de Souza, C.E. (1992a). Robust Hoo control for linear systems with norm-bounded time-varying uncertainties. IEEE Trans. Aut. Control, AC-26, 1188-1191. Xie, 1., and de Souza, C.E. (1992b). Hoc control and quadratic stabilization of systems with parameter uncertainty via output feedback. IEEE Trans. Aut. Control, AC-26, 1253-1256.

.. .€Nf,

r~ r [ _F1W

1-

7J

= [7Jl

7J2 .. . 7JNf ·

WF ] > 0 W -

(A3)

yields V ~ -cllxII 2 for an C > 0, which guarantees the overall system is quadratically stable. Since W is positive definite, the matrix inequality (A3) is equivalent to

rw- l r -

FW- l FT

2:: o.

(A4)

From the following Lemma 1, (A4) is satisfied if and only if r -G is a semi-M-matrix for irreducible G. This completes the proof. I Lemma 1 (Araki, 1977) Let C be an irreducible matrix with non-negative elements and D be a diagonal matrix with positive elements. The matrix D - C is a semi-M-matrix if and only if there is a diagonal matrix with positive elements W such that the matrix DW D - CTWC is positive semidefinite.

APPENDIX B Proof of Theorem 3:

Using (3) and the following inequalities

APPENDIX A

+ cAiEIiEAi

Proof of Theorem 1:

(;.; Pi HAiHIi Pi

Given a quadratic function V;(Xi) = xT PiXi, its time-derivative corresponding to the closed-loop system (6) is given by using (8) as

CB~iPiHB2iHTJ2iPi

2:: ~A[ Pi + Pi ~Ai

+ cB2;l
Vi = xT {Pi [Ai + ~Ai(t)] + [Ai + ~Ai(t)]T Pi} Xi +2xT Pi[Bli + ~Bli(t)]Vi ~ +XiPi[Bli + ~Bli(t)hl[Bli + ~Bli(t)f PiXi TT-Xi [Cli + ~Cli(t)] [Cli + ~Cli(t)]Xi -cixT Xi

~ -1lt} -

+ 2x[ P;[E li + ~Eli(t)]Vi rd - ciX[Xi + 2€dlvdl

it can be seen that the positive-definite solution Pi to the Riccati equation (11) satisfies (8) . I

(AI) 502