Robust H2 Control of Norm-Bounded Uncertain Continuous-Time Systems - An LMI Approach

2004 IEEE International Symposium on Computer Aided Conuol Systems Design Taipei, Taiwan, September 2-4,2004

Robust

Hz

Control of Norm-Bounded Uncertain

Continuous-Time System---An LMI Approach

Cheng-Te Lai, Chun-Hsiung Fang, Shih-Wei Kau, and Chin-Hshg Lee

Department of Electrical Engineering National Kaohsiung University of Applied Sciences

41 5 Chien-Kung Road, Kaohsiung 807, TAIWAN

Tel:

886-7-3814526 ext. 5521

Fax: 886-7-3834652 Email: [email protected] Abstract

-

Thispaper considers robust H: controlproblem

of continuous-time systems with time-varying norm- bounded uncertainties. A necessary and suflcient condition f o r the analysis of robust H, control is derived in

the form of linear matrix inequalities (LMI). Using the

analysis result, a state feedback confroller and a dynamic output feedback controller are designed respectively so

that the closed-loop uncertain system is quadratically stable and all its transfer matrices have Hrnorm bounded

by a prescribed value. With LMI manipulations, hvo

necessaly and su$cient conditions f o r the solvability of above design problems are addressed.

Keywords: Linear matrix inequalities, uncertain systems, robust H2 control, quadratic stability, norn-bounded uncertainties.

1 Introduction

In recent years, linear matrix inequalities (LMI) techniques [1,2], have become an essential tool for the analysis and synthesis of control systems in the area of robust control. This is because LMI techniques offer the advantage of operational simplicity in regard to classical approaches which involve cumbersome material of Riccati equations. Furthermore, based on the interior-point algorithms, solving LMIs nowadays can be performed very efficiently.

Robust control of nom-bounded uncertain systems has attracted considerable attention in the past ten years [3,4,5].

In

contrast to the rapid developments of robust H, control [6,7,8,9,10,1 I], the progress of robust H2 control is relatively small [12,13]. This is because the synthesis for robust Hz performance of uncertain systems is much harder (see p.230, [14]). In this paper we give a necessary and sufficient condition to analyze the problem of robust

H2 control of norm-hounded uncertain systems. The

condition is then applied to synthesize robust H2

controllers for systems with norm-bounded uncertainties. Two kinds of feedback control frameworks are considered static state feedback and dynamic output feedback. For the former case, an optimal controller that minimizes HZ performance of closed-loop systems for all uncertainties can be designed by the proposed approach.

The paper is organized as follows: problem formulation and some preliminary lemmas are stated in Section 2. In Section 3, main results of the paper are presented. Section 4 contains hvo numerical examples for demonstrating the ideas behind. Finally, some concludiug remarks in Section 5 end the paper.

The notation used throughout the paper is fairly standard. M>0 (or M<O) means

M

is symmetric and positive (or negative) defmite. and Tr(Z) stand for transpose of M and trace of Z, respectively. I, denotes the identity matrix with dimension n, and we simply use I to represent an identity matrix with proper dimension.

IIMII,

stands for 2-norm of M.

2 Preliminaries

Consider the following uncertain system i ( t ) = ( A + A A ) x ( t ) + ( B + A E ) w ( t )

(1)

where x E R" is the state variable, IV E R' the exogenous input, and z e R' the controlled output. The uncertain matrices AA and AB are described by

where H I , GI and G2 are known matrices of appropriate dimension and A(t) is a time-varying nom-bounded matrix satisfying

. ( t ) = C x ( t )

q = ~ , 4 ( t ) ~ ,

G I

(2)

114(f)ii, 1 (3)

For a system described by

i ( r ) =

A X ( f ) + B w ( f )

. ( t ) = (4)

its transfer function from w to z is denoted by

T ,

.

An important result for Hz control of system (4) is stated in Lemma 1.

Lemma 1 [1,12] Let y > 0 be given. The system (4) is stable with

llZ;$

< y if and only if there exist Q > 0 and

Z > 0 such that

Tr (Z) < 1. (7) Next lemma will be used to prove the main result in next section.

Lemma 2 [I51 Let

f = f ' ,

L

, and R be matrices of proper dimension. Then

? + i h l i + ( L h l i ) T < O

V A with11A112<1 (8)

if and only if there exists a scalar & > 0 such that

l+&i'?

+ & . ' R T R < 0 . (9) 3 Main Results

In

this section, a necessary and sufficient condition is firstly derived for the analysis of robust H2 control of

system (1). Based on the analysis result, a state feedback controller and a dynamic output feedback controller are designed respectively to achieve robust H2 performance of closed-loop systems.

3.1 Robust H2 control analysis

According to Lemma 1, an important defmition for robust H2 control is given first.

Definition 1 Let y > 0 be given. The system ( I ) is said to he quadratically stable with IlT-ll: < y if there exist Q > 0 and Z > 0 such that

< o

(10)

r

( A + A A ) Q + Q ( A + M ) ~ ( B + A B ) ( B + A B Y

-Y

1

[C"Q

]>'': Tr ( Z ) < 1 (12)

hold for all allowable AA and AB.

If the uncertainties can be represented as in (Z), according to Definition 1, a necessary and sufficient LMI condition for system (1) to be quadratically stable with

lk.X[:

y is stated as follows

Theorem 1 Let y > O be given. The system ( I ) is quadratically stable witb 1lT-U: < y if and only if there exist

Q > O , Z>O,andascalar p > O suchthat

rAQ+QAr B QG:

&

T r ( Z ) < 1. (15)

Proof: From D e f ~ t i o n 1, we only need to show that (13) is equivalent to (10). With (2), (10) can he rewritten as

By Lemma 2, the inequality (16) holds for all A satisfying

11A1I2

<

1 if and only if there exists a scalar p > 0 such that

01

By Schw complement, (18) is equivalent to (13).

0

Remark 1 Since Q > 0 , it is easy to verify that exist

Q > 0 , Z > 0 , and a scalar ,u > 0 such that (13)-( 15) hold if and only if there exist X > 0 , Z > 0 , and a scalar p > 0 such that

rArX+m

xB

C:

uxH,l

Tr ( Z ) < 1

3.2 Robust control design

For the design problem, consider the following 001111-

i(t)=(A+A4)x(t)+(B,+M,)w(I)+(B,+AB2)u(l) bounded uncertain system

$ ( I ) = C , x ( t ) + D , * U ( l ) (22)

Y ( ~ ) = ( G

+ 4 4 ) + ( ~ , ,

+ ~ ) w ( r ) + ( ~ , ,

+ ~ & ) 4 )

where U E R ~ is the control input and y t R " is the measured ouhmt. The uncertainties are described hv

5 Conclusions

This paper presents necessruy and sufficient conditions for robust Hz control analysis and design of continuous- time systems with time-varying norm-bounded uncertainties. The conditions are expressed in the form of LMI. Thus they are simple and computationally tractable. Based on the proposed result, other performance requirements such as pole-clustering and mixed

H , IH, control can be easily incorporated. Although only continuous-time systems as discussed in this paper, the result can be easily extended to the discrete-time cases. 6 Acknowledgments

This work was supported by National Science Council of Taiwan, under Grant No. NSC 92-2213-E-151-007. References

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1992.

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Appendix

Proposition 1 Let

X

> 0 be a solution to (42) and (43), and assume it can be partitioned as in (45)

Then, Without loss of generality, R may be assumed nonsingular. Furthermore, assume

Then the submatrix

S

is nonsingular, too.

Proof: Suppose R is singular, then there always exists a small p > 0 such that the matrix

2

defmed below

has R being nonsingular. Note that p can be chosen small enough so that the LMI (42) and (43) won’t be violated when X is replaced by

2 ,

Therefore, starting from any

solution

to

(42) and

(43), which does

have

some singular

R,

we can always fmd a very close solution that will meet the nonsingularity requirement on R. By matrix inversion formula

where Y

P

P - RV’R‘ , It is clear that S is nonsingular if

R is nonsingular.

0