An Improvement on Robust H∞ Control for Uncertain Continuous-Time Descriptor Systems

An Improvement on Robust H∞ Control for Uncertain Continuous-Time Descriptor Systems 271

An Improvement on Robust H

∞

Control for Uncertain

Continuous-Time Descriptor Systems

Hung-Jen Lee, Shih-Wei Kau, Yung-Sheng Liu, Chun-Hsiung Fang*, Jian-Liung Chen, Ming-Hung Tsai, and Li Lee*

Abstract: This paper proposes a new approach to solve robust H∞ control problems for

uncertain continuous-time descriptor systems. Necessary and sufficient conditions for robust

H∞ control analysis and design are derived and expressed in terms of a set of LMIs. In the

proposed approach, the uncertainties are allowed to appear in all system matrices. Furthermore, a couple of assumptions that are required in earlier design methods are not needed anymore in the present one. The derived conditions also include several interesting results existing in the literature as special cases.

Keywords: Descriptor systems, H∞ control, LMI, robust control, uncertainties.

1. INTRODUCTION

It is well known that the descriptor system (also referred to singular systems, or generalized state-space systems, or implicit systems, or semistate systems in the literature) described by the following model ( ) ( ) ( ) ( ) ( ) ( ) Ex t Ax t Bu t y t Cx t Du t = + = +  (1) has higher capability in describing a physical system. In (1), the matrix n n

E

∈\

× may be singular. Assume

rank(E)=r and denote by p the degree of the

characteristic polynomial sE− A. For descriptor systems, it is interesting to note that 0≤ ≤ ≤p r n. The system (1) is termed to be regular and impulse-free if p=r and termed to be admissible if it is

p =rand all roots of sE− =A 0 are Hurwitz stable.

Descriptor-system models are often more convenient and natural than standard state-space models in the description of interconnected large-scalar systems [3], economic systems [12], electrical network [14], power systems [1], chemical processes [9], and so on [10]. This is the reason why descriptor systems have attracted much interest in recent years [4-13].

The H∞ control problem of descriptor systems has

been addressed by several researchers. For instance, to solve H∞ control problem, the concept of J-spectral

factorization and(J,J’)-spectral factorization had been extended to descriptor systems in [7] and [15]. Based on the generalized algebraic Riccati equation, necessary and sufficient conditions for H∞ control of

continuous-time and discrete-time descriptor systems were given in [8] and [18], respectively. Recently, because of the numerical efficiency of LMI, the H∞

control problem of descriptor systems was resolved by using LMI approaches [13,5-20]. When descriptor systems contain uncertainties, the robust H∞ control

result currently available in the literature is very limited. Reference [6] proposed a necessary and sufficient LMI-based condition for robust H∞ control

of uncertain descriptor systems. Based on it and under some assumptions including the admissibility of nominal system, necessary and sufficient GARI-based conditions are developed to solve the state feedback and the dynamic output feedback synthesis problems. However, as indicated in [16], all results of [6] are only sufficient due to an incorrect proof of the necessary statement. Differently, an LMI-based approach is proposed in [16] to tackle exactly the same problem as [6]. However, all results obtained in [16] are still sufficient only.

In this paper, a new LMI approach is proposed for __________

Manuscript received June 29, 2005; revised January 25, 2006; accepted February 20, 2006. Recommended by Editorial Board member Seung-Bok Choi under the direction of past Editor-in-Chief Myung Jin Chung. This work was supported by National Science Council of Taiwan under Grant No. NSC-92-2213-E-110-024 and NSC-93-2745- E-151-001.

Hung-Jen Lee, Shih-Wei Kau, Chun-Hsiung Fang, and Yung-Sheng Liu are with the Department of Electrical and Electronics Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road, Kaohsiung 807, Taiwan (e-mails: {hjlee, shiewkau, chfang}@cc.kuas.edu.tw, [email protected]).

Jian-Liung Chen, Ming-Hung Tsai, and Li Lee are with the Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung 804, Taiwan (e-mails: {jlchen, mhtsai, leeli}@mail.ee.nsysu.edu.tw).

* Corresponding authors. ,

272 Hung-Jen Lee, Shih-Wei Kau, Yung-Sheng Liu, Chun-Hsiung Fang, Jian-Liung Chen, Ming-Hung Tsai, and Li Lee solving the same problem mentioned above. There are

four major contributions in this paper. (I) Necessary and sufficient conditions for robust H∞ control are

derived. Before this presentation, only sufficient conditions for the same problem were obtained. (II) No assumption as needed in [6] is required. (III) The system model considered in this paper is more general since all system matrices are allowed to have uncertainties. In [6,16], only the state matrix contains uncertainties. (IV) The present result includes the major result of [13,18] as special cases.

2. PROBLEM FOMULATION

Consider an uncertain continuous-time descriptor system ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) w u z zw zu y yw yu Ex t A x t B w t B u t z t C x t D w t D u t y t C x t D w t D u t Δ Δ Δ Δ Δ Δ Δ Δ Δ = + + = + + = + +  (2) where ( ) n

x t ∈ \ is the state vector, ( )_{w t} ∈\ the mw

exogenous input, ( )_{u t} ∈ \mu the control input,

( ) qz

z t ∈ \ the controlled output, and _{y t}( )∈ \qy the

measured output. Assume the system matrices A_Δ, , w B _Δ B_uΔ, C_zΔ, D_zwΔ, D_zuΔ, C_yΔ, D_ywΔ,andD_yuΔ are described as 1 2 1 2 3 3 , w u w u z zw zu z zw zu y yw yu y yw yu A B B A B B C D D C D D C D D C D D H H J J J H ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ +_{⎢ ⎥}Δ ⎡_⎣ ⎤_⎦ ⎢ ⎥ ⎣ ⎦ + + + + + + + + +  (3) where n n, A∈ \ × n mw, w B ∈ \ × n mu, u B ∈ \ × C_z∈ , z q ×n \ qz mw, zw D ∈ \ × qz mu, zu D ∈ \ × C_y∈ \qy×n, y w q m yw D ∈ \ × and qy mu yu

D ∈ \ × are constant matrices representing the nominal system. ₁ n s,

H ∈ \ × H₂∈ , z q ×s \ ₃ qy s, H ∈ \ × J₁∈ \s n× , _J2∈ \s m× w, and 3 u s m

J

∈ \

× provide structure information of uncer-tainties. _{Δ ∈ \}s s× _{is a norm-bounded uncertain matrix}

satisfying

.

T s

I

Δ Δ ≤

(4) References [6,16] considered the robust H∞ control

problem of the following special system

(

1 1

)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) w u z zu y yw Ex t A H J x t B w t B u t z t C x t D u t y t C x t D w t = + Δ + + = + = +  (5)

in which uncertainties appear only on the state matrix. Next definition and lemma are directly quoted from [6].

Definition 1 [6, Definition 2.5]: Given

γ

>0, the unforced system (5) (i.e. u(t) = 0) is stated to be quadratically admissible with disturbance attenuation

γ

for all uncertainties

Δ

if there exists a nonsin- gular matrix X such that for all

Δ

(

1 1

)

(

1 1

)

2 0, 1 0. T T T T T T T w w z z E X X E A H J X X A H J X B B X C C γ = ≥ + Δ + + Δ + + < (6)

Lemma 1 [6, Lemma 2.6]: Consider the system in (5) and a prescribed scalar

γ

>0. Assume _{Δ Δ ≤} T

2 s

I

ρ where

ρ

is a given real number. Then (6) holds for all

Δ

if and only if there exists a nonsingular matrix Y, independent of

Δ

, such that

[

1

]

2 1 1 1 0, 1 0. T T T w T T T w _T z T T z E Y Y E B A Y Y A Y B H Y H C C J J γ γ γ ρ ρ = ≥ ⎡ ⎤ + + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ +_{⎣ ⎢ ⎥}< ⎦ ⎣ ⎦ (7)

As mentioned in [16] that, actually, Lemma 1 is only sufficient because an obvious argument error appears in the proof of necessity. More precisely, the inequality (20) of [6] can’t be as claimed to be derived from substituting (19) into (17a) in [6]. The following simple example shows a contradiction between Definition 1 and Lemma 1. Let γ =1, ρ= and E =1

[

]

1 0 1.2 0 1 , , , 1 0 , 0 0 A 0 1 Bw 0 Cz − ⎡ ⎤ ₌⎡ ⎤ ₌⎡ ⎤ ₌ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ H1=

[

]

[

]

1 0 , 1 0 , 1 1 . 1 J ⎡ ⎤ = ∈ − ⎢ ⎥ ⎣ ⎦ + Note that 1.2 0 0 0.2 X= ⎢⎡ ⎤_⎥ − ⎣ ⎦

satisfies (6) for allΔ ∈ −

[

1, 1

]

. Hence, by Definition 1, the corresponding system is quadratically admissible with disturbance attenuation 1. However, to check feasibility of (7), by letting 1 2 3 4

y

y

Y

y

y

⎡

⎤

= ⎢

_⎣

⎥

_⎦

in (7).

The first condition of (7) implies

y

₁

≥

0

and

2 0

y = and the second condition of (7) gives , , , , , ,

An Improvement on Robust H∞ Control for Uncertain Continuous-Time Descriptor Systems 273 2 2 1 1 3 3 4 2 3 4 4 4 2.4 2 (1 ) 0 (1 ) 2 y y y y y y y y y ⎡ − + + + ⎤ _< ⎢ _{+ +} ⎥ ⎣ ⎦ ,

which, by Schur complement and some simple algebra, is equivalent to 2 4 2 4 0 y + y < and 2 2 3 1 1 2 4 4 2.4 2 2 y y y y y − + < + or 2 4 2 4 0 y + y < and

(

)

2 2 ₃ 1 2 4 4 1.2 0.56 2 y y y y − + < + . Since it is impossible to find three real numbers y1, y3,

and y4 to satisfy the above two inequalities simul-

taneously, the inequality (7) has no solution at all. This obviously indicates the result of [6] is incorrect. Since all the other results in [6] are based on Lemma 1, they are only sufficient, too.

The goal of this paper is to derive necessary and sufficient LMI-based conditions for robust H∞ control

of (2), which is more general than (5). The new conditions are applied to design two types of controllers so that the closed-loop system is quadratically admissible with disturbance attenuation

γ

. For solving the robust H∞ control problem of (2),

Definition 1 is extended to a more general case as follows.

Definition 2: Given

γ

>0, the unforced uncertain descriptor system (2) (i.e. u(t) = 0) is said to be quadratically admissible with disturbance attenuation

γ

for all uncertainties

Δ

satisfying (4) if there exists a nonsingular matrix P such that for all

Δ

0 T T E P=P E≥ ,

(8)

(

)

T T T T w z zw A P_Δ +P A_Δ+ P B ₊+C D_{+ +} (9)

(

₂

) (

1

)

0. w T T T T m zw zw w zw z z z I D D B P D C C C γ − _{+ +} − ₊ + _{+ +} + _{+ +}<

Next Lemma plays a key role in the development of next section.

Lemma 2 [19]: Given appropriate dimensional matrices X, Y, and a symmetric matrix Z, then

0

T T T

Z + XΔ +Y Y Δ X < for all

Δ

satisfying T

I

Δ Δ ≤ if and only if there exists a scalar

ε

>

0

such that

1 ₀

T T

Z+εXX +ε−Y Y < .

3. MAIN RESULTS

In this section, two necessary and sufficient LMI-based conditions for robust H∞ analysis and design of

system (2) are derived, respectively.

3.1. Robust H∞ analysis

First, the result of robust H∞ control analysis of (2)

is presented.

Theorem 1: The unforced uncertain continuous-time descriptor system (2) is quadratically admissible with disturbance attenuation

γ

for allΔif and only if there exists a nonsingular matrix P and a scalar

0

ε

>

satisfying 0 T T X E =EX ≥ ,

(10) 1 1 2 2 1 1 2 1 2 1 2 2 2 0 0 0 w z T T T w T w m T z zw T T T T T z T T zw T q s X A AX H H B B I C X H H D J X J X C H H X J D J I H H I ε γ ε ε ε ε ⎡ _{+ +} ⎢ ⎢ − ⎢ ⎢ + ⎢ ⎢⎣ ⎤ + ⎥ ⎥ < ⎥ − + ⎥ ⎥ − _⎥⎦ . (11)

Proof: By congruence and settingX−1= , (10) :P

becomes (8) and (11) is equivalent to 1 1 2 2 1 1 2 1 2 1 2 2 2 0 0 0 w z T T T T T w T w m T z zw T T T T z T T zw T q s A P P A P H H P P B B P I C H H P D J J C P H H J D J I H H I ε γ ε ε ε ε ⎡ _{+ +} ⎢ ⎢ − ⎢ ⎢ + ⎢ ⎢⎣ ⎤ + ⎥ ⎥ < ⎥ − + ⎥ ⎥ − _⎥⎦ ,

which can be represented further into _A+

ε

_HH T

1_{J J}T ₀

ε

− +  < , where 2 _, w z T T T T w z T T w m zw z zw q A P P A P B C A B P I D C D I

γ

⎡ ₊ ⎤ ⎢ ⎥ =_⎢ − _⎥ ⎢ ₋ ⎥ ⎢ ⎥ ⎣ ⎦ 

[

]

1 1 2 2 0 , 0 T P H H J J J H ⎡ ⎤ ⎢ ⎥ =_{⎢ ⎥} = ⎢ ⎥ ⎣ ⎦   _. By Lemma 2, we obtain

0

T T T

A



+ Δ +

H J

 

J



Δ

H



<

,

An Improvement on Robust H∞ Control for Uncertain Continuous-Time Descriptor Systems 279

Let 1

Y  X− and partition Y as in (29). By Proposition 1, we have the following results.

Proposition 2: Yi, i =1, 2, 3, 4 are nonsingular.

Proof: Since X and X are invertible, by the ₄

matrix inversion formulas, we have

1 _{1 1 1} 1 2 2 4 1 2 1 1 1 1 1 3 4 4 3 4 4 3 2 4 3 4 X X X X Y Y X X X X X X X X X Y Y − _{− − −} − − − − − ⎡ ϒ −ϒ ⎤ ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ _{− ϒ + ϒ} ⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ where 1 1 2 4 3 X X X− X ϒ − . Since X2, X3, X4, and

ϒ

are all nonsingular, the above equality implies that

Y1, Y2, and Y3 are nonsingular. Finally, since X1 and X4

are nonsingular, Y4 can be rewritten as

(

)

1

1 1 1 1 1

4 4 4 3 2 4 4 3 1 2 .

Y =X− +X− X ϒ− X X− = X −X X− X − Therefore, Y₄ is nonsingular, too.

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Hung-Jen Lee was born in Taipei,

Taiwan in 1955. He received his Bachelor degree in Electronics Engineering from National Chiao-Tung University in 1977 and his Master degree in Electrical Engineer-ing from National Taiwan University in 1982. Since 1984, he has been an Instructor of Department of Electronic Engineering at National Kaohsiung University of Applied Sciences. His research interests are in the areas of circuit analysis and database web application.

280 Hung-Jen Lee, Shih-Wei Kau, Yung-Sheng Liu, Chun-Hsiung Fang, Jian-Liung Chen, Ming-Hung Tsai, and Li Lee

Shih-Wei Kau was born in Taichung,

Taiwan, in 1950. He received his diploma from the National Kaohsiung Normal University in 1974. He got research scholarship of DSE for Industry control at the Mamhann University, German in 1980 and 1981. Currently, he is working torward his Ph.D. at the Strathclyde University, UK. From 1974 to 1979 he served as a Teaching Assistant of Electrical Engineering Department at National Kaohsiung Institute of Technology (NKIT). In 1982 he returned to NKIT and served as an Instructor of EE and also the Chair of Computer center at NKIT. Now, he is working at National Kaohsiung University of Applied Sciences. His research interests are in the areas of neural genetic application in industry, control system integrated in remote control using neuron chip, PLC control system etc.

Yung-Sheng Liu was born in

Hsin-Chu, Taiwan, in 1980. He received his B.S. degree from Department of Electrical Engineering, Naitonal Yunlin University of Science and Technology, Yunlin, Taiwan, in 2002. He received his M.S. degree in the Electrical Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan, in 2004. Currently, he is working for Ingrasys Technology Inc., Tao-Yuan, Taiwan. His research interests include fuzzy control, linear system control, and singular systems.

Chun-Hsiung Fang was born in

Tainan, Taiwan, in 1963. He received his diploma from the National Kaohsiung Institute of Technology in 1983, the M.S. degree from National Taiwan University in 1987, and the Ph.D. degree from National Sun Yat-SenUniversityin 1997, all in Electrical Engineering. From 1987 to 1990, he served as an Instructor of Electrical Engineering Department at National Kaohsiung Institute of Technology and was promoted to be an Associate Professor in 1990. Since 1993, he has been a Full Professor. Currently, he is a Professor of Electrical Engineering Department at National Kaohsiung University of Applied Sciences. From 2000 to 2001, he was a Visiting Scholar at the University of Maryland, College Park, Maryland. His research interests are in the areas of robust control, singular systems, and fuzzy control.

Jian-Liung Chen received the B.S.,

M.S. degree in Automatic Control Engineering from the Feng Chia University, Taichung, Taiwan, in 1993, 1996, respectively, and the Ph.D. degree in Electrical Engineering from the National Sun Yat-Sen University in 2003. Currently, he is an Assistant Professor of the Department of Electrical Engineering, Kao-Yuan University, Lu-Chu Hsiang, Kaohsiung, Taiwan, where has been since 2005. His research interests include LMI approach in robust control and descriptor system theory.

Ming-Hung Tsai received the B.S.

degree in Electronic Engineering from the Feng Chia University, Taichung, Taiwan, in 2000, and the M.S. degree in Electrical Engineering from the National Sun Yat-Sen University in 2002. Currently, he joined the CNet Technology Inc., Hsin-Chu City, Taiwan, and is presently a Hardware Engineer. His recent research interests are in wireless communication systems, RF circuit design, and networking design.

Li Lee received the B.S. degree in

Control Engineering from the National Chiao Tung University, Taiwan, in 1978, the M.S. degree in Electrical Engineering from the National Cheng Kung University, Taiwan, in 1984, and the Ph.D. degree in Electrical Engineering from the University of Maryland, College Park, Maryland, in 1992. After graduation, he joined the Department of Electrical Engineering, National Sun Yat-Sen University. His research interests are in dynamical system theory and robust control design.