Design of Robust Quadratic Finite-Horizon Optimal Static Output Feedback Controllers for Linear Uncertain Singular Systems

544 IEEE SYSTEMS JOURNAL, VOL. 3, NO. 4, DECEMBER 2009

Design of Robust Quadratic Finite-Horizon Optimal

Static Output Feedback Controllers for Linear

Uncertain Singular Systems

Shinn-Horng Chen, Wen-Hsien Ho, and Jyh-Horng Chou, Senior Member, IEEE

Abstract—By complementarily fusing the robust stabilizability

condition, the orthogonal-functions approach (OFA) and the

hy-brid Taguchi-genetic algorithm (HTGA), an integrative method is

proposed in this paper to design the robust-stable and

quadratic-optimal static output feedback controller such that i) the linear

singular control system with structured parameter uncertainties is

regular, impulse-free and asymptotically stable and ii) a quadratic

finite-horizon integral performance index for the linear nominal

singular control system can be minimized. Based on some essential

properties of matrix measures, a new sufficient condition is

pre-sented for ensuring that the linear singular system with structured

and quadratically-coupled structured parameter uncertainties is

regular, impulse free and asymptotically stable. By using the OFA

and the robust stabilizability condition, the dynamic-optimization

problem for the robust-stable and quadratic-optimal static output

feedback control design of the linear uncertain singular system is

transformed into a static-constrained-optimization problem

repre-sented by the algebraic equations with constraint of robust

stabi-lizability condition; thus greatly simplifying the robust-stable and

quadratic-optimal static output feedback control design problem

of the linear uncertain singular system. Then, for the

static-con-strained-optimization problem, the HTGA is employed to find the

robust-stable and quadratic-optimal static output feedback

con-troller of the linear uncertain singular control system. One design

example of the robust-stable and quadratic-optimal static output

feedback controller for a mass-spring-damper mechanical system

with structured parameter uncertainties is given to demonstrate

the applicability of the proposed integrative approach.

Index Terms—Robust stabilizability, singular systems, static

output

feedback

control,

parameter

uncertainties,

orthog-onal-functions approach, genetic algorithm.

I. I

I

N recent years, there has been a growing interest in the

system-theoretic problems of singular systems due to

the extensive applications of singular systems in large-scale

Manuscript received November 30, 2008; revised September 23, 2009. First published December 15, 2009; current version published January 27, 2010. This work was supported in part by the National Science Council of Taiwan under Grants NSC 96-2221-E-151-055, NSC 96-2628-E-327-004-MY3, and NSC 97-2221-E-037-003.

S. H. Chen is with the Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung City, Taiwan.

W. H. Ho is with the Department of Medical Information Management, Kaoh-siung Medical University, KaohKaoh-siung City, Taiwan (e-mail: whho@kmu.edu. tw).

J. H. Chou is with the Institute of System Information and Control, National Kaohsiung First University of Science and Technology, Kaohsiung City, Taiwan (e-mail: choujh@ccms.nkfust.edu.tw).

Digital Object Identifier 10.1109/JSYST.2009.2037358

systems, circuits, power systems, economics, control theory,

robots, mechanical systems, and other areas [1], [2]. Recently,

several results and methods for the analysis and design of linear

singular control systems have been presented in the literatures

(see, for example, [1]–[10] and references therein). On the

other hand, it is well-known that an approximate system model

is always used in practice and sometimes the approximate error

should be covered by introducing parameter uncertainties in

control system analysis and design. Besides, in practice, it is

not always possible to have full access to the state variables

and only the partial information through a measured output is

available. Therefore, the study on the analysis and design of

linear uncertain singular systems with static output feedback

control is more realistic in control engineering. The complexity

of the static output feedback control problem has been proved

to be NP-hard [11]. Geromel et al. [12] have analyzed the

solv-ability of a necessary and sufficient condition for the existence

of a stabilizing static output feedback gain for continuous-time

linear systems. They have also provided simple procedures to

determine a suboptimal solution to the

control problem by

means of static output feedback control [12]. In recent years,

there are a few works studying the robust stability and

stabi-lizability problems for the linear uncertain singular systems

with static output feedback control in the literatures [4], [6],

[13]–[18]. Only robust stabilization is often not enough in

con-trol design. For the concon-trol systems design, it is often of interest

to synthesize a quadratic-optimal controller such that the

con-trol objective of minimizing a quadratic finite-horizon integral

performance criterion is achieved [19], [20]. However, to the

authors’ best knowledge, there are no literatures to investigate

the issue of designing the robust-stable and quadratic-optimal

static output feedback controllers for the linear uncertain

singular systems, where a quadratic finite-horizon integral

performance index is minimized.

The purpose of this paper is to propose an integrative

opti-mization method to design the robust-stable and

quadratic-op-timal finite-horizon static output feedback controller such that

i) the linear singular systems with structured parameter

uncer-tainties is regular, impulse-free and asymptotically stable and

ii) a quadratic finite-horizon integral performance index for the

linear nominal singular control system can be minimized. The

proposed integrative method complementarily fuses the

orthog-onfunctions approach (OFA), the hybrid Taguchi-genetic

al-gorithm (HTGA) and the robust stabilizability condition, where

the robust stabilizability condition is derived in this paper for

ensuring that the linear uncertain singular control system is

CHEN et al.: ROBUST QUADRATIC FINITE-HORIZON OPTIMAL STATIC OUTPUT FEEDBACK CONTROLLERS 549

Fig. 2. Control forces of the designed example.

Remark 3: For some engineering problems, the performance

of finite-time horizon is considered for the performance

require-ment of both transient and steady-state responses [19]. In

gen-eral, the optimal linear-quadratic control over a finite-horizon

requires a time-varying feedback gain matrix

. It is not easy

to obtain and to implement the time-varying feedback gain

ma-trix

. But, by using the proposed integrative method, the

designed robust optimal quadratic finite-horizon static output

feedback controller having constant gain can be easily obtained

and implemented.

V. C

The robust-stable and quadratic-optimal static output

feed-back control problem of the linear uncertain singular control

systems has been investigated in this paper. By using the OFA,

an algebraic algorithm is presented in this paper to solve the

linear nominal singular feedback dynamic equation. Then, the

presented algebraic algorithm is complementarily fused with

the HTGA to design the robust-stable and quadratic-optimal

static output feedback controller of the linear uncertain singular

system such that the control objective of directly minimizing a

quadratic integral performance index subject to the constraint

of robust stabilizablity criterion can be achieved. The quadratic

integral performance index is also converted into the algebraic

form by using the OFA. Since, by using the OFA and the

proposed robust stabilizablity criterion, the

dynamic-opti-mization problem for the robust-stable and quadratic-optimal

static output feedback control design of the linear uncertain

singular control system can be replaced by a

static-param-eter-constrained-optimization problem represented by the

algebraic equations with constraint of robust stabilizablity

criterion, and since the proposed algorithm only involves the

algebraic computation, the design procedures of robust-stable

and quadratic-optimal static output feedback controller for the

linear uncertain singular system may be either greatly reduced

or much simplified accordingly. Therefore, this proposed

inte-grative approach facilitates the design task of the robust-stable

and quadratic-optimal static output feedback singular control

system. The illustrative example regarding control problems of

two DOF uncertain mass-spring-damper mechanical system has

shown that the proposed integrative approach is effective for

designing the robust-stable and quadratic-optimal static output

feedback controller of the linear uncertain singular system.

A

The authors thank the reviewers and the associate editor for

their constructive comments and suggestions.

R

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Shinn-Horng Chen received the B.S. and M.S.

degrees in mechanical engineering from the National Taiwan University of Science and Technology, Taipei, in 1987 and 1989, respectively, and the Ph.D. degree in mechanical and mechatronic engineering from the National Sun Yat-Sen University, Kaoh-siung City, Taiwan, in 1996.

He is currently a Professor of mechanical engi-neering with the National Kaohsiung University of Applied Sciences, Kaohsiung City, where he was a Professor and the Chairman with the Mechanical Engineering Department from October 2006 to July 2009. His research interests include the areas of robust control, vibration control, and optimal design.

Wen-Hsien Ho received the B.S. degree in marine

engineering from the National Taiwan Ocean Uni-versity, Keelung, Taiwan, in 1991, the B.S. degree in industrial and information management from the National Cheng-Kung University, Tainan, Taiwan, in 1998, and the M.S. degree in mechanical and automa-tion engineering and the Ph.D. degree in engineering science and technology from the National Kaohsiung First University of Science and Technology, Kaoh-siung City, Taiwan, in 2002 and 2006, respectively.

He is currently an Associate Professor with the Department of Medical Information Management, Kaohsiung Medical Uni-versity, Kaohsiung City. His research interests include intelligent systems and control, computational intelligence and methods, robust control, and quality engineering.

Jyh-Horng Chou (M’04–SM’04) received the B.S.

and M.S. degrees in engineering science from the National Cheng-Kung University, Tainan, Taiwan, in 1981 and 1983, respectively, and the Ph.D. degree in mechatronic engineering from the National Sun Yat-Sen University, Kaohsiung City, Taiwan, in 1988.

He is currently a Professor, the Vice President, and the Acting President of the National Kaohsiung First University of Science and Technology, Kaohsiung City, Taiwan. His current research interests include intelligent systems and control, computational intelligence and methods, au-tomation technology, robust control, and quality engineering. He is an Editorial Member or an Associate Editor of nine international journals.