Optimal Output Feedback Control for Linear Uncertain Systems Using LMI-Based Approach and Genetic Algorithm

2006

IEEE

International

Conference

on

Systems,

Man,

and

Cybernetics

October 8-11, 2006,

Taipei,

Taiwan

Optimal

Output Feedback Control for Linear Uncertain

Systems

Using LMI-Based Approach and Genetic Algorithm

Shinn-Homg Chen, Wen-Hsien

Ho

and

Jyh-Homg Chou,

Senior

Member,

IEEE

Abstract This paper considers the robust-optimal design problems of output feedback controllers for linear systems with both time-varying elemental (structured) and norm-bounded (unstructured) parameter uncertainties. A new sufficient condition is proposed in terms of linear matrix inequalities (LMIs) for ensuring that the linear output feedback systems with both time-varying elemental and norm-bounded parameter uncertainties are asymptotically stable, where the mixed quadratically-coupled parameter uncertainties are directly considered in the problem formulation. A numerical example is given to show that the presentedsufficient condition is less conservative than the existing one reported recently. Then, by integrating the hybrid Taguchi-genetic algorithm (HTGA) and the proposed LMI-based sufficient condition, a new integrative approach is presented to find the output feedback controllers of the linear systems with both time-varying elemental and norm-bounded parameter uncertainties such that the control objective of minimizing a quadratic integral performance criterion subject to the stability robustness constraint is achieved. A design example of the robust-optimal output feedback controller for the AFTI/F-16 aircraft control system with the time-varying elemental parameter uncertainties is given to demonstrate the applicability of the proposed new integrative approach.

1. INTRODUCTION

In general, a mathematical description is only an approximation of the actual physical system and deals with fixed nominal parameters. Usually, these parameters are not known exactly due to imperfect identification or measurement, aging of components and/or changes inenvironmental conditions. Thus, it is almost impossible to get an exact model for the system due to the existence of various parameter uncertainties. Here, we consider linear state-space systems with time-varying uncertain parameters in the system matrix, input matrix, and output matrix. Because the output feedback controller design is usually based on the nominal values of these system matrices, it is interesting to know whether the closed-loop system remains asymptotically stable in the presence of time-varying Manuscript received March 30, 2006. This work wassupported bythe National Science Council, Taiwan,Republic ofChina, under grant number NSC 94-221 8-E327-001.

S. H. Chen is with the Department ofMechanical Engineering, National Kaohsiung University of Applied Sciences, 415 Chien-Kung Road, Kaohsiung 807, Taiwan, R.O.C. (e-mail:

shchen@cc.kuas.edu.tw).

W. H. Ho is with the Institute of Engineering Science and Technology, National Kaohsiung First University of Science and Technology, 1 University Road, Yenchao, Kaohsiung 824, Taiwan,

R.O.C.(e-mail: wenhsienl

102gyahoo.com.tw).

J. H. Chou is with the Institute of Engineering Science and Technology, National Kaohsiung First University of Science and Technology, I University Road, Yenchao, Kaohsiung 824, Taiwan,

R.O.C. (phone: +886-7- 6011000; fax: +886-7-6011066; e-mail: choujh(ccms.nkfust.edu.tw).

uncertain parameters. Applying those existing robust stability analysisresults [1]-[13] tosolve this problem is not easy, in that after output feedback, there will be coupled terms of parameters in the closed-loop system matrix because of the uncertainparameters in bothinput and output matrices [14]-[15]. Though we may regard thesecoupled terms as new independentparameters ifwe insist on using thoseexisting robuststability analysis results, literatures [14] and [16]have showed that a conservative analysis conclusion may be reached. Therefore, literatures [14] and [16] investigated the robust stability problem of linear systems with constant output feedback in the presence oftime-varying elemental (structured) parameteruncertainties by directly considering the coupled terms in the problem formulation. Su and Fong [14] used the Lyapunov method to analyze the robust stability of linear continuous-time systems with quadratically-coupledelemental uncertainties. Tseng etal. [16]applied thestructuredsingularvaluetechniquetosolve the robuststability analysis problem of linearcontinuous-time systems withquadratically-coupledelemental uncertainties. Inaddition,it is well known that an approximate system model is always used in practice and sometimesthe approximationerrorshouldbecovered by introducing both elemental (structured) and norm-bounded

(unstructured) parameter uncertainties in control system analysis

anddesign [17]. That is,it is not unusual that at times we haveto

dealwith a system simultaneously consistingof two parts: onepart has only the elemental parameteruncertainties, and the other part has the norm-bounded parameter uncertainties. Therefore, very recently, based on the Lyapunov approach and some essential properties of matrix measures, Chen and Chou [15] proposed a sufficient conditionto studytheproblem ofstabilityrobustness for linear output feedback systems with bothtime-varying elemental

and norm-boundedparameteruncertainties bydirectly considering

the mixed quadratically-coupled uncertainties (i.e., quadratically-coupled elemental, quadratically-coupled

norm-bounded, and coupled elemental and norm-bounded uncertaintiesappearingtogether)intheproblemformulation. Here it

shouldbe noted thatonlythe articles in[14]-[16] studiedthe robust

stability of linear continuous-time systems with output feedback controllers and time-varying uncertain parameters by

directly

considering thecoupled terms intheproblem formulation. Thatis,

the research on the stability robustness of linear continuous-time systemswith outputfeedback controllersandtime-varying uncertain

parametersbydirectlyconsideringthecoupledtermsintheproblem

formulation is considerably rare and almost embryonic. Besides,

here it should be also notedthat, for the case ofonly considering

time-varyingelementalparameteruncertainties,ChenandChou[15] haveshown that their sufficient condition is less conservative than those in [14] and[16].

On the other hand, only robust stability is often notenoughin control design. Theoptimaltrackingperformance is also considered in many practical control engineering applications. Hence, one of the most important study issues for linear continuous-time output feedback control systems withtime-varying uncertain parameters is to find the output feedback controllers that minimize an

H,

performanceindex (i.e., the integral of the squared error(ISE) or the integral of thetime-weighted squared error (ITSE)) subject to the stability robustness constraint, where the quadratically-coupled

parameteruncertaintiesaredirectlyconsidered in the robuststability

problem formulation. But, to the authors' best knowledge, the mentioned-abovestudyissuehasnotbeen discussed in the literature. Therefore, the purpose of this paper is to propose a newmethodfor

findingthe outputfeedback controllersofthe linear continuous-time systems with both time-varying elemental and norm-bounded

parameter uncertainties such that the control objective of minimizing an

H,

performance index subject to the stability

robustness constraintisachieved. Theproposednewmethodisan

integrative approach which integrates the linear-matrix-inequality (LMl)technique and thehybridTaguchi-genetic algorithm (HTGA),

where the LMI technique is used to derive the robust stability condition and the mixed quadratically-coupled parameter

uncertaintiesaredirectly consideredinthe robuststability problem formulation. The reasonwhythe HTGA isappliedinthis paper is thatChou andhisassociateshave shownthat the HTGA mayobtain better andmorerobust results than thoseexisting improved genetic algorithms reportedin theliterature[1 8]-[19].

This paper isorganizedasfollows.InSection 2, an LMI-based

robuststabilitycondition ispresentedfor the linear continuous-time systems with output feedback controllers as well as both

time-varying elementaland norm-bounded parameteruncertainties by directly considering the mixed quadratically-coupled

uncertainties (i.e., quadratically-coupled elemental, quadratically-coupled norm-bounded, and coupled elemental and

norm-bounded uncertainties appearing together) in the problem

formulation. A numericalexampleand the conservatismcomparison

between the proposed LMI-based sufficient condition and the

non-LMI-basedsufficient conditionproposedin[15] arealsogiven

inthissection. The HTGA forthe mixed H2I LMI robust-optimal

output feedback controllers design is described in Section 3. A

design example of the AFTI/F-16 aircraft control system is also given in this section for demonstrating the applicability of the

proposed new integrative method. Finally, Section 4 offers some

conclusions.

Il.

STABILITY ROBUSTNESS ANALYSIS

Consider thelinearuncertain system withthestate-spacemodel

x~(t)

=

A(t)x(t)+B(t)u(t)

,(1)

y(t)

=

Q(tx(t

'

(2)

where

x(t)

cR' isthe statevector, y(t)ERP isthe output vector, u(t) ERq is the

input

vector,and

in in

A(t)

A +s

(t)A

+

A(t),

B(t)=B +

,cj

(t)Bj

+B(t),

j=l j=l

m

and C(t)=

CO

+±

ZCj (t)Cj

+C(t) (3)

Jl

are the system matrix, the input matrix and the output matrix, respectively, in which

8j

(t)

(

ej<

8j

(t)<£ / , and j=1,

2,...,

m)

are the time-varying elemental uncertainties;

Aj

,

Bi

and

CJ

(

j=1,2,...,m

)

are,

respectively,

the

given

nxn, nxq and pxn constantmatrices whichareprescribed priortodenotethe linearly

dependent information on time-varying elemental uncertainties

8£/(t)

S; m is the number of

independent time-varying

uncertain parameters. The time-varying norm-bounded uncertain matrices

A(t), B(t) and

C(t)

areassumedtobe

bounded,

i.e.,

A(t) <

A3,

||B(t)||<

,2

, and ||C(t) <

p,8

(4)

where /3, /32' and /3 are non-negative real constant number, and

111

denotes any matrixnorm.

In this paper, we only discuss the static output feedback controllers. Thus, the closed-loop system equation of the linear uncertain system canbe expressed as

x(t)

LA

+ Z

_,(t)Ej

+Z

jZi (t)Ck(t)Eok

+

F(t)

x(t),

(5)

j=i j=l k=l

where K denotes the outputfeedback gain matrix,

AO AO

+BoKC0,

Ej

=ABj

+B(KCj

+±BKC0,

Elk =-(B KC +

BkKCj),

and

F(t)

A(t)

+

BoKC(t)

+

B(t)KC,

+

B(t)KC(t)

+

Z£j

(t)(B1KC(t)+B(t)KC ).

j=l

From Eqs. (4)and (6), we can get that

||F(t)II

<

K

, inwhich

(6)

(7)

m

18

=A,9

+

9211KCO

11

+

9311BOKI|

+,92,3

II|K||I

+

J:X

(3

JIB.,KII

+

8211KC ||),

and jmax{|e, j} Therefore, the norm-bounded uncertain

matrix F(t) canbe expressed by

F(t)=

A(t)

(8)

where

A(t)

is an unknown matrix function which is bounded by

A(t)

EQ:=

{A(t)IIA(t)II

<1,theelementDfA(t)areLebesgurmeasurabl4

The robuststability problemtobe studiedcanbedescribedas:

given an output feedback gain matrix K for making

A.

a stable

matrix, determine the condition such that the linear closed-loop uncertainsystemin(5)isstill asymptotically stable.

Inwhatfollows, under the assumption thatanoutputfeedback gain matrix K has been specified to make

AO

a stable matrix in

advance, by using the Lyapunov approach, we present a new

sufficient condition interms of linearmatrix inequalities (LMIs), which canbe efficiently solved by using thenumerically efficient

convex programming algorithms [20] for ensuring that the linear

closed-loop uncertainsystemin (5)remainsasymptotically stable.

Theorem:

The linear closed-loop uncertain system in (5) remains

asymptotically stable, if, for the specified output feedback gain

matrix K and for a specified constant a with 0<a

<1,

there

exist asymmetric positive definite matrix P andaconstant r>0

suchthatthe following LMIsaresimultaneously satisfied:

UP±PU ±TI

mi P <0,

(h

and thesamematrix P makes the

following inequality

hold:

(10)

Is m mn

)

"Ej(t)(0

+

I

I

- (t)c,

(t),4j,

<I X

Ji=ll j=l k=l

where

i-1,2,...,2;

A

=aAo

A, (1

-a)A,;

Ui

= ;(t)Ej,8(t)

=,/j

or8j;

4137

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Figure 1. Convergence of H2 tracking performance (J for the

AFTI/F-1 6control systeminExample2.

2.5 2 1.5 rw 0.5 05 105 0 2 4 6 10 12 14 16 18 20 Time(sec)

Figure 2. Desired output response (dash line) and actual output

response (solid line)ofthe uncertain AFTI/F-1 6 aircraft controlsysteminExample2.

4141

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