Robust D-stability of Generalized State-Space System with One-parameter Uncertainties

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Proceedings of the 1999 IEEE

International Conference on Contml Applications

Kohalo Coast-Island of Hawa'l, Hawal'l. USA August 22-21.1999

TuM4-3

2:40

Robust 27-Stability of Generalized State-Space

Systems with One-Parameter Uncertainties

1 Chun-Hsiung Fang, Chun-Lin

Lu

Lin Hong,

and

Shih-Wei

Kau

Li

Lee

Department

of

Electrical Engineering

Department

of

Electrical Engineering

National Kaohsiung Institute

of

Technology

Kaohsiung 807, TAIWAN

National Sun Yat-Sen University

Kaohsiung

804,

TAIWAN

chfangQrnai1.ee.nkit.edu.tw

[email protected]

Abstract

The robust D-stability problem for generalized state- space systems with uncertainties in the form of one- parameter family of matrices is investigated in this p a per. The maximal bounds of perturbations for simul- taneously preserving the regularity, impulse-immunity, and D-stability are analytically derived.

1 Introduction and notations

The uncertain generalized state-space model considered in this paper is described by [l]

The following notations will be used throughout the paper. C-,C+, and

CO

denote the sets {s E C : Re(s)

<

0}, {s E G : Re(s)

>

O}, and {s E C : %e(s) = 0}, respectively. The set of roots of. JsP -

&I

= 0 is denoted by A(P,&). ( a , b ) stands for an open interval. q ( M )

stands for the i t h nonzero singular value of matrix M . M" means complex conjugate (no transpose) of matrix

M . X$,,(M) and X;,,(M) represent the largest positive and the smallest negative real elenients in A ( I , M ) ,

respectively. If there is no positive real element in

A ( / , M ) , set X$,,(M) = O+. Set X;,,(M) = 0- if

A(/, M ) has no negative real elements. I p stands for the identity matrix with dimension p . The empty set is denoted by

0. fB

and 8 mean the Kronecker sum and the Kronecker product.

E i ( t )

=

( A

+

A ) z ( t ) ₍₁₎

where E ,

A , A

E Cnxn and the matrix E may be singular. For brevity, we may use the pair { E , A

+

A} t o represent the system (1) and, hence, the nominal sys- tem (i.e.

A

= 0) is equivalently denoted by the Pair

{ E ,

A } .

In this paper, a class of structured perturbations is considered. Specifically, tlic A in (1) is in the form of one-parameter f a m i l y of matrices 1111

2 Preliminaries

For solving the problem, three intervals are defined firstly:

(klmi,, klmaz): the largest interval such that the pair

where k is a real parameter and

Hi

t Cnx", i =

1 , 2 , . . .

,

q , are given matrices indicating different perturbation directions. The uncertainty (2) covers a va- riety class of perturbations 1111. Assume the nominal system { E , A } is regular and impulse-free, and all

fi-

nite poles of { E , A } lie in the regions D (termed D-

stability in the literature [12J). The considered problem is to find the maximal allowable interval for k such that { E , A

+

A} is regular, impulse-free, and D-stable.

The

related results of this problem can be found in

[4, 8, l l , 12). However, there is no paper discussing the problem mentioned above.

TAIWAN under contract No. NSC-88-2213-E151-00L1. 0-7803-5446-XI99 $10.00 0 1999 IEEE

(kzmin, kzmol.): the largest interval such that the pair

{ E , A

+

CE1

P H , } is impulse-free for all k E

( h m i n ,

k ~ ~ ~ ~ ) ;

(klrnin, k3mor): the largest subinterval within (kzmi,,

I C Z , , , ~ ~ ) such that A(E,

A+CL1

ki",)

c

D for all k t (kimin, k3maz).

It has been shown that (kz,,,, kz,,,)

c

(!qmi,, klmaS)

[a,

101. SO, according to the above definitions, the de- sired interval is actually equal to (k3m,nr k3maz). Using the siiigular value decomposition, (kz,,,, kzmoz) can be easily obtained from E , A , and Hi [3,

Ql.

Therefore, we will focus on finding (k3,,,inl k3moz).

work w&5 supported by the National

science

council of 1078

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by (24) is of the form (1) where 5 Conclusions 0 0 0 1 0 0 -

0

1 - 1 0

0 0 0 0

0 1 0 0 0 0 0 0 ’ 0 0 0 0 0 0 0 0 0 0 0 0

E =

A = ‘ 0 0 0 0 0 0 - 0 0 0 0 0 0 0 0

0

0 0 0

-el,

o

o o

0 0

0 -ez2

0 o o

o

0

0 - c z 0 0 0

’

and

A

= kH1

+

k2H2

+

k3H3 with 0 0 0

0 0 0 -

0

0 0 0

0

0 0 0 4 1 3

0

0 0

0 4 2 3 0 0 0 0 0 0 - c ~ o o

-

o H3 =

H I

= ‘ - T i -TI -0.2Ti 0 0 0 0 7 2 0 0 0 0

0

o a o

-ell 0 0 0 0 0

0

-e21 0 0 0 0 0 0 - c l 0 0 0

T h e maximal perturbaton bounds for poleclustered in- side the specified regions for uncertain generalized state- space systems are analytically obtained. Two regions which are commanly used in control system design are discussed.

References

[l] L. Dai, Singular Control Systems - Lecture notes in control and information sciences, Springer-Verlag, Berlin, 1989.

[2]

C.-H.

Fang and F.-R. Chang, “Analysis of stability robustness for generalized state-space systems with structured perturbations,” Systems & Control Letters, vol. 21, no. 2, pp. 109-114, 1993.

[3] C.-H. Fang, “Robust stability of uncertain generalized state-space systems,” P h D Dissertation, National Sun Yat-Sen University, Taiwan, 1997.

[4]

M. Fu

and

B.

R. Barmish, “Maximal unidirec- tional perturbation bounds for stability of polynomials and matrices,” Systems & Control Letters, vol. 11, pp. 173-179, 1988.

[5]

R.

A. Horn and

C. R.

Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 1991. [6] L. Lee and

C.-H.

Fang, “An improved bound for stability robustness of uncertain generalized state-space systems,” Proc. of 1994 ACC, pp. 240-241.

[7]

L.

Lee and

C.-H. Fang,

“A simple method to cal- culate the exact bound

for

robust stability of uncertain discrete-time generalized state-space systems,” Proc. of 34th

CDC,

pp. 1348-1353, 1995.

[8]

L.

Lee, C.-H. Fang, and J.-G. Hsieh, “Exact uni- directional perturbation bounds for robust stability of uncertain generalized state-space systems : Continuous- time case,“ Automatica, vol. 33, no. 10, pp.1923-1927, 1997.

[9]

L.

Lee and

C.-H.

Fang, “An L F T appraoch to robust stability of uncertain generalized state-space systems”, Index. 548, 1998,

[lo] L. Qiu and

E.

J. Davison “The stability robustness of generalized eigenvalues,” IEEE Rans. Automat.

Control, vol. 37, no. 6, pp.886-891, 1992.

[ll] L. Saydy, A. L. Tits, and

H.

Abed, “Guardian maps and the generalized stability of parametrized fam- ilies of matrices and polynomials,” Math. of Control, Signals, and Systems, vol. 3, pp. 345-371, 1990. {-0,0172, -0.008,0,0.297,0.307,0.788,0.6283}. [12]

C. B.

Soh, “Stability robustness measures of state-

space models,” Int. J. Syst. Science, vol. 22, no. 10, pp. 1867-1884, 1991.

Thus (k3min,k3mor) = (-58.0199,1.5916).

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113)

G. C.

Verghese,

B. C.

Levy, and

T.

Kailath,

“A

generalized state-space for singular systems,” IEEE

naris Automat. Control, vol.

26,

no. 4, pp.811-831, 1981.

1141

K.

Zhou,

J. C.

Doyle, and

K. Glover, Robust

and

Optimal Control, Prentice-Hall,

New Jersey, 1996.

I

t-

“

Figure 3: T h e open left half region with angle=&

Figure 1: An open conic sector region, 0

5

81

5 5,

0

5

82

5

4

Figure 4:

A

region with certain relative stability

Figure 2: The open left half region with a n g l e d 1

Figure 5: An open disk D ( h , r )

Figure 6: An autonomous circuit example for uncertain generalized state-space models