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PERTURBATION BOUNDS FOR ROOT-CLUSTERING IN A CIRCULAR REGION OF LINEAR CONTINUOUS SYSTEMS

Abduldmir A. Abdul-Wahab Department of Technical Affairs

Ministry of Communications P.O. Box 31407 Kuwait

M.A. Zohdy Department of Electrical and Systems Engineering

Oakland University Rochester, MI 48309

Abstract A sufficient condition is formulated for a perturbed continuous system to guarantee that all the eigenvalues of the nominal system will remain in a circle in the left half complex-plane. An illustrative example is included.

1. Introduction The problem of maintaining the stability of a nominally stable system subjected to perturbations has been considered by many researchers. Different stability robustness measures are developed in the literature of control theory. Explicit bounds on the perturbation of a linear continuous system to maintain stability are given by Patel and Toda [3] via the Lyapunov stability analysis. Less conservative bounds for structured perturbations are obtained by Yedavalli [4]. Improved meausres of stability robustness are derived by Yedavalli and Liang [5] to reduce the conservatism in these bounds by using a state transformation. Zhou and Khargonekar [6] assumed the perturbations in the various elements of the system matrix are dependent on only a small number of uncertain parameters which may vary independently. In this paper, we extend the idea of finding robustness measure bounds for stability in the presence of perturbations to eigenvalue clustering in a circular region in the left half plane (LHP) such that the eigenvalues of the perturbed system stay in that region in spite of perturbations.

2. Problem Development It has been shown in [2] that the eigenvalues of a square matrix A lie within a specified circle of radius r and with origin -a in the complex-plane if and only if

has a positive definite symmetric solution P for any arbitrary positive definite symmetric matrix Q, where a is any real constant. Equation (1) leads to

a ATP + aP A +ATP A +(a2 - r2)P = -r2 Q. ( 2)

PI (3) A ~ P + P A = -[LQ + - A ~ P A+-

= -T. (4)

Thus 2 1 (a 2- r2)

a a a

Now consider root-clustering of a linear continuous system in a circular region in the left half complex plane. Thus T is a symmetric positive definite matrix for a symmetric positive definite matrix P when A is a stable matrix, where a > 0 and -a+r 0. Theorem: The eigenvalues of the perturbed continuous system

remain in the circular region of the eigenvalues of the nominal stability matrix A, in the left half complex plane of radius rand with origin at -a, if

4 t P '4 x(t)tr[ x(t)l ( 5 )

where P is the symmetric positivedefinite solution of (I), 11.11 is any vector norm, hmin is the minimum eigenvalue, and hmax is the maximum eigenvalue. Proof: Define the Lyapunov function

V(x(t))= xT(t)Px(t). Thus,

(7)

V(x(t))=;r (t)Px(t)+xT(t)P;c(t) (8)

+x(t)m x(t)l (9)

=-xT(t)T x(t)+2(x(t)Pr[ x(t)])E (10)

=xT(t)[ATP+PA]x(t)+TT[ x(t)]Px(t)

where P>o is the solution of equation (1) necessary for root clustering of the stability matrix A in a circular region of radius rand origin at -a. A sufficient condition for the eigenvalues of a continuous system to lie in the LHP is that the Lyapunov function V(x(t))>O satisfies

V(x(t))<O v x(tP0. (11) Suppose that the sufficient condition of the theorem is satisfied. Then after some manipulation, we obtain

and therefore all the eigenvalues of the perturbed system (5) lie in the specified circular region in the LHP.

V(x(t))=-xT(t)[ T-Amin(T)In ]x(t) (12)

747

Consider now the perturbation T[x(t)] is linear in x(t). Then the perturbation matrix T[x(t)] becomes

and equation (5) becomes

The sufficient condition (6) becomes

Remark 1: In the case E denotes some given positive bound on all elements of the perturbation matrix E(t), the sufficient condition in (6) can be replaced by

P E I- n

r[ x(t)l=E(t)x(t) (13)

x(t )=(A+E(t))x(t). (14)

IIECt)ll~ P. (15)

( 16)

3. Example Consider the continuous perturbed system described by

with x(t)=(A+E(t))x(t) (14)

A=[" -3 -4 '1 The eigenvalues of A are -1 and -3. Consider root clustering inside a circular region with r=2 and origin at -a=-2. Solving (1) with Q=12, we obtain

4.533 2.133 '+.133 2.41

with the eigenvalues 5.852 and 1.082. Thus, p =0.224. Therefore, a sufficient condition for the eigenvalues of the perturbed system to remain in the specified circular region is llE(t)1160.224 and the upper bound E on the elements of the perturbation matrix E(t) is €60. I 12. Repeating the above example with r =5 and a=5, we obtain with Q=12,

6.653 1.261 '=[ 1.261 1.424 1

with the eigenvalues 6.941 and 1.136. Thus p=0.394. So, the upper bound on all the elements of E(t) is ~~0 .197 .

4.Conclusion The idea of stability robustness of a continuous system has been extended to rootclustering inside a circular region in the left half complex- plane. Bounds were obtained for weakly structured perturbations. Bounds can be derived for highly structured perturbations, dependent variations and interval matrices [ l ] .

References [ I ] A.A. Abdul-Wahab, "Perturbation Bounds for Root Clustering of Linear Continuous-Time Systems," Int. J. Systems Science, vol. 22, no. 5, pp. 921-930, May 1991. [2] K. Furuta and S.B. Kim, "Pole Assignment in a Specified Disk," IEEE Trans.Automat. Contr., vol. AC-32, no. 5, pp. 423427, May 1987. (31 R.V. Patel and M. Toda, "Quantitative Measures of Robustness for Multivariable Systems," Proc. Joint Automatic Control Conference, 1980

[4] R.K. Yedavalli, "Perturbation Bounds for Robust Stability in Linear State Space Models," Int. J. Contr., vol. 42, pp. 1507-1 51 7, 1985. [5] R.K. Yedavalli and Z. Liang, "Reduced Conservatism in Stability Robustness Bounds by State Transformation," IEEE Trans. Automat. Contr., vol. AC-31, pp. 863-866, 1986. [6] K. Zhou and P.P. Khargonekar, "Stability Robustness Bounds for Linear State-Space Models with Structured Uncertainity," IEEE Trans. Automat. Contr.. vol. AC-32, no. 7, pp. 621-623, 1987.

I.

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Circular region in the left half of the complex plane.

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