In this paper a one-parameter family of perturbations is considered for an uncertain singular system which is nominally regular, impulse-free and stable. Based on linear fractional transformations, two different approaches in time-domain and frequency-domain are provided to derive a closed form solution for the maximal bounds under which regularity, impulse-immunity and stability are robustly preserved. The approaches can be successfully applied to both continuouddiscrete-time singular systems. L INTRODUCTIONThe system-theoretic problem of singular systems has been an active research area in the recent years. Several control problems such as pole placement, eigenstructure assignment, the controllability/observability properties, compensator and observer design, have been investigated for linear singular systems [1]-[5]. Recently, there has been a growing interest in solving the robust control problem of uncertain singular systems [6]-[ 151.Consider an uncertain linear singular system described by the equation of the form (1) Ek(t) = ( A + A)x(t)for the continuous-time cases; and the formfor the discrete-time cases, where x E%" and E, A ,with E possibly singular of rank r, 0 < r I n .By the definitions for singular systems [I], [16], 1171, the uncertain singular system (1) or (2) is said to be regular if the characteristic polynomial det(;lE -A -A) is not identically zero, impulse-free (i.e. without dynamical infinite modes) if rank(@= deg(det(;lE -A -A)), and stable iE all the roots of equation d e t ( E -A -A) = 0 have negative real parts for the continuous-time cases, or have moduli less than 1 for the discrete-time cases. It is worth noting when the robust stability problem of an uncertain singular system is investigated, the properties of regularity as well as impulse-immunity should be considered simultaneously [ 81, [ 91, [ IS].In this paper the robust regularity, impulse-immunity and stability (RZ.8.) of system (1) or (2) with A belonging to a one-parameter family of perturbations described as A:= kHl + k 2 H 2 +.a --+ k'H,will be investigated, where k is a real parameter and H,, j = 1,2,..-, I, are specified matrices to denote different directional perturbations. To simpllfy notation, the matrix pair (E,A +A) is used to stand for the system (1) or (2). In particular, A = k H l , the maximal unidirectional perturbation bounds for robust stability of systems ( E , A +~H , ) have been investigated in [11], [12], [15]. Moreover, if E=I, the stability issue of systems ( I , A + A)have been addressed in [ 191 based on guardian map.Assume that the nominal system ( E , A ) is regular, impulse-free and stable for the uncertain system (E,A +A), the goal of this paper is to search for the maximal interval (k;,", k;-) with k;," < 0 and k;m > 0 such that system (1) or (2) is guaranteed to be still regular, impulse-free and stable. The same issue was also considered by Fang and Lee [13], whose basic idea is to transform a stability robustness problem to a rank robustness problem. However, the extension from continuous-...
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