ISSN:
1433-0490
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
Notes:
Abstract In this paper we consider the problem of uniform asymptotic stabilization of neutral delay differential systems by a suitable choice of linear feedback law, where the controller is static and at timet 0 has full knowledge ofx(t 0) ∈ ℝ n as well asx(t 0 − d) for finitely many delays present in the system. Using complex analysis in several variables in a Banach algebra context we present a solution to this problem in the form of a sufficient condition for the existence of such a stabilizing feedback gain. This condition is a weak form of (algebraic) reachability together with a condition, which we call resolvability, on the solution of an associated algebraic Riccati equation. In the case of commensurable delays our results apply to neutral systems having aD-operator which is stable in the sense of Cruz and Hale. In the non-commensurate case, our results hold for systems with aD-operator satisfying a stronger condition on the spectrum of the evolution equation, which we call formal stability. A graphical criterion, in the spirit of the multi-dimensional Nyquist theory, is presented for formal stability ofD. We find these results especially interesting in light of recent work [39] which shows that, for a special class of systems, formal stability is necessary for stabilization by a feedback gain of the type considered here. This, in fact, gives a counterexample [39] to the well known condition of Pandolfi [40]. Included among our results is an extension to the neutral case of the result in [11] which states that if the pointwise Kronecker indices are constant then the system is feedback equivalent to a delay-free system. As corollaries to this result we obtain some existing results [33] on canonical forms for time-delay systems, and, in addition, exhibit a large class of systems which are resolvable.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01744436
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