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A several complex variables approach to feedback stabilization of linear neutral delay-differential systems

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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) ∈ ℝn 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.

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Authors and Affiliations

  1. Department of Mathematics and Division of Applied Sciences, Harvard University, Cambridge, MA, USA

    Christopher I. Byrnes

  2. School of Electrical Engineering, Cornell University, Ithaca, NY, USA

    Mark W. Spong

  3. Department of Systems Science and Mathematics, Washington University, St. Louis, MO, USA

    Tzyh-Jong Tarn

Authors
  1. Christopher I. Byrnes
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  2. Mark W. Spong
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  3. Tzyh-Jong Tarn
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Additional information

Research partially supported by NASA under Grant NSG-2265 and NSF under Grant ENG-79-09459.

Research partially supported by the NSF under Grants ECS-8017184 an INT-7902976 at Washington University and ECS-8214262 at Cornell University.

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Byrnes, C.I., Spong, M.W. & Tarn, TJ. A several complex variables approach to feedback stabilization of linear neutral delay-differential systems. Math. Systems Theory 17, 97–133 (1984). https://doi.org/10.1007/BF01744436

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  • DOI: https://doi.org/10.1007/BF01744436

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