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Stability independent of delay using rational functions. (English) Zbl 1175.93195
Summary: This paper is concerned with the problem of assessing the stability of linear systems with a single time-delay. Stability analysis of linear systems with time-delays is complicated by the need to locate the roots of a transcendental characteristic equation. In this paper we show that a linear system with a single time-delay is stable independent of delay if and only if a certain rational function parameterized by an integer $$k$$ and a positive real number $$T$$ has only stable roots for any finite $$T\geq 0$$ and any $$k\geq 2$$. We then show how this stability result can be further simplified by analyzing the roots of an associated polynomial parameterized by a real number $$\delta$$ in the open interval (0,1). The paper is closed by showing counterexamples where stability of the roots of the rational function when $$k=1$$ is not sufficient for stability of the associated linear system with time-delay. We also introduce a variation of an existing frequency-sweeping necessary and sufficient condition for stability independent of delay which resembles the form of a generalized Nyquist criterion. The results are illustrated by numerical examples.

##### MSC:
 93D20 Asymptotic stability in control theory 93C15 Control/observation systems governed by ordinary differential equations 93C05 Linear systems in control theory
##### Keywords:
linear systems; time-delays; stability independent of delay
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##### References:
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