Stability independent of delay using rational functions.

*(English)*Zbl 1175.93195Summary: 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 |

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\textit{F. O. Souza} et al., Automatica 45, No. 9, 2128--2133 (2009; Zbl 1175.93195)

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