Abstract
In 1963 a paper written by Pontryagin in 1942 is submitted to the American Mathematical Society Translation, [PON 63] in which he presents a generalization for the Hermite-Bieler theorem. Pontryagin's generalization quickly gains popularity and in the same year is referenced in the well known manual by Bellman and Cooke [BEL 69]. Silva, Datta and Bhattacharyya [SIL 00], [SIL 01] have recently made extensive use of this result, as they were investigating the stability of continuous-time delay systems with a PID controller in the feedback path and a constant uncertain communication delay. They propose a numerical plane sweeping algorithm that characterizes the stability region in the PID parameter space. This article first presents an extension of Pontryagin's theorem. Then, based on the results of Silva's team, this new formulation is applied to derive an explicit closed-form expression for the stability region in the PID controller parameter space.
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