A Conformal Mapping and Interpolation Approach to Delay Margin for General Unstable Plants

This article proposes a new approach to studying the delay margin problem for general unstable linear time-invariant (LTI) systems, which addresses the question: What is the largest stabilizable delay range over all LTI controllers? The proposed approach converts the delay margin to the robust stabilization for LTI systems involving a positive uncertain parameter. The conformal mapping and Nevanlinna–Pick interpolation are employed to develop a unified numerical procedure for computing upper and lower bounds on the delay margin, and for designing robustly stabilizing controllers, aimed at maximizing the stabilizability range of the delay parameter, promised by the delay margin. The results obtained in this article show that the proposed new approach in this article provides an effective solution method to tackle the delay margin and applies to general unstable linear systems of finite order. The obtained upper and lower bounds of the delay margin are close to each other, illustrated by several typical linear systems and numerical examples.