An analytic interpolation approach to stability margins with emphasis on time delay

TL;DR

This paper introduces a new analytic interpolation method to more accurately determine stability margins, especially maximum delay margins, by incorporating frequency-dependent shifts, improving bounds in robust stabilization.

Contribution

It presents a novel analytic interpolation approach with frequency-dependent shifts to obtain sharper delay margin bounds and integrate gain and phase margin constraints.

Findings

Developed a method for sharper delay margin bounds.

Integrated gain and phase margin constraints.

Enhanced understanding of Nyquist plot constraints at different frequencies.

Abstract

Unlike the situation with gain and phase margins in robust stabilization, the problem to determine an exact maximum delay margin is still an open problem, although extensive work has been done to establish upper and lower bounds. The problem is that the corresponding constraints in the Nyquist plot are frequency dependent, and encircling the point $s=-1$ has to be done at sufficiently low frequencies, as the possibility to do so closes at higher frequencies. In this paper we present a new method for determining a sharper lower bound by introducing a frequency-dependent shift. The problem of finding such a bound simultaneously with gain and phase margin constraints is also considered. In all these problems we take an analytic interpolation approach.