Fundamental bounds on delay margin: When is a delay system stabilizable?

Abstract

This paper concerns the stabilization of linear systems subject to unknown, possibly time-varying delays. Drawing upon analytic interpolation and rational approximation techniques, we develop fundamental bounds on the delay margin, with which the delay plant is guaranteed to be stabilizable by a controller. Our contribution is threefold. First, for a single-input single-output system with an arbitrary number of plant unstable poles and nonminimum phase zeros, we provide an explicit, computationally efficient bound on the delay margin, which requires computing only the largest real eigenvalue of a constant matrix. Second, for multi-input multi-output systems, we show that estimates on the variation ranges of multiple delays can be obtained by solving LMI problems, and further, by computing the radius of delay variations. Third, we show that with appropriate care, these bounds and estimates can be extended to systems subject to time-varying delays. When specialized to more specific cases, e.g., to plants with one unstable pole and one nonminimum phase zero, our results give rise to analytical expressions exhibiting explicit dependence of the bounds and estimates on the pole and zero, thus demonstrating how fundamentally unstable poles and nonminimum phase zeros may limit the range of delays over which a plant can be stabilized.