Delay Margin for Robust Stabilization of LTI Delay Systems

Robust stabilization of linear time-invariant (LTI) systems subject to uncertain, possibly time-varying delays is considered in this chapter. The fundamental issue under investigation, referred to as the delay margin problem, addresses the question: What is the largest range of delay such that there exists a single LTI feedback controller capable of stabilizing robustly all the plants for delays within that range? We study this longstanding, open problem by employing analytic interpolation and rational approximation techniques. Fundamental bounds on the delay margin are obtained, within which the delay plant is guaranteed to be stabilizable by a certain LTI output feedback controller. For single-input single-output (SISO) systems 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. When specialized to more specific cases, e.g., to plants with one unstable pole but possibly multiple nonminimum phase zeros, the bound gives rise to simple analytical expressions demonstrating how fundamentally unstable poles and nonminimum phase zeros may limit the range of delays over which a plant can be robustly stabilized by a LTI controller. The result is then extended to systems subject to time-varying delays.