On Delay Margin Bounds of Discrete-Time Systems

This paper studies the delay margin problem of discrete-time, linear time-invariant (LTI) systems. For general LTI plants with multiple unstable poles and nonminimum phase zeros, we employ analytic function interpolation and rational approximation techniques to derive bounds on the delay margin. Readily computable and explicit lower bounds are found by computing the real eigenvalues of a constant matrix, and LTI controllers can be synthesized based on the H-infinity control theory to achieve the bounds. For first-order unstable plants, we also obtain bounds achievable by PID controllers, which are of interest to PID control design and implementation. It is worth noting that unlike its continuous-time counterpart, the discretetime delay margin problem being considered herein constitutes a simultaneous stabilization problem, which is known to be rather difficult. While previous work on the discrete-time delay margin led to negative results, the bounds developed in this paper provide instead a guaranteed range of delays within which the delay plants can be robustly stabilized, and in turn solve the special class of simultaneous stabilization problems in question.