Robust instability analysis of discrete-time LTI systems via phase change rate maximization and its applications

The problem of phase change rate (PCR) maximization for discrete-time transfer functions is examined and shown to facilitate the search for a minimum-norm stable controller to stabilize an unstable system — an effort that is closely related to robust instability analysis. The latter arises in the study of sustained oscillatory phenomena in nonlinear systems and other practical applications. This paper formulates the PCR maximization problem for discrete-time systems subject to constraints at the unique peak gain frequency, and provides a step-by-step solution to the problem. A constant or a first-order all-pass function are shown to be the optimal solution to PCR maximization and can be viewed as the worst-case strongly stabilizing perturbation, thereby providing a sufficient condition for determining the robust instability radius — an effective measure of instability — of an unstable system. Two real-world applications are presented to illustrate the utility of our results. The first is the strong stabilization of a magnetic levitation system using a minimum-effort digital controller for improved energy efficiency. The second involves quantifying the robustness of the oscillatory behavior associated with neural spike generations in the FitzHugh–Nagumo model subject to perturbations.