On Time Delay Margin Estimation for Adaptive Control and Robust Modification Adaptive Laws

This paper presents methods for estimating time delay margin for adaptive control of input delay systems with almost linear structured uncertainty and a step input reference command signal. The bounded linear stability analysis method seeks to represent an adaptive law by a locally bounded linear approximation within a time window. The time delay margin of this input delay system represents a local stability measure and is computed analytically by three methods: Padé approximation, Lyapunov-Krasovskii method with sum-ofsquares optimization, and matrix measure method. These methods are applied to the standard model-reference adaptive control, σ -modification adaptive law, and optimal control modification adaptive law. The windowing analysis results in non-unique estimates of the time delay margin since it is dependent on the length of a time window and parameters which vary from one time window to the next. The optimal control modification adaptive law overcomes this limitation in that, as the adaptive gain tends to infinity and if the matched uncertainty is linear, then the closed-loop input delay system tends to a linear time-invariant system. A lower bound of the time delay margin of this system can then be estimated uniquely without the need for the windowing analysis. Simulation results demonstrates the feasibility of the bounded linear stability method for time delay margin estimation.