Adaptive Stabilization of Uncertain Linear System with Stochastic Delay by PDE Full-State Feedback

This article focuses on the equilibrium regulation problem of a class of uncertain linear systems with stochastic input delay, where the delay is modeled as a Markov process with finite states. Our approach is based on the partial differential equation backstepping method. We first consider the Markov delay with finite known states. An adaptive controller is designed to achieve global almost sure asymptotic convergence. Then, the Markov delay with finite unknown states is discussed. Owing to the unknown Markov delay resulting in the single distributed actuator state being unmeasured, the update law is redesigned, and only the local almost sure asymptotic stability is proved. In the control design, we adopt a constant-time-horizon-prediction-based control law to robustly compensate for the stochastic delay, which requires that the constant be close enough to all the Markov process states. Then, through Lyapunov functional analysis, we prove the almost sure boundedness of all the signals. Moreover, stochastic Barbalat's lemma is applied to realize equilibrium regulation. Finally, to show the effectiveness of our results, a numerical example is given.