This paper focuses on the equilibrium regulation problem of a class of uncertain linear systems existing stochastic input delay, where the delay is modeled as a Markov process with finite states. Our approach is based on the partial differential equation (PDE) 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 of the Markov process states. Then, through Lyapunov functional analysis, we prove the almost sure boundedness of all the signals. Moreover, stochastic Barbălat's lemma is applied to realize equilibrium regulation. Finally, to show the effectiveness of our results, a numerical example is given.
ASJC Scopus subject areas
- コンピュータ サイエンスの応用