Robustness of reaction–diffusion PDEs predictor-feedback to stochastic delay perturbations

Abstract

This paper studies the robustness of a PDE backstepping delay-compensated boundary controller for a reaction–diffusion partial differential equation (PDE) with respect to a nominal delay subject to stochastic error disturbance. The stabilization problem under consideration involves random perturbations modeled by a finite-state Markov process that further obstruct the actuation path at the controlled boundary of the infinite-dimension plant. This scenario is useful to describe several actuation failure modes in process control. Employing the recently introduced infinite-dimensional representation of the state of an actuator subject to stochastic input delay for ODEs (Ordinary Differential Equations), we convert the stochastic input delay into r+1 unidirectional advection PDEs, where r corresponds to the number of jump states. Our stability analysis assumes full-state measurement of the spatially distributed plant’s state and relies on a hyperbolic–parabolic PDE-PDE cascade representation of the plant plus actuator dynamics. Integrating the plant and the nominal stabilizing boundary control action, all while considering probabilistic delay disturbances, we establish the proof of mean-square exponential stability as well as the well-posedness of the closed-loop system when random phenomena weaken the nominal actuator compensating effect. Our proof is based on the Lyapunov method, the theory of infinitesimal operator for stability, and C0-semigroup theory for well-posedness. Our stability result refers to the L2-norm of the plant state and the H2-norm of the actuator state. Furthermore, our study presents a qualitative analysis of the maximum deviation of the stochastic disturbance relative to the nominal delay, which is expressed as a function of the severity of the plant instability, namely, the magnitude of the reactivity coefficientand the known upper bound of the transition rate of the underlying stochastic perturbations. Extensive simulation results illustrate the viability of the robustness study.