Hybrid Control of Parabolic PDEs: Handling Faults of Constrained Control Actuators

Abstract

This work proposes a hybrid control methodology that integrates feedback and switching for fault-tolerant constrained control of linear parabolic partial differential equations (PDEs) for which the spectrum of the spatial differential operator can be partitioned into a finite “slow” set and an infinite stable “fast” complement. Modal decomposition techniques are initially used to derive a finite-dimensional system (set of ordinary differential equations (ODEs) in time) that captures the dominant dynamics of the PDE. The ODE system is then used as the basis for the integrated synthesis, via Lyapunov techniques, of a stabilizing nonlinear feedback controller together with a switching law that orchestrates the switching between the admissible control actuator con- figurations, based on their constrained regions of stability, in a way that respects actuator constraints and maintains closed-loop stability in the event of actuator failure. Precise conditions that guarantee stability of the constrained closed-loop PDE system under actuator switching are provided. The proposed method is applied to the problem of stabilizing an unstable, spatially unifrom steady-state of a linear parabolic PDE under constraints and actuator failures.