Detection, isolation and management of actuator faults in parabolic PDEs under uncertainty and constraints

This paper presents a methodology for the design of integrated robust fault detection and isolation (FDI) and fault-tolerant control (FTC) architecture for transport- reaction processes modeled by nonlinear parabolic partial differential equations (PDEs) with time-varying uncertain variables, actuator constraints and faults. The design is based on an approximate, finite-dimensional system that captures the dominant dynamic modes of the PDE. Initially, an invertible coordinate transformation, obtained with judicious actuator placement, is used to transform the approximate system into an equivalent form where the evolution of each dominant mode is excited directly by only one actuator and decoupled from the rest. For each mode, a robustly stabilizing bounded feedback controller that achieves an arbitrary degree of asymptotic attenuation of the effect of uncertainty is then synthesized and its constrained stability region is explicitly characterized in terms of the constraints, actuator locations and the size of uncertainty. A key idea in the controller synthesis is to shape the healthy closed-loop response of each mode in a prescribed fashion that decouples the effects of uncertainty and other modes on its dynamics, thus allowing (1) the derivation of performance-based FDI rules for each actuator, and (2) an explicit characterization of the state-space regions where FDI can be performed under uncertainty and constraints. Following FDI, a switching law is derived to orchestrate actuator reconfiguration in a way that preserves robust closed- loop stability. Finally, the theoretical results are demonstrated using a diffusion-reaction process example.