Robust actuator fault isolation and management in constrained uncertain parabolic PDE systems

Abstract

This paper presents a methodology for the design of an integrated robust fault detection and isolation (FDI) and fault-tolerant control (FTC) architecture for distributed parameter systems 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 dynamics of the PDE system. Initially, an invertible coordinate transformation–obtained through judicious actuator placement–is used to transform the approximate system into a form where the evolution of each state is excited directly by only one actuator. For each state, 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, the actuator locations and the size of the uncertainty. A key idea in the controller synthesis is to shape the fault-free closed-loop response of each state in a prescribed fashion that facilitates the derivation of (1) dedicated FDI residuals and thresholds for each actuator, and (2) an explicit characterization of the state-space regions where FDI can be performed under uncertainty and constraints. A switching law is then derived to orchestrate actuator reconfiguration in a way that preserves robust closed-loop stability following FDI. Precise FDI rules and control reconfiguration criteria that account for model reduction errors are derived for the implementation of the FDI-FTC structure on the distributed parameter system. Finally, the results are demonstrated using a tubular reactor example.