Robust Fault Diagnosis of Uncertain One-dimensional Wave Equations

Unlike its Ordinary Differential Equation (ODE) counterpart, fault diagnosis of Partial Differential Equations (PDE) has received limited attention in existing literature. The main difficulty in PDE fault diagnosis arises from the spatio-temporal evolution of the faults, as opposed to temporal-only fault dynamics in ODE systems. In this work, we develop a fault diagnosis scheme for one-dimensional wave equations. A key aspect of this fault diagnosis scheme is to distinguish the effect of uncertainties from faults. The scheme consists of a PDE observer whose output error is treated as a fault indicating residual signal. Furthermore, a threshold on the residual signal is utilized to infer fault occurrence. The convergence properties of the PDE observer and residual signal are analyzed via Lyapunov stability theory. The threshold is designed based on the uncertain residual dynamics and the upper bound of the uncertainties. Simulation studies are performed to illustrate the effectiveness of the proposed fault diagnosis scheme.