Fault detection and estimation for a class of PIDE systems based on boundary observers

In this paper, we propose a simultaneous state estimation and fault estimation approach for a class of first-order hyperbolic partial integral differential equation systems. Specifically, we consider the multiplicative boundary actuator and sensor faults, ie, unknown fault parameters multiplying by the boundary input or boundary state (ie, output). As a consequence, two difficulties arise immediately: (1) simultaneous estimation of both plant state and faults is a nonlinear problem due to the multiplication between fault parameters and plant signals; (2) no prior information is available to determine the type (actuator or sensor) of faults. To overcome these difficulties, this paper develops adaptive fault parameter update laws and embeds the resulting laws into the plant state observer design. First, we propose new approaches to estimate actuator fault and sensor fault, respectively. Next, we develop a novel method to simultaneously estimate actuator and sensor faults. The proposed observer and update laws, designed using only one boundary measurement, ensure both state estimation and fault parameter estimation. By choosing appropriate Lyapunov functions, we prove that the estimates of state and fault parameters converge to an arbitrarily small neighborhood of their true values. Numerical simulations are used to demonstrate the effectiveness of the proposed estimation approaches.