Fault Detection for a One-Dimensional Wave PDE System

Abstract

In this paper, we design a fault diagnosis scheme for a class of variable coefficient wave equation (an overhead crane system), which is composed of an observer and its output error is treated as residual signal. When the system is in a healthy state, the output residual signal decays exponentially. Due to the existence of external disturbance, the residual is not zero in the state without fault. Therefore, we further design a reasonable threshold which is based on the upper bound of the residual dynamics and external disturbance to reduce the influence of external disturbance and determine whether the system fault occurs. The convergence properties of partial differential equation (PDE) observer and residual signal are analyzed by Lyapunov stability theory. Finally, the effectiveness of this fault diagnosis is illustrated by simulation and we can judge whether this overhead crane system fails by this fault diagnosis scheme.

Design of adaptive fault‐tolerant controllers for perturbed nonlinear ODE‐PDE cascaded systems

“…In order to solve the tracking control problems for a class of nonlinear ODE‐PDE cascaded systems, Cao and Liu 16 developed an adaptive FTC scheme so that the tracking error of ODE and the states of PDE were asymptotically stable. Zhang 35 designed a fault diagnosis scheme for a class of perturbed ODE‐PDE systems with known upper bound so that the residual signal was bounded. Xing and Liu 25 presented an adaptive neural network based robust uniformly stable controller for cascade ODE‐string systems with matched uncertainties and perturbations and subjected to unknown actuator faults.…”

“…Noted also that dynamic equations of the plant in References 5,21‐24,27,28,31‐33 are only governed by linear PDE; and the dynamic equations in References 9,15 are linear ODE‐PDE. The controllers may not be applied to systems with unmatched perturbations; these include 5,7,8,10‐16,19‐21,23‐25,27‐30,33,36 The upper bounds of perturbations must be known beforehand; these include 7,14,15,17,20,21,24,27,28,35 The possibility of actuator's failure were not considered in the design of control schemes, these include 5,6,8‐14,17‐24,26‐34,37,38 …”

“…According to the previous limitations 1–4, the motivation and the originality of this research is clear that, in this article, a FTC scheme is proposed for a class of nonlinear ODE‐PDE cascaded systems with matched and unmatched disturbances. Not only the dynamic equations one considered in this article are more extended than those in the literature, 5‐38 but also the control scheme one proposed is able to simultaneously eliminate the limitations (1)‐(4) mentioned above. In addition, the proposed control scheme provides the following advantages: Both adaptive mechanisms and perturbation estimator are utilized so that the upper bounds of perturbations and perturbation estimation errors are not required to be known beforehand. The unknown loss of effectiveness of actuator's degradation can be time‐varying variables. The controlled states of ODE and PDE of the cascaded system are able to converge to zero within a finite time. …”

“…The upper bounds of perturbations must be known beforehand; these include. 7,14,15,17,20,21,24,27,28,35 4. The possibility of actuator's failure were not considered in the design of control schemes, these include.…”