Robust Active Fault Diagnosis for Linear Stochastic Systems Within Bayesian Decision Framework

Abstract

This paper is concerned with the active fault diagnosis (AFD) problem for a class of linear stochastic systems with unknown prior probabilities of modes. In AFD within the Bayesian decision framework, the unknown prior probabilities typically make diagnosis performance compromised. A novel robust AFD approach is developed to settle the AFD problem under uncertainties in prior probabilities of modes for linear stochastic systems. The input design goal is to minimize the misdiagnosis probability on the basis of the robustness for uncertainties in prior probabilities of modes. The input design problem is characterized as a min-max optimization problem. As there is no closed-form expression for the misdiagnosis probability, its a novel upper bound family is deduced as an alternative. The extensively utilized Bhattacharyya upper bound is proven to be included in this upper bound family. The tightness of the upper bounds in the proposed family is analyzed. A novel two-layer optimization method is proposed to solve the input design problem to global optimality. Two numerical examples are presented to illustrate the effectiveness and superiority of the developed robust AFD approach. Note to Practitioners—The majority of the existing stochastic AFD results are developed within the Bayesian decision framework, where prior probabilities of system modes are required to be known. However, the prior probabilities are difficult to obtain in practice, which makes these AFD approaches impractical. A major challenge in stochastic AFD is to evaluate the probability of misdiagnosis due to the presence of multivariate integrals in its expression. An extensively utilized solution is to exploit its Bhattacharyya upper bound as an alternative. Unfortunately, there is looseness in the Bhattacharyya upper bound under certain circumstances, which is likely to make the misdiagnosis probability evaluated inaccurately and render diagnosis performance unsatisfactory. In this paper, the authors develop a novel robust AFD approach with a new upper bound family on the misdiagnosis probability to improve diagnosis performance for linear stochastic systems with unknown prior probabilities of modes. The developed AFD approach is robust for uncertainties in prior probabilities of modes. The input design problem is characterized as a min-max optimization problem, which is solved to global optimality by a novel two-layer optimization method. The upper bounds in the proposed family are tighter than the Bhattacharyya upper bound under certain conditions. For nonlinear stochastic systems with unknown prior probabilities of modes, the proposed robust AFD may not be applicable. In future research, we will focus on the robust AFD problem for nonlinear stochastic systems with unknown prior probabilities of modes.