Confidence set-based analysis of minimal detectable fault under hybrid Gaussian and bounded uncertainties

Abstract

This paper proposes a confidence set-based computational method of minimal detectable fault (MDF) based on the confidence set-separation condition between the healthy and faulty residual sets for discrete linear time-invariant systems. The state-estimation-error dynamics for the analysis of MDF under hybrid random and bounded uncertainties is divided into two sub-dynamics. The first sub-dynamics is only affected by the bounded uncertainties. Under the precondition of Schur stability, an outer-approximation of minimal robust positively invariant set with any given precision is obtained. While the second sub-dynamics is only affected by the random uncertainties following the Gaussian distributions. It is proved that the behavior of the second sub-dynamics at steady stage also follows a certain Gaussian distribution, which can be bounded by confidence zonotopes given a proper confidence level. MDF for actuator and sensor faults can be obtained by solving a non-convex optimization problem to minimize the magnitude of fault subjected to the residual set-separation constraints, which is equivalent to compute a distance from the origin to hyperplane along a fixed direction by exploiting the geometric property. At the end of this paper, a two-link manipulator model and a vehicle model are used to verify the effectiveness of our proposed method.