Performance metric and analytical gain optimality for set-based robust fault detection

Abstract

This paper proposes a new performance metric for set-based robust fault detection on linear time-invariant systems, which quantitatively characterizes all faults that are unguaranteed to be detected by a specific set-based fault detection method at next time instant. Based on this metric, a novel online optimal design method is proposed for the set-theoretic unknown input observer, where the optimal parameters of the observer are obtained by analytically solving an optimization problem with bilinear matrix inequalities. Under a mild assumption, it is demonstrated that the observer is internally stable under the proposed optimal design. Moreover, the proposed online optimal design method is directly related to an algebra Riccati equation as time approaches infinity, and hence an offline optimal design method is further proposed to reduce the computational complexity. Under the proposed optimal designs, it is demonstrated that the observer can detect faults more timely and with much less computational complexity. At the end of this paper, an electric-circuit example and a high-dimensional numerical example are used to illustrate the effectiveness of the proposed methods.