A set based probabilistic approach to threshold design for optimal fault detection

Traditional deterministic robust fault detection threshold designs, such as the norm-based or limit-checking method, are plagued by high conservativeness, which leads to poor fault detection performance. On one side they are ill-suited at tightly bounding the healthy residuals of uncertain nonlinear systems, as such residuals can take values in arbitrarily shaped, possibly non-convex regions. On the other hand, they must be robust even to worst-case, rare values of the modeling and measurement uncertainties. In order to maximize performance of detection, we propose two innovative ideas. First, we introduce threshold sets, parametrized in a way to bound arbitrarily well the residuals produced in healthy condition by an observer-based residual generator. Secondly, we formulate a chance-constrained cascade optimization problem to determine such a set, leading to optimal detection performance of a given class of faults, while guaranteeing robustness in a probabilistic sense. We then provide a computationally tractable framework by using randomization techniques, and a simulation analysis where a well-known three-tank benchmark system is considered.