Closed-loop input design for guaranteed fault diagnosis using set-valued observers

Active fault diagnosis (AFD) can be used to improve the diagnosability of faults by injecting a suitably designed input into a process. When faults are described as discrete switches between linear systems with uncertainties bounded within zonotopes, an optimal open-loop input guaranteeing diagnosis within a specified time horizon can be computed efficiently by solving a Mixed Integer Quadratic Program (MIQP). In this article, the constrained zonotope (CZ) set representation recently developed by the authors is used to extend the MIQP approach to general polytopic uncertainties without sacrificing efficiency. Next, this approach is combined with a CZ-based set-valued observer in a moving horizon framework to achieve rigorous closed-loop AFD. This method can greatly accelerate diagnosis relative to the open-loop approach, but requires online optimization. To reduce the online cost, we propose a method for solving the open-loop problem explicitly with respect to past measurements and inputs, which requires only observability of the nominal and faulty models. The effectiveness of the proposed approaches is demonstrated through several numerical examples.