Asymptotic behavior and solution approximation of Active Robust fault detection for closed-loop systems

The Parseval Theorem together with the Lagrange multiplier method is proposed to design the infinite horizon input signal (auxiliary signal) for active failure detection such that its use enables us to detect the faults in a multiple-model framework. We consider the asymptotic behavior of the robust fault detection problem and its stationary optimal solutions for a linear uncertain system controlled by a linear feedback. The optimization criterion considered in this paper is a worst case quadratic cost, which is the same cost used for the design of the controller in practice. A complete frequency analysis of the solution, the optimal signals in the stationary case, is provided. The optimal costs for finite horizon problems converge to the optimal infinite horizon cost as the horizon increases. Solving the problem of infinite horizon active fault detection for discrete-time systems can be used as an approximation to long interval finite horizon problems.