Minimum system sensitivity study of linear discrete time systems for fault detection (Q460396)

Summary: Fault detection is a critical step in the fault diagnosis of modern complex systems. An important notion in fault detection is the smallest gain of system sensitivity, denoted as \(\mathcal H_-\) index, which measures the worst fault sensitivity. This paper is concerned with characterizing \(\mathcal H_-\) index for linear discrete time systems. First, a necessary and sufficient condition on the lower bound of \(\mathcal H_-\) index in finite time horizon for linear discrete time-varying systems is developed. It is characterized in terms of the existence of solution to a backward difference Riccati equation with an inequality constraint. The result is further extended to systems with unknown initial condition based on a modified \(\mathcal H_-\) index. In addition, for linear time-invariant systems in infinite time horizon, based on the definition of the \(\mathcal H_-\) index in frequency domain, a condition in terms of algebraic Riccati equation is developed. In comparison with the well-known bounded real lemma, it is found that \(\mathcal H_-\) index is not completely dual to \(\mathcal H_\infty\) norm. Finally, several numerical examples are given to illustrate the main results.