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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 ℋ− index, which measures the worst fault sensitivity. This paper is concerned with characterizing ℋ− index for linear discrete time systems. First, a  necessary and sufficient condition on the lower bound of 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 ℋ− index. In addition, for linear time-invariant systems in infinite time horizon, based on the definition of the 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 ℋ− index is not completely dual to ℋ∞ norm. Finally, several numerical examples are given to illustrate the main results

Minimum System Sensitivity Study of Linear Discrete Time Systems for Fault Detection