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

Abstract EN

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.