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Mathematical Problems in Engineering
Volume 2013, Article ID 276987, 13 pages
Research Article

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

Institute for Aerospace Studies, University of Toronto, Toronto, ON, Canada M3H 5T6

Received 2 September 2012; Accepted 15 January 2013

Academic Editor: Bin Jiang

Copyright © 2013 Xiaobo Li and Hugh H. T. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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