A non-probabilistic measure of reliability of linear systems based on expansion of convex models

Abstract

In this paper we develop a rigorous quantitative alternative to the probabilistic theory of reliability. The intuitive concept of reliability which underlies our analysis is that a system is reliable if it is robust with respect to uncertainty. That is, a system is reliable if it can tolerate a large amount of uncertainty before failure can occur. Conversely, a system is unreliable if it is fragile with respect to uncertainty. We model uncertainty with non-probabilistic convex models, and measure the amount of uncertainty with the expansion parameters of the convex models. The measure of reliability developed here is the amount of uncertainty the system can tolerate before failure. We consider linear dynamic systems with uncertain inputs and uncertain output failure states. The reliability of these systems hinges on the disjointness of convex sets. Using the hyperplane separation theorem for convex sets, we introduce a concept of modal reliability.