Stability of Linear Systems With Interval Time Delays Excluding Zero

Abstract

The stability of linear systems with multiple, time-invariant, independent and uncertain delays is investigated. Each delay is assumed to reside within a known interval excluding zero. A delay-free sufficient comparison system is formed by replacing the delay elements with parameter-dependent filters, satisfying certain properties. It is shown that robust stability of this finite dimensional parameter-dependent comparison system, guarantees stability of the original time-delay system. This result is novel in the sense that it does not require any a priori knowledge regarding stability of the time-delay system for some fixed delay. When the parameter-dependent filters are formed in a particular manner using Pade approximations, an upper bound on the degree-of-conservatism of the comparison system may be obtained, which is independent of the time-delay system considered. With this, it is shown that the conservatism of this comparison system may be made arbitrarily small. A linear matrix ineqaulity (LMI) formulation is presented for analysis of the stability of the parameter-dependent comparison system. In the single-delay case, an eigenvalue criterion is also available for stability analysis which incurs no additional conservatism