An Improved Stability Method for Linear Systems with Fast-Varying Delays

Eugenii Shustin, Emilia Fridman

Published 2005-11-15, updated 2006-02-27Version 2

Stability of linear systems with uncertain bounded time-varying delays is studied under assumption that the nominal delay values are not equal to zero. An input-output approach to stability of such systems is known to be based on the bound of the $L_2$-norm of a certain integral operator. There exists a bound on this operator in two cases: in the case where the delay derivative is not greater than 1 and in the case without any constraints on the delay derivative. In the present note we fill the gap between the two cases by deriving an operator bound which is an increasing and continuous function of the delay derivative upper bound $d \geq 1$. For $d\to \infty$ the new bound corresponds to the second case and improves the existing bound. As a result, improved frequency-domain and time-domain stability criteria are erived for systems with the delay derivative greater than 1.