Synthesis of Razumikhin and Lyapunov–Krasovskii approaches to stability analysis of time-delay systems

Abstract

a b s t r a c tIn this paper, a necessary and sufficient condition for the exponential stability of linear systems with several time-delays is presented. Such a condition is based on the construction of quadratic lower bounds for the Lyapunov-Krasovskii functionals on the special Razumikhin-type set of functions. The result reveals a constructive procedure for the stability analysis whose application is illustrated with examples.

Exponential estimates for solutions of linear systems with distributed delay

“…It uses the functionals referred in [4]. Although there are no quadratic lower bounds for these functionals, recently it was shown that such bounds exist on some special set of functions (see Theorem 1) [9]. The latter result allows to apply the functionals to the robustness analysis.…”

Stability Analysis of Linear Time-Delay Systems with Two Incommensurate Delays

“…The "proximity" between the related system with commensurate delays and the original one is defined by the condition whether the time derivative of the modified functional is negative definite or not. Such modification allows to extend the results from [10,11,14] to the systems with incommensurate delays and to formulate the constructive procedure for the exponential stability and instability analysis of such systems which is the main contribution of the chapter.…”

“…In papers [10,11,14], we apply the Lyapunov -Krasovskii functionals for stability analysis using a special Razumikhin-type condition. It turns out that the stability criterion can be expressed in terms of the fact that the functional admits the quadratic lower bound on the functions satisfying this condition.…”

filtering for non‐linear systems with stochastic sensor saturations and Markov time delays: the asymptotic stability in probability

“…It is well known that time delays are frequently encountered in many practical systems and may deteriorate the system performances if they are not appropriately coped with in the design procedure [23,24,25,26]. The filtering problem with time-delays has been investigated for a variety of systems such as two-dimensional systems [27] and neural networks [28].…”