The ISS framework for time-delay systems: a survey

Abstract

At the occasion of Eduardo D. Sontag's 70 th birthday, we provide here an overview of the tools available to study input-to-state stability (ISS) and related notions for time-delay systems. After a hopefully pedagogical presentation of the main differences with respect to the finite-dimensional theory, we review basic stability concepts for input-free time-delay systems, as well as instruments to guarantee them in practice, including the Lyapunov-Krasosvkii, Lyapunov-Razumikhin, and Halanay approaches. We then consider the influence of inputs through the notions of ISS, integral ISS, and input-to-output stability and provide both Lyapunov-like and solutions-based characterizations of these properties. We also show how these notions can be helpful for the stability analysis of interconnected systems, whether in cascade or in feedback form. We finally provide a list of questions which remain open until now.

On the design and the digital implementation of observer‐based controllers for tracking of nonlinear time‐delay systems

“…In the following, some useful notions and results very helpful for the presentation of the proposed control methodology are introduced. In particular, the notion of ISS and some useful results related to such a notion 28,29,31,32 are properly revised in order to deal with tracking control problems. Let us consider a nonlinear time-delay system described by the following RFDE…”

“…1. the function w is locally absolutely continuous in [0, b) 29 ; 2. the upper right-hand derivative of the function w,…”

“…A continuous function 𝛽 ∶ R + × R + → R + is of class  if, for each fixed t ≥ 0, the function s → 𝛽(s, t) is of class  and, for each fixed s ≥ 0, the function t → 𝛽(s, t) is of class . For positive integers n, l, m, for a function f ∶  n ×  l ×  m → R n , and for a locally Lipschitz functional V ∶  n → R + , the derivative (upper right-hand Dini directional derivative in the case Δ = 0, and derivative in Driver's form in the case Δ > 0 28,29 )…”

Optimal Dichotomy of Temporal Scales and Boundedness/ Stability of Time – Varying Multidimensional Nonlinear Systems

“…Consequently, it has been encountered in a deluge number of monographs, [8], [9], [12]- [14], [16], [18], [19], [21],23], [25], [28] and research papers, see the review papers [2], [3], [15], [20] and further references therein, which develop stability criteria for some types of system with time delay. Advancements in this area were achieved primarily through extensions of the Lyapunov methodology to systems with delays that were initially commenced by N. Krasovskii [24] and Razumikhin [33], and also through the applications of Halanay's inequality [17], see also [12], 13] and additional references therein, and Bohl-Perron theorem, [4], [5], [14] and some other techniques.…”

“…These problems become more apparent in the applications of the Lyapunov methodology to the assessment of the boundedness of solutions to nonhomogeneous VNDS with variable coefficients that were rarely addressed theretofore, see, e.g., [1]. Still, the application of the techniques used for the analysis of the input-to-state stability has account of some of the traits in the behavior of such systems, see recent review paper [3] and additional references therein. The problems concerned with dynamic behavior of some scalar nonlinear systems with delay and time-varying coefficients were studied as well, see, e.g., [6] and [7], and additional references therein.…”