Robust stability analysis for linear systems with distributed delays: A time‐domain approach

Abstract

This work is devoted to the robust stability analysis of linear systems with distributed delays. The approach is based on recent results in the Lyapunov-Krasovskii framework, where a Lyapunov-Krasovskii functional with prescribed negative definite derivative is applied. The stability conditions obtained in this work, are simple inequalities which depend on the so-called delay Lyapunov matrix. The cases of matrix parameters and delays uncertainties are addressed, accurately detecting exact stability bounds when applied in an iterative manner. Our method is used successfully to tackle the challenging problem of robust predictor-based stabilization of systems with state and input delays.

Contribuciones al estudio de sistemas lineales con retardos: el enfoque de funcionales de tipo completo

“…Cabe recalcar que Alexandrova and Zhabko (2018) introdujeron una metodología de análisis de robustez basada en la funcional v 0 (x t ) que permite reducir el conservatismo. Este enfoque se extendió al caso de sistemas con retardos distribuidos (Juárez et al, 2020a). Jarlebring et al (2011) mostraron que la norma H 2 para sistemas de tipo retardado y neutral se puede caracterizar en términos de la matriz de Lyapunov (vea también Sumacheva and Kharitonov (2014)).…”

Vladimir Kharitonov: Achievements in research

“…The results on complete type functionals embodied in Kharitonov's monograph 62 allow the solution of several issues: analyze the robust stability with respect to time-invariant, time-varying or even nonlinear disturbances, 62,63,88,[108][109][110][111][112][113] determine the critical values of the parameters at which the stability is lost, 62,[114][115][116][117][118][119] compute the  2 norm, 84,85,120 estimate the domain of attraction, [121][122][123][124][125] compute the quadratic performance indices, 62,77,126 synthesize suboptimal control laws. [127][128][129][130][131][132] Another application is, of course, the stability analysis of individual systems based on Theorem 3.…”

Stabilisation of distributed time-delay systems: a smoothed spectral abscissa optimisation approach

“…Within the first approach, one can find, for instance, the characterization of stability crossing curves, (Morȃrescu, Niculescu, & Gu, 2007), or the quasi-continuous pole placement methodology, (Michiels, Vyhlídal, & Zítek, 2010), whereas within the second approach, one mostly finds results based on the solution of Linear Matrix Inequalities (LMIs), see, e.g., (Chen & Zheng, 2007;Feng, Nguang, & Perruquetti, 2020;Feng, Nguang, & Seuret, 2019;Liu, Fridman, Johansson, & Xia, 2016;Liu, Seuret, Xia, Gouaisbaut, & Ariba, 2019;Xie, Fridman, & Shaked, 2001;Zheng & Frank, 2002). Results based on the so-called delay Lyapunov matrix are also reported in (Juárez, Alexandrova, & Mondié, 2020), (Mondié, Ochoa, & Ochoa, 2011), (Egorov, Cuvas, & Mondié, 2017). It is important to mention that despite the stabilization algorithms based on LMIs are predominant in the literature, they lead to conservative results as they are based on sufficient conditions for stability.…”