New stability conditions for systems with distributed delays

Oren Solomon, Emilia Fridman

Research output: Contribution to journalArticlepeer-review

Abstract

In the present paper, sufficient conditions for the exponential stability of linear systems with infinite distributed delays are presented. Such systems arise in population dynamics, in traffic flow models, in networked control systems, in PID controller design and in other engineering problems. In the early Lyapunov-based analysis of systems with distributed delays (Kolmanovskii & Myshkis, 1999), the delayed terms were treated as perturbations, where it was assumed that the system without the delayed term is asymptotically stable. Later, for the case of constant kernels and finite delays, less conservative conditions were derived under the assumption that the corresponding system with the zero-delay is stable (Chen & Zheng, 2007). We will generalize these results to the infinite delay case by extending the corresponding Jensen's integral inequalities and Lyapunov-Krasovskii constructions. Our main challenge is the stability conditions for systems with gamma-distributed delays, where the delay is stabilizing, i.e. the corresponding system with the zero-delay as well as the system without the delayed term are not asymptotically stable. Here the results are derived by using augmented Lyapunov functionals. Polytopic uncertainties in the system matrices can be easily included in the analysis. Numerical examples illustrate the efficiency of the method. Thus, for the traffic flow model on the ring, where the delay is stabilizing, the resulting stability region is close to the theoretical one found in Michiels, Morarescu, and Niculescu (2009) via the frequency domain analysis.

Original languageEnglish
Pages (from-to)3467-3475
Number of pages9
JournalAutomatica
Volume49
Issue number11
DOIs
StatePublished - Nov 2013

Keywords

  • Gamma-distributed delay
  • Infinite time-delay
  • Lyapunov-Krasovskii method

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