Hierarchical conditions for exponential stabilisation of time-varying delay systems using Bessel–Legendre inequalities and Finsler's lemma

Abstract

This paper addresses the exponential stabilisation of time-varying delay systems by means of Lyapunov–Krasovskii approaches, Bessel–Legendre inequalities and Linear Matrix Inequalities (LMIs). The key aspect of the proposed control synthesis method is that the LMI conditions are proved for the first time to be hierarchical under time-varying delays, that is to say, a progressive conservatism reduction is achieved as long as the degree of Legendre polynomials is increased. To this end, Finsler's lemma has been properly applied to decouple the Lyapunov matrices from the controller gain. Finally, simulation results are provided to illustrate the effectiveness of the proposed method.

Keywords:

• Time delay system
• Bessel–Legendre inequalities
• Lyapunov–Krasovskii method
• linear matrix inequality
• Finlser's lemma

Data availability statement

Authors agree to make data and materials supporting the results or analyses presented in their paper available upon reasonable request.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Agencia Estatal de Investigación and Ministerio de Ciencia e Innovación [grant number PID2020-116585GB-I00].

Notes on contributors

Antonio González-Sorribes

Antonio González-Sorribes received the Telecommunications Engineer degree in 2001 and his Ph.D. in Automation and Industrial Informatics from Universitat Politècnica de València (UPV) in 2012. He was a postdoctoral researcher at the Laboratory of Industrial and Human Automation control, Mechanical engineering and Computer Science, CNRS, UMR 8201, Valenciennes, France from 2013 to 2014. Currently, he works as Associate Professor at the Department of System Engineering and Automation at the Universitat Politècnica de València (Spain). His research interests are within the broad area of time delay systems, robust control, networked control systems, multirobot systems and process control applications.