Input-to-state stability for discrete-time switched systems by using Lyapunov functions with relaxed constraints

Huijuan Li; Huijuan Li

doi:10.3934/math.20231576

AIMS Mathematics

2023, Volume 8, Issue 12: 30827-30845. doi: 10.3934/math.20231576

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Research article

Input-to-state stability for discrete-time switched systems by using Lyapunov functions with relaxed constraints

Huijuan Li ^,

School of Mathematics and Physics, China University of Geosciences (Wuhan), Wuhan, 430074, China

Received: 10 September 2023 Revised: 23 October 2023 Accepted: 06 November 2023 Published: 15 November 2023
MSC : 34D05, 37B25, 93C10, 93C55, 93D20

In this paper, input-to-state stability (ISS) is investigated for discrete-time time-varying switched systems. For a switched system with a given switching signal, the less conservative assumptions for ISS are obtained by using the defined weak multiple ISS Lyapunov functions (WMISSLFs). The considered switched system may contain some or all subsystems which do not possess ISS. Besides, for an ISS subsystem the introduced Lyapunov function could be increasing along the trajectory of the subsystem without input at some moments. Then for a switched system under any switching signal, the relaxed sufficient constraints for ISS are attained by using the defined weak common ISS Lyapunov functions. For this case, each subsystem of the considered system must be ISS. The proposed function may be increasing along the trajectory of each ISS subsystem of the considered system without input at some instants. The relationship between WMISSLFs for a switched system and the defined weak multiple Lyapunov functions for this switched system without input is set up. Three numerical examples are investigated to display the usefulness of the principal outcomes. According to the main conclusions, an intermittent controller is applied to ensure ISS for a discrete-time disturbed Chua's chaotic system.
- input-to-state stability,
- ISS Lyapunov function,
- switched system,
- intermittent control
Citation: Huijuan Li. Input-to-state stability for discrete-time switched systems by using Lyapunov functions with relaxed constraints[J]. AIMS Mathematics, 2023, 8(12): 30827-30845. doi: 10.3934/math.20231576

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Abstract

In this paper, input-to-state stability (ISS) is investigated for discrete-time time-varying switched systems. For a switched system with a given switching signal, the less conservative assumptions for ISS are obtained by using the defined weak multiple ISS Lyapunov functions (WMISSLFs). The considered switched system may contain some or all subsystems which do not possess ISS. Besides, for an ISS subsystem the introduced Lyapunov function could be increasing along the trajectory of the subsystem without input at some moments. Then for a switched system under any switching signal, the relaxed sufficient constraints for ISS are attained by using the defined weak common ISS Lyapunov functions. For this case, each subsystem of the considered system must be ISS. The proposed function may be increasing along the trajectory of each ISS subsystem of the considered system without input at some instants. The relationship between WMISSLFs for a switched system and the defined weak multiple Lyapunov functions for this switched system without input is set up. Three numerical examples are investigated to display the usefulness of the principal outcomes. According to the main conclusions, an intermittent controller is applied to ensure ISS for a discrete-time disturbed Chua's chaotic system.

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