Characterising the stabilisability for second-order linear switched systems

Abstract

The stabilisability of second-order linear switched systems is the topic focused on in a number of papers, however, some of its fundamental characteristics are still unrevealed. In this article, regarding a pair of subsystems, we are interested in characterising this problem from both the state-triggering and time-domain viewpoints and in revealing the inherent connection between them. To this end, we first use polar coordinates to represent the geometric property of switching and the dynamical behaviours caused by it, and then put the geometrised switching control into correspondence with the implicit stabilisation mechanism behind it. Doing this in a strong way relies on classifying switching control into two types intuitively, namely, chattering switching and spinning switching, and then clarifying their distinction and properties rigorously. Furthermore, the uniform convergence of the forced trajectory is shown by presenting estimation on its decay rate; and the limit cycles and sliding motions generated by switching are also accounted for. The results are illustrated by elaborate examples.

Keywords:

• second-order switched systems
• exponential stabilisation
• limit cycles
• sliding motions

Acknowledgements

The author would like to thank the anonymous reviewers for their helpful and insightful comments for improving this article. This work is supported by the National Natural Science Foundation of China under Grant Nos. 61104119 and 60974027. The author is supported by Reserve Talents of Universities Overseas Research Program of Heilongjiang.