Region stability and stabilisation of switched linear systems with multiple equilibria

ABSTRACT

This paper studies stability and stabilisation issues of switched linear time-invariant systems with stable/unstable multiple equilibria. Investigation of such switched systems is motivated by a switching economic system. The well-known common Lyapunov function method is shown to be ineffecctive in analysing region stability of switched systems with multiple equilibria via a counterexample. When every subsystem has an equilibrium point and all multiple equilibria pairwise differ, this paper proposes some sufficient conditons for region stability/instability of such switched systems with respect to a region containing all multiple equilibria under arbitrary quasi-periodical switchings. These novel results imply that there may exist stable limit cycles of such switched systems. Based on the stability results, a global asymptotic region-stabilising controller, quasi-periodical switching path, and corresponding algorithm are all designed for such switched control systems. Several illustrative examples demonstrate the effectiveness and practicality of our new results.

KEYWORDS:

• Switched linear systems
• n-dimensional systems
• region stability and stabilisation
• switching controllers
• multiple equilibria

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 61374031], [grant number 61522306]; The Natural Science Foundation of Shandong Province of China [grant number ZR2017MF071], [grant number ZR2017MA018]; The Doctoral Scientific Research Foundation of Shandong Technology and Business University [grant number BS201617].