Stabilising switching law to minimise return ratio with two linear planar subsystems

Abstract

This article deals with the stabilisation problem for the switched system composed of two planar linear subsystems via characterising the construction of the optimally arranged switching surfaces. To this end, we first explain the stabilisation mechanism in terms of return ratio and define the most stable trajectory by means of minimising the return ratio. Then we identify the most stable trajectory according to the geometric construction that its turning points constitute the subspaces where the fields of two subsystems are linearly dependent. This observation renders a constructive way to determine the optimally arranged switching surfaces and, therefore, enables us to examine the stabilisability in a necessary and sufficient sense. Finally, we use classical examples to demonstrate the theoretical results.

Keywords:

• planar dynamical systems
• switched systems
• optimal switching law
• stabilisation

Acknowledgements

The author would like to thank the anonymous referees for their valuable comments that helped in improving the article.

Additional information

Funding

This work is supported by Reserve Talents of Universities Overseas Research Program of Heilongjiang.