Necessary and sufficient conditions for regional stabilisability of generic switched linear systems with a pair of planar subsystems

Abstract

In this article, the regional stabilisability issues of a pair of planar LTI systems are investigated through the geometrical approach, and easily verifiable necessary and sufficient conditions are derived. The main idea of the article is to characterise the best case switching signals based upon the variations of the constants of the integration of the subsystems. The conditions are generic as all possible combinations of the subsystem dynamics are considered.

Keywords:

• switched linear systems
• stabilisability
• geometrical approach
• best case analysis

Acknowledgements

The authors would like to thank the AE and anonymous reviewers for their constructive comments which have helped improve the presentation of the result in this article.

Notes

Notes

1. If θ _e ∈ (θ*, θ), the Cauchy principal value of the improper integral is introduced as , which is also bounded because .

2. If k _A = 1, any vector in the phase plane is the eigenvector of A, which contradicts Assumption 2 since B have two real eigenvectors.

3. Note that β = 0 in S ₁₁ and α = 0 in S ₁₂ have been excluded by Assumption 2.