Exponential stability of switched block triangular systems under arbitrary switching

Abstract

In this paper, exponential stability of continuous-time and discrete-time switched $k\times k$ block triangular systems under arbitrary switching is studied. Firstly, under the assumption that all subsystem matrices are Hurwitz and a family of those corresponding block diagonal matrices is commutative, we prove that a continuous-time switched linear system is exponentially stable under arbitrary switching. Next, under the assumption that all subsystem matrices are Hurwitz and all those block diagonal matrices are normal, it is shown that the same switched system is exponentially stable under arbitrary switching. Further, under similar conditions we prove that a discrete-time switched linear system is exponentially stable under arbitrary switching. After that, illustrative numerical examples of the obtained results are also given. Finally, we prove that $3\times 3$ normal matrices have nine parameter representations which are useful for numerical examples (in the Appendix).

Keywords:

• Switched systems
• block triangular systems
• exponential stability
• arbitrary switching
• normal matrices

2020 Mathematics Subject Classifications:

• 93C30
• 93D05
• 93D23

Acknowledgments

The authors would like to thank the editor-in-chief, the handling editor and the reviewer for their constructive comments and useful advices.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported in part by JSPS KAKENHI [grant number 19K04443].