Stability of simultaneously block triangularisable switched systems with partial state reset

ABSTRACT

We study the stability of a certain class of switched systems where discontinuous jumps (resets) on some of the state components are allowed, at the switching instants. It is known that, if all components of the state are available for reset, the system can be stabilisable by an adequate choice of resets. However, this question may have negative answer if there are forbidden state components for reset. We give a sufficient condition for the stabilisability of a switched system, under arbitrary switching, by partial state reset in terms of a block simultaneous triangularisability condition. Based on this sufficient condition, we show that the particular class of systems with partially commuting stable system matrices is stabilisable by partial state reset. We also provide an algorithm that allows testing whether a switched system belongs to this particular class of systems.

KEYWORDS:

• Switched system
• stability
• partial reset
• simultaneous block triangularisation

Disclosure statement

The authors have no potential conflict of interest to declare.

Notes

1. The situation where R^{(q, p)}₂₂ are not invertible raises interesting questions that will be investigated in due time.

2. [X, Y] stands for the traditional commutator of two square matrices X and Y, i.e. [X, Y] = XY − YX.

Additional information

Funding

This work was supported by Portuguese funds from The Portuguese Foundation for Science and Technology (FCT), through the Center for Research and Development in Mathematics and Applications (CIDMA), within project UID/MAT/04106/2013, and the Center of Mathematics and Applications (CMA), within project UID/MAT/00212/2013. This work was also supported by Feder funds through COMPETE2020 – Programa Operacional Competitividade e Internacionalização (POCI), [Project POCI-01-0145-FEDER-0 06933 – SYSTEC – Research Center for Systems and Technologies].