A note on the stability of bimodal systems in with discontinuous vector fields

Abstract

This paper presents the necessary and sufficient conditions for global asymptotic stability of a class of bimodal piecewise linear systems in ${\mathbb{R}}^3$. The approach being used allows the vector field to be discontinuous on the switching plane. It is shown that the discontinuity of the vector field plays a crucial role in global asymptotic stability. This point is illustrated by examples, where the change in the discontinuity of the vector field (without changing the eigenvalues of subsystems) can cause the system to be globally asymptotically stable or unstable.

Keywords:

• piecewise linear systems
• bimodal systems
• asymptotic stability
• discontinuous vector field