Stabilization of rank-deficient continuous-time switched affine systems

TL;DR

This paper addresses the challenge of stabilizing rank-deficient continuous-time switched affine systems by designing a state-dependent switching function that guarantees global asymptotic stability, with practical validation on a dc motor control system.

Contribution

It introduces a novel switching control method for rank-deficient systems, ensuring stability despite the higher-dimensional equilibrium set and singular convex combinations.

Findings

The proposed switching function guarantees global asymptotic stability.

Local exponential stability of the equilibrium point is achieved.

Validation on a practical dc motor system demonstrates effectiveness.

Abstract

This paper treats the global stabilization problem of continuous-time switched affine systems that have rank-deficient convex combinations of their dynamic matrices. For these systems, the already known set of attainable equilibrium points has higher dimensionality than in the full-rank case due to the existence of what we define as singular equilibrium points. Our main goal is to design a state-dependent switching function to ensure global asymptotic stability of a chosen point inside this set with conditions expressed in terms of linear matrix inequalities. For this class of systems, global exponential stability is generally impossible to be guaranteed. Hence, the proposed switching function is shown to ensure global asymptotic and local exponential stability of the desired equilibrium point. The position control and the velocity control with integral action of a dc motor driven by an h-bridge fed via a boost converter are used for validation. This practical application example is composed of eight subsystems, and all possible convex combinations of the dynamic matrices are singular.