Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems

TL;DR

This paper develops new stability and stabilization conditions for linear positive impulsive and switched systems, using copositive Lyapunov functions and linear programming, applicable to uncertain systems and control design.

Contribution

It introduces a lifting approach with a clock-variable to make stability conditions affine in system matrices, enabling effective control design for impulsive and switched systems.

Findings

Proposes finite and semi-infinite linear programming conditions for stability.

Introduces a lifting approach for uncertain systems and control design.

Provides numerical examples demonstrating the effectiveness of the methods.

Abstract

Several results regarding the stability and the stabilization of linear impulsive positive systems under arbitrary, constant, minimum, maximum and range dwell-time are obtained. The proposed stability conditions characterize the pointwise decrease of a linear copositive Lyapunov function and are formulated in terms of finite-dimensional or semi-infinite linear programs. To be applicable to uncertain systems and to control design, a lifting approach introducing a clock-variable is then considered in order to make the conditions affine in the matrices of the system. The resulting stability and stabilization conditions are stated as infinite-dimensional linear programs for which three asymptotically exact computational methods are proposed and compared with each other on numerical examples. Similar results are then obtained for linear positive switched systems by exploiting the possibility of reformulating a switched system as an impulsive system. Some existing stability conditions are retrieved and extended to stabilization using the proposed lifting approach. Several examples are finally given for illustration.