Dwell-time stability and stabilization conditions for linear positive impulsive and switched systems
This work provides a unified framework for analyzing and controlling positive impulsive and switched systems with dwell-time constraints, extending existing results to stabilization and uncertain systems.
The paper develops stability and stabilization conditions for linear positive impulsive and switched systems under various dwell-time constraints, using linear copositive Lyapunov functions and a lifting approach to handle uncertainties and control design. The conditions are formulated as infinite-dimensional linear programs with three asymptotically exact computational methods proposed and compared.
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.