A computational technique characterizing the asymptotic stabilizability of planar linear switched systems with unstable modes

摘要

In this paper a new computational technique for characterizing the asymptotic stabilizability of linear switched systems on the plane is presented. Switching between a finite number of unstable LTI systems is studied. A ray-gridding idea is used which allows the construction of sufficiently dense subdivisions of the state space into conic regions, whose boundaries are candidate switching domains. Progressive refinements of the partition add flexibility in the search for a solution. Necessary and sufficient conditions for the asymptotic stabilizability of linear switched systems are developed by constructing polyhedral Lyapunov-like functions (PLF) associated to the convergence of an iterative algorithm. Therefore, it is shown that the existence of a PLF is equivalent to the asymptotic stabilizability of linear switched systems with unstable modes. The extension of the results to higher dimensions and the computational complexity associated with their implementation is also discussed.