On Existence of Stabilizing Switching Laws within a Class of Unstable Linear Systems

The equivalence of two conditions, condition (3) and condition (4) stated in Problem Statement section, regardingthe existence of stabilizing switching laws between two unstable linear systems first appeared in (Feron 1996). Although Feron never published this result, it has been referenced in almost everysurvey on switched systems; see, for example, (Liberzon and Morse 1999). This paper proposes another way to prove the equivalenceof two conditions regarding the existence of stabilizing switching laws between two unstable linearsystems. One is effective for theoretical derivation, while the other is implementable, and a class ofstabilizing switching laws have been explicitly constructed by Wicks et al. (1994). With the help of theequivalent relation, a condition for the existence of controllers and stabilizing switching laws betweentwo unstabilizable linear control systems is then proposed. Then, the study is further extended to theissue concerning the construction of quadratically stabilizing switching laws among unstable linearsystems and unstabilizable linear control systems. The obtained results are employed to study theexistence of control laws and quadratically stabilizing switching laws within a class of unstabilizablelinear control systems. The numerical examples are illustrated and simulated to show the feasibility andeffectiveness of the proposed methods