Stabilization and Uniformity of Controllability of Switched Linear Systems with Unstable Subsystems

In this paper, the stabilization problem of switched linear time-varying systems with unstable subsystems is studied. Firstly, the condition of uniformity of controllability for all subsystems is introduced, under which the subsystems are exponentially stable and their state transition matrices are uniformly bounded. Secondly, the activation times of unstable and stable subsystems are defined in the form of ratio inequalities, effectively compensating for the state divergence caused by unstable subsystems. Finally, a programmable exponential stabilization scheme for the considered switched system is proposed by utilizing the average dwell time method. A numerical example is given to verify the effectiveness of all the results.