The fastest stabilization of second-order switched systems with all modes unstable via an optimal state-dependent switching rule revisited

In this paper, an optimal state-dependent switching rule design method is proposed for the fastest asymptotic stabilization of second-order switched systems, wherein all eigenvalues of each subsystem are positive real parts. First, the definition of an optimal invertible transformation is proposed based on the physical meaning of a vibrational system with one degree-of-freedom. Then, the formulas of both the optimal invertible transformations and the optimal switching lines are calculated. In this way, the designed optimal state-dependent switching rule can minimize the energy increment of unstable subsystem operation and maximize the energy loss of system switching simultaneously, achieving the fastest asymptotic stability. Moreover, the critical stability condition for a general switched system is investigated, demonstrating that this state-dependent switching rule is optimal. Finally, three examples are provided to verify the effectiveness and superiority of the results.