State-dependent Switching Stabilization of Fourth-order Switched Systems With Fully Unstable Subsystems

This paper presents a state-dependent switching law to achieve asymptotic stability of a fourth-order switched system, in which each subsystem has two pairs of conjugate characteristic roots with positive real parts. Firstly, a linear vibration system of two-degrees-of-freedom is introduced to match the standard fourth-order linear subsystem. An energy function of each subsystem is then defined as its kinetic and potential energies. Secondly, an invertible transformation is introduced to convert a general fourth-order subsystem into a standard one, thereby constructing its energy function with definite physical meaning. Thirdly, in order to reasonably define an energy ratio function between two subsystems, an intermediate subsystem is introduced so that an identical invertible transformation can be used to convert the subsystem and the intermediate subsystem into standard fourth-order ones simultaneously. Based on the obtained energy ratio function, a state-dependent switching law is constructed to maximize the energy loss in a switching loop, thus stabilizing the fourth-order switched system at a fast speed. In the end, the effectiveness of the proposed method is illustrated by four numerical simulations.