State-dependent switching rule design based on polar coordinate method: A case of hybrid unstable subsystem composition

This article investigates switched systems with three types of hybrid unstable subsystems: focus–node, focus–saddle, and node–saddle combinations. The objective is to develop suitable state-dependent switching rules to achieve rapid asymptotic stabilization of the system state to the origin. The proposed switching rule design follows three main steps. First, each subsystem is reformulated in polar coordinates. By analyzing the derivative of the radial distance with respect to the polar angle, a radial increment integral criterion is established to quantify system convergence behavior under both rotational and chattering switching mechanisms. Second, an in-depth analysis is conducted on the intrinsic characteristics of different unstable subsystem combinations. Region partitioning of the phase plane and switching analysis of phase trajectories are carried out based on the relative motion among trajectories. Finally, by leveraging the dynamic properties of subsystem phase trajectories, the most suitable switching mechanism is selected to fully utilize the advantageous features of each subsystem. The corresponding switching lines are then determined by comparing the convergence benefits of the active subsystems under each switching mechanism. Simulation results demonstrate that the resulting state-dependent switching strategy significantly improves convergence rate of the switched system while reducing conservatism.