Asymptotic stability analysis of planar switched homogeneous systems

The main purpose of this study is to provide a novel approach to asymptotic stability analysis of planar switched systems. These switched systems are generated by a family of two-dimensional autonomous subsystems with generalized homogeneous right-hand sides. Here, we consider arbitrary numbers of subsystems with the same dilation coefficients and possibly different degrees of homogeneity. First, based on homogeneity and graphical characteristics, we introduce the new concept of quasi-polar representation form for two-dimensional vector fields. Then using quasi-polar representation form, we define homogeneous rotational direction and growth rate of state trajectories, by which we present sufficient conditions for global asymptotic stability of planar switched homogeneous (PSH) systems. To show the asymptotic stability of such systems, Lyapunov stability and convergence of system's response are proved. Unidirectional and bidirectional stabilizing switching laws are also probed for PSH systems. Simulation results show the applicability of the approach.