On Stability of Switched Homogeneous Nonlinear Systems

In this paper, the problem of stability of switched homogenous systems is addressed. First of all, if there is a quadratic Lyapunov function such that nonlinear homogenous systems are asymptotically stable, a matrix Lyapunov-like equation is obtained for a stable nonlinear homogenous system using semi-tensor product of matrices, and Lyapunov equation of linear system is just its particular case. Following the previous results, a sufficient condition is obtained for stability of switched nonlinear homogenous systems, and a switching law is designed by partition of state space. In particular, a constructive approach is provided to avoid chattering phenomena which is caused by the switching rule. Then for planer switched homogeneous systems, an LMI approach to stability of planer switched homogeneous systems is presented. As such, it is easily verified by computer as possibly as that of linear system. An example is given to illustrate that candidate common Lyapunov function is a key point for design of switching law.