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This paper addresses the exponential stability issue of discrete-time switched nonlinear systems by means of the linearization approach. Generally speaking, the linearization method requires that the vector field of the original nonlinear system is continuously differentiable. Since the vector field of a switched system does not meet the differentiable condition (discontinuous at switching instants), the linearization method is seldom used in the context of switched systems. In the present paper, it is assumed that the vector field of each subsystem is continuously differentiable. With this assumption, two Lyapunov theorems are proposed to verify the exponential and asymptotic stabilities of the considered system. Applying these results and using the linearized system, a multiple homogeneous polynomial Lyapunov function (HPLF) is adapted to establish the exponential stability of switched nonlinear systems. Moreover, it is shown that the conservatism of the stability condition reduces as the degree of the HPLF increases.
Liu et al. (Mon,) studied this question.