Stabilization of nonlinear systems via designed center manifold

This paper addresses the problem of local state feedback stabilization of a class of nonlinear systems with nonminimum phase zero dynamics. A new technique, namely, the Lyapunov function with homogeneous derivative along solution curves was developed to test the approximate stability of the dynamics on the center manifold. A set of convenient sufficient conditions are provided to test the negativity of the homogeneous derivatives. Using these conditions and assuming the zero dynamics has stable and center linear parts, a method is proposed to design controls such that the dynamics on the designed center manifold of the closed-loop system is approximately stable. It is proved that using this method, the first variables in each of the integral chains of the linearized part of the system do not affect the approximation order of the dynamics on the center manifold. Based on this fact, the concept of injection degree is proposed. According to different kinds of injection degrees certain sufficient conditions are obtained for the stabilizability of the nonminimum phase zero dynamics. Corresponding formulas are presented for the design of controls.