1372 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO. 9, SEPTEMBER 2001 Stabilization of Nonlinear Systems via Designed Center Manifold Daizhan Cheng and Clyde Martin Abstract—This paper addresses the problem of the 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 has been 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 dy- namics 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. Index Terms—Approximate stability, center manifold, injection degree, Lyapunov function with homogeneous derivative, zero dy- namics. I. INTRODUCTION S TABILIZATION is one of the basic tasks in control design. The asymptotic stability and stabilization of nonlinear sys- tems have received significant attention [18]–[24]. The center manifold approach has been developed to solve the problem [1], [2], [12], [18], [24]. In [1], [2], some special nonlinear controls are designed to stabilize some particular control systems. The method used there is basically a case-by-case study. For con- trol systems in normal form, assume the center manifold has minimum phase, then a quasi-linear feedback can be used to stabilize linearly controllable variables. We refer to [3]–[6] for minimum phase method and its applications. Based on these pioneer works, this paper proposes a proce- dure to produce a state feedback to stabilize nonminimum phase zero dynamics. The designed state feedback control ensures that the dynamics on the designed center manifold of the closed-loop Manuscript received October 1, 1998; revised November 5, 1999, September 1, 2000, and February 10, 2001. Recommended by Associate Editor L. Y. Wang. This work was supported by the Chinese National Science Foundation under Grants G69774008, G59837270, and G1998020308, and by the National Key Project of China. The work of the second author was supported in part by Grants from the U.S. National Science Foundation. D. Cheng is with the Institute of Systems Science, Chinese Academy of Sci- ences, Beijing 100080, P. R. China (e-mail: dcheng@iss03.iss.ac.cn). C. Martin is with the Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409 USA (e-mail: martin@math.ttu.edu). Publisher Item Identifier S 0018-9286(01)08825-0. system is approximately stable. To obtain the desirable proper- ties, we combine the center manifold method with Lyapunov function method. Motivated by the works on stabilization of homogeneous vector field [13]–[17], we propose a new method, namely, that of a Lyapunov function with homogeneous derivatives along solution curves. This Lyapunov function is used to test the approximate stability of a dynamics with odd degree approxi- mating systems, where degree means the polynomial degree. It is particularly suitable for testing the dynamics on a designed center manifold of a closed-loop system, because the degrees of the approximate system of the dynamics on the center manifold may be converted by certain state feedback controls to have odd degree. In this way, the method is applicable to a large class of nonlinear systems with stable and center zero dynamics. To avoid counting the order of smoothness, through this paper the systems and all other objects involved are assumed to be , or as smooth as required, on a neighborhood of the origin. We motivate this work by means of a practical problem: con- sider the stabilization of an airplane via a designed center man- ifold. We may find some useful observations from this example for design of both the center manifold and the stabilizing con- trols. The following example is basically taken from [7], with a modification that the speed is assumed to be dependent on al- titude when the atmospheric resistance is taken into considera- tion. Example 1.1 [7]: Denote an airplane’s altitude in meters by . Assume that the body of the plane is slanted radians with respect to the horizontal and that the ground speed is . Also, assume the flight path forms an angle of radians with the hor- izontal and is small. The system is described as (1.1) where is a constant representing a natural oscillation frequency and and are positive constants. The problem we address is altitude tracking: i.e., a target altitude , where . Set , , and assume . We have with . Then the system (1.1) is transformed into a standard form as (1.2) 0018–9286/01$10.00 © 2001 IEEE