1. Feedback Stabilization of Nonlinear Singularly Perturbed Systems MENG Bo JING Yuanwei SHEN Chao College of Information Science and Engineering, Northeastern University, Shenyang 110004 2. Problem Formulation Consider the singularly perturbed system It can be divided into a fast and a slow sub-systems. This process is called two- time scale separation in singular perturbation theory. Assume is invertible in. Setting, the system becomes Since the equation admits a unique solution, we have where This reduced model is called quasi-steady model ， and is also the slow- subsystem. From the original system, we can define a fast time scale Then the original system takes the form Similarly setting ， we have the system which is called boundary layer model, and is also the fast subsystem. Assumption 1 There exists a analytical co-vector field, such that the matrix is Hurwitz uniformly stable in. The new variable is the quasi-steady state of the fast dynamic, that is Once the fast states hit the given manifold ， the slow system takes the form Assumption 2 The relative degree of the slow system is, that is, for all, there is, and is an involutive distribution, matrix is nonsingular in Taking each order Lie derivative of to perform new coordinate transform- ation of the state, then we get Brunowsky standard form of the system Stability Analysis Assumption 3 Equilibrium of zero dynamic is exponentially stable. The Lyapunov function of the slow system satisfies The Lyapunov function of the boundary layer system satisfies And the selected Lyapunov function for the overall system is We can get the following result Fig.1 state trajectories of the closed-loop system 1. Introduction The feedback stabilization of singularly perturbed system is considered. Base on singular perturbation and exact linearization theory of differential geometry, the slow sub-system is transformed to Brunowsky standard model. A Lyapunov function for the overall system is established through that of the slow system and the boundary layer system. By calculating its derivative along the state trajectory of the original system, the upper bound of perturbed para meter is given to obtain the condition of asymptotically stability. Simulation results show the effectiveness and feasibility of the proposed controller. 4. Simulations The final control law is The simulation figures are 3. Main Results Design of Feedback Controller Consider the controller in the following form Under this control law ， the original system takes the form Performing a standard two-time scale decomposition, we have the fast system Introduce a new vector, where Fig.1 shows the state trajectories of the closed-loop system under the state feedback controller. The initial values are,,. The trajectory of hit slow manifold rapidly in approximate vertical direction along the fast manifold. It embodies that the motion time of boundary layer system is relatively transitory ( ), and all states converge to the origin along the slow manifold. Contact email: mengbo_422@126.com Paper ID: A9-155