Second order SM Regulator for Nonlinear Non Minimum Phase Systems Marcos I. Galicia* Alexander G. Loukianov* Bernardino Castillo-Toledo* Jorge Rivera** * CINVESTAV, Unidad Guadalajara, Av. Científica 1145, Zapopan, 45015, Jalisco, México (Tel: (+52)-33-3777-3600; e-mails:[mgalicia] [louk][toledo]@gdl.cinvestav.mx). ** CUCEI, Department of Electronic, Universidad de Guadalajara, Guadalajara, 44430, Jalisco, México. ( e-mail:jorge.rivera@cucei.udg.mx) Abstract: This work presents an approach to solve the output regulation problem for a class of nonlinear non minimum phase systems. Based on decomposition block control technique and sliding mode (SM) control, we propose a sliding manifold which is designed such that the zero dynamics become asymptotically stable on this manifold. To enforce the SM motion on the designed manifold a super- twisting SM algorithm is used. The effectiveness of the proposed methodology is verified via the design of a torque tracking controller for an induction motor. 1. INTRODUCTION Roughly speaking, the regulator problem, consists in designing a continuous state or error feedback controller such that the output of a system tracks a reference signal possibly in the presence of a disturbance signal, that is in the classical setup. A solution for the linear setting of the problem was presented in (Francis, 1977), based on the existence of a solution for a set of algebraic matrix equations. In the nonlinear framework, it was shown in (Isidori et al., 1990) that the solution can be posed in terms of the solution of a set of nonlinear differential equations, which represents a generalization of the Francis conditions. These equations become known as the Francis-Isidori-Byrnes (FIB) equations. Basically, the regulator solution can be viewed as finding a steady-state manifold on which the output tracking error map is zero, and which can be made attractive and invariant by a feedback. An alternative approach to deal with this problem is the use of the sliding mode technique to decompose and simplify the regulator design procedure and impose robustness properties (Utkin, 1992), (Elmali et al., 1992). The underlying idea is to design a sliding surface on which the dynamics of the system are constrained to evolve by means of a discontinuous control law, instead of designing a continuous stabilizing feedback, as in the case of the classical regulator problem. The sliding manifold contains the steady- state surface, and the dynamics of the systems tend along the sliding manifold, to the steady-state behavior. In the full information case, a static state feedback sliding mode regulator design have been studied in (Elmali et al., 1993), and (Castillo-Toledo et al., 1995). To overcome the limitation of the full information knowledge, a dynamic discontinuous error feedback strategy have been designed in (Edwards et al., 1998) for linear systems, and in (Sira-Ramirez, 1993), and (Loukianov et al., 2004) for some classes of nonlinear systems. Considering the state of the exosystem is accessible, a dynamic error feedback regulator has been proposed in (Bonivento et al., 2001) for a class of nonlinear non minimum phase systems with unitary relative degree. In this work, we address the problem of output sliding mode regulation, as well as the asymptotically stabilization of the zero dynamics for a general case of nonlinear systems without restriction on the relative degree. The paper is organized as follows. In Section 2, the regulator problem statement is described. Section 3 presents the SM regulation problem for nonlinear systems. Section 4 introduces a class of non minimum phase nonlinear system presented in the perturbed Nonlinear Block Controllable form (NBC-form) with residual dynamics. In Section 5, a sliding manifold on which the zero dynamics become asymptotically stable is designed and second order SM algorithm is applied to ensure the designed manifold be attractive. An adaptive algorithm for the stabilization of the residual dynamics is described in Section 6. Section 7 presents an application of the proposed method to design an induction motor torque controller. Simulation results are presented in Section 8. 2. PROBLEM STATEMENT Consider the following system subject to disturbance () () () () (,) t t     x fx Bxu Dxw gx  (1) ()  y hx where n   x X R is the state vector, m  u U R is the control vector, the vector field () fx and the columns of () Bx and () Dx are smooth and bounded mappings of class f f) , 0 [ C , the vector () gx is the unknown disturbance, p  w W R is a vector of external known disturbances generated by an external system described by ( )  w  w  . (2) It is assumed (0) 0  f , (0)=0 h , (0)=0  and () rank m  Bx . Preprints of the 18th IFAC World Congress Milano (Italy) August 28 - September 2, 2011 Copyright by the International Federation of Automatic Control (IFAC) 11055