# VSC - Variable Structure Control of Nonlinear Multivariable...

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Variable Structure Control of Nonlinear Multivariable Systems: A Tutorial This paper presents, in a tutorial manner, the design of variable structure control WSC) systems for a class of multivariable nonlin- ear time varying systems. By the use of the Utkin-DraienoviC “method of equivalent control” and generalized Lyapunov stabil- ity concepts, VSC design is described in a unified manner. Com- plications that arise due to multiple inputs are then described and several approaches useful in overcoming these complications are then developed. After this, the paper investigates recent developments and the kinship of VSC and the deterministic approach to the control of uncertain systems. All points are illustrated by numerical exam- ples. In addition, the recent VSC applications literature is sur- veyed. I. INTRODUCTION Variable Structure Control (VSC) is a viable high-speed switching feedback control (for example, the gains in each feedback path switch between two values according to some rule). This variable structure control law provides an effective and robust means of controlling nonlinear plants. It has its roots in relay and bang-bang control theory. The advances in computer technology and high-speed switch- ing circuitry, have made the practical implementation of VSC a reality and of increasing interest to control engineers (see References). Essentially, VSC utilizes a high-speed switching control law to drive the nonlinear plant’s state trajectory onto a specified and user-chosen surface in the state space (called the sliding or switching surface), and to maintain the plant’s state trajectory on this surface for all subsequent time. This surface is called the switching surface because if the state trajectory of the plant is “above” the surface a control path has one gain and a different gain if the trajectory drops ”below” the surface. The plant dynamics restricted to this surface represent the controlled system’s behavior. By proper design of the sliding surface, VSC attains the con- Manuscript received June 3,1987; revised November 16, 1987. R. A. DeCarlo and S. H. Zak arewith the School of Electrical Engi- neering, Purdue University, West Lafayette, IN 47907, USA. G. P. Matthews is with the Electrical and Electronics Engineering Department, General Motors Research Laboratories, Warren, MI 48090, USA. IEEE Log Number 8719082. ventional goals of control such as stabilization, tracking, regulation, etc. The purpose of this paper is to furnish quick readable access to key design techniques in VSC (scattered through- out the literature) for a class of nonlinear time-varying sys- tems. Because of the paper’s tutorial nature, the presen- tation includes only several of the basic forms of the many VSC design methods for multivariable, nonlinear, time- varying systems. These basic forms often need tweaking before application.

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## This note was uploaded on 02/07/2011 for the course MECH 646 taught by Professor Danielnassrallah during the Fall '10 term at American University of Beirut.

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VSC - Variable Structure Control of Nonlinear Multivariable...

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