In this introductory chapter attention will be focused upon the regulator, where the aim of the design is to regulate the system state to zero. Much of the theory may be applied with suitable modifications to model-following and tracking control systems. In this chapter some of the basic features and properties of variable structure control have been discussed. Numerous applications to a wide variety of practical problem are listed. The deterministic control of uncertain time-varying systems control is achieved using nonlinear feedback control functions, which operate effectively over a specified magnitude range of a class of system parameter variations, without the need for on-line identification of the values of the parameters. This control philosophy contrasts sharply with stochastic adaptive control systems in which the control law is altered whilst system parameter values are calculated using online identification algorithms.

In this chapter we have outlined an approach to hyperplane design in variable structure systems based on the assignment of the eigenstructure in the sliding mode. In order to clarify the number of degrees of freedom in this process and to simplify the numerical implementation of the design procedures, a particular canonical form for the system has been proposed. A control scheme to drive the state into the sliding mode has been outlined, along with a method of reducing the chattering component of the control. A computer-aided design package, VASSYD, has been developed by the authors to assist the designer.

The technique of variable structure model-following control system design has been discussed. The design objective is to force the error between the model and the plant to zero as time tends to infinity. The initial condition of the plant and model need not be equal although the matching conditions must be satisfied for a perfect model-following.A great deal of the work in the field of variable structure flight control system design to date has concentrated upon solving the state-regulator problem and indeed some very robust control schemes have been designed.

This chapter shows how a broad class of uncertain mechanical systems (including robots) can be forced to behave like a chosen and therefore perfectly known reference model. This model can be used to calculate off-line some particular trajectories or to design some model based control schemes. The presented results are well suited for applications where the robot has to repeatedly perform certain tasks, e.g. in assembly operations with very high precision requirements. A nonlinear reference model and a simple regulator with variable structure control is used. The advantages of using a nonlinear reference model are twofold.

A systematic approach to the design of VSC systems has been given. Using the hyperstability theory, the stability of the global system has been established and the existence of the sliding modes. Simple control laws can be used and high speed of response is obtainable by forcing the system with maximum allowable signals, e.g. by means of relay or unit-vector laws. A simple way to gain confidence with synthesis is an extensive and accurate simulation before implementing the control system. To this end it is very useful to realise effective CAD packages for variable structure control system design.

This chapter considers the problem of obtaining memoryless stabilising feedback controllers for uncertain dynamical systems described by ordinary differential equations. Various classes of controllers are presented. The design of all these controllers is based on Lyapunov theory. The results to obtain tracking controllers for a general class of uncertain mechanical systems was utilised. These controllers are illustrated by application to a model of the Manutec r ³ robot which has an uncertain payload. Before proceeding with the problem, some basic notions and results for ordinary differential equations is introduced.

The conditions that allow the construction of a composite control for the practical stabilisation of a singularly perturbed uncertain system have been analysed. Even though the system has been considered nominally linear, the presence of the uncertainties couples the fast and the slow dynamics of the system and, in consequence, the controller must contain a nonlinear term that allows the separate design of the fast and the slow controllers. It has been found that the feedback gains of the slow variable, besides having a lower bound (as is usual in the control of uncertain systems), also possess an upper bound that depends mainly on the structural coupling between the slow and the fast parts of the system. This upper bound prevents the variables which have been con sidered as having slow dynamics from becoming fast under the action of a high gain feedback.

With the development of CSAC the control system designer can, for all orders of system, 'fine tune' an initial VSC system design and therefore improve the speed of response of the system whilst in the sliding mode. The CSAC possesses a degree of robustness since it is based upon robust VSC theory. Chatter motion associated with ideal sliding motion in an ideal VSC system is eliminated as the CSAC system is developed from SmVSC, and CSAC is easily implemented. By introducing the CSAC system, the VSC system is now split into three distinct phases: the attainment (hitting) of the sliding mode, the sliding mode itself and then an improvement (adaptation) of the response in that sliding mode, should that be required. There are many degrees of freedom available in prescribing the rate of adaptation in the multivariable CSAC system and this degree greatly increases as the order of the reduced state space of the sliding mode increases. Further investigation into CSAC and its applications should provide additional interesting results.

In sliding mode control theory the major attention has been paid to finite dimensional systems described by ordinary differential equations. However, mathematical models of a wide range of processes in modern technology are partial differential equations, integro-differential equations and equations with delays. Attempts at theoretical generalisations and applications of sliding modes to control of infinite-dimensional plants show that control scientists and engineers are faced the challenge of increased complexity. Practically all the concepts of discontinuous control theory should be completely revised. Even the basic concepts discontinuity surface, sliding mode, component-wise design should be clarified or reintroduced. In the examples in this chapter we have touched upon infinite-dimensional systems mainly with distributed control.