Deterministric Control of Uncertain Systems
Includes sections on: Sliding mode control with switching command devices. Hyperplane design and CAD of variable structure control systems. Variable structure controllers for robots. The hyperstability approach to VSCS design. Nonlinear continuous feedback for robust tracking. Control of uncertain systems with neglected dynamics. Control of infinite dimensional plants.
Inspec keywords: self-adjusting systems; stability; feedback; nonlinear control systems; uncertain systems; variable structure systems; control system synthesis; continuous systems
Other keywords: variable structure control; state observation; model following control; hyperplane design; sliding mode control; continuous selfadaptive control; nonlinear composite control; hyperstability approach; subspace attractivity; deterministic control; nonlinear continuous feedback control; uncertain system control; output feedback application
Subjects: Nonlinear control systems; Self-adjusting control systems; Stability in control theory; Control system analysis and synthesis methods; Fuzzy control; Multivariable control systems
- Book DOI: 10.1049/PBCE040E
- Chapter DOI: 10.1049/PBCE040E
- ISBN: 9780863411700
- e-ISBN: 9781849193412
- Page count: 380
- Format: PDF
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Front Matter
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1 An introduction to variable structure control
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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.
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2 Sliding mode control with switching command devices
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In this chapter this second type of variable structure control system is described, beginning with scalar control systems. The basic structure, the sliding conditions and the state equations in the sliding mode will be considered. The pole assignment method will be used to design the switching strategy. Afterwards the sliding mode domain and the switching frequency will be studied. These theoretical considerations will be illustrated with a practical example, the position control of a DC drive. Next, for multivariable systems, the sliding mode conditions, the design of the multivariable switching strategy with a pole assignment method and robustness, i.e. invariance properties with regard to parameter variations, will be described.
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3 Hyperplane design and CAD of variable structure control systems
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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.
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4 Subspace attractivity and invariance: Ultimate attainment of prescribed dynamic behaviour
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The outline of the chapter is as follows. We begin with a brief resume of relevant concepts and results from the theory of differential inclusions. The class of uncertain systems to be considered in then made precise. We continue with a treatment of the variable structure systems concept of an invariant subspace ℒ (with prescribed dynamic behaviour therein) and construct a discontinuous feedback strategy which renders if globally finite-time attractive (thereby ensuring ultimate attainment of prescribed dynamic behaviour). The approach is essentially that of Ryan & Corless (1984) (with origins in Corless & Leitmann, 1981), subsequently recast in a differential inclusion setting by Goodall & Ryan (1986, 1988). Finally, our results are extended to problems of tracking and model following.
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5 Model-following control of time-varying and nonlinear avionics systems
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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.
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6 Canonical formulation and general principles of variable structure controllers
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This chapter has essentially two goals: first it introduces a new variable structure controller using a canonical system representation, orthogonal sliding hyperplanes, and a straightforward synthesis and analysis procedure. The second goal is more didactically oriented. Some basic facts of VSC will become evident from the special formulation. In first section it has been shown that the canonical formulation and the rank conditions are equivalent. In second section the new controller is presented which consists of two distinct terms. The first is linear state feedback which produces a pole-zero cancellation leaving only integrators in the transfer matrix of the known part of the system. The second term, a relay type controller, forces the initially nonzero outputs to zero and stabilises the pole-zero cancellation in the presence of parameter disturbances. In third section some interesting remarks on the nature of sliding variable structure systems are listed. Fourth section shows that the presented regulator can be modified in a straightforward way in order to accomplish the requirements of model-following VSC . The numerical example demonstrates the synthesis and analysis procedure.
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7 Variable structure controllers for robots
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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.
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8 Applications of output feedback in variable structure control
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Most of the techniques for the design of Variable Structure Control Systems (VSCS) for both scalar and multivariable systems that is presented in the literature assumes that either the state vector is directly measurable or that an observer is used to reconstruct the state. In systems which do not have the full state vector available or that have a set of output measurements y, the usual technique for VSCS design of equivalent control cannot be applied. This chapter discusses the design of VSCS for the case of output feedback. This will be followed by a section extending the ideas to the output feedback case.
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9 The hyperstability approach to VSCS design
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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.
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10 Nonlinear continuous feedback control for robust tracking
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A robust tracking control system has been presented. The model, affected by bounded parameter uncertainties, has a structure often encountered in the analysis of complex mechanical systems. The continuous nonlinear control law which has been proposed (and which is an approximation of discontinuous laws) ensures the attractiveness of a certain region of the state space, while leaving the possibility of choosing the rate of convergence toward any ball of ultimate boundedness or just a decoupled structure of the controller. The method of analysis and design merges ideas of the VSC and Lyapunov approaches. Some of those ideas have been developed here and enriched for the tracking problem.
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11 Deterministic control of uncertain systems. A Lyapunov theory approach
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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.
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12 Control of uncertain systems with neglected dynamics
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This chapter discusses the control of uncertain system with negative dynamics. The prototype for the class of systems considered in this chapter consists of a dynamical process controlled by a feedback law acting on state data generated by sensor and implemented via actuator.
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13 Nonlinear composite control of a class of nominally linear singularly perturbed uncertain systems
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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.
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14 Some extensions of variable structure control theory for the control of nonlinear systems
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The main goal of this chapter is to describe new tools in the analysis of discontinuous control systems and to show, with some significant control problems, how the VSC approach yields efficient and robust control, compared with other techniques used to deal with the same problems.
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15 Continuous self-adaptive control using variable structure design techniques
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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.
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16 State observation of nonlinear control systems via the method of Lyapunov
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In this chapter we concentrate our attention on the problem of state observation of nonlinear dynamical systems whose nonlinearities/uncertainties are bounded. The main tool used in the design of observers for such systems is the Lyapunov method.
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17 Control of infinite-dimensional plants
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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.
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Back Matter
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