A.S.I. Zinober

Continuous approximation of variable structure control

A straightforward modification of the theory of ‘sliding’ variable structure control systems is used to design a control law that is a continuous arbitrarily close approximation, within a predefined subspace, to the discontinuous variable structure control law. Linear combinations of eigenvectors, and certain null spaces, are used to analyse the state trajectory under the influence of the continuous control law. Comparisons are drawn and the inherent robustness properties of variable structure control systems are retained, while the undesirable chatter motion of the sliding mode is eliminated. The accuracy of the approximation to the idealized case can easily be calculated. The general results are illustrated by the simulation of a time-invariant fourth-order multivariable system with two controls

Analysis and Design of Variable Structure Systems Using a Geometric Approach

Multivariable variable structure systems in the sliding mode are studies using a geometric approach. The properties of system order reduction and disturbance rejection are proved using projector theory. New design methods for the choice of switching hyperplanes are derived for the closed-loop eigenvalue/eigenvector assignment problem.

Paper: Nonlinear adaptive model-following control

The use of discontinuous control in adaptive model-following continuous control systems with nonlinear time-varying plant is described. The first approach is based on the theory of variable structure systems. Hyperstability theory is the basis of the second scheme; the third one results by combining the previous approaches. The link among the three approaches is explored by studying the error dynamics. Practical implementation and applications are discussed.

Sliding mode state observation for non-linear systems

In this paper a class of non-linear systems with uncertainty and a Lipschitz part is considered. The non-linear part satisfies the Lipschitz condition, whilst the uncertain part is a bounded function. A sliding mode observer is designed to yield feedforward compensation input to stabilize the error estimation system with and without a Lipschitz non-linearity. Sufficient conditions for stability of the Thau observer are also proposed. These conditions ensure the stability of the non-linear observer by selecting a suitable observer gain matrix. In general, when the system contains unmatched uncertainty, ultimate boundedness is obtained. The ultimate boundedness radius depends upon the defined ‘distance of the unmatched uncertainty’. Some sufficient conditions may ensure the stability of a non-linear observer for systems with uncertainties.

Quasi-Continuous Higher Order Sliding-Mode Controllers for Spacecraft-Attitude-Tracking Maneuvers

This paper studies higher order sliding-mode-control laws to deal with some spacecraft-attitude-tracking problems. Quasi-continuous second- and third-order sliding controllers and differentiators are applied to quaternion-based spacecraft-attitude-tracking maneuvers. A class of linear sliding manifolds is selected as a function of angular velocities and quaternion errors. The second method of Lyapunov is used to show that tracking is achieved globally. An example of multiaxial attitude-tracking maneuvers is presented, and simulation results are included to verify and compare the practical usefulness of the various controllers.

Brief Sliding mode control of boost and buck-boost power converters using method of stable system centre

Nonminimum phase tracking control is studied for boost and buck-boost power converters. The sliding mode controller is designed to track directly a causal voltage tracking profile given by an exogenous system. The nonminimum phase output tracking problem is reduced to a state tracking problem. Bounded state tracking profiles are generated by equations of stable system centre. Numerical examples demonstrate the effectiveness of the sliding mode control design.

Brief Transfer equivalence and realization of nonlinear higher order input-output difference equations

Two fundamental modelling problems in nonlinear discrete-time control systems are studied using the language of differential forms. The discrete-time nonlinear single-input single-output systems to be studied are described by input-output (i/o) difference equations, i.e. a high order difference equation relating the input, the output and a finite number of their time shifts. A new definition of equivalence is introduced which generalizes the notion of transfer equivalence well known for the linear case. Our definition is based upon the notion of an irreducible differential form of the system and was inspired by the analogous definition for continuous-time systems. The second problem to be addressed is the realization problem. The i/o difference equation is assumed to be in the irreducible form so that one can obtain an accessible and observable realization. Necessary and sufficient conditions are given for the existence of a (local) state-space realization of the irreducible i/o difference equation. These conditions are formulated in terms of the integrability of certain subspaces of one-forms, classified according to their relative degree. The sufficiency part of the proof gives a constructive procedure (up to finding the integrating factors and integration of the set of one-forms) for obtaining a locally observable and accessible state-space system. If the system is not in the irreducible form, one has first to apply the reduction procedure to transform the system into the irreducible form.

Robust hyperplane design in multivariable variable structure control systems

The problem of designing a stable sliding mode giving robust performance of a variable structure control system is studied. A set of specified eigenvalues is assigned to the closed-loop feedback system during the sliding mode. The design philosophy seeks to build on the known robustness and invariance properties associated with the use of discontinuous (or continuous) non-linear controls, by assigning the eigenvectors associated with the sliding-mode eigenvalues in a manner that leads to a robust control scheme. A previously developed canonical form for the hyperplane design problem is again employed. The design technique centres on a recently published eigenvalue assignment algorithm, and is illustrated by the inclusion of an example.

Sliding mode time-delay systems

The sliding mode control of time-delay systems is considered. Time-delay sliding system stability is studied for the cases of some information about the delay and also lack of information. The sliding surface is delay-independent as for the traditional sliding surface and the reaching condition is achieved by applying a conventional discontinuous control.

Deterministic control of uncertain systems

In the deterministic control of uncertain multivariable systems robust closed-loop control is achieved for a specified magnitude range of a class of parameter variations and disturbances. These nonlinear time-varying systems employ nonlinear control functions. The two main approaches are Lyapunov control and variable structure control. The development of the theory of this latter class of control systems will be briefly reviewed here and particular reference will be made to the eigenstructure assignment approach. The use of a CAD package VASSYD allows for straightforward design and simulation studies. Desirable robustness and invariance properties are readily obtained.