Continuous approximation of variable structure control

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Abstract
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