Sliding Mode Variable Structure Control Of Delay Systems

Variable structure control has been extensively studied since 1950. Robustness on the changes of dynamics and parameters, and disturbance perturbation is its predominant features. If the phase trajectory of sliding mode is limited to a subspace lower than original systems, then the order of the corresponding differential equations also descends. When the system has some conditions such as matches of uncertainty or interference, there exist invariance nature of sliding mode on the interference or the changes of parameters. These characteristics of variable structure control have gained increasing attention from the field of control. And the problem of variable structure control of multi-variable linear systems and nonlinear systems has been widely studied and many outstanding results have been obtained. Although some investigations have been got for variable structure of the delay systems, its development is rather slow and its development is in initial stage, especially for the multi-variable delay systems, the systems with multi-delay controllers and the neutral systems with delays.Based on Lyapunov functional method, stability theory of Razumikhin, lemma of Schur and linear matrix inequalities(LMI), the problem of variable structure control was studied for the uncertain linear system with multiple delays, the system with multi-delay controllers, the neutral system with delays, and the neutral system with nonlinear perturbation in this paper. When we considered the uncertain with changes multi-delay systems, a simple and easily realized variable structure controller was designed by introducing transformation and utilizing Schur complement, Lyapunov functional method and tendency law. When investigating variable structure control of uncertain systems with multiple delays, we indirectly got variable structure control of the systems by considering the effects of the uncertain terms and the multiple delays. Furthermore, it was proved that the system can get tangent plane and asymptotic stability of sliding mode in a finite time under this control. By using the tendency law, the corresponding control for the neutral systems with delays was given. And then, a sufficient condition for asymptotic stability of sliding mode was described in terms of linear matrix inequality. Finally, based on variable structure control and the general theory of the delay systems, we chosen the switching function to construct the law of variable structure control when the system was subjected to nonlinear perturbation. At the same time, the asymptotic stability of sliding mode was proved by constructing reasonable Lyapunov function and shown the reaching conditions.