Variable Structure Control for Singularly Perturbed Linear Continuous Systems With Matched Disturbances

A variable structure system can be studied using the singular perturbation theory. The discontinuous control that leads to a finite-time reaching of the sliding surface creates fast-time transients analogous to the stable boundary layer dynamics of a singularly perturbed system. As the sliding mode is attained, the slow-time dynamics prevails, just as that of a singularly perturbed system after the boundary layer dynamics fades away. In this technical note, the problem of sliding mode control for singularly perturbed systems in the presence of matched bounded external disturbances is investigated. A composite sliding surface is constructed from solutions of algebraic Lyapunov equations which are derived from both the fast and the slow subsystems. The resultant sliding motion ensures Lyapunov stability with disturbance rejection. Two proposed schemes that ensure the asymptotic stability of the system are presented. The effectiveness of the proposed methods is demonstrated in a numerical example of a magnetic tape control system.