Variable structure control for a class of systems with mismatched uncertainties

The problem of Variable Structure Control (VSC) for a class of systems with mismatched uncertainties is studied in this paper. The systems considered in this paper can be transformed into two staged sub-systems. The first subsystem is a linear system with uncertainties U1, and its inputs are the state variables of the second sub-system. The second sub-system, having the same dimension as the inputs, is a nonlinear system with uncertainties U2 and its inputs are the control inputs for the original system. In fact, U1 and U2 are the mismatched uncertainties and matched uncertainties of the original system, respectively. Based on the quadratic stabilization for the first sub-system, the desired values of the second state variables are synthesized by Linear Matrix Inequality (LMI) technique, and the switching function vector is constructed based on the errors between the values of the second state variables and their desired values. Furthermore, the variable structure controller is derived by the reachable condition for the sliding-mode such that the closed-loop system can enter and retain the sliding-mode. Meanwhile, the closed-loop system has the stable sliding-mode motion.