Robust Control Using Sliding Mode for a Class of Under-Actuated Systems With Mismatched Uncertainties

Based on the methodology of sliding mode, this paper presents a robust controller for a class of under-actuated systems with mismatched uncertainties. Such a system consists of a nominal system and the mismatched uncertainties. The structural characteristic of the nominal system is that it is made up of several subsystems. Based on this characteristic, the hierarchical structure of the sliding mode surfaces is designed for the nominal system as follows. Firstly, the nominal system is divided into several subsystems and the sliding mode surface of every subsystem is defined. Secondly, the sliding mode surface of one subsystem is selected as the first layer sliding mode surface. The first layer sliding mode surface is then to construct the second layer sliding mode surface with the sliding mode surface of another subsystem. This process continues till the sliding mode surfaces of all the subsystems are included. For dealing with the mismatched uncertainties, a lumped sliding mode compensator is designed at the last layer sliding mode surface. The asymptotic stability of every layer sliding mode surface and the sliding mode surface of each subsystem is proven theoretically by Barbalat´s lemma. Simulation results show the validity of this robust control method through stabilization control of a double inverted pendulums system with mismatched uncertainties.