An Integral Sliding Mode Control Design for a Class of Underactuated Motion Systems

In this paper we deal with one of the most challenging control problems - controlling a class of underactuated systems with uncertainties. A synthesized integral sliding mode controller (ISMC) is proposed. The ISMC, constructed with a suitably chosen integral sliding surface, is able to completely nullify the influence from any matched factors, especially the matched uncertainties. As a consequence, a sliding manifold is generated, in which the controller design can focus on the unmatched factors only. A unique advantage of the ISMC is its preservation of the same number of actuators in the sliding manifold as the original system. From practice, a linear state-feedback controller, simple and smooth, is found adequate in stabilizing the sliding manifold in a wide range around the equilibrium. However, it is extremely difficult to verify the effectiveness of such a linear state feedback for underactuated systems in the presence of unmatched factors such as unmatched uncertainties. A main contribution of this work is to explore the design issue and effectiveness of the linear controller for the underactuated systems. First we reformulate the unmatched factors into several representative forms, then the linear matrix inequality approach is employed to design the feedback gains and maximize the stability region concurrently. As an illustrative example, the synthesized ISMC is applied to a unicycle plus inverse pendulum system, and the results validate the effectiveness of the proposed controller.