LMI-based synthesis of a robust saturated controller for an underactuated mechanical system subject to motion constraints

Abstract

In this paper, we suggest a control synthesis approach for the robust stabilization of an underactuated mechanical system, the Inertial Wheel Inverted Pendulum (IWIP), with norm-bounded parametric uncertainties and subject to both motion constraints and actuator saturation. A state-feedback controller is adopted to achieve such robust stabilization of the IWIP at its upright position, while the problem of the motion constraints is addressed by considering the saturation effect of the control input. Our design methodology of the robust saturated state-feedback control law is realized within the framework of Linear Matrix Inequalities (LMIs). We show first that the synthesis problem of such controller is written in terms of Bilinear Matrix Inequalities (BMIs), which are hardly tractable numerically. Then, in order to overcome this obstacle, we use some judicious mathematical tools to convert these BMIs into LMIs. Moreover, we consider the problem of enlarging the domain of attraction by computing the largest attractive invariant ellipsoid for the uncertain constrained nonlinear dynamics of the underactuated IWIP under the saturated state-feedback control law. Simulation results demonstrate the effectiveness and the robustness of the proposed robust saturated state-feedback controller towards norm-bounded parametric uncertainties while constraints on the system motion and the control input are respected.