On estimation of the domain of attraction for sliding mode control of underactuated nonlinear systems

SUMMARY A system is considered underactuated if the number of the actuator inputs is less than the number of degrees of freedom for the system. Sliding mode control for underactuated systems has been shown to be an effective way to achieve system stabilization. It involves exponentially stable sliding surfaces so that when the closed-loop system trajectory reaches the surface, it moves along the surface while converging to the origin. In this paper, a general framework that provides sufficient conditions for asymptotic stabilization of underactuated nonlinear systems using sliding mode control in the presence of system uncertainties is presented. Specifically, it is shown that the closed-loop system trajectories reach the sliding surface in finite time, and a constructive methodology to determine exponential stability of the closed-loop system on the sliding surface is developed, which ensures asymptotic stability of the overall closed-loop system. Furthermore, the aforementioned framework provides the basis to determine an estimate of the domain of attraction for the closed-loop system with uncertainties. Finally, the results developed in the paper are experimentally validated using a linear inverted pendulum testbed to show a good match between the actual domain of attraction of the upward equilibrium state of the pendulum and its analytical estimate.Copyright © 2012 John Wiley & Sons, Ltd.