A nonlinear controller with guaranteed uniqueness of Filippov’s solution is presented for controlling a two-link non-fixed-base inverted pendulum. The control system is described by differential equations with discontinuous right-hand sides, which violates the requirements of the conventional solution theories to ordinary differential equations, i.e., the vector fields must be at least Lipschitz continuous. It has been shown that Lyapunov’s second method can be used for such non-smooth systems directly under the condition of existence and uniqueness of Filippov’s solution. For this non-smooth control system with three discontinuity surfaces, the uniqueness of the solution is studied using Filippov’s solution concept. The system itself is a nonlinear, non-autonomous dynamic system without an isolated equilibrium point, which violates the assumption of Lyapunov’s stability theory. To analyze the stability of the control system, a Lyapunov-like function is constructed, which satisfies all the requirements for a Lyapunov function. Such a function can serve as an upper bound of the region in which the pendulum can be stabilized. Simulation results are presented to validate the proposed approach.
On Control of a Two-Link Non-Fixed-Base Inverted Pendulum With Guaranteed Uniqueness
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Zeng, H, Wu, Q, & Sepehri, N. "On Control of a Two-Link Non-Fixed-Base Inverted Pendulum With Guaranteed Uniqueness." Proceedings of the ASME 2004 International Mechanical Engineering Congress and Exposition. Dynamic Systems and Control, Parts A and B. Anaheim, California, USA. November 13–19, 2004. pp. 1391-1397. ASME. https://doi.org/10.1115/IMECE2004-61239
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