Ball on a beam: stabilization under saturated input control with large basin of attraction

TL;DR

This paper develops a control strategy for stabilizing two underactuated beam-and-ball systems, explicitly considering input saturation, and demonstrates how to maximize their basins of attraction through simulation.

Contribution

It introduces a novel control law that expands the basin of attraction close to the controllability domain for both systems, including a new circular beam-and-ball system.

Findings

Control law effectively stabilizes systems with saturated inputs.

Basin of attraction can be made arbitrarily close to the controllability domain.

Simulation confirms the control law's efficiency for both systems.

Abstract

This article is devoted to the stabilization of two underactuated planar systems, the well-known straight beam-and-ball system and an original circular beam-and-ball system. The feedback control for each system is designed, using the Jordan form of its model, linearized near the unstable equilibrium. The limits on the voltage, fed to the motor, are taken into account explicitly. The straight beam-and-ball system has one unstable mode in the motion near the equilibrium point. The proposed control law ensures that the basin of attraction coincides with the controllability domain. The circular beam-and-ball system has two unstable modes near the equilibrium point. Therefore, this device, never considered in the past, is much more difficult to control than the straight beam-and-ball system. The main contribution is to propose a simple new control law, which ensures by adjusting its gain parameters that the basin of attraction arbitrarily can approach the controllability domain for the linear case. For both nonlinear systems, simulation results are presented to illustrate the efficiency of the designed nonlinear control laws and to determine the basin of attraction.