A phase-portrait stability analysis for a reaction wheel pendulum using a generalized backstepping control approach

Abstract

This research presents a stability analysis of a reaction wheel pendulum (RWP) using the phase-portrait method. To derive the general control law that stabilizes the RWP system in the upper vertical position, a comprehensive backstepping control design is provided. This control is formulated in a generalized manner for a two-dimensional dynamic system, with the key advantage of guaranteeing asymptotic stability via Lyapunov's stability theorem. In this approach, the upper vertical position transitions from an unstable saddle point in open loop to a stable node in closed loop. The backstepping control design, when applied to the RWP system, incorporates a polynomial-based controller combined with a trigonometric function. A qualitative comparison with existing literature, including passivity- and Lyapunov-based control designs, confirms the generalization capabilities of the proposed backstepping controller. The phase-portrait analysis was conducted using MATLAB version 2024$b$.