Adaptive Fuzzy Control for Uncertain Underactuated Systems with Unknown Control Direction

Abstract

This paper proposes two adaptive fuzzy control laws for a class of second-order underactuated mechanical systems (UMSs) characterized by unknown nonlinear dynamics and uncertain control direction. The control design begins by formulating an ideal sliding mode control (ISMC) when the system dynamics are exactly known, which can achieve predefined control. Objectives precise convergence of the system outputs to desired values and boundedness of all closed-loop signals. Subsequently, since the actual dynamics are unknown, a fuzzy system with adjustable parameters is used to approximate this ideal controller while preserving the predefined objectives. The key contribution of this work lies in addressing the challenge of unknown control direction through two distinct approaches. The first approach extends the use of the Nussbaum-type function, commonly used in fully actuated systems, to estimate the control direction for UMSs. In the second approach, the control direction is modeled as an unknown constant gain, and an adaptive law is developed to estimate it without the need for the Nussbaum function, offering a more streamlined solution. The stability properties of the closed-loop system for both control strategies are proven using the Lyapunov method. Simulation results and a comparative analysis of the two control laws, alongside recent works, are presented to demonstrate the effectiveness and robustness of the proposed strategies