Fuzzy-Based Controller Synthesis and Optimization for Underactuated Mechanical Systems With Nonholonomic Servo Constraints

This article investigates the trajectory tracking problem of underactuated mechanical systems (UMSs) with companion nonholonomic servo constraints and uncertainties. For such motion tasks, the existing approaches in the literature attempt unrealistically to furnish a reliable closed-form solution, rendering it difficult to have high-quality tracking performance with theoretical support. In addition, the uncertainties typically pose substantial difficulty in the controller synthesis. Here, by invoking the methodology of fuzzy sets, the uncertainties in the UMSs are elegantly represented; and with this, the formulation becomes such that a closer link between the uncertain dynamical model of the UMSs and the real world is established. The reference trajectories are regarded appropriately as servo constraints, and subsequently an adaptive robust controller is designed to accomplish the trajectory tracking task from a specific viewpoint of servo constraint tracking. As supported by rigorous proofs, the closed-form solution to the proposed controller is obtained with guaranteed Lyapunov stability. Leveraging on the closed-form solution, the global optimizer to the controller gain parameter can be determined, which is shown to exhibit several important properties including existence and uniqueness. Finally, a numerical example is presented to demonstrate the effectiveness of the designed method.