Dynamically Scaled Immersion and Invariance Adaptive Control for Euler–Lagrange Mechanical Systems

Abstract

This paper addresses adaptive control of specific Euler–Lagrange systems: rigid-body attitude control and the -link robot manipulator. For each problem, the model parameters are unknown but the lower bound of the smallest eigenvalue of the inertia matrix is assumed to be known. The dynamic scaling immersion and invariance adaptive controller is proposed to stabilize the system without employing a filter for the regressor matrix. A scalar scaling factor is instead implemented to overcome the integrability obstacle that arises in immersion and invariance adaptive control design. First, a filter-free controller is proposed for the attitude problem such that the rate feedback gain is proportional to the square of the scaling factor in the tracking error dynamics. The gain is then shown to be bounded through state feedback while achieving stabilization of the tracking error. A similar approach for the dynamic gain design is applied to a filter-dependent controller, where a filter for the angular rate is used to build a parameter estimator. The proposed design is also applied to robot manipulator systems. Spacecraft attitude and two-link planar robot tracking problems are considered to demonstrate the performance of the controllers through simulations.