Immersion and Invariance Adaptive Control Through Polynomial Adaptation1

Abstract

In conventional immersion and invariance (I&I) adaptive control design, control parameter adaptation is typically linear with respect to the parameter error-induced perturbation, resulting in quadratic rate dissipation of energy associated with the off-the-manifold variable. Departing from such a convention, this article contributes a novel strategy—polynomial adaptation. As the name suggests, control parameter adaptation in this approach takes the form of a general polynomial in relation to the perturbation. Accordingly, this new design induces polynomial rate energy dissipation, which is faster than the quadratic one in the conventional scheme, thereby enhancing closed-loop control performance. The theoretical underpinnings of the new approach are demonstrated through the design of an I&I adaptive tracking control law for a general nth-order, single-input–single-output, parametrically uncertain, nonlinear system in the controllable canonical form. In addition, a numerical study of the proposed method on the second-order forced Duffing oscillator shows its improved transient performance in comparison to a baseline controller developed with the standard I&I adaptive control technique.