Adaptive Controller Design for Tracking and Disturbance Attenuation in Parametric Strict-Feedback Nonlinear Systems

The authors develop a systematic procedure for obtaining robust adaptive controllers that achieve asymptotic tracking and disturbance attenuation for a class of nonlinear systems that are described in the parametric strict-feedback form and are subject to additional exogenous disturbance inputs. Their approach to adaptive control is performance-based, where the objective for the controller design is not only to find an adaptive controller, but also to construct an appropriate cost functional, compatible with desired asymptotic tracking and disturbance attenuation specifications, with respect to which the adaptive controller is “worst case optimal.” In this respect, they also depart from the standard worst case (robust) controller design paradigm where the performance index is fixed priori. Three main ingredients of the paper are the backstepping methodology, worst case identification schemes, and singular perturbations analysis. Under full state measurements, closedform expressions have been obtained for an adaptive controller and the corresponding value function, where the latter satisfies a Hamilton–Jacobi–Isaacs equation (or inequality) associated with the underlying cost function, thereby leading to satisfaction of a dissipation inequality for the former. An important by-product of the analysis is the finding that the adaptive controllers that meet the dual specifications of asymptotic tracking and disturbance attenuation are generally not certainty-equivalent, but are asymptotically so as the measure quantifying the designer’s confidence in the parameter estimate goes to infinity. To illustrate the main results, the authors include a numerical example involving a third-order system.