A nonparametric learning framework for nonlinear robust output regulation

A nonparametric learning solution framework is proposed for the global nonlinear robust output regulation problem. We first extend the assumption that the steady-state generator is linear in the exogenous signal to the more relaxed assumption that it is polynomial in the exogenous signal. Additionally, a nonparametric learning framework is proposed to eliminate the construction of an explicit regressor, as required in the adaptive method, which can potentially simplify the implementation and reduce the computational complexity of existing methods. With the help of the proposed framework, the robust nonlinear output regulation problem can be converted into a robust non-adaptive stabilization problem for the augmented system with integral input-to-state stable (iISS) inverse dynamics. Moreover, a dynamic gain approach can adaptively raise the gain to a sufficiently large constant to achieve stabilization without requiring any a priori knowledge of the uncertainties appearing in the dynamics of the exosystem and the system. Furthermore, we apply the nonparametric learning framework to globally reconstruct and estimate multiple sinusoidal signals with unknown frequencies without the need for adaptive parametric techniques. An explicit nonlinear mapping can directly provide the estimated parameters, which will exponentially converge to the unknown frequencies. Finally, a feedforward control design is proposed to solve the linear output regulation problem using the nonparametric learning framework. Two simulation examples are provided to illustrate the effectiveness of the theoretical results.