A nonparametric learning framework for nonlinear robust output regulation

This paper proposes a nonparametric learning solution framework for a generic internal model design of nonlinear robust output regulation. The global robust output regulation problem for a class of nonlinear systems with output feedback subject to a nonlinear exosystem can be tackled by constructing a linear generic internal model, provided that a continuous nonlinear mapping exists. An explicit continuous nonlinear mapping was constructed recently in [1] under the assumption that the steady-state generator is linear in the exogenous signal. We further relax such an assumption to a relaxed assumption that the steady-state generator is polynomial in the exogenous signal. A nonparametric learning framework is proposed to solve a linear time-varying equation to make the nonlinear continuous mapping always exist. With the help of the proposed framework, the nonlinear robust 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. We further apply the nonparametric learning framework to globally reconstruct and estimate multiple sinusoidal signals with unknown frequencies without using adaptive techniques. An explicit nonlinear mapping can directly provide the estimated parameters, which will exponentially converge to the unknown frequencies. As a result, a feedforward control design is proposed to solve the output regulation using our nonparametric learning framework.