Output Reachability of Chen-Fliess series: A Newton-Raphson Approach

The optimization of Chen-Fliess series allows addressing the problem of the computation of reachable sets of nonlinear affine control systems. This provides an input-output approach to reachability analysis. The Newton-Raphson method is one of the most important line search numerical algorithms. The method is based on the computation of the Hessian and the second-order Taylor approximation of the function of interest. Currently, only first-order optimization methods, such as gradient descent, exist for Chen-Fliess series. In this paper, the framework of differential languages is introduced to allow a systematic description of higher-order derivatives of Chen-Fliess series. This is achieved by defining the derivative operation of a word in a monoid that coincides with the Gâteaux derivative of a Chen-Fliess series. In this context, the Hessian of a Chen-Fliess series, its second-order Taylor approximation, and the second-order optimization condition are provided for Chen-Fliess series. Then the Newton-Raphson algorithm is adapted to Chen-Fliess series optimization to compute reachable sets. Illustrative examples and simulations are presented throughout the paper.