Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit‐time optimal control Part II: Numerical approach

Abstract

AbstractThis paper continues the study (Yegorov, Dower, and Grüne et al.) and develops a curse‐of‐dimensionality‐free numerical approach to feedback stabilization, whose theoretical foundation was built in Yegorov et al. and involved the characterization of control Lyapunov functions (CLFs) via exit‐time optimal control. First, we describe an auxiliary linearization‐based technique for the construction of a local CLF and discuss how to determine its appropriate sublevel set that can serve as the terminal set in the exit‐time optimal control problem leading to a global or semi‐global CLF. Next, the curse of complexity is addressed with regard to the approximation of CLFs and associated feedback strategies in high‐dimensional regions. The goal is to enable for efficient model predictive control implementations with essentially faster (though less accurate) online policy updates than in case of solving direct or characteristics‐based nonlinear programming problems for each sample instant. We propose a computational approach that combines gradient enhanced modifications of the Kriging and inverse distance weighting frameworks for scattered grid interpolation. It in particular allows for convenient offline inclusion of new data to improve obtained approximations (machine learning can be used to select relevant new sparse grid nodes). Moreover, our method is designed so as to a priori return proper values of the CLF interpolant and its gradient on the entire terminal set of the considered exit‐time optimal control problem. Supporting numerical simulation results are also presented.