Excitation for Adaptive Optimal Control of Nonlinear Systems in Differential Games

TL;DR

This paper establishes conditions under which polynomial transformations of signals satisfy the Persistent Excitation condition, crucial for convergence in adaptive control, and demonstrates improved performance in differential games.

Contribution

It provides new theoretical conditions for PE preservation under polynomial transformations and validates them through a control method outperforming probing noise.

Findings

Proposed conditions ensure polynomial-transformed signals are PE.

The excitation method accelerates convergence in differential games.

Method outperforms probing noise in numerical experiments.

Abstract

This work focuses on the fulfillment of the Persistent Excitation (PE) condition for signals which result from transformations by means of polynomials. This is essential e.g. for the convergence of Adaptive Dynamic Programming algorithms due to commonly used polynomial function approximators. As theoretical statements are scarce regarding the nonlinear transformation of PE signals, we propose conditions on the system state such that its transformation by polynomials is PE. To validate our theoretical statements, we develop an exemplary excitation procedure based on our conditions using a feedforward control approach and demonstrate the effectiveness of our method in a nonzero-sum differential game. In this setting, our approach outperforms commonly used probing noise in terms of convergence time and the degree of PE, shown by a numerical example.