Excitation for Adaptive Optimal Control of Nonlinear Systems in Differential Games

This article 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 feed-forward 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.