Iterative Learning and Extremum Seeking for Repetitive Time-Varying Mappings

TL;DR

This paper introduces a novel extremum seeking control method combined with iterative learning control to efficiently track time-varying optimizers, analyzing convergence and error bounds through a modified Lie bracket system.

Contribution

It proposes an online integral iterative learning control with a forgetting factor and analyzes its convergence, integrating it with extremum seeking for time-varying optimization.

Findings

Convergence of the iterative learning extremum seeking system is established.

Tracking error can be made arbitrarily small by increasing frequency.

The method effectively tracks time-varying optimizers within finite time.

Abstract

In this paper, we develop an extremum seeking control method integrated with iterative learning control to track a time-varying optimizer within finite time. The behavior of the extremum seeking system is analyzed via an approximating system - the modified Lie bracket system. The modified Lie bracket system is essentially an online integral-type iterative learning control law. The paper contributes to two fields, namely, iterative learning control and extremum seeking. First, an online integral type iterative learning control with a forgetting factor is proposed. Its convergence is analyzed via $k$-dependent (iteration- dependent) contraction mapping in a Banach space equipped with $\lambda$-norm. Second, the iterative learning extremum seeking system can be regarded as an iterative learning control with "control input disturbance." The tracking error of its modified Lie bracket system can be shown uniformly bounded in terms of iterations by selecting a sufficiently large frequency. Furthermore, it is shown that the tracking error will finally converge to a set, which is a $\lambda$-norm ball. Its center is the same with the limit solution of its corresponding "disturbance-free" system (the iterative learning control law); and its radius can be controlled by the frequency.