Iterative Learning and Extremum Seeking for Repetitive Time-Varying Mappings

In this paper, we develop an iterative learning control method integrated with extremum seeking control to track a time-varying optimizer within finite time horizon. 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 <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>-dependent (iteration-dependent) contraction mapping in a Banach space equipped with so called <inline-formula> <tex-math notation="LaTeX">$\lambda$</tex-math></inline-formula>-norm. Second, the iterative learning extremum seeking system can be interpreted as an iterative learning control with the approximation error as “disturbance”. The tracking error of its modified Lie bracket system can be shown uniformly bounded in terms of iterations by selecting a sufficiently large dither frequency. Furthermore, it is shown that the tracking error will eventually converge to a set. The center of the set corresponds to the limit solution of the “disturbance-free” system, and its radius can be controlled by the frequency.