Modifying and optimizing the inverse of the frequency response circulant matrix as an iterative learning control compensator

Feedback control systems do not do what you ask. The concept of bandwidth is defined to tell what components of a command are reasonably well handled. Iterative Learning Control (ILC) seeks to converge to zero error following any given finite time desired trajectory as iterations progress. The approach can be used to achieve high precision tracking in spacecraft sensors performing repeated highly accurate sensor scanning. ILC asks for zero error for a finite time tracking maneuver, containing initial transients each iteration. The purpose of this paper is to create a method of designing ILC compensators based on steady state frequency response, and have the ILC converge to zero error in spite of transients and bandwidth. In this work the inverse of the circulant matrix of Markov parameters is used as a learning gain matrix. One can show that this matrix gives the steady state frequency response of the system at the finite number of frequencies observable in the finite data sequence of an iteration or run. Methods are used to adjust the steady state frequency response gains to address the transient part of the error signal. Numerical simulations compare the design approach to common time domain ILC design approaches, and one observes much faster convergence.