Basic learning control law designs are summarized, and conditions for convergence of the learning process are developed when several different choices of filter frequency cutoffs are used for robustification. This paper presents a cliff harmonic-frequency filter with a sharp frequency cutoff and a weighted harmonic-frequency filter with frequency weighting, that are applied each iteration in iterative learning control. Filter matrices based on the state-space model of a finite-difference digital filter are derived for Matlab’s and Gustafsson’s forward and backward filtering which are commonly called filtfilt methods. Furthermore, filter matrices for Matlab’s and Gustafsson’s filtfilts are revised to make the input convergence matrix to be monotonically stable. Numerical examples are used to demonstrate the effectiveness of the harmonic-frequency filters comparing with Matlab’s and Gustafsson’s filtfilt methods based on Butterworth filters of different orders and cutoff frequencies. It is found that the singular values of filter matrices are related to the squared amplitude of the Butterworth filter at harmonic frequencies.

总结了基本的学习控制律设计，并在使用几种不同的滤波器截止频率选择进行鲁棒化时开发了学习过程收敛的条件。本文提出了一种具有陡峭频率截止的悬崖谐波频率滤波器和一种具有频率加权的加权谐波频率滤波器，它们在迭代学习控制中的每次迭代中应用。基于有限差分数字滤波器的状态空间模型的滤波器矩阵被推导出用于 Matlab 和 Gustafsson 的前向和后向滤波，这通常称为 filtfilt 方法。此外，Matlab 和 Gustafsson 的 filtfilts 的滤波器矩阵被修改，使输入收敛矩阵单调稳定。数值例子用于证明谐波频率滤波器与基于不同阶数和截止频率的巴特沃斯滤波器的 Matlab 和 Gustafsson 的 filtfilt 方法相比的有效性。发现滤波器矩阵的奇异值与谐波频率下巴特沃斯滤波器的幅度平方有关。