Convergence characteristics of PD-type iterative learning control in discrete frequency domain

On the basis that a Dirichlet-type signal over a finite time period can be expanded in a Fourier series consisting of fundamental-frequency sinusoidal and cosine waves plus a sequence of higher-frequency harmonic waves, this paper investigates the convergence characteristics of the first- and second-order proportional-derivative-type iterative learning control schemes for repetitive linear time-invariant systems in discrete spectrum. By deriving the properties of the Fourier coefficients in a complex form with respect to the linear time-invariant dynamics and adopting Parseval's Energy Equality, the average energy of the tracking error signal over the finite operation time interval is converted into a quarter of a summation of the fundamental spectrum plus the harmonic spectrums. By means of analyzing the feature of discrete frequency-wise spectrum of the tracking error, sufficient and necessary conditions for monotone convergence with respect to the first-order iterative learning control scheme is deduced together with convergence of the second-order learning scheme is discussed. Numerical simulations manifest the validity and the effectiveness.