## Abstract

In this paper, conventional first- and second-order proportional-derivative-type (PD-type) iterative learning control (ILC) updating algorithms are discussed for a type of nonlinear time-invariant system. On the basis of the Bellman–Gronwall inequality, the convergence is derived in the sense that the tracking error is measured in the form of the Lebesgue-p norm. This analysis shows that, under an appropriate condition, the first-order PD-type ILC updating law is monotonically convergent while the second-order law is convergent and the monotonicity is guaranteed after finite iterations. Further, by analysing the characteristic polynomial of the second-order PD-type ILC updating law, an argument about the comparison of convergence speed in terms of Qp-factor is made. The argument clarifies that the second-order PD-type ILC law can be Qp faster, equivalent or slower than the first-order law, depending upon different sets of the learning gains. Numerical simulations are conducted to show their validity and effectiveness.