Convergence characteristics of proportional-type iterative learning control in the sense of Lebesgue-p norm

This article studies convergence characteristics of first- and second-order P-type (proportional-type) iterative learning control laws for a class of partially known linear time-invariant systems with a direct feed-through term. In the study, the tracking error is measured in the sense of Lebesgue-p norm, based on which the concepts of the Q_p factor and the convergence speed are specified. For the first-order updating law, the monotone convergence is achieved by means of the generalised Young inequality of convolution integral. Moreover, for the second-order updating rule, the convergence is also verified and its speed is compared with the first-order law. Through analysis, it is quantitatively noted that both the system dynamics and the learning gains affect the convergence. It is also observed that the iterative learning process with the second-order law can be Q_p-faster, Q_p-equivalent or Q_p-slower than the system with the first-order rule, depending on the selected learning gains. Numerical simulation manifests the validity and the effectiveness.