Convergence properties concerning Lebesgue-p norm of iterative learning control for a class of fractional differential systems

This paper investigates the monotonic convergence and speed comparison of first-and second-order proportional-a-order-integral-derivative-type ((PID)-D-a-type) iterative learning control (ILC) schemes for a linear time-invariant (LTI) system, which is governed by the fractional differential equation with order a ? (1, 2). By introducing the Lebesgue-p (L-P) norm and utilizing the property of the Mittag-Leffler function and the boundedness feature of the fractional integration operator, the sufficient condition for the monotonic convergence of the first-order updating law is strictly analyzed. Therewith, the sufficient condition of the second-order learning law is established using the same means as the first one. The obtained results objectively reveal the impact of the inherent attributes of system dynamics and the constructive mode of the ILC rule on convergence. Based on the sufficient condition of first/second-order updating law, the convergence speed of first-and second-order schemes is determined quantitatively. The quantitative result demonstrates that the convergence speed of second-order law can be faster than the first one when the learning gains and weighting coefficients are properly selected. Finally, the effectiveness of the proposed methods is illustrated by the numerical simulations.