Focused on the problem that the convergence speed of traditional iterative learning control algorithm used in linear regular systems is slow, a kind of fast iterative learning control algorithm was designed for a class of linear regular systems. Compared with the traditional P-type iterative learning control algorithm, the algorithm increases tracking error at neighboring two iterations generated from last difference signal and present difference signal. And the convergence of the algorithm was proven by using Yong inequality of convolutional inference in the sense of Lebesgue- p norm. The results show the tracking error of the system will converge to zero with infinite iterations. The convergence condition is also given. Compared with P-type iterative learning control, the proposed algorithm can fasten the convergence and avoid the shortcomings of using λ norm to measure the tracking error. Simulation further testifies the validity and effectiveness.