Convergence characteristics of fractional‐order PID‐type iterative learning control for nonlinear systems with arbitrary initial state

This paper presents first‐ and second‐order fractional‐order PID‐type iterative learning control schemes for a class of nonlinear systems, incorporating mechanisms for initial state learning. Based on an iterative approach, we derive a generalized Gronwall inequality that accounts for singularities. We then use the generalized Gronwall and integral norm inequalities to establish sufficient conditions for monotonic convergence in the sense of Lebesgue‐p norm. The convergence speed is analyzed by examining the characteristic polynomials of the two learning rules. Our analysis shows that both system dynamics and learning gains influence convergence. Additionally, it is observed that the second‐order learning law can exhibit varying convergence speeds compared with the first‐order scheme, depending on the choice of control algorithm factors. Numerical simulations confirm the validity and effectiveness of the proposed approach.