PID-Type Fractional Order Iterative Learning Control with Initial State Shift

针对分数阶线性连续时不变系统的初值问题，提出了一种基于初值学习的PI1−αDα型迭代学习控制算法。在 λ 范数的意义下，对控制算法的收敛性条件进行了严格证明。理论分析表明，在此算法的作用下，随着迭代次数的增加，能够实现系统输出对期望输出的精确跟踪，保证了跟踪误差的收敛性。相比传统的PID型算法，该算法解决了算法中要求每一次迭代初值都相同的限制，消除了随机初值对系统的影响。数值仿真验证了所提算法的有效性和正确性。For the initial value problem of fractional-order linear continuous-time invariant systems, a PI1−αDα iterative learning control algorithm based on initial value learning is proposed, in the sense of λ norm, a rigorous proof of convergence conditions for the control algorithm is established. Theoretical analysis demonstrates that as the number of iterations increases, this algorithm achieves precise tracking of the desired system output and guarantees convergence of tracking errors. Compared with traditional PID-type algorithms, the proposed method overcomes the constraint of requiring identical initial values in each iteration inherent to conventional control algorithms, effectively eliminating the impact of random initial values on system performance. Numerical simulations validate the effectiveness and correctness of the proposed algorithm.