具有初态偏移的分数阶PID型迭代学习控制
PID-Type Fractional Order Iterative Learning Control with Initial State Shift
摘要: 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。
Abstract: 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.
文章引用：高帆. 具有初态偏移的分数阶PID型迭代学习控制[J]. 理论数学, 2025, 15(7): 61-69. https://doi.org/10.12677/pm.2025.157207

参考文献

[1] Gao, S., Song, Q., Jiang, H., Shen, D. and Lv, Y. (2025) Decentralized Learning Control for High-Speed Trains with Unknown Time-Varying Speed Delays. Applied Mathematical Modelling, 137, Article 115711. [Google Scholar] [CrossRef] 
[2] Xiao, T. (2024) Iterative Dissipativity of Partial Difference Equation Dynamics in Open-Loop Iterative Learning Control Mode. Mathematics, 12, Article 3128. [Google Scholar] [CrossRef] 
[3] Jin, X. (2016) Iterative Learning Control for Non-Repetitive Trajectory Tracking of Robot Manipulators with Joint Position Constraints and Actuator Faults. International Journal of Adaptive Control and Signal Processing, 31, 859-875. [Google Scholar] [CrossRef] 
[4] Kim, D.I. and Kim, S. (1996) An Iterative Learning Control Method with Application for CNC Machine Tools. IEEE Transactions on Industry Applications, 32, 66-72.
[5] Yang, D.R., Lee, K.S., Ahn, H.J., et al. (2003) Experimental Application of a Quadratic Optimal Iterative Learning Control Method for Control of Wafer Temperature Uniformity in Rapid Thermal Processing. IEEE Transactions on Semiconductor Manufacturing, 16, 36-44. [Google Scholar] [CrossRef] 
[6] Gorinevsky, D. (2002) Loop Shaping for Iterative Control of Batch Processes. IEEE Control Systems, 22, 55-65.
[7] Mezghani, M., Roux, G., Cabassud, M., Le Lann, M.V., Dahhou, B. and Casamatta, G. (2002) Application of Iterative Learning Control to an Exothermic Semibatch Chemical Reactor. IEEE Transactions on Control Systems Technology, 10, 822-834. [Google Scholar] [CrossRef] 
[8] Saab, S.S. (2004) A Stochastic Iterative Learning Control Algorithm with Application to an Induction Motor. International Journal of Control, 77, 144-163. [Google Scholar] [CrossRef] 
[9] Trubin, M.V. and Yurkevich, V.D. (2024) Iterative Learning Control for an Electrohydraulic Strength Test Bench. Optoelectronics, Instrumentation and Data Processing, 60, 420-425. [Google Scholar] [CrossRef] 
[10] He, Z. and Chen, Z. (2024) Iterative Learning Control for Non-Normal and Biased Measured Targets. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 238, 1450-1458. [Google Scholar] [CrossRef] 
[11] Zhang, C., Li, S. and Zhang, Z. (2024) Industrial Robot Arm Dynamic Modeling Simulation and Variable-Gain Iterative Learning Control Strategy Design. Journal of Mechanical Science and Technology, 38, 3729-3739. [Google Scholar] [CrossRef] 
[12] Zhao, H., Wang, F., Zhang, Y., Zheng, Z., Ma, J. and Fu, C. (2024) Fractional-Order Modeling and Stochastic Dynamics Analysis of a Nonlinear Rubbing Overhung Rotor System. Fractal and Fractional, 8, Article 643. [Google Scholar] [CrossRef] 
[13] Bensidhoum, T. and Bouakrif, F. (2020) Adaptive P-Type Iterative Learning Radial Basis Function Control for Robot Manipulators with Unknown Varying Disturbances and Unknown Input Dead Zone. International Journal of Robust and Nonlinear Control, 30, 4075-4094. [Google Scholar] [CrossRef] 
[14] Zhang, X., Xu, W.B. and Lu, W.R. (2021) The Control Strategy of Manipulator Based on Fractional-Order Iterative Learning. Automatic Control and Computer Sciences, 55, 368-376. [Google Scholar] [CrossRef] 
[15] Wu, H., Huang, J., Wu, K., Lopes, A.M. and Chen, L. (2024) Precise Tracking Control via Iterative Learning for One-Sided Lipschitz Caputo Fractional-Order Systems. Mathematical Biosciences and Engineering, 21, 3095-3109. [Google Scholar] [CrossRef] [PubMed]
[16] Luo, D., Wang, J. and Shen, D. (2020) Iterative Learning Control for Locally Lipschitz Nonlinear Fractional-Order Multi-Agent Systems. Journal of the Franklin Institute, 357, 6671-6693. [Google Scholar] [CrossRef] 
[17] Luo, Y., Liu, C. and Liu, G. (2019) Consensus Tracking by Iterative Learning Control for Linear Heterogeneous Multiagent Systems Based on Fractional-Power Error Signals. Algorithms, 12, 185. [Google Scholar] [CrossRef] 
[18] Ghasemi, I., Ranjbar Noei, A. and Sadati, J. (2016) Sliding Mode Based Fractional-Order Iterative Learning Control for a Nonlinear Robot Manipulator with Bounded Disturbance. Transactions of the Institute of Measurement and Control, 40, 49-60. [Google Scholar] [CrossRef] 
[19] Chen, Y.Q. and Moore, K.L. (2001) On D α-Type Iterative Learning Control. Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, 4-7 December 2001, 4451-4456.
[20] Zhao, Y., Li, Y. and Liu, H. (2022) Fractional-Order Iterative Learning Control with Nonuniform Trial Lengths. International Journal of Control, Automation and Systems, 20, 3167-3176. [Google Scholar] [CrossRef] 
[21] Aydin, M. and Mahmudov, N.I. (2022) Iterative Learning Control for Impulsive Fractional Order Time‐Delay Systems with Nonpermutable Constant Coefficient Matrices. International Journal of Adaptive Control and Signal Processing, 36, 1419-1438. [Google Scholar] [CrossRef] 
[22] Wang, G.X., Wang, R., Yi, D.H., et al. (2024) Iterative Learning Formation Control via Input Sharing for Fractional-Order Singular Multi-Agent Systems with Local Lipschitz Nonlinearity. Fractal and Fractional, 8, 347. [Google Scholar] [CrossRef] 
[23] Gao, Y.H., Lou, W.D., Lu, H.L., et al. (2020) Consensus Control of Multi-Agent Robot System with State Delay Based on Fractional-Order Iterative Learning Control Algorithm. Journal Européen des Systèmes Automatisés, 53, 771-779. [Google Scholar] [CrossRef] 
[24] Liu, Q. and Tian, S.P. (2022) Iterative Learning Control Analysis for Linear Fractional-Order Singular Systems. International Journal of Control, Automation and Systems, 20, 3951-3959. [Google Scholar] [CrossRef] 
[25] Wang, Y.Q., Zhou, F.G., Yin, L., et al. (2021) Iterative Learning Control for Fractional Order Linear Systems with Time Delay Based on Frequency Analysis. International Journal of Control, Automation and Systems, 19, 1588-1596. [Google Scholar] [CrossRef] 
[26] Lan, Y.H. and Cui, Z.M. (2018) ILC with Initial State Learning for Fractional Order Linear Distributed Parameter Systems. Algorithms, 11, Article 85. [Google Scholar] [CrossRef] 
[27] Kumar, S., Yadav, P. and Singh, K.V. (2024) Product Integration Techniques for Fractional Integro-Differential Equations. Mathematical Methods in the Applied Sciences, 48, 2833-2858. [Google Scholar] [CrossRef] 
[28] Gao, Z.Y., Liu, S.D. and Wang, J.R. (2017) PID-Type Iterative Learning Control for Impulsive Ordinary Differential Equations. Journal of Applied Mathematics and Computing, 54, 41-55. [Google Scholar] [CrossRef] 

为你推荐