优化迭代学习控制算法及其收敛性分析
|
摘要
迭代学习控制从最初解决机器手运动的控制问题发展为解决很多具有周期特性的工程对象的控制问题,其利用实际输出和期望输出的偏差对当前的控制输入信号进行学习修正的思想,越来越得到了控制工程领域的广泛接受和高度重视。由于迭代学习控制方法所具有的不过分依赖模型的特点和实际应用中的良好效果,已经成为了现代智能控制技术中的一种重要控制方法。
为解决传统参数优化迭代学习控制算法只适用于正定模型的局限性,本文提出一种高阶参数优化迭代学习控制算法。该算法利用多次迭代误差的信息,建立参数优化目标函数得到最优迭代学习律。在被控对象不满足正定性的条件下,仍保证跟踪误差单调收敛于零。此外,还提出了基于奇异值分解的PID型参数优化迭代学习控制算法,根据范数性能指标建立目标函数,通过对系统模型矩阵的奇异值分解得到学习增益矩阵,使算法应用于模型为非正定的系统时仍然保证闭环跟踪误差单调收敛至零。该算法还将PID型控制器引入到参数优化迭代学习控制算法的设计中,提高了学习速度。
针对非线性系统跟踪控制问题,本文将拟Broyden法和参数优化迭代学习控制方法结合,提出一种新的优化迭代学习控制算法,利用拟Broyden法对系统的Jacobian矩阵进行迭代近似计算,并通过目标函数对学习增益进行优化。该算法不仅能够简化传统牛顿法中求逆计算所带来的复杂性,而且具有单调递减的特性和全局收敛性。
针对一类特殊采样点的轨迹终端跟踪控制问题,本文提出了一种改进的牛顿法迭代学习控制算法,该算法具有单调收敛的性能和较大的收敛范围。利用一个间歇反应的生成物产量跟踪实例对算法的有效性进行了验证。
本文将迭代学习优化控制方案应用到一类具有扰动和初始状态误差等不确定性的非线性离散系统中。从理论上证明了对于无扰动情况,算法能够保证系统跟踪误差以几何速度一致收敛于零,系统控制输入收敛到期望输入轨迹；对于具有扰动和初始状态误差的非线性离散系统,给出了该算法具有BIBO鲁棒稳定性的充分条件。
本文还给出了P型迭代学习控制在带有扰动的一般性非线性系统具有鲁棒性的条件。从理论上证明了在初始状态误差和状态、输出扰动有界的情况下,系统输出能够收敛于期望轨迹的邻域内；在各种干扰消除时系统输出能够一致收敛于期望轨迹。
Iterative learning control (ILC), which is firstly proposed in the movement control of manipulator, is a technique for improving the tracking performance of systems or processes that operate repetitively over a fixed time. ILC utilizes the deviation between actual output and desired output to modify current control input signal by learning, this control method has brought ILC widely acceptance and highly attention in the field of control engineering. The ILC algorithm does not rely on the repeatable system model very much and has good tracking accuracy in practice. Due to these characteristics, ILC has become an important control method of modern intelligent control technology.
Traditional parameter iterative learning control algorithm ensures the tracking error converge monotonically to zero only when the original plant is positive. In order to solving this limitation, a high-order parameter optimal iterative learning control algorithm is introduced in this paper. The proposed algorithm is established by a quadratic performance index with the tracking errors from earlier trials to construct optimal learning law. The proposed algorithm can guarantee the tracking error converge monotonically to zero even the relaxation system is non-positive.
A PID parameter optimal iterative learning control algorithm based on singular value decomposition is also given in this paper. The proposed algorithm establishes the norm performance index and obtains learning gain matrix by applying singular value decomposition to the original plant, and ensure the closed-loop tracking errors of this algorithm converge monotonously to zero even the original plant is non-positive. Furthermore, a PID controller is added to the design of ILC algorithm to improve learning efficiency.
Focuses on the tracking problems of dynamic nonlinear system, a novel parameter optimal Broyden-like method based iterative learning control scheme is proposed in this paper. The Broyden-like method is used to iteratively calculate the approximation of the Jacobian matrix, and the parameter-optimal method is used to optimize the learning gain of the algorithm. Compared with traditional Newton-method, the proposed algorithm can avoid the calculation of the inverse of the Jacobian matrix of system. Furthermore, the algorithm has the properties of global and monotonic convergence.
A class of specific target tracking problems for non-linear systems with the reference trajectory sampled only in some specific times is discussed. Accordingly, a modified Newton-method-based iterative learning control scheme is suggested to solve this problem. It is proved theoretically that the algorithm can ensure the closed loop system to have global and monotonic convergence. A case study of final product tracking problem of a mixed, liquid-phase batch reactor is given to demonstrate the effectiveness and tracking accuracy of the proposed algorithm.
In order to solve robust iterative learning control problem of a class of discrete-time nonlinear systems with initial state errors, external disturbances and output noises, a novel norm-optimal based robust iterative learning control algorithm is proposed in this paper. It is proven theoretically that in the absence of these disturbances, the algorithm can guarantee the tracking error of the closed-loop system uniformly converging to zero geometrically. In the presence of these disturbances and uncertainties, the sufficient condition of robust BIBO stability for the proposed algorithm is given.
Finally, this paper discus the robust stability condition of P-type iterative learning control algorithm for a class of nonlinear dynamic discrete system. It is proved that in the presence of state, output disturbances and initial uncertainties, the system output can converge to the neighbor domain of desired trajectories. In the absence of these disturbances and uncertainties, the system output can converge to the desired trajectories uniformly.
为解决传统参数优化迭代学习控制算法只适用于正定模型的局限性,本文提出一种高阶参数优化迭代学习控制算法。该算法利用多次迭代误差的信息,建立参数优化目标函数得到最优迭代学习律。在被控对象不满足正定性的条件下,仍保证跟踪误差单调收敛于零。此外,还提出了基于奇异值分解的PID型参数优化迭代学习控制算法,根据范数性能指标建立目标函数,通过对系统模型矩阵的奇异值分解得到学习增益矩阵,使算法应用于模型为非正定的系统时仍然保证闭环跟踪误差单调收敛至零。该算法还将PID型控制器引入到参数优化迭代学习控制算法的设计中,提高了学习速度。
针对非线性系统跟踪控制问题,本文将拟Broyden法和参数优化迭代学习控制方法结合,提出一种新的优化迭代学习控制算法,利用拟Broyden法对系统的Jacobian矩阵进行迭代近似计算,并通过目标函数对学习增益进行优化。该算法不仅能够简化传统牛顿法中求逆计算所带来的复杂性,而且具有单调递减的特性和全局收敛性。
针对一类特殊采样点的轨迹终端跟踪控制问题,本文提出了一种改进的牛顿法迭代学习控制算法,该算法具有单调收敛的性能和较大的收敛范围。利用一个间歇反应的生成物产量跟踪实例对算法的有效性进行了验证。
本文将迭代学习优化控制方案应用到一类具有扰动和初始状态误差等不确定性的非线性离散系统中。从理论上证明了对于无扰动情况,算法能够保证系统跟踪误差以几何速度一致收敛于零,系统控制输入收敛到期望输入轨迹；对于具有扰动和初始状态误差的非线性离散系统,给出了该算法具有BIBO鲁棒稳定性的充分条件。
本文还给出了P型迭代学习控制在带有扰动的一般性非线性系统具有鲁棒性的条件。从理论上证明了在初始状态误差和状态、输出扰动有界的情况下,系统输出能够收敛于期望轨迹的邻域内；在各种干扰消除时系统输出能够一致收敛于期望轨迹。
Iterative learning control (ILC), which is firstly proposed in the movement control of manipulator, is a technique for improving the tracking performance of systems or processes that operate repetitively over a fixed time. ILC utilizes the deviation between actual output and desired output to modify current control input signal by learning, this control method has brought ILC widely acceptance and highly attention in the field of control engineering. The ILC algorithm does not rely on the repeatable system model very much and has good tracking accuracy in practice. Due to these characteristics, ILC has become an important control method of modern intelligent control technology.
Traditional parameter iterative learning control algorithm ensures the tracking error converge monotonically to zero only when the original plant is positive. In order to solving this limitation, a high-order parameter optimal iterative learning control algorithm is introduced in this paper. The proposed algorithm is established by a quadratic performance index with the tracking errors from earlier trials to construct optimal learning law. The proposed algorithm can guarantee the tracking error converge monotonically to zero even the relaxation system is non-positive.
A PID parameter optimal iterative learning control algorithm based on singular value decomposition is also given in this paper. The proposed algorithm establishes the norm performance index and obtains learning gain matrix by applying singular value decomposition to the original plant, and ensure the closed-loop tracking errors of this algorithm converge monotonously to zero even the original plant is non-positive. Furthermore, a PID controller is added to the design of ILC algorithm to improve learning efficiency.
Focuses on the tracking problems of dynamic nonlinear system, a novel parameter optimal Broyden-like method based iterative learning control scheme is proposed in this paper. The Broyden-like method is used to iteratively calculate the approximation of the Jacobian matrix, and the parameter-optimal method is used to optimize the learning gain of the algorithm. Compared with traditional Newton-method, the proposed algorithm can avoid the calculation of the inverse of the Jacobian matrix of system. Furthermore, the algorithm has the properties of global and monotonic convergence.
A class of specific target tracking problems for non-linear systems with the reference trajectory sampled only in some specific times is discussed. Accordingly, a modified Newton-method-based iterative learning control scheme is suggested to solve this problem. It is proved theoretically that the algorithm can ensure the closed loop system to have global and monotonic convergence. A case study of final product tracking problem of a mixed, liquid-phase batch reactor is given to demonstrate the effectiveness and tracking accuracy of the proposed algorithm.
In order to solve robust iterative learning control problem of a class of discrete-time nonlinear systems with initial state errors, external disturbances and output noises, a novel norm-optimal based robust iterative learning control algorithm is proposed in this paper. It is proven theoretically that in the absence of these disturbances, the algorithm can guarantee the tracking error of the closed-loop system uniformly converging to zero geometrically. In the presence of these disturbances and uncertainties, the sufficient condition of robust BIBO stability for the proposed algorithm is given.
Finally, this paper discus the robust stability condition of P-type iterative learning control algorithm for a class of nonlinear dynamic discrete system. It is proved that in the presence of state, output disturbances and initial uncertainties, the system output can converge to the neighbor domain of desired trajectories. In the absence of these disturbances and uncertainties, the system output can converge to the desired trajectories uniformly.
引文
[1]UCHIYAMA M. Formulation of high-speed motion pattern of a mechanical arm by trial [J]. Transactions of the Society of Instrumentation and Control Engineers,1978, 14(6):706-712.
[2]ARIMOTO S, KAWAMURA S, MIYAZAKI F, et al. Learning control theory for dynamical systems[C].24th IEEE Conference on Decision and Control. Fort Lauderdale, Florida, USA,1985:1375-1380.
[3]MOORE KL. Iterative learning control for deterministic systems[M]. London: Springer,1993.
[4]MOORE KL. Iterative learning control-an expository overview[J]. Applied and Computational Controls, Signal Proceeding and Circuits,1998,1:151-214.
[5]李新忠,简林柯,何钺.迭代学习控制的研究及应用[J].机床与油压,1997,6：3-5.
[6]刘慧,许晓鸣,张仲俊.迭代学习控制的回顾与展望[J].机器人,1996,18(6)：347-379.
[7]谢振东,谢胜利,刘永清.非线性系统学习控制理论的发展与展望[J].控制理论与应用,2000,17(1)：4-8.
[8]BIEN Z, XU J X. Iterative learning control-Analysis, Design, Integration and Applications[M]. Netherlands:Kluwer Academic Publishers,1998.
[9]CHEN Y, WEN C. Iterative learning control-Convergence, robustness and applications[M]. London:Springer,1999.
[10]林辉,王林.迭代学习控制理论[M].西安：西北工业大学出版社,1998.
[11]MOORE K L, XU J X. Editorial:Special Issue on iterative learning control[J], international Journal of Control,2000,73(10):819-823.
[12]XU J X, TAN Y. Linear and nonlinear iterative learning control [M]. London:Springer, 2003.
[13]THARAYIL M, ALLEYNE A G. A survey of iterative learning control [J]. Control Systems, 26(3):96-144,2006.
[14]YANG Q C, MOORE K L. Iterative Learning Control:Brief Survey and Categorization [J]. Systems, Man, and Cybernetics,2007,37(6):1099-1121.
[15]WANG YOUQING, GAO FURONG, FRANCIS J, et al. Survey on iterative learning control, repetitive control, and run-to-run control [J]. Journal of Process Control,2009,19 (10):1589-1600.
[16]ARIMOTO S, KAWAMURA S, MIYAZAKI F. Bettering operation of robots by learning[J]. Journal of Robotic Systems,1984,1(2):123-140.
[17]KAWAMURA S, MIYAZAKI F, ARIMOTO S. Realization of robot motion based on a learning method[J]. IEEE Transactions on System, Man, and Cybernetics,1998,18(1): 126-134.
[18]NANIWA T, ARIMOTO S. Learning control for robot tasks under geometric end point constrains[J]. IEEE Transactions on Robotic and Automation,1995,11(3):432-441.
[19]ARIMOTO S. Learning control theory for robotic motion[J]. International Journal of Adaptive Control and Signal Processing.1990,4(6):543-564.
[20]皮道映,孙优贤.离散非线性时变系统开闭环PI型迭代学习控制律及其收敛性[J].自动化学报,1998,24(5)：636-639.
[21]LEE H S, BIEN Z. Study on robustness of iterative learning control with non-zero initial error[J]. International Journal of Control,1996,64(3):345-359.
[22]孙明轩,黄宝健,张学智.非线性系统的PD型迭代学习控制[J].自动化学报,1998,24(5)：711-714.
[23]BIEN Z, HUN K M. High-order P-type iterative learning controller using current iteration tracking error[J]. International Journal of Control,1997,68(2): 331-342.
[24]PARK K H. An average operator-based iterative learning control for variable initial state error[J]. IEEE Transactions on Automatic Control,2005,50(6):865-869.
[25]CHEN Y Q, Wang C Y, SUN M. A robust high-order P-type iterative learning controller using current iteration tracking error[J]. International Journal of Control.1997, 68(2):345-353.
[26]CHEN Y Q, GONG Z M, WEN C Y. Analysis of high-order iterative learning control algorithm for uncertain nonlinear systems with state delays [J]. Automatica,1998, 34(3):1867-1880.
[27]Xu J X, Tan Y. Robust optimal design and convergence properties analysis of iterative learning control approaches[J]. Automatica,2002,38(11):1867-1880.
[28]RUAN X, BIEN Z Z, WANG Q. Convergence characteristics of proportional-type iterative learning control in the sense of Lebesgue-p norm[J]. IET Control Theory and Applications,2012,6(5):1-8.
[29]RUAN X, BIEN Z, WANG Q. Convergence properties of iterative learning control processes in the sense of Lebesgue-p norm[J]. Asian Journal of control,2012,14 (4): 1095-1107.
[30]LEE K H, BIEN Z. Initial condition problem of learning control [J]. IEE Proceedings of Control Theory and Applications, Part D,1991,138(6):525-528.
[31]LEE H S, BIEN Z. Study on robustness of iterative learning control with non-zero initial error[J]. International Journal of Control,1996,64(3):345-359.
[32]PARK K H, BIEN Z, HWANG D H. A study on the robustness of a PID-type iterative learning controller against initial state error[J]. International Journal of Systems Science,1999,30(1):49-59.
[33]PRAK K H, BIEN Z. A generalized iterative learning controller against initial state error[J]. International Journal of control,2000,70(10):871-881.
[34]PRAK K H. An average operator-based iterative learning control for variable initial state error[J]. IEEE Transactions on Automatic control,2005,50(6):865-869.
[35]RUAN X, BIEN Z. Pulse compensation for PD-type iterative learning control against initial state shift[J]. International Journal of Systems science,2012,43(11): 2172-2184.
[36]PARK K H, BIEN Z. A Study On Iterative Learning Control With Adjustment Of Learning Interval For Monotone Convergence In The Sense Of sup-norm[J]. Asian Journal of Control,2002,4(1):111-118.
[37]PARK H, BIEN Z. Intervalized iterative learning control for monotonic convergence in the sense of sup-norm[J]. International Journal of Control,2005,78(15): 1218-1227.
[38]RUAN X, BIEN Z Z, WANG Q. Convergence characteristics of proportional-type iterative learning control in the sense of Lebesgue-norm[J]. Control Theory & Applications, IET,2012,6(5):707-714.
[39]RUAN X, BIEN Z Z, WANG Q. Convergence Properties of Iterative Learning Control Processes in the Sense of the Lebesgue-Norm[J]. Asian Journal of Control,2012, 14(4):1095-1107.
[40]KUREK J. The general state-space model for a two dimensional linear digital systems[J]. IEEE Transition of Automatic Control,1985,30(6):600-602.
[41]KACZOREK T. Two-Dimensional Linear Systems[M]. London:Springer,1985:32-35.
[42]GALKOWSKI K, ROGERS E, GRAMACKI A, et al. Higher order discretisation methods for a class of 2-D continuous-discrete linear systems[J]. Circuits, Devices and Systems,1999,146(6):315-320.
[43]MENG D Y, JIA Y M, DU J P, et al. Feedback iterative learning control for time-delay systems based on 2D analysis approach[J]. Journal of Control Theory and Applications,2010,8(4):457-463.
[44]DYMKOV M, GAISHUN I, ROGERS E, et al. Control theory for a class of 2D continuous-discrete linear systems[J]. International Journal of Control,2004, 77(9):847-860.
[45]LI X D, CHOW T W S, H, J K L.2-D system theory based iterative learning control for linear continuous systems with time delays [J]. Circuits and Systems I:Regular Papers, IEEE Transactions on,2005,52(7):1421-1430.
[46]SHI J, GAO F R, WU T J. Robust design of integrated feedback and iterative learning control of a batch process based on a 2D Roesser system [J]. Journal of Process Control,2005,15(8):907-924.
[47]SHI J, GAO F R. Integrated Design and Structure Analysis of Robust Iterative Learning Control System Based on a Two-Dimensional Model[J]. Industrial & Engineering Chemistry Research,2005,44(21):8095-8105.
[48]HLADOWSKI L, GALKOWSKI K, CAI Z L, ROGERS E, et al. Experimentally supported 2D systems based iterative learning control law design for error convergence and performance. Control Engineering Practice,2010,18(4):339-348.
[49]TOGAI M, YAMANO 0. Analysis and design of an optimal learning control scheme for industrial robots:a discrete system approach[C]. Proc. of 24th IEEE Conf. on Decision and-Control, Ft. Lauderdale, Florida, USA,1985:1399-1404.
[50]TAO K M, KOSUT R L, ARAL G. Learning feed-forward control [C]. Proc. Of the American Control Conference, Baltimore, Maryland, USA,1994:2575-2579.
[51]GORINEVSKY D M. Direct learning of feed-forward control for manipulator path tracking[C]. Proc. of IEEE International Symp. on Intelligent Control, Glasgow, United Kingdom,1992:42-47.
[52]AMANN N, OEWNS D H, ROGERS E. Iterative Learning Control using Optimal Feedback and Feed forward Actions[J]. International Journal of control,1996,65(2): 277-293.
[53]AMANN N, OEWNS D H, ROGERS E. Iterative Learning Control for Discrete-time Systems with Exponential Rate of Convergence[J]. Proceedings of the IEE Control Theory Applications,1996,143(2):217-224.
[54]AMANN, N, OEWNS D H, ROGERS E. Predictive Optimal Iterative Learning Control [J]. International Journal of Control,1998,69(2):203-226.
[55]OWENS D H, FENG K. Parameter Optimisation in Iterative Learning Control[J]. International Journal of Control,2003,76(11):1059-1069.
[56]HATONEN J, OWENS D H, FENG K. Basis Functions and Parameter Optimisation in High order Iterative Learning Control[J]. Automatica,2006,42(2):287-294.
[57]HARTE T J, HATONEN J, OEWNS D H. Discrete time Inverse Model-based Iterative Learning Control:Stability, Monotonicity and Robustness[J]. International Journal of Control,2005,78(8):577-586.
[58]OEWNS D H, HATONEN J, DALEY, S. Robust Monotone Gradient-based Discrete-time Iterative Learning Control[J]. International Journal of Robust and Nonlinear Control,2009,19(6):634-661.
[59]OWENS D H. Multivariable norm optimal and parameter optimal iterative learning control:a unified formulation[J]. International Journal of Control, 2012,85 (8):1010-1025.
[60]OWENS D H, HATONEN J. Iterative Learning Control-an Optimisation Paradigm[J]. IFAC Annual Reviews in Control,2005,29(1):57-70.
[61]PANG B, SHAO C, Modified Newton-based ILC solving specific target tracking problems for non-linear batch reactor[J]. International Journal of Modelling, Identification and Control,2013,18(2):158-165.
[62]SON T D, ANN H S. Terminal iterative learning control with multiple intermediate pass points[C]. American Control Conference, San Francisco, California, USA,2011: 3651-3656.
[63]BARTON K L, ALLEYNE A G. A norm optimal approach to time-varying ILC with application to a multi-axis robotic testbed[J]. Control Systems Technology, IEEE Transactions on,2011,19(1):166-180.
[64]VAN DE WIJDEVEN J, DONKERS T, BOSGRA 0. Iterative Learning Control for uncertain systems:Robust monotonic convergence analysis[J]. Automatica,2009,45(10): 2383-2391.
[65]CHU B, OWENS D H. Accelerated norm-optimal iterative learning control algorithms using successive projection[J]. International Journal of Control,2009,82(8): 1469-1484.
[66]WANG D, LIU D, WEI Q. Finite-horizon neuro-optimal tracking control for a class of discrete-time nonlinear systems using adaptive dynamic programming approach[J]. Neurocomputing,2012,78(1):14-22.
[67]OWENS D H, FREEMAN C T, VAN DINH T. Norm-optimal iterative learning control with intermediate point weighting:Theory, algorithms, and experimental evaluation [J]. Control Systems Technology, IEEE Transactions on,2013,21(3):999-1007.
[68]XU J. Analysis of iterative learning control for a class of nonlinear discrete-time systems[J]. Automatica,1997,33(10):1905-1907.
[69]CHIEN C J. A discrete iterative learning control for a class of nonlinear time-varying systems[J]. IEEE transactions on Automatic Control,1998, 43(5):748-752.
[70]Wang D. Convergence and robustness of discrete time nonlinear systems with iterative learning control[J]. Automatica,1998,34(11):1445-1448.
[71]CHEN C K, HWANG J. Iterative learning control for position tracking of a pneumatic actuated X-Y table [J]. Control Engineering Practice,2005,13(12):1455-1461.
[72]XIONG Z, ZHANG J. Batch-to-batch optimal control of nonlinear batch processes based on incrementally updated models[J]. IEE Proceedings:Control theory and applications,2004,15(2):158-165.
[73]CHIEN C. A discrete iterative learning control for a class of nonlinear time-varying systems[J].IEEE Transactions on Automatic Control,1998,43(5):748-752.
[74]CHOI C H, JEONG G M. Perfect tracking for maximum-phase nonlinear systems by iterative learning control[J]. International Journal of Systems Science, 2001,32(9):1177-1183.
[75]史忠科.连续非线性系统的迭代学习控制方法控制理论与应用[J],1997,14(6)：878-882.
[76]谢胜利,谢振东,韦岗.非线性离散系统的迭代学习控制方法[J].华南理工大学学报,1998,26(2)：7-13.
[77]谢胜利,谢振东,田森平.非线性系统的迭代学习控制及其算法实现[J].控制理论与应用,2002,19(2)：187-172.
[78]林辉,王林.非线性系统闭环P型迭代学习控制的收敛性[J].控制理论与应用,1995,12(6)：742-746.
[79]皮道映,孙优贤.离散非线性系统开闭环P型迭代学习控制律及其收敛性[J].控制理论与应用,1997,14(2)：157-161.
[80]HEINZINGER G, FENWICK D, PADEN B, et al. Robust leaning control[C]. Proceedings of the 28th IEEE Conference on Decision and Control. Tampa, Florida, USA,1989: 436-440.
[81]ARIMOTO S. Stability and robustness of PID feedback control for robot manipulators of sensory capability[C]. The first Int. Symp of Robotics Research. USA:MIT press, 1984:783-799.
[82]WANG D. Convergence and robustness of discrete time nonlinear systems with iterative learning control[J]. Automatica,1998,34(11):1445-1448.
[83]CHEN Y, WEN C, SUN M. A robust high-order P-type iterative learning controller using current iteration tracking error[J]. International Journal of Control,1997, 68(2):331-342.
[84]CHIEN C J. A discrete iterative learning control for a class of nonlinear time-varying systems[J]. Automatic Control, IEEE Transactions on,1998,43(5): 748-752.
[85]HEINZINGER G, FENWICK D, PADEN B, et al. Stability of learning control with disturbances and uncertain initial conditions[J]. Automatic Control, IEEE Transactions on,1992,37(1):110-114.
[86]SAAB S S, VOGT W G, MICKLE M H. Learning control algorithms for tracking "slowly" varying trajectories[J]. Systems, Man, and Cybernetics, Part B:Cybernetics, IEEE Transactions on,1997,27(4):657-670.
[87]皮道映,张政江,孙优贤.非线性系统开闭环PI型迭代学习控制的鲁棒性[J].浙江大学学报(工学版),2001,35(5)：479-482.
[88]王建国,阮小娥.工业过程迭代学习控制算法的鲁棒性[J].工程数学学报,2004,21(1)：127-130.
[89]PARK K H. A study on the robustness of a PID-type iterative learning controller against initial state error[J]. International Journal of Systems Science,1999, 30(1):49-59.
[90]齐蓉,林辉.基于迭代学习的PID控制器研究[J].航空计算技术,2003,33(3)：82-85.
[91]TAYEBI A, ZAREMBA M B. Robust iterative learning control design is straightforward for uncertain LTI systems satisfying the robust performance condition[J]. Automatic Control, IEEE Transactions on,2003,48(1):101-106.
[92]CHIEN C J, LIU J S. A P-type iterative learning controller for robust output tracking of nonlinear time-varying systems [J]. International Journal of Control, 1996,64(2):319-334.
[93]XU J X, Viswanathan B. Adaptive robust iterative learning control with dead zone scheme[J]. Automatica,2000,36(1):91-99.
[94]XU J X, QU Z. Robust iterative learning control for a class of nonlinear systems [J]. Automatica,1998,34(8):983-988.
[95]MOON J H, DOH T Y, CHUNG M J. A robust approach to iterative learning control design for uncertain systems[J]. Automatica,1998,34(8):1001-1004.
[96]DOH T Y. Robust iterative learning control with current feedback for uncertain linear systems [J]. International journal of systems science,1999,30(1):39-47.
[97]WANG D. A simple iterative learning controller for manipulators with flexible joints[J]. Automatica,1995,31(9):1341-1344.
[98]WANG D, SOH Y C, CHEAH C C. Robust motion and force control of constrained manipulators by learning[J]. Automatica,1995,31(2):257-262.
[99]GAO F, YANG Y, SHAO C. Robust iterative learning control with applications to injection molding process[J]. Chemical Engineering Science,2001,56(24): 7025-7034.
[100]SHAO C, GAO F R, YANG Y. Robust stability of optimal iterative learning control and application to injection molding machine[J]. Acta Automatica Sinica,2003, 29(1):72-79.
[101]LI H J, HAO X H, XU W T. A Fast Parameter Optimal Iterative Learning Control Algorithm[C]. International Conference on Embedded Software and Systems Symposia, Chengdu, SiChuan, China,2008:375-379.
[102]HAO X H, OWENS D H, DALEY S. Proportional difference type iterative learning control algorithm based on parameter optimization[C]. Control and Decision Conference, Yantai, Shandong, China,2008:3136-3141.
[103]OWENS D H, TOMAS-RODRIGUEZ M, DALEY S. Limit sets and switching strategies in parameter-optimal iterative learning control[J]. International Journal of Control,2008,81(4):626-640.
[104]HATOEN J J, OWENS D H. MOORE L. An algebraic approach to iterative learning control[J]. International Journal of Control,2004,77(1):45-54.
[105]LIN T, OWENS D H, H?T?NEN J. Newton method based iterative learning control for discrete non-linear systems [J]. International Journal of Control,2006,79(10): 1263-1276.
[106]BLASCHKE B, NEUBAUER A, SCHERZER 0. On convergence rates for the iteratively regularized Gauss-Newton method[J]. IMA Journal of Numerical Analysis,1997, 17(3):421-436.
[107]HINTERMULLER M, ITO K, KUNISCH K. The primal-dual active set strategy as a semi smooth Newton method [J]. SIAM Journal on Optimization,2002,13(3):865-888.
[108]LIN T, OWENS D H. Monotonic Newton method based ILC with parameter optimization for non-linear systems[J]. International Journal of Control,2007,80(8): 1291-1298.
[109]BROYDEN C G. A class of methods for solving nonlinear simultaneous equations [J]. Mathematics of computation,1965,19(92):577-593.
[110]VOLCKAERT M, VAN MULDERS A, SCHOUKENS J, et al. Model based nonlinear iterative learning control:A constrained Gauss-Newton approach[C].17th Mediterranean Conference on Control and Automation, Thessaloniki, Greece.2009:718-723.
[111]HANBA S. Globally convergent Jacobian-free nonlinear equation solvers based on non-monotone norm descent conditions and a modified line search technique[J]. Optimization Methods & Software,2010,25(5):819-837.
[112]林琳,黄南天,高兴泉.基于DFP校正拟牛顿法的傅里叶神经网络拟牛顿法的傅里叶神经网络[J]. Computer Engineering,2012,38(10):144-147.
[113]ORTEGA J M, RHEINBOLDT W C. Iterative solution of nonlinear equations in several variables[M]. London:Academic Press,1970:56-60.
[114]AVRACHENKOV K E. Iterative learning control based on quasi-Newton methods[C]. Proceedings of the 37th IEEE Conference on Decision and Control. Tampa, Florida, USA,1998:170-174.
[115]HU M, DU H, LING S F, et al. Motion control of an electrostrictive actuator[J]. Mechatronics,2004,14(2):153-161.
[116]DU H, HU M, XIE J, et al. Control of an electrostrictive actuator using Newton's method[J]. Precision engineering,2005,29(3):375-380.
[117]李庆扬,莫孜中,祁力群.非线性方程组的数值解法[M].北京：科学出版社,1999：117.
[118]GOLDBERG D E. Genetic algorithm in search [J]. Optimization and Machine Learning, 1989.
[119]MARTINEZ J M. Practical quasi-Newton methods for solving nonlinear systems[J]. Journal of Computational and Applied Mathematics,2000,124(1):97-121.
[120]LU N, WANG F, GAO F, et al. Statistical modeling and. online monitoring for batch processes[J]. Acta Automatica Sinica,2006,32(3):400-410.
[121]XIONG Z H, ZHANG J, Wang X, et al. Integrated tracking control strategy for batch processes using a batch-wise linear time-varying perturbation model[J]. IET Control Theory & Applications,2007,1(1):179-188.
[122]LU N, GAO F. Stage-based process analysis and quality prediction for batch processes[J]. Industrial & engineering chemistry research,2005,44(10): 3547-3555.
[123]LEE K S, LEE J H. Iterative learning control-based batch process control technique for integrated control of end product properties and transient profiles of process variables[J]. Journal of Process Control,2003,13(7):607-621.
[124]LEE K S, CHIN I S, LEE H J, et al. Model predictive control technique combined with iterative learning for batch processes[J]. AIChE Journal,1999, n45(10): 2175-2187.
[125]XIONG Z, XU Y, DONG J, et al. Neural network based iterative learning control for product qualities in batch processes[J]. International Journal of Modelling, Ide tification and Control,2010,11(1):107-114.
[126]AGARWAL P K, CHAND S. Fuzzy logic control of three-pole active magnetic bearing system [J]. International Journal of Modelling, Identification and Control,2011, 12(4):395-411.
[127]XU J X, CHEN Y, LEE T H, et al. Terminal iterative learning control with an application to RTPCVD thickness control [J]. Automatica,1999,35(9):1535-1542.
[128]GAUTHIER G, BOULET B. Terminal iterative learning control design with singular value decomposition decoupling for thermoforming ovens[C]. American Control Conference, St. Louis, Missouri, USA,2009:1640-1645.
[129]XIONG Z, ZHANG J. A batch-to-batch iterative optimal control strategy based on recurrent neural network models[J]. Journal of Process Control,2005,15:11-21.
[130]LUCIBELLO P, PANZIERI S, Ulivi G. Repositioning control of a two-link flexible arm by learning[J]. Automatica,1997,33(4):579-590.
[131]HOU Z, WANG Y, YIN C, et al. Terminal iterative learning control based station stop control of a train[J]. International Journal of Control,2011,84(7): 1263-1274.
[132]王兴华.关于牛顿法的收敛域[J].科学通报数理化专辑,1980,1(1)：40-42.
[133]HATONEN J J, HARTE T J, OWENS D H, et al. A new robust iterative learning control algorithm for application on a gantry robot[C]. Proceedings of Emerging Technologies and Factory Automation, Lisbon, Portugal,2003:305-312.
[134]COTT B J, Macchietto S. Temperature control of exothermic batch reactors using generic model control[J]. Industrial & engineering chemistry research,1989, 28(8):1177-1184.
[135]CHO W, EDGAR T F, LEE J. Iterative learning dual-mode control of exothermic batch reactors[J]. Control Engineering Practice,2008,16(10):1244-1249
[136]李星毅,绪远,王轶.非线性系统开闭环PID型迭代学习控制算法的鲁棒性[J].计算机工程与设计,2008,29(24)：6253-6254.
[137]MENG D, JIA Y, DU J, et al. Robust discrete-time iterative learning control for nonlinear systems with varying initial state shifts[J]. Automatic Control, IEEE Transactions on,2009,54(11):2626-2631.
[138]HATZILOS V, OWENS D H. A Genetic Algorithm Based Optimization Method for Iterative learning Control Systems[C]. Proceedings of 4th Workshop on robot mutation and control, Bukowy Dworek, Poland,2002.
[139]HATZILOS V, HATONEN J, OWENS D H. Genetic algorithms in norm-optimal linear and non-linear iterative learning control [J]. International Journal of Control,2004, 77(2),188-197.
[140]PRATAP D K, AGARWAL S, MEYARIVAN S. A fast and elitist multi objective genetic algorithm:NSGA-II[J]. Evolutionary Computation,2002,6(2):182-197.
[141]NGUYEN D H, BANJERDPONGCHAI D. A convex optimization approach to robust iterative learning control for linear systems with time-varying parametric uncertainties[J]. Automatica,2011,47(9):2039-2043.
[142]VAN DE WIJDEVEN J, DONKERS T, BOSGRA 0. Iterative Learning Control for uncertain systems:Robust monotonic convergence analysis[J]. Automatica,2009,45(10): 2383-2391.
[2]ARIMOTO S, KAWAMURA S, MIYAZAKI F, et al. Learning control theory for dynamical systems[C].24th IEEE Conference on Decision and Control. Fort Lauderdale, Florida, USA,1985:1375-1380.
[3]MOORE KL. Iterative learning control for deterministic systems[M]. London: Springer,1993.
[4]MOORE KL. Iterative learning control-an expository overview[J]. Applied and Computational Controls, Signal Proceeding and Circuits,1998,1:151-214.
[5]李新忠,简林柯,何钺.迭代学习控制的研究及应用[J].机床与油压,1997,6：3-5.
[6]刘慧,许晓鸣,张仲俊.迭代学习控制的回顾与展望[J].机器人,1996,18(6)：347-379.
[7]谢振东,谢胜利,刘永清.非线性系统学习控制理论的发展与展望[J].控制理论与应用,2000,17(1)：4-8.
[8]BIEN Z, XU J X. Iterative learning control-Analysis, Design, Integration and Applications[M]. Netherlands:Kluwer Academic Publishers,1998.
[9]CHEN Y, WEN C. Iterative learning control-Convergence, robustness and applications[M]. London:Springer,1999.
[10]林辉,王林.迭代学习控制理论[M].西安：西北工业大学出版社,1998.
[11]MOORE K L, XU J X. Editorial:Special Issue on iterative learning control[J], international Journal of Control,2000,73(10):819-823.
[12]XU J X, TAN Y. Linear and nonlinear iterative learning control [M]. London:Springer, 2003.
[13]THARAYIL M, ALLEYNE A G. A survey of iterative learning control [J]. Control Systems, 26(3):96-144,2006.
[14]YANG Q C, MOORE K L. Iterative Learning Control:Brief Survey and Categorization [J]. Systems, Man, and Cybernetics,2007,37(6):1099-1121.
[15]WANG YOUQING, GAO FURONG, FRANCIS J, et al. Survey on iterative learning control, repetitive control, and run-to-run control [J]. Journal of Process Control,2009,19 (10):1589-1600.
[16]ARIMOTO S, KAWAMURA S, MIYAZAKI F. Bettering operation of robots by learning[J]. Journal of Robotic Systems,1984,1(2):123-140.
[17]KAWAMURA S, MIYAZAKI F, ARIMOTO S. Realization of robot motion based on a learning method[J]. IEEE Transactions on System, Man, and Cybernetics,1998,18(1): 126-134.
[18]NANIWA T, ARIMOTO S. Learning control for robot tasks under geometric end point constrains[J]. IEEE Transactions on Robotic and Automation,1995,11(3):432-441.
[19]ARIMOTO S. Learning control theory for robotic motion[J]. International Journal of Adaptive Control and Signal Processing.1990,4(6):543-564.
[20]皮道映,孙优贤.离散非线性时变系统开闭环PI型迭代学习控制律及其收敛性[J].自动化学报,1998,24(5)：636-639.
[21]LEE H S, BIEN Z. Study on robustness of iterative learning control with non-zero initial error[J]. International Journal of Control,1996,64(3):345-359.
[22]孙明轩,黄宝健,张学智.非线性系统的PD型迭代学习控制[J].自动化学报,1998,24(5)：711-714.
[23]BIEN Z, HUN K M. High-order P-type iterative learning controller using current iteration tracking error[J]. International Journal of Control,1997,68(2): 331-342.
[24]PARK K H. An average operator-based iterative learning control for variable initial state error[J]. IEEE Transactions on Automatic Control,2005,50(6):865-869.
[25]CHEN Y Q, Wang C Y, SUN M. A robust high-order P-type iterative learning controller using current iteration tracking error[J]. International Journal of Control.1997, 68(2):345-353.
[26]CHEN Y Q, GONG Z M, WEN C Y. Analysis of high-order iterative learning control algorithm for uncertain nonlinear systems with state delays [J]. Automatica,1998, 34(3):1867-1880.
[27]Xu J X, Tan Y. Robust optimal design and convergence properties analysis of iterative learning control approaches[J]. Automatica,2002,38(11):1867-1880.
[28]RUAN X, BIEN Z Z, WANG Q. Convergence characteristics of proportional-type iterative learning control in the sense of Lebesgue-p norm[J]. IET Control Theory and Applications,2012,6(5):1-8.
[29]RUAN X, BIEN Z, WANG Q. Convergence properties of iterative learning control processes in the sense of Lebesgue-p norm[J]. Asian Journal of control,2012,14 (4): 1095-1107.
[30]LEE K H, BIEN Z. Initial condition problem of learning control [J]. IEE Proceedings of Control Theory and Applications, Part D,1991,138(6):525-528.
[31]LEE H S, BIEN Z. Study on robustness of iterative learning control with non-zero initial error[J]. International Journal of Control,1996,64(3):345-359.
[32]PARK K H, BIEN Z, HWANG D H. A study on the robustness of a PID-type iterative learning controller against initial state error[J]. International Journal of Systems Science,1999,30(1):49-59.
[33]PRAK K H, BIEN Z. A generalized iterative learning controller against initial state error[J]. International Journal of control,2000,70(10):871-881.
[34]PRAK K H. An average operator-based iterative learning control for variable initial state error[J]. IEEE Transactions on Automatic control,2005,50(6):865-869.
[35]RUAN X, BIEN Z. Pulse compensation for PD-type iterative learning control against initial state shift[J]. International Journal of Systems science,2012,43(11): 2172-2184.
[36]PARK K H, BIEN Z. A Study On Iterative Learning Control With Adjustment Of Learning Interval For Monotone Convergence In The Sense Of sup-norm[J]. Asian Journal of Control,2002,4(1):111-118.
[37]PARK H, BIEN Z. Intervalized iterative learning control for monotonic convergence in the sense of sup-norm[J]. International Journal of Control,2005,78(15): 1218-1227.
[38]RUAN X, BIEN Z Z, WANG Q. Convergence characteristics of proportional-type iterative learning control in the sense of Lebesgue-norm[J]. Control Theory & Applications, IET,2012,6(5):707-714.
[39]RUAN X, BIEN Z Z, WANG Q. Convergence Properties of Iterative Learning Control Processes in the Sense of the Lebesgue-Norm[J]. Asian Journal of Control,2012, 14(4):1095-1107.
[40]KUREK J. The general state-space model for a two dimensional linear digital systems[J]. IEEE Transition of Automatic Control,1985,30(6):600-602.
[41]KACZOREK T. Two-Dimensional Linear Systems[M]. London:Springer,1985:32-35.
[42]GALKOWSKI K, ROGERS E, GRAMACKI A, et al. Higher order discretisation methods for a class of 2-D continuous-discrete linear systems[J]. Circuits, Devices and Systems,1999,146(6):315-320.
[43]MENG D Y, JIA Y M, DU J P, et al. Feedback iterative learning control for time-delay systems based on 2D analysis approach[J]. Journal of Control Theory and Applications,2010,8(4):457-463.
[44]DYMKOV M, GAISHUN I, ROGERS E, et al. Control theory for a class of 2D continuous-discrete linear systems[J]. International Journal of Control,2004, 77(9):847-860.
[45]LI X D, CHOW T W S, H, J K L.2-D system theory based iterative learning control for linear continuous systems with time delays [J]. Circuits and Systems I:Regular Papers, IEEE Transactions on,2005,52(7):1421-1430.
[46]SHI J, GAO F R, WU T J. Robust design of integrated feedback and iterative learning control of a batch process based on a 2D Roesser system [J]. Journal of Process Control,2005,15(8):907-924.
[47]SHI J, GAO F R. Integrated Design and Structure Analysis of Robust Iterative Learning Control System Based on a Two-Dimensional Model[J]. Industrial & Engineering Chemistry Research,2005,44(21):8095-8105.
[48]HLADOWSKI L, GALKOWSKI K, CAI Z L, ROGERS E, et al. Experimentally supported 2D systems based iterative learning control law design for error convergence and performance. Control Engineering Practice,2010,18(4):339-348.
[49]TOGAI M, YAMANO 0. Analysis and design of an optimal learning control scheme for industrial robots:a discrete system approach[C]. Proc. of 24th IEEE Conf. on Decision and-Control, Ft. Lauderdale, Florida, USA,1985:1399-1404.
[50]TAO K M, KOSUT R L, ARAL G. Learning feed-forward control [C]. Proc. Of the American Control Conference, Baltimore, Maryland, USA,1994:2575-2579.
[51]GORINEVSKY D M. Direct learning of feed-forward control for manipulator path tracking[C]. Proc. of IEEE International Symp. on Intelligent Control, Glasgow, United Kingdom,1992:42-47.
[52]AMANN N, OEWNS D H, ROGERS E. Iterative Learning Control using Optimal Feedback and Feed forward Actions[J]. International Journal of control,1996,65(2): 277-293.
[53]AMANN N, OEWNS D H, ROGERS E. Iterative Learning Control for Discrete-time Systems with Exponential Rate of Convergence[J]. Proceedings of the IEE Control Theory Applications,1996,143(2):217-224.
[54]AMANN, N, OEWNS D H, ROGERS E. Predictive Optimal Iterative Learning Control [J]. International Journal of Control,1998,69(2):203-226.
[55]OWENS D H, FENG K. Parameter Optimisation in Iterative Learning Control[J]. International Journal of Control,2003,76(11):1059-1069.
[56]HATONEN J, OWENS D H, FENG K. Basis Functions and Parameter Optimisation in High order Iterative Learning Control[J]. Automatica,2006,42(2):287-294.
[57]HARTE T J, HATONEN J, OEWNS D H. Discrete time Inverse Model-based Iterative Learning Control:Stability, Monotonicity and Robustness[J]. International Journal of Control,2005,78(8):577-586.
[58]OEWNS D H, HATONEN J, DALEY, S. Robust Monotone Gradient-based Discrete-time Iterative Learning Control[J]. International Journal of Robust and Nonlinear Control,2009,19(6):634-661.
[59]OWENS D H. Multivariable norm optimal and parameter optimal iterative learning control:a unified formulation[J]. International Journal of Control, 2012,85 (8):1010-1025.
[60]OWENS D H, HATONEN J. Iterative Learning Control-an Optimisation Paradigm[J]. IFAC Annual Reviews in Control,2005,29(1):57-70.
[61]PANG B, SHAO C, Modified Newton-based ILC solving specific target tracking problems for non-linear batch reactor[J]. International Journal of Modelling, Identification and Control,2013,18(2):158-165.
[62]SON T D, ANN H S. Terminal iterative learning control with multiple intermediate pass points[C]. American Control Conference, San Francisco, California, USA,2011: 3651-3656.
[63]BARTON K L, ALLEYNE A G. A norm optimal approach to time-varying ILC with application to a multi-axis robotic testbed[J]. Control Systems Technology, IEEE Transactions on,2011,19(1):166-180.
[64]VAN DE WIJDEVEN J, DONKERS T, BOSGRA 0. Iterative Learning Control for uncertain systems:Robust monotonic convergence analysis[J]. Automatica,2009,45(10): 2383-2391.
[65]CHU B, OWENS D H. Accelerated norm-optimal iterative learning control algorithms using successive projection[J]. International Journal of Control,2009,82(8): 1469-1484.
[66]WANG D, LIU D, WEI Q. Finite-horizon neuro-optimal tracking control for a class of discrete-time nonlinear systems using adaptive dynamic programming approach[J]. Neurocomputing,2012,78(1):14-22.
[67]OWENS D H, FREEMAN C T, VAN DINH T. Norm-optimal iterative learning control with intermediate point weighting:Theory, algorithms, and experimental evaluation [J]. Control Systems Technology, IEEE Transactions on,2013,21(3):999-1007.
[68]XU J. Analysis of iterative learning control for a class of nonlinear discrete-time systems[J]. Automatica,1997,33(10):1905-1907.
[69]CHIEN C J. A discrete iterative learning control for a class of nonlinear time-varying systems[J]. IEEE transactions on Automatic Control,1998, 43(5):748-752.
[70]Wang D. Convergence and robustness of discrete time nonlinear systems with iterative learning control[J]. Automatica,1998,34(11):1445-1448.
[71]CHEN C K, HWANG J. Iterative learning control for position tracking of a pneumatic actuated X-Y table [J]. Control Engineering Practice,2005,13(12):1455-1461.
[72]XIONG Z, ZHANG J. Batch-to-batch optimal control of nonlinear batch processes based on incrementally updated models[J]. IEE Proceedings:Control theory and applications,2004,15(2):158-165.
[73]CHIEN C. A discrete iterative learning control for a class of nonlinear time-varying systems[J].IEEE Transactions on Automatic Control,1998,43(5):748-752.
[74]CHOI C H, JEONG G M. Perfect tracking for maximum-phase nonlinear systems by iterative learning control[J]. International Journal of Systems Science, 2001,32(9):1177-1183.
[75]史忠科.连续非线性系统的迭代学习控制方法控制理论与应用[J],1997,14(6)：878-882.
[76]谢胜利,谢振东,韦岗.非线性离散系统的迭代学习控制方法[J].华南理工大学学报,1998,26(2)：7-13.
[77]谢胜利,谢振东,田森平.非线性系统的迭代学习控制及其算法实现[J].控制理论与应用,2002,19(2)：187-172.
[78]林辉,王林.非线性系统闭环P型迭代学习控制的收敛性[J].控制理论与应用,1995,12(6)：742-746.
[79]皮道映,孙优贤.离散非线性系统开闭环P型迭代学习控制律及其收敛性[J].控制理论与应用,1997,14(2)：157-161.
[80]HEINZINGER G, FENWICK D, PADEN B, et al. Robust leaning control[C]. Proceedings of the 28th IEEE Conference on Decision and Control. Tampa, Florida, USA,1989: 436-440.
[81]ARIMOTO S. Stability and robustness of PID feedback control for robot manipulators of sensory capability[C]. The first Int. Symp of Robotics Research. USA:MIT press, 1984:783-799.
[82]WANG D. Convergence and robustness of discrete time nonlinear systems with iterative learning control[J]. Automatica,1998,34(11):1445-1448.
[83]CHEN Y, WEN C, SUN M. A robust high-order P-type iterative learning controller using current iteration tracking error[J]. International Journal of Control,1997, 68(2):331-342.
[84]CHIEN C J. A discrete iterative learning control for a class of nonlinear time-varying systems[J]. Automatic Control, IEEE Transactions on,1998,43(5): 748-752.
[85]HEINZINGER G, FENWICK D, PADEN B, et al. Stability of learning control with disturbances and uncertain initial conditions[J]. Automatic Control, IEEE Transactions on,1992,37(1):110-114.
[86]SAAB S S, VOGT W G, MICKLE M H. Learning control algorithms for tracking "slowly" varying trajectories[J]. Systems, Man, and Cybernetics, Part B:Cybernetics, IEEE Transactions on,1997,27(4):657-670.
[87]皮道映,张政江,孙优贤.非线性系统开闭环PI型迭代学习控制的鲁棒性[J].浙江大学学报(工学版),2001,35(5)：479-482.
[88]王建国,阮小娥.工业过程迭代学习控制算法的鲁棒性[J].工程数学学报,2004,21(1)：127-130.
[89]PARK K H. A study on the robustness of a PID-type iterative learning controller against initial state error[J]. International Journal of Systems Science,1999, 30(1):49-59.
[90]齐蓉,林辉.基于迭代学习的PID控制器研究[J].航空计算技术,2003,33(3)：82-85.
[91]TAYEBI A, ZAREMBA M B. Robust iterative learning control design is straightforward for uncertain LTI systems satisfying the robust performance condition[J]. Automatic Control, IEEE Transactions on,2003,48(1):101-106.
[92]CHIEN C J, LIU J S. A P-type iterative learning controller for robust output tracking of nonlinear time-varying systems [J]. International Journal of Control, 1996,64(2):319-334.
[93]XU J X, Viswanathan B. Adaptive robust iterative learning control with dead zone scheme[J]. Automatica,2000,36(1):91-99.
[94]XU J X, QU Z. Robust iterative learning control for a class of nonlinear systems [J]. Automatica,1998,34(8):983-988.
[95]MOON J H, DOH T Y, CHUNG M J. A robust approach to iterative learning control design for uncertain systems[J]. Automatica,1998,34(8):1001-1004.
[96]DOH T Y. Robust iterative learning control with current feedback for uncertain linear systems [J]. International journal of systems science,1999,30(1):39-47.
[97]WANG D. A simple iterative learning controller for manipulators with flexible joints[J]. Automatica,1995,31(9):1341-1344.
[98]WANG D, SOH Y C, CHEAH C C. Robust motion and force control of constrained manipulators by learning[J]. Automatica,1995,31(2):257-262.
[99]GAO F, YANG Y, SHAO C. Robust iterative learning control with applications to injection molding process[J]. Chemical Engineering Science,2001,56(24): 7025-7034.
[100]SHAO C, GAO F R, YANG Y. Robust stability of optimal iterative learning control and application to injection molding machine[J]. Acta Automatica Sinica,2003, 29(1):72-79.
[101]LI H J, HAO X H, XU W T. A Fast Parameter Optimal Iterative Learning Control Algorithm[C]. International Conference on Embedded Software and Systems Symposia, Chengdu, SiChuan, China,2008:375-379.
[102]HAO X H, OWENS D H, DALEY S. Proportional difference type iterative learning control algorithm based on parameter optimization[C]. Control and Decision Conference, Yantai, Shandong, China,2008:3136-3141.
[103]OWENS D H, TOMAS-RODRIGUEZ M, DALEY S. Limit sets and switching strategies in parameter-optimal iterative learning control[J]. International Journal of Control,2008,81(4):626-640.
[104]HATOEN J J, OWENS D H. MOORE L. An algebraic approach to iterative learning control[J]. International Journal of Control,2004,77(1):45-54.
[105]LIN T, OWENS D H, H?T?NEN J. Newton method based iterative learning control for discrete non-linear systems [J]. International Journal of Control,2006,79(10): 1263-1276.
[106]BLASCHKE B, NEUBAUER A, SCHERZER 0. On convergence rates for the iteratively regularized Gauss-Newton method[J]. IMA Journal of Numerical Analysis,1997, 17(3):421-436.
[107]HINTERMULLER M, ITO K, KUNISCH K. The primal-dual active set strategy as a semi smooth Newton method [J]. SIAM Journal on Optimization,2002,13(3):865-888.
[108]LIN T, OWENS D H. Monotonic Newton method based ILC with parameter optimization for non-linear systems[J]. International Journal of Control,2007,80(8): 1291-1298.
[109]BROYDEN C G. A class of methods for solving nonlinear simultaneous equations [J]. Mathematics of computation,1965,19(92):577-593.
[110]VOLCKAERT M, VAN MULDERS A, SCHOUKENS J, et al. Model based nonlinear iterative learning control:A constrained Gauss-Newton approach[C].17th Mediterranean Conference on Control and Automation, Thessaloniki, Greece.2009:718-723.
[111]HANBA S. Globally convergent Jacobian-free nonlinear equation solvers based on non-monotone norm descent conditions and a modified line search technique[J]. Optimization Methods & Software,2010,25(5):819-837.
[112]林琳,黄南天,高兴泉.基于DFP校正拟牛顿法的傅里叶神经网络拟牛顿法的傅里叶神经网络[J]. Computer Engineering,2012,38(10):144-147.
[113]ORTEGA J M, RHEINBOLDT W C. Iterative solution of nonlinear equations in several variables[M]. London:Academic Press,1970:56-60.
[114]AVRACHENKOV K E. Iterative learning control based on quasi-Newton methods[C]. Proceedings of the 37th IEEE Conference on Decision and Control. Tampa, Florida, USA,1998:170-174.
[115]HU M, DU H, LING S F, et al. Motion control of an electrostrictive actuator[J]. Mechatronics,2004,14(2):153-161.
[116]DU H, HU M, XIE J, et al. Control of an electrostrictive actuator using Newton's method[J]. Precision engineering,2005,29(3):375-380.
[117]李庆扬,莫孜中,祁力群.非线性方程组的数值解法[M].北京：科学出版社,1999：117.
[118]GOLDBERG D E. Genetic algorithm in search [J]. Optimization and Machine Learning, 1989.
[119]MARTINEZ J M. Practical quasi-Newton methods for solving nonlinear systems[J]. Journal of Computational and Applied Mathematics,2000,124(1):97-121.
[120]LU N, WANG F, GAO F, et al. Statistical modeling and. online monitoring for batch processes[J]. Acta Automatica Sinica,2006,32(3):400-410.
[121]XIONG Z H, ZHANG J, Wang X, et al. Integrated tracking control strategy for batch processes using a batch-wise linear time-varying perturbation model[J]. IET Control Theory & Applications,2007,1(1):179-188.
[122]LU N, GAO F. Stage-based process analysis and quality prediction for batch processes[J]. Industrial & engineering chemistry research,2005,44(10): 3547-3555.
[123]LEE K S, LEE J H. Iterative learning control-based batch process control technique for integrated control of end product properties and transient profiles of process variables[J]. Journal of Process Control,2003,13(7):607-621.
[124]LEE K S, CHIN I S, LEE H J, et al. Model predictive control technique combined with iterative learning for batch processes[J]. AIChE Journal,1999, n45(10): 2175-2187.
[125]XIONG Z, XU Y, DONG J, et al. Neural network based iterative learning control for product qualities in batch processes[J]. International Journal of Modelling, Ide tification and Control,2010,11(1):107-114.
[126]AGARWAL P K, CHAND S. Fuzzy logic control of three-pole active magnetic bearing system [J]. International Journal of Modelling, Identification and Control,2011, 12(4):395-411.
[127]XU J X, CHEN Y, LEE T H, et al. Terminal iterative learning control with an application to RTPCVD thickness control [J]. Automatica,1999,35(9):1535-1542.
[128]GAUTHIER G, BOULET B. Terminal iterative learning control design with singular value decomposition decoupling for thermoforming ovens[C]. American Control Conference, St. Louis, Missouri, USA,2009:1640-1645.
[129]XIONG Z, ZHANG J. A batch-to-batch iterative optimal control strategy based on recurrent neural network models[J]. Journal of Process Control,2005,15:11-21.
[130]LUCIBELLO P, PANZIERI S, Ulivi G. Repositioning control of a two-link flexible arm by learning[J]. Automatica,1997,33(4):579-590.
[131]HOU Z, WANG Y, YIN C, et al. Terminal iterative learning control based station stop control of a train[J]. International Journal of Control,2011,84(7): 1263-1274.
[132]王兴华.关于牛顿法的收敛域[J].科学通报数理化专辑,1980,1(1)：40-42.
[133]HATONEN J J, HARTE T J, OWENS D H, et al. A new robust iterative learning control algorithm for application on a gantry robot[C]. Proceedings of Emerging Technologies and Factory Automation, Lisbon, Portugal,2003:305-312.
[134]COTT B J, Macchietto S. Temperature control of exothermic batch reactors using generic model control[J]. Industrial & engineering chemistry research,1989, 28(8):1177-1184.
[135]CHO W, EDGAR T F, LEE J. Iterative learning dual-mode control of exothermic batch reactors[J]. Control Engineering Practice,2008,16(10):1244-1249
[136]李星毅,绪远,王轶.非线性系统开闭环PID型迭代学习控制算法的鲁棒性[J].计算机工程与设计,2008,29(24)：6253-6254.
[137]MENG D, JIA Y, DU J, et al. Robust discrete-time iterative learning control for nonlinear systems with varying initial state shifts[J]. Automatic Control, IEEE Transactions on,2009,54(11):2626-2631.
[138]HATZILOS V, OWENS D H. A Genetic Algorithm Based Optimization Method for Iterative learning Control Systems[C]. Proceedings of 4th Workshop on robot mutation and control, Bukowy Dworek, Poland,2002.
[139]HATZILOS V, HATONEN J, OWENS D H. Genetic algorithms in norm-optimal linear and non-linear iterative learning control [J]. International Journal of Control,2004, 77(2),188-197.
[140]PRATAP D K, AGARWAL S, MEYARIVAN S. A fast and elitist multi objective genetic algorithm:NSGA-II[J]. Evolutionary Computation,2002,6(2):182-197.
[141]NGUYEN D H, BANJERDPONGCHAI D. A convex optimization approach to robust iterative learning control for linear systems with time-varying parametric uncertainties[J]. Automatica,2011,47(9):2039-2043.
[142]VAN DE WIJDEVEN J, DONKERS T, BOSGRA 0. Iterative Learning Control for uncertain systems:Robust monotonic convergence analysis[J]. Automatica,2009,45(10): 2383-2391.