Journal of Advances in Applied Mathematics

JAAM
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Volume 5, Number 1, January 2020

The Application of Zhang-Gradient Method for Iterative Learning Control
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KB)PP.
20-25

, Pub. Date:December 12, 2019
DOI:

10.22606/jaam.2020.51003
**Author(s)**
Zhang Qunli

**Affiliation(s)**
Department of Mathematical and Statistical, Heze University, Heze, Shandong 274015, P. R. China

**Abstract**
The novel sufficient conditions for nonlinear systems without and with time-delay, whose
initial state are zero or not, are studied using the -norm, Zhang-gradient method and retarded
Gronwall-like inequality. An examples is shown the effectiveness of the mentioned technique.

**Keywords**
Iterative learning control, Zhang-gradient method, tracking error; convergence, time
delay

**References**
- [1] S.Arimoto, S.Kawamura, and F.Miyazaki. Bettering operation of robots by learning, Journal of Robotic Systems,
1(2),pp.123-140,1984.
- [2] H.-S. Lee and Z. Bien, A note on convergence property of iterative learning controller with respect to sup norm,
Automatica,vol.33,no.8,pp.1591–1593,1997.
- [3] Y. Chen, Z. Gong, and C. Wen, Analysis of a high-order iterative learning control algorithm for uncertain
nonlinear systems with state delays, Automatica,vol.34,no.3,pp.345–353, 1998.
- [4] Sheng Liu, Changkui Xu, and Lanyong Zhang. Robust Course Keeping Control of a Fully Submerged Hydrofoil
Vessel without Velocity Measurement: An Iterative Learning Approach, Mathematical Problems in Engineering,
Article ID 7979438, Volume 2017, 14 pages ,2017.
- [5] Dongqi Ma, Hui Lin. An Accelerating Iterative Learning Control Based on an Adjustable Learning Interval,
Journal of Control Science and Engineering, Article ID 1731676, Volume 2017,6 pages, 2017.
- [6] R.H.Chi,Z.S.Hou,andJ.X.Xu. AdaptiveILCforaclass of discrete-time systems with iteration-varying trajectory
and random initial condition, Automatica, vol.44,no.8,pp.2207–2213, 2008.
- [7] Leila Noueili, Wassila Chagra, and Moufida Ksouri. New Iterative Learning Control Algorithm Using Learning
Gain Based on σ Inversion for Nonsquare Multi-Input Multi-Output Systems, Modelling and Simulation in
Engineering , Article ID 4195938, Volume 2018, 9 pages ,2018.
- [8] Lei Li. Lebesgue-p Norm Convergence Analysis of PDα-Type Iterative Learning Control for Fractional-Order
Nonlinear Systems,Discrete Dynamics in Nature and Society , Article ID 5157267, Volume 2018,10 pages, 2018.
- [9] Xiongfeng Deng, Xiuxia Sun, and Shuguang Liu. Consensus Learning Control for Leader-Following Nonlinear
Multiagent Systems with Control Delay, Wireless Communications and Mobile Computing, Article ID 2035683,
Volume 2019,10 pages,2019.
- [10] Xiongfeng Deng, Xiuxia Sun, Shuguang Liu, and Boyang Zhang. Leader-Following Consensus for Second-Order
Nonlinear Multiagent Systems with Input Saturation via Distributed Adaptive Neural Network Iterative
Learning Control, Complexity, Research Article (13 pages), Article ID 9858504, Volume 201913 pages, 2019.
- [11] Xiaoli Li, Jian Liu, Linkun Wang, Kang Wang, and Yang Li. Welding Process Tracking Control Based on Multiple Model Iterative Learning Control, Mathematical Problems in Engineering, Article ID 6137352, Volume
2019, 9 pages, 2019.
- [12] J. Li and J. Li. Iterative learning control approach for a kind of heterogeneous multi-agent systems with
distributed initial state learning, Applied Mathematics and Computation, vol.265,pp.1044–1057, 2015.
- [13] L. Yan and J. Wei. Fractional order nonlinear systems with delay in iterative learning control, Applied
Mathematics and Computation,vol.257,pp.546–552,2015.
- [14] H. Cai, Y. Huang, J. Du, T. Tang, D. Zuo, and J. Li. Iterative learning control with extended state observer for
telescope sys-tem, Mathematical Problems in Engineering, vol.2015,Article ID 701510, 8 pages, 2015.
- [15] Zhang Qunli. The effect of initial state error for nonlinear systems with delay via iterative learning control,
Advances in Mathematical Physics, Volume 2016, Article ID 461950, 6pages,2016.
- [16] J.Cao, Y.Wan. Matrix measure strategies for stability and synchronization of inertial BAM neural network with
time delay, Neural Netw.53,pp.165-172,2014.
- [17] S.Das, A.Acharya, I.Pan. Simulation studies on the design of optimum PID controller to suppress chaotic
oscillations in a family of Lorenz-like multi-wing attractors , Math.Comput. Simul,100,pp.72-87,2014.
- [18] C.Yi, Y.Zhang, D.Guo. A new type of recurrent neural networks for real-time solution of Lyapunov equation
with time-varying coefficient matrices, Math. Comput. Simul, 92,pp.40-52,2013.
- [19] Y.Zhang,M.Li, Y.Yin,L.Jin, X.Yu. Controller design of nonlinear system for fully trackable and paths by
combining ZD and GD, in:Proc.25th Control and Decision Conference,pp.209-214,2013.
- [20] Y.Zhang, C.Yi, D.Guo, J.Zheng. “Comparison on Zhang neural dynamics and gradient-based neural dynamics
for online solution of nonlinear time-varying equation”, Neural Comput. Appl. 20,pp.1-7,2011.
- [21] S.S.Ge,F.Hong,andT.H.Lee, Robust adaptive control of nonlinear systems with unknown time delays,
Automatica, vol.41,no.7,pp.1181–1190,2005.
- [22] C. C. Hua, G. Feng, and X. P. Guan. Robust controller design of a class of nonlinear time delay systems via
backstepping method, Automatica,vol.44,no.2,pp.567–573,2008.
- [23] X. D. Ye. Adaptive stabilization of time-delay feedforward nonlinear systems, Automatica,vol.47,no.5,pp.950–
955,2011.
- [24] J. Na. Adaptive prescribed performance control of nonlinear systems with unknown dead zone, International
Journal of Adaptive Control and Signal Processing, vol.27, no.5,pp.426–446, 2013.
- [25] Z.-Y. Sun and Y.-G. Liu. Adaptive control design for a class of uncertain high-order nonlinear systems with time
delay, Asian Journal of Control,vol.17,no.2,pp.535–543,2015.
- [26] Qunli Zhang. Synchronization of multi-chaotic systems via ring impulsive control, Control theory &
Applications,vol.27,no.2, pp. 226–232, 2010.
- [27] Qunli Zhang. Synchronization of multi-chaotic systems with ring and chain intermittent connections, Applied
Mechanics and Materials,vol.241–244,pp.1081–1087,2013.
- [28] Q.Zhang. A class of vector Lyapunov functions for stability analysis of nonlinear impulsive differential systems,
Mathematical Problems in Engineering, vol. 2014, Article ID 649012, 9 pages, 2014.
- [29] Q. Zhang. Matrix measure with application in quantized synchronization analysis of complex networks with
delayed time via the general intermittent control, Annals of Applied Mathematics,vol.31,no.1,pp.115–126,2015.
- [30] M.X.Sun and B.J.Huang. Iterative learning control , National Defence Industry Press, Beijing, China,1999.
- [31] Zhang Y., Ma W., Cai B. From Zhang neural network to Newton iteration for matrix inversion, IEEE Trans.
Circuits Syst. I, Regul. Pap,56(7),pp1405-1415,2009.
- [32] Zhang Y., Yi C. Zhang neural network and neural-dynamic method, Nova Science Publishers, New York,2011.
- [33] R.P.Agarwal, S.deng, and W. Zhang. Generalization of a retarded Gronwall-like inequality and its applications,
Applied Mathematics and Computation,165(3),pp.599-612,2005.