A parametric learning based robust iterative learning control (ILC) scheme is applied to the time varying delay multiple-input and multiple-output (MIMO) linear systems. The convergence conditions are derived by using the and linear matrix inequality (LMI) approaches, and the convergence speed is analyzed as well. A practical identification strategy is applied to optimize the learning laws and to improve the robustness and performance of the control system. Numerical simulations are illustrated to validate the above concepts. 1. Introduction Learning mechanism enables the human beings to master skills, while the experiences gained from practices play important roles in this procedure. It is expected that the learning mechanism can also be introduced to machines, which enables them to achieve satisfactory performance from previous acquired input-output information. The method of ILC was firstly applied to control manipulators at high speed which is proposed by Uchiyama [1]. In 1984, Arimoto [2] published the first English paper of ILC for accurate tracking of robot trajectories. The basic idea of ILC is utilizing the information of the previous iteration to realize perfect tracking without exact knowledge of the system parameters, and a typical ILC scheme is shown as in Figure 1. In the recent three decades, many kinds of learning laws are utilized which can be mainly divided by two categories: the linear learning laws and the nonlinear learning laws. For example, the linear learning laws include but are not limited to the parametric learning law [3, 4], the robust learning law [5], the high-order learning law [6, 7], the PD type learning law [8, 9], and so on [10]. On the other hand, the Newton learning law and the Secant learning law belong to the nonlinear ones [11, 12] which have faster convergence speed comparing to some linear cases. Moreover, the control objectives are mainly focused on the linear continuous and discrete forms [13, 14] and the nonlinear systems with relative degree one [15] or the quasilinear forms [16] and so forth [17–27]. Figure 1: The basic idea of ILC. The time delay systems are ubiquitous in real world control problems [28] such as networked control systems, chemical processes, hydraulic, and rolling mill systems. The time delay affects the system performance in a large scale. Serious performance degradation and even instability can be led by time delay [29]. For decades, considerable efforts have been paid to assure the robust performance of time delay systems in both theories and applications [30]. The research of time
References
[1]
M. Uchiyama, “Formulation of high-speed motion of a mechanical arm by trial,” Transactions of SICE, vol. 14, no. 6, pp. 706–712, 1978.
[2]
S. Arimoto, S. Kawamura, and F. Miyazaki, “Bettering operation of robots by learning,” Journal of Robotic Systems, vol. 1, no. 2, pp. 123–140, 1984.
[3]
A. Tayebi, “Analysis of two particular iterative learning control schemes in frequency and time domains,” Automatica, vol. 43, no. 9, pp. 1565–1572, 2007.
[4]
C. K. Yin, J. X. Xu, and Z. S. Hou, “An ILC scheme for a class of nonlinear continuous-time systems with time-iteration-varying parameters subject to second-order internal model,” Asian Journal of Control, vol. 13, no. 1, pp. 126–135, 2011.
[5]
H.-S. Ahn, L.-M. Kevin, and Y.-Q. Chen, Iterative Learning Controlrobustness and Monotonic Convergence for Interval Systems, Communications and Control Engineering, Springer, 2007.
[6]
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.
[7]
Z. Bien and K. M. Huh, “Higher-order iterative learning control algorithm,” IEE Proceedings D: Control Theory and Applications, vol. 136, no. 3, pp. 105–112, 1989.
[8]
J. X. Xu, K. Abidi, X. L. Niu, and D. Q. Huang, “Sampled-data iterative learning control for a piezoelectric motor,” in Proceedings of the 21st IEEE International Symposium on Industrial Electronics (ISIE '12), pp. 899–904, Hangzhou, China, May 2012.
[9]
S. Dominik and A. Harald, “Robust iterative learning control for nonlinear systems with measurement disturbances,” in Proceedings of the American Control Conference, pp. 5484–5489, Montreal, Canada, 2012.
[10]
Y. Chen, C. Wen, and M.-X. Sun, “A robust high-order P-type iterative learning controller using current iteration tracking error,” International Journal of Control, vol. 68, no. 2, pp. 331–342, 1997.
[11]
J. Xu and Y. Tan, “On the P-type and Newton-type ILC schemes for dynamic systems with non-affine-in-input factors,” Automatica, vol. 38, no. 8, pp. 1237–1242, 2002.
[12]
J.-X. Xu and Y. Tan, Linear a nd Nonlinear Iterative Learning Control, vol. 291 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2003.
[13]
L.-J. Zhang, D.-Z. Cheng, and J.-B. Liu, “Stabilization of switched linear systems,” Asian Journal of Control, vol. 5, no. 4, pp. 476–483, 2003.
[14]
X. Bu, F. Yu, Z. Hou, and F. Wang, “Iterative learning control for a class of linear discrete-time switched systems,” Acta Automatica Sinica, vol. 39, no. 9, pp. 1564–1569, 2013.
[15]
M. X. Sun and Q. Z. Yan, “Error tracking of iterative learning control systems,” Acta Automatica Sinica, vol. 39, no. 3, pp. 251–262, 2013.
[16]
W. H. Moase and C. Manzie, “Fast extremum-seeking for Wiener-Hammerstein plants,” Automatica, vol. 48, no. 10, pp. 2433–2443, 2012.
[17]
D. Meng, Y. Jia, J. Du, and J. Zhang, “On iterative learning algorithms for the formation control of nonlinear multi-agent systems,” Automatica, vol. 50, no. 1, pp. 291–295, 2014.
[18]
K. Delchev, “Iterative learning control for nonlinear systems: a bounded-error algorithm,” Asian Journal of Control, vol. 15, no. 2, pp. 453–460, 2013.
[19]
R. H. Chi, Z. S. Hou, S. T. Jin, and D. W. Wang, “Discrete-time adaptive ILC for non-parametric uncertain nonlinear systems with iteration-varying trajectory and random initial condition,” Asian Journal of Control, vol. 15, no. 2, pp. 562–570, 2013.
[20]
D. Shen and H. Chen, “A Kiefer-Wolfowitz algorithm based iterative learning control for Hammerstein-WIEner systems,” Asian Journal of Control, vol. 14, no. 4, pp. 1070–1083, 2012.
[21]
X. Ruan, Z. Z. Bien, and Q. Wang, “Convergence properties of iterative learning control processes in the sense of the Lebesgue- norm,” Asian Journal of Control, vol. 14, no. 4, pp. 1095–1107, 2012.
[22]
X.-H. Bu, Z.-S. Hou, F.-S. Yu, and Z.-Y. Fu, “Iterative learning control for a class of non-linear switched systems,” IET Control Theory & Applications, vol. 7, no. 3, pp. 470–481, 2013.
[23]
H. Sun, Z. Hou, and D. Li, “Coordinated iterative learning control schemes for train trajectory tracking with overspeed protection,” IEEE Transactions on Automation Science and Engineering, vol. 10, no. 2, pp. 323–333, 2013.
[24]
S.-P. Yang, J.-X. Xu, D.-Q. Huang, and Y. Tan, “Optimal iterative learning control design for multi-agent systems consensus tracking,” Systems & Control Letters, vol. 69, pp. 80–89, 2014.
[25]
B. Kim, T. Lee, Y. Kim, and H. Ahn, “Iterative learning control for spatially interconnected systems,” Applied Mathematics and Computation, vol. 237, pp. 438–445, 2014.
[26]
Y. Song, Y. Li, X. Wang, X. Ma, and J. H. Ruan, “An improved reinforcement learning algorithm for cooperative behaviors of mobile robots,” Journal of Control Science and Engineering, vol. 2014, Article ID 270548, 8 pages, 2014.
[27]
J. Peng and Y. Liu, “Adaptive robust quadratic stabilization tracking control for robotic system with uncertainties and external disturbances,” Journal of Control Science and Engineering, vol. 2014, Article ID 715250, 10 pages, 2014.
[28]
H. Wu, “Decentralised adaptive robust control of uncertain large-scale non-linear dynamical systems with time-varying delays,” IET Control Theory & Applications, vol. 6, no. 5, pp. 629–640, 2012.
[29]
A. R. Fioravanti, C. Bonnet, and S. Niculescu, “A numerical method for stability windows and unstable root-locus calculation for linear fractional time-delay systems,” Automatica, vol. 48, no. 11, pp. 2824–2830, 2012.
[30]
C.-Y. Kao, “On stability of discrete-time LTI systems with varying time delays,” IEEE Transactions on Automatic Control, vol. 57, no. 5, pp. 1243–1248, 2012.
[31]
H.-B. Zhang, H. Zhong, and C.-Y. Dang, “Delay-dependent decentralized filtering for discrete-time nonlinear interconnected systems with time-varying delay based on the T-S fuzzy model,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 3, pp. 431–443, 2012.
[32]
J. Doyle, “Analysis of feedback systems with structured uncertainties,” Control Theory and Applications, vol. 129, no. 6, pp. 242–250, 1982.
[33]
X. Jin and J. Xu, “Iterative learning control for output-constrained systems with both parametric and nonparametric uncertainties,” Automatica, vol. 49, no. 8, pp. 2508–2516, 2013.
[34]
B. Xu, X. Huang, D. Wang, and F. Sun, “Dynamic surface control of constrained hypersonic flight models with parameter estimation and actuator compensation,” Asian Journal of Control, vol. 16, no. 1, pp. 162–174, 2014.
[35]
D. Meng and Y. Jia, “Anticipatory approach to design robust iterative learning control for uncertain time-delay systems,” Asian Journal of Control, vol. 13, no. 1, pp. 38–53, 2011.
[36]
F. Le, I. Markovsky, C. T. Freeman, and E. Rogers, “Identification of electrically stimulated muscle models of stroke patients,” Control Engineering Practice, vol. 18, no. 4, pp. 396–407, 2010.
[37]
S. Toshiharu, “Identification of linear continuous-time systems based on iterative learning control,” in Recent Advances in Learning and Control, pp. 205–218, 2008.
[38]
F. Atsushi and O. Shinsuke, “Parameter identification of continuous-time systems using iterative learning control,” International Journal of Control, Automation and Systems, vol. 9, no. 2, pp. 203–210, 2011.
[39]
M. C. Campi, T. Sugie, and F. Sakai, “An iterative identification method for linear continuous-time systems,” IEEE Transactions on Automatic Control, vol. 53, no. 7, pp. 1661–1669, 2008.
[40]
L. Liu, K. K. Tan, and T. H. Lee, “SVD-based accurate identification and compensation of the coupling hysteresis and creep dynamics in piezoelectric actuators,” Asian Journal of Control, vol. 16, no. 1, pp. 59–69, 2014.
[41]
F. Sakai and T. Sugie, “An identification method for MIMO continuous-time systems via iterative learning control concepts,” Asian Journal of Control, vol. 13, no. 1, pp. 64–74, 2011.
[42]
J. Bendtsen and K. Trangbaek, “Closed-loop identification for control of linear parameter varying systems,” Asian Journal of Control, vol. 16, no. 1, pp. 40–49, 2014.
[43]
Z.-S. Hou and S.-T. Jin, “A novel data-driven control approach for a class of discrete-time nonlinear systems,” IEEE Transactions on Control Systems Technology, vol. 19, no. 6, pp. 1549–1558, 2011.
[44]
T.-Y. Doh, J. R. Ryoo, and D. E. Chang, “Robust iterative learning controller design using the performance weighting function of feedback control systems,” International Journal of Control, Automation and Systems, vol. 12, no. 1, pp. 63–70, 2014.
[45]
O. Gaye, L. Autrique, Y. Orlov, E. Moulay, S. Brémond, and R. Nouailletas, “ stabilization of the current profile in tokamak plasmas via an LMI approach,” Automatica, vol. 49, no. 9, pp. 2795–2804, 2013.
[46]
X.-Z. Jin, G.-H. Yang, X.-H. Chang, and W.-W. Che, “Robust faulttolerant control with adaptive compensation,” Acta Automatica Sinca, vol. 39, no. 1, pp. 31–42, 2013.
[47]
Y.-Q. Ye, D.-W. Wang, B. Zhang, and Y.-G. Wang, “Simple LMI based learning control design,” Asian Journal of Control, vol. 11, no. 1, pp. 74–77, 2009.
[48]
D. Meng, Y. Jia, J. Du, and F. Yu, “Delay-dependent conditions for monotonic convergence of uncertain ILC systems: an LMI approach,” in Proceedings of the 49th IEEE Conference on Decision and Control (CDC '10), pp. 6955–6960, Atlanta, Ga, USA, December 2010.
[49]
L. Zhai, G.-H. Tian, F.-Y. Zhou, and Y. Li, “The robust iterative learning control of networked control systems with varying references,” in Proceeding of the 25th Chinese Control and Decision Conference (CCDC '13), pp. 19–24, Guiyang, China, May 2013.
[50]
L. Zhai, G. Tian, F. Zhou, and Y. Li, “A frequency analysis of time delayed iterative learning control system,” in Proceedings of the 32nd Chinese Control Conference (CCC '13), pp. 256–261, Xi’an, China, July 2013.
[51]
M. Wu, Y. He, and J.-H. She, Stability Analysis and Robust Control of Time-Delay Systems, Springer, Berlin, Germany, 2010.
[52]
M. Wu, Y. He, J.-H. She, and G.-P. Liu, “Delay-dependent criteria for robust stability of time-varying delay systems,” Automatica, vol. 40, no. 8, pp. 1435–1439, 2004.
