Abstract
This paper deals with the problem of iterative learning control algorithm for a class of mixed distributed parameter systems. Here, the considered distributed parameter systems are composed of mixed hyperbolic-parabolic partial differential equations. The domain of the system is divided into two parts in which the system is hyperbolic and parabolic, respectively, with transmission conditions at the interface. According to the characteristics of the systems, iterative learning control laws are proposed for such mixed hyperbolic-parabolic distributed parameter systems based on P-type learning scheme. Using the contraction mapping method, it is shown that the scheme can guarantee the output tracking errors on L 2 space converge along the iteration axis. A simulation example illustrates the effectiveness of the proposed method.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
S. Arimoto, S. Kawamura, and F. Miyazaki, “Bettering operation of robots by learning,” Journal of Robotic Systems, vol. 1, no. 2, pp.123–140, Summer 1984. [click]
J. X. Xu, “Analysis iterative learning control for a class of nonlinear discrete-time systems,” Automatica, vol. 33, no. 10, pp. 1905–1907, October 1997. [click]
D. H. Owens, “Multivariable norm optimal and parameter optimal iterative learning control: a unified formulation,” International Journal of Control, vol. 85, no. 8, pp. 1010–1025, August 2012. [click]
M. Sun and D. Wang. “Sampled-data iterative learning control for nonlinear systems with arbitrary relative degree,” Automatica, vol. 37, no. 2, pp. 283–289, February 2001. [click]
M. X. Sun, D. W. Wang, and Y. Y. Wang, “Varying-order iterative learning control against perturbed initial conditions,” Journal of The Franklin Institute, vol. 347, no. 8, pp. 1526–1549, October 2010.
D. A. Bristow, M. Tharayil, and A. G. Alleyne, “A survey of iterative learning control: A learning-method for highperformance tracking control,” IEEE Control Syst.Mag., vol. 26, no. 3, pp. 96–114, June 2006. [click]
G. Bastin and J. M. Coron, “On boundary feedback stabilization of non-uniform linear 2×2 hyperbolic systems over a bounded interval,” Systems Control Letters, vol. 60, no. 11, pp. 900–906, November 2011. [click]
S. X. Tang and C. K. Xie, “State and output feedback boundary control for a coupled PDE-ODE system,” Systems Control Letters, vol. 60, no. 8, pp. 540–545, August 2011. [click]
R. Vazquez and M. Krstic, “A closed-form feedback controller for stabilization of the linearized 2-D Navier-Stokes poiseuille system,” IEEE Transactions on Automatic Control, vol. 52, no. 12, pp. 2298–2312, December 2007.
Z. H. Qu, “An iterative learning algorithm for boundary control of a stretched moving string,” Automatica, vol. 38, no. 5, pp. 821–827, May 2002. [click]
D. Q. Huang, J. X. Xu, X. F. Li, C. Xu, and M. Yu, “Dtype anticipatory iterative learning control for a class of inhomogeneous heat equations,” Automatica, vol. 49, no. 8, pp. 2397–2408, August 2013. [click]
D. Q. Huang, X. F. Li, J. X. Xu, C. Xu, and W. He, “Iterative learning control of inhomogeneous distributed parameter systems-frequency domain design and analysis,” Systems Control Letters, vol. 72, pp. 22–29, October 2014. [click]
Y. V. Orlov, “Discontinuous unit feedback control of uncertain infinite-dimensional systems,” IEEE Transactions on Automatic Control, vol. 45, no. 5, pp. 834–843, May 2000.
A. Pisano, Y. Orlov, and E. Usai, “Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques,” SIAM J. Control Optim, vol. 49, no. 2, pp. 363–382, March 2011. [click]
X. J. Fan, S. P Tian, and H. P. Tian, “Iterative learning control of distributed parameter system based on geometric analysis,” Proc. of the Eighth International Conference on Machine Learning and Cybernetics, pp. 3673–3677, 2009.
J. L. Kang, “A Newton-type iterative learning algorithm of output tracking control for uncertain nonlinear distributed parameter systems,” Proc. of the 33rd Chinese Control Conference, pp. 8901–8905, 2014.
X. S. Dai and S. P. Tian, “Iterative learning control for distributed parameter systems with time-delay,” Proc. of 2011 Chinese Control and Decision Conference, pp. 2304–2307,2011.
C. Xu, R. Arastoo, and E. Schuster, “On Iterative learning control of parabolic distributed parameter systems,” Proc. of 17 th Mediterranean Conference on Control and Automation, pp. 510–515, 2009.
J. Choi, B. J. Seo, and K. S. Lee, “Constrained digital regulation of hyperbolic PDE: a learning control approach,” Korean Journal of Chemical Engineering, vol. 18, no. 5, pp. 606–611, September 2001. [click]
Q. Fu, “Iterative learning control for second order nonlinear hyperbolic distributed parameter systems,” Journal of Systems Science and Mathematical Sciences (in Chinese), vol. 34, no. 3, pp. 284–293, March 2014.
Q. Fu, “Iterative learning control for irregular distributed parameter systems,” Control and Decision (in Chinese), vol. 31, no. 1, pp. 114–122, January 2016.
G. P. Boswell, H. Jacobs, F. A. Davidson, G. M. Gadd, and K. Ritz, “A positive numerical scheme for a mix-type partial differential equation model for fungal growth,” Applied Mathematics and Computation, vol. 138, no. 2,3, pp. 321–340, June 2003. [click]
R. K. Kenneyd and D. Lee, “A finite-difference solution to a mix-type partial differential equation: an ocean dynamic motion model,” Mathl Comput.Modelling, vol. 10, no. 2, pp. 75–85, February 1988.
O. Terlyga, H. Bellout, and F. Bloom, “Global existence, uniqueness, and stability for a nonlinear hyperbolicparabolic problem in pulse combustion,” Acta Mathematica Scientia, vol. 32B, no. 1, pp. 41–74, January 2012.
M. Raoofi and K. Zumbrun, “Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems,” Journal of Differential equations, vol. 246, no. 4, pp. 1539–1567, February 2009. [click]
T. Nguyen and K. Zumbrun, “Long time stability of large-amplitude noncharacteristic boundary layers for hyperbolic-parabolic systems,” J. Math. Pures Appl., vol. 92, no. 6, pp. 547–598, December 2009. [click]
A. Ashyralyev and H. A. Yurtsever, “A note on the second order of accuracy difference schemes for hyperbolicparabolic equations,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 517–537, July 2005.
A. Ashyralyev and Y. Ozdemir, “Stability of difference schemes for hyperbolic-parabolic equations,” Computers and Mathematics with Applications, vol. 50, no. 8–9, pp. 1443–1476, October-November 2005. [click]
A. Ashyralyev and Y. Ozdemir, “On numerical solutions for hyperbolic-parabolic equations with multipoint nonlocal boundary condition,” Journal of The Franklin Institute, vol. 351, no. 2, pp. 602–630, February 2014. [click]
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Associate Editor Young Ik Son under the direction of Editor Yoshito Ohta. This work was supported by the National Natural Science Foundation of China (No. 11371013). The authors would like to express their gratitude to the editor and the anonymous referees for their valuable suggestions that have greatly improved the quality of the paper.
Qin Fu received his Ph.D degree in Control Theory and Control Engineering from Nanjing University of Science and Technology, China in 2009. He is currently an associate professor of Suzhou University of Science and Technology. His research interests include decentralized control, robust control, and iterative learning control.
Wei-Guo Gu received his M.S. degree in fundamental mathematics from Suzhou University of Science and Technology, China in 2015. He is currently a computer programmer. His research interests include iterative learning control and feed-back control.
Pan-Pan Gu is currently pursuing a M.S. in School of Mathematics and Physics, Suzhou University of Science and Technology, China. His research interest is iterative learning control.
Jian-Rong Wu received his B.S. degree in Mathematics from Nanjing Normal University, China in 1985, and his M.S. and Ph.D degrees in Mathematics from Harbin Institue Technology, China, in 1988 and 2000, respectively. He is currently a professor of Suzhou University of Science and Technology. His research interests are in the areas of fuzzy systems and singular systems etc.
Rights and permissions
About this article
Cite this article
Fu, Q., Gu, WG., Gu, PP. et al. Iterative learning control for a class of mixed hyperbolic-parabolic distributed parameter systems. Int. J. Control Autom. Syst. 14, 1455–1463 (2016). https://doi.org/10.1007/s12555-015-0256-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-015-0256-z