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An adaptive modeling method for time-varying distributed parameter processes with curing process applications

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Abstract

Curing processes are nonlinear distributed parameter systems (DPS) with time-varying spatiotemporal dynamics. However, existing data-driven modeling methods have only considered time-varying dynamics in the time direction and paid less attention to those in the spatial direction. This has led to poor modeling accuracy for nonlinear DPS with time-varying spatiotemporal dynamics. In this paper, we propose an adaptive modeling method to estimate the distribution model for this kind of DPS. An adaptive time/space separation is first developed to decompose the time/space coupling dynamics. Time-varying spatial basis functions are then constructed, which can represent time-varying dynamics in the spatial direction. An adaptive T–S fuzzy modeling method is further developed for online learning of unknown dynamics derived from the data. This modeling can adapt to real-time spatiotemporal variation after the time/space synthesis since it utilizes time-varying spatiotemporal dynamics. Finally, curing experiments successfully test and demonstrate the effectiveness of the proposed method.

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References

  1. Jiang, M., Deng, H.: Optimal combination of spatial basis functions for the model reduction of nonlinear distributed parameter systems. Commun. Nonlinear Sci. Numer. Simul. 17(12), 5240–5248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Deng, H., Li, H.X.: Spectral-approximation-based intelligent modeling for distributed thermal processes. IEEE Trans. Control Syst. Technol. 13(5), 686–700 (2005)

    Article  Google Scholar 

  3. Li, H.X., Deng, H., Zhong, J.: Model-based integration of control and supervision for one kind of curing process. IEEE Trans. Electron. Packag. Manuf. 27(3), 117–186 (2004)

    Article  Google Scholar 

  4. Banerjee, S., Cole, J.V., Jensen, K.F.: Nonlinear model reduction strategies for rapid thermal processing systems. IEEE Trans. Semicond. Manuf. 11(2), 266–275 (1998)

    Article  Google Scholar 

  5. Hisung, J.C., Pearson, R.A.: Processing diagrams for polymeric die attach adhesives. In Proceedings \(47{{\rm th}}\) IEEE Electronic Components Technology Conference, pp. 536–543 (1997)

  6. Aling, H., Banerjee, S., Bangia, A.K., Cole, V., Ebert, J., Emani-Naeini, A., Jensen, K.F., Kevrekidis, I.G., Shvartsman, S.: Nonlinear model reduction for simulation and control of rapid thermal processing. In: Proceedings of the 1977 American Control Conference, Albuquerque, New Mexico, pp. 2233–2238 (1997)

  7. Perkins, R.H., Riley, Terrence J., Gyurcsik, R.S.: Thermal uniformity and stress minimization during rapid thermal processes. IEEE Trans. Semicond. Manuf. 8(3), 272–279 (1995)

    Article  Google Scholar 

  8. Butkovskij, A.G.: Optimal Control of Distributed Parameter Systems. Nauka, Moscow (1965). (in Russian)

    Google Scholar 

  9. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  10. Ray, W.H.: Advanced Process Control. McGraw-Hill, New York (1981)

    Google Scholar 

  11. Aggelogiannaki, E., Sarimveis, H.: Nonlinear model predictive control for distributed parameter systems using data driven artificial neural network models. Comput. Chem. Eng. 32(6), 1225–1237 (2008)

    Article  Google Scholar 

  12. Deng, H., Li, H.X.: Hybrid intelligence based modeling for nonlinear distributed parameter process with applications to the curing process. IEEE Int. Conf. Syst. Man Cybern. 4, 3506–3511 (2003)

    Google Scholar 

  13. Park, H.M., Cho, D.H.: The use of the Karhunen–Loeve decomposition for the modeling of distributed parameter systems. Chem. Eng. Sci. 51, 81–98 (1996)

    Article  Google Scholar 

  14. Wang, J.W., Li, H.X., Wu, H.N.: Distributed proportional plus second-order spatial derivative control for distributed parameter systems subject to spatiotemporal uncertainties. Nonlinear Dyn. 76, 2041–2058 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wu, C.J.: Nonlinear Galerkin optimal truncated low-dimensional dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 1(1), 36–42 (1996)

    Article  Google Scholar 

  16. Ding, Q., Zhang, K.P.: Order reduction and nonlinear behaviors of a continuous rotor system. Nonlinear Dyn. 67, 251–262 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Soderstrom, T., Stoica, P.: System Identification. Prentice-Hall, London (1989)

    MATH  Google Scholar 

  18. Guo, L.Z., Billings, S.A.: Sate-space reconstruction and spatiotemporal prediction of lattice dynamical systems. IEEE Trans. Autom. Control 52(4), 622–632 (2007)

    Article  MathSciNet  Google Scholar 

  19. Coca, D., Billings, S.A.: Identification of finite dimensional models of infinite dimensional dynamical systems. Automatica 38(11), 1851–1865 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lacarbonara, W., Nayfeh, A.H., Kreider, W.: Experimental validation of reduction methods for nonlinear vibrations of distributed-parameter systems: analysis of a buckled beam. Nonlinear Dyn. 17, 95–117 (1998)

    Article  MATH  Google Scholar 

  21. Nayfeh, A., Lacarbonara, W.: On the discretization of distributed-parameter systems with quadratic and cubic nonlinearities. Nonlinear Dyn. 13, 203–220 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Qi, C.K., Li, H.X.: Kernel-based spatiotemporal multimodeling for nonlinear distributed parameter industrial processes. Ind. Eng. Chem. Res. 51, 13205–13218 (2012)

  23. Christofides, P.D.: Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-reaction Processes. Birkha User, Boston (2001)

    Book  MATH  Google Scholar 

  24. Shvartsman, S.Y., Theodoropoulos, C., Rico-Martinez, R., Kevrekidis, I.G., Titi, E.S., Mountziaris, T.J.: Order reduction for nonlinear dynamic models of distributed reacting systems. J. Process Control 10, 177–184 (2000)

    Article  Google Scholar 

  25. Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1993)

    MATH  Google Scholar 

  26. Zhang, J.Z., Liu, Y., Feng, P.H.: Approximate inertial manifolds of burgers equation approached by nonlinear Galerkin’s procedure and its application. Commun. Nonlinear Sci. Numer. Simul. 16(12), 4666–4670 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Qi, C.K.: Modeling of nonlinear distributed parameter system for industrial thermal processes. Ph.D. thesis, City University of Hong Kong (2009)

  28. Fernandez-Berdaguer, E.M., Santos, J.E., Sheen, D.: An iterative procedure for estimation of variable coefficients in a hyperbolic system. Appl. Math. Comput. 76(2–3), 213–250 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Müller, T.G., Timmer, J.: Parameter identification techniques for partial differential equations. Int. J. Bifurc. Chaos 14(6), 2053–2060 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ding, L., Gustafsson, T., Johansson, A.: Model parameter estimation of simplified linear models for a continuous paper pulp digester. J. Process Control 17(2), 115–127 (2007)

    Article  Google Scholar 

  31. Sirovich, L.: New Perspectives in Turbulence, 1st edn. Springer, New York (1991)

    Book  MATH  Google Scholar 

  32. Gay, D.H., Ray, W.H.: Identification and control of distributed parameter systems by means of the singular value decomposition. Chem. Eng. Sci. 50(10), 1519–1539 (1995)

  33. Coca, D., Billings, S.A.: Identification of finite dimensional models of infinite dimensional dynamical systems. Automatica 38(11), 1851–1865 (2002)

  34. Liu, Z., Li, H.X.: A spatiotemporal estimation method for temperature distributed in lithium ion batteries. IEEE Trans. Ind. Inform. (2014). doi:10.1109/YII.2014.2341955

  35. Qi, C.K., Li, H.X.: Hybrid Karhunen-Loève/neural modeling for a class of distributed parameter systems. Int. J. Intell. Syst. Technol. Appl. 4(1–2), 141–160 (2008)

    Google Scholar 

  36. Ghazal, M., Mohammad, J.Y.: Predictive control of uncertain nonlinear parabolic PDE systems using a Galerkin/neural-network-based model. Commun. Nonlinear Sci. Numer. Simul. 1, 388–404 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Li, H.X., Qi, C.K., Yu, Y.G.: A Spatio-temporal Volterra modeling approach for a class of nonlinear distributed parameter processes. J. Process Control 9(7), 1126–1142 (2009)

    Article  Google Scholar 

  38. Qi, C.K., Li, H.X.: A Time/space separation based Hammerstein modeling approach for nonlinear distributed parameter processes. Comput. Chem. Eng. 33(7), 1247–1260 (2009)

    Article  Google Scholar 

  39. Qi, C.K., Li, H.X.: A Karhunen-Loève decomposition based Wiener modeling approach for nonlinear distributed parameter processes. Ind. Eng. Chem. Res. 47(12), 4184–4192 (2008)

    Article  Google Scholar 

  40. Qi, C.K., Li, H.X.: A LS-SVM modeling approach for nonlinear distributed parameter processes. In: Proceedings of the 7th World Congress on Intelligent Control and Automation, Chongqing, China, June 25–27, pp. 569–574 (2008)

  41. Takagi, T., Sugeno, M.: Fuzzy identification of systems and its applications to modeling and control. IEEE Trans. Syst. Man Cybern. SMC–15(1), 116–132 (1985)

    Article  MATH  Google Scholar 

  42. Wang, Y.H., Fan, Y.Q., Wang, Q.Y., Zhang, Y.: Adaptive fuzzy synchronization for a class of chaotic systems with unknown nonlinearities and disturbances. Nonlinear Dyn. 69, 1167–1176 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Chapman, Alan J.: Fundamentals of Heat Transfer. Macmillan Publishing Company, New York (1987)

    Google Scholar 

  44. Tanaka, K., Wang, H.O.: Fuzzy Control Systems Design and Analysis. Wiley, New York (2001)

    Book  Google Scholar 

  45. Holme, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures. Dynamical Systems, and Symmetry. Cambridge University Press, New York (1996)

    Book  MATH  Google Scholar 

  46. Hao, Y.: Fuzzy Control and Modeling: Analytical Foundations and Applications, 1st edn. Wiley, New York (2000)

    Google Scholar 

  47. Qi, C.K., Li, H.X.: Nonlinear dimension reduction based neural modeling for distributed parameter processes. Chem. Eng. Sci. 64(19), 4164–4170 (2009)

    Article  Google Scholar 

Download references

Acknowledgments

This project is partially supported by the National Basic Research Program (973) of China (2011CB706802), National Natural Science Foundation of China (51205420), Program for New Century Excellent Talents in University (NCET-13-0593), and Hunan Provincial Natural Science Foundation of China (14JJ3011).

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  1. State Key Laboratory of High Performance Complex Manufacturing, School of Mechanical and Electrical Engineering, Central South University, Changsha, 410083, Hunan, China

    XinJiang Lu, Wei Zou & MingHui Huang

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  1. XinJiang Lu
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  2. Wei Zou
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Correspondence to XinJiang Lu.

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Lu, X., Zou, W. & Huang, M. An adaptive modeling method for time-varying distributed parameter processes with curing process applications. Nonlinear Dyn 82, 865–876 (2015). https://doi.org/10.1007/s11071-015-2201-3

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