Advertisement

Nonlinear Dynamics

, Volume 95, Issue 1, pp 73–86 | Cite as

System identification of distributed parameter system with recurrent trajectory via deterministic learning and interpolation

  • Xunde DongEmail author
  • Cong Wang
  • Qigui Yang
  • Wenjie Si
Original Paper
  • 202 Downloads

Abstract

In the paper, we propose a novel approach to identify distributed parameter system (DPS) with recurrent state trajectory. Different from existing literature, the system dynamics rather than parameters or structure of DPS is identified in the study. Due to the infinite-dimensional feature of DPS, the partial differential equation describing the DPS is first approximated by a set of ordinary differential equations. By employing finite difference method, the spatial derivatives at a set of spatial points are approximated. Then, the DPS dynamics at the set of spatial points is identified via deterministic learning. With the identification results, a mechanism based on interpolation method is proposed to approximate the DPS dynamics at any other spatial point. That is, we can accurately identify the DPS dynamics at any spatial point. Numerical results involving the identification of an important mathematical physics equation are presented to illustrate the validity of the approach.

Keywords

System identification System dynamics Deterministic learning Radial basis function neural network 

Notes

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Ucinski, D.: Optimal Measurement Methods for Distributed Parameter System Identification. CRC Press, Boca Raton (2004)zbMATHCrossRefGoogle Scholar
  2. 2.
    Schlacher, K., Schöberl, M.: Modelling, analysis and control of distributed parameter systems. Math. Comput. Model. Dyn. Syst. 17(1), 1–2 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Gonzalez-Garcia, R., Rico-Martinez, R., Wolf, W., Lubke, M., Eiswirth, M., Anderson, J.S., Kevrekidis, I.G.: Characterization of a two-parameter mixed-mode electrochemical behavior regime using neural networks. Phys. D Nonlinear Phenom. 151(1), 27–43 (2001)zbMATHCrossRefGoogle Scholar
  4. 4.
    Krischer, K., Rico-Martinez, R., Kevrekidis, I., Rotermund, H., Ertl, G., Hudson, J.: Model identification of a spatiotemporally varying catalytic reaction. AIChE J. 39(1), 89–98 (1993)CrossRefGoogle Scholar
  5. 5.
    Li, H.X., Qi, C.: Modeling of distributed parameter systems for applications synthesized review from time–space separation. J Process Control 20(8), 891–901 (2010)CrossRefGoogle Scholar
  6. 6.
    Coca, D., Billings, S.A.: Direct parameter identification of distributed parameter systems. Int. J. Syst. Sci. 31(1), 11–17 (2000)zbMATHCrossRefGoogle Scholar
  7. 7.
    Banks, H.T., Kunisch, K.: Estimation Techniques for Distributed Parameter Systems. Springer, Berlin (2012)zbMATHGoogle Scholar
  8. 8.
    Guo, L., Billings, S.A., Coca, D.: Identification of partial differential equation models for a class of multiscale spatio-temporal dynamical systems. Int. J. Control 83(1), 40–48 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cheng, J., Nakagawa, J., Yamamoto, M., Yamazaki, T.: Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse probl. 25(11), 115002 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Garvie, M.R., Trenchea, C.: Identification of space-time distributed parameters in the Gierer–Meinhardt reaction–diffusion system. SIAM J. Appl. Math. 74(1), 147–166 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Wang, Q., Feng, D., Cheng, D.: Parameter identification for a class of abstract nonlinear parabolic distributed parameter systems. Comput. Math. Appl. 48(12), 1847–1861 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Fang, H., Wang, J., Feng, E., Li, Z.: Parameter identification and application of a distributed parameter coupled system with a movable inner boundary. Comput. Math. Appl. 62(11), 4015–4020 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Li, L., Liu, W., Han, B.: Dynamical level set method for parameter identification of nonlinear parabolic distributed parameter systems. Commun. Nonlinear Sci. Numer. Simul. 17(7), 2752–2765 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Li, H.X., Qi, C.: Spatio-Temporal Modeling of Nonlinear Distributed Parameter Systems: A Time/space Separation Based Approach, vol. 50. Springer, Berlin (2011)zbMATHCrossRefGoogle Scholar
  15. 15.
    Gonzalez-Garcia, R., Rico-Martinez, R., Kevrekidis, I.: Identification of distributed parameter systems: a neural net based approach. Comput. Chem. Eng. 22, S965–S968 (1998)CrossRefGoogle Scholar
  16. 16.
    Guo, L., Billings, S.A.: Identification of partial differential equation models for continuous spatio-temporal dynamical systems. IEEE Trans. Circuits Syst. II Express Br. 53(8), 657–661 (2006)CrossRefGoogle Scholar
  17. 17.
    Li, H.X., Qi, C., Yu, Y.: A spatio-temporal volterra modeling approach for a class of distributed industrial processes. J. Process Control 19(7), 1126–1142 (2009)CrossRefGoogle Scholar
  18. 18.
    Rannacher, R., Vexler, B.: A priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements. SIAM J. Control Optim. 44(5), 1844–1863 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Wouwer, A.V., Renotte, C., Queinnec, I., Bogaerts, P.: Transient analysis of a wastewater treatment biofilter–distributed parameter modelling and state estimation. Math. Comput. Modell. Dyn. Syst. 12(5), 423–440 (2006)zbMATHCrossRefGoogle Scholar
  20. 20.
    Orlov, Y., Bentsman, J.: Adaptive distributed parameter systems identification with enforceable identifiability conditions and reduced-order spatial differentiation. IEEE Trans. Autom. Control 45(2), 203–216 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Chen, W., Wen, C., Hua, S., Sun, C.: Distributed cooperative adaptive identification and control for a group of continuous-time systems with a cooperative pe condition via consensus. IEEE Trans. Autom. Control 59(1), 91–106 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Chang, J.: Identification of variable coefficients for vibrating systems by boundary control and observation. J. Control Theory Appl. 6(2), 127–132 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Hong, K.S., Bentsman, J.: Direct adaptive control of parabolic systems: algorithm synthesis and convergence and stability analysis. IEEE Trans. Autom. Control 39(10), 2018–2033 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Demetriou, M., Rosen, I.: On the persistence of excitation in the adaptive estimation of distributed parameter systems. IEEE Trans. Autom. Control 39(5), 1117–1123 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Wang, C., Hill, D.J.: Learning from neural control. IEEE Trans. Neural Netw. 17(1), 130–146 (2006)CrossRefGoogle Scholar
  26. 26.
    Wang, C., Hill, D.J.: Deterministic learning and rapid dynamical pattern recognition. IEEE Trans. Neural Netw. 18(3), 617–630 (2007)CrossRefGoogle Scholar
  27. 27.
    Wang, C., Hill, D.J.: Deterministic Learning Theory for Identification, Recognition, and Control. CRC Press, Boca Raton (2009)Google Scholar
  28. 28.
    Shil’nikov, L.P.: Methods of Qualitative Theory in Nonlinear Dynamics, Part I, vol. 5. World Scientific, Singapore (2001)CrossRefGoogle Scholar
  29. 29.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112. Springer, Berlin (2013)Google Scholar
  30. 30.
    Ralston, A., Rabinowitz, P.: A First Course in Numerical Analysis. Courier Corporation, Chelmsford (2012)zbMATHGoogle Scholar
  31. 31.
    Lam, N.S.N.: Spatial interpolation methods: a review. Am. Cartogr. 10(2), 129–150 (1983)CrossRefGoogle Scholar
  32. 32.
    Caruso, C., Quarta, F.: Interpolation methods comparison. Comput. Math. Appl. 35(12), 109–126 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM, Philadelphia (2007)zbMATHCrossRefGoogle Scholar
  34. 34.
    Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, vol. 37. Springer, Berlin (2010)zbMATHGoogle Scholar
  35. 35.
    Zhou, G., Wang, C.: Deterministic learning from control of nonlinear systems with disturbances. Prog. Nat. Sci. 19(8), 1011–1019 (2009)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Hohmann, A., Deuflhard, P.: Numerical Analysis in Modern Scientific Computing: An Introduction, vol. 43. Springer, Berlin (2012)zbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Automation Science and EngineeringSouth China University of TechnologyGuangzhouChina
  2. 2.School of MathematicaSouth China University of TechnologyGuangzhouChina

Personalised recommendations