Advertisement

Nonlinear Dynamics

, Volume 77, Issue 3, pp 935–949 | Cite as

A linear approach to generalized minimum variance controller design for MIMO nonlinear systems

  • Yousef AlipouriEmail author
  • Javad Poshtan
Original Paper

Abstract

Designing minimum variance controllers (MVC) for nonlinear systems is confronted with many difficulties. The methods which are able to identify MIMO nonlinear systems are scarce, and linear models are not accurate in modeling nonlinear systems. In this paper, Vector ARX (VARX) models are proposed for designing MVC and generalized minimum variance controller (GMVC) for linear and nonlinear systems, and the accuracy of these models in approximating the nonlinear MIMO system is studied. However, the VARX is a linear model. It is shown that this model can identify some kinds of nonlinear systems with any desired accuracy. Therefore, the controller designed by the VARX is accurate, even for these nonlinear systems. The proposed controller is tested on a both linear system and a nonlinear four-tank benchmark process. In spite of the simplicity of designing GMVCs for the VARX models, the results show that the proposed method is accurate and implementable.

Keywords

Generalized minimum variance controller VARX model  MIMO nonlinear system Discrete Taylor series Four-tank benchmark system Interactor matrix method 

References

  1. 1.
    Seborg, D.E., Edgar, T.F., Mellichamp, D.A.: Process Dynamics and Control. Wiley, New York (2004)Google Scholar
  2. 2.
    Shinskey, F.G.: Process-Control Systems: Application, Design, and Tuning. McGraw Hill, New York (1996)Google Scholar
  3. 3.
    Bialkowski, W.L.: Dreams versus reality: a view from both sides of the gap. Pulp & Paper Canada 94, 19–27 (1993)Google Scholar
  4. 4.
    Jelali, M.: Control System Performance Monitoring Assessment, Diagnosis and Improvement of Control Loop Performance in Industrial Automation. Springer, London (2010)Google Scholar
  5. 5.
    Astrom, K.J.: Introduction to Stochastic Control Theory. Academic Press, New York (1970)Google Scholar
  6. 6.
    Martensson, J., Rojas, C.R., Hjalmarsson, H.: Conditions when minimum variance control is the optimal experiment for identifying a minimum variance controller. Automatica 47, 578–583 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    DeVries, W., Wu, S.: Evaluation of process control effectiveness and diagnosis of variation in paper basis weight via multivariate time-series analysis. IEEE Trans. Automat. Control 23, 702–708 (1978)CrossRefGoogle Scholar
  8. 8.
    Huang, B., Shah, S.L.: Practical issues in multivariable feedback control performance assessment, In: Proc IFAC ADCHEM, Banff, Canada pp. 429–434 (1997)Google Scholar
  9. 9.
    Huang, B., Shah, S.L.: Performance Assessment of Control Loops. Springer, London (1999)CrossRefGoogle Scholar
  10. 10.
    Rogozinski, M., Paplinski, A., Gibbard, M.: An algorithm for calculation of nilpotent interactor matrix for linear multivariable systems. IEEE Trans. Automat. Control 32, 234–237 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Huang, B., Shah, S., Badmus, L., Vishnubhotla, A.: Control performance assessment: an enterprise asset management solution. www.matrikon.com/download/products/lit/processdoctor_pa_eam.pdf (1999)
  12. 12.
    Huang, B.: Multivariate Statistical Methods for Control Loop Performance Assessment, PhD Thesis, University of Alberta, Canada, (1997)Google Scholar
  13. 13.
    Huang, B., Ding, S.X., Thornhill, N.: Practical solutions to multivariate feedback control performance assessment problem: reduced a priori knowledge of interactor matrices. J. Process Control 15, 573–583 (2005)CrossRefGoogle Scholar
  14. 14.
    Kadali, R., Huang, B.: Multivariate controller performance assessment without interactor matrix—a subspace approach. IEEE Trans. Control Syst. Technol. 15(1), 65–74 (2007)CrossRefGoogle Scholar
  15. 15.
    Xia, H., Majecki, P., Ordys, A., Grimble, M.: Performance assessment of MIMO systems under partial information. Proceeding of the 2004 American Control Conference Boston, Massachusetts, (2004)Google Scholar
  16. 16.
    Florakis, K.A., Fassois, S.D., Hemez, F.M.: MIMO LMS.ARMAX identification of vibrating structures. Part II: a critical assessment. Mech. Syst. Signal Process. 15(4), 737–758 (2001)CrossRefGoogle Scholar
  17. 17.
    Filipovic, V.: Decentralized stochastic minimum variance controller. In: Proceedings of the 10th Mediterranean Conference on Control and Automation - MED2002 Lisbon, Portugal (2002)Google Scholar
  18. 18.
    Kolodziej, J.R., Mook, D.J.: Model determination for nonlinear state-based system identification. Nonlinear Dyn. 63(4), 735–753 (2011)CrossRefGoogle Scholar
  19. 19.
    Harris, T.J., Yu, W.: Controller assessment for a class of non-linear systems. J. Process Control 17, 607–619 (2007)CrossRefGoogle Scholar
  20. 20.
    Liu, X.J., Rosano, F.L.: Generalized minimum variance control of steam-boiler temperature using neuro-fuzzy approach. In: Proceedings of the 5’WorId Congress on Intelligent Control and Automation, Hangzhou, P.R. China, 15–19 June (2004)Google Scholar
  21. 21.
    Zhang, Z., Hu, L.S., Zhang, X.L.: Performance assessment of nonlinear control systems based on fuzzy modeling. In: IEEE International Conference on Fuzzy Systems, Jeju Island, South Korea, 20–24 August (2009)Google Scholar
  22. 22.
    Wen, G.X., Liu, Y.J.: Adaptive fuzzy-neural tracking control for uncertain nonlinear discrete-time systems in the NARMAX form. Nonlinear Dyn. 66(4), 745–753 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Horvath, C., Leeflang, P.S., Wieringa, J.E., Wittink, C.R.: Competitive reaction and feedback effects based VARX models of pooled store data. Int. J. Res. Mark. 22(4), 415–426 (2005)CrossRefGoogle Scholar
  24. 24.
    Hios, J.D., Fassois, S.D.: Identification of a global model describing the temperature effects on the dynamics of a smart composite beam. Proceedings of ISMA, Leuven, Belgium, paper ID 230, (2006)Google Scholar
  25. 25.
    Sakellariou, J.S., Fassois, S.D.: A functional pooling framework for the identification of systems under multiple operating conditions. In: Proceedings of the 15th Mediterranean Control Conference, Athens, Greece, (2007)Google Scholar
  26. 26.
    Chiuso, A.: The role of vector autoregressive modeling in predictor-based subspace identification. Automatica 43(6), 1034–1048 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Grimble, M.J.: Generalized minimum variance control law revisited. Optim Control Appl Methods 9(1), 63–77 (1988)Google Scholar
  28. 28.
    Ostermark, R.: Genetic hybrid tuning of VARMAX and state space algorithms. Soft Comput. 14, 91–99 (2010)CrossRefGoogle Scholar
  29. 29.
    Penm, J., Penm, J., Terrell, R.D.: The recursive fitting of subset VARX model. J. Time Ser. Anal. 14(6), 603–619 (1993)Google Scholar
  30. 30.
    Ljung, L.: System identification: theory for the user. Prentice Hall, New Jersey (1987)zbMATHGoogle Scholar
  31. 31.
    Johansson, K.H.: The quadruple-tank process: a multivariable laboratory process with an adjustable zero. IEEE Trans. Control Syst. Technol. 8(3), 456–465 (2000)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Gatzke, E.P., Meadows, E.S., Wang, C., Doyle, F.J.: Model based control of a four-tank system. Comput. Chem. Eng. 24(2–7), 1503–1509 (2000)CrossRefGoogle Scholar
  33. 33.
    Wang, Y., Wang, G.: Decoupling control based on dynamic surface control for MIMO nonlinear systems. In: Fourth IEEE International Conference on Information and Computing (2011)Google Scholar
  34. 34.
    Xu, N.S., Bai, Y.F., Zhang, L.: A novel high-order associative memory system via discrete Taylor series. IEEE Trans Neural Netw 14(4), 734–747 (2003). doi: 10.1109/TNN.2003.811700 Google Scholar
  35. 35.
    Nava, J., Kreinovich, V.: Equivalence of Gian-Carlo Rota Poset approach and Taylor aeries approach extended to variant ligands. J. Uncertain Syst. 5(2), 111–118 (2011)Google Scholar
  36. 36.
    Einar, H., Phillips, R.S.: Functional Analysis and Semi-Groups, vol. 31, pp. 300–327. AMS Colloquium Publications, Providence (1957) Google Scholar
  37. 37.
    Feller, W.: An Introduction to Probability Theory and its Applications, vol. 2, 3rd edn. Wiley, New York (1971)zbMATHGoogle Scholar
  38. 38.
    Xia, H., Majecki, P., Ordys, A., Grimble, M.J.: Performance assessment of MIMO systems based on I/O delay information. J. Process Control 16, 373–383 (2006)CrossRefGoogle Scholar
  39. 39.
    Alipouri, Y., Poshtan, J.: Designing robust minimum variance controller by discrete slide mode controller approach. ISA Transaction, accepted paper (2012)Google Scholar
  40. 40.
    Parikh, N.N., Patwardhan, S.C., Gudi, R.D.: Closed loop identification of quadruple tank system using an improved indirect approach. In: 8th IFAC Symposium on Advanced Control of Chemical Processes, Furama Riverfront, Singapore, pp. 10–13 July (2012)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Electrical Engineering DepartmentUniversity of Science and TechnologyTehranIran

Personalised recommendations