Advertisement

Application of Gaussian Processes to the Modelling and Control in Process Engineering

  • Juš Kocijan
  • Alexandra Grancharova
Chapter
Part of the Studies in Computational Intelligence book series (SCI, volume 561)

Abstract

Many engineering systems can be characterized as complex since they have a nonlinear behaviour incorporating a stochastic uncertainty. It has been shown that one of the most appropriate methods for modelling of such systems is based on the application of Gaussian processes (GPs). The GP models provide a probabilistic non-parametric modelling approach for black-box identification of nonlinear stochastic systems. This chapter reviews the methods for modelling and control of complex stochastic systems based on GP models. The GP-based modelling method is applied in a process engineering case study, which represents the dynamic modelling and control of a laboratory gas–liquid separator. The variables to be controlled are the pressure and the liquid level in the separator and the manipulated variables are the apertures of the valves for the gas flow and the liquid flow. GP models with different regressors and different covariance functions are obtained and evaluated. A selected GP model of the gas–liquid separator is further used to design an explicit stochastic model predictive controller to ensure the optimal control of the separator.

Keywords

Covariance Function Inverse Model Model Predictive Control Internal Model Control Gaussian Process Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Alessio, A., Bemporad, A.: A survey on explicit model predictive control. In: Magni, L., Raimondo, D.M., Allgöwer, F. (eds.) Nonlinear Model Predictive Control: Towards New Challenging Applications. Lecture Notes in Control and Information Sciences, vol. 384, pp. 345–369. Springer, Berlin (2009)CrossRefGoogle Scholar
  2. 2.
    Ažman, K., Kocijan, J.: Application of Gaussian processes for black-box modelling of biosystems. ISA Trans. 46, 443–457 (2007)CrossRefGoogle Scholar
  3. 3.
    Ažman, K., Kocijan, J.: Non-linear model predictive control for models with local information and uncertainties. Trans. Inst. Meas. Control 30(5), 371–396 (2008)Google Scholar
  4. 4.
    Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38, 3–20 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Cannon, M., Couchman, P., Kouvaritakis, B.: MPC for stochastic systems. In: Findeisen, R., Allgöwer, F., Biegler, L.T. (eds.) Assessment and Future Directions of Nonlinear Model Predictive Control. Lecture Notes in Control and Information Sciences, vol. 358, pp. 255–268. Springer, Berlin (2007)CrossRefGoogle Scholar
  6. 6.
    Couchman, P., Cannon, M., Kouvaritakis, B.: Stochastic MPC with inequality stability constraints. Automatica 42, 2169–2174 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Couchman, P., Kouvaritakis, B., Cannon, M.: LTV models in MPC for sustainable development. Int. J. Control 79, 63–73 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Deisenroth, M., Peters, J., Rasmussen, C.: Approximate dynamic programming with Gaussian processes. In: Proceedings of American Control Conference (ACC), pp. 4480–4485. Seattle, WA (2008)Google Scholar
  9. 9.
    Deisenroth, M., Rasmussen, C.: Bayesian inference for efficient learning in control. In: Proceedings of Multidisciplinary Symposium on Reinforcement Learning (MSRL). Montreal, Canada (2009)Google Scholar
  10. 10.
    Deisenroth, M., Rasmussen, C.: Efficient reinforcement learning for motor control. In: Proceedings of the 10th International Ph.D. Workshop on Systems and Control: A Young Generation Viewpoint. Hluboka nad Vltavou, Czech Republic (2009)Google Scholar
  11. 11.
    Deisenroth, M., Rasmussen, C., Peters, J.: Model-based reinforcement learning with continuous states and actions. In: Proceedings of the European Symposium on Artificial Neural Networks (ESANN), pp. 19–24. Bruges, Belgium (2008)Google Scholar
  12. 12.
    Deisenroth, M.P.: Efficient reinforcement learning using gaussian processes. Ph.D. thesis, Karlsruhe Institute of Technology, Karlsruhe (2010)Google Scholar
  13. 13.
    Deisenroth, M.P., Rasmussen, C.E., Peters, J.: Gaussian process dynamic programming. Neurocomputing 72(7–9), 1508–1524 (2009)CrossRefGoogle Scholar
  14. 14.
    Engel, Y., Szabo, P., Volkinshtein, D.: Learning to control an Octopus arm with Gaussian process temporal difference methods. In: Weiss, Y., Schoelkopf, B., Platt, J. (eds.) Advances in Neural Information Processing Systems, vol. 18, pp. 347–354. MIT Press, Cambridge (2006)Google Scholar
  15. 15.
    Fiacco, A.V.: Introduction to sensitivity and stability analysis in nonlinear programming. Academic Press, New York (1983)Google Scholar
  16. 16.
    Filatov, N., Unbehauen, H.: Survey of adaptive dual control methods. IEE Proc. Control Theory Appl. 147(1), 119–128 (2000)CrossRefGoogle Scholar
  17. 17.
    Girard, A., Murray-Smith, R.: Gaussian processes: prediction at a noisy input and application to iterative multiple-step ahead forecasting of time-series. In: Murray-Smith, R., Shorten, R. (eds.) Switching and Learning in Feedback Systems. Lecture Notes in Computer Science, vol. 3355, pp. 158–184. Springer, Berlin (2005)CrossRefGoogle Scholar
  18. 18.
    Grancharova, A., Johansen, T.A.: Approaches to explicit nonlinear model predictive control with reduced partition complexity. In: Proceedings of European Control Conference, pp. 2414–2419. Budapest, Hungary (2009)Google Scholar
  19. 19.
    Grancharova, A., Johansen, T.A.: Explicit Nonlinear Model Predictive Control: Theory and Applications. Lecture Notes in Control and Information Sciences, vol. 429. Springer, Berlin (2012)Google Scholar
  20. 20.
    Grancharova, A., Johansen, T.A., Tøndel, P.: Computational aspects of approximate explicit nonlinear model predictive control. In: Findeisen, R., Allgöwer, F., Biegler, L.T. (eds.) Assessment and Future Directions of Nonlinear Model Predictive Control. Lecture Notes in Control and Information Sciences, vol. 358, pp. 181–190. Springer, Berlin (2007)CrossRefGoogle Scholar
  21. 21.
    Grancharova, A., Kocijan, J.: Stochastic predictive control of a thermoelectric power plant. In: Proceedings of the International Conference Automatics and Informatics ‘07, pp. I-13–I-16. Sofia (2007)Google Scholar
  22. 22.
    Grancharova, A., Kocijan, J.: Explicit stochastic model predictive control of gas–liquid separator based on Gaussian process model. In: Proceedings of the International Conference on Automatics and Informatics, pp. B-85–B-88. Sofia, Bulgaria (2011)Google Scholar
  23. 23.
    Grancharova, A., Kocijan, J., Johansen, T.A.: Explicit stochastic nonlinear predictive control based on Gaussian process models. In: Proceedings of European Control Conference (ECC), pp. 2340–2347. Kos, Greece (2007)Google Scholar
  24. 24.
    Grancharova, A., Kocijan, J., Johansen, T.A.: Explicit stochastic predictive control of combustion plants based on Gaussian process models. Automatica 44(4), 1621–1631 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Gregorčič, G., Lightbody, G.: Gaussian processes for internal model control. In: Rakar, A. (ed.) Proceedings of 3rd International Ph.D. Workshop on Advances in Supervision and Control Systems, a Young Generation Viewpoint, pp. 39–46. Strunjan, Slovenia (2002)Google Scholar
  26. 26.
    Gregorčič, G., Lightbody, G.: From multiple model networks to the Gaussian processes prior model. In: Proceedings of IFAC ICONS Conference, pp. 149–154. Faro (2003)Google Scholar
  27. 27.
    Gregorčič, G., Lightbody, G.: Internal model control based on Gaussian process prior model. In: Proceedings of the 2003 American Control Conference, ACC 2003, pp. 4981–4986. Denver, CO (2003)Google Scholar
  28. 28.
    Gregorčič, G., Lightbody, G.: Gaussian process approaches to nonlinear modelling for control. In: Intelligent Control Systems Using Computational Intelligence Techniques, IEE Control Series, vol. 70, pp. 177–217. IEE, London (2005)Google Scholar
  29. 29.
    Gregorčič, G., Lightbody, G.: Gaussian process internal model control. Int. J. Syst. Sci. 1–16 (2011). http://www.tandfonline.com/doi/abs/10.1080/00207721.2011.564326
  30. 30.
    Isermann, R., Lachman, K.H., Matko, D.: Adaptive Control Systems. Systems and Control Engineering. Prentice Hall International, New York (1992)Google Scholar
  31. 31.
    Johansen, T.A.: Approximate explicit receding horizon control of constrained nonlinear systems. Automatica 40, 293–300 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Ko, J., Klein, D.J., Fox, D., Haehnel, D.: Gaussian processes and reinforcement learning for identification and control of an autonomous blimp. In: Proceedings of the International Conference on Robotics and Automation, pp. 742–747. Rome (2007)Google Scholar
  33. 33.
    Kocijan, J.: Control algorithms based on Gaussian process models: a state-of-the-art survey. In: Proceedings of the Special International Conference on Complex Systems: Synergy of Control, Communications and Computing—COSY 2011 (2011)Google Scholar
  34. 34.
    Kocijan, J.: Dynamic GP models: an overview and recent developments. In: Recent Researches in Applied Mathematics and Economics: Proceedings of the 6th International Conference on Applied Mathematics, Simulation, Modelling, (ASM’12), pp. 38–43. Vougliameni, Greece (2012)Google Scholar
  35. 35.
    Kocijan, J., Girard, A., Banko, B., Murray-Smith, R.: Dynamic systems identification with Gaussian processes. Math. Comput. Modell. Dyn. Syst. 11(4), 411–424 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Kocijan, J., Grancharova, A.: Gaussian process modelling case study with multiple outputs. C R Acad Bulg. Sci. 63(4), 601–608 (2010)Google Scholar
  37. 37.
    Kocijan, J., Leith, D.J.: Derivative observations used in predictive control. In: Proceedings of IEEE Melecon Conference, vol. 1, pp. 379–382. Dubrovnik (2004)Google Scholar
  38. 38.
    Kocijan, J., Likar, B.: Gas–liquid separator modelling and simulation with Gaussian-process models. Simul. Model. Pract. Theory 16(8), 910–922 (2008)CrossRefGoogle Scholar
  39. 39.
    Kocijan, J., Murray-Smith, R.: Nonlinear predictive control with Gaussian process model. In: Murray-Smith, R., Shorten, R. (eds.) Switching and Learning in Feedback Systems. Lecture Notes in Computer Science, vol. 3355, pp. 185–200. Springer, Heidelberg (2005)Google Scholar
  40. 40.
    Kocijan, J., Murray-Smith, R., Rasmussen, C., Girard, A.: Gaussian process model based predictive control. In: Proceedings of 4th American Control Conference (ACC 2004), pp. 2214–2218. Boston, MA (2004)Google Scholar
  41. 41.
    Kocijan, J., Murray-Smith, R., Rasmussen, C.E., Likar, B.: Predictive control with Gaussian process models. In: Proceedings of IEEE Region 8 EUROCON 2003: Computer as a Tool, vol. A, pp. 352–356. Ljubljana (2003)Google Scholar
  42. 42.
    Kouvaritakis, B., Cannon, M., Couchman, P.: MPC as a tool for sustainable development integrated policy assessment. IEEE Trans. Autom. Control 51, 145–149 (2006)CrossRefMathSciNetGoogle Scholar
  43. 43.
    Kraft, D.: On converting optimal control problems into nonlinear programming problems. In: Schittkowski, K. (ed.) Computational Mathematical Programming, NATO ASI Series, vol. F15, pp. 261–280. Springer, Berlin (1985)Google Scholar
  44. 44.
    Likar, B., Kocijan, J.: Predictive control of a gas–liquid separation plant based on a Gaussian process model. Comput. Chem. Eng. 31(3), 142–152 (2007)CrossRefGoogle Scholar
  45. 45.
    Maciejowski, J.M.: Predictive Control with Constraints. Pearson Education Limited, Harlow (2002)Google Scholar
  46. 46.
    Mayne, D.Q., Rawlings, J.B., Rao, C.V., Scokaert, P.O.M.: Constrained model predictive control: stability and optimality. Automatica 36, 789–814 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  47. 47.
    Murray-Smith, R., Sbarbaro, D.: Nonlinear adaptive control using nonparametric Gaussian process prior models. In: Proceedings of IFAC 15th World Congress. Barcelona (2002)Google Scholar
  48. 48.
    Murray-Smith, R., Sbarbaro, D., Rasmussen, C., Girard, A.: Adaptive, cautious, predictive control with Gaussian process priors. In: Proceedings of 13th IFAC Symposium on System Identification. Rotterdam, Netherlands (2003)Google Scholar
  49. 49.
    Nguyen-Tuong, D., Peters, J.: Learning robot dynamics for computed torque control using local Gaussian processes regression. In: Symposium on Learning and Adaptive Behaviors for Robotic Systems, pp. 59–64 (2008)Google Scholar
  50. 50.
    Nguyen-Tuong, D., Peters, J., Seeger, M., Schoelkopf, B.: Learning inverse dynamics: a comparison. In: Proceedings of the European Symposium on Artificial Neural Networks (ESANN), pp. 13–18. Bruges, Belgium (2008)Google Scholar
  51. 51.
    Nguyen-Tuong, D., Seeger, M., Peters, J.: Computed torque control with nonparametric regression models. In: Proceedings of the 2008 American Control Conference, ACC 2008, p. 6. Seattle, Washington (2008)Google Scholar
  52. 52.
    Nguyen-Tuong, D., Seeger, M., Peters, J.: Real-time local GP model learning, vol. 264. From Motor Learning to Interaction Learning in Robots, pp. 193–207. Springer, Heidelberg (2010)Google Scholar
  53. 53.
    Norgaard, M., Ravn, O., Poulsen, N.K., Hansen, L.K.: Neural Networks for Modelling and Control of Dynamic Systems: A Practitioner’s Handbook. Advanced Textbooks in Control and Signal Processing. Springer, London (2000)CrossRefGoogle Scholar
  54. 54.
    Palm, R.: Multiple-step-ahead prediction in control systems with Gaussian process models and TS-fuzzy models. Eng. Appl. Artif. Intell. 20(8), 1023–1035 (2007)CrossRefGoogle Scholar
  55. 55.
    Petelin, D., Kocijan, J.: Control system with evolving Gaussian process model. In: Proceedings of IEEE Symposium Series on Computational Intelligence, SSCI 2011. IEEE, Paris (2011)Google Scholar
  56. 56.
    Pistikopoulos, E.N., Georgiadis, M.C., Dua, V.: Multi-parametric model-based control. Wiley-VCH, Weinheim (2007)Google Scholar
  57. 57.
    Rasmussen, C.E., Deisenroth, M.P.: Probabilistic inference for fast learning in control. In: Recent Advances in Reinforcement Learning. Lecture Notes on Computer Science, vol. 5323, pp. 229–242. Springer, Berlin (2008)Google Scholar
  58. 58.
    Rasmussen, C.E., Kuss, M.: Gaussian processes in reinforcement learning. In: S. Thurn, L. Saul, B. Schoelkopf (eds.) Advances in Neural Information Processing Systems Conference, vol. 16, pp. 751–759. MIT Press, Cambridge (2004)Google Scholar
  59. 59.
    Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)zbMATHGoogle Scholar
  60. 60.
    Sbarbaro, D., Murray-Smith, R.: An adaptive nonparametric controller for a class of nonminimum phase non-linear system. In: Proceedings of IFAC 16th World Congress. Prague, Czech Republic (2005)Google Scholar
  61. 61.
    Sbarbaro, D., Murray-Smith, R.: Self-tuning control of nonlinear systems using Gaussian process prior models. In: Murray-Smith, R., Shorten, R. (eds.) Switching and Learning in Feedback Systems. Lecture Notes in Computer Science, vol. 3355, pp. 140–157. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  62. 62.
    Sbarbaro, D., Murray-Smith, R., Valdes, A.: Multivariable generalized minimum variance control based on artificial neural networks and Gaussian process models. In: International Symposium on Neural Networks. Springer, New York (2004)Google Scholar
  63. 63.
    Wittenmark, B.: Adaptive dual control. In: Control Systems, Robotics and Automation, Encyclopedia of Life Support Systems (EOLSS), Developed Under the Auspices of the UNESCO. Eolss Publishers, Oxford (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Systems and ControlJožef Stefan InstituteLjubljanaSlovenia
  2. 2.Centre for Systems and Information TechnologiesUniversity of Nova GoricaNova GoricaSlovenia
  3. 3.Institute of System Engineering and RoboticsBulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations