A Vector Quantization Approach to Scenario Generation for Stochastic NMPC

  • Graham C. Goodwin
  • Jan Østergaard
  • Daniel E. Quevedo
  • Arie Feuer
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 384)


This paper describes a novel technique for scenario generation aimed at closed loop stochastic nonlinear model predictive control. The key ingredient in the algorithm is the use of vector quantization methods.We also show how one can impose a tree structure on the resulting scenarios. Finally, we briefly describe how the scenarios can be used in large scale stochastic nonlinear model predictive control problems and we illustrate by a specific problem related to optimal mine planning.


Scenario generation closed loop control stochastic nonlinear model predictive control vector quantization 


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  1. 1.
    Goodwin, G.C., Serón, M.M., Mayne, D.Q.: Optimization opportunities in mining, metal and mineral processing. Annual Reviews in Control 32(1), 17–32 (2008)CrossRefGoogle Scholar
  2. 2.
    Mayne, D.Q., Rawlings, J.B., Rao, C.V., Scokaert, P.O.M.: Constrained model predictive control: Optimality and stability. Automatica 36, 789–814 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Goodwin, G.C., Serón, M.M., De Doná, J.A.: Constrained Control & Estimation – An Optimization Perspective. Springer, London (2005)Google Scholar
  4. 4.
    Camacho, E.F., Bordons, C.: Model Predictive Control. Springer, New York (1999)Google Scholar
  5. 5.
    Rojas, C.R., Goodwin, G.C., Serón, M.M.: Open-cut mine planning via closed-loop receding horizon optimal control. In: Sánchez-Peña, R., Quevedo, J., Puig Cayuela, V. (eds.) Identification and control: The gap between theory and practice. Springer, Heidelberg (2007)Google Scholar
  6. 6.
    Dreyfus, S.E.: Some types of optimal control of stochastic systems. J. SIAM, Series A: Control 2(1), 120–134 (1964)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Bertsekas, D.P.: Dynamic programming and suboptimal control: A survey from adp to mpc. European J. Contr. 11(4–5), 310–334 (2005)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Cannon, M., Kouvaritakis, B., Wu, X.: Stochastic predictive control with probabilistic constraints. Automatica (to appear)Google Scholar
  9. 9.
    Pflug, G.C.: Optimization of Stochastic Models: The Interface Between Simulation and Optimization. Kluwer Academic Publishers, Boston (1996)zbMATHGoogle Scholar
  10. 10.
    Muñoz de la Peña, D., Bemporad, A., Alamo, T.: Stochastic programming applied to model predictive control. In: Proc. IEEE Conf. Decis. Contr. and Europ. Contr. Conf., Seville, Spain, pp. 1361–1366 (December 2005)Google Scholar
  11. 11.
    Pflug, G.C.: Scenario tree generation for multiperiod financial optimization by optimal discretization. Math. Program., Ser. B 89, 251–271 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Mirkov, R., Pflug, G.C.: Tree approximations of stochastic dynamic programs. SIAM Journal on Optimization 18(3), 1082–1105 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hochreiter, R., Pflug, G.C.: Financial scenario generation for stochastic multi-stage decision processes as facility location problems. Ann. Oper. Res. 152(1), 257–272 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Gülpınar, N., Rustem, B., Settergren, R.: Simulation and optimization approaches to scenario tree generation. Journal of Economic Dynamics & Control 28, 1291–1315 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kuhn, D.: Aggregation and discretization in multistage stochastic programming. Math. Program., Ser. A 113, 61–94 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Gersho, A., Gray, R.M.: Vector Quantization and Signal Compression. Kluwer Academic Publishers, Dordrecht (1992)zbMATHGoogle Scholar
  17. 17.
    Gray, R.M., Neuhoff, D.: Quantization. IEEE Trans. Inf. Theory 44(6), 2325–2383 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Lloyd, S.P.: Least squares quantization in PCM (unpublished Bell Laboratories technical note) (1957)Google Scholar
  19. 19.
    Linde, Y., Buzo, A., Gray, R.M.: An algorithm for vector quantizer design. IEEE Trans. Commun. 28, 84–95 (1980)CrossRefGoogle Scholar
  20. 20.
    Gray, R., Kieffer, J., Linde, Y.: Locally optimal block quantizer design. Information and Control 45, 178–198 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    MacQueen, J.: Some methods for classification and analysis of multivariate observations. In: Proc. 5th Berkeley Symp., vol. 1, pp. 281–297 (1967)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Graham C. Goodwin
    • 1
  • Jan Østergaard
    • 1
  • Daniel E. Quevedo
    • 1
  • Arie Feuer
    • 2
  1. 1.School of Electrical Engineering and Computer ScienceThe University of NewcastleAustralia
  2. 2.The TechnionHaifaIsrael

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