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Stochastic Model Predictive Control

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Handbook of Model Predictive Control

Part of the book series: Control Engineering ((CONTRENGIN))

Abstract

Stochastic Model Predictive Control (SMPC) accounts for model uncertainties and disturbances based on their probabilistic description. This chapter considers several formulations and solutions of SMPC problems and discusses some examples and applications in this diverse, complex, and growing field.

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Notes

  1. 1.

    For notational convenience, the control and prediction horizons are considered to be identical.

  2. 2.

    Hereafter we use the common notation in predictive control to differentiate prediction time instances t + k from time t at which predictions are made.

References

  1. Bavdekar, V., Mesbah, A.: Stochastic nonlinear model predictive control with joint chance constraints. In: Proceedings of the 10th IFAC Symposium on Nonlinear Control Systems, Monterey, pp. 276–281 (2016)

    Google Scholar 

  2. Bellman, R.E.: Dynamic Programming. Princeton University Press, New Jersey (1957)

    MATH  Google Scholar 

  3. Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)

    Book  Google Scholar 

  4. Bernardini, D., Bemporad, A.: Scenario-based model predictive control of stochastic constrained linear systems. In: Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, pp. 6333–6338 (2009)

    Google Scholar 

  5. Bernardini, D., Bemporad, A.: Stabilizing model predictive control of stochastic constrained linear systems. IEEE Trans. Autom. Control 57(6), 1468–1480 (2012)

    Article  MathSciNet  Google Scholar 

  6. Bertsekas, D.P.: Dynamic Programming and Optimal Control. Athena Scientific, Belmont (2000)

    Google Scholar 

  7. Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53, 464–501 (2011)

    Article  MathSciNet  Google Scholar 

  8. Bichi, M., Ripaccioli, G., Di Cairano, S., Bernardini, D., Bemporad, A., Kolmanovsky, I.: Stochastic model predictive control with driver behavior learning for improved powertrain control. In: Proceedings of the 49th IEEE Conference on Decision and Control, pp. 6077–6082. IEEE, Piscataway (2010)

    Google Scholar 

  9. Birge, J., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)

    MATH  Google Scholar 

  10. Calafiore, G.C., Fagiano, L.: Robust model predictive control via scenario optimization. IEEE Trans. Autom. Control 58(1), 219–224 (2013)

    Article  MathSciNet  Google Scholar 

  11. Cameron, R.H., Martin, W.T.: The orthogonal development of non-linear functionals in series of fourier-hermite functionals. Ann. Math. 48, 385–392 (1947)

    Article  MathSciNet  Google Scholar 

  12. Cannon, M., Kouvaritakis, B., Ng, D.: Probabilistic tubes in linear stochastic model predictive control. Syst. Control Lett. 58(10), 747–753 (2009)

    Article  MathSciNet  Google Scholar 

  13. Cannon, M., Kouvaritakis, B., Wu, X.: Probabilistic constrained MPC for multiplicative and additive stochastic uncertainty. IEEE Trans. Autom. Control 54(7), 1626–1632 (2009)

    Article  MathSciNet  Google Scholar 

  14. Cannon, M., Kouvaritakis, B., Rakovic, S.V., Cheng, Q.: Stochastic tubes in model predictive control with probabilistic constraints. IEEE Trans. Autom. Control 56(1), 194–200 (2011)

    Article  MathSciNet  Google Scholar 

  15. Chen, Z.: Bayesian filtering: from Kalman filters to particle filters, and beyond. Statistics 182, 1–69 (2003)

    Article  Google Scholar 

  16. Di Cairano, S., Bernardini, D., Bemporad, A., Kolmanovsky, I.V.: Stochastic MPC with learning for driver-predictive vehicle control and its application to hev energy management. IEEE Trans. Control Syst. Technol. 22(3), 1018–1031 (2014)

    Article  Google Scholar 

  17. Fagiano, L., Khammash, M.: Nonlinear stochastic model predictive control via regularized polynomial chaos expansions. In: Proceedings of the 51st IEEE Conference on Decision and Control, Maui, pp. 142–147 (2012)

    Google Scholar 

  18. Geletu, A., Klöppel, M., Zhangi, H., Li, P.: Advances and applications of chance-constrained approaches to systems optimisation under uncertainty. Int. J. Syst. Sci. 44, 1209–1232 (2013)

    Article  MathSciNet  Google Scholar 

  19. Ghanem, R., Spanos, P.: Stochastic finite elements – a spectral approach. Springer, New York (1991)

    Book  Google Scholar 

  20. Goulart, P.J., Kerrigan, E.C., Maciejowski, J.M.: Optimization over state feedback policies for robust control with constraints. Automatica 42, 523–533 (2006)

    Article  MathSciNet  Google Scholar 

  21. Hoblit, F.M.: Gust Loads on Aircraft: Concepts and Applications. AIAA, Reston (1988)

    Book  Google Scholar 

  22. Hokayem, P., Cinquemani, E., Chatterjee, D., Ramponi, F., Lygeros, J.: Stochastic receding horizon control with output feedback and bounded controls. Automatica 48, 77–88 (2012)

    Article  MathSciNet  Google Scholar 

  23. Kang, J., Raghunathan, A.U., Di Cairano, S.: Decomposition via ADMM for scenario-based model predictive control. In: American Control Conference (ACC), pp. 1246–1251. IEEE, Piscataway (2015)

    Google Scholar 

  24. Kim, K.K., Braatz, R.D.: Generalised polynomial chaos expansion approaches to approximate stochastic model predictive control. Int. J. Control 86, 1324–1337 (2013)

    Article  MathSciNet  Google Scholar 

  25. Kothare, M.V., Balakrishnan, V., Morari, M.: Robust constrained model predictive control using linear matrix inequalities. Automatica 32(10), 1361–1379 (1996)

    Article  MathSciNet  Google Scholar 

  26. Kouvaritakis, B., Cannon, M.: Model Predictive Control: Classical, Robust and Stochastic. Springer, Cham (2015)

    Google Scholar 

  27. Kouvaritakis, B., Cannon, M.: Stochastic model predictive control. Encyclopedia of Systems and Control, pp. 1350–1357. Springer, Berlin (2015)

    Google Scholar 

  28. Kouvaritakis, B., Cannon, M., Raković, S.V., Cheng, Q.: Explicit use of probabilistic distributions in linear predictive control. Automatica 46(10), 1719–1724 (2010)

    Article  MathSciNet  Google Scholar 

  29. Kumar, P.R., Varaiya, P.: Stochastic Systems: Estimation, Identification, and Adaptive Control. SIAM, Philadelphia (2016)

    MATH  Google Scholar 

  30. Mahalanobis, P.C.: On the generalized distance in statistics. Proc. Natl. Inst. Sci. 2, 49–55 (1936)

    MATH  Google Scholar 

  31. Mayne, D.Q., Seron, M.M., Raković, S.: Robust model predictive control of constrained linear systems with bounded disturbances. Automatica 41(2), 219–224 (2005)

    Article  MathSciNet  Google Scholar 

  32. Mayne, D.Q., Raković, S., Findeisen, R., Allgöwer, F.: Robust output feedback model predictive control of constrained linear systems. Automatica 42(7), 1217–1222 (2006)

    Article  MathSciNet  Google Scholar 

  33. Mesbah, A.: Stochastic model predictive control: an overview and perspectives for future research. IEEE Control Syst. 36(6), 30–44 (2016)

    Article  MathSciNet  Google Scholar 

  34. Mesbah, A.: Stochastic model predictive control with active uncertainty learning: a survey on dual control. Annu. Rev. Control. 45, 107–117 (2018)

    Article  Google Scholar 

  35. Navarro, M., Witteveen, J., Blom, J.: Polynomial chaos expansion for general multivariate distributions with correlated variables (2014, Preprint). arXiv 1406.5483

    Google Scholar 

  36. Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17, 969–996 (2006)

    Article  MathSciNet  Google Scholar 

  37. Paulson, J.A., Mesbah, A.: An efficient method for stochastic optimal control with joint chance constraints for nonlinear systems. Int. J. Robust Nonlinear Control (2017, https://onlinelibrary.wiley.com/doi/abs/10.1002/rnc.3999)

  38. Paulson, J.A., Streif, S., Mesbah, A.: Stability for receding-horizon stochastic model predictive control. In: Proceedings of the American Control Conference, Chicago, pp. 937–943 (2015)

    Google Scholar 

  39. Paulson, J.A., Buehler, E.A., Mesbah, A.: Arbitrary polynomial chaos for uncertainty propagation of correlated random variables in dynamic systems. In: Proceedings of the IFAC World Congress, Toulouse, pp. 3607–3612 (2017)

    Google Scholar 

  40. Rakovic, S.V., Kouvaritakis, B., Cannon, M., Panos, C., Findeisen, R.: Parameterized tube model predictive control. IEEE Trans. Autom. Control 57(11), 2746–2761 (2012)

    Article  MathSciNet  Google Scholar 

  41. Raković, S.V., Kouvaritakis, B., Findeisen, R., Cannon, M.: Homothetic tube model predictive control. Automatica 48(8), 1631–1638 (2012)

    Article  MathSciNet  Google Scholar 

  42. Raković, S.V., Levine, W.S., Açıkmeşe, B.: Elastic tube model predictive control. In: American Control Conference (ACC), pp. 3594–3599. IEEE, Piscataway (2016)

    Google Scholar 

  43. Ripaccioli, G., Bernardini, D., Di Cairano, S., Bemporad, A., Kolmanovsky, I.: A stochastic model predictive control approach for series hybrid electric vehicle power management. In: American Control Conference, Baltimore, pp. 5844–5849 (2010)

    Google Scholar 

  44. Schildbach, G., Fagiano, L., Frei, C., Morari, M.: The scenario approach for stochastic model predictive control with bounds on closed-loop constraint violations. Automatica 50(12), 3009–3018 (2014)

    Article  MathSciNet  Google Scholar 

  45. Scokaert, P.O., Mayne, D.: Min-max feedback model predictive control for constrained linear systems. IEEE Trans. Autom. Control 43(8), 1136–1142 (1998)

    Article  MathSciNet  Google Scholar 

  46. Van Hessem, D.H., Scherer, C.W., Bosgra, O.H.: LMI-based closed-loop economic optimization of stochastic process operation under state and input constraints. In: Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, pp. 4228–4233 (2001)

    Google Scholar 

  47. Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)

    Article  MathSciNet  Google Scholar 

  48. Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The second author would like to acknowledge Mr. Nan Li of the University of Michigan for the assistance and helpful comments.

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Correspondence to Ilya V. Kolmanovsky .

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Mesbah, A., Kolmanovsky, I.V., Di Cairano, S. (2019). Stochastic Model Predictive Control. In: Raković, S., Levine, W. (eds) Handbook of Model Predictive Control. Control Engineering. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-77489-3_4

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  • DOI: https://doi.org/10.1007/978-3-319-77489-3_4

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  • Publisher Name: Birkhäuser, Cham

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