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Real-Time Implementation of Explicit Model Predictive Control

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Handbook of Model Predictive Control

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

Abstract

This chapter explains the synthesis of explicit MPC feedback laws that allow for real-time implementation on hardware with limited computational and storage properties. Four methods are introduced. The first one replaces the potentially complex explicit MPC controller by a simpler feedback law by exploiting the geometry of explicit solutions. The second method reduces the storage footprint of explicit MPC by a complete elimination of critical regions, replaced by a direct evaluation of optimality conditions. The common denominator of both methods is that they preserve optimality while considerably reducing the complexity. The third method trades lower complexity for suboptimality while simultaneously minimizing the performance loss. Finally, a method for designing stabilizing explicit MPC controllers for control of nonlinear systems is introduced.

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Notes

  1. 1.

    If \(G_{\mathcal{A}}\) does not have a full row rank, it is always possible to identify a subset of \(\mathcal{A}\) such that all rows of \(G_{\mathcal{A}}\) are linearly independent, see, e.g., [37].

References

  1. Ahmadi, A.A., Hall, G.: DC decomposition of nonconvex polynomials with algebraic techniques. arxiv.org (2015)

    Google Scholar 

  2. Baldoni, V., Berline, N., De Loera, J.A., Köppe, M., Vergne, M.: How to integrate a polynomial over a simplex. Math. Comput. 80(273), 297 (2010)

    Article  MathSciNet  Google Scholar 

  3. Baotić, M.: Optimal Control of Piecewise Affine Systems – a Multi-parametric Approach. Dr. sc. thesis, ETH Zurich, Zurich, Switzerland, March 2005

    Google Scholar 

  4. Baotić, M.: Polytopic computations in constrained optimal control. Automatica 50(3–4), 119–134 (2009)

    Google Scholar 

  5. Bemporad, A.: Hybrid Toolbox - User’s Guide. New Society, Gabriola (2003)

    Google Scholar 

  6. Bemporad, A.: A multiparametric quadratic programming algorithm with polyhedral computations based on nonnegative least squares. IEEE Trans. Autom. Control 60(11), 2892–2903 (2015)

    Article  MathSciNet  Google Scholar 

  7. Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.: The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20 (2002)

    Article  MathSciNet  Google Scholar 

  8. Blanchini, F., Miani, S.: Set-Theoretic Methods in Control. Birkhauser, Boston (2008)

    MATH  Google Scholar 

  9. Borrelli, F., Bemporad, A., Morari, M.: Geometric algorithm for multiparametric linear programming. J. Optim. Theory Appl. 118(3), 515–540 (2003)

    Article  MathSciNet  Google Scholar 

  10. Borrelli, F., Bemporad, A., Morari, M.: Predictive Control for Linear and Hybrid Systems. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  11. Christophersen, F.J., Kvasnica, M., Jones, C.N., Morari, M.: Efficient evaluation of piecewise control laws defined over a large number of polyhedra. In: Antsaklis, P.J., Tzafestas, S.G. (eds.) Proceedings of the European Control Conference ECC ’07, pp. 2360–2367 (2007)

    Google Scholar 

  12. Domahidi, A., Zeilinger, M., Morari, M., Jones, C.: Learning a feasible and stabilizing explicit model predictive control law by robust optimization. In: 2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), pp. 513–519 (2011)

    Google Scholar 

  13. Domahidi, A., Ullmann, F., Morari, M., Jones, C.: Learning near-optimal decision rules for energy efficient building control. In: 2012 IEEE 51st Annual Conference on Decision and Control (CDC), pp. 7571–7576 (2012)

    Google Scholar 

  14. Dórea, C.E.T., Hennet, J.C.: (A, B)-invariant polyhedral sets of linear discrete-time systems. J. Optim. Theory Appl. 103(3), 521–542 (1999)

    Google Scholar 

  15. Drgoňa, J., Klaučo, M., Janeček, F., Kvasnica, M.: Optimal control of a laboratory binary distillation column via regionless explicit MPC. Comput. Chem. Eng. 96, 139–148 (2017)

    Article  Google Scholar 

  16. Grieder, P., Morari, M.: Complexity reduction of receding horizon control. In: IEEE Conference on Decision and Control, Maui, December 2003, pp. 3179–3184

    Google Scholar 

  17. Gupta, A., Bhartiya, S., Nataraj, P.: A novel approach to multiparametric quadratic programming. Automatica 47(9), 2112–2117 (2011)

    Article  MathSciNet  Google Scholar 

  18. Herceg, M., Kvasnica, M., Jones, C., Morari, M.: Multi-parametric toolbox 3.0. In: 2013 European Control Conference, pp. 502–510 (2013)

    Google Scholar 

  19. Herceg, M., Mariethoz, S., Morari, M: Evaluation of piecewise affine control law via graph traversal. In: 2013 European Control Conference (ECC), pp. 3083–3088. IEEE, Piscataway (2013)

    Google Scholar 

  20. Holaza, J., Takács, B., Kvasnica, M., Di Cairano, S.: Nearly optimal simple explicit MPC controllers with stability and feasibility guarantees. Optim. Control Appl. Methods 35(6), 667–684 (2015)

    Article  MathSciNet  Google Scholar 

  21. Jones, C.N., Morari, M.: Polytopic approximation of explicit model predictive controllers. IEEE Trans. Autom. Control 55(11), 2542–2553 (2010)

    Article  MathSciNet  Google Scholar 

  22. Klaučo, M., Drgoňa, J., Kvasnica, M., Di Cairano, S.: Building temperature control by simple MPC-like feedback laws learned from closed-loop data. In: Preprints of the 19th IFAC World Congress Cape Town (South Africa) August 24–August 29, 2014, pp. 581–586 (2014)

    Google Scholar 

  23. Korda, M., Jones, C.N.: Stability and performance verification of optimization-based controllers. Automatica 78, 34–45 (2017)

    Article  MathSciNet  Google Scholar 

  24. Korda, M., Mezić, I: Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. arXiv preprint arXiv:1611.03537 (2017)

    Google Scholar 

  25. Kvasnica, M., Fikar, M.: Clipping-based complexity reduction in explicit MPC. IEEE Trans. Autom. Control 57(7), 1878–1883 (2012)

    Article  MathSciNet  Google Scholar 

  26. Kvasnica, M., Löfberg, J., Fikar, M.: Stabilizing polynomial approximation of explicit MPC. Automatica 47(10), 2292–2297 (2011)

    Article  MathSciNet  Google Scholar 

  27. Lasserre, J.B.: Moments, Positive Polynomials and Their Applications, 1st edn. Imperial College Press, London (2009)

    Book  Google Scholar 

  28. Löfberg, J.: YALMIP: a toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei (2004)

    Google Scholar 

  29. MathWorks: Statistics and machine learning toolbox: User’s guide (r2016b). https://www.mathworks.com/help/pdf_doc/stats/stats.pdf, September 2016

  30. Milne, G.W.: Grumman f-14 benchmark control problem solution using BLKLAB. In: IEEE Control Systems Society Workshop on Computer-Aided Control System Design, December 1989, pp. 94–101

    Google Scholar 

  31. MOSEK ApS: The MOSEK optimization toolbox for MATLAB manual (2016)

    Google Scholar 

  32. Oberdieck, R., Diangelakis, N.A., Papathanasiou, M., Nascu, I., Pistikopoulos, E.: Pop-parametric optimization toolbox. Ind. Eng. Chem. Res. 55(33), 8979–8991 (2016)

    Article  Google Scholar 

  33. Oishi, Y.: Direct design of a polynomial model predictive controlle. IFAC Proceedings Volumes 45(13), 633–638 (2012)

    Article  Google Scholar 

  34. Oishi, Y.: Simplified approaches to polynomial design of model predictive controllers. In: 2013 IEEE International Conference on Control Applications (CCA), pp. 960–965 (2013)

    Google Scholar 

  35. Parisini, T., Zoppoli, R.: A receding-horizon regulator for nonlinear systems and a neural approximation. Automatica 31(10), 1443–1451 (1995)

    Article  MathSciNet  Google Scholar 

  36. Primbs, J.A.: The analysis of optimization based controllers. Automatica 37(6), 933–938 (2001)

    Article  MathSciNet  Google Scholar 

  37. Spjøtvold, J., Tøndel, P., Johansen, T.A.: A Method for Obtaining Continuous Solutions to Multiparametric Linear Programs. In: IFAC World Congress, Prague (2005)

    Google Scholar 

  38. Tøndel, P., Johansen, T.A., Bemporad, A.: Evaluation of piecewise affine control via binary search tree. Automatica 39(5), 945–950 (2003)

    Article  MathSciNet  Google Scholar 

  39. Wen, Ch., Ma, X., Ydstie, B.E.: Analytical expression of explicit MPC solution via lattice piecewise-affine function. Automatica 45(4), 910–917 (2009)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

M. Kvasnica, J. Holaza, and P. Bakarac gratefully acknowledge the contribution of the Slovak Research and Development Agency under the project APVV 15-0007.

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Correspondence to Michal Kvasnica .

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Kvasnica, M., Jones, C.N., Pejcic, I., Holaza, J., Korda, M., Bakaráč, P. (2019). Real-Time Implementation of Explicit 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_17

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  • DOI: https://doi.org/10.1007/978-3-319-77489-3_17

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