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Nonlinear Model Predictive Control pp 345–369Cite as

A Survey on Explicit Model Predictive Control

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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 384))

Abstract

Explicit model predictive control (MPC) addresses the problem of removing one of the main drawbacks of MPC, namely the need to solve a mathematical program on line to compute the control action. This computation prevents the application of MPC in several contexts, either because the computer technology needed to solve the optimization problem within the sampling time is too expensive or simply infeasible, or because the computer code implementing the numerical solver causes software certification concerns,especially in safety critical applications.

Explicit MPC allows one to solve the optimization problem off-line for a given range of operating conditions of interest. By exploiting multiparametric programming techniques, explicit MPC computes the optimal control action off line as an “explicit” function of the state and reference vectors, so that on-line operations reduce to a simple function evaluation. Such a function is piecewise affine in most cases, so that the MPC controller maps into a lookup table of linear gains.

In this paper we survey the main contributions on explicit MPC appeared in the scientific literature. After recalling the basic concepts and problem formulations of MPC, we review the main approaches to solve explicit MPC problems, including a novel and simple suboptimal practical approach to reduce the complexity of the explicit form. The paper concludes with some comments on future research directions.

This work was partially supported by the European Commission under the HYCON Network of Excellence, contract number FP6-IST-511368.

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References

  1. Acevedo, J., Pistikopoulos, E.N.: A multiparametric programming approach for linear process engineering problems under uncertainty. Ind. Eng. Chem. Res. 36, 717–728 (1997)

    Article  Google Scholar 

  2. Adler, I., Monteiro, R.D.C.: A geometric view of parametric linear programming. Algorithmica 8(2), 161–176 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alessio, A., Bemporad, A.: Feasible mode enumeration and cost comparison for explicit quadratic model predictive control of hybrid systems. In: 2nd IFAC Conference on Analysis and Design of Hybrid Systems, Alghero, Italy, pp. 302–308 (2006)

    Google Scholar 

  4. Allwright, J.C., Papavasiliou, G.C.: On linear programming and robust model-predictive control using impulse-responses. Systems & Control Letters 18, 159–164 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Baotić, M.: An efficient algorithm for multi-parametric quadratic programming. Technical Report AUT02-05, Automatic Control Institute, ETH, Zurich, Switzerland (2002)

    Google Scholar 

  6. Baotić, M., Borrelli, F., Bemporad, A., Morari, M.: Efficient on-line computation of constrained optimal control. SIAM Journal on Control and Optimization 47(5), 2470–2489 (2008)

    Article  MathSciNet  Google Scholar 

  7. Bemporad, A.: Multiparametric nonlinear integer programming and explicit quantized optimal control. In: Proc. 42nd IEEE Conf. on Decision and Control, Maui, Hawaii, USA, pp. 3167–3172 (2003)

    Google Scholar 

  8. Bemporad, A.: Efficient conversion of mixed logical dynamical systems into an equivalent piecewise affine form. IEEE Trans. Automatic Control 49(5), 832–838 (2004)

    Article  MathSciNet  Google Scholar 

  9. Bemporad, A.: Hybrid Toolbox – User’s Guide (January 2004), http://www.dii.unisi.it/hybrid/toolbox

  10. Bemporad, A.: Model-based predictive control design: New trends and tools. In: Proc. 45th IEEE Conf. on Decision and Control, San Diego, CA, pp. 6678–6683 (2006)

    Google Scholar 

  11. Bemporad, A., Borrelli, F., Morari, M.: Piecewise linear optimal controllers for hybrid systems. In: American Control Conference, Chicago, IL, pp. 1190–1194 (June 2000)

    Google Scholar 

  12. Bemporad, A., Borrelli, F., Morari, M.: Model predictive control based on linear programming — The explicit solution. IEEE Trans. Automatic Control 47(12), 1974–1985 (2002)

    Article  MathSciNet  Google Scholar 

  13. Bemporad, A., Borrelli, F., Morari, M.: Min-max control of constrained uncertain discrete-time linear systems. IEEE Trans. Automatic Control 48(9), 1600–1606 (2003)

    Article  MathSciNet  Google Scholar 

  14. Bemporad, A., Filippi, C.: Suboptimal explicit receding horizon control via approximate multiparametric quadratic programming. Journal of Optimization Theory and Applications 117(1), 9–38 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Bemporad, A., Filippi, C.: An algorithm for approximate multiparametric convex programming. Computational Optimization and Applications 35(1), 87–108 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Bemporad, A., Fukuda, K., Torrisi, F.D.: Convexity recognition of the union of polyhedra. Computational Geometry: Theory and Applications 18, 141–154 (2001)

    MATH  MathSciNet  Google Scholar 

  17. Bemporad, A., Morari, M.: Control of systems integrating logic, dynamics, and constraints. Automatica 35(3), 407–427 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Besselmann, T., Löfberg, J., Morari, M.: Explicit model predictive control for systems with linear parameter-varying state transition matrix. In: Proc. 17th IFAC World Congress, Seoul, Corea (2008)

    Google Scholar 

  20. Borrelli, F., Baotić, M., Bemporad, A., Morari, M.: Dynamic programming for constrained optimal control of discrete-time linear hybrid systems. Automatica 41(10), 1709–1721 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Borrelli, F., Bemporad, A., Fodor, M., Hrovat, D.: A hybrid approach to traction control. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, pp. 162–174. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  22. Borrelli, F., Bemporad, A., Morari, M.: A geometric algorithm for multi-parametric linear programming. Journal of Optimization Theory and Applications 118(3), 515–540 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Di Cairano, S., Bemporad, A., Kolmanovsky, I., Hrovat, D.: Model predictive control of magnetically actuated mass spring dampers for automotive applications. International Journal of Control 80(11), 1701–1716 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Campo, P.J., Morari, M.: Robust model predictive control. In: American Control Conference, vol. 2, pp. 1021–1026 (1987)

    Google Scholar 

  25. Christophersen, F.J., Kvasnica, M., Jones, C.N., Morari, M.: Efficient evaluation of piecewise control laws defined over a large number of polyhedra. In: Proc. European Control Conf., Kos, Greece, pp. 2360–2367 (2007)

    Google Scholar 

  26. Christophersen, F.J., Zeilinger, M.N., Jones, C.N., Morari, M.: Controller complexity reduction for piecewise affine systems through safe region elimination. In: IEEE Conference on Decision and Control, New Orleans, LA, pp. 4773–4778 (2007)

    Google Scholar 

  27. Dua, V., Pistikopoulos, E.N.: An algorithm for the solution of multiparametric mixed integer linear programming problems. Annals of Operations Research 1, 123–139 (2000)

    Article  MathSciNet  Google Scholar 

  28. Filippi, C.: On the geometry of optimal partition sets in multiparametric linear programming. Technical Report 12, Department of Pure and Applied Mathematics, University of Padova, Italy (June 1997)

    Google Scholar 

  29. Gal, T.: Postoptimal Analyses, Parametric Programming, and Related Topics, 2nd edn. de Gruyter, Berlin (1995)

    Google Scholar 

  30. Gal, T., Greenberg, H.J. (eds.): Advances in Sensitivity Analysis and Parametric Programming. International Series in Operations Research & Management Science, vol. 6. Kluwer Academic Publishers, Dordrecht (1997)

    MATH  Google Scholar 

  31. Gal, T., Nedoma, J.: Multiparametric linear programming. Management Science 18, 406–442 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  32. Gass, S.I., Saaty, T.L.: The computational algorithm for the parametric objective function. Naval Research Logistics Quarterly 2, 39–45 (1955)

    Article  MathSciNet  Google Scholar 

  33. Geyer, T., Torrisi, F.D., Morari, M.: Optimal complexity reduction of polyhedral piecewise affine systems. Automatica 44, 1728–1740 (2008)

    Article  MATH  Google Scholar 

  34. Geyer, T., Torrisi, F.D., Morari, M.: Efficient Mode Enumeration of Compositional Hybrid Models. In: Maler, O., Pnueli, A. (eds.) HSCC 2003. LNCS, vol. 2623, pp. 216–232. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  35. Grieder, P., Kvasnica, M., Baotić, M., Morari, M.: Low complexity control of piecewise affine systems with stability guarantee. In: American Control Conference, Boston, MA (2004)

    Google Scholar 

  36. Grieder, P., Morari, M.: Complexity reduction of receding horizon control. In: Proc. 42th IEEE Conf. on Decision and Control, Maui, Hawaii, USA, pp. 3179–3184 (2003)

    Google Scholar 

  37. Ingimundarson, A., Ocampo-Martinez, C., Bemporad, A.: Model predictive control of hybrid systems based on mode-switching constraints. In: Proc. 46th IEEE Conf. on Decision and Control, New Orleans, LA, pp. 5265–5269 (2007)

    Google Scholar 

  38. Johansen, T.A.: On multi-parametric nonlinear programming and explicit nonlinear model predictive control. In: Proc. 41th IEEE Conf. on Decision and Control, Las Vegas, Nevada, USA, pp. 2768–2773 (December 2002)

    Google Scholar 

  39. Johansen, T.A.: Approximate explicit receding horizon control of constrained nonlinear systems. Automatica 40, 293–300 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  40. Johansen, T.A., Grancharova, A.: Approximate explicit constrained linear model predictive control via orthogonal search tree. IEEE Trans. Automatic Control 58(5), 810–815 (2003)

    Article  MathSciNet  Google Scholar 

  41. Johansen, T.A., Jackson, W., Schrieber, R., Tøndel, P.: Hardware synthesis of explicit model predictive controllers. IEEE Trans. Contr. Systems Technology 15(1), 191–197 (2007)

    Article  Google Scholar 

  42. Johansen, T.A., Petersen, I., Slupphaug, O.: On explicit suboptimal LQR with state and input constraints. In: Proc. 39th IEEE Conf. on Decision and Control, Sydney, Australia, pp. 662–667 (December 2000)

    Google Scholar 

  43. Jones, C., Morari, M.: Multiparametric linear complementarity problems. In: Proc. 45th IEEE Conf. on Decision and Control, San Diego, CA, pp. 5687–5692 (2006)

    Google Scholar 

  44. Jones, C.N., Baric, M., Morari, M.: Multiparametric linear programming with applications to control. European Journal of Control 13, 152–170 (2007)

    Article  Google Scholar 

  45. Jones, C.N., Grieder, P., Raković, S.V.: A logarithmic-time solution to the point location problem for parametric linear programming. Automatica 42(12), 2215–2218 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  46. Jones, C.N., Kerrigan, E.C., Maciejowski, J.M.: Lexicographic perturbation for multiparametric linear programming with applications to control. Automatica 43(10), 1808–1816 (2007)

    Article  MATH  Google Scholar 

  47. Kerrigan, E.C., Maciejowski, J.M.: Feedback min-max model predictive control using a single linear program: Robust stability and the explicit solution. Int. J. Robust Nonlinear Control 14(4), 395–413 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  48. Kvasnica, M., Grieder, P., Baotić, M.: Multi Parametric Toolbox, MPT (2006), http://control.ee.ethz.ch/~mpt/

  49. Lincoln, B., Rantzer, A.: Relaxing dynamic programming. IEEE Trans. Automatic Control 51(8), 1249–1260 (2006)

    Article  MathSciNet  Google Scholar 

  50. Mangasarian, O.L., Rosen, J.B.: Inequalities for stochastic nonlinear programming problems. Operations Research 12, 143–154 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  51. Mayne, D.Q., Rakovíc, S.V., Kerrigan, E.C.: Optimal control and piecewise parametric programming. In: Proc. European Control Conf., Kos, Greece, pp. 2762–2767 (2007)

    Google Scholar 

  52. Mayne, D.Q.: Control of constrained dynamic systems. Technical Report EEE/C&P/DQM/9/2001, Imperial College, London, U.K. (September 2001)

    Google Scholar 

  53. Mayne, D.Q., Rakovic, S.: Optimal control of constrained piecewise affine discrete time systems using reverse transformation. In: Proc. 41th IEEE Conf. on Decision and Control, Las Vegas, Nevada, USA, pp. 1546–1551 (December 2002)

    Google Scholar 

  54. Mayne, D.Q., Rawlings, J.B., Rao, C.V., Scokaert, P.O.M.: Constrained model predictive control: Stability and optimality. Automatica 36(6), 789–814 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  55. Muñoz de la Peña, D., Alamo, T., Bemporad, A., Camacho, E.F.: A dynamic programming approach for determining the explicit solution of MPC controllers. In: Proc. 43th IEEE Conf. on Decision and Control, Paradise Island, Bahamas, pp. 2479–2484 (2004)

    Google Scholar 

  56. Muñoz de la Peña, D., Alamo, T., Ramírez, D.R., Camacho, E.F.: Min-max model predictive control as a quadratic program. IET Control Theory & Applications 1(1), 328–333 (2007)

    Article  MathSciNet  Google Scholar 

  57. Muñoz de la Peña, D., Bemporad, A., Filippi, C.: Robust explicit MPC based on approximate multi-parametric convex programming. IEEE Trans. Automatic Control 51(8), 1399–1403 (2006)

    Article  Google Scholar 

  58. Orchard-Hays, W.: Notes on linear programming (part 6): the Rand code for the simplex method (sx4). Technical Report 1440, Rand Corporation (1955)

    Google Scholar 

  59. Pannocchia, G., Rawlings, J.B., Wright, S.J.: Fast, large-scale model predictive control by partial enumeration. Automatica 43, 852–860 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  60. Propoi, A.I.: Use of linear programming methods for synthesizing sampled-data automatic systems. Automation and Remote Control 24(7), 837–844 (1963)

    Google Scholar 

  61. Ramírez, D.R., Camacho, E.F.: Piecewise affinity of min-max mpc with bounded additive uncertainties and a quadratic criterion. Automatica 42(2), 295–302 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  62. Schechter, M.: Polyhedral functions and multiparametric linear programming. Journal of Optimization Theory and Applications 53(2), 269–280 (1987)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  64. Seron, M.M., DeDoná, J.A., Goodwin, G.C.: Global analytical model predictive control with input constraints. In: Proc. 39th IEEE Conf. on Decision and Control, Sydney, Australia, pp. 154–159 (2000)

    Google Scholar 

  65. Spjøtvold, J., Kerrigan, E.C., Jones, C.N., Tøndel, P., Johansen, T.A.: On the facet-to-facet property of solutions to convex parametric quadratic programs. Automatica 42(12), 2209–2214 (2006)

    Article  MathSciNet  Google Scholar 

  66. Spjøtvold, J., Rakovic, S.V., Tøndel, P., Johansen, T.A.: Utilizing reachability analysis in point location problems. In: Proc. 45th IEEE Conf. on Decision and Control, San Diego, CA, pp. 4568–4569 (2006)

    Google Scholar 

  67. Tøndel, P., Johansen, T.A.: Complexity reduction in explicit linear model predictive control. In: Proc. 15th IFAC World Congress, Barcelona, Spain (2002)

    Google Scholar 

  68. Tøndel, P., Johansen, T.A., Bemporad, A.: An algorithm for multi-parametric quadratic programming and explicit MPC solutions. Automatica 39(3), 489–497 (2003)

    Article  MathSciNet  Google Scholar 

  69. 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 

  70. Tøndel, P., Johansen, T.A., Bemporad, A.: Further results on multiparametric quadratic programming. In: Proc. 42th IEEE Conf. on Decision and Control, Maui, Hawaii, USA, pp. 3173–3178 (2003)

    Google Scholar 

  71. Torrisi, F.D., Bemporad, A.: HYSDEL — A tool for generating computational hybrid models. IEEE Trans. Contr. Systems Technology 12(2), 235–249 (2004)

    Article  MathSciNet  Google Scholar 

  72. Wang, Y., Jones, C.N., Maciejowski, J.: Efficient point location via subdivision walking with application to explicit MPC. In: Proc. European Control Conf., Kos, Greece, pp. 447–453 (2007)

    Google Scholar 

  73. Witsenhausen, H.S.: A min-max control problem for sampled linear systems. IEEE Trans. Automatic Control 13(1), 5–21 (1968)

    Article  MathSciNet  Google Scholar 

  74. Zafiriou, E.: Robust model predictive control of processes with hard constraints. Computers & Chemical Engineering 14(4/5), 359–371 (1990)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Information Engineering, University of Siena, Italy

    Alessandro Alessio & Alberto Bemporad

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  1. Alessandro Alessio
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  2. Alberto Bemporad
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Editors and Affiliations

  1. Dipartimento di Informatica e Sistemistica, Universita’ di Pavia, via Ferrata 1, 27100 Pavia, Italy

    Lalo Magni

  2. Automatic Control Laboratory, ETH Zurich , ETL I 24.2, Physikstrasse 3, CH-8092, Zürich, Switzerland

    Davide Martino Raimondo

  3. Institute for Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, 70550, Stuttgart, Germany

    Frank Allgöwer

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Alessio, A., Bemporad, A. (2009). A Survey on Explicit Model Predictive Control. In: Magni, L., Raimondo, D.M., Allgöwer, F. (eds) Nonlinear Model Predictive Control. Lecture Notes in Control and Information Sciences, vol 384. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01094-1_29

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  • DOI: https://doi.org/10.1007/978-3-642-01094-1_29

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