Approximation Properties of Receding Horizon Optimal Control

Survey Article

Abstract

In this survey, receding horizon control is presented as a method for obtaining approximately optimal solutions to infinite horizon optimal control problems by iteratively solving a sequence of finite horizon optimal control problems. We investigate conditions under which we can obtain mathematically rigorous approximation results for this approach. A key ingredient of our analysis is the so-called turnpike property of optimal control problems.

Keywords

Optimal control Receding horizon control Model predictive control Turnpike property dissipativity 

Mathematics Subject Classification

93C10 93D15 49M37 

References

  1. 1.
    Altmüller, N.: Model predictive control for partial differential equations. Ph.D. thesis, Universität Bayreuth (2014) Google Scholar
  2. 2.
    Altmüller, N., Grüne, L.: Distributed and boundary model predictive control for the heat equation. GAMM-Mitt. 35, 131–145 (2012) MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Altmüller, N., Grüne, L.: A comparative stability analysis of Neumann and Dirichlet boundary MPC for the heat equation. In: Proceedings of the 1st IFAC Workshop on Control of Systems Modeled by Partial Differential Equations, CPDE 2013, pp. 161–166 (2013) Google Scholar
  4. 4.
    Altmüller, N., Grüne, L., Worthmann, K.: Receding horizon optimal control for the wave equation. In: Proceedings of the 49th IEEE Conference on Decision and Control, CDC2010, Atlanta, Georgia, pp. 3427–3432 (2010) Google Scholar
  5. 5.
    Amrit, R., Rawlings, J.B., Angeli, D.: Economic optimization using model predictive control with a terminal cost. Annu. Rev. Control 35, 178–186 (2011) CrossRefGoogle Scholar
  6. 6.
    Angeli, D., Amrit, R., Rawlings, J.B.: On average performance and stability of economic model predictive control. IEEE Trans. Autom. Control 57, 1615–1626 (2012) MathSciNetCrossRefGoogle Scholar
  7. 7.
    Annunziato, M., Borzì, A.: Optimal control of probability density functions of stochastic processes. Math. Model. Anal. 15, 393–407 (2010) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Annunziato, M., Borzì, A.: A Fokker-Planck control framework for multidimensional stochastic processes. J. Comput. Appl. Math. 237, 487–507 (2013) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Azmi, B., Kunisch, K.: On the stabilizability of the Burgers’ equation by receding horizon control. Preprint, TU Graz, 2015. http://www.uni-graz.at/~kunisch/papers/KK_291.pdf5
  10. 10.
    Bertsekas, D.P.: Dynamic Programming and Optimal Control, vols. 1 and 2. Athena Scientific, Belmont (1995) MATHGoogle Scholar
  11. 11.
    Boccia, A., Grüne, L., Worthmann, K.: Stability and feasibility of state constrained MPC without stabilizing terminal constraints. Syst. Control Lett. 72, 14–21 (2014) MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Brock, W.A., Mirman, L.: Optimal economic growth and uncertainty: the discounted case. J. Econ. Theory 4, 479–513 (1972) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Byrnes, C.I., Lin, W.: Losslessness, feedback equivalence, and the global stabilization of discrete-time nonlinear systems. IEEE Trans. Autom. Control 39, 83–98 (1994) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Carlson, D.A., Haurie, A.B., Leizarowitz, A.: Infinite Horizon Optimal Control—Deterministic and Stochastic Systems, 2nd edn. Springer, Berlin (1991) CrossRefMATHGoogle Scholar
  15. 15.
    Chen, H., Allgöwer, F.: A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34, 1205–1217 (1998) MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Choi, H., Hinze, M., Kunisch, K.: Instantaneous control of backward-facing step flows. Appl. Numer. Math. 31, 133–158 (1999) MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Damm, T., Grüne, L., Stieler, M., Worthmann, K.: An exponential turnpike theorem for dissipative discrete time optimal control problems. SIAM J. Control Optim. 52, 1935–1957 (2014) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Diehl, M., Amrit, R., Rawlings, J.B.: A Lyapunov function for economic optimizing model predictive control. IEEE Trans. Autom. Control 56, 703–707 (2011) MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dorfman, R., Samuelson, P.A., Solow, R.M.: Linear Programming and Economic Analysis. Dover Publications, New York (1987). Reprint of the 1958 original MATHGoogle Scholar
  20. 20.
    Faulwasser, T., Bonvin, D.: On the design of economic NMPC based on approximate turnpike properties. In: Proceedings of the 54rd IEEE Conference on Decision and Control, CDC 2015, pp. 4964–4970 (2015) Google Scholar
  21. 21.
    Faulwasser, T., Korda, M., Jones, C.N., Bonvin, D.: Turnpike and dissipativity properties in dynamic real-time optimization and economic MPC. In: Proceedings of the 53rd IEEE Conference on Decision and Control, CDC 2014, pp. 2734–2739 (2014) CrossRefGoogle Scholar
  22. 22.
    Findeisen, R., Allgöwer, F.: An introduction to Nonlinear Model Predictive Control. In: 21st Benelux Meeting on Systems and Control, Veldhoven, The Netherlands, pp. 119–141 (2002) Google Scholar
  23. 23.
    Fleig, A., Grüne, L.: Estimates on the minimal stabilizing horizon length in Model Predictive Control for the Fokker-Planck equation. Preprint, Universität Bayreuth, 2016, submitted for publication Google Scholar
  24. 24.
    Grüne, L.: Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems. SIAM J. Control Optim. 48, 1206–1228 (2009) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Grüne, L.: NMPC without terminal constraints. In: Proceedings of the IFAC Conference on Nonlinear Model Predictive Control, NMPC’12, pp. 1–13 (2012) Google Scholar
  26. 26.
    Grüne, L.: Economic receding horizon control without terminal constraints. Automatica 49, 725–734 (2013) MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Grüne, L., Müller, M.A.: On the relation between strict dissipativity and the turnpike property. Syst. Control Lett. 90, 45–53 (2016). doi: 10.1016/j.sysconle.2016.01.003 CrossRefMATHGoogle Scholar
  28. 28.
    Grüne, L., Panin, A.: On non-averaged performance of economic MPC with terminal conditions. In: Proceedings of the 54th IEEE Conference on Decision and Control, CDC 2015, Osaka, Japan pp. 4332–4337 (2015) Google Scholar
  29. 29.
    Grüne, L., Pannek, J.: Nonlinear Model Predictive Control. Theory and Algorithms. Springer, London (2011) CrossRefMATHGoogle Scholar
  30. 30.
    Grüne, L., Pannek, J., Seehafer, M., Worthmann, K.: Analysis of unconstrained nonlinear MPC schemes with time varying control horizon. SIAM J. Control Optim. 48, 4938–4962 (2010) MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Grüne, L., Rantzer, A.: On the infinite horizon performance of receding horizon controllers. IEEE Trans. Autom. Control 53, 2100–2111 (2008) MathSciNetCrossRefGoogle Scholar
  32. 32.
    Grüne, L., Semmler, W., Stieler, M.: Using nonlinear model predictive control for dynamic decision problems in economics. J. Econ. Dyn. Control 60, 112–133 (2015) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Grüne, L., Stieler, M.: Asymptotic stability and transient optimality of economic MPC without terminal conditions. J. Process Control 24, 1187–1196 (2014) CrossRefGoogle Scholar
  34. 34.
    Hinze, M.: Instantaneous closed loop control of the Navier-Stokes system. SIAM J. Control Optim. 44, 564–583 (2005) MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Hinze, M., Kunisch, K.: On suboptimal control strategies for the Navier-Stokes equations. In: Control and Partial Differential Equations, Marseille-Luminy, 1997. ESAIM Proc., vol. 4, pp. 181–198. Soc. Math. Appl. Indust., Paris (1998) Google Scholar
  36. 36.
    Hinze, M., Volkwein, S.: Analysis of instantaneous control for the Burgers equation. Nonlinear Anal. 50, 1–26 (2002) MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Hundhammer, R., Leugering, G.: Instantaneous control of vibrating string networks. In: Grötschel, M., Krumke, S.O., Rambau, J. (eds.) Online Optimization of Large Scale Systems, pp. 229–249. Springer, Berlin (2001) CrossRefGoogle Scholar
  38. 38.
    Ito, K., Kunisch, K.: Receding horizon optimal control for infinite dimensional systems. ESAIM Control Optim. Calc. Var. 8, 741–760 (2002). A tribute to J.L. Lions MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Kaganovich, M.: Efficiency of sliding plans in a linear model with time-dependent technology. The Review of Economic Stuidies 52, 691–702 (1985) CrossRefMATHGoogle Scholar
  40. 40.
    Keerthi, S.S., Gilbert, E.G.: Optimal infinite horizon feedback laws for a general class of constrained discrete-time systems: stability and moving horizon approximations. J. Optim. Theory Appl. 57, 265–293 (1988) MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Lincoln, B., Rantzer, A.: Relaxing dynamic programming. IEEE Trans. Autom. Control 51, 1249–1260 (2006) MathSciNetCrossRefGoogle Scholar
  42. 42.
    Mayne, D.Q.: An apologia for stabilising terminal conditions in model predictive control. Int. J. Control 86, 2090–2095 (2013) MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    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) MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    McKenzie, L.W.: Optimal economic growth, turnpike theorems and comparative dynamics. In: Handbook of Mathematical Economics. Handbooks in Econom., vol. III, pp. 1281–1355. North-Holland, Amsterdam (1986) Google Scholar
  45. 45.
    Mohammadi, M., Borzì, A.: Analysis of the Chang-Cooper discretization scheme for a class of Fokker-Planck equations. J. Numer. Math. 23, 271–288 (2015) MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Müller, M.A., Grüne, L.: Economic model predictive control without terminal constraints for optimal periodic behavior. Automatica, to appear Google Scholar
  47. 47.
    Porretta, A., Zuazua, E.: Long time versus steady state optimal control. SIAM J. Control Optim. 51, 4242–4273 (2013) MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Primak, S., Kontorovich, V., Lyandres, V.: Stochastic Methods and Their Applications to Communications. Wiley, Hoboken (2004) CrossRefMATHGoogle Scholar
  49. 49.
    Propoĭ, A.: Application of linear programming methods for the synthesis of automatic sampled-data systems. Avtom. Telemeh. 24, 912–920 (1963) MathSciNetGoogle Scholar
  50. 50.
    Protter, P.E.: Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability, vol. 21. Springer, Berlin (2005) Google Scholar
  51. 51.
    Rawlings, J.B., Mayne, D.Q.: Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison (2009) Google Scholar
  52. 52.
    Reble, M., Allgöwer, F.: Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems. Automatica 48, 1812–1817 (2011) MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Risken, H.: The Fokker-Planck Equation, 2nd edn. Springer Series in Synergetics, vol. 18. Springer, Berlin (1989) CrossRefMATHGoogle Scholar
  54. 54.
    Schulze Darup, M., Cannon, M.: A missing link between nonlinear MPC schemes with guaranteed stability. In: Proceedings of the 54rd IEEE Conference on Decision and Control, CDC 2015, pp. 4977–4983 (2015) Google Scholar
  55. 55.
    Trélat, E., Zuazua, E.: The turnpike property in finite-dimensional nonlinear optimal control. J. Differ. Equ. 258, 81–114 (2015) MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Tuna, S.E., Messina, M.J., Teel, A.R.: Shorter horizons for model predictive control. In: Proceedings of the 2006 American Control Conference, Minneapolis, Minnesota, USA, pp. 863–868 (2006) Google Scholar
  57. 57.
    von Neumann, J.: A model of general economic equilibrium. Rev. Econ. Stud. 13, 1–9 (1945) CrossRefGoogle Scholar
  58. 58.
    Willems, J.C.: Dissipative dynamical systems. I. General theory. Arch. Ration. Mech. Anal. 45, 321–351 (1972) MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Willems, J.C.: Dissipative dynamical systems. II. Linear systems with quadratic supply rates. Arch. Ration. Mech. Anal. 45, 352–393 (1972) MathSciNetCrossRefMATHGoogle Scholar
  60. 60.
    Worthmann, K.: Stability analysis of unconstrained receding horizon control schemes. Ph.D. thesis, Universität Bayreuth (2011) Google Scholar
  61. 61.
    Worthmann, K., Mehrez, M.W., Zanon, M., Mann, G.K.I., Gosine, R.G., Diehl, M.: Regulation of differential drive robots using continuous time MPC without stabilizing constraints or costs. In: Proceedings of the 5th IFAC Conference on Nonlinear Model Predictive Control, NMPC 2015, Seville, Spain. IFAC-PapersOnLine, vol. 48, pp. 129–135 (2015). Google Scholar
  62. 62.
    Worthmann, K., Reble, M., Grüne, L., Allgöwer, F.: The role of sampling for stability and performance in unconstrained nonlinear model predictive control. SIAM J. Control Optim. 52, 581–605 (2014) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Deutsche Mathematiker-Vereinigung and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of BayreuthBayreuthGermany

Personalised recommendations