Advertisement

Vietnam Journal of Mathematics

, Volume 47, Issue 4, pp 897–929 | Cite as

Exponentially Convergent Receding Horizon Strategy for Constrained Optimal Control

  • Wanting Xu
  • Mihai AnitescuEmail author
Original Article
  • 6 Downloads

Abstract

Receding horizon control has been a widespread method in industrial control engineering as well as an extensively studied subject in control theory. In this work, we consider a lag L receding horizon strategy that applies the initial L optimal controls from each quadratic program to each receding horizon. We investigate a discrete-time and time-varying linear-quadratic optimal control problem that includes a nonzero reference trajectory and constraints on both state and control. We prove that, under boundedness and controllability conditions, the solution obtained by the receding horizon strategy converges to the solution of the full problem interval exponentially fast in the length of the receding horizon for some lag L. The exponential rate of convergence provides a systematic way of choosing the receding horizon length given a desired accuracy level. We illustrate our theoretical findings using a small, synthetic production cost model with real demand data.

Keywords

Constrained optimal control Receding horizon control Sensitivity analysis 

Mathematics Subject Classification (2010)

49N10 49N35 49Q12 

Notes

Acknowledgements

We thank Prof. V. Zavala for pointing us to references about stability issues in rolling horizon control. We thank the anonymous referee of [30] who suggested that we look into RHC as well as the referee of this paper for important comments about our assumptions and scope relative to [11]. This material was based upon work supported by the U.S. Department of Energy, Office of Science, under Contract DE-AC02-06CH11347. M. A. acknowledges partial NSF funding through awards FP061151-01-PR and CNS-1545046.

References

  1. 1.
    Bellingham, J., Richards, A., How, J.P.: Receding horizon control of autonomous aerial vehicles. In: American Control Conference, 2002. Proceedings of the 2002, vol. 5, pp. 3741–3746. IEEE (2002)Google Scholar
  2. 2.
    Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. 1. Athena Scientific, Belmont (1995)zbMATHGoogle Scholar
  3. 3.
    Bhattacharya, R., Balas, G.J., Kaya, A., Packard, A.: Nonlinear receding horizon control of F-16 aircraft. In: American Control Conference, 2001. Proceedings of the 2001, vol. 1, pp. 518–522. IEEE (2001)Google Scholar
  4. 4.
    Biegler, L.T., Zavala, V.M.: Large-scale nonlinear programming using IPOPT: an integrating framework for enterprise-wide dynamic optimization. Comput. Chem. Eng. 33, 575–582 (2009)CrossRefGoogle Scholar
  5. 5.
    Boccia, A., Grüne, L., Worthmann, K.: Stability and feasibility of state constrained MPC without stabilizing terminal constraints. Syst. Control Lett. 72, 14–21 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2013)zbMATHGoogle Scholar
  7. 7.
    Dunbar, W.B., Caveney, D.S.: Distributed receding horizon control of vehicle platoons: Stability and string stability. IEEE Trans. Autom. Control 57, 620–633 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Franz, R., Milam, M., Hauser, J.: Applied receding horizon control of the Caltech Ducted Fan. In: American Control Conference, 2002. Proceedings of the 2002, vol 5, pp. 3735–3740. IEEE (2002)Google Scholar
  9. 9.
    Grüne, L.: Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems. SIAM J. Control Optim. 48, 1206–1228 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Grüne, L., Pannek, J.: Nonlinear Model Predictive Control: Theory and Algorithms. Communications and Control Engineering. Springer, London (2011)CrossRefGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Interconnection, P.: Estimated hourly load data. http://www.pjm.com/markets-and-operations/energy/real-time/loadhryr.aspx. Accessed: 2017-07-23
  13. 13.
    Jadbabaie, A., Hauser, J.: On the stability of receding horizon control with a general terminal cost. IEEE Trans. Autom. Control 50, 674–678 (2005)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jadbabaie, A., Yu, J., Hauser, J.: Unconstrained receding-horizon control of nonlinear systems. IEEE Trans. Autom. Control 46, 776–783 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kwon, W., Pearson, A.: A modified quadratic cost problem and feedback stabilization of a linear system. IEEE Trans. Autom. Control 22, 838–842 (1977)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kwon, W.H., Bruckstein, A.M., Kailath, T.: Stabilizing state-feedback design via the moving horizon method. Int. J. Control 37, 631–643 (1983)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kwon, W.H., Han, S.H.: Receding Horizon Control: Model Predictive Control for State Models. Springer, London (2006)Google Scholar
  19. 19.
    Kwon, W.H., Lee, Y.S., Han, S.H.: General receding horizon control for linear time-delay systems. Automatica 40, 1603–1611 (2004)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Lubin, M., Dunning, I.: Computing in operations research using Julia. INFORMS J. Comput. 27, 238–248 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mayne, D.Q., Michalska, H.: Receding horizon control of nonlinear systems. IEEE Trans. Autom. Control 35, 814–824 (1990)MathSciNetCrossRefGoogle Scholar
  22. 22.
    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)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nocedal, J., Wright, S.: Numerical Optimization. Springer-Verlag, New York (2006)zbMATHGoogle Scholar
  24. 24.
    Primbs, J.A., Nevistić, V.: Feasibility and stability of constrained finite receding horizon control. Automatica 36, 965–971 (2000)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rawlings, J.B., Muske, K.R.: The stability of constrained receding horizon control. IEEE Trans. Autom. Control 38, 1512–1516 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Reble, M., Allgöwer, F.: Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems. Automatica 48, 1812–1817 (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Richalet, J., Rault, A., Testud, J.L., Papon, J.: Model predictive heuristic control: Applications to industrial processes. Automatica 14, 413–428 (1978)CrossRefGoogle Scholar
  28. 28.
    Sethi, S., Sorger, G.: A theory of rolling horizon decision making. Ann. Oper. Res. 29, 387–415 (1991)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Sideris, A., Rodriguez, L.A.: A Riccati Approach to Equality Constrained Linear Quadratic Optimal Control. In: Proceeding of the 2010 American Control Conference, pp. 5167–5172 (2010)Google Scholar
  30. 30.
    Xu, W., Anitescu, M.: Exponentially accurate temporal decomposition for long-horizon linear-quadratic dynamic optimization. SIAM J. Optim. 28, 2541–2573 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Zhang, W., Hu, J., Abate, A.: On the value functions of the discrete-time switched LQR problem. IEEE Trans. Autom. Control 54, 2669–2674 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of ChicagoChicagoUSA
  2. 2.Mathematics and Computer Science DivisionArgonne National LaboratoryLemontUSA

Personalised recommendations