Advertisement

CONTROLO 2016 pp 143-153 | Cite as

Sampled–Data Model Predictive Control Using Adaptive Time–Mesh Refinement Algorithms

  • Luís Tiago Paiva
  • Fernando A. C. C. Fontes
Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 402)

Abstract

We address sampled–data nonlinear Model Predictive Control (MPC) schemes, in particular we address methods to efficiently and accurately solve the underlying continuous-time optimal control problems (OCP). In nonlinear OCPs, the number of discretization points is a major factor affecting the computational time. Also, the location of these points is a major factor affecting the accuracy of the solutions. We propose the use of an algorithm that iteratively finds the adequate time–mesh to satisfy some pre–defined error estimate on the obtained trajectories. The proposed adaptive time–mesh refinement algorithm provides local mesh resolution considering a time–dependent stopping criterion, enabling an higher accuracy in the initial parts of the receding horizon, which are more relevant to MPC. The results show the advantage of the proposed adaptive mesh strategy, which leads to results obtained approximately as fast as the ones given by a coarse equidistant–spaced mesh and as accurate as the ones given by a fine equidistant–spaced mesh.

Keywords

Predictive control Nonlinear systems Optimal control Real–time optimization Continuous–time systems Adaptive algorithms Time–mesh refinement Sampled-data systems 

Notes

Acknowledgments

Research carried out while the 2nd author was a visiting scholar at Texas A&M University, College Station, USA. The support from Texas A&M and FEDER/COMPETE2020-POCI/FCT funds through grants POCI-01-0145-FEDER-006933 - SYSTEC and PTDC/EEI-AUT/2933/2014 is acknowledged.

References

  1. 1.
    Betts, J.T.: Practical methods for optimal control using nonlinear programming. SIAM (2001)Google Scholar
  2. 2.
    Betts, J.T., Huffman, W.P.: Mesh refinement in direct transcription methods for optimal control. Optim. Control Appl. Methods 19(1), 1–21 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Caldeira, A., Fontes, F.: Model predictive control for path-following of nonholonomic systems. In: IFAC (ed.) Proceedings of the 10th Portuguese Conference in Automatic Control. pp. 374–379 (2010)Google Scholar
  4. 4.
    Chen, H., Allgöwer, F.: A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34(10), 1205–1217 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Findeisen, R., Imsland, L., Allgöwer, F., Foss, B.: Towards a sampled-data theory for nonlinear model predictive control. In: Kang, W., Xiao, M., Borges, C. (eds.) New Trends in Nonlinear Dynamics and Control, and their applications. Lecture Notes in Control and Information Sciences, vol. 295, pp. 295–311. Springer Verlag, Berlin (2003)CrossRefGoogle Scholar
  6. 6.
    Fontes, F.A.C.C., Magni, L.: Min-max model predictive control of nonlinear systems using discontinuous feedbacks. IEEE Trans. Autom. Control 48, 1750–1755 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Fontes, F.A.C.C., Magni, L., Gyurkovics, E.: Sampled-data model predictive control for nonlinear time-varying systems: Stability and robustness. In: Allgöwer, F., Findeisen, R., Biegler, L. (eds.) Assessment and Future Directions of Nonlinear Model Predictive Control, Lecture Notes in Control and Information Systems, vol. 358, pp. 115–129. Springer Verlag (2007)Google Scholar
  8. 8.
    Fontes, F., Fontes, D., Caldeira, A.: Model Predictive Control of Vehicle Formations, Lecture Notes in Control and Information Sciences, vol. 381. Springer-Verlag (2009)Google Scholar
  9. 9.
    Fontes, F.A.C.C.: A general framework to design stabilizing nonlinear model predictive controllers. Syst. Control Lett. 42(2), 127–143 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fontes, F.A., Pereira, F.L.: Model predictive control of impulsive dynamical systems. In: Proceedings of the 4th IFAC conference on nonlinear model predictive control, NMPC’12. IFAC Proceedings Volumes (IFAC-PapersOnline), vol. 4, pp. 305–310. Noordwijkerhout; Netherlands (2012)Google Scholar
  11. 11.
    Grüne, L., Nesic, D., Pannek, J.: Model predictive control for nonlinear sampled-data systems. In: F. Allgöwer, L. Biegler, R.F.e. (eds.) Assessment and Future Directions of Nonlinear Model Predictive Control (NMPC05), Lecture Notes in Control and Information Sciences, vol. 358, pp. 105–113. Springer Verlag, Heidelberg (2007)Google Scholar
  12. 12.
    Grüne, L., Pannek, J.: Nonlinear model predictive control. Springer (2011)Google Scholar
  13. 13.
    Grüne, L., Palma, V.G.: Robustness of performance and stability for multistep and updated multistep MPC schemes. Discret. Contin. Dyn. Syst. 35(9), 4385–4414 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lazar, M., Allgower, F., Van den Hof, P., Cott, B. (eds.): 4th IFAC Conference on Nonlinear Model Predictive Control, NMPC’12. IFAC Proceedings Volumes (IFAC-PapersOnline), IFAC, Noordwijkerhout; Netherlands (2012)Google Scholar
  15. 15.
    Magni, L., Scattolini, R.: Model predictive control of continuous-time nonlinear systems with piecewise constant control. IEEE Trans. Autom. Control 49, 900–906 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mayne, D.Q., Michalska, H.: Receding horizon control of nonlinear systems. IEEE Trans. Autom. Control 35, 814–824 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Michalska, H., Mayne, D.Q.: Robust receding horizon control of constrained nonlinear systems. IEEE Trans. Autom. Control 38, 1623–1633 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Paiva, L.T.: Optimal control in constrained and hybrid nonlinear system: Solvers and interfaces. Faculdade de Engenharia, Universidade do Porto, Technical Report (2013)Google Scholar
  19. 19.
    Paiva, L.T.: Numerical Methods in Optimal Control and Model Predictive Control. Ph.D., Universidade do Porto, Porto, Portugal (Dec 2014)Google Scholar
  20. 20.
    Paiva, L.T., Fontes, F.A.: Adaptive time-mesh refinement in optimal control problems with state constraints. Discret. Contin. Dyn. Syst. 35(9), 4553–4572 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Pannocchia, G., Rawlings, J., Mayne, D., Mancuso, G.: Whither discrete time model predictive control? IEEE Trans. Autom. Control 60(1), 246–252 (2015)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Patterson, M.A., Hager, W.W., Rao, A.V.: A ph mesh refinement method for optimal control. Optimal Control Applications and Methods (Feb 2014)Google Scholar
  23. 23.
    Pereira, F.L., Fontes, F.A., Aguiar, A.P., de Sousa, J.B.: An optimization-based framework for impulsive control systems. In: Developments in Model-Based Optimization and Control, pp. 277–300. Springer (2015)Google Scholar
  24. 24.
    Prodan, I., Olaru, S., Fontes, F.A., Pereira, F.L., de Sousa, J.B., Maniu, C.S., Niculescu, S.I.: Predictive control for path-following. from trajectory generation to the parametrization of the discrete tracking sequences. In: Developments in Model-Based Optimization and Control, pp. 161–181. Springer (2015)Google Scholar
  25. 25.
    Prodan, I., Olaru, S., Fontes, F.A., Stoica, C., Niculescu, S.I.: A predictive control-based algorithm for path following of autonomous aerial vehicles. In: 2013 IEEE International Conference on Control Applications (CCA), pp. 1042–1047. IEEE (2013)Google Scholar
  26. 26.
    Raković, S.V., Kouramas, K.: Invariant approximations of the minimal robust positively invariant set via finite time Aumann integrals. In: 2007 46th IEEE Conference on Decision and Control, pp. 194–199. IEEE (2007)Google Scholar
  27. 27.
    Rawlings, J.B., Mayne, D.Q.: Model Predictive Control: Theory and Design. Nob Hill Pub. (Aug 2009)Google Scholar
  28. 28.
    Rucco, A., Aguiar, A.P., Fontes, F.A., Pereira, F.L., de Sousa, J.B.: A model predictive control-based architecture for cooperative path-following of multiple unmanned aerial vehicles. In: Developments in Model-Based Optimization and Control, pp. 141–160. Springer (2015)Google Scholar
  29. 29.
    Zhao, Y., Tsiotras, P.: Density functions for mesh refinement in numerical optimal control. J. Guid. Control Dyn. 34(1), 271–277 (2011)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Luís Tiago Paiva
    • 1
  • Fernando A. C. C. Fontes
    • 1
  1. 1.Systec–ISR, Faculdade de EngenhariaUniversidade do PortoPortoPortugal

Personalised recommendations