Skip to main content
SpringerLink
Log in

Nonlinear Programming Formulations for Nonlinear and Economic Model Predictive Control

  • Chapter
  • First Online:
Handbook of Model Predictive Control

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

Abstract

We present a framework for constructing robust nonlinear model predictive controllers (NMPCs) with either tracking or economic objectives. For this, we explore properties of nonlinear programming problems (NLPs) that arise in the formulation of NMPC subproblems and show their influence on stability and robustness properties. In particular, NLPs that satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ), the constant rank constraint qualification (CRCQ), and generalized strong second order sufficient conditions (GSSOSC) have solutions that are continuous with respect to perturbations of the problem data. These are important prerequisites for nominal and robust stability of NMPC controllers. Moreover, we show that ensuring these properties is possible through reformulation of the NLP subproblem for NMPC, through the addition of ℓ 1 penalty terms. We also show how these properties extend beyond tracking objective functions to economic NMPC (eNMPC), a more general dynamic optimization problem, where further reformulation is required for stability guarantees. We present and discuss the relative merits of three alternative methods for stabilizing eNMPC: objective regularization based on the full state-space, objective regularization based on a reduced set of states, and the addition of a stabilizing constraint. Finally, we demonstrate these eNMPC formulations on a continuously stirred tank reactor (CSTR) as well as a pair of coupled distillation columns.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Amrit, R., Rawlings, J.B., Biegler, L.T.: Optimizing process economics online using model predictive control. Comput. Chem. Eng. 58, 334–343 (2013)

    Article  Google Scholar 

  2. Angeli, D.: Handbook of Model Predictive Control. Chapter Economic Model Predictive Control, p. yyy. Birkhüser, Basel (2018)

    Google Scholar 

  3. Angeli, D., Amrit, R., Rawlings, J.: On average performance and stability of economic model predictive control. IEEE Trans. Autom. Control 57(7), 1615–1626 (2012)

    Article  MathSciNet  Google Scholar 

  4. Biegler, L.T., Yang, X., Fischer, G.G.: Advances in sensitivity-based nonlinear model predictive control and dynamic real-time optimization. J. Process Control 30, 104–116 (2015)

    Article  Google Scholar 

  5. Chen, H., Allgöwer, F.: A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica 34, 1205–1217 (1998)

    Article  MathSciNet  Google Scholar 

  6. Diehl, M., Amrit, R., Rawlings, J.: A Lyapunov function for economic optimizing model predictive control. IEEE Trans. Autom. Control 56(3), 703–707 (2011)

    Article  MathSciNet  Google Scholar 

  7. Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic, New York (1983)

    MATH  Google Scholar 

  8. Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL: A Modeling Language for Mathematical Programming. Duxbury Press, Pacific Grove, Belmont (2002)

    MATH  Google Scholar 

  9. Gauvin, J.: A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Math. Program. 12(1), 136–138 (1977)

    Article  MathSciNet  Google Scholar 

  10. Gopalakrishnan, A., Biegler, L.T.: Economic nonlinear model predictive control for the periodic optimal operation of gas pipeline networks. Comput. Chem. Eng. 52, 90–99 (2013)

    Article  Google Scholar 

  11. Griffith, D., Zavala, V., Biegler, L.: Robustly stable economic NMPC for non-dissipative stage costs. J. Process Control 57, 116–126 (2017)

    Article  Google Scholar 

  12. Grüne, L.: Economic receding horizon control without terminal constraints. Automatica 49, 725–734 (2013)

    Article  MathSciNet  Google Scholar 

  13. Heidarinejad, M., Liu, J., Christofides, P.: Economic model predictive control of nonlinear process systems using lyapunov techniques. AIChE Journal 58(3), 855–870 (2011)

    Article  Google Scholar 

  14. Huang, R., Harinath, E., Biegler, L.T.: Lyapunov stability of economically-oriented NMPC for cyclic processes. J. Process Control 21, 501–509 (2011)

    Article  Google Scholar 

  15. Huang, R., Harinath, E., Biegler, L.T.: Economicaly-oriented nonlinear model predictive control for energy applications. J. Process Control 21(4), 501–509 (2011)

    Article  Google Scholar 

  16. Janin, R.: Directional derivative of the marginal function in nonlinear programming. In: Fiacco, A.V. (ed.) Sensitivity, Stability and Parametric Analysis. Mathematical Programming Studies, vol. 21, pp. 110–126. Springer, Berlin (1984)

    Chapter  Google Scholar 

  17. Jäschke, J., Yang, X., Biegler, L.T.: Fast economic model predictive control based on NLP-sensitivities. J. Process Control 24, 1260–1272 (2014)

    Article  Google Scholar 

  18. Jiang, Z.P., Wang, Y.: Input-to-state stability for discrete-time nonlinear systems. Automatica 37, 857–869 (2001)

    Article  MathSciNet  Google Scholar 

  19. Keerthi, S.S., Gilbert, E.G.: Optimal infinite-horizon feedback laws for general class of constrained discrete-time systems: stability and moving-horizon approximations. IEEE Trans. Autom. Control 57, 265–293 (1988)

    MathSciNet  MATH  Google Scholar 

  20. Kojima, M.: Strongly stable stationary solutions in nonlinear programming. In: Robinson, S.M. (ed.) Analysis and Computation of Fixed Points. Academic, New York (1980)

    Google Scholar 

  21. Leer, R.B.: Self-optimizing control structures for active constraint regions of a sequence of distillation columns. Master’s thesis, Norweign University of Science and Technology (2012)

    Google Scholar 

  22. Limon, D., Alamo, T., Raimondo, D., Pe\(\widetilde{\mbox{ n}}\) a, D., Bravo, J., Ferramosca, A., Camacho, E.: Input-to-state stability: a unifying framework for robust model predictive control. In: Magni, L., Raimondo, D., Allgöwer, F. (eds.) Nonlinear Model Predictive Control: Towards New Challenging Applications. Springer, Berlin (2009)

    Google Scholar 

  23. Magni, L., Scattolini, R.: Robustness and robut design of mpc for nonlinear discrete-time systems. In: Findeisen, R., Allgöwer, F., Biegler, L.T. (eds.) Assessment and Future Directions of Nonlinear Model Predictive Control, pp. 239–254. Springer, Berlin (2007)

    Chapter  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Nocedal, J., Wright, S.: Numerical Optimization. Operations Research and Financial Engineering, 2nd edn. Springer, New York (2006)

    Google Scholar 

  26. Pannocchia, G., Rawlings, J.B., Wright, S.J.: Conditions under which supboptimal nonlinear MPC is inherently robust. Syst. Control Lett. 60, 747–755 (2011)

    Article  Google Scholar 

  27. Ralph, D., Dempe, S.: Directional derivatives of the solution of a parametric nonlinear program. Math. Program. 70(1–3), 159–172 (1995)

    MathSciNet  MATH  Google Scholar 

  28. Rawlings, J.B., Mayne, D.Q.: Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison (2009)

    Google Scholar 

  29. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)

    Article  MathSciNet  Google Scholar 

  30. Skogestad, S.: Dynamics and control of distillation columns: a tutorial introduction. Chem. Eng. Res. Des. 75(A), 539–562 (1997)

    Article  Google Scholar 

  31. Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)

    Article  MathSciNet  Google Scholar 

  32. Würth, L., Rawlings, J.B., Marquardt, W.: Economic dynamic real-time optimization and nonlinear model predictive control on infinite horizons. In: International Symposium on Advanced Control of Chemical Process, Istanbul (2009)

    Google Scholar 

  33. Yang, X.: Advanced-multi-step and economically oriented nonlinear model predictive control. PhD thesis, Carnegie Mellon University (2015)

    Google Scholar 

  34. Yang, X., Biegler, L.T.: Advanced-multi-step nonlinear model predictive control. J. Process Control 23, 1116–1128 (2013)

    Article  Google Scholar 

  35. Yang, X., Griffith, D., Biegler, L.: Nonlinear programming properties for stable and robust NMPC. In: 5th IFAC Conference on Nonlinear Model Predictive Control, IFAC-PapersOnLine 48(23), 388–397 (2015)

    Google Scholar 

  36. Yu, M.: Model reduction and nonlinear model predictive control of large-scale distributed parameter systems with applications in solid sorbent-based CO2 capture. PhD thesis, Carnegie Mellon University (2017)

    Google Scholar 

  37. Zavala, V.M.: A multiobjective optimization perspective on the stability of economic MPC. In: 9th International Symposium on Advanced Control of Chemical Processes, pp. 975–981 (2015)

    Google Scholar 

  38. Zavala, V., Biegler, L.: The advanced step NMPC controller: optimality, stability and robustness. Automatica 45, 86–93 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chemical Engineering Department, Carnegie Mellon University, Pittsburgh, PA, USA

    Mingzhao Yu, Devin W. Griffith & Lorenz T. Biegler

Authors
  1. Mingzhao Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Devin W. Griffith
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lorenz T. Biegler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lorenz T. Biegler .

Editor information

Editors and Affiliations

  1. Independent Researcher, London, UK

    Saša V. Raković

  2. Department of Electrical and Computer Engineering, University of Maryland, College Park, MD, USA

    William S. Levine

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Yu, M., Griffith, D.W., Biegler, L.T. (2019). Nonlinear Programming Formulations for Nonlinear and Economic 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_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-77489-3_20

  • Published:

  • Publisher Name: Birkhäuser, Cham

  • Print ISBN: 978-3-319-77488-6

  • Online ISBN: 978-3-319-77489-3

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation