Advertisement

Basic MPC Formulation

  • Gergely Takács
  • Boris Rohal’-Ilkiv
Chapter

Abstract

Model predictive control (MPC) is a modern optimization based control strategy, which has been predominantly used for applications with slow dynamics so far. The implementation of MPC for fast systems such as active vibration attenuation (AVC) requires a firm theoretical knowledge of the method, its issues and limitations. This chapter is aimed at the reader with little or no prior contact with model predictive control. Discussing basic concepts like prediction, penalty, cost and optimization helps the reader to get up to speed using the most straightforward approach possible. Taking the basic building blocks of MPC and assembling them, one should ultimately understand how the optimal, constrained, quadratic programming-based MPC algorithm works. The chapter starts with an introduction of the fundamental idea behind model predictive control, followed by a review of the historic development of this field of control engineering. Next, generating a sequence of future states based on the current measurement and a state-space model is presented. Assembling a numerical indicator of the quality of control from the grounds up is just as important as predicting the sequence of future states, thus essentials of creating a quadratic cost are also presented. With the cost function available, one is able to create a simple MPC control law in the absence of constraints. Finally, the formulation of process constraints is described, and the resulting quadratic programming (QP) optimization problem is reviewed. The mathematical problem of quadratic programming is presented through an example, along with two basic solution algorithms for QP. The chapter is finished by the discussion of the important infinite horizon cost dual-mode MPC algorithm.

Keywords

Cost Function Model Predictive Control Linear Quadratic Prediction Horizon Interior Point Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Agachi PS, Nagy Z, Cristea MV, Imre-Lucaci A (2006) Model based control: case studies in process engineering, 1st edn. Wiley-VCH, WeinheimCrossRefGoogle Scholar
  2. 2.
    Åström KJ, Wittenmark B (1973) On self tuning regulators. Automatica 9:185–199zbMATHCrossRefGoogle Scholar
  3. 3.
    Belavý C (2009) Teória Automatického Riadenia II: Návody na cvičenia, 1st edn. Slovenská vysoká škola technická v Bratislave: Strojnícka Fakulta, Bratislava (Theory of automatic control II: seminar guide) In Slovak languageGoogle Scholar
  4. 4.
    Bitmead RR, Gevers M, Wertz V (1990) Adaptive optimal control: the thinking man’s GPC. Prentice Hall, San FranciscozbMATHGoogle Scholar
  5. 5.
    Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgezbMATHGoogle Scholar
  6. 6.
    Camacho EF, Bordons C (1995) Model predictive control in the process industry, advances in industrial control, vol 104. Springer, LondonCrossRefGoogle Scholar
  7. 7.
    Camacho EF, Bordons C (2007) Model predictive control, 2nd edn. Springer, LondonGoogle Scholar
  8. 8.
    Cannon M (2005) Model predictive control, Lecture notes, Michaelmas Term 2005 (4 Lectures). Course code 4ME44. University of Oxford, OxfordGoogle Scholar
  9. 9.
    Cannon M, Kouvaritakis B (2005) Optimizing prediction dynamics for robust MPC. IEEE Trans Autom Control 50(11):1892–1897. doi:  10.1109/TAC.2005.858679 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Clarke DW, Gawthrop PJ (1979) A self-tuning controller. In: IEE proceedings Part D 123:633–640Google Scholar
  11. 11.
    Clarke DW, Zhang L (1987) Long-range predictive control using weighting-sequence models. In: IEE Proceedings Part D 134(3):187–195Google Scholar
  12. 12.
    Clarke DW, Mohtadi C, Tuffs PS (1987) Generalized predictive control, Part I: The basic algorithm. Automatica 23(2):137–148zbMATHCrossRefGoogle Scholar
  13. 13.
    Clarke DW, Mohtadi C, Tuffs PS (1987) Generalized predictive control, Part II: extensions and interpretations. Automatica 23(2):149–160zbMATHCrossRefGoogle Scholar
  14. 14.
    Cutler CR, Ramaker BC (1980) Dynamic matrix control—a computer control algorithm. In: Automatic control conference, San FranciscoGoogle Scholar
  15. 15.
    de Keyser RMC, Cauwenberghe ARV (1985) Extended prediction self-adaptive control. In: IFAC symposium on identification and system parameter estimation, Yorkshire pp 1317–1322Google Scholar
  16. 16.
    Demircioğlu H, Gawthrop PJ (1991) Continuous-time generalised predictive control. Automatica 27(1):55–74MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Ferreau HJ (2006) An online active set strategy for fast solution of parametric quadratic programs with applications to predictive engine control. Master’s thesis, University of HeidelbergGoogle Scholar
  18. 18.
    Ferreau HJ (2011) qpOASES—Online active set strategy, Leuven. Available: http://www.qpoases.org
  19. 19.
    Ferreau HJ, Bock HG, Diehl M (2008) An online active set strategy to overcome the limitations of explicit MPC. Int J Robust Nonlinear Control 18(8):816–830MathSciNetCrossRefGoogle Scholar
  20. 20.
    Fletcher R (2000) Practical methods of optimization. Wiley, New YorkGoogle Scholar
  21. 21.
    Fuller CR, Elliott SJ, Nelson PA (1996) Active Control of Vibration, 1st edn. Academic Press, San FranciscoGoogle Scholar
  22. 22.
    Grafixar / morgueFile (2008) Oil refinery in Texas City, Texas. In agreement with the morgueFile free license: free to use for commercial work without attribution. Online, http://www.morguefile.com/archive/display/212157
  23. 23.
    Grossner J, Kouvaritakis B, Rossiter J (1997) Cautious stable predictive control: a guaranteed stable predictive control algorithm with low input activity and good robustness. Int J Control 67(5):675–697CrossRefGoogle Scholar
  24. 24.
    Inman DJ (2006) Vibration with control. Wiley, ChichesterCrossRefGoogle Scholar
  25. 25.
    Karas A (2002) Stabilizujúce prediktívne riadenie systémov s obmedzeniami. PhD thesis, Slovak University of Technology in Bratislava, Bratislava (Stabilizing predictive control of systems with constraints.) In Slovak languageGoogle Scholar
  26. 26.
    Karas A, Rohal’-Ilkiv B, Belavý C (2007) Praktické aspekty prediktívneho riadenia, 1st edn. Slovak University of Technology in Bratislava / Slovenská E-Akadémia, Bratislava (Practical aspects of predictive control.) In Slovak language.Google Scholar
  27. 27.
    Kinnaert M (1989) Adaptive generalized predictive controller for MIMO systems. J Process Control 50(1):161–172MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kvasnica M (2009) Real-time model predictive control via multi-parametric programming: theory and tools, 1st edn. VDM Verlag, SaarbrückenGoogle Scholar
  29. 29.
    Kwon WH, Byun DG (1989) Receding horizon tracking control as a~predictive control and its stability properties. Int J Control 50(5):1807–1824MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Kwon WH, Pearson AE (1975) On the stabilisation of a discrete constant linear system. IEEE Trans Autom Control 20(6):800–801MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Kwon WH, Pearson AE (1978) On feedback stabilization of time-varying discrete linear systems. IEEE Trans Autom Control 23:479–481MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Kwon WH, Choi HH, Byun DG, Noh SB (1992) Recursive solution of generalized predictive control and its equivalence to receding horizon tracking control. Automatica 28(6):1235–1238MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Lee JH, Morari M, Garcia CE (1994) State space interpretation of model predictive control. Automatica 30(4):707–717MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Maciejowski JM (2000) Predictive control with constraints, 1st edn. Prentice Hall, New JerseyGoogle Scholar
  35. 35.
    Morari M, Lee JH (1999) Model predictive control: past, present and future. Comput Chem Eng 23(4):667–682CrossRefGoogle Scholar
  36. 36.
    Mosca E (1994) Optimal, predictive and adaptive control, 1st edn. Prentice Hall, New JerseyGoogle Scholar
  37. 37.
    Nesterov Y, Nemirovskii A (1994) Interior point polynomial methods in convex programming, studies in applied mathematics, vol 13. SIAM, PennsylvaniaCrossRefGoogle Scholar
  38. 38.
    Ordys AW, Clarke DW (1993) A state-space description for GPC controllers. Int J Syst Sci 24(9):1727–1744MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Pistikopoulos EN, Georgiadis MC, Dua V (eds) (2007) Multi-parametric model-based control, vol 2., 1st edn. Wiley-VCH, WeinheimGoogle Scholar
  40. 40.
    Potra FA, Wright SJ (1997) Primal-dual interior-point methods. SIAM, PennsylvaniaGoogle Scholar
  41. 41.
    Preumont A (2002) Vibration control of active structures. 2nd edn. Kluwer Academic Publishers, DordrechtzbMATHGoogle Scholar
  42. 42.
    Preumont A, Seto K (2008) Active control of structures. 3rd edn. Wiley, ChichesterCrossRefGoogle Scholar
  43. 43.
    Qin SJ, Badgwell TA (1997) An overview of industrial model predictive control technology. In: Proceedings of chemical process control—V. Tahoe City. In: AIChE symposium series, vol 93, pp 232–256Google Scholar
  44. 44.
    Qin SJ, Badgwell TA (1999) An overview of nonlinear model predictive control applications. In: Zheng FAA (ed) Nonlinear model predictive control, Birkhauser Verlag, pp 369–392Google Scholar
  45. 45.
    Qin SJ, Badgwell TA (2003) A survey of industrial model predictive control technology. Control Eng Pract 11(7):733–764. doi:  10.1016/S0967-0661(02)00186-7, http://www.sciencedirect.com/science/article/B6V2H-47BX35T-1/2/1e355f78abeb6d9ee76d726330e7ca54
  46. 46.
    Rawlings JB, Muske KR (1993) The stability of constrained receding horizon control. IEEE Trans Autom Cont 38(10):1512–1516MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Richalet J, Rault A, Testud JL, Papon J (1978) Model predictive heuristic control: application to industrial processes. Automatica 14(2):413–428CrossRefGoogle Scholar
  48. 48.
    Rossiter JA (2003) Model-based predictive control: a practical approach. 1st edn. CRC Press, FloridaGoogle Scholar
  49. 49.
    Scokaert POM, Rawlings JB (1996) Infinite horizon linear quadratic control with constraints. In: Proceedings of IFAC’96 World Congress, San Francisco, vol 7a-04 1, pp 109–114Google Scholar
  50. 50.
    Shah SL, Mohtadi C, Clarke DW (1987) Multivariable adaptive zrol without a prior knowledge of the delay matrix. Syst Control Lett 9:295–306MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Soeterboek R (1992) Predictive control—a unified approach. Prentice Hall, New YorkzbMATHGoogle Scholar
  52. 52.
    The Mathworks (2011) Matlab optimization toolbox v6.0 (R2011a). Software. The Mathworks Inc., Natick. Available at http://www.mathworks.com/products/optimization/
  53. 53.
    Ydstie BE (1984) Extended horizon adaptive control. In: Proceedings of 9th IFAC World Congress, BudapestGoogle Scholar
  54. 54.
    Yoon TW (1994) Robust adaptive predictive control. PhD thesis, Department of Engineering Science, Oxford University, OxfordGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  • Gergely Takács
    • 1
  • Boris Rohal’-Ilkiv
    • 1
  1. 1.Faculty of Mechanical Engineering, Institute of Automation, Measurement and Applied InformaticsSlovak University of Technology in BratislavaBratislava 1Slovakia

Personalised recommendations