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Constrained Model Predictive Control of Processes with Uncertain Structure Modeled by Jump Markov Linear Systems

  • Jens TonneEmail author
  • Olaf Stursberg
Chapter
  • 839 Downloads
Part of the Mathematical Engineering book series (MATHENGIN)

Abstract

Linear systems with abrupt changes in its structure, e.g. caused by component failures of a production system, can be modelled by the use of jump Markov linear systems (JMLS). This chapter proposes a finite horizon model predictive control (MPC) approach for discrete-time JMLS considering input constraints as well as constraints for the expectancy of the state trajectory. For the expected value of the state as well as a quadratic cost criterion, recursive prediction schemes are formulated, which consider dependencies on the input trajectory explicitly. Due to the proposed prediction scheme, the MPC problem can be formulated as a quadratic program (QP) exhibiting low computational effort compared to existing approaches. The resulting properties concerning stability as well as computational complexity are investigated and demonstrated by illustrative simulation studies.

Keywords

Markov Jump Linear Systems (JMLS) Model Predictive Control (MPC) Discrete-time JMLS Quadratic Cost Criterion Finite Horizon MPC 
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.
    Costa OLV, Fragoso MD, Marques RP (2005) Discrete-time Markov jump linear systems. Probability and its applications. Springer, New YorkGoogle Scholar
  2. 2.
    Maciejowski JM (2002) Predictive control with constraints. Prentice-Hall, New JerseyGoogle Scholar
  3. 3.
    Costa OLV, Filho EOA (1996) Discrete-time constrained quadratic control of Markovian jump linear systems. In: Conference on Decision Control 2:1763–1764Google Scholar
  4. 4.
    Costa OLV, Filho EOA, Boukas EK, Marques RP (1999) Constrained quadratic state feedback control of discrete-time Markovian jump linear systems. Automatica 35(4):617–626MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    do Val JBR, Başar T (1999) Receding horizon control of jump linear systems and a macroeconomic policy problem. J Econ Dyn Control 23(8):1099–131, 1999Google Scholar
  6. 6.
    Vargas AN, do Val JBR, Costa EF (2004) Receding horizon control of Markov jump linear systems subject to noise and unobserved state chain. In: Conference on decision and control, vol 4, pp 4381–4386Google Scholar
  7. 7.
    Park B-G, Kwon WH (2002) Robust one-step receding horizon control of discrete-time Markovian jump uncertain systems. Automatica 38(7):1229–1235MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Vargas AN, Furloni W, do Val JBR (2006) Constrained model predictive control of jump linear systems with noise and non-observed Markov state. In: American control conferenceGoogle Scholar
  9. 9.
    Vargas AN, Furloni W, do Val JBR (2007) Control of Markov jump linear systems with state and input constraints: a necessary optimality condition. In: 3rd IFAC symposium on system, structure and control, vol 3. pp 250–255Google Scholar
  10. 10.
    Vargas AN, Furloni W, do Val JBR (2013) Second moment constraints and the control problem of Markov jump linear systems. Numer Linear Algebra Appl 20(2):357–368Google Scholar
  11. 11.
    Blackmore L, Bektassov A, Ono M, Williams BC (2007) Robust, optimal predictive control of jump Markov linear systems using particles. In: Hybrid systems: computation and control, Lecture notes in computer science, vol 4416. Springer, New York, pp 104–117Google Scholar
  12. 12.
    Blackmore L, Ono M, Bektassov A, Williams BC (2010) A probabilistic particle-control approximation of chance-constrained stochastic predictive control. IEEE Trans Robot, 26(3):502–517Google Scholar
  13. 13.
    Yin Y, Shi Y, Liu F (2013) Constrained model predictive control on convex polyhedron stochastic linear parameter varying systems. Int. Journal of Innovative Computing. Inf Control 9(10):4193–4204Google Scholar
  14. 14.
    Yin Y, Liu Y, Karimi HR (2014) A simplified predictive control of constrained Markov jump system with mixed uncertainties. Abstr Appl Anal Special Issue:1–7Google Scholar
  15. 15.
    Lu J, Li D, Xi Y (2012) Constrained MPC of uncertain discrete-time Markovian jump linear systems. In: 31st Chinese control conference, pp 4131–4136Google Scholar
  16. 16.
    Song Y, Liu S, Wei G (2015) Constrained robust distributed model predictive control for uncertain discrete-time Markovian jump linear system. J Frankl Inst 352(1):73–92MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Dombrovskii VV, Dombrovskii DV, Lyashenko EA (2005) Predictive control of random-parameter systems with multiplicative noise. Application to investment portfolio optimization. Autom Remote Control 66(4):583–595MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Dombrovskii VV, Yu T (2011) Ob"edko. Predictive control of systems with Markovian jumps under constraints and its application to the investment portfolio optimization. Autom Remote Control 72(5):989–1003MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Yan Z, Wang J (2013) Stochastic model predictive control of Markov jump linear systems based on a two-layer recurrent neural network. In: IEEE international conference on information and automation, pp 564–569Google Scholar
  20. 20.
    Bernardini D, Bemporad A (2012) Stabilizing model predictive control of stochastic constrained linear systems. IEEE Trans Autom Control 57(6):1468–1480Google Scholar
  21. 21.
    Patrinos P, Sopasakis P, Sarimveis H, Bemporad A (2014) Stochastic model predictive control for constrained discrete-time Markovian switching systems. Automatica 50(10):2504–2514MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lu J, Xi Y, Li D, Cen L (2014) Probabilistic constrained stochastic model predictive control for Markovian jump linear systems with additive disturbance. In: 19th IFAC world congress, vol 19, pp 10469–10474Google Scholar
  23. 23.
    Chitraganti S, Aberkane S, Aubrun C, Valencia-Palomo G, Dragan V (2014) On control of discrete-time state-dependent jump linear systems with probabilistic constraints: a receding horizon approach. Syst Control Lett 74:81–89Google Scholar
  24. 24.
    Tonne J, Jilg M, Stursberg O (2015) Constrained model predictive control of high dimensional jump Markov linear systems. In: American control conference pp 2993–2998Google Scholar
  25. 25.
    Jerez JL, Kerrigan EC, Constantinides GA (2011) A condensed and sparse QP formulation for predictive control. In: 50th IEEE conference on decision and control, pp 5217–5222Google Scholar
  26. 26.
    Seber GAF, Lee AJ (2003) Linear regression analysis. Wiley series in probability and statistics. 2nd edn. Wiley, New YorkGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Volkswagen AGBaunatalGermany
  2. 2.Department of Electrical Engineering and Computer Science, Institute of Control and System TheoryUniversity of KasselKasselGermany

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