Robust Model Predictive Control for Uncertain Positive Time-delay Systems

  • Junfeng ZhangEmail author
  • Haoyue Yang
  • Miao Li
  • Qian Wang
Regular Papers Control Theory and Applications


This paper proposes the robust model predictive control of positive time-delay systems with interval and polytopic uncertainties, respectively. The model predictive control framework consists of linear constraint, linear performance index, linear Lyapunov function, linear programming algorithm, and cone invariant set. By virtue of matrix decomposition technique, robust model predictive controllers of interval and polytopic positive systems with multiple state delays are designed, respectively. A multi step control strategy is utilized and a cone invariant set is constructed. Linear programming is used for the corresponding MPC conditions. Finally, a numerical example is given to verify the effectiveness of the proposed design.


Linear programming robust model predictive control time delay uncertain positive systems 


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  1. [1]
    T. Kaczorek, Positive 1D and 2D Systems, Springer–Verlag, London, 2002.CrossRefzbMATHGoogle Scholar
  2. [2]
    L. Farina and S. Rinaldi, Positive Linear Systems: Theory and Applications, Wiley, New York, 2000.CrossRefzbMATHGoogle Scholar
  3. [3]
    A. Khanafer, T. Basar, and B. Gharesifard, “Stability of epidemic models over directed graphs: A positive systems approach,” Automatica, vol. 74, pp. 126–134, December 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    R. Shorten, F. Wirth, and D. Leith, “A positive systems model of TCP–like congestion control: asymptotic results,” IEEE/ACM Trans. Net., vol. 14,no. 3, pp. 616–629, June 2006.Google Scholar
  5. [5]
    M. Ait Rami and F. Tadeo, “Controller synthesis for positive linear systems with bounded controls,” IEEE Trans. Circuits Syst. II Expr. Briefs, vol. 54, no. 2, pp. 151–155, February 2007.CrossRefGoogle Scholar
  6. [6]
    M. Ait Rami, F. Tadeo, and A. Benzaouia, “Control of constrained positive discrete systems,” Proc. American Control Conf., Marriott Marquis, New York, USA, pp. 5851–5856, 2007.Google Scholar
  7. [7]
    O. Mason and R. Shorten, “On linear copositive Lyapunov functions and the stability of switched positive linear systems,” IEEE Trans. Autom. Control, vol. 52, no. 7, pp. 1346–1349, July 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    F. Knorn, O. Mason, and R. Shorten, “On linear co–positive Lyapunov functions for sets of linear positive systems,” Automatica, vol. 45, no. 8, pp. 1943–1947, August 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    C. Briat, “Robust stability and stabilization of uncertain linear positive systems via integral linear constraints: L1–and L¥–gains characterization,” Int. J. Robust Nonlinear Control, vol. 23, no. 17, pp. 1932–1954, November 2013.CrossRefzbMATHGoogle Scholar
  10. [10]
    A. Rantzer, “Scalable control of positive systems,” Europ. J. Control, vol. 24, pp. 72–80, July 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    A. Benzaouia, A. Hmamed, F. Mesquine, M. Benhayoun, and F. Tadeo, “Stabilization of continuous–time fractional positive systems by using a Lyapunov function,” IEEE Trans. Autom. Control, vol. 59, no. 8, pp. 2203–2208, August 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    X. Zhao, L. Zhang, P. Shi, and M. Liu, “Stability of switched positive linear systems with average dwell time switching,” Automatica, vol. 48, no. 6, pp. 1132–1137, June 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    J. Lian and J. Liu, “New results on stability of switched positive systems: an average dwell–time approach,” IET Control Theory Appl., vol. 7, no. 12, pp. 1651–1658, August 2013.MathSciNetCrossRefGoogle Scholar
  14. [14]
    E. Hernandez–Vargas, P. Colaneri, R. Middleton, and F. Blanchini, “Discrete–time control for switched positive systems with application to mitigating viral escape,” Int. J. Robust Nonlinear Control, vol. 21, no. 10, pp. 1093–1111, July 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    X. Ding and X. Liu, “Stability analysis for switched positive linear systems under state–dependent switching,” Int. J. Control Autom. Syst., vol. 15, pp. 481–488, April 2017.CrossRefGoogle Scholar
  16. [16]
    Y. Ebihara, D. Peaucelle, and D. Arzelier, “L1 gain analysis of linear positive systems and its application,” The 50th IEEE Confer. Decision Control Europ. Control Confer., pp. 4029–4034, 2011.CrossRefGoogle Scholar
  17. [17]
    X. Liu, “Constrained control of positive systems with delays,” IEEE Trans. Autom. Control, vol. 54, no. 7, pp. 1596–1600, July 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    X. Liu, W. Yu, and L. Wang, “Stability analysis of positive systems with bounded time–varying delays,” IEEE Trans. Circuits Syst. II Expr. Briefs, vol. 56, no. 7, pp. 600–604, July 2009.CrossRefGoogle Scholar
  19. [19]
    M. Busowicz, “Robust stability of positive continuous–time linear systems with delays,” Int. J. Applied Math. Computer Science, vol. 20, no. 4, pp. 665–670, December 2010.MathSciNetCrossRefGoogle Scholar
  20. [20]
    P. H. A. Ngoc, “On a class of positive linear differential equations with infinite delay,” Syst. Control Lett., vol. 60, no. 12, pp. 1038–1044, December 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    P. H. A. Ngoc, “Stability of positive differential systems with delay,” IEEE Trans. Autom. Control, vol. 58, no. 1, pp. 203–209, Januray 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    J. Shen and J. Lam, “L¥–gain analysis for positive systems with distributed delays,” Automatica, vol. 50, no. 1, pp. 175–179, January 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    J. Shen and S. Chen, “Stability and L¥–gain analysis for a class of nonlinear positive systems with mixed delays,” Int. J. Robust Nonlinear Control, vol. 27, no. 1, pp. 39–49, January 2017.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Y. Ebihara, D. Peaucelle, D. Arzelier, and F. Gouaisbaut, “Dominant pole analysis of stable time–delay positive systems,” IET Control Theory Appl., vol. 8, no. 17, pp. 1963–1971, November 2014.MathSciNetCrossRefGoogle Scholar
  25. [25]
    Y. Wang, J. Zhang, and M. Liu, “Exponential stability of impulsive positive systems with mixed time–varying delays,” IET Control Theory Appl., vol. 8, no. 15, pp. 1537–1542, October 2014.MathSciNetCrossRefGoogle Scholar
  26. [26]
    V.S. Bokharaie and O. Mason, “On delay–independent stability of a class of nonlinear positive time–delay systems,” IEEE Trans. Autom. Control, vol. 59, no. 7, pp. 1974–1977, July 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    W. Qi and X. Gao, “Positive L1–gain filter design for positive continuous–time Markovian jump systems with partly known transition rates,” Int. J. Control Autom. Syst., vol. 14, pp. 1413–1420, December 2016.CrossRefGoogle Scholar
  28. [28]
    D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica, vol. 36, no. 6, pp. 789–814, June 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    S. Qin and T.A. Badgwell, “A survey of industrial model predictive control technology,” Control Engineer. Practice, vol. 11, no. 7, pp. 733–764, July 2003.CrossRefGoogle Scholar
  30. [30]
    M. V. Kothare, V. Balakrishnan, and M. Morari, “Robust constrained model predictive control using linear matrix inequalities,” Automatica, vol. 32, no. 10, pp. 1361–1379, October 1996.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    E. F. Camacho and C. Bordons, Model predictive control in the process industry, Springer Science Business Media, 2012.Google Scholar
  32. [32]
    D. He, L. Wang, and J. Sun, “On stability of multiobjective NMPC with objective prioritization,” Automatica, vol. 57, pp. 189–198, July 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    D. He, S. Yu, and L. Ou, “Lexicographic MPC with multiple economic criteria for constrained nonlinear systems,” J. Franklin Inst., vol. 355, no. 2, pp. 753–773, January 2018.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    R. Yang, G. Liu, P. Shi, C. Thomas, and M. V. Basin, “Predictive output feedback control for networked control systems,” IEEE Trans. Industrial Electr., vol. 61, no. 1, pp. 512–520, Januray 2014.CrossRefGoogle Scholar
  35. [35]
    J. H. Lee, “Model predictive control: Review of the three decades of development,” Int. J. Control Autom. Syst., vol. 9, no. 3, pp. 415–424, June 2011.CrossRefGoogle Scholar
  36. [36]
    Y. Xi, D. Li, and S. Lin, “Model predictive control–status and challenges,” Acta Autom. Sinica, vol. 39, no. 3, pp. 222–236, March 2013.MathSciNetCrossRefGoogle Scholar
  37. [37]
    D. Q. Mayne, “Model predictive control: Recent developments and future promise,” Automatica, vol. 50, no. 12, pp. 2967–2986, December 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    J. Zhang, X. Cai, W. Zhang, and Z. Han, “Robust model predictive control with ℓ1–gain performance for positive systems,” J. Frankl. Instit., vol. 352, no. 7, pp. 2831–2846, July 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    J. Zhang, X. Zhao, Y. Zuo, and R. Zhang, “Linear programming–based robust model predictive control for positive systems,” IET Control Theory Appl., vol. 10, no. 15, pp. 1789–1797, October 2016.MathSciNetCrossRefGoogle Scholar
  40. [40]
    J. Zhang, X. Jia, R. Zhang, and S. Fu, “Parameterdependent Lyapunov function based model predictive control for positive systems and its application in urban water management,” Proc. of the 36th Chinese Control Conf., Dalian, July 26th–28th, pp. 4573–4578, 2017.Google Scholar
  41. [41]
    S. C. Jeong and P. G. Park, “Constrained MPC algorithm for uncertain time–varying systems with state–delay,” IEEE Trans. Autom. Control, vol. 50, no. 2, pp. 257–263, February 2005.MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    B. Ding and B. Huang, “Constrained robust model predictive control for time–delay systems with polytopic description,” Int. J. Control, vol. 80, no. 4, pp. 509–522, February 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    B. Ding, “Robust model predictive control for multiple time delay systems with polytopic uncertainty description,” Int. J. Control, vol. 83, no. 9, pp. 1844–1857, August 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    D. Li and Y. Xi, “Constrained feedback robust model predictive control for polytopic uncertain systems with time delays,” Int. J. Syst. Sci., vol. 42, no. 10, pp. 1651–1660, October 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    S. Olaru and S. I. Niculescu, “Predictive control for linear systems with delayed input subject to constraints,” IFAC Proceedings Volumes, vol. 41, no. 2, pp. 11208–11213, 2008.CrossRefGoogle Scholar
  46. [46]
    Y. Shi, T. Chai, H. Wang, and C. Su, “Delay–dependent robust model predictive control for time–delay systems with input constraints,” In American Control Conf., pp. 4880–4885, 2009.Google Scholar
  47. [47]
    M. A. F. Martins, A. S. Yamashita, B. F. Santoro, and D. Odloak, “Robust model predictive control of integrating time delay processes,” J. Process Control, vol. 23, no. 7, pp. 917–932, August 2013.CrossRefGoogle Scholar
  48. [48]
    I. Škrjanc, S. BlažiŠ, S. Oblak, and J. Richalet, “An approach to predictive control of multivariable time–delayed plant: Stability and design issues,” ISA Trans., vol. 43, no. 4, pp. 585–595, October 2004.CrossRefGoogle Scholar
  49. [49]
    C. Ocampo–Martinez, V. Puig, G. Cembrano, R. Creus, and M. Minoves, “Improving water management efficiency by using optimization–based control strategies: The Barcelona case study,” Water Sci. Technol. Water Supply, vol. 9, no. 5, pp. 565–575, December 2009.CrossRefGoogle Scholar
  50. [50]
    C. Ocampo–Martinez, V. Puig, G. Cembrano, and J. Quevedo, “Application of predictive control strategies to the management of complex networks in the urban water cycle,” IEEE Control Syst. Magazine, vol. 33, no. 1, pp. 15–41, January 2013.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Junfeng Zhang
    • 1
    • 2
    Email author
  • Haoyue Yang
    • 1
  • Miao Li
    • 1
  • Qian Wang
    • 1
  1. 1.School of AutomationHangzhou Dianzi University310018China
  2. 2.Key Laboratory of System Control and Information ProcessingMinistry of Education of ChinaShanghaiChina

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