Circuits, Systems, and Signal Processing

, Volume 38, Issue 5, pp 2000–2022 | Cite as

On Reduced-Order Linear Functional Interval Observers for Nonlinear Uncertain Time-Delay Systems with External Unknown Disturbances

  • Dinh Cong Huong
  • Mai Viet ThuanEmail author


In this paper, we consider the problem of designing reduced-order linear functional interval observers for nonlinear uncertain time-delay systems with external unknown disturbances. Given bounds on the uncertainties, we design two reduced-order linear functional state observers in order to compute two estimates, an upper one and a lower one, which bound the unmeasured linear functions of state variables. Conditions for the existence of a pair of reduced-order linear functional observers are presented, and they are translated into a linear programming problem in which the observers’ matrices can be effectively computed. Finally, the effectiveness of the proposed design method is supported by four examples and simulation results.


Reduced-order observers Interval observers Uncertain models Biological systems 



The authors sincerely thank the anonymous reviewers for their constructive comments that helped improve the quality and presentation of this paper. This work was completed while the authors were visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). We would like to thank the Institute for its support and hospitality. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.01-2017.300.


  1. 1.
    M. Ait Rami, F. Tadeo, U. Helmke, Positive observers for linear positive systems, and their implications. Int. J. Control 84(4), 716–725 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    M. Ait Rami, M. Schönlein, J. Jordan, Estimation of linear positive systems with unknown time-varying delays. Eur. J. Control 19(3), 179–187 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. Ben-Israel, T.N.E. Greville, Generalized Inverses Theory and Applications (Springer, Berlin, 2003)zbMATHGoogle Scholar
  4. 4.
    J. Blesa, V. Puig, Y. Bolea, Fault detection using interval LPV models in an open-flow canal. Control Eng. Pract. 18(5), 460–470 (2010)CrossRefGoogle Scholar
  5. 5.
    J. Blesa, D. Rotondo, V. Puig, F. Nejjaria, FDI and FTC of wind turbines using the interval observer approach and virtual actuators/sensors. Control Eng. Pract. 24, 138–155 (2014)CrossRefGoogle Scholar
  6. 6.
    S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM Studies in Applied Mathematics (SIAM, Philadelphia, 1994)CrossRefzbMATHGoogle Scholar
  7. 7.
    J. Chen, R.J. Patton, Robust Model-Based Fault Diagnosis for Dynamic Systems (Kluwer, Boston, 1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Z. Chen, Z. Cao, Q. Huang, S.L. Campbell, Decentralized observer-based reliable control for a class of interconnected Markov Jumped time-delay system subject to actuator saturation and failure. Circuits Syst. Signal Process. (2018). MathSciNetGoogle Scholar
  9. 9.
    M. Darouach, Linear functional observers for systems with delays in state variables. IEEE Trans. Autom. Control 46(3), 491–496 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    D. Efimov, W. Perruquetti, J.P. Richard, Interval estimation for uncertain systems with time-varying delays. Int. J. Control 86(10), 1777–1787 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    D. Efimov, S. Li, Y. Hu, S. Muldoon, H. Javaherian, V.O. Nikiforov, Application of interval observers to estimation and control of air–fuel ratio in a direct injection engine, in Proceedings of the ACC, Chicago (2015)Google Scholar
  12. 12.
    D. Efimov, T. Raïssi, Design of interval observers for uncertain dynamical systems. Autom. Remote Control 77, 191–225 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    L. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications (Wiley, New York, 2000)CrossRefzbMATHGoogle Scholar
  14. 14.
    G. Goffaux, M. Remy, A.V. Wouwer, Continuous–discrete confidence interval observer-application to vehicle positioning. Inf. Fusion 14(4), 541–550 (2013)CrossRefGoogle Scholar
  15. 15.
    J.L. Gouzé, A. Rapaport, M.Z. Hadj-Sadok, Interval observers for uncertain biological systems. Ecol. Model. 133(1–2), 45–56 (2000)CrossRefGoogle Scholar
  16. 16.
    J.K. Hale, S.M.V. Lunel, Introduction to Functional Differential Equations (Springer, New York, 1993)CrossRefzbMATHGoogle Scholar
  17. 17.
    W.M. Haddad, V. Chellaboina, T. Rajpurohit, Dissipativity theory for nonnegative and compartmental dynamical systems with time delay. IEEE Trans. Autom. Control 49(5), 747–751 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    M.Z. Hadj-Sadok, J.L. Gouzé, Estimation of uncertain models of activated sludge processes with interval observers. J. Process Control 11(3), 299–310 (2001)CrossRefGoogle Scholar
  19. 19.
    M. Hou, P. Zitek, R.J. Patton, An observer design for linear time-delay systems. IEEE Trans. Autom. Control 47(1), 121–125 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    D.C. Huong, M.V. Thuan, State transformations of time-varying delay systems and their applications to state observer design. Discrete Contin. Dyn. Syst. Ser. S 10(3), 413–444 (2017)MathSciNetGoogle Scholar
  21. 21.
    T. Kaczorek, Fractional descriptor observers for fractional descriptor continuous-time linear system. Arch. Control Sci. 24, 27–37 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    A. Krener, A. Isidori, Linearization by output injection and nonlinear observers. Syst. Control Lett. 3(1), 47–52 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    A. Krener, W. Respondek, Nonlinear observers with linearization error dynamics. SIAM J. Control Optim. 23(2), 197–216 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    P. Li, J. Lam, Positive state-bounding observer for positive interval continuous-time systems with time delay. Int. J. Robust Nonlinear Control 22(11), 1244–1257 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    S. Li, Z. Xiang, Stabilisation of a class of positive switched nonlinear systems under asynchronous switching. Int. J. Syst. Sci. 48, 1537–1547 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    S. Li, Z. Xiang, Stochastic stability analysis and \(L_{\infty }\)-gain controller design for positive Markov jump systems with time-varying delays. Nonlinear Anal. Hybrid Syst. 22, 31–42 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    S. Li, Z. Xiang, H. Lin, H.R. Karimi, State estimation on positive Markovian jump systems with time-varying delay and uncertain transition probabilities. Inf. Sci. 369, 251–266 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Z. Liu, L. Zhao, H. Xiao, C. Gao, Adaptive \(H_{\infty }\) integral sliding mode control for uncertain singular time-delay systems based on observer. Circuits Syst. Signal Process. 36(11), 4365–4387 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    D.G. Luenberger, Introduction to Dynamic Systems: Theory, Models and Applications (Wiley, New York, 1979)zbMATHGoogle Scholar
  30. 30.
    F. Mazenc, O. Bernard, Interval observers for linear time-invariant systems with disturbances systems. Automatica 47, 140–147 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    F. Mazenc, S. Niculescu, O. Bernard, Exponentially stable interval observers for linear systems with delay. SIAM J. Control Optim. 50(1), 286–305 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    M. Moisan, O. Bernard, J.L. Gouzé, Near optimal interval observers bundle for uncertain bioreactors. Automatica 45(1), 291–295 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    M. Moisan, O. Bernard, Robust interval observers for global Lipschitz uncertain chaotic systems. Syst. Control Lett. 59(11), 687–694 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    J.D. Murray, Mathematical Biology (Springer, Berlin, 1990)zbMATHGoogle Scholar
  35. 35.
    P.T. Nam, H. Trinh, P.N. Pathirana, Reachable set bounding for nonlinear perturbed time-delay systems: the smallest bound. Appl. Math. Lett. 43, 68–71 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    B. Olivier, J.L. Gouzé, Closed loop observers bundle for uncertain biotechnological models. J. Process Control 14(7), 765–774 (2004)CrossRefGoogle Scholar
  37. 37.
    P. Pepe, Z.-P. Jiang, A Lyapunov–Krasovskii methodology for ISS and iISS of time-delay systems. Syst. Control Lett. 55, 1006–1014 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    V. Puig, A. Stancu, T. Escobet, F. Nejjaria, J. Quevedoa, R.J. Patton, Passive robust fault detection using interval observers: application to the DAMADICS benchmark problem. Control Eng. Pract. 14, 621–633 (2006)CrossRefGoogle Scholar
  39. 39.
    L. Qian, Q. Lu, J. Bai, Z. Feng, Dynamics of a prey-dependent digestive model with state-dependent impulsive control. Int. J. Bifurc. Chaos 22, 1250092 (2012). (11 pages) CrossRefzbMATHGoogle Scholar
  40. 40.
    C.R. Rao, Calculus of generalized inverses of matrices part I: general theory. Sankhya Ser. A 29, 317–342 (1967)MathSciNetzbMATHGoogle Scholar
  41. 41.
    A. Rapaport, J.L. Gouzé, Practical observers for uncertain affine output injection systems, in European Control Conference, CD-Rom, Karlsruhe, 31 August–3 September (1999), pp. 1505–1510Google Scholar
  42. 42.
    Z. Shu, J. Lam, H. Gao, B. Du, L. Wu, Positive observers and dynamic outputfeedback controllers for interval positive linear systems. IEEE Trans. Circuits Syst. I Regul. Pap. 55(10), 3209–3222 (2008)MathSciNetCrossRefGoogle Scholar
  43. 43.
    H. Trinh, T. Fernando, Functional Observers for Dynamical Systems (Springer, Berlin, 2012)CrossRefzbMATHGoogle Scholar
  44. 44.
    H. Trinh, D.C. Huong, L.V. Hien, S. Nahavandi, Design of reduced-order positive linear functional observers for positive time-delay systems. IEEE Trans. Circuits Syst. II Exp. Briefs 64(5), 555–559 (2017)CrossRefGoogle Scholar
  45. 45.
    S. Yin, S.X. Ding, X. Xia, H. Luo, A review on basic data-driven approaches for industrial process monitoring. IEEE Trans. Ind. Electron. 61(11), 6418–6428 (2014)CrossRefGoogle Scholar
  46. 46.
    S. Yin, X. Li, H. Gao, O. Kaynak, Data-based techniques focused on modern industry: an overview. IEEE Trans. Ind. Electron. 62(1), 657–667 (2015)CrossRefGoogle Scholar
  47. 47.
    V.D. Yurkevich, Multi-channel control system design for a robot manipulator based on the time-scale method. Optoelectron. Instrum. Data Process 52, 196–202 (2016)CrossRefGoogle Scholar
  48. 48.
    H. Zhang, Y. Shi, J. Wang, On energy-to-peak filtering for nonuniformly sampled nonlinear systems: a Markovian jump system approach. IEEE Trans. Fuzzy Syst. 22(1), 212–222 (2014)CrossRefGoogle Scholar
  49. 49.
    Y. Zhao, Z. Feng, Desynchronization in synchronous multi-coupled chaotic neurons by mix-adaptive feedback control. J. Biol. Dyn. 7, 1–10 (2013)MathSciNetCrossRefGoogle Scholar
  50. 50.
    J. Zheng, B. Cui, State estimation of chaotic Lurie systems via communication channel with transmission delay. Circuits Syst. Signal Process. 37(10), 4568–4583 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department for Management of Science and Technology DevelopmentTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.Department of Mathematics and InformaticsThainguyen University of SciencesThainguyenVietnam

Personalised recommendations