Robust functional observer for stabilising uncertain fuzzy systems with time-delay

  • Syed Imranul IslamEmail author
  • Peng Shi
  • Cheng-Chew Lim
Original Paper


This paper presents a new technique for stabilising a Takagi–Sugeno (T-S) fuzzy system with time-delay and uncertainty. A robust fuzzy functional observer is employed to design a controller when the system states are not measurable. The model uncertainty is norm bounded, and the time-delay is time-varying but bounded. The parallel distributed compensation method is applied for defining the fuzzy functional observer to design this controller. The proposed procedure reduces the observer order to the dimension of the control input. Improved stability conditions are provided for the observer compared with the existing results of functional observer-based stabilisation of T-S fuzzy models. Lyapunov–Krasovskii functionals are used to construct delay-dependent stability conditions as linear matrix inequalities. The solution of these inequalities is used for calculating the observer parameters. The sensitivity of the estimation error to the model uncertainty is reduced by minimising the \(L_2\) gain. The new design method developed is illustrated and verified using two examples.


Takagi–Sugeno fuzzy model Functional observer Time-delay Robust controller design 



This work was partially supported by the National Nature Science Foundation of China (61773131, U1509217), and the Australian Research Council (DP170102644).


  1. Boukal Y, Zasadzinski M, Darouach M, Radhy N (2016) Robust functional observer design for uncertain fractional-order time-varying delay systems. In: American control conference (ACC). IEEE, Boston, pp 2741–2746Google Scholar
  2. Chen B, Liu X (2005) Delay-dependent robust \(H_{\infty }\) control for T-S fuzzy systems with time delay. IEEE Trans Fuzzy Syst 13(4):544–556CrossRefGoogle Scholar
  3. Chen SM, Chang YC (2011) Weighted fuzzy rule interpolation based on GA-based weight-learning techniques. IEEE Trans Fuzzy Syst 19(4):729–744CrossRefGoogle Scholar
  4. Chen SM, Munif A, Chen GS, Chuan HL, Kuo BC (2012) Fuzzy risk analysis based on ranking generalized fuzzy numbers with different left heights and right heights. Expert Syst Appl 39(7):6320–6334CrossRefGoogle Scholar
  5. Darouach M (2000) Existence and design of functional observers for linear systems. IEEE Trans Autom Control 45(5):940–943MathSciNetCrossRefGoogle Scholar
  6. Darouach M (2001) Linear functional observers for systems with delays in state variables. IEEE Trans Autom Control 46(3):491–496MathSciNetCrossRefGoogle Scholar
  7. Fadali MS (2005) Fuzzy functional observers for dynamic TSK systems. In: The 14th IEEE International conference on fuzzy systems. IEEE, Reno, pp 389–394Google Scholar
  8. Feng G (2006) A survey on analysis and design of model-based fuzzy control systems. IEEE Trans Fuzzy Syst 14(5):676–697CrossRefGoogle Scholar
  9. Guerra TM, Vermeiren L (2004) LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi–Sugeno’s form. Automatica 40(5):823–829MathSciNetCrossRefGoogle Scholar
  10. Ha QP, Trinh H, Nguyen HT, Tuan HD (2003) Dynamic output feedback sliding-mode control using pole placement and linear functional observers. IEEE Trans Ind Electron 50(5):1030–1037CrossRefGoogle Scholar
  11. Islam SI, Lim CC, Shi P (2018a) Functional observer based controller for stabilizing Takagi–Sugeno fuzzy systems with time-delays. J Frankl Inst 355(8):3619–3640MathSciNetCrossRefGoogle Scholar
  12. Islam SI, Lim CC, Shi P (2018b) Functional observer-based fuzzy controller design for continuous nonlinear systems. Int J Syst Sci 49(5):1047–1060MathSciNetCrossRefGoogle Scholar
  13. Jiang B, Karimi HR, Kao Y, Gao C (2018) A novel robust fuzzy integral sliding mode control for nonlinear semi-markovian jump T-S fuzzy systems. IEEE Trans Fuzzy Syst.
  14. Krokavec D, Filasov A (2012) A reduced-order TS fuzzy observer scheme with application to actuator faults reconstruction. Math Probl Eng 2012:1–25MathSciNetGoogle Scholar
  15. Krokavec D, Filasov A (2014) On reduced-order fuzzy observer for one class of bilinear Takagi–Sugeno systems. In: 2014 European control conference (ECC). IEEE, Strasbourg, pp 981–986Google Scholar
  16. Lai YF, Chen MY, Chiang HS (2018) Constructing the lie detection system with fuzzy reasoning approach. Granul Comput 3(2):169–176CrossRefGoogle Scholar
  17. Li J, Niemann D, Wang HO, Tanaka K (1999) Parallel distributed compensation for Takagi–Sugeno fuzzy models: multiobjective controller design. In: American control conference, vol 3. IEEE, San Diego, pp 1832–1836Google Scholar
  18. Liu H, Zhang L (2018) Fuzzy rule-based systems for recognition-intensive classification in granular computing context. Granul Comput. CrossRefGoogle Scholar
  19. Liu X, Zhang Q (2003) New approaches to \(H_{\infty }\) controller designs based on fuzzy observers for T-S fuzzy systems via LMI. Automatica 39(9):1571–1582MathSciNetCrossRefGoogle Scholar
  20. Ma XJ, Sun ZQ (2001) Analysis and design of fuzzy reduced-dimensional observer and fuzzy functional observer. Fuzzy Sets Syst 120:35–63MathSciNetCrossRefGoogle Scholar
  21. Mohajerpoor R, Abdi H, Nahavandi S (2016) A new algorithm to design minimal multi-functional observers for linear systems. Asian J Control 18(3):842–857MathSciNetCrossRefGoogle Scholar
  22. Nguang SK, Shi P (2003) \(H_{\infty }\) fuzzy output feedback control design for nonlinear systems: an LMI approach. IEEE Trans Fuzzy Syst 11(3):331–340CrossRefGoogle Scholar
  23. Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New YorkzbMATHGoogle Scholar
  24. Shi P, Boukas EK, Shi Y, Agarwal RK (2003) Optimal guaranteed cost control of uncertain discrete time-delay systems. J Comput Appl Math 157(2):435–451MathSciNetCrossRefGoogle Scholar
  25. Shojaei F, Arefi MM, Khayatian A, Karimi HR (2018) Observer-based fuzzy adaptive dynamic surface control of uncertain nonstrict feedback systems with unknown control direction and unknown dead-zone. IEEE Trans Syst Man Cybern Syst.
  26. Sun C, Wang F, He X (2017) Robust fault-tolerant control for fuzzy delay systems with unmeasurable premise variables via uncertain system approach. Int J Innov Comput Inf Control 13(3):823–846Google Scholar
  27. Takagi T, Sugeno M (1985) Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern SMC–15(1):116–132CrossRefGoogle Scholar
  28. Tanaka K, Sugeno M (1992) Stability analysis and design of fuzzy control systems. Fuzzy Sets Syst 45(2):135–156MathSciNetCrossRefGoogle Scholar
  29. Tanaka K, Ikeda T, Wang HO (1998) Fuzzy regulators and fuzzy observers: relaxed stability conditions and LMI-based designs. IEEE Trans Fuzzy Syst 6(2):250–265CrossRefGoogle Scholar
  30. Teh PS, Trinh H (2012) Partial state and unknown input estimation for time-delay systems. Int J Syst Sci 43(4):748–763MathSciNetCrossRefGoogle Scholar
  31. Tran HM, Trinh H, Nam PT (2015) Functional observer-based fault detection of time-delay systems via an LMI approach. In: Australian control conference (AUCC). IEEE, Gold Coast, pp 194–199Google Scholar
  32. Trinh H, Fernando T (2007) On the existence and design of functional observers for linear systems. In: International conference on mechatronics and automation. IEEE, Harbin, pp 1974–1979Google Scholar
  33. Wang G, Li Y, Li X (2017) Approximation performance of the nonlinear hybrid fuzzy system based on variable universe. Granul Comput 2(2):73–84. CrossRefGoogle Scholar
  34. Wang H, Chen SM (2008) Evaluating students’ answerscripts using fuzzy numbers associated with degrees of confidence. IEEE Trans Fuzzy Syst 16(2):403–415CrossRefGoogle Scholar
  35. Wang HO, Tanaka K, Grifln M (1995) Parallel distributed compensation of nonlinear systems by Takagi–Sugeno fuzzy model. In: IEEE international conference on fuzzy systems, vol 2. IEEE, Yokohama, pp 531–538Google Scholar
  36. Wang HO, Tanaka K, Griffin MF (1996) An approach to fuzzy control of nonlinear systems: stability and design issues. IEEE Trans Fuzzy Syst 4(1):10CrossRefGoogle Scholar
  37. Wang Y, Karimi HR, Lam H, Shen H (2018) An improved result on exponential stabilization of sampled-data fuzzy systems. IEEE Trans Fuzzy Syst.
  38. Wu ZG, Shi P, Su H, Chu J (2012) Reliable \(H_\infty\) control for discrete-time fuzzy systems with infinite-distributed delay. IEEE Trans Fuzzy Syst 20(1):22–31CrossRefGoogle Scholar
  39. Yordanova S, Yankov V, Jain L (2017) MIMO fuzzy logic supervisor-based adaptive control using the example of coupled-tanks levels control. Int J Innov Comput Inf Control 13(2):453–470Google Scholar
  40. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353CrossRefGoogle Scholar
  41. Zhang J, Shi P, Qiu J, Nguang SK (2015) A novel observer-based output feedback controller design for discrete-time fuzzy systems. IEEE Trans Fuzzy Syst 23(1):7Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Electrical and Electronic EngineeringThe University of AdelaideAdelaideAustralia

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