Soft Computing

, Volume 23, Issue 4, pp 1407–1419 | Cite as

Lyapunov–Krasovskii stable T2FNN controller for a class of nonlinear time-delay systems

  • Sehraneh GhaemiEmail author
  • Kamel Sabahi
  • Mohammad Ali Badamchizadeh
Methodologies and Application


In this paper, a type-2 fuzzy neural network (T2FNN) controller has been designed for a class of nonlinear time-delay systems using the feedback error learning (FEL) approach. In the FEL strategy, the T2FNN controller is in the feedforward path to overcome the nonlinearity and time delay and a classical controller is in the feedback path to guarantee the stability of the controlled system. Using the Lyapunov–Krasovskii stability theorem, the adaptation rules for training of T2FNN controller have been achieved in a way that, in the presence of the unknown disturbance and time-varying delay, the tacking error becomes zero. In the proposed stability criteria and adaptation laws, since just the training error is utilized, i.e., the mathematical model of the system or its parameters is not needed, the overall training and control algorithm is computationally simple. In the present study, the effect of delay has been considered in tuning the T2FNN parameters and, therefore, the performance of the proposed controller has been improved. The proposed strategy has been applied to systems with time-varying input delay and measurement noise and compared with indirect type-1 fuzzy sliding controller. The effectiveness of the proposed controller is shown by some simulation results.


Nonlinear time-delay system Type-2 fuzzy neural network controller Lyapunov–Krasovskii functional Measurement noise and stability 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of this paper.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Abid H, Toumi A (2016) Adaptive fuzzy sliding mode controller for a class of SISO nonlinear time-delay systems. Soft Comput 20:649–659CrossRefzbMATHGoogle Scholar
  2. Abiyev RH, Kaynak O (2010) Type 2 fuzzy neural structure for identification and control of time-varying plants. IEEE Trans Ind Electron 57:4147–4159CrossRefGoogle Scholar
  3. Allouche M, Dahech K, Chaabane M (2017) Multiobjective maximum power tracking control of photovoltaic systems: T-S fuzzy model-based approach. Soft Comput 1–12.
  4. Almaraashi M, John R, Hopgood A, Ahmadi S (2016) Learning of interval and general type-2 fuzzy logic systems using simulated annealing: theory and practice. Inf Sci 360:21–42CrossRefGoogle Scholar
  5. Almohammadi K, Hagras H, Alghazzawi D, Aldabbagh G (2016) A zSlices-based general type-2 fuzzy logic system for users-centric adaptive learning in large-scale e-learning platforms. Soft Comput 21:6859–6880CrossRefGoogle Scholar
  6. Amador-Angulo L, Castillo O (2016) A new fuzzy bee colony optimization with dynamic adaptation of parameters using interval type-2 fuzzy logic for tuning fuzzy controllers. Soft Comput 22:571–594CrossRefGoogle Scholar
  7. Antão R, Mota A, Martins RE (2016) Model-based control using interval type-2 fuzzy logic systems. Soft Comput 22(2):607–620Google Scholar
  8. Arefi MM, Zarei J, Karimi HR (2014) Adaptive output feedback neural network control of uncertain non-affine systems with unknown control direction. J Frankl Inst 351:4302–4316MathSciNetCrossRefzbMATHGoogle Scholar
  9. Bhatia NP, Szegö GP (1970) Stability theory of dynamical systems. Springer, New YorkCrossRefzbMATHGoogle Scholar
  10. Biglarbegian M, Melek W, Mendel JM (2011) Design of novel interval type-2 fuzzy controllers for modular and reconfigurable robots: theory and experiments. IEEE Trans Ind Electr 58:1371–1384CrossRefGoogle Scholar
  11. Boyd SP, El Ghaoui L, Feron E, Balakrishnan V (1994) Linear matrix inequalities in system and control theory. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  12. Castillo O, Melin P, Alanis A, Montiel O, Sepúlveda R (2011) Optimization of interval type-2 fuzzy logic controllers using evolutionary algorithms. Soft Comput 15:1145–1160CrossRefGoogle Scholar
  13. Du H, Zhang N (2009) Controller design for time-delay systems using genetic algorithms. Eng Appl Artif Intell 22:397–400CrossRefGoogle Scholar
  14. Gao Q, Feng G, Xi Z, Wang Y, Qiu J (2014) Robust control of T-S fuzzy time-delay systems via a new sliding-mode control scheme. IEEE Trans Fuzzy Syst 22:459–465CrossRefGoogle Scholar
  15. Guo L, Gu H, Zhang D (2010) Robust stability criteria for uncertain neutral system with interval time varying discrete delay. Asian J Control 12:739–745MathSciNetCrossRefGoogle Scholar
  16. Hale JK, Lunel SMV (2013) Introduction to functional differential equations. Springer, BerlinzbMATHGoogle Scholar
  17. Ideta AM (2006) Stability of feedback error learning method with time delay. Neurocomputing 69:1645–1654CrossRefGoogle Scholar
  18. Kawato M, Furukawa K, Suzuki R (1987) A hierarchical neural-network model for control and learning of voluntary movement. Biol Cybern 57:169–185CrossRefzbMATHGoogle Scholar
  19. Khanesar MA, Kaynak O, Yin S, Gao H (2015a) Adaptive indirect fuzzy sliding mode controller for networked control systems subject to time-varying network-induced time delay. IEEE Trans Fuzzy Syst 23:205–214CrossRefGoogle Scholar
  20. Khanesar MA, Kayacan E, Reyhanoglu M, Kaynak O (2015b) Feedback error learning control of magnetic satellites using type-2 fuzzy neural networks with elliptic membership functions. IEEE Trans Cybern 45:858–868CrossRefGoogle Scholar
  21. Koo GB, Park JB, Joo YH (2014) Decentralized fuzzy observer-based output-feedback control for nonlinear large-scale systems: an LMI approach. IEEE Trans Fuzzy Syst 22:406–419CrossRefGoogle Scholar
  22. Kumbasar T (2014) A simple design method for interval type-2 fuzzy PID controllers. Soft Comput 18:1293–1304CrossRefGoogle Scholar
  23. LaSalle JP (1976) The stability of dynamical systems. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  24. Lin T-C, Roopaei M (2010) Based on interval type-2 adaptive fuzzy \(\text{ H }\infty \) tracking controller for SISO time-delay nonlinear systems. Commun Nonlinear Sci Numer Simul 15:4065–4075MathSciNetCrossRefzbMATHGoogle Scholar
  25. Liu Y-J, Tong S (2016) Barrier Lyapunov functions-based adaptive control for a class of nonlinear pure-feedback systems with full state constraints. Automatica 64:70–75MathSciNetCrossRefzbMATHGoogle Scholar
  26. Liu Y-J, Tong S, Li D-J, Gao Y (2015) Fuzzy adaptive control with state observer for a class of nonlinear discrete-time systems with input constraint. IEEE Trans Fuzzy Syst 24:1147–1158CrossRefGoogle Scholar
  27. Maldonado Y, Castillo O, Melin P (2013) Particle swarm optimization of interval type-2 fuzzy systems for FPGA applications. Appl Soft Comput 13:496–508CrossRefGoogle Scholar
  28. Marouf S, Esfanjani RM, Akbari A, Barforooshan M (2016) T-S fuzzy controller design for stabilization of nonlinear networked control systems. Eng Appl Artif Intell 50:135–141CrossRefGoogle Scholar
  29. Nakanishi J, Schaal S (2004) Feedback error learning and nonlinear adaptive control. Neural Netw 17:1453–1465CrossRefzbMATHGoogle Scholar
  30. Perez J, Valdez F, Castillo O, Melin P, Gonzalez C, Martinez G (2017) Interval type-2 fuzzy logic for dynamic parameter adaptation in the bat algorithm. Soft Comput 21:667–685CrossRefGoogle Scholar
  31. Poursamad A, Davaie-Markazi AH (2009) Robust adaptive fuzzy control of unknown chaotic systems. Appl Soft Comput 9:970–976CrossRefGoogle Scholar
  32. Richard J-P (2003) Time-delay systems: an overview of some recent advances and open problems. Automatica 39:1667–1694MathSciNetCrossRefzbMATHGoogle Scholar
  33. Ruan X, Ding M, Gong D, Qiao J (2007) On-line adaptive control for inverted pendulum balancing based on feedback-error-learning. Neurocomputing 70:770–776CrossRefGoogle Scholar
  34. Sabahi K, Ghaemi S, Pezeshki S (2014) Application of type-2 fuzzy logic system for load frequency control using feedback error learning approaches. Appl Soft Comput 21:1–11CrossRefGoogle Scholar
  35. Saravanakumar R, Ali MS, Hua M (2016) H\(\infty \) state estimation of stochastic neural networks with mixed time-varying delays. Soft Comput 20:3475–3487CrossRefzbMATHGoogle Scholar
  36. Sheng L, Ma X (2014) Stability analysis and controller design of interval type-2 fuzzy systems with time delay. Int J Syst Sci 45:977–993MathSciNetCrossRefzbMATHGoogle Scholar
  37. Singh M, Srivastava S, Hanmandlu M, Gupta J (2009) Type-2 fuzzy wavelet networks (T2FWN) for system identification using fuzzy differential and Lyapunov stability algorithm. Appl Soft Comput 9:977–989CrossRefGoogle Scholar
  38. Smith OJ (1959) A controller to overcome dead time. ISA J 6:28–33Google Scholar
  39. Tsai S-H, Chen Y-A, Lo J-C (2016) A novel stabilization condition for a class of T-S fuzzy time-delay systems. Neurocomputing 175:223–232CrossRefGoogle Scholar
  40. Yu Z, Li S (2014) Neural-network-based output-feedback adaptive dynamic surface control for a class of stochastic nonlinear time-delay systems with unknown control directions. Neurocomputing 129:540–547CrossRefGoogle Scholar
  41. Zhou Q, Wu C, Jing X, Wang L (2016) Adaptive fuzzy backstepping dynamic surface control for nonlinear Input-delay systems. Neurocomputing 199:58–65CrossRefGoogle Scholar
  42. Zhu Q, Zhang T, Yang Y (2012) New results on adaptive neural control of a class of nonlinear systems with uncertain input delay. Neurocomputing 83:22–30CrossRefGoogle Scholar
  43. Ziegler JG, Nichols NB (1942) Optimum settings for automatic controllers, trans. ASME, 64Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of Electrical and Computer EngineeringUniversity of TabrizTabrizIran
  2. 2.Faculty of Electrical Engineering, Mamaghan BranchIslamic Azad UniversityMamaghanIran

Personalised recommendations