Fuzzy Relational Matrix-Based Stability Analysis for First-Order Fuzzy Relational Dynamic Systems

Document Type : Research Paper


1 Electrical Engineering Department, Amirkabir University of Technology (AUT), Tehran, Iran

2 Electrical Engineering Department, Amirk- abir University of Technology (AUT), Tehran, Iran


In this paper, two sets of sufficient conditions are obtained to ensure the existence and stability of a unique equilibrium point of unforced first-order fuzzy relational dynamical systems by using two different approaches which are both based on the fuzzy relational matrix of the model.
In the first approach, the equilibrium point of the system is one of the centers of the related membership functions.
In the second approach, the equilibrium point of the system is the origin (the center of the middle membership function) and the behavior of the system, though can be nonlinear, is symmetric around the origin.
The results are approved by numerical examples.


A. Aghili Ashtiani and M. B. Menhaj,
{\it Construction and applications of a modified fuzzy relational model},
Journal of Intelligent \& Fuzzy Systems, DOI: 10.3233/IFS-130838, in press.
A. Aghili Ashtiani, P. Raja, S. K. Y. Nikravesh, {\it Various symmetries in matrix theory with application to modeling dynamic systems}, The Journal of Nonlinear Science and Applications (JNSA), {\bf7(1)} (2014), 63-69. 
A. Aghili Ashtiani and S. K. Y. Nikravesh,
{\it A new approach to stability analysis of fuzzy relational model of dynamic systems},
Iranian Journal of Fuzzy Systems, \textbf{9(1)} (2012), 39--48.
%A. Aghili Ashtiani, S. K. Y. Nikravesh, and P. Raja,
%{\it Centrally Symmetric, Plus Symmetric, and Row-wise Symmetric Matrices and their Properties},
%Proc. 5th Seminar on Linear Algebra and its Applications,
%October 28-30, Babolsar, Iran (2009) 13--16.
%A. Aghili Ashtiani and M. B. Menhaj,
%{\it Introducing a New Pair of Differentiable Fuzzy Norms and its Application to Fuzzy Relational Function Approximation},
%Proc. 10th Joint Conference on Information Sciences,
%Salt Lake City, USA: World Scientific Publishing Co. ISBN: 978-981-270-967-7 (2007) 1329--1336.
A. Aghili Ashtiani and M. B. Menhaj,
{\it Numerical solution of fuzzy relational equations based on smooth fuzzy norms},
Soft Computing, \textbf{14(6)} (2010), 545--557.
J. Chen and L. Chen,
{\it Study on stability of fuzzy closed-loop control systems},
Fuzzy Sets and Systems, \textbf{57(2)} (1993), 159--168.
%\bibitem{fkp} T. Furuhashi, H. Kakami, J. Peters and W. Pedrycz, A Stability Analysis of Fuzzy Control System Using a Generalized fuzzy Petri Net Model, IEEE World Congress on Computational Intelligence.
%\bibitem{hf} T. Hasegawa and T. Furuhashi, Stability analysis of fuzzy control systems simplified as a discrete system, Control and Cybernetics, 27, No. 4 (1998).
A. Kandel, Y. Luo and Y. Q. Zhang,
{\it Stability analysis of fuzzy control systems},
Fuzzy Sets and Systems, \textbf{105} (1999), 33--48.
%%\bibitem{kp} B. Kelkar and B. Postlethwaite, Enhancing the generality of fuzzy relational models for control, Fuzzy Sets and Systems 100 (1998) 117-129.
J. B. Kiszka, M. M. Gupta, and P. N. Nikiforuk,
{\it Energetic stability of fuzzy dynamic systems},
IEEE Transactions on Systems Man and Cybernetics, \textbf{15} (1985), 783--792.
C. Kolodziej and R. Priemer,
{\it Stability analysis of fuzzy systems},
Journal of the Franklin Institute, \textbf{336} (1999), 851--873.
%\bibitem{l} T. Leephakpreeda, Stability Analysis of a Fuzzy Control System, Thammasat Int. J. Sc. Tech., 2, No. l (1997).
\bibitem{Pedrycz1993FCFS} % Book
W. Pedrycz,
{\it Fuzzy control and fuzzy systems},
2'nd Ed., Research Studies Press Ltd., John Wiley \& Sons Inc., 1993.
W. Pedrycz,
{\it An identification algorithm in fuzzy relational systems},
Fuzzy Sets and Systems, \textbf{13} (1984), 153--167.
J. N. Ridley, I. S. Shaw, and J. J. Kruger,
{\it Probabilistic fuzzy model for dynamic systems},
Electronics Letters, \textbf{24} (1988), 890--892.
%E. Sanchez,
%{\it Resolution of Composite Fuzzy Relation Equations},
%Information and Control 30(1) (1976) 38--48.
A. A. Suratgar and S. K. Y. Nikravesh,
{\it A new method for linguistic modeling with stability analysis and applications},
Intelligent Automation and Soft Computing, \textbf{15(3)} (2009), 329--342.
%A. A. Suratgar and S. K. Y. Nikravesh,
%{\it Potential energy based stability analysis of fuzzy linguistic systems},
%Iranian Journal of Fuzzy Systems 2(1) (2005) 67--74.
%%A. A. Suratgar and S. K. Y. Nikravesh,
%%{\it A New Sufficient Condition for Stability of Fuzzy Systems},
%%Proceeding of Iranian Conference in Electrical Engineering, (2002) 441--445.
T. $\check{S}$ijak, S. Te$\check{s}$njak and O. Kuljaca,
{\it Stability analysis of fuzzy control system using describing function method},
Proc. 9th Mediterranean Conf. Control and Automation,
Dubrovnik, Croatia (2001).