2012
9
2
2
161
Cover vol. 9, no.2, June 2012
2
2
1

0
0
BEHAVIOR OF SOLUTIONS TO A FUZZY NONLINEAR
DIFFERENCE EQUATION
BEHAVIOR OF SOLUTIONS TO A FUZZY NONLINEAR
DIFFERENCE EQUATION
2
2
In this paper, we study the existence, asymptotic behavior of the positive solutions of a fuzzy nonlinear difference equation$$ x_{n+1}=frac{Ax_n+x_{n1}}{B+x_{n1}}, n=0,1,cdots,$$ where $(x_n)$ is a sequence of positive fuzzy number, $A, B$ are positive fuzzy numbers and the initial conditions $x_{1}, x_0$ are positive fuzzy numbers.
1
In this paper, we study the existence, asymptotic behavior of the positive solutions of a fuzzy nonlinear difference equation$$ x_{n+1}=frac{Ax_n+x_{n1}}{B+x_{n1}}, n=0,1,cdots,$$ where $(x_n)$ is a sequence of positive fuzzy number, $A, B$ are positive fuzzy numbers and the initial conditions $x_{1}, x_0$ are positive fuzzy numbers.
1
12
Qianhong
Zhang
Qianhong
Zhang
Guizhou Key Laboratory of Economics System Simulation, Guizhou
University of Finance and Economics, Guiyang, Guizhou 550004, P. R. China
Guizhou Key Laboratory of Economics System
China
zqianhong68@com.cn
Lihui
Yang
Lihui
Yang
Department of Mathematics, Hunan City University, Yiyang, Hunan
413000, P. R. China
Department of Mathematics, Hunan City University,
China
ll.hh.yang@gmail.com
Daixi
Liao
Daixi
Liao
Basic Science Department, Hunan Institute of Technology, Hengyang,
Hunan 421002, P. R. China
Basic Science Department, Hunan Institute
China
liaodaixizaici@sohu.com
Fuzzy difference equation
Boundedness
Persistence
Equilibrium point
stability
[bibitem{AbAl: Mfs}## S. Abbasbandy and M. Alavi, {it A method for solving fuzzy linear system}, Iranian Journal of Fuzzy Systems, {bf 2} (2005), 3743. ##bibitem{BeFr: Aam}## D. Benest and C. Froeschle, {it Analysis and modelling of discrete dynamical## systems}, Gordon and Breach Science Publishers, The Netherland, 1998. ## bibitem{DeKo: Abf}## E. Y. Deeba and A. De Korvin, {it Analysis of fuzzy difference equations of## a model of $CO_2$ level in the blood}, Appl. Math. Lett., {bf 12} (1999), 3340. ## bibitem{DeKo:Afd}## E. Y. Deeba, A. De Korvin and E. L. Koh, {it A fuzzy difference equation with an## application}, J. Difference Equation Appl., {bf 2} (1996), 365374. ## bibitem{DeLa: Nsc}## R. DeVault, G. Ladas and S. W. Schultz, {it Necessary and sufficient conditions## the boundedness of $x_{n+1}=A/x_n^p+B/x_{n1}^q$}, J. Difference## Equations Appl., {bf 3} (1998), 259266. ## bibitem{DeLa: Ors}## R. DeVault, G. Ladas and S. W. Schultz, {it On the recursive sequence $x_{n+1}=A/x_n+1/x_{n2}$}, Proc. Amer. Math. Soc.,## {bf 126} (1998), 32573261. ##bibitem{KoLa:Gbn}## V. L. Kocic and G. Ladas, {it Golobal behavior of nonlinear difference## equations of higher order with applications}, Kluwer Academic Publishers, 1993. ##bibitem{KuLa: Rde}## M. R. S. Kulenvic, G. Ladas and N. R. Prokup, {it A rational difference equation}, Comput. Math. Appl., {bf 41} (2001),## 671678. ## bibitem{LiSu: Drd}## W. Li and H. Sun, {it Dynamics of a rational difference equation}, Appl. Math. Comput.,## {bf 163} (2005), 577591. ## bibitem{PaPa: Ofd1}## G. Papaschinopoulos and B. K. Papadopoulos, {it On the fuzzy difference equation $x_{n+1}=A+B/x_n$},## Soft Comput., {bf 6} (2002), 456461. ##bibitem{PaPa:Ofd2}## G. Papaschinopoulos and B. K. Papadopoulos, {it On the fuzzy difference equation## $x_{n+1}=A+x_n/x_{nm}$}, Fuzzy Sets and Systems, {bf 129} (2002), 7381. ##bibitem{PaSc: Ost}## G. Papaschinopoulos and C. J. Schinas, {it On a systems of two nonlinear difference equation},## J. Math. Anal. Appl., {bf 219} (1998), 415426. ##bibitem{PaSc: Ofd}## G. Papaschinopoulos and C. J. Schinas, {it On the fuzzy difference equation $x_{n+1}=sum_{i=0}^{k1}A_i/x_{ni}^{p_i}+1/x_{nk}^{p_k}$},## J. Difference Equation Appl., {bf 6(7)} (2000), 8589. ## bibitem{PaSt: Bab}##G. Papaschinopoulos and G. Stefanidou, {it Boundedness and asymptotic##behavior of the solutions of a fuzzy difference equation}, Fuzzy Sets and Systems, {bf 140} (2003), 523539. ## bibitem{PhPu: Gai}## C. G. Philos, I. K. Purnaras and Y. G. Sficas, {it Global attractivity in a## nonlinear difference equation}, Appl. Math. Comput., {bf 62} (1994), 249258. ##bibitem{StPa: Fde}## G. Stefanidou and G. Papaschinopoulos, {it A fuzzy difference equation of a rational form}, J. Nonlin. Math. Phys.,## Supplement, {bf 12(2)} (2005), 300315. ##bibitem{WuZh: Epn}## C. Wu and B. Zhang, {it Embedding problem of noncompact fuzzy number space## $E^{sim}$}, Fuzzy Sets and Systems, {bf 105} (1999), 165169.##]
GENERALIZED FUZZY VALUED $theta$Choquet INTEGRALS
AND THEIR DOUBLENULL ASYMPTOTIC ADDITIVITY
GENERALIZED FUZZY VALUED CHOQUET INTEGRALS
AND THEIR DOUBLENULL ASYMPTOTIC ADDITIVITY
2
2
The generalized fuzzy valued $theta$Choquet integrals will beestablished for the given $mu$integrable fuzzy valued functionson a general fuzzy measure space, and the convergence theorems ofthis kind of fuzzy valued integral are being discussed.Furthermore, the whole of integrals is regarded as a fuzzy valuedset function on measurable space, the doublenull asymptoticadditivity and pseudodoublenull asymptotic additivity of thefuzzy valued set functions formed are studied when the fuzzymeasure satisfies autocontinuity from above (below).\
1
The generalized fuzzy valued $theta$Choquet integrals will beestablished for the given $mu$integrable fuzzy valued functionson a general fuzzy measure space, and the convergence theorems ofthis kind of fuzzy valued integral are being discussed.Furthermore, the whole of integrals is regarded as a fuzzy valuedset function on measurable space, the doublenull asymptoticadditivity and pseudodoublenull asymptotic additivity of thefuzzy valued set functions formed are studied when the fuzzymeasure satisfies autocontinuity from above (below).\
13
24
Guijun
Wang
Guijun
Wang
School of Mathematics Science, Tianjin Normal University, Tianjin
300387, China
School of Mathematics Science, Tianjin Normal
China
tjwgj@126.com
Xiaoping
Li
Xiaoping
Li
School of Management, Tianjin Normal University, Tianjin 300387,
China
School of Management, Tianjin Normal University,
China
lxpmath@126.com
Fuzzy measures
Fuzzy valued $theta$Choquet integrals
Autocontinuous from above (below)
Doublenull asymptotic additive
Pseudodoublenull asymptotic additive
[[1] C. Alaca, A new perspective to the mazurulam problem in 2fuzzy 2normed linear spaces,##Iranian Journal of Fuzzy Systems, 7(2) (2010), 109119.##[2] A. Maturo, On some structures of fuzzy numbers, Iranian Journal of Fuzzy Systems, 6(4)##(2009), 4959.##[3] E. Pasha, A. Saiedifar and B. Asady, The percentiles of fuzzy numbers and their applications,##Iranian Journal of Fuzzy Systems, 6(1) (2009), 2744.##[4] M. Sugeno, Theory of fuzzy integrals and its applications, Ph. D. Dissertation, Tokyo Institute##of Technology, 1974.##[5] Z. Y. Wang, The autocontinuity of set function and the fuzzy integral, J. Math. Anal. Appl.,##99 (1984), 195218.##[6] Z. Y. Wang, Asymptotic structural characteristics of fuzzy measure and their applications,##Fuzzy Sets and Systems, 16(2) (1985), 227290.##[7] Z. Y. Wang, J. K. George and W. Wang, Monotone set functions dened by Choquet integral,##Fuzzy Sets and Systems, 81(2) (1996), 241250.##[8] G. J. Wang and X. P. Li, On the convergence of fuzzy valued functional dened byintegrable##fuzzy valued functions, Fuzzy Sets and Systems, 107(2) (1999), 219226.##[9] G. J. Wang and X. P. Li, Autocontinuity and preservation of structural characteristics of##generalized fuzzy numbervalued Choquet integrals, Advances in Mathematics, (in Chinese),##34(1) (2005), 91100.##[10] G. J. Wang and X. P. Li, Pseudoautocontinuity and heredity of generalized fuzzy number##valued Choquet integrals, Systems Sci. Math. Sci., (in Chinese), 26(4) (2006), 126132.##[11] G. J. Wang and X. P. Li, On the CI average convergence for sequence of fuzzy valued##functions, Systems Sci. Math. Sci., (in Chinese), 29(2) (2009), 253262.##[12] G. Q. Zhang, Fuzzy Measure, Guiyang: Guizhou Technology Press, (in Chinese), 1998.##]
OPTIMIZED FUZZY CONTROL DESIGN OF AN
AUTONOMOUS UNDERWATER VEHICLE
OPTIMIZED FUZZY CONTROL DESIGN OF AN
AUTONOMOUS UNDERWATER VEHICLE
2
2
In this study, the roll, yaw and depth fuzzy control of an Au tonomous Underwater Vehicle (AUV) are addressed. Yaw and roll angles are regulated only using their errors and rates, but due to the complexity of depth dynamic channel, additional pitch rate quantity is used to improve the depth loop performance. The discussed AUV has four aps at the rear of the vehicle as actuators. Two rule bases and membership functions based on Mamdani type and Sugeno type fuzzy rule have been chosen in each loop. By invoking the normalized steepest descent optimization method, the optimum values for the membership function parameters are found. Though the AUV is a highly nonlinear system, the simulation of the designed fuzzy logic control system based on the equations of motion shows desirable behavior of the AUV spe cially when the parameters of the fuzzy membership functions are optimized.
1
In this study, the roll, yaw and depth fuzzy control of an Au tonomous Underwater Vehicle (AUV) are addressed. Yaw and roll angles are regulated only using their errors and rates, but due to the complexity of depth dynamic channel, additional pitch rate quantity is used to improve the depth loop performance. The discussed AUV has four aps at the rear of the vehicle as actuators. Two rule bases and membership functions based on Mamdani type and Sugeno type fuzzy rule have been chosen in each loop. By invoking the normalized steepest descent optimization method, the optimum values for the membership function parameters are found. Though the AUV is a highly nonlinear system, the simulation of the designed fuzzy logic control system based on the equations of motion shows desirable behavior of the AUV spe cially when the parameters of the fuzzy membership functions are optimized.
25
41
Behrooz
Raeisy
Behrooz
Raeisy
School of Electrical and Computer Engineering, Shiraz Univer
sity, Shiraz, Iran and Iranian Space Agency, Iranian Space Center, Mechanic Institute,
Shiraz, Iran, P.O. Box: 71555414
School of Electrical and Computer Engineering,
Iran
raeisy@shirazu.ac.ir
Ali Akbar
Safavi
Ali Akbar
Safavi
School of Electrical and Computer Engineering, Shiraz Univer
sity, Shiraz, Iran
School of Electrical and Computer Engineering,
Iran
safavi@shirazu.ac.ir
Ali Reza
Khayatian
Ali Reza
Khayatian
School of Electrical and Computer Engineering, Shiraz Uni
versity, Shiraz, Iran
School of Electrical and Computer Engineering,
Iran
khayatia@shirazu.ac.ir
Fuzzy optimized control
Autonomous underwater vehicle
Normalized steepest descent
Neural Network
[[1] G. Antonelli, S. Chiaverini, N. Sarkar and M. West, Adaptive control of an autonomous##underwater vehicle. experimental results on ODIN, IEEE Proceedings, International Sympo##sium on Computational Intelligence in Robotics and Automation, 1999. ##[2] A. Balasuriya and L. Cong, Adaptive fuzzy sliding mode controller for underwater vehicles,##IEEE Proceedings, The 4th international conference on control and automations (ICCA'03),##Canada, June 2003.##[3] T. BinaZadeh, A. R. Khayatian and P. KarimAghaee, Identification and control of 6 DOF underwater##variable mass object, 13th iranian conference on electrical engineering (ICEE2005),##Zanjan, Iran, 2005.##[4] J. Blakelock, Automatic control of aircraft and missiles, 2nd edition, Willy, February 1991.##[5] F. Dougherty, T. Sherman, G. Woolweaver and G. Lovell, An autonomous underwater vehicle##(AUV) flight control system using sliding mode control, Proceedings, OCEANS '88.##Baltimore, MD USA, Oct. 1988.##[6] T. Fossen and M. Blanke, Nonlinear output feedback control of underwater vehicle propellers##using feedback from estimated axial flow velocity, IEEE Journal of Oceanic Engineering, Apr##[7] T. Fossen, Guidance and control of ocean vehicles, John Wiley & Sons, 1994.##[8] J. S. Han, H. S. Kim and J. Neggers, Actions, norms, subactions and kernels of (fuzzy)##norms, Iranian Journal of Fuzzy Systems, 7(2) (2010), 141147.##[9] A. Hasankhani, A. Nazari and M. Sahelis, Some properties of fuzzy hilbert spaces and norm##of operators, Iranian Journal of Fuzzy Systems, 7(3) (2010), 129157.##[10] K. Ishii and T. Ura, An adaptive neuralnet controller system for an underwater vehicle,##Control Engineering Practice, Elsevierm, 8(2) (2000), 177184.##[11] J. S. R. Jang, C. T. Sun and E. Mizutani, Neurofuzzy and soft computing, Prentic Hall,##[12] N. E. Leonard and P. S. Krishnaprasad, Motion control of an autonomous underwater vehicle##with an adaptive feature, IEEE Proceedings of Autonomous Underwater Vehicle Technology,##AUV '94, Cambridge, MA, USA, Jul 1994.##[13] J. H. Li, P. M. Lee and S. J. Lee, Neural net based nonlinear daptive control for autonomous##underwater vehicles, IEEE international Conference on Robotics and Automation, May 2002.##[14] Y. Nakamura and S. Savant, Nonlinear tracking control of autonomous underwater vehicles,##IEEE Proceedings on Robotics and Automation, May 1992.##[15] T. Prestero, Development of a sixdegree of freedom simulation model for the REMUS autonomous##underwater vehicle, MTS/IEEE Conference and Exhibition, OCEANS, 2001.##[16] B. Raeisy, M. Kharati, A. A. Safavi and A. R. Khayatian, Equation of motion derivation of##variable mass underwater vehicle and 6DOF simulation with helping of neural network, 17th##Anual International Conference on Mechanical Engineering, Tehran, Iran, May 2009.##[17] B. Raeisy, A. A. Safavi and A. R. Khayatian, Fuzzy logic depth control of an autonomous##underwater vehicle and optimization of it with normalize steepened descent method, 17th##Anual International Conference on Mechanical Engineering, Tehran, Iran, May 2009.##[18] B. Raeisy, A. A. Safavi and A. R. Khayatian, Optimized fuzzy logic yaw and roll control of##an autonomous underwater vehicle, 8th Iranian Conference on Fuzzy System, Tehran, Iran,##October 2008.##[19] L. Rodrigues and P.Tavares, Sliding mode control of an AUV in the diving and steering##planes, MTS/IEEE Conference Proceedings, MG de Sousa Prado OCEANS'96, Sep 1996.##[20] N. Sey'edi and M. A. Mirjalili, Hydrodynamic stability coefficients calculation of submarine##and its weapons using added mass and misile DATCOM, 4th Conference of Underwater##Science and Technology (fcoust), Isfahan, May 2007.##[21] E. Shivanian and E. Khoram, Optimization of linear objective function subject to fuzzy relation##inequalities constraints with maxproduct compozition, Iranian Journal of Fuzzy Systems,##7(3) (2010), 5171.##[22] S. R.Vukelich, S. L. Stoy and M. E. Moore, Missile DATCOM user’s manual, Dought Aircraft##Company Inc., 1988.##[23] J. Wang and G. Lee, Selfadaptive recurrent neurofuzzy control of an autonomous underwater##vehicle, IEEE Transactions on Robotics and Automation, 19(2) (2003).##]
NONFRAGILE GUARANTEED COST CONTROL OF
TS FUZZY TIMEVARYING DELAY SYSTEMS WITH
LOCAL BILINEAR MODELS
NONFRAGILE GUARANTEED COST CONTROL OF
TS FUZZY TIMEVARYING DELAY SYSTEMS WITH
LOCAL BILINEAR MODELS
2
2
This paper focuses on the nonfragile guaranteed cost control problem for a class of TS fuzzy timevarying delay systems with local bilinear models. The objective is to design a nonfragile guaranteed cost state feedback controller via the parallel distributed compensation (PDC) approach such that the closedloop system is delaydependent asymptotically stable and the closedloop performance is no more than a certain upper bound in the presence of the additive controller gain perturbations. A sufficient condition for the existence of such nonfragile guaranteed cost controllers is derived via the linear matrix inequality (LMI) approach and the design problem of the fuzzy controller is formulated in term of LMIs. The simulation examples show that the proposed approach is effective.
1
This paper focuses on the nonfragile guaranteed cost control problem for a class of TS fuzzy timevarying delay systems with local bilinear models. The objective is to design a nonfragile guaranteed cost state feedback controller via the parallel distributed compensation (PDC) approach such that the closedloop system is delaydependent asymptotically stable and the closedloop performance is no more than a certain upper bound in the presence of the additive controller gain perturbations. A sufficient condition for the existence of such nonfragile guaranteed cost controllers is derived via the linear matrix inequality (LMI) approach and the design problem of the fuzzy controller is formulated in term of LMIs. The simulation examples show that the proposed approach is effective.
43
62
Junmin
Li
Junmin
Li
Department of Mathematics, Xidian University, 710071, Xi'an, P.R. China
Department of Mathematics, Xidian University,
China
jmli@mail.xidian.edu.cn
Guo
Zhang
Guo
Zhang
Department of Electrical Engineering and Automation, Luoyang Insti
tute of Science and Technology, Luoyang, 471023, P.R. China
Department of Electrical Engineering and
China
gzhang163163@163.com
Fuzzy control
Nonfragile control
Guaranteed cost control
Delaydependent
linear matrix inequality (LMI)
TS fuzzy bilinear model
[[1] B. Chen and X. P. Liu, Delaydependent robust Hinnite control for TS fuzzy systems with##time delay, IEEE Trans. Fuzzy Systems, 13 (2005), 238249.##[2] B. Chen, C. Lin, X. P. Liu and S. C. Tong, Guaranteed cost control of T{S fuzzy systems##with input delay, Int. J. Robust Nonlinear Control, 18 (2008), 12301256.##[3] B. Chen, X. P. Liu, S. C. Tong and C. Lin, Observerbased stabilization of T{S fuzzy systems##with input delay, IEEE Trans. Fuzzy Systems, 16 (2008), 652663.##[4] B. Chen, X. Liu, S. Tong and C. Lin, guaranteed cost control of TS fuzzy systems with state##and input delay, Fuzzy Sets and Systems, 158 (2007), 22512267.##[5] M. Chen, G. Feng, H. Ma and G. Chen, Delaydependent Hinnite lter design for discrete##time fuzzy systems with timevarying delays, IEEE Trans. Fuzzy Systems, 17 (2009), 604616.##[6] W. H. Chen, Z. H. Guan and X. M. Lu, Delaydependent output feedback guaranteed cost##control for uncertain timedelay systems, Automatica, 44 (2004), 12631268.##[7] J. Dong, Y. Wang and G. Yang, Control synthesis of continuoustime TS fuzzy systems##with local nonlinear models, IEEE Trans. Systems, Man, CyberneticsPart B, 39 (2009),##12451258.##[8] B. Z. Du, J. Lam and Z. Shu, Stabilization for state/input delay systems via static and##integral output feedback, Automatica, 46 (2010), 20002007.##[9] D. L. Elliott, Bilinear systems in Encyclopedia of Electrical Engineering, New York: Wiley,##[10] H. J. Gao, J. Lam and Z. D. Wang, Discrete bilinear stochastic systems with timevarying##delay: stability analysis and control synthesis, Chaos, Solitons and Fractals, 34 (2007), 394##[11] H. J. Gao, X. Liu and J. Lam, Stability analysis and stabilization for discretetime fuzzy##systems with timevarying delay, IEEE Trans. Systems, Man, CyberneticsPart B, 39 (2009),##[12] D. W. C. Ho and Y. Niu, Robust fuzzy design for nonlinear uncertain stochastic systems via##slidingmode control, IEEE Trans Fuzzy Systems, 15 (2007), 350358.##[13] H. L. Huang and F. G. Shi, Robust H1 control for TCS timevarying delay systems with##norm bounded uncertainty based on LMI approach, Iranian Journal of Fuzzy Systems, 6##(2009), 114.##[14] L. H. Keel and S. P. Bhattacharryya, Robust, fragile, or optimal, IEEE Trans. Automatic##Control, 42 (1997), 10981105.##[15] J. H. Kim, Delaydependent robust and nonfragile guaranteed cost control for uncertain##singular systems with timevarying state and input delays, International Journal of Control,##Automation and Systems, 7 (2009), 357364.##[16] F. Leibfritz, An LMIbased algorithm for designing suboptimal static H2/Hinnite output##feedback controllers, SIAM J Control Optimization, 57 (2001), 17111735.##[17] J. M. Li, G. Zhang and C. Du, Robust Hinnity control for a class of multiple input fuzzy##bilinear systems with uncertainties, Control Theory and Applications, 26 (2009), 12981302.##[18] L. Li and X. D. Liu, New approach on robust stability for uncertain T{S fuzzy systems with##state and input delays, Chaos, Solitons and Fractals, 40 (2009), 23292339.##[19] T. H. S. Li and S. H. Tsai, TS fuzzy bilinear model and fuzzy controller design for a class##of nonlinear systems, IEEE Trans. Fuzzy Systems, 15 (2007), 494505.##[20] T. H. S. Li, S. H. Tsai and et al, Robust Hinnite fuzzy control for a class of uncertain##discrete fuzzy bilinear systems, IEEE Trans. Systems, Man, CyberneticsPart B, 38 (2008),##[21] R. R. Mohler, Bilinear control processes, New York: Academic, 1973.##[22] C. T. Pang and Y. Y. Lur, On the stability of TakagiSugeno fuzzy systems with timevarying##uncertainties, IEEE Trans. Fuzzy Systems, 16 (2008), 162170.##[23] R. E. Precup, S. Preitl, J. K. Tar, M. L. Tomescu, M. Takacs, P. Korondi and P. Baranyi,##Fuzzy control systems performance enhancement by iterative learning control, IEEE Trans.##Industrial Electronics, 55 (2008), 34613475.##[24] K. Tanaka and H. O. Wang, Fuzzy control systems design and analysis: a linear matrix##inequality approach, John Wiley and Sons, 2001.##[25] S. H. Tsai and T. H. S. Li, Robust fuzzy control of a class of fuzzy bilinear systems with##timedelay, Chaos, Solitons and Fractals, 39 (2007), 20282040.##[26] R. J. Wang, W. W. Lin and W. J. Wang, Stabilizability of linear quadratic state feedback##for uncertain fuzzy timedelay systems, IEEE Trans. Systems, Man, CyberneticsPart B, 34##(2004), 12881292.##[27] H. N. Wu and H. X. Li, New approach to delaydependent stability analysis and stabilization##for continuoustime fuzzy systems with timevarying delay, IEEE Trans. Fuzzy Systems, 15##(2007), 482493.##[28] D. D. Yang and K. Y. Cai, Reliable guaranteed cost sampling control for nonlinear timedelay##systems, Mathematics and Computers in Simulation, 80 (2010), 20052018.##[29] G. H. Yang and J. L. Wang, Nonfragile Hinnite control for linear systems with multiplica##tive controller gain variations, Automatica, 37 (2001), 727737.##[30] G. H. Yang, J. L. Wang and C. Lin, Hinnite control for linear systems with additive##controller gain variations, Int. J Control, 73 (2000), 15001506.##[31] J. S. Yee, G. H. Yang and J. L. Wang, Nonfragile guaranteed cost control for discretetime##uncertain linear systems, Int. J Systems Science, 32 (2001), 845853.##[32] K. W. Yu and C. H. Lien, Robust Hinnite control for uncertain T{S fuzzy systems with##state and input delays, Chaos, Solitons and Fractals, 37 (2008), 150156.##[33] D. Yue and J. Lam, Nonfragile guaranteed cost control for uncertain descriptor systems with##timevarying state and input delays, Optimal Control Applications and Methods, 26 (2005),##[34] B. Y. Zhang, S. S. Zhou and T. Li, A new approach to robust and nonfragile Hinnitecontrol##for uncertain fuzzy systems, Information Sciences, 177 (2007), 51185133.##[35] J. Zhang, Y. Xia and R. Tao, New results on Hinnite ltering for fuzzy timedelay systems,##IEEE Trans. Fuzzy Systems, 17 (2009), 128137.##[36] J. H. Zhang, P. Shi and J. Q. Qiu, Nonfragile guaranteed cost control for uncertain stochastic##nonlinear timedelay systems, Journal of the Franklin Institute, 3462009676690.##[37] S. S. Zhou, J. Lam and W. X. Zheng, Control design for fuzzy systems based on relaxed##nonquadratic stability and Hinnite performance conditions, IEEE Trans. Fuzzy Systems,##15 (2007), 188198.##[38] S. S. Zhou and T. Li, Robust stabilization for delayed discretetime fuzzy systems via basis##dependent LyapunovKrasovskii function, Fuzzy Sets and Systems, 151 (2005), 139153.##]
Statistical Convergence and Strong $p$Ces`{a}ro Summability of Order $beta$
in Sequences of Fuzzy Numbers
Statistical Convergence and Strong $p$Ces`{a}ro Summability of Order $beta$
in Sequences of Fuzzy Numbers
2
2
In this study we introduce the concepts of statistical convergence of order$beta$ and strong $p$Ces`{a}ro summability of order $beta$ for sequencesof fuzzy numbers. Also, we give some relations between the statisticalconvergence of order $beta$ and strong $p$Ces`{a}ro summability of order$beta$ and construct some interesting examples.
1
In this study we introduce the concepts of statistical convergence of order$beta$ and strong $p$Ces`{a}ro summability of order $beta$ for sequencesof fuzzy numbers. Also, we give some relations between the statisticalconvergence of order $beta$ and strong $p$Ces`{a}ro summability of order$beta$ and construct some interesting examples.
63
73
H.
Altinok
H.
Altinok
Department of Mathematics, Firat University, 23119, Elazig, Turkey
Department of Mathematics, Firat University,
Turkey
hifsialtinok@yahoo.com
Y.
Altin
Y.
Altin
Department of Mathematics, Firat University, 23119, Elazig, Turkey
Department of Mathematics, Firat University,
Turkey
yaltin23@yahoo.com
M.
Isik
M.
Isik
Department of Statistics, Firat University, 23119, Elazig, Turkey
Department of Statistics, Firat University,
Turkey
misik63@yahoo.com
Fuzzy number
Statistical convergence
Cesro summability
[bibitem{altin2}##Y. Altin, M. Et and M. Bac{s}ari r, textit{On some##generalized difference sequences of fuzzy numbers}, Kuwait J. Sci. Eng.,## {bf 34(1A)} (2007), 114. ##bibitem{altinok}H. Altinok, R. c{C}olak and M. Et, $lambda##$textit{Difference sequence spaces of fuzzy numbers}, Fuzzy Sets and##Systems, textbf{160(21)} (2009), 31283139. ##bibitem{Aytar}S. Aytar and S. Pehlivan, textit{Statistically convergence##of sequences of fuzzy numbers and sequences of }$alpha$textit{cuts},##International J. General Systems, (2007), 17. ##bibitem{burgin}M. Burgin, textit{Theory of fuzzy limits}, Fuzzy Sets and##Systems, textbf{115} (2000), 433443. ##bibitem{Connor}J. S. Connor, textit{The statistical and strong }%##$p$textit{Ces`{a}ro convergence of sequences}, Analysis, textbf{8(12)} (1988), 4763. ##bibitem{colakb}R. c{C}olak, textit{Statistical convergence of order##}$alpha,$ Modern Methods in Analysis and its Applications, Anamaya Publ.##New Delhi, India, (2010), 121129. ##bibitem{Diamond}P. Diamond and P. Kloeden, textit{Metric spaces of fuzzy##sets}, Fuzzy Sets and Systems, textbf{35(2)} (1990), 241249. ##bibitem{Fang}J. X. Fang and H. Hung, textit{On the level convergence of##a sequence of fuzzy numbers}, Fuzzy Sets and Systems, textbf{147} (2004), 417435. ##bibitem{Fast}H. Fast, textit{Sur la convergence statistique}, Colloquium##Math., textbf{2} (1951), 241244. ##bibitem{Fridy}J. A. Fridy, textit{On statistical convergence}, Analysis,##textbf{5(4)} (1985), 301313. ##bibitem{gadjiev}A. D. Gadjiev and C. Orhan, textit{Some approximation##theorems via statistical convergence}, Rocky Mountain J. Math., textbf{32(1)} (2002), 129138. ##bibitem{hancl}J. Hanv{c}l, L. Miv{s}'{i}k and J. T'{o}th,##textit{Cluster points of sequences of fuzzy real numbers}, Soft Computing,##textbf{14(4)} (2010), 399404. ##bibitem{kumar}V. Kumar and K. Kumar, textit{On the ideal convergence of##sequences of fuzzy numbers}, Information Sciences, textbf{178(24)} (2008), 46704678. ##bibitem{kwon}J. S. Kwon, textit{On statistical and }$p$%##textit{Ces`{a}ro convergence of fuzzy numbers}, Korean J. Comput. &Appl.##Math., textbf{7(1)} (2000), 195203. ##bibitem{Matloka}M. Matloka, textit{Sequences of fuzzy numbers}, BUSEFAL,##textbf{28} (1986), 2837. ##bibitem{mur}M. Mursaleen and M. Bac{s}ari r, textit{On some new##sequence spaces of fuzzy numbers}, Indian J. Pure Appl. Math., textbf{34(9)} (2003), 13511357. ##bibitem{Nuray}F. Nuray and E. Savac{s}, textit{Statistical convergence##of sequences of fuzzy real numbers}, Math. Slovaca, textbf{45(3)} (1995), 269273. ##bibitem{Schoenberg}I. J. Schoenberg, textit{The integrability of certain##functions and related summability methods}, Amer. Math. Monthly, textbf{66} (1959), 361375. ##bibitem{talo}"{O}. Talo and F. Bac{s}ar, textit{On the space }%##$bv_{p}(F)$textit{ of sequences of }$p$textit{bounded variation of fuzzy##numbers}, Acta Math. Sin. Engl. Ser., textbf{24(7)} (2008), 12051212. ##bibitem{Tri}B. C. Tripathy and A. J. Dutta, textit{On fuzzy##realvalued double sequence space }$_{2}ell_{F}^{p}$textit{, Math. Comput.##Modelling}, {bf 46(910)} (2007), 12941299.##]
A MODIFICATION ON RIDGE ESTIMATION FOR FUZZY
NONPARAMETRIC REGRESSION
A MODIFICATION ON RIDGE ESTIMATION FOR FUZZY
NONPARAMETRIC REGRESSION
2
2
This paper deals with ridge estimation of fuzzy nonparametric regression models using triangular fuzzy numbers. This estimation method is obtained by implementing ridge regression learning algorithm in the La grangian dual space. The distance measure for fuzzy numbers that suggested by Diamond is used and the local linear smoothing technique with the cross validation procedure for selecting the optimal value of the smoothing param eter is fuzzi ed to t the presented model. Some simulation experiments are then presented which indicate the performance of the proposed method.
1
This paper deals with ridge estimation of fuzzy nonparametric regression models using triangular fuzzy numbers. This estimation method is obtained by implementing ridge regression learning algorithm in the La grangian dual space. The distance measure for fuzzy numbers that suggested by Diamond is used and the local linear smoothing technique with the cross validation procedure for selecting the optimal value of the smoothing param eter is fuzzi ed to t the presented model. Some simulation experiments are then presented which indicate the performance of the proposed method.
75
88
Rahman
Farnoosh
Rahman
Farnoosh
School of Mathematics, Iran University of Science and Tech
nology, Narmak, Tehran16846, Iran
School of Mathematics, Iran University of
Iran
rfarnoosh@iust.ac.ir
Javad
Ghasemian
Javad
Ghasemian
School of Mathematics, Iran University of Science and Technol
ogy, Narmak, Tehran16846, Iran
School of Mathematics, Iran University of
Iran
jghasemian@iust.ac.ir, jghasemian@gmail.com
Omid
Solaymani Fard
Omid
Solaymani Fard
School of Mathematics and Computer Science, Damghan Uni
versity, Damghan, Iran
School of Mathematics and Computer Science,
Iran
osfard@du.ac.ir, omidsfard@gmail.com
Fuzzy regression
Ridge estimation
Fuzzy nonparametric regression
Local linear smoothing
[[1] C. B. Cheng and E. S. Lee, Fuzzy regression with radial basis function networks, Fuzzy Sets##and Systems, 119 (2001), 291301.##[2] P. Diamond, Fuzzy least squares, Information Sciences, 46 (1988), 141157.##[3] N. R. Draper and H. Smith, Applied Regression Analysis, Wiley, New York, 1980.##[4] H. Drucker, C. Burges, L. Kaufman, A. Smola and V. N. Vapnik, Support vector regression##machines, in: M. C. Mozer, M. I. Jordan, T. Petsche, Eds., Advances in Neural Information##Processing Systems, MIT Press, Cambridge, MA, 9 (1996), 155162.##[5] O. S. Fard and A. V. Kamyad, Modied kstep method for solving fuzzy initial value problems,##Iranian Journal of Fuzzy Systems, 8(1) (2011), 4963.##[6] J. Fan and I. Gijbels, Local Polynomial Modeling and Its Applications, Chapman & Hall,##London, 1996.##[7] W. Hardle, Applied Nonparametric Regression, Cambridge University Press, New York, 1990.##[8] J. D. Hart, Nonparametric Smoothing and Lackoft Tests, SpringerVerlag, New York, 1997.##[9] T. J. Hastie and R. J. Tibshirani, Generalized Additive Models, Chapman & Hall, London,##[10] D. H. Hong and C. Hwang, Support vector fuzzy regression machines, Fuzzy Sets and Systems,##138 (2003), 271281.##[11] D. H. Hong, C. Hwang and C. Ahn, Ridge estimation for regression models with crisp inputs##and Gaussian fuzzy output, Fuzzy Sets and Systems, 142 (2004), 307319. ##[12] A. E. Hoerl and R. W. Kennard, Ridge regression: biased estimates for nonorthogonal prob##lems, Technometrics, 12 (1970), 5567.##[13] H. Ishibuchi and H. Tanaka, Fuzzy regression analysis using neural networks, Fuzzy Sets and##Systems, 50 (1992), 257265.##[14] H. Ishibuchi and H. Tanaka, Fuzzy neural networks with interval weights and its application##to fuzzy regression analysis, Fuzzy Sets and Systems, 57 (1993), 2739.##[15] B. Kim and R. R. Bishu, Evaluation of fuzzy linear regression models by comparing mem##bership functions, Fuzzy Sets and Systems, 100 (1998), 343352.##[16] R. X. Liu, J. Kuang, Q. Gong and X. L. Hou, Principal component regression analysis with##SPSS, Computer Methods and Programs in Biomedicine, 71 (2003), 141147.##[17] S. Pourahmad, S. M. T. Ayatollahi and S. M. Taheri, Fuzzy logistic regression: a new possi##bilistic model and its application in clinical vague status, Iranian Journal of Fuzzy Systems,##8 (2011), 117.##[18] H. Shakouri G and R. Nadimi, A novel fuzzy linear regression model based on a nonequality##possibility index and optimum uncertainty, Applied Soft Computing, 9 (2009), 590598.##[19] C. Saunders, A. Gammerman and V. Vork, Ridge regression learning algorithm in dual vari##able, Proceedings of the 15th International Conference on Machine Learning, (1998), 515521.##[20] N. Wang, W. X. Zhang and C. L. Mei, Fuzzy nonparametric regression based on local linear##smoothing technique, Information Sciences, 177 (2007), 38823900.##[21] M. S. Yang and C. H. Ko, On a class of fuzzy cnumbers clustering procedures for fuzzy data,##Fuzzy Sets and Systems, 84 (1996), 4960.##]
Delaydependent robust stabilization and $H_{infty}$
control for uncertain stochastic TS fuzzy systems with multiple
time delays
Delaydependent robust stabilization and $H_{infty}$
control for uncertain stochastic TS fuzzy systems with multiple
time delays
2
2
In this paper, the problems of robust stabilization and$H_{infty}$ control for uncertain stochastic systems withmultiple time delays represented by the TakagiSugeno (TS) fuzzymodel have been studied. By constructing a new LyapunovKrasovskiifunctional (LKF) and using the bounding techniques, sufficientconditions for the delaydependent robust stabilization and $H_{infty}$ control scheme are presented in terms of linear matrixinequalities (LMIs). By solving these LMIs, a desired fuzzycontroller can be obtained which can be easily calculated byMatlab LMI control toolbox. Finally, a numerical simulation isgiven to illustrate the effectiveness of the proposed method.
1
In this paper, the problems of robust stabilization and$H_{infty}$ control for uncertain stochastic systems withmultiple time delays represented by the TakagiSugeno (TS) fuzzymodel have been studied. By constructing a new LyapunovKrasovskiifunctional (LKF) and using the bounding techniques, sufficientconditions for the delaydependent robust stabilization and $H_{infty}$ control scheme are presented in terms of linear matrixinequalities (LMIs). By solving these LMIs, a desired fuzzycontroller can be obtained which can be easily calculated byMatlab LMI control toolbox. Finally, a numerical simulation isgiven to illustrate the effectiveness of the proposed method.
89
111
T.
Senthilkumar
T.
Senthilkumar
Department of Mathematics, Gandhigram Rural InstituteDeemed
University, Gandhigram  624 302, Tamilnadu, India
Department of Mathematics, Gandhigram Rural
India
tskumar2410@gmail.com
P.
Balasubramaniam
P.
Balasubramaniam
Department of Mathematics, Gandhigram Rural Institute
Deemed University, Gandhigram  624 302, Tamilnadu, India
Department of Mathematics, Gandhigram Rural
India
balugru@gmail.com
TakagiSugeno (TS) fuzzy systems
Robust $H_{infty}$ control
Stochastic system
Linear matrix inequalities (LMIs)
Multiple time delays
LyapunovKrasovskii functional (LKF)
[bibitem{BR} M. Basin and A. Rodkina, {it On delaydependent stability for a class of## nonlinear stochastic systems with multiple state delays}, Nonlinear## Anal., {bf 68} (2008), 21472157. ##bibitem{BL} E. K. Boukas and Z. K. Liu, {it Deterministic and stochastic timedelay## systems}, Birkhäuser, Boston, 2002. ##bibitem{BGFB} S. Boyd, L. E. Ghauoi, E. Feron and V. Balakrishnan, {it Linear matrix inequalities in## system and control theory}, SIAM, Philadelphia, PA, 1994. ##bibitem{CF} Y. Y. Cao and P. M. Frank, {it Stability analysis and synthesis of nonlinear## timedelay systems via linear TakagiSugeno fuzzy models},## Fuzzy Sets and Systems, {bf 124} (2001), 213229. ##bibitem{CRF} S. G. Cao, N. W. Rees and G. Feng, {it Stability analysis and design## for a class of continuoustime fuzzy control systems},## Int. J. Control, {bf 64} (1996), 10691087. ##bibitem{CGL} W. H. Chen, Z. H. Guan and X. Lu, {it Delaydependent exponential stability## of uncertain stochastic systems with multiple delays: an LMI approach},## Syst. Control. Lett., {bf 54} (2005), 547555. ##bibitem{CL} B. Chen and X. Liu, {it Delaydependent robust $H_{infty}$ control for TS## fuzzy systems with time delay}, IEEE Trans. Fuzzy Syst., {bf 13} (2005), 544556. ##bibitem{CLLL} B. Chen, X. Liu, C. Lin and K. Liu, {it Robust $H_{infty}$ control of## TakagiSugeno fuzzy systems with state and input time delays},## Fuzzy Sets and Systems, {bf 160} (2009), 403422. ##bibitem{F} G. Feng, {it A survey on analysis and design of modelbased fuzzy control systems},## IEEE Trans. Fuzzy Syst., {bf 14} (2006), 676697. ##bibitem{GLW} H. Gao, J. Lam and C. Wang, {it Robust energytopeak filter design for## stochastic timedelay systems}, Syst. Control. Lett., {bf 55} (2006), 101111. ##bibitem{GC} X. P. Guan and C. L. Chen, {it Delaydependent guaranteed cost control for## TS fuzzy systems with time delays}, IEEE Trans. Fuzzy Syst.,## {bf 12} (2004), 236249. ##bibitem{HP} L. V. Hien and V. N. Phat, {it Robust stabilization of linear polytopic control## systems with mixed delays}, Acta Math. Vietnamica, {bf 35} (2010), 427438. ##bibitem{HPSIAM} D. Hinrichsen and A. J. Pritchard, {it Stochastic $H_{infty}$}, SIAM J. Control## Optim., {bf 36} (1998), 15041538. ##bibitem{H} F. H. Hsiao, {it Robust $H_{infty}$ fuzzy control of nonlinear systems with## multiple time delays}, Int. J. Syst. Sci., {bf 38} (2007), 351360. ##bibitem{HY} L. Hu and A. Yang, {it Fuzzy modelbased control of nonlinear stochastic## systems with timedelay}, Nonlinear Anal. TMA., {bf 71} (2009), 28552865. ##bibitem{HH} H. Huang and D. W. C. Ho, {it Delaydependent robust control of uncertain## stochastic fuzzy systems with timevarying delay}, IET Control Theory## Appl., {bf 1} (2007), 10751085. ##bibitem{HSIJFS} H. L. Huang and F. G. Shi, {it Robust control for TS timevarying delay## systems with norm bounded uncertainty based on LMI approach},## Iranian Journal of Fuzzy Systems, textbf{6} (2009), 114. ##bibitem{KM} V. B. Kolmanovskii and A. D. Myshkis, {it Applied theory of functional## differential equations}, Dordrecht, The Netherlands: Kluwer, 1992. ##bibitem{LWL} C. Lin, Q. G. Wang and T. H. Lee, {it Improvement on observerbased## $H_{infty}$ control for TS fuzzy systems}, Automatica, {bf 41} (2005), 16511656. ##bibitem{M} X. Mao, {it Stochastic differential equations and their applications},## Horwood, Chichester, 1997. ##bibitem{MPKL} Y. S. Moon, P. Park, W. H. Kwon and Y. S. Lee, {it Delaydependent robust## stabilization of uncertain statedelayed systems},## Int. J. Control, {bf 74} (2001), 14471455. ##bibitem{P} V. N. Phat, {it Memoryless $H_{infty}$ controller design for switched## nonlinear systems with mixed timevarying delays}, Int. J. Control, {bf 82} (2009), 18891898. ##bibitem{SN} P. Shi and S. K. Nguang, {it $H_{infty}$ output feedback control of fuzzy system models## under sampled measurements}, Comput. Math. Appl., {bf 46} (2003), 705717. ##bibitem{TIW} K. Tanaka, T. Ikeda and H. O. Wang, {it Fuzzy regulators and fuzzy## observers: relaxed stability conditions and LMIbased designs},## IEEE Trans. Fuzzy Syst., {bf 6} (1998), 250265. ##bibitem{TS} T. Takagi and M. Sugeno, {it Fuzzy identification systems and it's application to modeling## and control}, IEEE Trans. Syst. Man Cybern., {bf 15} (1985), 116132. ##bibitem{WHL} Z. Wang, D. W. C. Ho and X. Liu, {it A note on the robust stability of## uncertain stochastic fuzzy systems with timedelays},## IEEE Trans. Syst. Man, Cybern. A, {bf 34} (2004), 570576. ##bibitem{WTG} H. O. Wang, K. Tanaka and M. F. Griffn, {it An approach to fuzzy control## of nonlinear systems: stability and design issues},## IEEE Trans. Fuzzy Syst., {bf 4} (1996), 1423. ##bibitem{WXS} Y. Wang, L. Xie and C. E. de Souza, {it Robust control of a class of## uncertain nonlinear system}, Syst. Control. Lett., {bf 19} (1992), 139149. ##bibitem{XC} S. Xu and T. Chen, {it Robust $H_{infty}$ control for uncertain stochastic## systems with state delay}, IEEE Trans. Automat.## Contr., {bf 47} (2002), 20892094. ##bibitem{XL} S. Xu and J. Lam, {it Robust $H_{infty}$ control for uncertain discrete## timedelay fuzzy systems via output feedback controllers},## IEEE Trans. Fuzzy Syst., {bf 13} (2005), 8293. ##bibitem{YGS} R. Yang, H. Gao and P. Shi, {it Delaydependent robust $H_{infty}$ control## for uncertain stochastic timedelay systems},## Int. J. Robust Nonlinear Control, textbf{20} (2010), 18521865. ##bibitem{YH} Z. Yi and P. A. Heng, {it Stability of fuzzy control systems with bounded## uncertain delays}, IEEE Trans. Fuzzy Syst., {bf 10} (2002), 9297. ##bibitem{Y} J. Yoneyama, {it New robust stability conditions and design of robust## stabilizing controllers for TakagiSugeno fuzzy timedelay systems},## IEEE Trans. Fuzzy Syst., {bf 15} (2007), 828839. ##bibitem{ZGLD} Y. Zhao, H. Gao, J. Lam and B. Du, {it Stability and stabilization of## delayed TS fuzzy systems: a delay partitioning approach},## IEEE Trans. Fuzzy Syst., {bf 17} (2009), 750762. ##bibitem{ZGW} H. Zhang, Q. Gong and Y. Wang, {it Delaydependent robust $H_{infty}$## control for uncertain fuzzy hyperbolic systems with multiple delays},## Progr. Natur. Sci., {bf 18} (2008), 97104. ##bibitem{ZLL} H. Zhang, S. Lun and D. Liu, {it Fuzzy $H_{infty}$ filter design for a## class of nonlinear discretetime systems with multiple time delays},## IEEE Trans. Fuzzy Syst., {bf 15} (2007), 453469. ##bibitem{ZWL} H. Zhang, Y. Wang and D. Liu, {it Delaydependent guaranteed cost control## for uncertain stochastic fuzzy systems with multiple time delays},## IEEE Trans. Syst., Man, Cybern. B, Cybern., {bf 38} (2008), 126140. ##bibitem{ZXZZ} B. Zhang, S. Xu, G. Zong and Y. Zou, {it Delaydependent stabilization## for stochastic fuzzy systems with time delays}, Fuzzy Sets and Systems, {bf 158} (2007), 22382250. ##bibitem{ZFLX} S. Zhou, G. Feng, J. Lam and S. Xu, {it Robust $H_{infty}$ control for## discretetime fuzzy systems via basisdependent Lyapunov functions},## Information Sciences, {bf 174} (2005), 197217. ##bibitem{ZL} S. Zhou and T. Li, {it Robust stabilization for delayed discretetime fuzzy## systems via basisdependent LyapunovKrasovskii function},## Fuzzy Sets and Systems, {bf 151}(2005), 139153.##]
ON GENERALIZED FUZZY MULTISETS AND THEIR USE IN
COMPUTATION
ON GENERALIZED FUZZY MULTISETS AND THEIR USE IN
COMPUTATION
2
2
An orthogonal approach to the fuzzification of both multisets and hybridsets is presented. In particular, we introduce $L$multifuzzy and$L$fuzzy hybrid sets, which are general enough and in spirit with thebasic concepts of fuzzy set theory. In addition, we study the properties ofthese structures. Also, the usefulness of these structures is examined inthe framework of mechanical multiset processing. More specifically, weintroduce a variant of fuzzy P~systems and, since simplefuzzy membrane systems have been introduced elsewhere, we simply extendpreviously stated results and ideas.
1
An orthogonal approach to the fuzzification of both multisets and hybridsets is presented. In particular, we introduce $L$multifuzzy and$L$fuzzy hybrid sets, which are general enough and in spirit with thebasic concepts of fuzzy set theory. In addition, we study the properties ofthese structures. Also, the usefulness of these structures is examined inthe framework of mechanical multiset processing. More specifically, weintroduce a variant of fuzzy P~systems and, since simplefuzzy membrane systems have been introduced elsewhere, we simply extendpreviously stated results and ideas.
113
125
Apostolos
Syropoulos
Apostolos
Syropoulos
Greek Molecular Computing Group, 366, 28th October St.,
GR67100 Xanthi, Greece
Greek Molecular Computing Group, 366, 28th
Greece
asyropoulos@yahoo.com
Lfuzzy sets
Fuzzy Multisets
Computability
P Systems
[[1] M. Akay, M. Cohen and D. Hudson, Fuzzy sets in life sciences, Fuzzy Sets and Systems, 90##(1997), 219224.##[2] S. Anelli, E. Damiani, O. D'Antona and D. E. Loeb, Getting results with negative thinking,##LACES 05A9517, eprint arXiv:math.CO/9502214.##[3] D. A. Baum, Individuality and the existence of species through time, Systematic Biology,##47(4) (1989), 641653.##[4] G. Berry and G. Boudol, The chemical abstract machine, Theoretical Comput. Sci., 96##(1992), 217248.##[5] W. Blizard, The development of multiset theory, Modern Logic, 1 (1991), 319352.##[6] R. De Nicola and S. A. Smolka, Concurrency: theory and practice, ACM Computing Surveys##28A, 4es (1996), 52.##[7] J. Goguen, Lfuzzy sets, Journal of Mathematical Analysis and Application, 18 (1967), 145##[8] D. E. Knuth, The art of computer programming, Seminumerical Algorithms, Addisson##Wesley, 2 (1981).##[9] D. Loeb, Sets with a negative number of elements, Advances in Mathematics, 91 (1992),##[10] S. Miyamoto, Fuzzy multisets and their generalizations, In Multiset Processing, C. Calude,##G. Paun, G. Rozenberg and A. Salomaa, eds., in Lecture Notes in Computer Science. Springer##Verlag, Berlin, 2235 (2001), 225235.##[11] S. Miyamoto, Data structure and operations for fuzzy multisets, In Transactions on Rough##Sets II: Rough Sets and Fuzzy Sets, J. F. Peters, A. Skowron, D. Dubois, J. GrzymalaBusse##and M. Inuiguchi, eds., in Lecture Notes in Computer Science. SpringerVerlag, Berlin, 3135##(2004), 189200.##[12] G. Paun, Membrane computing: an introduction, SpringerVerlag, Berlin, 2002.##[13] J. R. Searle, Mind: a brief introduction, Oxford University Press, Oxford, UK, 2004.##[14] L. A. Stein, Challenging the computational metaphor: implications for how we think, Cyber##netics & Systems, 30(6) (1999), 473507.##[15] A. Syropoulos, Mathematics of multisets, In Multiset Processing, C. Calude, G. Paun,##G. Rozenberg and A. Salomaa, eds., in Lecture Notes in Computer Science. SpringerVerlag,##Berlin, 2235 (2001), 347358.##[16] A. Syropoulos, Fuzzifying P systems, The Computer Journal, 49(5) (2006), 619628.##[17] A. Syropoulos, Fuzzy chemical abstract machines, CoRR abs/0903.3513, 2009.##[18] A. Syropoulos, On nonsymmetric multifuzzy sets, Critical Review IV, (2010), 35{41.##[19] A. Tzouvaras, The linear logic of multisets, Logic Journal of the IGPL, 6(6) (1998), 901916.##[20] A. Tzouvaras, The logic of multisets continued: the case of disjunction, Studia Logica, 75##(2003), 287304.##[21] S. Vickers, Topology via logic, Cambridge Tracts in Theoretical Computer Science, Cambridge##University Press, Cambridge, U. K., 6 (1990).##[22] K. Weihrauch, Computable analysis: an introduction, SpringerVerlag, Berlin/Heidelberg,##[23] R. R. Yager, On the theory of bags, Int. J. General Systems, 13 (1986), 2337.##[24] X. Zheng and K. Weihrauch, The arithmetical hierarchy of real numbers, Mathematical Logic##Quarterly, 47(1) (2001), 5165.##]
GLOBAL ROBUST STABILITY CRITERIA FOR TS FUZZY
SYSTEMS WITH DISTRIBUTED DELAYS AND TIME
DELAY IN THE LEAKAGE TERM
GLOBAL ROBUST STABILITY CRITERIA FOR TS FUZZY
SYSTEMS WITH DISTRIBUTED DELAYS AND TIME
DELAY IN THE LEAKAGE TERM
2
2
The paper is concerned with robust stability criteria for Takagi Sugeno (TS) fuzzy systems with distributed delays and time delay in the leakage term. By exploiting a model transformation, the system is converted to one of the neutral delay system. Global robust stability result is proposed by a new LyapunovKrasovskii functional which takes into account the range of delay and by making use of some inequality techniques. Based on the interval timevarying delays, new stability criteria are obtained in terms of linear matrix inequalities (LMIs). Finally, three numerical examples and their simulations are given to show the effectiveness and advantages of our results.
1
The paper is concerned with robust stability criteria for Takagi Sugeno (TS) fuzzy systems with distributed delays and time delay in the leakage term. By exploiting a model transformation, the system is converted to one of the neutral delay system. Global robust stability result is proposed by a new LyapunovKrasovskii functional which takes into account the range of delay and by making use of some inequality techniques. Based on the interval timevarying delays, new stability criteria are obtained in terms of linear matrix inequalities (LMIs). Finally, three numerical examples and their simulations are given to show the effectiveness and advantages of our results.
127
146
P.
Balasubramaniam
P.
Balasubramaniam
Department of Mathematics, Gandhigram Rural InstituteDeemed
University, Gandhigram  624 302, Tamilnadu, India
Department of Mathematics, Gandhigram Rural
India
balugru@gmail.com
S.
Lakshmanan
S.
Lakshmanan
Department of Mathematics, Gandhigram Rural InstituteDeemed
University, Gandhigram  624 302, Tamilnadu, India
Department of Mathematics, Gandhigram Rural
India
lakshm@gmail.com
R.
Rakkiyappan
R.
Rakkiyappan
Department of Mathematics, Bharathiar University, Coimbatore 
641 046, Tamilnadu, India
Department of Mathematics, Bharathiar University,
India
rakkigru@gmail.com
Delaydependent stability
Linear matrix inequality
Lyapunov–Krasovskii functional
TS fuzzy systems
[bibitem{BGB} S. Boyd, L. E. Ghaoui, E. Feron and V. Balakrishnan, textit{Linear##matrix inequalities in systems and control theory}, Philadelphia:##SIAM, 1994.##bibitem{4} M. Chadli and A. E. Hajjaji, textit{A observerbased robust fuzzy##control of nonlinear systems with parametric uncertainties},##Fuzzy Sets and Systems, textbf{157} (2006), 12791281 ##bibitem{CLT} B. Chen, X. P. Liu and S. C. Tong, textit{New delaydependent stabilization conditions of TS systems with constant delay}, Fuzzy Sets and Systems, textbf{158} (2007), 22092224. ##bibitem{3} H. Gassara, A. E. Hajjaji and M. Chaabane, textit{Robust control of##TS fuzzy systems with timevarying delay using new approach},##Int. J. Robust Nonlinear Control, textbf{20} (2010), 1566–1578 ##bibitem{iranian3} R. Ghanbari, N. MahdaviAmiri and R. Yousefpour, textit{Exact and approximate solutions of fuzzy lr linear systems: new algorithms using a least squares model and the abs approach}, Iranian Journal of Fuzzy##Systems, textbf{7} (2010), 118. ##bibitem{BK} K. Gopalsamy, textit{Stability and oscillations in delay differential##equations of population dynamics}, Kluwer Academic Publishers,##Dordrecht, 1992. ##bibitem{leak1} K. Gopalsamy, textit{Leakage delays in BAM},##J Math. Anal. Appl., textbf{325} (2007), 11171132. ##bibitem{XPC} X. P. Guan and C. L. Chen, textit{Delaydependent guaranteed cost control for TS fuzzy systems with time delays}, IEEE Trans. Fuzzy Syst., textbf{12} (2004) 236249. ##bibitem{Unc2} Q. L. Han, textit{Robust stability for a class of linear systems with##timevarying delay and nonlinear perturbations}, Comput.##Math. Appl., textbf{47} (2004), 12011209. ##bibitem{F33} F. H. Hsiao, textit{Robust $H_{infty}$ fuzzy control of nonlinear##systems with multiple time delays}, In. J.##Syst. Sci., textbf{38} (2007), 351360. ##bibitem{In2} X. Jiang and Q. L. Han, textit{Delaydependent robust stability for##uncertain linear systems with interval timevarying delay},##Automatica, textbf{42} (2006), 10591065. ##bibitem{Unc1} O. M. Kwon, J. H. Park and S. M. Lee, textit{On stability criteria for##uncertain delaydifferential systems of neutral type with##timevarying delays}, Appl. Math. Comput.,##textbf{197} (2008), 864873. ##bibitem{leak3} X. Li, X. Fu, P. Balasubramaniam and R. Rakkiyappan, textit{Existence,##uniqueness and stability analysis of recurrent neural networks##with time delay in the leakage term under impulsive##perturbations}, Nonlinear Anal. Real World Appl.,##textbf{11} (2010), 40924108. ##bibitem{leak2} C. Li and T. Huang, textit{On the stability of## nonlinear systems with leakage delay}, J. Franklin Inst., textbf{346} (2009), 366377. ## bibitem{Comp1} C. G. Li, H. J. Wang and X. F. Liao, textit{Delaydependent robust stability of##uncertain fuzzy systems with timevarying delays}, IEE##Proce. Control Theory Appl., textbf{151} (2004), 417421. ##bibitem{Comp2} C. H. Lien, textit{Further results on delaydependent robust stability of##uncertain fuzzy systems with timevarying delay}, Chaos Solitons## Fractals, textbf{28} (2006), 422427. ##bibitem{Comp3}C. H. Lien, K. W. Yu, W. D. Chen, Z. L. Wan and Y. J. Chung, textit{Stability criteria for uncertain TakagiSugeno fuzzy systems with interval timevarying delay}, IET Control Theory Appl., textbf{1} (2007), 746769. ##bibitem{F31} F. Liu, M. Wu, Y. He and R. Yokoyama, textit{New delaydependent##stability criteria for TS fuzzy systems with timevarying delay},##Fuzzy Sets and Systems, textbf{161} (2010), 20332042. ##bibitem{iranian1}M. Meidani, G. Habibagahi and S. Katebi, textit{An aggregated fuzzy reliability index for slope stability analysis}, Iranian Journal of Fuzzy Systems, textbf{1} (2004), 1731. ##bibitem{In1} C. Peng and Y. C. Tian, textit{Delaydependent robust stability criteria##for uncertain systems with interval timevarying delay}, J.## Comput. Appl. Math., textbf{214} (2008), 480494. ##bibitem{F3} C. Peng, D. Yue, T. C. Yang and E. G. Tian, textit{On delaydependent##approach for robust stability and stabilization of TS fuzzy##systems with constant delay and uncertainties}, IEEE Trans. Fuzzy Syst., textbf{17} (2009), 11431156. ##bibitem{Unc3} W. Qian, S. Cong, Y. Sun and S. Fei, textit{Novel robust stability criteria for##uncertain systems with timevarying delay}, Appl. Math.## Comput., textbf{215} (2009), 866872. ##bibitem{F4} F. O. Souza, L. A. Mozelli and R. M. Palhares,##textit{On stability and stabilization of TS fuzzy timedelayed##systems}, IEEE Trans. Fuzzy Syst., textbf{17} (2009), 14501455. ##bibitem{In4} J. Sun, G. P. Liu, J. Chen and D. Rees, textit{Improved##delayrangedependent stability criteria for linear systems with##timevarying delays}, Automatica, textbf{46} (2010), 466470 . ##bibitem{triple3} J. Sun and G. P. Liu, textit{A new delaydependent stability criterion##for timedelay systems}, Asian J. Control, textbf{11} (2009), 427431. ##bibitem{triple1} J. Sun, G. P. Liu, J. Chen and D. Rees, textit{Improved stability criteria##for neural networks with timevarying delay}, Phys. Lett. A, textbf{373} (2009), 342348. ##bibitem{iranian2} A. A. Suratgar and S. K. Y. Nikravesh, textit{Potential energy based stability analysis of fuzzy linguistic systems}, Iranian Journal of Fuzzy Systems, textbf{2} (2005), 6574. ##bibitem{1} T. Takagi and M. Sugeno, textit{Fuzzy identification of systems and its##application tomodeling and control}, IEEE Trans. Syst. Man, textbf{15} (1985), 116132 . ##bibitem{2} K. Tanaka and M. Sano, textit{A robust stabilization problem of fuzzy##control systems and its application to backing up control of a##trucktrailer}, IEEE Trans. Fuzzy Syst.,##textbf{2} (1994), 119134 ##bibitem{F1} E. Tian and C. Peng, textit{Delaydependent stability analysis and##synthesis of uncertain TS fuzzy systems with timevarying delay},##Fuzzy Sets and Systems, textbf{157} (2006), 544559. ##bibitem{Unc4} M. Wu, Y. He, J. H. She and G. P. Liu, textit{Delaydependent criteria for##robust stability of timevarying delay systems}, Automatica,##textbf{40} (2004), 14351439. ##bibitem{F2} J. Yoneyama, textit{New delaydependent approach to robust stability##and stabilization for TakagiSugeno fuzzy timedelay systems},##Fuzzy Sets and Systems, textbf{158} (2007), 22252237. ##bibitem{F32} J. Yoneyama, textit{Robust stability and stabilizing##controller design of fuzzy systems with discrete and distributed##delays}, Information Sciences, textbf{178} (2008), 19351947. ##bibitem{Unc11} D. Yue, E. Tian, Y. Zhang and C. Peng, textit{Delaydistributiondependent##robust stability of uncertain systems with timevarying delay},##Int. J. Robust Nonlinear Control, textbf{19} (2009), 377393. ##bibitem{In3} W. Zhang, X. S. Cai and Z. Z. Han, textit{Robust stability criteria for##systems with interval timevarying delay and nonlinear##perturbations}, J. Comput. Appl. Math., textbf{234} (2010), 174180.##]
$L$ordered Fuzzifying Convergence Spaces
$L$ordered Fuzzifying Convergence Spaces
2
2
Based on a complete Heyting algebra, we modify the definition oflatticevalued fuzzifying convergence space using fuzzy inclusionorder and construct in this way a Cartesianclosed category, calledthe category of $L$ordered fuzzifying convergence spaces, in whichthe category of $L$fuzzifying topological spaces can be embedded.In addition, two new categories are introduced, which are called thecategory of principal $L$ordered fuzzifying convergence spaces andthat of topological $L$ordered fuzzifying convergence spaces, andit is shown that they are isomorphic to the category of$L$fuzzifying neighborhood spaces and that of $L$fuzzifyingtopological spaces respectively.
1
Based on a complete Heyting algebra, we modify the definition oflatticevalued fuzzifying convergence space using fuzzy inclusionorder and construct in this way a Cartesianclosed category, calledthe category of $L$ordered fuzzifying convergence spaces, in whichthe category of $L$fuzzifying topological spaces can be embedded.In addition, two new categories are introduced, which are called thecategory of principal $L$ordered fuzzifying convergence spaces andthat of topological $L$ordered fuzzifying convergence spaces, andit is shown that they are isomorphic to the category of$L$fuzzifying neighborhood spaces and that of $L$fuzzifyingtopological spaces respectively.
147
161
Wenchao
Wu
Wenchao
Wu
Department of Mathematics, Ocean University of China, 266100 Qing
dao, P. R. China
Department of Mathematics, Ocean University
China
wuwenchao107@163.com
Jinming
Fang
Jinming
Fang
Department of Mathematics, Ocean University of China, 266100 Qing
dao, P. R. China
Department of Mathematics, Ocean University
China
jinmingfang@163.com
$L$fuzzifying topology
$L$ordered fuzzifying convergence structure
$L$filter
Cartesianclosed category
Reflective subcategory
[bibitem{1} H. Boustique, R. N. Mohapatra and G. Richardson, {it Latticevalued fuzzy interior operators}, Fuzzy Sets and Systems, {bf 160} (2009), 29472955.##bibitem{2} H. Boustique and G. Richardson, {it A note on regularity}, Fuzzy Sets and Systems, {bf 162} (2011), 6466.##bibitem{3} J. Fang, {it Stratified $L$ordered convergence structures}, Fuzzy Sets and Systems, {bf 161} (2010), 21302149.##bibitem{4} P. V. Flores, R. N. Mohapatra and G. Richardson, {it Latticevalued spaces: fuzzy convergence}, Fuzzy Sets and Systems, {bf 157} (2006), 27062714.##bibitem{5} P. V. Flores and G. Richardson, {it Latticevalued convergence: diagonal axioms}, Fuzzy Sets and Systems, {bf 159} (2008), 25202528.##bibitem{6} U. H"{o}hle, {it Characterization of $L$topologies by $L$valued neighborhoods}, Chapter 5, In:##Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory, The##Handbooks of Fuzzy Sets Series, (U. H"{o}hle, S. E. Rodabaugh,##eds.), Kluwer Academic Publishers, Boston, Dordrecht,##London, {bf3} (1999), 389432.##bibitem{7} U. H"{o}hle and A. P. v{S}ostak, {it Axiomatic foundations of fixedbasis fuzzy topology},##In: Mathematics of Fuzzy Sets: Logic, Topology, and Measure Theory,##The Handbooks of Fuzzy Sets Series, (U. H"{o}hle, S. E. Rodabaugh,##eds.), Kluwer Academic Publishers, Boston, Dordrecht,##London, {bf3} (1999), 123173.##bibitem{8} G. J"{a}ger, {it A category of $L$fuzzy convergence spaces}, Quaestiones Mathematicae, {bf 24} (2001), 501517.##bibitem{9} G. J"{a}ger, {it Subcategories of latticevalued convergence spaces}, Fuzzy Sets and Systems, {bf 156} (2005), 124.##bibitem{10} G. J"{a}ger, {it Pretopological and topological latticevalued convergence spaces}, Fuzzy Sets and Systems, {bf 158} (2007), 424435.##bibitem{11} G. J"{a}ger, {it Fischer's diagonal condition for latticevalued convergence spaces}, Quaestiones Mathematicae, {bf 31} (2008), 1125.##bibitem{12} R. Lowen, {it Convergence in fuzzy topological spaces}, Gen. Top. Appl., {bf 10} (1979), 147160.##bibitem{13} K. C. Min, {it Fuzzy limit spaces}, Fuzzy Sets and Systems, {bf 32} (1989), 343357.##bibitem{14} L. Xu, {it Characterizations of fuzzifying topologies by some limit structures}, Fuzzy Sets and Systems, {bf 123} (2001), 169176.##bibitem{15} W. Yao, {it On manyvalued stratified $L$fuzzy convergence spaces}, Fuzzy Sets and Systems, {bf 159} (2008), 25032519.##bibitem{16} W. Yao, {it On $L$fuzzifying convergence spaces}, Iranian Journal of Fuzzy Systems, {bf 6}textbf{(1)} (2009), 6380.##bibitem{17} M. S. Ying, {it A new approach to fuzzy topology (I)}, Fuzzy Sets and Systems, {bf 39} (1991), 303321.##bibitem{18} D. Zhang, {it On the reflectivity and coreflectivity of $L$fuzzifying topological spaces in $L$topological##spaces}, Acta Mathematica Sinica (English Series), {bf##18}textbf{(1)} (2002), 5568.##bibitem{19} D. Zhang, {it $L$fuzzifying topologies as $L$topologies}, Fuzzy Sets and Systems, {bf 125} (2002), 135144.##bibitem{20} D. Zhang and L. Xu, {it Categories isomorphic to bf FNS}, Fuzzy Sets and Systems, {bf 104} (1999), 373380.##]
Persiantranslation vol. 9, no.2, June 2012
2
2
1

165
174