System decomposition-based stability criteria for Takagi-Sugeno fuzzy uncertain stochastic delayed neural networks in quaternion field

R. Sriraman; R. Samidurai; V. C. Amritha; G. Rachakit; Prasanalakshmi Balaji; R. Sriraman; R. Samidurai; V. C. Amritha; G. Rachakit; Prasanalakshmi Balaji

doi:10.3934/math.2023587

AIMS Mathematics

2023, Volume 8, Issue 5: 11589-11616. doi: 10.3934/math.2023587

Previous Article Next Article

Research article Special Issues

System decomposition-based stability criteria for Takagi-Sugeno fuzzy uncertain stochastic delayed neural networks in quaternion field

1.
Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, Tamil Nadu-603 203, India
2.
Department of Mathematics, Thiruvalluvar University, Vellore, Tamil Nadu-632 115, India
3.
Department of Mathematics, National Institute of Technology Warangal, Telangana-506004, India
4.
Department of Mathematics, Faculty of Science, Maejo University, Chiang Mai-50290, Thailand
5.
Department of Computer Science, King Khalid University, Abha-62529, Saudi Arabia

Received: 01 January 2023 Revised: 04 March 2023 Accepted: 09 March 2023 Published: 16 March 2023
MSC : 92B20, 93D05, 93D20, 37H30, 03E72

Stochastic disturbances often occur in real-world systems which can lead to undesirable system dynamics. Therefore, it is necessary to investigate stochastic disturbances in neural network modeling. As such, this paper examines the stability problem for Takagi-Sugeno fuzzy uncertain quaternion-valued stochastic neural networks. By applying Takagi-Sugeno fuzzy models and stochastic analysis, we first consider a general form of Takagi-Sugeno fuzzy uncertain quaternion-valued stochastic neural networks with time-varying delays. Then, by constructing suitable Lyapunov-Krasovskii functional, we present new delay-dependent robust and global asymptotic stability criteria for the considered networks. Furthermore, we present our results in terms of real-valued linear matrix inequalities that can be solved in MATLAB LMI toolbox. Finally, two numerical examples are presented with their simulations to demonstrate the validity of the theoretical analysis.
- quaternion-valued neural networks,
- robust stability,
- stochastic disturbance,
- Lyapunov-Krasovskii functional,
- Takagi-Sugeno fuzzy
Citation: R. Sriraman, R. Samidurai, V. C. Amritha, G. Rachakit, Prasanalakshmi Balaji. System decomposition-based stability criteria for Takagi-Sugeno fuzzy uncertain stochastic delayed neural networks in quaternion field[J]. AIMS Mathematics, 2023, 8(5): 11589-11616. doi: 10.3934/math.2023587

Related Papers:

Abstract

Stochastic disturbances often occur in real-world systems which can lead to undesirable system dynamics. Therefore, it is necessary to investigate stochastic disturbances in neural network modeling. As such, this paper examines the stability problem for Takagi-Sugeno fuzzy uncertain quaternion-valued stochastic neural networks. By applying Takagi-Sugeno fuzzy models and stochastic analysis, we first consider a general form of Takagi-Sugeno fuzzy uncertain quaternion-valued stochastic neural networks with time-varying delays. Then, by constructing suitable Lyapunov-Krasovskii functional, we present new delay-dependent robust and global asymptotic stability criteria for the considered networks. Furthermore, we present our results in terms of real-valued linear matrix inequalities that can be solved in MATLAB LMI toolbox. Finally, two numerical examples are presented with their simulations to demonstrate the validity of the theoretical analysis.

References

[1]	J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. U.S.A, 79 (1982), 2554–2558. https://doi.org/10.1073/pnas.79.8.2554 doi: 10.1073/pnas.79.8.2554
[2]	M. A. Cohen, S. Grossberg, Absolute stability of global pattern formation and parallel memory storage by competitive neural networks, IEEE Trans. Syst. Man Cybern., 13 (1983), 815–826. https://doi.org/10.1109/TSMC.1983.6313075 doi: 10.1109/TSMC.1983.6313075
[3]	L. Chua, L. Yang, Cellular neural networks: Applications, IEEE Trans. Circuits Syst., 35 (1988), 1273–1290. https://doi.org/10.1109/31.7601 doi: 10.1109/31.7601
[4]	H. Huang, J. Cao, On global asymptotic stability of recurrent neural networks with time-varying delays, Appl. Math. Comput., 142 (2003), 143–154. https://doi.org/10.1016/S0096-3003(02)00289-8 doi: 10.1016/S0096-3003(02)00289-8
[5]	B. Kosko, Bidirectional associative memories, IEEE Trans. Syst. Man Cybern., 18 (1988), 49–60. https://doi.org/10.1109/21.87054 doi: 10.1109/21.87054
[6]	S. Blythe, X. R. Mao, X. X. Liao, Stability of stochastic delay neural networks, J. Franklin Inst., 338 (2001), 481–495. https://doi.org/10.1016/S0016-0032(01)00016-3 doi: 10.1016/S0016-0032(01)00016-3
[7]	R. Yang, Z. Zhang, P. Shi, Exponential stability on stochastic neural networks with discrete interval and distributed delays, IEEE Trans. Neural Netw., 21 (2010), 169–175. https://doi.org/10.1109/TNN.2009.2036610 doi: 10.1109/TNN.2009.2036610
[8]	X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, (1997).
[9]	L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, (1972).
[10]	A. Friedman, Stochastic Differential Equations and Applications, Academic Press, New York, (1976).
[11]	D. Yang, X. Li, Robust stability analysis of stochastic switched neural networks with parameter uncertainties via state-dependent switching law, Neurocomputing, 452 (2021), 813–819. https://doi.org/10.1016/j.neucom.2019.11.120 doi: 10.1016/j.neucom.2019.11.120
[12]	G. Liu, S. X. Yang, Y. Chai, W. Feng, W. Fu, Robust stability criteria for uncertain stochastic neural networks of neutral-type with interval time-varying delays, Neural Comput. Appl., 22 (2013), 349–359. https://doi.org/10.1007/s00521-011-0696-1 doi: 10.1007/s00521-011-0696-1
[13]	Z. Meng, Z. Xiang, Stability analysis of stochastic memristor-based recurrent neural networks with mixed time-varying delays, Neural Comput. Appl., 28 (2017), 1787–1799. https://doi.org/10.1007/s00521-015-2146-y
[14]	W. Xie, Q. Zhu, F. Jiang, Exponential stability of stochastic neural networks with leakage delays and expectations in the coefficients, Neurocomputing, 173 (2016), 1268–1275. https://doi.org/10.1016/j.neucom.2015.08.086 doi: 10.1016/j.neucom.2015.08.086
[15]	K. Zhong, S. Zhu, Q. Yang, Further results for global exponential stability of stochastic memristor-based neural networks with time-varying delays, Int. J. Syst. Sci., 47 (2016), 3573–3580. https://doi.org/10.1080/00207721.2015.1095955 doi: 10.1080/00207721.2015.1095955
[16]	G. Sun, Y. Zhang, Exponential stability of impulsive discrete-time stochastic BAM neural networks with time-varying delay, Neurocomputing, 131 (2014), 323–330. https://doi.org/10.1016/j.neucom.2013.10.010 doi: 10.1016/j.neucom.2013.10.010
[17]	Q. Song, Z. Zhao, Y. Liu, F. E. Alsaadi, Mean-square input-to-state stability for stochastic complex-valued neural networks with neutral delay, Neurocomputing, 470 (2022), 269–277. https://doi.org/10.1016/j.neucom.2021.10.117 doi: 10.1016/j.neucom.2021.10.117
[18]	Y. Cao, R. Sriraman, N. Shyamsundarraj, R. Samidurai, Robust stability of uncertain stochastic complex-valued neural networks with additive time-varying delays, Math. Comput. Simul., 171 (2020), 207–220. https://doi.org/10.1016/j.matcom.2019.05.011 doi: 10.1016/j.matcom.2019.05.011
[19]	R. Sriraman, Y. Cao, R. Samidurai, Global asymptotic stability of stochastic complex-valued neural networks with probabilistic time-varying delays, Math. Comput. Simul., 171 (2020), 103–118. https://doi.org/10.1016/j.matcom.2019.04.001 doi: 10.1016/j.matcom.2019.04.001
[20]	D. Liu, S. Zhu, W. Chang, Mean square exponential input-to-state stability of stochastic memristive complex-valued neural networks with time varying delay, Int. J. Syst. Sci., 48 (2017), 1966–1977. https://doi.org/10.1080/00207721.2017.1300706 doi: 10.1080/00207721.2017.1300706
[21]	W. Gong, J. Liang, X. Kan, L. Wang, A. M. Dobaie, Robust state estimation for stochastic complex-valued neural networks with sampled-data, Neural Comput. Appl., 31 (2019), 523–542. https://doi.org/10.1007/s00521-017-3030-8 doi: 10.1007/s00521-017-3030-8
[22]	D. L. Lee, Relaxation of the stability condition of the complex-valued neural networks, IEEE Trans. Neural Netw., 12 (2001), 1260–1262. https://doi.org/10.1109/72.950156 doi: 10.1109/72.950156
[23]	B. Zhou, Q. Song, Boundedness and complete stability of complex-valued neural networks with time delay, IEEE Trans. Neural Netw. Learn. Syst., 24 (2013), 1227–1238. https://doi.org/10.1109/TNNLS.2013.2247626 doi: 10.1109/TNNLS.2013.2247626
[24]	A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Philos. Soc., 85 (1979), 199–225. https://doi.org/10.1017/S0305004100055638
[25]	T. Isokawa, T. Kusakabe, N. Matsui, F. Peper, Quaternion neural network and its application, Knowl. Based Intell. Eng. Syst., 2774 (2003), 318–324. https://doi.org/10.1007/978-3-540-45226-3-44 doi: 10.1007/978-3-540-45226-3-44
[26]	Y. Liu, Y. Zheng, J. Lu, J. Cao, L. Rutkowski, Constrained quaternion-variable convex optimization: A quaternion-valued recurrent neural network approach, IEEE Trans. Neural Netw. Learn. Syst., 31 (2020), 1022–1035. https://doi.org/10.1109/TNNLS.2019.2916597 doi: 10.1109/TNNLS.2019.2916597
[27]	A. B. Greenblatt, S. S. Agaian, Introducing quaternion multi-valued neural networks with numerical examples, Inf. Sci., 423 (2018), 326–342. https://doi.org/10.1016/j.ins.2017.09.057 doi: 10.1016/j.ins.2017.09.057
[28]	Q. Song, X. Chen, Multistability analysis of quaternion-valued neural networks with time delays, IEEE Trans. Neural Netw. Learn. Syst., 29 (2018), 5430–5440. https://doi.org/10.1109/TNNLS.2018.2801297 doi: 10.1109/TNNLS.2018.2801297
[29]	J. Wang, T. Li, X. Luo, Y. Q. Shi, S. K. Jha, Identifying computer generated images based on quaternion central moments in color quaternion wavelet domain, IEEE Trans. Circuits Syst. Video Technol., 29 (2018), 2775–2785. https://doi.org/10.1109/TCSVT.2018.2867786 doi: 10.1109/TCSVT.2018.2867786
[30]	Y. Liu, D. Zhang, J. Lou, J. Lu, J. Cao, Stability analysis of quaternion-valued neural networks: Decomposition and direct approaches, IEEE Trans. Neural Netw. Learn. Syst., 29 (2018), 4201–4211. https://doi.org/10.1109/TNNLS.2017.2755697 doi: 10.1109/TNNLS.2017.2755697
[31]	Z. Xia. Y. Liu, J. Lu, J. Cao, L. Rutkowski, Penalty method for constrained distributed quaternion-variable optimization, IEEE Trans. Cybern., 51 (2021), 5631–5636. https://doi.org/10.1109/TCYB.2020.3031687 doi: 10.1109/TCYB.2020.3031687
[32]	Y. Wang, K. I. Kou, C. Zou, Y. Y. Tang, Robust sparse representation in quaternion space, IEEE Trans. Image Process., 30 (2021), 3637–3649. https://doi.org/10.1109/TIP.2021.3064193 doi: 10.1109/TIP.2021.3064193
[33]	H. Chen, T. Wang, J. Cao, P. P. Vidal, Y. Yang, Dynamic quaternion extreme learning machine, IEEE Trans. Circuits Syst. II: Exp. Briefs, 68 (2021), 3012–3016. https://doi.org/10.1109/TCSII.2021.3067014 doi: 10.1109/TCSII.2021.3067014
[34]	R. Sriraman, G. Rajchakit, C. P. Lim, P. Chanthorn, R. Samidurai, Discrete-time stochastic quaternion-valued neural networks with time delays: An asymptotic stability analysis, Symmetry, 12 (2020), 936. https://doi.org/10.3390/sym12060936 doi: 10.3390/sym12060936
[35]	J. Shu, B. Wu, L. Xiong, Stochastic stability criteria and event-triggered control of delayed Markovian jump quaternion-valued neural networks, Appl. Math. Comput., 420 (2022), 126904. https://doi.org/10.1016/j.amc.2021.126904 doi: 10.1016/j.amc.2021.126904
[36]	Q. Song, R. Zeng, Z. Zhao, Y. Liu, F. E. Alsaadi, Mean-square stability of stochastic quaternion-valued neural networks with variable coefficients and neutral delays, Neurocomputing, 471 (2022), 130–138. https://doi.org/10.1016/j.neucom.2021.11.033 doi: 10.1016/j.neucom.2021.11.033
[37]	C. Li, J. Cao, A. Kashkynbayev, Synchronization in quaternion-valued neural networks with delay and stochastic impulses, Neural Process. Lett., 54 (2022), 691–708. https://doi.org/10.1007/s11063-021-10653-0 doi: 10.1007/s11063-021-10653-0
[38]	U. Humphries, G. Rajchakit, P. Kaewmesri, P. Chanthorn, R. Sriraman, R. Samidurai, C. P. Lim, Stochastic memristive quaternion-valued neural networks with time delays: An analysis on mean square exponential input-to-state stability, Mathematics, 8 (2020), 815. https://doi.org/10.3390/math8050815 doi: 10.3390/math8050815
[39]	T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern., 15 (1985), 116–132. https://doi.org/10.1109/TSMC.1985.6313399 doi: 10.1109/TSMC.1985.6313399
[40]	K. Tanaka, H. O. Wang, Fuzzy Control Systems Design and Analysis: A Linear Matrix Inequality Approach, Wiley, New York, (2001).
[41]	C. K. Ahn, Delay-dependent state estimation for T-S fuzzy delayed Hopfield neural networks, Nonlinear Dyn., 61 (2010), 483–489. https://doi.org/10.1007/s11071-010-9664-z doi: 10.1007/s11071-010-9664-z
[42]	R. Li, J. Cao, Passivity and dissipativity of fractional-order quaternion-valued fuzzy memristive neural networks: Nonlinear scalarization approach, IEEE Trans. Cybern., 52 (2022), 2821–2832. https://doi.org/10.1109/TCYB.2020.3025439 doi: 10.1109/TCYB.2020.3025439
[43]	B. Liu, P. Shi, Delay-range-dependent stability for fuzzy BAM neural networks with time-varying delays, Phys. Lett. A, 373 (2009), 1830–1838. https://doi.org/10.1016/j.physleta.2009.03.044 doi: 10.1016/j.physleta.2009.03.044
[44]	R. Sriraman, P. Vignesh, V. C. Amritha, G. Rachakit, P. Balaji, Direct quaternion method-based stability criteria for quaternion-valued Takagi-Sugeno fuzzy BAM delayed neural networks using quaternion-valued Wirtinger-based integral inequality, AIMS Math., 8 (2023), 10486–10512. https://doi.org/10.3934/math.2023532 doi: 10.3934/math.2023532
[45]	J. Jian, P. Wan, Global exponential convergence of fuzzy complex-valued neural networks with time-varying delays and impulsive effects, Fuzzy Sets Syst., 338 (2018), 23–39. https://doi.org/10.1016/j.fss.2017.12.001 doi: 10.1016/j.fss.2017.12.001
[46]	R. Sriraman, N. Asha, Global stability of Clifford-valued Takagi-Sugeno fuzzy neural networks with time-varying delays and impulses, Kybernetika, 58 (2022), 498–521. https://doi.org/10.14736/kyb-2022-4-0498 doi: 10.14736/kyb-2022-4-0498
[47]	Y. Cao, S. Ramajayam, R. Sriraman, R. Samidurai, Leakage delay on stabilization of finite-time complex-valued BAM neural network: Decomposition approach, Neurocomputing, 463 (2021), 505–513. https://doi.org/10.1016/j.neucom.2021.08.056
[48]	R. Samidurai, R. Sriraman, S. Zhu, Leakage delay-dependent stability analysis for complex-valued neural networks with discrete and distributed time-varying delays, Neurocomputing, 338 (2021), 262–273. https://doi.org/10.1016/j.neucom.2019.02.027
[49]	F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl., 251 (1997), 21–57. https://doi.org/10.1016/0024-3795(95)00543-9 doi: 10.1016/0024-3795(95)00543-9
[50]	C. Pradeep, A. Chandrasekar, R. Murugesu, R. Rakkiyappan, Robust stability analysis of stochastic neural networks with Markovian jumping parameters and probabilistic time-varying delays, Complexity, 21 (2016), 59–72. https://doi.org/10.1002/cplx.21630 doi: 10.1002/cplx.21630
[51]	X. Chen, Z. Li, Q. Song, J. Hu, Y. Tan, Robust stability analysis of quaternion-valued neural networks with time delays and parameter uncertainties, Neural Netw., 91 (2017), 55–65. https://doi.org/10.1016/j.neunet.2017.04.006 doi: 10.1016/j.neunet.2017.04.006
[52]	P. G. Park, J. W. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica, 47 (2011), 235–238. https://doi.org/10.1016/j.automatica.2010.10.014 doi: 10.1016/j.automatica.2010.10.014

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)