Skip to main content
SpringerLink
Log in

Zeroing Neural Network Based on Neutrosophic Logic for Calculating Minimal-Norm Least-Squares Solutions to Time-Varying Linear Systems

  • Published:
Neural Processing Letters Aims and scope Submit manuscript

Abstract

This paper presents a dynamic model based on neutrosophic numbers and a neutrosophic logic engine. The introduced neutrosophic logic/fuzzy adaptive Zeroing Neural Network dynamic is termed NSFZNN and represents an improvement over the traditional Zeroing Neural Network (ZNN) design. The model aims to calculate the matrix pseudo-inverse and the minimum-norm least-squares solutions of time-varying linear systems. The improvement of the proposed model emerges from the advantages of neutrosophic logic over fuzzy and intuitionistic fuzzy logic in solving complex problems associated with predictions, vagueness, uncertainty, and imprecision. We use neutrosphication, de-fuzzification, and de-neutrosophication instead of fuzzification and de-fuzzification exploited so far. The basic idea is based on the known advantages of neutrosophic systems compared to fuzzy systems. Simulation examples and engineering applications on localization problems and electrical networks are presented to test the efficiency and accuracy of the proposed dynamical system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Ansari AQ (2017). Keynote speakers: From fuzzy logic to neutrosophic logic: a paradigme shift and logics. pp 11–15, Jaipur, India. IEEE

  2. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets Syst 20(1):87–96

    MathSciNet  MATH  Google Scholar 

  3. Ben-Israel A (1986) Generalized inverses of matrices: a perspective of the work of Penrose. Math Proc Camb Philos Soc 100(3):407

    MathSciNet  MATH  Google Scholar 

  4. Ben-Israel A (2002) The Moore of the Moore–Penrose inverse. Electron J Linear Algebra 9:150–157

    MathSciNet  MATH  Google Scholar 

  5. Christianto V, Smarandache F (2019) A review of seven applications of neutrosophic logic: in cultural psychology, economics theorizing, conflict resolution, philosophy of science, etc. Multidiscip Sci J 2:128–137

    Google Scholar 

  6. Dai J, Chen Y, Xiao L, Jia L, He Y (2022) Design and analysis of a hybrid GNN–ZNN model with a fuzzy adaptive factor for matrix inversion. IEEE Trans Ind Inform 18(4):2434–2442

    Google Scholar 

  7. Dean P, Porrill J (1998) Pseudo-inverse control in biological systems: a learning mechanism for fixation stability. Neural Netw 7–8:1205–1218

    Google Scholar 

  8. Dempster AG, Cetin E (2016) Interference localization for satellite navigation systems. Proc IEEE 104(6):1318–1326

    Google Scholar 

  9. Deng Y, Ren Z, Kong Y, Bao F, Dai Q (2017) A hierarchical fused fuzzy deep neural network for data classification. IEEE Trans Fuzzy Syst 25(4):1006–1012

    Google Scholar 

  10. Feng S, Wu H (2017) Hybrid robust boundary and fuzzy control for disturbance attenuation of nonlinear coupled ode-beam systems with application to a flexible spacecraft. IEEE Trans Fuzzy Syst 25(5):1293–1305

    Google Scholar 

  11. Hu Z, Xiao L, Li K, Li K, Li J (2021) Performance analysis of nonlinear activated zeroing neural networks for time-varying matrix pseudoinversion with application. Appl Soft Comput 98:106735

    Google Scholar 

  12. Huang F, Zhang X (2006) An improved Newton iteration for the weighted Moore-Penrose inverse. Appl Math Comput 174(2):1460–1486

    MathSciNet  MATH  Google Scholar 

  13. Huang H, Fu D, Xiao X, Ning Y, Wang H, Jin L, Liao S (2020) Modified Newton integration neural algorithm for dynamic complex-valued matrix pseudoinversion applied to mobile object localization. IEEE Trans Ind Inf 17(4):2432–2442

    Google Scholar 

  14. Jia L, Xiao L, Dai J, Cao Y (2021) A novel fuzzy-power zeroing neural network model for time-variant matrix Moore-Penrose inversion with guaranteed performance. IEEE Trans Fuzzy Syst 29(9):2603–2611

    Google Scholar 

  15. Jia L, Xiao L, Dai J, Qi Z, Zhang Z, Zhang Y (2021) Design and application of an adaptive fuzzy control strategy to zeroing neural network for solving time-variant QP problem. IEEE Trans Fuzzy Syst 29(6):1544–1555

    Google Scholar 

  16. Katsikis VN, Mourtas SD, Stanimirović PS, Zhang Y (2022) Solving complex-valued time-varying linear matrix equations via QR decomposition with applications to robotic motion tracking and on angle-of-arrival localization. IEEE Trans Neural Netw Learn Syst 33(8):3415–3424

    MathSciNet  Google Scholar 

  17. Katsikis VN, Stanimirović PS, Mourtas SD, Xiao L, Karabasević D, Stanujkić D (2022) Zeroing neural network with fuzzy parameter for computing pseudoinverse of arbitrary matrix. IEEE Trans Fuzzy Syst 30(9):3426–3435

    Google Scholar 

  18. Li W, Ma X, Luo J, Jin L (2021) A strictly predefined-time convergent neural solution to equality-and inequality-constrained time-variant quadratic programming. IEEE Trans Syst Man Cybern Syst 51(7):4028–4039

    Google Scholar 

  19. Liao B, Zhang Y (2014) Different complex ZFs leading to different complex ZNN models for time-varying complex generalized inverse matrices. IEEE Trans Neural Netw Learn Syst 25(9):1621–1631

    Google Scholar 

  20. Lin J, Lin CC, Lo HS (2009) Pseudo-inverse Jacobian control with grey relational analysis for robot manipulators mounted on oscillatory bases. J Sound Vib 326(3–5):421–437

    Google Scholar 

  21. Liu X, Yu Y, Zhong J, Wei Y (2012) Integral and limit representations of the outer inverse in Banach space. Linear Multilinear Algebra 60:333–347

    MathSciNet  MATH  Google Scholar 

  22. Mallik S, Mohanty S, Mishra BS (2022) Biologically Inspired Techniques in Many Criteria Decision Making, volume 271 of Smart Innovation, Systems and Technologies, chapter Neutrosophic Logic and its Scientific Applications. Springer, Singapore

  23. Naik J, Dash S, Dash P, Bisoi R (2018) Short term wind power forecasting using hybrid variational mode decomposition and multi-kernel regularized pseudo inverse neural network. Renew Energy 118:180–212

    Google Scholar 

  24. Noroozi A, Oveis AH, Hosseini SM, Sebt MA (2018) Improved algebraic solution for source localization from TDOA and FDOA measurements. IEEE Wirel Commun Lett 7(3):352–355

    Google Scholar 

  25. Penrose R (1956) On a best approximate solutions to linear matrix equations. Proc Cambr Philos Soc 52:17–19

    MATH  Google Scholar 

  26. Precup R-E, Tomescu M-L, Dragos C-A (2014) Stabilization of Rössler chaotic dynamical system using fuzzy logic control algorithm. Int J Gen Syst 43(5):413–433

    MathSciNet  MATH  Google Scholar 

  27. Salama AA, Alhasan KF, Elagamy HA, Smarandache F (2021) Neutrosophic dynamic set. Neutrosophic Knowl 3

  28. Smarandache F (1998) Neutrosophy: Neutrosophy probability, set and logic. American Research Press, Analytic Synthesis and Synthetic Analysis

    MATH  Google Scholar 

  29. Smarandache F (2001) A unifying field in logics: Neutrosophic logic, neutrosophic set, neutrosophic probability and statistics (fourth edition). arXiv:math/0101228

  30. Smarandache F (2003) Neutrosophic logic—generalization of the intuitionistic fuzzy logic. arXiv:math/0303009

  31. Stanimirovic PS, Katsikis VN, Jin L, Mosic D (2021) Properties and computation of continuous-time solutions to linear systems. Appl Math Comput 405:126242

    MathSciNet  MATH  Google Scholar 

  32. Stanimirović PS, Katsikis VN, Li S (2019) Integration enhanced and noise tolerant ZNN for computing various expressions involving outer inverses. Neurocomputing 329:129–143

    Google Scholar 

  33. Stanimirović PS, Katsikis VN, Zhang Z, Li S, Chen J, Zhou M (2020) Varying-parameter Zhang neural network for approximating some expressions involving outer inverses. Optim Methods Softw 35(6):1304–1330

    MathSciNet  MATH  Google Scholar 

  34. Stanimirović PS, Stojanović I, Katsikis VN, Pappas D, Zdravev Z (2015) Application of the least squares solutions in image deblurring. Math Probl Eng, 2015. Article ID 298689

  35. Tan Z, Hu Y, Xiao L, Chen K (2019) Robustness analysis and robotic application of combined function activated RNN for time-varying matrix pseudo inversion. IEEE Access 7:33434–33440

    Google Scholar 

  36. Wang A, Liu L, Qiu J, Feng G (2019) Event-triggered robust adaptive fuzzy control for a class of nonlinear systems. IEEE Trans Fuzzy Syst 27(8):1648–1658

    Google Scholar 

  37. Wang F, Chen B, Liu X, Lin C (2018) Finite-time adaptive fuzzy tracking control design for nonlinear systems. IEEE Trans Fuzzy Syst 26(3):1207–1216

    Google Scholar 

  38. Wang H, Li J, Liu H (2006) Practical limitations of an algorithm for the singular value decomposition as applied to redundant manipulators. In: 2006 IEEE conference on robotics, automation and mechatronics, pp 1–6

  39. Xiao L, Liao B, Li S, Zhang Z, Ding L, Jin L (2018) Design and analysis of FTZNN applied to the real-time solution of a nonstationary Lyapunov equation and tracking control of a wheeled mobile manipulator. IEEE Trans Ind Inform 14(1):98–105

    Google Scholar 

  40. Xiao L, Zhang Y (2014) From different Zhang functions to various ZNN models accelerated to finite-time convergence for time-varying linear matrix equation. Neural Process Lett 39(3):309–326

    Google Scholar 

  41. Yu F, Liu L, Xiao L, Li K, Cai S (2019) A robust and fixed-time zeroing neural dynamics for computing time-variant nonlinear equation using a novel nonlinear activation function. Neurocomputing 350:108–116

    Google Scholar 

  42. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    MATH  Google Scholar 

  43. Zhang Y, Ge SS (2005) Design and analysis of a general recurrent neural network model for time-varying matrix inversion. IEEE Trans Neural Netw 16(6):1477–1490

    Google Scholar 

  44. Zhang Y, Li F, Yang Y, Li Z (2012) Different Zhang functions leading to different Zhang-dynamics models illustrated via time-varying reciprocal solving. Appl Math Model 36(9):4502–4511

    MathSciNet  MATH  Google Scholar 

  45. Zhang Y, Yang Y, Tan N, Cai B (2011) Zhang neural network solving for time-varying full-rank matrix Moore–Penrose inverse. Computing 92(2):97–121

    MathSciNet  MATH  Google Scholar 

  46. Zhang Z, Fu Z, Zheng L, Gan M (2018) Convergence and robustness analysis of the exponential-type varying gain recurrent neural network for solving matrix-type linear time-varying equation. IEEE Access 6:57160–57171

    Google Scholar 

  47. Zhang Z, Yan Z (2020) An adaptive fuzzy recurrent neural network for solving non-repetitive motion problem of redundant robot manipulators. IEEE Trans Fuzzy Syst 28(4):684–691

    Google Scholar 

  48. Zhang Z, Zheng L, Qiu T, Deng F (2020) Varying-parameter convergent-differential neural solution to time-varying overdetermined system of linear equations. IEEE Trans Autom Control 65(2):874–881

    MathSciNet  MATH  Google Scholar 

  49. Zhou J, Zhu Y, Li XR, You Z (2002) Variants of the Greville formula with applications to exact recursive least squares. SIAM J Matrix Anal Appl 24(1):150–164

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Predrag Stanimirović is supported by the Ministry of Education, Science and Technological Development, Republic of Serbia, Contract No. 451-03-68/2020-14/200124. Predrag Stanimirović is supported by the Science Fund of the Republic of Serbia, #GRANT No 7750185, Quantitative Automata Models: Fundamental Problems and Applications—QUAM. This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Grant No. 075-15-2022-1121).

Author information

Authors and Affiliations

  1. Department of Economics, Division of Mathematics-Informatics and Statistics-Econometrics, National and Kapodistrian University of Athens, Sofokleous 1 Street, 10559, Athens, Greece

    Vasilios N. Katsikis & Spyridon D. Mourtas

  2. University of Athens Master of Business Administration, National and Kapodistrian University of Athens, Athens, Greece

    Vasilios N. Katsikis

  3. Administration of Businesses and Organizations Department, Hellenic Open University, 26335, Patras, Greece

    Vasilios N. Katsikis

  4. Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, Niš, 18000, Serbia

    Predrag S. Stanimirović

  5. Laboratory “Hybrid Methods of Modelling and Optimization in Complex Systems”, Siberian Federal University, Prosp. Svobodny 79, Krasnoyarsk, Russia, 660041

    Predrag S. Stanimirović & Spyridon D. Mourtas

  6. Hunan Provincial Key Laboratory of Intelligent Computing and Language Information Processing, Changsha, 410081, China

    Lin Xiao

  7. Technical Faculty at Bor, University of Belgrade, Vojske Jugoslavije 12, Bor, 19210, Serbia

    Dragiša Stanujkić

  8. Faculty of Applied Management, Economics and Finance, University Business Academy in Novi Sad, Jevrejska 24, Belgrade, 11000, Serbia

    Darjan Karabašević

Authors
  1. Vasilios N. Katsikis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Predrag S. Stanimirović
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Spyridon D. Mourtas
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lin Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Dragiša Stanujkić
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Darjan Karabašević
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vasilios N. Katsikis.

Ethics declarations

Conflicts of interest

The authors Vasilios N. Katsikis, Predrag S. Stanimirović, Spyridon D. Mourtas, Lin Xiao, Dragiša Stanujkić and Darjan Karabašević of the paper entitled “Zeroing Neural Network based on Neutrosophic Logic for Calculating Minimal-norm Least-squares Solutions to Time-varying Linear Systems,” declare that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Katsikis, V.N., Stanimirović, P.S., Mourtas, S.D. et al. Zeroing Neural Network Based on Neutrosophic Logic for Calculating Minimal-Norm Least-Squares Solutions to Time-Varying Linear Systems. Neural Process Lett 55, 8731–8753 (2023). https://doi.org/10.1007/s11063-023-11175-7

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11063-023-11175-7

Keywords

Search

Navigation