A Non-linear and Noise-Tolerant ZNN Model and Its Application to Static and Time-Varying Matrix Square Root Finding

Abstract

Based on the indefinite error-monitoring function, we propose a novel Zhang neural network (ZNN) model called NNT-ZNN with two properties of nonlinear and noise-tolerant for the time-varying and static matrix square root finding in this paper. Compared to the existing models associated with the square matrix root finding, the NNT-ZNN model proposed in this study fully takes error caused by possible noise on ZNN hardware implementation into account. Under the background that the large model-implementation error, the model still has the ability to converge to the theoretical square root of the given matrix with simulative results illustrated in the paper. For the purpose of comparison, the ZNN model proposed by Zhang et al. is also introduced. Beyond that, the corresponding convergence results of the NNT-ZNN model corresponding to various activation functions, are also shown via time-varying and static positive definite matrix. In the end, the experiments are simulated with MATLAB, which further verifies the availability, effectiveness of the proposed NNT-ZNN model, and robustness against unknown noise.

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References

  1. 1.

    Zhang Y, Jin L, Ke Z (2012) Superior performance of using hyperbolic sine activation functions in ZNN illustrated via time-varying matrix square roots finding. Comput Sci Inf Syst 9(4):1603–1625

    Article  Google Scholar 

  2. 2.

    Higham NJ (1997) Stable iterations for the matrix square root. Numer Algorithms 15(2):227–242

    MathSciNet  Article  Google Scholar 

  3. 3.

    Meini B (2004) The matrix square root from a new functional perspective: theoretical results and computational issues. SIAM J Matrix Anal Appl 26(2):362–376

    MathSciNet  Article  Google Scholar 

  4. 4.

    Long J, Hu X, Zhang L (2008) Newton’s method with exact line search for the square root of a matrix. In: International symposium on nonlinear dynamics. https://doi.org/10.1088/1742-6596/96/1/012034

  5. 5.

    Zhang Y, Yang Y (2008) Simulation and comparison of Zhang neural network and gradient neural network solving for time-varying matrix square roots. In: Proceedings of the 2nd international symposium on intelligent information technology application, pp 966–970

  6. 6.

    Zhang Y, Ke Z, Xu P, Yi C (2010) Time-varying square roots finding via Zhang dynamics versus gradient dynamics and the former’s link and new explanation to Newton–Raphson iteration. Inf Process Lett 110(24):1103–1109

    MathSciNet  Article  Google Scholar 

  7. 7.

    Zhang Y, Yang Y, Cai B, Guo D (2012) Zhang neural network and its application to Newton iteration for matrix square root estimation. Neural Comput Appl 21(3):453–460

    Article  Google Scholar 

  8. 8.

    Zhang Y, Li W, Guo D, Ke Z (2013) Different Zhang functions leading to different ZNN models illustrated via time-varying matrix square roots finding. Expert Syst Appl 40(11):4393–4403

    Article  Google Scholar 

  9. 9.

    Zhang Y, Chen D, Guo D, Liao B, Wang Y (2015) On exponential convergence of nonlinear gradient dynamics system with application to square root finding. Nonlinear Dyn 79(2):983–1003

    MathSciNet  Article  Google Scholar 

  10. 10.

    Xiao L (2017) A finite-time convergent Zhang neural network and its application to real-time matrix square root finding. Neural Comput Appl 10:1–8

    Google Scholar 

  11. 11.

    Zhang Y (2005) Revisit the analog computer and gradient-based neural system for matrix inversion. In: IEEE international symposium on intelligent control, pp 1411–1416

  12. 12.

    Zhang Y, Chen K, Tan H-Z (2009) Performance analysis of gradient neural network exploited for online time-varying matrix inversion. IEEE Trans Autom Control 54(8):1940–1945

    MathSciNet  Article  Google Scholar 

  13. 13.

    Zhang Y, Yang Y, Ruan G (2011) Performance analysis of gradient neural network exploited for online time-varying quadratic minimization and equality-constrained quadratic programming. Neurocomputing 74(10):1710–1719

    Article  Google Scholar 

  14. 14.

    Zhang Y, Yi C, Guo D, Zheng J (2011) Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation. Neural Comput Appl 20(1):1–7

    Article  Google Scholar 

  15. 15.

    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

    Article  Google Scholar 

  16. 16.

    Guo D, Zhang Y (2012) Zhang neural network, Getz–Marsden dynamic system, and discrete-time algorithms for time-varying matrix inversion with application to robots’ kinematic control. Neurocomputing 97:22–32

    Article  Google Scholar 

  17. 17.

    Jin L, Zhang Y, Li S, Zhang Y (2017) Noise-tolerant ZNN models for solving time-varying zero-finding problems: a control-theoretic approach. IEEE Trans Autom Control 62(2):992–997

    MathSciNet  Article  Google Scholar 

  18. 18.

    Zhang Y, Li Z (2009) Zhang neural network for online solution of time-varying convex quadratic program subject to time-varying linear-equality constraints. Phys Lett A 373(18):1639–1643

    Article  Google Scholar 

  19. 19.

    Xiao L, Zhang Y (2012) Two new types of Zhang neural networks solving systems of time-varying nonlinear inequalities. IEEE Trans Circuits Syst I Regul Pap 59(10):2363–2373

    MathSciNet  Article  Google Scholar 

  20. 20.

    Li S, Chen S, Liu B (2013) Accelerating a recurrent neural network to finite-time convergence for solving time-varying Sylvester equation by using a sign-bi-power activation function. Neural Process Lett 37(2):189–205

    Article  Google Scholar 

  21. 21.

    Jin L, Zhang Y, Li S (2016) Integration-enhanced zhang neural network for real-time-varying matrix inversion in the presence of various kinds of noises. IEEE transactions on neural networks and learning systems 27(12):2615–2627

    Article  Google Scholar 

  22. 22.

    Xiao L, Liao B, Li S, Chen K (2018) Nonlinear recurrent neural networks for finite-time solution of general time-varying linear matrix equations. Neural Netw 98:102–113

    Article  Google Scholar 

  23. 23.

    Xiao L (2017) Accelerating a recurrent neural network to finite-time convergence using a new design formula and its application to time-varying matrix square root. J Frankl Inst 354(13):5667–5677

    MathSciNet  Article  Google Scholar 

  24. 24.

    Xiao L (2017) A finite-time recurrent neural network for solving online time-varying Sylvester matrix equation based on a new evolution formula. Nonlinear Dyn 90(3):1581–1591

    MathSciNet  Article  Google Scholar 

  25. 25.

    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

    Article  Google Scholar 

  26. 26.

    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

    Article  Google Scholar 

  27. 27.

    Prokhorov DV (2006) Training recurrent neurocontrollers for robustness with derivative-free Kalman filter. IEEE Trans Neural Netw 17(6):1606–1616

    Article  Google Scholar 

  28. 28.

    Dini DH, Mandic DP (2012) Class of widely linear complex Kalman filters. IEEE Trans Neural Netw Learn Syst 23(5):775–786

    Article  Google Scholar 

  29. 29.

    He W, Nie S, Meng T, Liu Y-J (2017) Modeling and vibration control for a moving beam with application in a drilling riser. IEEE Transactions on Control Systems Technology 25(3):1036–1043

    Article  Google Scholar 

  30. 30.

    Liu Y-J, Li S, Tong S, Chen CP (2017) Neural approximation-based adaptive control for a class of nonlinear nonstrict feedback discrete-time systems. IEEE Trans Neural Netw Learn Syst 28(7):1531–1541

    MathSciNet  Article  Google Scholar 

  31. 31.

    He W, Ge W, Li Y, Liu Y-J, Yang C, Sun C (2017) Model identification and control design for a humanoid robot. IEEE Trans Syst Man Cybern Syst 47(1):45–57

    Article  Google Scholar 

  32. 32.

    Liu Y-J, Lu S, Li D, Tong S (2017) Adaptive controller design-based ABLF for a class of nonlinear time-varying state constraint systems. IEEE Trans Syst Man Cybern Syst 47(7):1546–1553

    Article  Google Scholar 

  33. 33.

    Li X, Yu J, Li S, Ni L (2018) A nonlinear and noise-tolerant ZNN model solving for time-varying linear matrix equation. Neurocomputing 317:70–78. https://doi.org/10.1016/j.neucom.2018.07.067

    Article  Google Scholar 

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Correspondence to Jiguo Yu.

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This work is supported by NSF of China under Grants 61672321, 61832012, 61771289 and 61373027.

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Li, X., Yu, J., Li, S. et al. A Non-linear and Noise-Tolerant ZNN Model and Its Application to Static and Time-Varying Matrix Square Root Finding. Neural Process Lett 50, 1687–1703 (2019). https://doi.org/10.1007/s11063-018-9953-y

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Keywords

  • Non-linear and noise-tolerant ZNN (NNT-ZNN)
  • Static and time-varying matrix square root finding
  • Constant noise
  • Random noise
  • Activation functions
  • Residual error