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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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
Higham NJ (1997) Stable iterations for the matrix square root. Numer Algorithms 15(2):227–242
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Prokhorov DV (2006) Training recurrent neurocontrollers for robustness with derivative-free Kalman filter. IEEE Trans Neural Netw 17(6):1606–1616
Dini DH, Mandic DP (2012) Class of widely linear complex Kalman filters. IEEE Trans Neural Netw Learn Syst 23(5):775–786
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
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
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
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
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
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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
- Non-linear and noise-tolerant ZNN (NNT-ZNN)
- Static and time-varying matrix square root finding
- Constant noise
- Random noise
- Activation functions
- Residual error