×

On exponential convergence of nonlinear gradient dynamics system with application to square root finding. (English) Zbl 1345.65066

Summary: Gradient dynamics systems and their exponential convergence theories are investigated in this paper. Differing from widely considered linear gradient dynamics system (LGDS), a class of nonlinear gradient dynamics system (NGDS) is investigated with the exponential convergence analyzed. As an application to scalar square root finding, by defining six different square-based nonnegative error-monitoring functions (i.e., energy functions), six different NGDSs are theoretically designed and proposed in the form of first-order differential equations. Moreover, inspired by the exponential convergence theory of the LGDS, for each of the six proposed NGDSs, the corresponding exponential convergence theory is proved rigorously based on Lyapunov theory. Numerical verification and comparison further illustrate the efficacy of the proposed six NGDSs, in which the main differences and respective usages, as well as the application background and condition, are discussed in detail.

MSC:

65P99 Numerical problems in dynamical systems
34D08 Characteristic and Lyapunov exponents of ordinary differential equations
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Liu, W., Xiao, J., Li, L., Wu, Y., Lu, M.: Effects of gradient coupling on amplitude death in nonidentical oscillators. Nonlinear Dyn. 69, 1041-1050 (2012) · doi:10.1007/s11071-012-0325-2
[2] Zhang, Y.: A set of nonlinear equations and inequalities arising in robotics and its online solution via a primal neural network. Neurocomputing 70, 513-524 (2006) · doi:10.1016/j.neucom.2005.11.006
[3] Chen, X., Zhao, G., Mei, F.: A fractional gradient representation of the Poincar equations. Nonlinear Dyn. 73, 579-582 (2013) · Zbl 1281.70021
[4] Chen, J., Zhang, Y., Ding, R.: Gradient-based parameter estimation for input nonlinear systems with ARMA noises based on the auxiliary model. Nonlinear Dyn. 72, 865-871 (2013) · Zbl 1284.93224
[5] Fang, D., Qian, C.: The regularity criterion for 3D Navier-Stokes equations involving one velocity gradient component. Nonlinear Anal. Theory Methods Appl. 78, 86-103 (2013) · Zbl 1254.35177 · doi:10.1016/j.na.2012.09.019
[6] Zhang, Y., Shi, Y., Chen, K., Wang, C.: Global exponential convergence and stability of gradient-based neural network for online matrix inversion. Appl. Math. Comput. 215, 1301-1306 (2009) · Zbl 1194.65056 · doi:10.1016/j.amc.2009.06.048
[7] Zhang, Y., Chen, Z., Chen, K.: Convergence properties analysis of gradient neural network for solving online linear equations. Acta Autom. Sin. 35, 1136-1139 (2009)
[8] Ramezani, S.: Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory. Nonlinear Dyn. 73, 1399-1421 (2013) · Zbl 1281.74014
[9] Yang, C., Li, J., Li, Z.: Trajectory planning and optimized adaptive control for a class of wheeled inverted pendulum vehicle models. IEEE Trans. Cybern. 43, 24-35 (2012) · doi:10.1109/TSMCB.2012.2198813
[10] Li, Z., Yang, C., Tang, Y.: Decentralised adaptive fuzzy control of coordinated multiple mobile manipulators interacting with non-rigid environments. IET Control Theory Appl. 7, 397-410 (2013) · doi:10.1049/iet-cta.2011.0334
[11] Ding, F., Shi, Y., Chen, T.: Gradient-based identification methods for hammerstein nonlinear ARMAX models. Nonlinear Dyn. 45, 31-43 (2005) · Zbl 1134.93321
[12] Mead, C.: Analog VLSI and Neural Systems. Addison-Wesley, Boston (1989) · Zbl 0715.68002 · doi:10.1007/978-1-4613-1639-8
[13] Zhang, Y., Ma, W., Li, K., Yi, C.: Brief history and prospect of coprocessors. Chin. Sci. Technol. Inf. 13, 115-117 (2008) · Zbl 1011.33005
[14] Zhang, Y., Ke, Z., Xu, P., Yi, C.: 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, 1103-1109 (2010) · Zbl 1380.65086 · doi:10.1016/j.ipl.2010.09.013
[15] Chen, Y., Yi, C., Zhong, J.: Linear simultaneous equations’ neural solution and its application to convex quadratic programming with equality-constraint. J. Appl. Math. 2013, 1-6 (2013)
[16] Cardosoa, J., Kenney, C., Leite, F.: Computing the square root and logarithm of a real P-orthogonal matrix. Appl. Numer. Math. 46, 173-196 (2003) · Zbl 1030.65032 · doi:10.1016/S0168-9274(03)00033-3
[17] Hernández, M., Romero, N.: Accelerated convergence in Newton’s method for approximating square roots. J. Comput. Appl. Math. 177, 225-229 (2005) · Zbl 1062.65051 · doi:10.1016/j.cam.2004.09.025
[18] Boros, G., Moll, V.: The double square root, Jacobi polynomials and Ramanujan’s Master Theorem. J. Comput. Appl. Math. 130, 337-344 (2001) · Zbl 1011.33005 · doi:10.1016/S0377-0427(99)00372-6
[19] Li, J., Wang, X., Yao, Z.: Heat flow for the square root of the negative Laplacian for unit length vectors. Nonlinear Anal. Theory Methods Appl. 68, 83-96 (2008) · Zbl 1155.35470 · doi:10.1016/j.na.2006.10.033
[20] Livings, D., Dance, S., Nichols, N.: Unbiased ensemble square root filters. Physica D 237, 1021-1028 (2008) · Zbl 1147.93413 · doi:10.1016/j.physd.2008.01.005
[21] Cichocki, A., Unbehauen, R.: Neural Networks for Optimization and Signal Processing. Wiley, Chichester (1993) · Zbl 0824.68101
[22] Zhang, Y., Leithead, W.E.: Exploiting Hessian matrix and trust-region algorithm in hyperparameters estimation of Gaussian process. Appl. Math. Comput. 171, 1264-1281 (2005) · Zbl 1097.65019 · doi:10.1016/j.amc.2005.01.113
[23] Majerski, S.: Square-rooting algorithms for high-speed digital circuits. IEEE Trans. Comput. C-34, 724-733 (1985) · Zbl 0565.68040 · doi:10.1109/TC.1985.1676618
[24] Chisci, L., Zappa, G.: Square-root Kalman filtering of descriptor systems. Syst. Control Lett. 19, 325-334 (1992) · Zbl 0768.93081 · doi:10.1016/0167-6911(92)90071-Y
[25] Takahashi, D.: Implementation of multiple-precision parallel division and square root on distributed-memory parallel computers. Proceedings of the International Workshop on Parallel Processing, pp. 229-235 (2000)
[26] Trivedi, K.S., Ercegovac, M.D.: On-line algorithms for division and multiplication. IEEE Trans. Comput. C-26, 681-687 (1977) · Zbl 0406.68040 · doi:10.1109/TC.1977.1674901
[27] Tan, K.G.: The theory and implementation of high-radix division. Proceedings of the 4th IEEE Symposium on Computer Arithmetic, pp. 183-189 (1978)
[28] Yang, C., Wu, C., Zhang, P.: Estimation of Lyapunov exponents from a time series for n-dimensional state space using nonlinear mapping. Nonlinear Dyn. 69, 1493-1507 (2012) · doi:10.1007/s11071-012-0364-8
[29] Xiao, L., Zhang, Y.: Two new types of Zhang neural networks solving systems of time-varying nonlinear inequalities. IEEE Trans. Circuits Syst. I(59), 2363-2373 (2012) · Zbl 1468.65076
[30] Guo, Z., Huang, L.: Generalized Lyapunov method for discontinuous systems. Nonlinear Anal. Theory Methods Appl. 71, 3083-3092 (2009) · Zbl 1182.34008 · doi:10.1016/j.na.2009.01.220
[31] Zhang, Y., Chen, K., Tan, H.: Performance analysis of gradient neural network exploited for online time-varying matrix inversion. IEEE Trans. Autom. Contr. 54, 1940-1945 (2009) · Zbl 1367.92010 · doi:10.1109/TAC.2009.2023779
[32] Dabrowski, A.: Estimation of the largest Lyapunov exponent from the perturbation vector and its derivative dot product. Nonlinear Dyn. 67, 283-291 (2012) · Zbl 1242.37057 · doi:10.1007/s11071-011-9977-6
[33] Zhang, Y., Ma, W., Cai, B.: From Zhang neural network to Newton iteration for matrix inversion. IEEE Trans. Circuits Syst. Regul. Pap. 56, 1405-1415 (2009) · Zbl 1468.65026 · doi:10.1109/TCSI.2008.2007065
[34] Zhang, Y., Yi, C., Ma, W.: Comparison on gradient-based neural dynamics and Zhang neural dynamics for online solution of nonlinear equations. Proceedings of the 3rd International Symposium, pp. 269-279 (2008) · Zbl 1147.93413
[35] Zhang, Y., Ge, S.S.: Design and analysis of a general recurrent neural network model for time-varying matrix inversion. IEEE Trans. Neural Netw. 16, 1477-1490 (2005) · doi:10.1109/TNN.2005.857946
[36] Shanblatt, M.A.: A simulink-to-FPGA implementation tool for enhanced design flow. Proceedings of the IEEE International Conference on Microelectronic Systems Education, pp. 89-90 (2005)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.