On exponential convergence of nonlinear gradient dynamics system with application to square root finding

详细信息    查看全文

• 作者：Yunong Zhang (1) (2) (3)
  Dechao Chen (1) (2) (3)
  Dongsheng Guo (1) (2) (3)
  Bolin Liao (1) (2) (3) (4)
  Ying Wang (1) (2) (3)
  
  1. School of Information Science and Technology ; Sun Yat-sen University (SYSU) ; Guangzhou ; 510006 ; China
  2. The SYSU-CMU Shunde International Joint Research Institute ; Shunde ; 528300 ; China
  3. Key Laboratory of Autonomous Systems and Networked Control ; Ministry of Education ; Guangzhou ; 510640 ; China
  4. College of Information Science and Engineering ; Jishou University ; Jishou ; 416000 ; China
  
• 关键词：Nonlinear gradient dynamics system ; Exponential convergence ; Scalar square root finding ; Energy function ; Lyapunov theory
• 刊名：Nonlinear Dynamics
• 出版年：2015
• 出版时间：January 2015
• 年：2015
• 卷：79
• 期：2
• 页码：983-1003
• 全文大小：482 KB
• 参考文献：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鈥?050 (2012) CrossRef
  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鈥?24 (2006) CrossRef
  3. Chen, X., Zhao, G., Mei, F.: A fractional gradient representation of the Poincar equations. Nonlinear Dyn. 73, 579鈥?82 (2013)
  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鈥?71 (2013)
  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鈥?03 (2013) CrossRef
  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鈥?306 (2009) CrossRef
  7. Zhang, Y., Chen, Z., Chen, K.: Convergence properties analysis of gradient neural network for solving online linear equations. Acta Autom. Sin. 35, 1136鈥?139 (2009)
  8. Ramezani, S.: Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory. Nonlinear Dyn. 73, 1399鈥?421 (2013)
  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鈥?5 (2012) CrossRef
  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鈥?10 (2013) CrossRef
  11. Ding, F., Shi, Y., Chen, T.: Gradient-based identification methods for hammerstein nonlinear ARMAX models. Nonlinear Dyn. 45, 31鈥?3 (2005)
  12. Mead, C.: Analog VLSI and Neural Systems. Addison-Wesley, Boston (1989) CrossRef
  13. Zhang, Y., Ma, W., Li, K., Yi, C.: Brief history and prospect of coprocessors. Chin. Sci. Technol. Inf. 13, 115鈥?17 (2008)
  14. Zhang, Y., Ke, Z., Xu, P., Yi, C.: Time-varying square roots finding via Zhang dynamics versus gradient dynamics and the former鈥檚 link and new explanation to Newton-Raphson iteration. Inf. Process. Lett. 110, 1103鈥?109 (2010) CrossRef
  15. Chen, Y., Yi, C., Zhong, J.: Linear simultaneous equations鈥?neural solution and its application to convex quadratic programming with equality-constraint. J. Appl. Math. 2013, 1鈥? (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鈥?96 (2003) CrossRef
  17. Hern谩ndez, M., Romero, N.: Accelerated convergence in Newton鈥檚 method for approximating square roots. J. Comput. Appl. Math. 177, 225鈥?29 (2005) CrossRef
  18. Boros, G., Moll, V.: The double square root, Jacobi polynomials and Ramanujan鈥檚 Master Theorem. J. Comput. Appl. Math. 130, 337鈥?44 (2001) CrossRef
  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鈥?6 (2008) CrossRef
  20. Livings, D., Dance, S., Nichols, N.: Unbiased ensemble square root filters. Physica D 237, 1021鈥?028 (2008) CrossRef
  21. Cichocki, A., Unbehauen, R.: Neural Networks for Optimization and Signal Processing. Wiley, Chichester (1993)
  22. Zhang, Y., Leithead, W.E.: Exploiting Hessian matrix and trust-region algorithm in hyperparameters estimation of Gaussian process. Appl. Math. Comput. 171, 1264鈥?281 (2005) CrossRef
  23. Majerski, S.: Square-rooting algorithms for high-speed digital circuits. IEEE Trans. Comput. C鈥?4, 724鈥?33 (1985) CrossRef
  24. Chisci, L., Zappa, G.: Square-root Kalman filtering of descriptor systems. Syst. Control Lett. 19, 325鈥?34 (1992) CrossRef
  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鈥?35 (2000)
  26. Trivedi, K.S., Ercegovac, M.D.: On-line algorithms for division and multiplication. IEEE Trans. Comput. C鈥?6, 681鈥?87 (1977) CrossRef
  27. Tan, K.G.: The theory and implementation of high-radix division. Proceedings of the 4th IEEE Symposium on Computer Arithmetic, pp. 183鈥?89 (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鈥?507 (2012) CrossRef
  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鈥?373 (2012)
  30. Guo, Z., Huang, L.: Generalized Lyapunov method for discontinuous systems. Nonlinear Anal. Theory Methods Appl. 71, 3083鈥?092 (2009) CrossRef
  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鈥?945 (2009) CrossRef
  32. Dabrowski, A.: Estimation of the largest Lyapunov exponent from the perturbation vector and its derivative dot product. Nonlinear Dyn. 67, 283鈥?91 (2012) CrossRef
  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鈥?415 (2009) CrossRef
  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鈥?79 (2008)
  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鈥?490 (2005) CrossRef
  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鈥?0 (2005)
  
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X

文摘

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.