Abstract
Different from gradient-based neural networks, a special kind of recurrent neural network has recently been proposed by Zhang et al for real-time solution of time-varying problems. In this paper, we generalize such a design method to solving online the time-varying Sylvester equation. In comparison with gradient-based neural networks, the resultant Zhang neural network for solving time-varying Sylvester equation is designed based on a matrix-valued error function, instead of a scalar-valued error function. It is depicted in an implicit dynamics, instead of an explicit dynamics. Furthermore, Zhang neural network globally exponentially converges to the exact solution of the time-varying Sylvester equation. Simulation results substantiate the theoretical analysis and demonstrate the efficacy of Zhang neural network on time-varying problem solving, especially when using a power-sigmoid activation function.
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References
Zhang, Y.: Analysis and Design of Recurrent Neural Networks and Their Applications to Control and Robotic Systems. Ph.D. Thesis, Chinese University of Hong Kong (2002)
Zhang, Y., Jiang, D., Wang, J.: A Recurrent Neural Network for Solving Sylvester Equation with Time-Varying Coefficients. IEEE Transactions on Neural Networks 13, 1053–1063 (2002)
Manherz, R.K., Jordan, B.W., Hakimi, S.L.: Analog Methods for Computation of the Generalized Inverse. IEEE Transactions on Automatic Control 13, 582–585 (1968)
Jang, J., Lee, S., Shin, S.: An Optimization Network for Matrix Inversion. Neural Information Processing Systems, pp. 397–401, American Institute of Physics, New York (1988)
Wang, J.: A Recurrent Neural Network for Real-Time Matrix Inversion. Applied Mathematics and Computation 55, 89–100 (1993)
Zhang, Y.: Revisit the Analog Computer and Gradient-Based Neural System for Matrix Inversion. In: Proceedings of IEEE International Symposium on Intelligent Control, pp. 1411–1416. IEEE Computer Society Press, Los Alamitos (2005)
Zhang, Y.: Towards Piecewise-Linear Primal Neural Networks for Optimization and Redundant Robotics. In: Proceedings of IEEE International Conference on Networking, Sensing and Control, pp. 374–379. IEEE Computer Society Press, Los Alamitos (2006)
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)
Zhang, Y., Ge, S.S.: A General Recurrent Neural Network Model for Time-Varying Matrix Inversion. In: Proceedings of the 42nd IEEE Conference on Decision and Control, pp. 6169–6174. IEEE Computer Society Press, Los Alamitos (2003)
Zhang, Y., Ge, S.S.: Design and Analysis of a General Recurrent Neural Network Model for Time-Varying Matrix Inversion. IEEE Transactions on Neural Networks 16, 1477–1490 (2005)
Steriti, R.J., Fiddy, M.A.: Regularized Image Reconstruction Using SVD and a Neural Network Method for Matrix Inversion. IEEE Transactions on Signal Processing 41, 3074–3077 (1993)
Mead, C.: Analog VLSI and Neural Systems. Addison-Wesley, Reading, MA (1989)
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming – Theory and Algorithms. Wiley, New York (1993)
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Zhang, Y., Fan, Z., Li, Z. (2007). Zhang Neural Network for Online Solution of Time-Varying Sylvester Equation. In: Kang, L., Liu, Y., Zeng, S. (eds) Advances in Computation and Intelligence. ISICA 2007. Lecture Notes in Computer Science, vol 4683. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74581-5_30
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DOI: https://doi.org/10.1007/978-3-540-74581-5_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-74580-8
Online ISBN: 978-3-540-74581-5
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