Abstract
This paper presents a new kind of recurrent neural network proposed by Zhang et al. for solving online Lyapunov equation with time-varying coefficient matrices. Global exponential convergence could be achieved by such a recurrent neural network when solving the time-varying problems in comparison with gradient neural networks (GNN). MATLAB simulation of both neural networks for the real-time solution of time-varying Lyapunov equation is then investigated through several important techniques. Computer-simulation results substantiate the theoretical analysis and demonstrate the efficacy of such a Zhang neural network (ZNN) on time-varying Lyapunov equation solving, especially when using power-sigmoid activation functions.
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Phat, V.N., Niamsup, P.: Stability of Linear Time-Varying Delay Systems and Applications to Control Problems. Journal of Computational and Applied Mathematics 194, 343–356 (2006)
Subbotina, N.N.: The Value Functions of Singularly Perturbed Time-Optimal Control Problems in the Framework of Lyapunov Functions Method. Mathematical and Computer Modelling 45, 1284–1293 (2007)
Li, Y., Liu, K.J.R.: Static and Dynamic Convergence Behavior of Adaptive Blind Equalizers. IEEE Transactions on Signal Processing 44, 2736–2745 (1996)
Delorenzo, M.L.: Sensor and Actuator Selection for Large Space Structure Control. Journal of Guidance, Control, and Dynamics 13, 249–257 (1990)
Robert, B., Kushner, H.J.: Control of Mobile Communication Systems with Time-Varying Channels Viastability Methods. IEEE Transactions on Automatic Control 49, 1954–1962 (2004)
Zhang, Y., Wang, J.: A Dual Neural Network for Constrained Joint Torque Optimization of Kinematically Redundant Manipulators. IEEE Transactions on Systems, Man, and Cybernetics, Part B 32, 654–662 (2002)
Fernando, K.V., Nicholson, H.: Solution of Lyapunov Equation for the State Matrix. Electronics Letters 17, 204–205 (1981)
Sreeram, V., Agathoklis, P.: Solution of Lyapunov Equation with System Matrix in Companion Form. IEE Proceedings–D 138, 529–534 (1991)
Zhang, Y., Leithead, W.E., Leith, D.J.: Time-series Gaussian Process Regression Based on Toeplitz Computation of O(N 2) Operations and O(N)-Level Storage. In: 44th IEEE Conference on Decision and Control, pp. 3711–3716. IEEE Press, Seville (2005)
Leithead, W.E., Zhang, Y.: O(N 2)-Operation Approximation of Covariance Matrix Inverse in Gaussian Process Regression Based on Quasi-Newton BFGS Methods. Communications in Statistics - Simulation and Computation 36, 367–380 (2007)
Zhang, Y., Wang, J.: Global Exponential Stability of Recurrent Neural Networks for Synthesizing Linear Feedback Control Systems via Pole Assignment. IEEE Transactions on Neural Networks 13, 633–644 (2002)
Zhang, Y.: Revisit the Analog Computer and Gradient-Based Neural System for Matrix Inversion. In: 20th IEEE International Symposium on Intelligent Control, pp. 1411–1416. IEEE Press, Cyprus (2005)
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., 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)
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)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis, pp. 239–297. Cambridge University Press, Cambridge (1991)
Ding, F., Chen, T.: Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations. IEEE Transactions on Automatic Control 50, 1216–1221 (2005)
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Zhang, Y., Yue, S., Chen, K., Yi, C. (2008). MATLAB Simulation and Comparison of Zhang Neural Network and Gradient Neural Network for Time-Varying Lyapunov Equation Solving. In: Sun, F., Zhang, J., Tan, Y., Cao, J., Yu, W. (eds) Advances in Neural Networks - ISNN 2008. ISNN 2008. Lecture Notes in Computer Science, vol 5263. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87732-5_14
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DOI: https://doi.org/10.1007/978-3-540-87732-5_14
Publisher Name: Springer, Berlin, Heidelberg
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