两类Hopfield型神经网络的稳定性分析
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摘要
近年来,Hopfield型神经网络得到了广泛地研究,并且成功地应用于信号处理、模式识别、组合优化等领域。由于具有时滞的Hopfield型神经网络在解决动态图像处理等问题上有较重要的应用,因此,具有时滞的Hopfield型神经网络的研究更受到许多学者的关注.
神经网络的稳定性是一个关键问题,有些应用要求神经网络模型是全局渐近稳定的,即对应的动力系统的每一个解收敛到平衡点。论文在已有文献的基础上,通过对稳定性理论和神经网络的研究,基于Lyapunov泛函的思想,选取一个适合神经网络模型的Lyapunov函数,并通过一些处理矩阵不等式的技巧,对选取的Lyapunov函数求导放缩,最终得到一个保守性更好的线性矩阵不等式的结果,包括时滞神经网络全局渐近稳定和全局指数稳定。近些年神经网络稳定性方面的研究成为热点,已有的文献对时滞细胞神经网络稳定性的研究大都限于时滞无关的结果,如何建立时滞相关的稳定性判据就成了热点问题。本文对时滞相关进行了研究,得出了一些有益的结论,同时论文中的时滞相关的结果还可以推出时滞无关的结果。最后对二阶Hopfield型神经网络稳定性进行了分析,这是对高阶神经网络稳定性研究的初步探讨。在数值算例中,通过Matlab中的LMI工具箱和Simulink图像仿真器做出数值仿真与图像仿真,验证了结果的正确性。
In recent years, the Hopfield neural network has been widely studied, and has already been successfully applied to signal processing,pattern recognition,combination optimization and so on. Because the Hopfield neural network with time delays has important application in solving the problem of dynamic image processing Therefore, A study on Hopfield neural network with time delays has been paid close attention by many scholars.
Stability is a key issue in applications of neural networks.In some applications, it is desirable that the network possesses a unique and globally asymptotically stable equilibrium point for every external input. In researching this thesis, we choose a Lyapunov function to suit neural networks, through in theory of stability and neural networks, and then dispose the matrix with some skills. We get a better liner matrix inequality result, including global asymptotically stability and global exponential stability of time delays neural network. In these years, stability analysis of neural network turns to be hotspot, but most of existing results derived in the literature are independent of the delay parameters, how to propose a delay-dependent stability criterion for delayed cellular neural networks is a hot problem. In this thesis we present delay-dependent stability criterion, so as to get the maximum numerical delay and some results independent of the delay parameters easily. And more, stability analysis of a second order neural network of Hopfield in this paper is a new task, which is an experiment in stability analysis of high order neural network. At last, we present numerical examples to illustrate the validity of the results.
神经网络的稳定性是一个关键问题,有些应用要求神经网络模型是全局渐近稳定的,即对应的动力系统的每一个解收敛到平衡点。论文在已有文献的基础上,通过对稳定性理论和神经网络的研究,基于Lyapunov泛函的思想,选取一个适合神经网络模型的Lyapunov函数,并通过一些处理矩阵不等式的技巧,对选取的Lyapunov函数求导放缩,最终得到一个保守性更好的线性矩阵不等式的结果,包括时滞神经网络全局渐近稳定和全局指数稳定。近些年神经网络稳定性方面的研究成为热点,已有的文献对时滞细胞神经网络稳定性的研究大都限于时滞无关的结果,如何建立时滞相关的稳定性判据就成了热点问题。本文对时滞相关进行了研究,得出了一些有益的结论,同时论文中的时滞相关的结果还可以推出时滞无关的结果。最后对二阶Hopfield型神经网络稳定性进行了分析,这是对高阶神经网络稳定性研究的初步探讨。在数值算例中,通过Matlab中的LMI工具箱和Simulink图像仿真器做出数值仿真与图像仿真,验证了结果的正确性。
In recent years, the Hopfield neural network has been widely studied, and has already been successfully applied to signal processing,pattern recognition,combination optimization and so on. Because the Hopfield neural network with time delays has important application in solving the problem of dynamic image processing Therefore, A study on Hopfield neural network with time delays has been paid close attention by many scholars.
Stability is a key issue in applications of neural networks.In some applications, it is desirable that the network possesses a unique and globally asymptotically stable equilibrium point for every external input. In researching this thesis, we choose a Lyapunov function to suit neural networks, through in theory of stability and neural networks, and then dispose the matrix with some skills. We get a better liner matrix inequality result, including global asymptotically stability and global exponential stability of time delays neural network. In these years, stability analysis of neural network turns to be hotspot, but most of existing results derived in the literature are independent of the delay parameters, how to propose a delay-dependent stability criterion for delayed cellular neural networks is a hot problem. In this thesis we present delay-dependent stability criterion, so as to get the maximum numerical delay and some results independent of the delay parameters easily. And more, stability analysis of a second order neural network of Hopfield in this paper is a new task, which is an experiment in stability analysis of high order neural network. At last, we present numerical examples to illustrate the validity of the results.
引文
1 L. O. Chua, L. Yang. Cellular Neural Networks. Theory IEEE Trans. Circuits System, 1988,35:1257-1272
2 T. Roska, T. Boros, P. Thiran. Detecting Simple Motion Using Cellular Neural Networks. Cellular Neural Networks and Their Applications, 1999:127-138
3 S. Q. Hu, J. Wang. Global Stability of a Class of Continuous-time Recurrent Neural Networks. Circuits and Systems I: Fundamental Theory and Applications, 2002,49(2):1334-1347
4 S. Arik. An Analysis of Global Asymptotic Stability of Delayed Cellular Neural Networks. Neural Networks,2002,13(5):1239-1242
5 S. Arik. A Note on the Global Stability of Dynamical Neural Networks. Circuits and Systems I: Fundamental Theory and applications, 2002,49(4):502-504
6 X. F. Liao, G. R. Chen, E. N. Sanchez. LMI-based Approach for Asymptotically Stability Analysis of Delayed Neural Networks. Circuits andSystems I: Fundamental Theory and applications,2002,49(7):1033-1039
7 S. Yerramalla, B. Cukic, E. Fuller. Lyapunov Stability Analysis of the Quantization Error for DCS Neural Networks. Neural Networks,2003,3:2412-2417
8 H. Z. Zheng, F. Q. Gu, Y. S. Sun. Asymptotic Stability of FeedbackNeural Networks with Time-Delay. Machine Learning and Cybernetics,2003,2:1113-1116
9 J. Cao, J. Wang. Global Asymptotic Stability of a General Class of Recurrent Neural Networks with Time-varying Delays. Circuits and Systems I: Fundamental Theory and Applications,2003,1(50):34-44
10 J. D. Hwang, F. H. Hsiao. Stability Analysis of Neural-network Interconnected Systems. Neural Networks,2003,1(14):201-208
11 T. Ensari, S. Arik, V. Tavsanoglu. Global Asymptotic Stability of a class of Neural Networks with Time Varying Delays. Circuits and Systems,2004,5:820-823
12 A. M. Base, S. Pilyugin, S. A. Wismuller. Stability Analysis of a Self-organizingNeural Network with Feed Forward And Feedback Dynamics,Neural Networks,2004, 25(2):1505-1509
13 D. K. Chaturvedi, O. P. Malik, P. K. Kalra. Generalized Neuron-based Adaptive Power System Stabilizer Generation. Transmission and Distribution IEE Proceedings,2004, 151(2):213-218
14 V. Singh. Robust Stability of Cellular Neural Networks with Delay: Linear MatrixInequality Approach. Control Theory and Applications,2004,15(1):125-129
15 V. Singh. Global Asymptotic Stability of Cellular Neural Networks with Unequal Delays: LMI Approach. Electronics Letters,2004,40:548-549
16 Z. G. Zeng, J, Wang, X. X. Liao. Stability Analysis of Delayed Cellular Neural Networks Described Using Cloning Templates. Circuits and Systems I,2004, 51:2313-2324
17 X. M. Li, L. H. Huang, J. H. Wu. A New Method of Lyapunov Functionals for Delayed Cellular Neural Networks. Circuits and Systems I,2004,11:2263-2270
18 Z. G. Zeng, J. Wang, X. X. Liao. Global Exponential Stability of a General Class of Recurrent Neural Networks with Time-varying Delays. Circuits and Systems I: Fundamental Theory and Applications,2003,10(50):1353-1358
19 J. D. Cao, J. Wang. Global Exponential Stability and Periodicity of Recurrent Neural Networks with Time Delays. Circuits and Systems,2005,52:920-931
20 Y. Chen. Global Asymptotic Stability of Delayed Cohen-Grasberg Neural Networks. IEEE Transactions on Circuits and Systems I,2005,6:114-120
21 S. Vimal. Comments on“Global Robust Stability of Delayed Neural Networks”. IEEE Transactions on Circuits and Systems I,2005,6:254-262
22 T. G. Chu, C. S. Zhang. New Necessary and Sufficient Conditions for Absolute Stability of Neural Networks. Control and Automation ICCA '05.International Conference,2005,1:593-598
23 W. H. Chen, W. X. Zheng. Global Asymptotic Stability of a Class of Neural Networks with Distributed Delays. Circuits and Systems I,2005,6:215-221
24 T. P. Chen. Global Exponential Stability of Delayed Hopfield Neural Networks. NeuralNetworks,2001,14(8):977-980
25 S.Q. Hu, J. Wang. Absolute Exponential Stability of a Class of Continuous-time Recurrent Neural Networks. Neural Networks,2003,14(1):35-45
26 L. Xia, M. Xia. LMI Conditions for global asymptotically stability of neural networks with discrete and distributed delays. ICIC Express Letters, vol. 2, no. 3, pp. 257-262,2008.
27 Z. G. Zeng, J. Wang. Complete Stability of Cellular Neural Networks with Time-varying Delays, IEEE Transactions on Circuits and Systems I,2005,6:15-24
28 S. Y. Xu, Lam James, Ho Daniel. New LMI Condition for Delay-Dependent Asymptotic Stability of Delayed Hopfield Neural Networks. IEEE Transactions on Circuits and Systems II,2005,6:104-110
29 S. Senan, S. Arik. New Results for Exponential Stability of Delayed Cellular Neural Networks. Circuits and Systems II,2005,52:154-158
30 Z. C. Yang, D. Y. Xu, J. Deng. New Conditions for Exponential Stability of Delay Impulsive Neural Networks, Electronics, Circuits and Systems ICECS 2004 Proceedings of the 11th IEEE International Conference,2004:226-229
31 J. Peng, Z. B. Xu. H. Qiao. A Critical Analysis on Global Convergence of Hopfield-type Neural Networks. Circuits and Systems I: Fundamental Theory and Applications,2005,4(52):804-814
32 S. Arik. Global Asymptotic Stability Analysis of Bidirectional Associative Memory Neural Networks with Time Delays. Neural Networks,2005,16(3):580-586
33 H. T. Lu. Comments on“A generalized LMI-based Approach to the Global Asymptotic Stability of Delayed Cellular Neural Networks”. Neural Networks,2005,16(3):778-779
34 H. Qi, L. Qi, X. Yang. Deriving Sufficient Conditions for Global Asymptotic Stability of Delayed Neural Networks via Nonsmooth Analysis-II. Neural Networks,2005, 6(16):1701-1706
35 Q. S. Liu, J. D. Cao, Y. S. Xia. A Delayed Neural Network for Solving Linear Projection Equations and Its Analysis. Neural Networks,2005,16:834-843
36 T. Ensari, S. Arik. Global Stability Analysis of Neural Networks with Multiple TimeVarying Delays. Automatic Control,2005,50:1781-1785
37 H. J. Zhang, Z. Teng. Existence and Global Exponential Stability of Almost Periodic Solution for Cellular Neural Networks with Variable Coefficients and Time-Varying Delays. Neural Networks,2005,16:1340-1351
38 H. T. Lu. Global Exponential Stability Analysis of Cohen-Grossberg Neural Networks. Circuits and Systems II,2005,52:476-479
39 X. Liu, K. L. Teo, B. Xu. Exponential Stability of Impulsive High-order Hopfield-type Neural Networks with Time-Varying Delays. Neural Networks,2005, 6(16):1329-1339
40 L. J. Zhang, L. G. Si. Globally Exponential stability of a Class of Neural Networks with Variable Delays. Mathematical Applicata. 2007, 20(2): 258-262
41 S. M. Zhong, X. Z. Liu. Exponential Stability and Periodicity of Cellular Neural Networks with Time Delay. Mathematical and Computer Modelling, 2007, 45 (9210) : 1231-1240.
42 J. WANG, J. G. LU. Global Exponential Stability of Fuzzy Cellular Neural Networks with Delays and Reaction Diffusion Terms.Chaos, Solitons & Fractals, 2008, 38 (3) : 878-885
43 KWONOM, P. H. Exponential Stability for Uncertain Cellular Neural Networks with Discrete and Distributed Time Varying Delay. Applied Mathematics and Computation, 2008, 203 (2) : 813-823.
44 KWONOM, P. H. Delay Dependent Stability for Uncertain Cellular Neural Networks with Discrete and Distribute Time Varying Delays. Journal of the Franklin Institute , 2008, 345 (7) : 766-778
45 J. X. LU, Y.C. MA. Mean Square Exponential Stability and Periodic Solutions of Stochastic Delay Cellular Neural Networks . Chaos, Soliton s & Fractals, 2008, 38 (5): 1323-1331
46 Z. X. LIU, S. LV, S. M. ZHONG. Novel Exponential Stability and Periodicity of Cellular Neural Networks. Application Research of Computers,2009,08:2879-2880
47 M. Cheng, C. C. Tsai. Hybrid Robust Tracking Control for a Mobile Manipulator via Sliding-mode Neural Network Mechatronics ICM. IEEE InternationalConference,2005:537-542
48 L. Xie. Stochastic Robust Stability Analysis for Markovian Jumping Neural Networks with Time Delays Networking. Sensing and Control,2005,19:923-928
49 T. Hayakawa, W. M. Haddad, N. Hovakimyan. Neural Network Adaptive Control for Nonlinear Uncertain Dynamical Systems with Asymptotic Stability Guarantees. American Control,2005,8:1301-1306
50 C. T. Hsu, M. S. Kang and C. S. Chen. Design of Adaptive Load Shedding by Artificial Neural Networks. Generation Transmission and Distribution,2005,6(3):415-421
51 V. Kharitonov, S. Mondie, J. Collado. Exponential Estimates for Neutral Time-Delay Systems: an LMI Approach. Automatic Cotrol,2005,50(5): 666-670
52廖晓昕.稳定性的理论、方法和应用.广州.华中科技大学出版社:2004
2 T. Roska, T. Boros, P. Thiran. Detecting Simple Motion Using Cellular Neural Networks. Cellular Neural Networks and Their Applications, 1999:127-138
3 S. Q. Hu, J. Wang. Global Stability of a Class of Continuous-time Recurrent Neural Networks. Circuits and Systems I: Fundamental Theory and Applications, 2002,49(2):1334-1347
4 S. Arik. An Analysis of Global Asymptotic Stability of Delayed Cellular Neural Networks. Neural Networks,2002,13(5):1239-1242
5 S. Arik. A Note on the Global Stability of Dynamical Neural Networks. Circuits and Systems I: Fundamental Theory and applications, 2002,49(4):502-504
6 X. F. Liao, G. R. Chen, E. N. Sanchez. LMI-based Approach for Asymptotically Stability Analysis of Delayed Neural Networks. Circuits andSystems I: Fundamental Theory and applications,2002,49(7):1033-1039
7 S. Yerramalla, B. Cukic, E. Fuller. Lyapunov Stability Analysis of the Quantization Error for DCS Neural Networks. Neural Networks,2003,3:2412-2417
8 H. Z. Zheng, F. Q. Gu, Y. S. Sun. Asymptotic Stability of FeedbackNeural Networks with Time-Delay. Machine Learning and Cybernetics,2003,2:1113-1116
9 J. Cao, J. Wang. Global Asymptotic Stability of a General Class of Recurrent Neural Networks with Time-varying Delays. Circuits and Systems I: Fundamental Theory and Applications,2003,1(50):34-44
10 J. D. Hwang, F. H. Hsiao. Stability Analysis of Neural-network Interconnected Systems. Neural Networks,2003,1(14):201-208
11 T. Ensari, S. Arik, V. Tavsanoglu. Global Asymptotic Stability of a class of Neural Networks with Time Varying Delays. Circuits and Systems,2004,5:820-823
12 A. M. Base, S. Pilyugin, S. A. Wismuller. Stability Analysis of a Self-organizingNeural Network with Feed Forward And Feedback Dynamics,Neural Networks,2004, 25(2):1505-1509
13 D. K. Chaturvedi, O. P. Malik, P. K. Kalra. Generalized Neuron-based Adaptive Power System Stabilizer Generation. Transmission and Distribution IEE Proceedings,2004, 151(2):213-218
14 V. Singh. Robust Stability of Cellular Neural Networks with Delay: Linear MatrixInequality Approach. Control Theory and Applications,2004,15(1):125-129
15 V. Singh. Global Asymptotic Stability of Cellular Neural Networks with Unequal Delays: LMI Approach. Electronics Letters,2004,40:548-549
16 Z. G. Zeng, J, Wang, X. X. Liao. Stability Analysis of Delayed Cellular Neural Networks Described Using Cloning Templates. Circuits and Systems I,2004, 51:2313-2324
17 X. M. Li, L. H. Huang, J. H. Wu. A New Method of Lyapunov Functionals for Delayed Cellular Neural Networks. Circuits and Systems I,2004,11:2263-2270
18 Z. G. Zeng, J. Wang, X. X. Liao. Global Exponential Stability of a General Class of Recurrent Neural Networks with Time-varying Delays. Circuits and Systems I: Fundamental Theory and Applications,2003,10(50):1353-1358
19 J. D. Cao, J. Wang. Global Exponential Stability and Periodicity of Recurrent Neural Networks with Time Delays. Circuits and Systems,2005,52:920-931
20 Y. Chen. Global Asymptotic Stability of Delayed Cohen-Grasberg Neural Networks. IEEE Transactions on Circuits and Systems I,2005,6:114-120
21 S. Vimal. Comments on“Global Robust Stability of Delayed Neural Networks”. IEEE Transactions on Circuits and Systems I,2005,6:254-262
22 T. G. Chu, C. S. Zhang. New Necessary and Sufficient Conditions for Absolute Stability of Neural Networks. Control and Automation ICCA '05.International Conference,2005,1:593-598
23 W. H. Chen, W. X. Zheng. Global Asymptotic Stability of a Class of Neural Networks with Distributed Delays. Circuits and Systems I,2005,6:215-221
24 T. P. Chen. Global Exponential Stability of Delayed Hopfield Neural Networks. NeuralNetworks,2001,14(8):977-980
25 S.Q. Hu, J. Wang. Absolute Exponential Stability of a Class of Continuous-time Recurrent Neural Networks. Neural Networks,2003,14(1):35-45
26 L. Xia, M. Xia. LMI Conditions for global asymptotically stability of neural networks with discrete and distributed delays. ICIC Express Letters, vol. 2, no. 3, pp. 257-262,2008.
27 Z. G. Zeng, J. Wang. Complete Stability of Cellular Neural Networks with Time-varying Delays, IEEE Transactions on Circuits and Systems I,2005,6:15-24
28 S. Y. Xu, Lam James, Ho Daniel. New LMI Condition for Delay-Dependent Asymptotic Stability of Delayed Hopfield Neural Networks. IEEE Transactions on Circuits and Systems II,2005,6:104-110
29 S. Senan, S. Arik. New Results for Exponential Stability of Delayed Cellular Neural Networks. Circuits and Systems II,2005,52:154-158
30 Z. C. Yang, D. Y. Xu, J. Deng. New Conditions for Exponential Stability of Delay Impulsive Neural Networks, Electronics, Circuits and Systems ICECS 2004 Proceedings of the 11th IEEE International Conference,2004:226-229
31 J. Peng, Z. B. Xu. H. Qiao. A Critical Analysis on Global Convergence of Hopfield-type Neural Networks. Circuits and Systems I: Fundamental Theory and Applications,2005,4(52):804-814
32 S. Arik. Global Asymptotic Stability Analysis of Bidirectional Associative Memory Neural Networks with Time Delays. Neural Networks,2005,16(3):580-586
33 H. T. Lu. Comments on“A generalized LMI-based Approach to the Global Asymptotic Stability of Delayed Cellular Neural Networks”. Neural Networks,2005,16(3):778-779
34 H. Qi, L. Qi, X. Yang. Deriving Sufficient Conditions for Global Asymptotic Stability of Delayed Neural Networks via Nonsmooth Analysis-II. Neural Networks,2005, 6(16):1701-1706
35 Q. S. Liu, J. D. Cao, Y. S. Xia. A Delayed Neural Network for Solving Linear Projection Equations and Its Analysis. Neural Networks,2005,16:834-843
36 T. Ensari, S. Arik. Global Stability Analysis of Neural Networks with Multiple TimeVarying Delays. Automatic Control,2005,50:1781-1785
37 H. J. Zhang, Z. Teng. Existence and Global Exponential Stability of Almost Periodic Solution for Cellular Neural Networks with Variable Coefficients and Time-Varying Delays. Neural Networks,2005,16:1340-1351
38 H. T. Lu. Global Exponential Stability Analysis of Cohen-Grossberg Neural Networks. Circuits and Systems II,2005,52:476-479
39 X. Liu, K. L. Teo, B. Xu. Exponential Stability of Impulsive High-order Hopfield-type Neural Networks with Time-Varying Delays. Neural Networks,2005, 6(16):1329-1339
40 L. J. Zhang, L. G. Si. Globally Exponential stability of a Class of Neural Networks with Variable Delays. Mathematical Applicata. 2007, 20(2): 258-262
41 S. M. Zhong, X. Z. Liu. Exponential Stability and Periodicity of Cellular Neural Networks with Time Delay. Mathematical and Computer Modelling, 2007, 45 (9210) : 1231-1240.
42 J. WANG, J. G. LU. Global Exponential Stability of Fuzzy Cellular Neural Networks with Delays and Reaction Diffusion Terms.Chaos, Solitons & Fractals, 2008, 38 (3) : 878-885
43 KWONOM, P. H. Exponential Stability for Uncertain Cellular Neural Networks with Discrete and Distributed Time Varying Delay. Applied Mathematics and Computation, 2008, 203 (2) : 813-823.
44 KWONOM, P. H. Delay Dependent Stability for Uncertain Cellular Neural Networks with Discrete and Distribute Time Varying Delays. Journal of the Franklin Institute , 2008, 345 (7) : 766-778
45 J. X. LU, Y.C. MA. Mean Square Exponential Stability and Periodic Solutions of Stochastic Delay Cellular Neural Networks . Chaos, Soliton s & Fractals, 2008, 38 (5): 1323-1331
46 Z. X. LIU, S. LV, S. M. ZHONG. Novel Exponential Stability and Periodicity of Cellular Neural Networks. Application Research of Computers,2009,08:2879-2880
47 M. Cheng, C. C. Tsai. Hybrid Robust Tracking Control for a Mobile Manipulator via Sliding-mode Neural Network Mechatronics ICM. IEEE InternationalConference,2005:537-542
48 L. Xie. Stochastic Robust Stability Analysis for Markovian Jumping Neural Networks with Time Delays Networking. Sensing and Control,2005,19:923-928
49 T. Hayakawa, W. M. Haddad, N. Hovakimyan. Neural Network Adaptive Control for Nonlinear Uncertain Dynamical Systems with Asymptotic Stability Guarantees. American Control,2005,8:1301-1306
50 C. T. Hsu, M. S. Kang and C. S. Chen. Design of Adaptive Load Shedding by Artificial Neural Networks. Generation Transmission and Distribution,2005,6(3):415-421
51 V. Kharitonov, S. Mondie, J. Collado. Exponential Estimates for Neutral Time-Delay Systems: an LMI Approach. Automatic Cotrol,2005,50(5): 666-670
52廖晓昕.稳定性的理论、方法和应用.广州.华中科技大学出版社:2004