几类递归神经网络的稳定性及其应用研究
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摘要
自从20世纪80年代,Hopfield首次提出了利用能量函数的概念来研究一类具有固定权值的神经网络的稳定性并付诸电路实现以来,关于这类具有固定权值神经网络稳定性的定性研究得到大量的关注。由于神经网络的各种应用取决于神经网络的稳定特性,所以,关于神经网络的各种稳定性的定性研究就具有重要的理论和实际意义。递归神经网络具有较强的优化计算能力,是目前神经计算应用最为广泛的一类神经网络模型。
本文基于线性矩阵不等式技术,研究了几类时变时滞递归神经网络的稳定性问题,以及利用递归神经网络在优化计算中的应用。
本文的主要内容及研究成果总结如下：
(1)对具有固定权值的递归神经网络的稳定性研究现状及其在优化计算中的应用进行了综述,概述了人工神经网络的产生与发展,详细介绍了几类常见递归神经网络及其稳定性的研究现状,以及递归神经网络在优化计算中的研究进展。然后,给出了本文所用到的一些基本概念和相关引理,最后概述了本论文各章节的具体安排。
(2)研究了一类时变时滞静态神经网络的全局指数稳定性问题。通过构造新的Lyapunov泛函得到了具有较小保守性的稳定条件；考虑到时滞对稳定性能的影响,分别建立了仅依赖时滞上界的稳定判据和完全依赖时滞变化率的稳定判据。所得到的稳定判据能够适应慢变时滞和快变时滞两种情况,具有适用范围宽、易于验证等特点。
(3)研究了时变时滞Cohen-Grossberg神经网络的全局指数稳定性问题。基于线性矩阵不等式(LMI)和Lyapunov泛函等技术,建立了时变时滞Cohen-Grossberg神经网络的指数稳定判据。考虑时滞变化率和时滞上界对稳定性能的不同影响,建立了相应的两类稳定判据。通过与现有文献结果的比较,证明所得结果具有小的保守性,易于验证等特点。
(4)考虑离散神经网络的动力学特点,研究了时变时滞离散双向联想记忆(BAM)神经网络的全局指数稳定性问题。通过线性矩阵不等式技术和新的离散Lyapunov泛函技术,提出了该网络一个新的全局指数稳定规则。通过增加xT(k-σM)和yT(k-τM)的信息,和神经元状态变化率的信息,降低了稳定性的保守性,且易于验证；最后,通过仿真例子验证所提结果的有效性。
(5)借助控制系统中对不确定性的描述,针对满足匹配不确定的两类时变时滞递归神经网络,即时变时滞Cohen-Grossberg神经网络和时变时滞离散BAM神经网络,对其进行了鲁棒稳定性研究。基于线性不等式技术和Lyapunov泛函,提出了多个新的全局鲁棒指数稳定条件。最后,通过仿真例子,对所提两类时滞递归神经网络的鲁棒稳定判据进行了验证。
(6)针对一类同时含有等式约束和不等式约束的混杂约束的非线性规划问题,利用神经网络具有内在大规模并行运算和快速收敛等特性,提出了一种非线性优化神经网络,既克服了采用罚函数方法的神经网络求解优化问题的缺陷,同时与引入松弛变量的优化神经网络相比,又具有电路实现简单、计算量小和收敛速度快等特点。此外,利用能量函数对神经网络的稳定性和收敛性进行了分析,进而保证所提出的神经网络具有全局稳定性。
(7)研究了等式约束下的二次规划问题。考虑到时滞的存在能够改变神经网络的拓扑结构这一现象,通过人为引入时滞来达到改变网络动态行为的目的,提出了一种变时滞Lagrange神经网络来求解其规划问题的最优解。通过不同方法得到的多个稳定判据,且定理7.2和定理7.3同时适应于慢时变时滞和快时变时滞,具有适用范围宽、保守性小和易于验证等特点。
Since Hopfield first introduced the concept of energy function to study the stability for a class of fixed-weight recurrent neural networks called Hopfield networks, was proposed in 1980's, and implemented the neural networks in circuits, the qualitative analysis on the stability of the equilibrium point for this class of recurrent neural networks has been intensively studied by many scholars. It is significantly important in theory and practice to qualitatively study the stability of neural networks because many applications of neural networks are dependent on the properties of stability. Recurrent neural networks have powerful computational capabilities, and are one of the most important types in neurocomputing.
In this paper, the stability problem for several classes of recurrent neural networks and application of recurrent neural networks in optimization problems are investigated on the basis of linear matrix inequality technique.
The main contents and contributions of this dissertation are summarized as follows.
(1) It is summarized systematically that present situations of study on the stability of fixed-weight recurrent neural networks and the applications in optimal computation. A review of the developing history of artificial neural networks is firstly presented. And it is presented of the several kinds of recurrent neural networks with fixed-weight, the kinds of delays and the effects of the delays on the neural networks, and research of recurrent neural networks in optimal computation. Then, some basic theories and definition used in the chapters are outlined. And the main work of this paper is also simply introduced.
(2) Exponential stability of a class of static neural networks with time varying delay is investigated. Different from the previous methods, a new Lyapunov functional useful differential inequality is utilized to prove the exponential stability of the concerned neural networks on the basis of linear matrix inequality approach. Considering the different effects of change rate of time varying delay, two exponential stability criteria are established, where one is dependent of the upper bound of time varying delay and the other is dependent on the change rate of time varying delay. The obtained results can be suitable of the cases of slow/fast time varying delay, easy to check, and have wide application ranges and less conservativeness.
(3) The global exponential stability is discussed for Cohen-Grossberg neural networks with time varying delays. On the basis of the linear matrix inequalitiy (LMI) technique, and Lyapunov functional method combined with the Bellman inequality and Jensen inequality technique, we have obtained two main conditions to ensure the global exponential stability of the equilibrium point for this system, one of which is dependent on the change rate of time varying delays, and the other is dependent on the upper bound of time varying delays. Remarks are made with other previous works to show that the proposed results are less restrictive than those given in the earlier literatures, easier to check in practice, and suitable of the cases of slow or fast time varying delays.
(4) Considering the dynamics of discrete-time neural networks be different from those of continuous-time ones, it is studied the global exponential stability for discrete-time bidirectional associative memory (BAM) neural networks with time varying delays. By the linear matrix inequality (LMI) technique and discrete Lyapunov functional combined with inequality techniques, a new global exponential stability criterion of the equilibrium point is obtained for this system. They are added to information on xT(k-σM),yT (k-τM) and information on neural states change rate. Thus, this Lyapunov functional is more general and less conservative. And the simulation example is used to demonstrate the effectiveness of our result.
(5) By the description of the uncertain in control system, several global robust exponential stability condtions are presented for two kinds of uncertain recurrent neural networks with time varying delays via linear matrix inequality (LMI) technique and Lyapunov functional. At last, the simulation examples are used to demonstrate the effectiveness of our results.
(6) The most important advantages of the neural networks are massively parallel processing and fast convergence, therefore, neural networks as a promising approach are emphasised and employed in optimization problems. This paper presents a new optimized neural networks for nonlinear convex programming problems with both quality and inequality constraints. The proposed neural network avoids the deficiency of the penalty function approach. Meanwhile, the present network needs less neuron than that of slack variables approach, which lead to simple circuit implementation, reduced computation burden and fast convergence. On the basis of energy function, the stability and convergence of the proposed neural network are analyzed, which guarantee the global stability of the equilibrium point of neural network.
(7) It is studied to solve quadratic programming problem with equality constrains. Delays can alter the topology structure of neural network, so we can change dynamic behavior of neural network by adding some delay state items, and do not change the equilibrium points of neural network. So a Lagrange network with time varying delay is proposed to solve quadratic programming problem with equality constrains. Using different methods, the three stability criteria are obtained. And Theorem 7.2 and Theorem 7.3 can be adaptable to either quickly changed or slowly changed time varying delay, which have wider applications, less conservativeness and are easy to check.
本文基于线性矩阵不等式技术,研究了几类时变时滞递归神经网络的稳定性问题,以及利用递归神经网络在优化计算中的应用。
本文的主要内容及研究成果总结如下：
(1)对具有固定权值的递归神经网络的稳定性研究现状及其在优化计算中的应用进行了综述,概述了人工神经网络的产生与发展,详细介绍了几类常见递归神经网络及其稳定性的研究现状,以及递归神经网络在优化计算中的研究进展。然后,给出了本文所用到的一些基本概念和相关引理,最后概述了本论文各章节的具体安排。
(2)研究了一类时变时滞静态神经网络的全局指数稳定性问题。通过构造新的Lyapunov泛函得到了具有较小保守性的稳定条件；考虑到时滞对稳定性能的影响,分别建立了仅依赖时滞上界的稳定判据和完全依赖时滞变化率的稳定判据。所得到的稳定判据能够适应慢变时滞和快变时滞两种情况,具有适用范围宽、易于验证等特点。
(3)研究了时变时滞Cohen-Grossberg神经网络的全局指数稳定性问题。基于线性矩阵不等式(LMI)和Lyapunov泛函等技术,建立了时变时滞Cohen-Grossberg神经网络的指数稳定判据。考虑时滞变化率和时滞上界对稳定性能的不同影响,建立了相应的两类稳定判据。通过与现有文献结果的比较,证明所得结果具有小的保守性,易于验证等特点。
(4)考虑离散神经网络的动力学特点,研究了时变时滞离散双向联想记忆(BAM)神经网络的全局指数稳定性问题。通过线性矩阵不等式技术和新的离散Lyapunov泛函技术,提出了该网络一个新的全局指数稳定规则。通过增加xT(k-σM)和yT(k-τM)的信息,和神经元状态变化率的信息,降低了稳定性的保守性,且易于验证；最后,通过仿真例子验证所提结果的有效性。
(5)借助控制系统中对不确定性的描述,针对满足匹配不确定的两类时变时滞递归神经网络,即时变时滞Cohen-Grossberg神经网络和时变时滞离散BAM神经网络,对其进行了鲁棒稳定性研究。基于线性不等式技术和Lyapunov泛函,提出了多个新的全局鲁棒指数稳定条件。最后,通过仿真例子,对所提两类时滞递归神经网络的鲁棒稳定判据进行了验证。
(6)针对一类同时含有等式约束和不等式约束的混杂约束的非线性规划问题,利用神经网络具有内在大规模并行运算和快速收敛等特性,提出了一种非线性优化神经网络,既克服了采用罚函数方法的神经网络求解优化问题的缺陷,同时与引入松弛变量的优化神经网络相比,又具有电路实现简单、计算量小和收敛速度快等特点。此外,利用能量函数对神经网络的稳定性和收敛性进行了分析,进而保证所提出的神经网络具有全局稳定性。
(7)研究了等式约束下的二次规划问题。考虑到时滞的存在能够改变神经网络的拓扑结构这一现象,通过人为引入时滞来达到改变网络动态行为的目的,提出了一种变时滞Lagrange神经网络来求解其规划问题的最优解。通过不同方法得到的多个稳定判据,且定理7.2和定理7.3同时适应于慢时变时滞和快时变时滞,具有适用范围宽、保守性小和易于验证等特点。
Since Hopfield first introduced the concept of energy function to study the stability for a class of fixed-weight recurrent neural networks called Hopfield networks, was proposed in 1980's, and implemented the neural networks in circuits, the qualitative analysis on the stability of the equilibrium point for this class of recurrent neural networks has been intensively studied by many scholars. It is significantly important in theory and practice to qualitatively study the stability of neural networks because many applications of neural networks are dependent on the properties of stability. Recurrent neural networks have powerful computational capabilities, and are one of the most important types in neurocomputing.
In this paper, the stability problem for several classes of recurrent neural networks and application of recurrent neural networks in optimization problems are investigated on the basis of linear matrix inequality technique.
The main contents and contributions of this dissertation are summarized as follows.
(1) It is summarized systematically that present situations of study on the stability of fixed-weight recurrent neural networks and the applications in optimal computation. A review of the developing history of artificial neural networks is firstly presented. And it is presented of the several kinds of recurrent neural networks with fixed-weight, the kinds of delays and the effects of the delays on the neural networks, and research of recurrent neural networks in optimal computation. Then, some basic theories and definition used in the chapters are outlined. And the main work of this paper is also simply introduced.
(2) Exponential stability of a class of static neural networks with time varying delay is investigated. Different from the previous methods, a new Lyapunov functional useful differential inequality is utilized to prove the exponential stability of the concerned neural networks on the basis of linear matrix inequality approach. Considering the different effects of change rate of time varying delay, two exponential stability criteria are established, where one is dependent of the upper bound of time varying delay and the other is dependent on the change rate of time varying delay. The obtained results can be suitable of the cases of slow/fast time varying delay, easy to check, and have wide application ranges and less conservativeness.
(3) The global exponential stability is discussed for Cohen-Grossberg neural networks with time varying delays. On the basis of the linear matrix inequalitiy (LMI) technique, and Lyapunov functional method combined with the Bellman inequality and Jensen inequality technique, we have obtained two main conditions to ensure the global exponential stability of the equilibrium point for this system, one of which is dependent on the change rate of time varying delays, and the other is dependent on the upper bound of time varying delays. Remarks are made with other previous works to show that the proposed results are less restrictive than those given in the earlier literatures, easier to check in practice, and suitable of the cases of slow or fast time varying delays.
(4) Considering the dynamics of discrete-time neural networks be different from those of continuous-time ones, it is studied the global exponential stability for discrete-time bidirectional associative memory (BAM) neural networks with time varying delays. By the linear matrix inequality (LMI) technique and discrete Lyapunov functional combined with inequality techniques, a new global exponential stability criterion of the equilibrium point is obtained for this system. They are added to information on xT(k-σM),yT (k-τM) and information on neural states change rate. Thus, this Lyapunov functional is more general and less conservative. And the simulation example is used to demonstrate the effectiveness of our result.
(5) By the description of the uncertain in control system, several global robust exponential stability condtions are presented for two kinds of uncertain recurrent neural networks with time varying delays via linear matrix inequality (LMI) technique and Lyapunov functional. At last, the simulation examples are used to demonstrate the effectiveness of our results.
(6) The most important advantages of the neural networks are massively parallel processing and fast convergence, therefore, neural networks as a promising approach are emphasised and employed in optimization problems. This paper presents a new optimized neural networks for nonlinear convex programming problems with both quality and inequality constraints. The proposed neural network avoids the deficiency of the penalty function approach. Meanwhile, the present network needs less neuron than that of slack variables approach, which lead to simple circuit implementation, reduced computation burden and fast convergence. On the basis of energy function, the stability and convergence of the proposed neural network are analyzed, which guarantee the global stability of the equilibrium point of neural network.
(7) It is studied to solve quadratic programming problem with equality constrains. Delays can alter the topology structure of neural network, so we can change dynamic behavior of neural network by adding some delay state items, and do not change the equilibrium points of neural network. So a Lagrange network with time varying delay is proposed to solve quadratic programming problem with equality constrains. Using different methods, the three stability criteria are obtained. And Theorem 7.2 and Theorem 7.3 can be adaptable to either quickly changed or slowly changed time varying delay, which have wider applications, less conservativeness and are easy to check.
引文
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