摘要
近年来,递归神经网络(RNNs)的鲁棒稳定性问题成为研究的一个热点。但是由于缺乏RNNs的统一模型,所以其鲁棒稳定性研究并没有统一的方法可循。标准神经网络模型(SNNM-Standard Neural Network Model)由一个线性动力学系统和有界激励函数构成的静态非线性算子连接而成,被表示为线性微分包含(LDI)的形式,便于利用线性矩阵不等式(LMI)技术进行分析。通过参数的设置,SNNM可用来表示大多数时滞(非时滞)RNNs,为不同的RNNs提供了一个统一的分析框架。本文首先研究了SNNM的鲁棒稳定性问题,并成功应用于对各种RNNs的稳定性分析中。另外大多数包含神经网络或T-S模糊模型的时滞(非时滞)智能系统也都可以转化为SNNM的形式,以便于用统一的方法进行控制器的综合。本文在鲁棒稳定性分析的基础上分别对SNNM的鲁棒镇定控制和保性能控制进行了研究,并在此框架下进行非线性系统鲁棒控制器的综合。主要工作如下:
●简单介绍了SNNM的结构及其表示形式。通过将多个不同类型的RNNs转化为SNNM的形式演示了转化过程及其常用转化技巧。进一步分析了SNNM的逼近能力。证明了SNNM可以以任意精度逼近非线性动力学系统,为其在系统辨识和控制中的应用,提供了重要理论依据。
●对具有范数有界不确定性的连续(离散)SNNM的鲁棒渐近稳定性及指数稳定性问题进行了研究。结合Lyapunov稳定性理论和S-方法,推导出了关于时滞(非时滞)SNNM鲁棒稳定性的判定准则,充分考虑了网络非线性激励函数的约束条件。所得到的判定准则被表示为LMI形式,易于求解。另一方面,将鲁棒指数稳定性问题转化为一个广义特征值问题(GEVP),除了可以判断网络是否指数稳定,还可以方便的估计最大指数收敛率,克服了以往方法中存在的不足。其结论被应用于对RNNs的鲁棒稳定性分析中,结果证明是简单且有效的。
●利用SNNM来描述包含神经网络或T-S模糊模型的时滞(或非时滞)智能系统,并在SNNM描述的基础上进行鲁棒镇定控制器和保性能控制器的设计。给出了连续(离散)SNNM的状态反馈鲁棒镇定控制器和保性能控制器以及输出反馈鲁棒镇定控制器和保性能控制器的设计方法,利用变量替换法及一些矩阵变换技巧,控制器方程可以通过求解一组LMIs得到,与以前方法相比更易于设计和分析。大部分基于神经网络或T-S模糊模型的智能系统都可以转化为SNNM,以便采用统一的方法来综合这些智能系统的控制器。
SNNM为RNNs的分析以及非线性系统的控制器综合提供了一个新的思路。同时SNNM具有较强的可扩展性,可望在以后进一步拓宽其应用的范围和深度。
Robust stability of recurrent neural networks (RNNs) has received much attention in the past decades. Because of the lack of a unified model of RNNs, however, there is no universal approach for this study. Standard neural network model (SNNM) is the interconnection of a linear dynamic system and a bounded static nonlinear operator, which is depicted as a linear differential inclusion (LDI) to be analyzed easily with linear matrix inequalities (LMI) technique. Most delayed or non-delayed RNNs can be transformed into SNNM to be analyzed in a unified way. The robust stability of SNNM is investigated detailed in this dissertation, and is used in the analysis of the robust stability of other RNNs. Moreover, most delayed (or nondelayed) intelligent systems composed of neural networks or T-S fuzzy models can be depicted as SNNM for controller synthesis. Robust control and guaranteed cost control of SNNM are studied in this dissertation. The main contributions of this dissertation are summarized as follows:
A brief introduction of SNNM is presented, and several different RNNs are transformed into SNNM to show the transformation procedure and techniques. The approximation capability of SNNM is analyzed. It is approved that SNNM can approximate any dynamic systems to any degree of accuracy. The approximation capability and the learn ability justify the use of SNNM in practical applications.
The robust asymptotical stability and exponential stability of continuous (discrete-time) SNNM with norm-bounded uncertainties are investigated. Applying the Lyapunov stability theory and the S-Procedure technique, some criteria for robust stability SNNM are derived. The criteria are presented as LMI. Especially, the criteria for robust exponential stability are formulated as a generalized eigenvalue problem (GEVP), which establish an estimation of the exponential convergence rate and improve the previous results. The numerical simulations show the effectiveness of our results.
SNNM is used to describe some delayed(or non-delayed) intelligent systems composed of neural networks and T-S fuzzy models, and design the robust stabilizing controller and guaranteed cost controller based on the description model. State-feedback and output-feedback robust controllers for SNNMs are designed to stabilize the closed-loop systems. The control design equations are formed as a set of LMIs. Most neural-network-based (or fuzzy) intelligent systems can be transformed in to the SNNMs for controller synthesis in a unified way.
SNNM provide a new approach for the analysis of RNNs and controller synthesis of nonlinear systems.
引文
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