Following the development of psychology, physics and neurophysiology, the study about neural networks had been gradually on the rise in the last century. And now it has become a new cross subject. Within the past decades, because it was widely applied in various fields, for example, pattern recognition, signal processing, network control field, the study of neural network quickly became a focus of academic circles. Such that the study about neural networks generates various models and many branches of different forms. Stability is very important among the performance of systems, such that the research on the stability of neural networks has become an important branch. In the application of neural network model, its stability will be affected by many factors, such as time delays, external disturbance and so on. It is now well known that many biological and artificial neural networks contain inherent time delays in signal transmission, owing to the finite speed of information processing and the time of measurement, which has an adverse impact on the performance of the system, for example oscillation, divergence and instability. In summary, this paper mainly studies the stability of delayed neural networks and the design of fault detection filter, except these we also studied the problem about state estimator. Lyapunov stability theory is a basic tool to study the stability of neural networks. Based on the stability theory, combined with free matrix method and some lemmas about inequalities, this paper studies the performance analysis of neural networks.1. The problem of delaydependent stability of discrete stochastic neural networks with timevarying delay is investigated. The timevarying delay is divided into two parts,the timevarying part and the constant part. And according to the decomposition of the situation, We construct a new LyapunovKrasovskii functional. By using the discrete Jesen inequality, a delay dependent global asymptotic stability criterion is obtained. And a numerical example is given to verify the feasibility and effectiveness of the results.2. We investigate the problem of delaydependent stability of discrete stochastic neural networks with timevarying delay. We use a new way to divide the timevarying delay. By extending the delay decomposition technique in the existing literatures, not only the timevarying delay is decomposed, but the probability distribution on each interval is given for the convenience of the proof. Then by constructing a novel Lyapunov function,and using the discrete Jesen inequality, a linear matrix inequality(LMI) is developed to establish sufficient conditions for the RNNs to be globally asymptotically stable in mean square. And a numerical example is given to verify the feasibility and effectiveness of the results. We also illustrate the superiority of our results to other literatures in the form of tables.3. We investigate the stability of stochastic discretetime neural networks(NNs) with discrete timevarying delays and the leakage delay. As the partition of timevarying delay and the leakage delay brought in discretetime system for the first time, we construct a novel LyapunovKrasovskii function based on stability theory. We also proof a new lemma in order to get a better result. Further more sufficient conditions are derived to guarantee the globally asymptotically stability of the equilibrium point. We also give a numerical example to illustrates the feasibility and effectiveness of the results.4. The estimation problem of discrete neural networks with discrete and distributed delays is investigated. We not only divided discrete timedelay, the distribution of delay is also divided. Combined with the new Lyapunov function, a new stability criterion and a estimator gain matrix is proposed in terms of Linear Matrix Inequalities(LMIs) condition.At the same time we list the feasible solutions in the example, in order to illustrates the feasibility and effectiveness of the results.5. Due to the requirements of the system performance in current society is increasing day by day, we study the problem of fault detection for stochastic systems with mixed delays and parameter uncertainties. Considering the system with discrete delays and distributed delays and random sequence, we transform the random sequence into uncertainty of coefficient matrix. Such that it can be processed by the existing conclusions and lemmas. The main idea is to construct some new Lyapunov functional for the fault detection dynamics. A new robustly asymptotically stable criterion for the systems is derived through linear matrix inequality(LMI) by introducing a comprehensive different LyapunovKrasovskii functional. Based on the stable criterion, the fault detection filter is designed in terms of linear matrix inequalities(LMIs) which can be easily checked in practice. In the numerical example, we will give the gain matrix of the fault detection filter, the trajectory of residual vector and the residual vector of evaluation function are also given in the form of images to illustrate the effectiveness of the conclusion. Not only these, from the images we can illustrate the residual vector r(k) is sensitive about the fault signals, but it has good robustness to external disturbance.
