几类随机微分系统的动力学行为研究
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摘要
随机微分系统一直以来都是广大学者研究和讨论的重点话题之一。最近涌现了许多关于随机系统的突破性成果，比如：关于随机系统的稳定性分析、鲁棒控制以及滤波设计等。本文主要讨论和研究了两大类随机系统的动力学行为，一类是关于离散随机神经网络系统的稳定性和耗散性的研究；另一类是关于带有马尔科夫跳跃系统的故障检测和无源控制分析。
首先，讨论了几类不确定的离散随机神经网络的指数稳定性问题。通过构造合适的Lyapunov泛函，结合随机稳定性理论以及自由权矩阵方法，导出了一些充分条件以保证随机神经网络是全局均方指数稳定的，通过和已有结果比较发现，所得结论极大地降低了系统的保守性。最后给出了几个数值例子验证了结论的有效性。众所周知，稳定性问题的研究主要是关于系统平衡点的分析，然而从实际情况来看，系统的平衡点有可能不稳定或是根本不存在平衡点，这时系统的耗散性概念就被引入了。当前，很少有作者考虑关于离散随机神经网络的耗散性问题，因此这类问题具有很大的提升空间。鉴于此，本文又针对离散随机神经网络系统的耗散问题进行了研究。通过结合凸组合理论，获得了一个关于不确定离散随机神经网络系统的耗散性准则。
其次，基于当前社会对高安全标准日益提高的要求，本文进一步研究了不确定的马尔科夫跳跃系统的故障检测滤波设计问题。通过使用一个可观测的故障检测滤波作为残差产生器，此问题最终归结为H∞滤波设计问题。特别的，本文使用两个不同的马尔科夫过程来描述系统矩阵模式和时滞模式，这不仅是理论研究的需要，也是实际应用的需要。通过使用倒凸的方法保证了结果的优越性。
最后，针对转移率部分知道的不确定的马尔科夫跳跃系统的无源控制问题进行了讨论。无源性在电力系统和非线性控制系统中扮演着重要角色，并且提供了一个有效的工具用来讨论系统的稳定性，这也是研究无源性的主要原因。通过结合简森不等式，得到了一个理想的无源控制器且保证了闭环系统是无源的。
Stochastic differential system has been one of the most important discussion topicsfor the scholar. Recently, both analysis and synthesis problem for stochastic systems havebeen extensively studied and a great number of important results have been reported, suchas stability analysis for stochastic systems, robust control and filter design problems andso on. In this dissertation, the dynamical behavior about two kinds of stochastic systemsis discussed, one is stability and dissipativity analysis for discrete-time stochastic neuralnetworks with time-varying delays; other one is robust fault detection and passification ofMarkovian jump systems.
Firstly, the problem of exponential stability for a class of uncertain discrete-timestochastic neural network with time-varying delays is investigated. By constructing asuitable Lyapunov-Krasovskii functional, combining the stochastic stability theory andthe free-weighting matrix method, some sufficient conditions are established to ensure thestochastic neural networks are globally exponential stability in the mean square, which areproved to be less conservative than previous results. Finally, some numerical examplesare given to demonstrate the effectiveness of the proposed results. To the best of ourknowledge, few authors have considered the problem on the dissipativity of discrete-timestochastic neural networks with time-varying delays and this problem also has much moreimprovement space. Motivated by the above discussions, the global exponential dissipa-tivity in mean for uncertain stochastic discrete-time neural networks is studied. By com-bining with the convex combination theory, a new sufficient condition for checking theglobal dissipativity of the addressed stochastic discrete-time neural networks is obtained.
Secondly, with the rising demand for higher safety and reliability standards in themodern industries, the robust fault detection filter design problems for uncertain nonlin-ear Markovian jump systems is studied. By using a observer-based fault detection filter asresidual generator, the robust fault detection filter design is formulated as an H∞-filteringproblem. Particularly, two different Markov processes are considered for modeling therandomness of system matrix and the state delay, which is not only theoretically inter-esting and challenging, but also very important in practical applications. By using a newconvex polyhedron technique, it will reflect the superiority of our conclusions.
Finally, the problem of delay-dependent passivity analysis and passification of un-certain Markovian jump systems with partially known transition rates is investigated. Thepassivity and passification problem has played an efficient role in both electrical networkand nonlinear control systems, and provides a nice tool for analyzing the stability of sys-tems, this is a reason for our research. By combining with Jensen inequality, a desiredpassification controller is designed, which guarantees that the closed-loop system is pas-sive.
首先，讨论了几类不确定的离散随机神经网络的指数稳定性问题。通过构造合适的Lyapunov泛函，结合随机稳定性理论以及自由权矩阵方法，导出了一些充分条件以保证随机神经网络是全局均方指数稳定的，通过和已有结果比较发现，所得结论极大地降低了系统的保守性。最后给出了几个数值例子验证了结论的有效性。众所周知，稳定性问题的研究主要是关于系统平衡点的分析，然而从实际情况来看，系统的平衡点有可能不稳定或是根本不存在平衡点，这时系统的耗散性概念就被引入了。当前，很少有作者考虑关于离散随机神经网络的耗散性问题，因此这类问题具有很大的提升空间。鉴于此，本文又针对离散随机神经网络系统的耗散问题进行了研究。通过结合凸组合理论，获得了一个关于不确定离散随机神经网络系统的耗散性准则。
其次，基于当前社会对高安全标准日益提高的要求，本文进一步研究了不确定的马尔科夫跳跃系统的故障检测滤波设计问题。通过使用一个可观测的故障检测滤波作为残差产生器，此问题最终归结为H∞滤波设计问题。特别的，本文使用两个不同的马尔科夫过程来描述系统矩阵模式和时滞模式，这不仅是理论研究的需要，也是实际应用的需要。通过使用倒凸的方法保证了结果的优越性。
最后，针对转移率部分知道的不确定的马尔科夫跳跃系统的无源控制问题进行了讨论。无源性在电力系统和非线性控制系统中扮演着重要角色，并且提供了一个有效的工具用来讨论系统的稳定性，这也是研究无源性的主要原因。通过结合简森不等式，得到了一个理想的无源控制器且保证了闭环系统是无源的。
Stochastic differential system has been one of the most important discussion topicsfor the scholar. Recently, both analysis and synthesis problem for stochastic systems havebeen extensively studied and a great number of important results have been reported, suchas stability analysis for stochastic systems, robust control and filter design problems andso on. In this dissertation, the dynamical behavior about two kinds of stochastic systemsis discussed, one is stability and dissipativity analysis for discrete-time stochastic neuralnetworks with time-varying delays; other one is robust fault detection and passification ofMarkovian jump systems.
Firstly, the problem of exponential stability for a class of uncertain discrete-timestochastic neural network with time-varying delays is investigated. By constructing asuitable Lyapunov-Krasovskii functional, combining the stochastic stability theory andthe free-weighting matrix method, some sufficient conditions are established to ensure thestochastic neural networks are globally exponential stability in the mean square, which areproved to be less conservative than previous results. Finally, some numerical examplesare given to demonstrate the effectiveness of the proposed results. To the best of ourknowledge, few authors have considered the problem on the dissipativity of discrete-timestochastic neural networks with time-varying delays and this problem also has much moreimprovement space. Motivated by the above discussions, the global exponential dissipa-tivity in mean for uncertain stochastic discrete-time neural networks is studied. By com-bining with the convex combination theory, a new sufficient condition for checking theglobal dissipativity of the addressed stochastic discrete-time neural networks is obtained.
Secondly, with the rising demand for higher safety and reliability standards in themodern industries, the robust fault detection filter design problems for uncertain nonlin-ear Markovian jump systems is studied. By using a observer-based fault detection filter asresidual generator, the robust fault detection filter design is formulated as an H∞-filteringproblem. Particularly, two different Markov processes are considered for modeling therandomness of system matrix and the state delay, which is not only theoretically inter-esting and challenging, but also very important in practical applications. By using a newconvex polyhedron technique, it will reflect the superiority of our conclusions.
Finally, the problem of delay-dependent passivity analysis and passification of un-certain Markovian jump systems with partially known transition rates is investigated. Thepassivity and passification problem has played an efficient role in both electrical networkand nonlinear control systems, and provides a nice tool for analyzing the stability of sys-tems, this is a reason for our research. By combining with Jensen inequality, a desiredpassification controller is designed, which guarantees that the closed-loop system is pas-sive.
引文
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