分布参数系统与It(?)型随机系统的稳定与滑模控制研究
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摘要
由泛函微分方程与偏微分方程的相互渗透而自然产生的分布参数系统,已经形成了一个重要的研究分支。目前对分布参数系统的研究,主要采取两种研究方法:一是直接分析方法,二是C_0-半群方法。在实际控制系统中往往会受到系统的状态时滞、参数的不确定性以及外部扰动的影响,而且该系统是无穷维的,具有高度的复杂性,这为系统的分析与设计带来很大的困难。因此建立一种适合工程设计人员简单易行的设计方法是十分必要的。
在实际过程中,随机因素是客观存在的,用确定性方法描述系统可能会丢失系统的某些特性,从而利用确定性系统理论的控制方法对某些系统实行控制时,常常会严重背离所期望的效果。因此,必须在系统中考虑随机因素的描述。例如,对于通讯中的信号传递、环境污染中的污染源的扩散、海洋工程中的物质信息测量等问题,都必须建立随机模型进行分析与研究。所以,随机系统模型对工程问题具有重要的实际意义。所谓随机系统,就是指用以描述随机因素作用下的时间过程的微分动力系统。
基于以上考虑,本文主要针对分布参数系统与It(o|^)型随机系统所面对的性质问题展开深入的研究和探讨,具体内容如下:
1.首先介绍了该课题的背景,包括分布参数系统与It(o|^)型随机系统滑模控制理论的历史与发展现状。并在此基础上,说明了本课题的研究意义。
2.对滞后不确定分布参数系统进行了指数渐近稳定性分析。利用辅助函数法、结合Green公式及Poincaré不等式以及常泛函微分方程中对滞后和不确定项的处理技巧,将集中参数系统的不等式分析方法成功地推广到了滞后不确定分布参数系统的稳定性分析上,并得到了滞后不确定分布参数系统为渐近稳定的充分条件。
3.对不确定时滞分布参数系统进行了滑模控制研究。在一定条件下设计了不确定分布参数系统的滑动模控制器,分析了在滑动模切换面上滑动模控制系统关于不确定量的不变性特征及其稳定性。
4.在均方稳定的意义下,对多时变时滞参数不确定性的It(o|^)型随机系统进行了指数稳定性分析。利用Lyapunov泛函和It(o|^)微分公式,给出了系统均方指数稳定的代数判据。最后给出的例子说明了该方法的可行性。
5.研究了不确定变时滞It(o|^)型随机系统的滑模控制问题。直接构造了随机系统的变结构控制律,运用It(o|^)微分公式、矩阵不等式和随机微分理论证明了滑动模运动的次可达性;最后运用推广的Halanay不等式给出了滑动模态均方稳定的充分条件。仿真实例说明了该方法的可行性。
最后一部分总结了论文的主要工作,并且对分布参数系统和随机系统的滑模控制问题进行了展望。
Distributed parameter systems have been become an important research branch, based on the permeation of functional differential equations with partial differential equations naturally. Now, there are two main methods to research the distributed parameter system. One is the direct analysis method; the other is the C 0?semigroup method. In practical control systems, the system could not exclude the influence of state time-delays, parameter perturbation or external disturbances, and distributed parameter systems are of infinite dimensions and high complexity. All of these bring up great difficulty for the analysis and design of the system. Therefore, it is necessary to design a simple and easy method, which is fit for the engineering designer.
In practice, stochastic factors exist objectively, and the system described by the method of certain systems may lose some characteristics. If the control method of certain system theories is taken to control some practical systems, the practical system will deviate from expected results seriously. Therefore, the stochastic factor has to be taken into account in the description of systems. For example, in respect of some questions of the signal transfer in communication, the diffuseness of pollution source in environment pollution, the measure of material information in ocean engineering and so on, the stochastic modeling must be constructed to analysis and research these problems. Therefore, the modeling of stochastic systems is of important practical meanings as to engineering problems. The so-called stochastic system is differential dynamic systems of the time process, which points to describe stochastic factors.
As it is considered above, the paper is concerned with the property problem of distributed parameter systems and It(o|^) type stochastic systems. The main work and research results of the paper are as follows:
1.Firstly, the paper introduces the topic background, including the history and the present development condition of sliding mode control theories in respect of distributed parameter systems and It(o|^) type stochastic systems. Based on this foundation, the research meaning of the paper is explained.
2.The exponential asymptotical stability analysis is carried out for the uncertain distributed parameter system with time-delays. By making use of the auxiliary functional method and combining with Green formula, Poincaréinequality and some processing techniques of time-delays and uncertainties in ordinary functional differential equations, the inequality analysis method of lumped parameter systems are successfully extended to the stability analysis of the uncertain distributed parameter system with time-delays. In addition, the sufficient condition of asymptotical stability for the uncertain distributed parameter systems with time-delays is obtained.
3.The sliding mode control of the uncertain distributed parameter system with time-delays is discussed. Under certain conditions, the sliding mode controller is designed. On sliding switch surface, the invariant characteristic of uncertain variables and the stability of the sliding mode control system are analyzed.
4.Under the meaning of mean square, the exponential asymptotical stability of the parameter uncertain It(o|^) type stochastic system with multi-time-varying delays is studied. By making use of the Lyapunov function and Ito? differential formula, the algebraic criteria of exponential stability are given in mean square. At last, the feasibility of the result in the paper is illustrated by an example.
5.The sliding mode is studied in respect of the uncertain It(o|^) type stochastic system with time-varying delays. Moreover, the variable structure controller is constructed directly. By using Ito? differential formula, matrix inequality and stochastic differential theory, the sub-reachability of the sliding mode is proved. At the end, a sufficient condition of the sliding mode stability is derived by using the extended Halanay’s inequality in mean square. A simulation example is given to illustrate the feasibility.
In the last section, the main work in the paper is summarized and the sliding mode control problem of distributed parameter systems and stochastic systems are prospected .
在实际过程中,随机因素是客观存在的,用确定性方法描述系统可能会丢失系统的某些特性,从而利用确定性系统理论的控制方法对某些系统实行控制时,常常会严重背离所期望的效果。因此,必须在系统中考虑随机因素的描述。例如,对于通讯中的信号传递、环境污染中的污染源的扩散、海洋工程中的物质信息测量等问题,都必须建立随机模型进行分析与研究。所以,随机系统模型对工程问题具有重要的实际意义。所谓随机系统,就是指用以描述随机因素作用下的时间过程的微分动力系统。
基于以上考虑,本文主要针对分布参数系统与It(o|^)型随机系统所面对的性质问题展开深入的研究和探讨,具体内容如下:
1.首先介绍了该课题的背景,包括分布参数系统与It(o|^)型随机系统滑模控制理论的历史与发展现状。并在此基础上,说明了本课题的研究意义。
2.对滞后不确定分布参数系统进行了指数渐近稳定性分析。利用辅助函数法、结合Green公式及Poincaré不等式以及常泛函微分方程中对滞后和不确定项的处理技巧,将集中参数系统的不等式分析方法成功地推广到了滞后不确定分布参数系统的稳定性分析上,并得到了滞后不确定分布参数系统为渐近稳定的充分条件。
3.对不确定时滞分布参数系统进行了滑模控制研究。在一定条件下设计了不确定分布参数系统的滑动模控制器,分析了在滑动模切换面上滑动模控制系统关于不确定量的不变性特征及其稳定性。
4.在均方稳定的意义下,对多时变时滞参数不确定性的It(o|^)型随机系统进行了指数稳定性分析。利用Lyapunov泛函和It(o|^)微分公式,给出了系统均方指数稳定的代数判据。最后给出的例子说明了该方法的可行性。
5.研究了不确定变时滞It(o|^)型随机系统的滑模控制问题。直接构造了随机系统的变结构控制律,运用It(o|^)微分公式、矩阵不等式和随机微分理论证明了滑动模运动的次可达性;最后运用推广的Halanay不等式给出了滑动模态均方稳定的充分条件。仿真实例说明了该方法的可行性。
最后一部分总结了论文的主要工作,并且对分布参数系统和随机系统的滑模控制问题进行了展望。
Distributed parameter systems have been become an important research branch, based on the permeation of functional differential equations with partial differential equations naturally. Now, there are two main methods to research the distributed parameter system. One is the direct analysis method; the other is the C 0?semigroup method. In practical control systems, the system could not exclude the influence of state time-delays, parameter perturbation or external disturbances, and distributed parameter systems are of infinite dimensions and high complexity. All of these bring up great difficulty for the analysis and design of the system. Therefore, it is necessary to design a simple and easy method, which is fit for the engineering designer.
In practice, stochastic factors exist objectively, and the system described by the method of certain systems may lose some characteristics. If the control method of certain system theories is taken to control some practical systems, the practical system will deviate from expected results seriously. Therefore, the stochastic factor has to be taken into account in the description of systems. For example, in respect of some questions of the signal transfer in communication, the diffuseness of pollution source in environment pollution, the measure of material information in ocean engineering and so on, the stochastic modeling must be constructed to analysis and research these problems. Therefore, the modeling of stochastic systems is of important practical meanings as to engineering problems. The so-called stochastic system is differential dynamic systems of the time process, which points to describe stochastic factors.
As it is considered above, the paper is concerned with the property problem of distributed parameter systems and It(o|^) type stochastic systems. The main work and research results of the paper are as follows:
1.Firstly, the paper introduces the topic background, including the history and the present development condition of sliding mode control theories in respect of distributed parameter systems and It(o|^) type stochastic systems. Based on this foundation, the research meaning of the paper is explained.
2.The exponential asymptotical stability analysis is carried out for the uncertain distributed parameter system with time-delays. By making use of the auxiliary functional method and combining with Green formula, Poincaréinequality and some processing techniques of time-delays and uncertainties in ordinary functional differential equations, the inequality analysis method of lumped parameter systems are successfully extended to the stability analysis of the uncertain distributed parameter system with time-delays. In addition, the sufficient condition of asymptotical stability for the uncertain distributed parameter systems with time-delays is obtained.
3.The sliding mode control of the uncertain distributed parameter system with time-delays is discussed. Under certain conditions, the sliding mode controller is designed. On sliding switch surface, the invariant characteristic of uncertain variables and the stability of the sliding mode control system are analyzed.
4.Under the meaning of mean square, the exponential asymptotical stability of the parameter uncertain It(o|^) type stochastic system with multi-time-varying delays is studied. By making use of the Lyapunov function and Ito? differential formula, the algebraic criteria of exponential stability are given in mean square. At last, the feasibility of the result in the paper is illustrated by an example.
5.The sliding mode is studied in respect of the uncertain It(o|^) type stochastic system with time-varying delays. Moreover, the variable structure controller is constructed directly. By using Ito? differential formula, matrix inequality and stochastic differential theory, the sub-reachability of the sliding mode is proved. At the end, a sufficient condition of the sliding mode stability is derived by using the extended Halanay’s inequality in mean square. A simulation example is given to illustrate the feasibility.
In the last section, the main work in the paper is summarized and the sliding mode control problem of distributed parameter systems and stochastic systems are prospected .
引文
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