【作者】 郭克尖；

【导师】 陈塑寰；

【作者基本信息】 吉林大学 ， 固体力学， 2004， 硕士

【摘要】 对于确定性结构的振动控制理论的研究已经比较完善。然而，在许多工程实际中，经常存在着与材料性质、几何特性、外力、初始条件、边界条件、结构部件接头等有关的误差或不确定性。并且产生这些误差或不确定性的原因有各种各样，有的是参数具有制造误差、安装误差或不确定性；有的是参数具有计算误差和测量误差；有的是在系统不同的工况下，参数具有不同的数值；有的是参数具有一定的变化区域；有的是参数无法精确测定等等。因此，对于具有参数不确定性的振动控制理论的研究是非常必要的。然而，以前的研究都是从数学角度来讨论不确定系统控制的鲁棒性和稳定性问题，很难应用于工程实际。目前，研究不确定性问题的数学模型主要有以下几种：概率模型。将不确定量看成随机变量或随机过程，利用概率论和统计方法研究不确定现象。区间模型。用区间变量来表示不确定变量，利用区间分析方法来解决问题。模糊模型。用模糊变量或模糊函数表示不确定变量，利用模糊统计方法来研究不确定问题。凸模型。将不确定量用带有约束的集合（如椭球）进行约束，然后利用各种优化方法来研究不确定现象。工程中最常用的模型是概率模型，把结构参数作为一个随机向量来处理。然而，由于实际问题中往往缺乏足够的实验数据来建立随机向量的概率密度函数，而使概率方法不能得到可靠的结果。而凸模型理论把不确定量作为约束集合（如多维椭球），需要较少的信息，而能得到较为可靠的结果。本文应用凸（集）模型理论来描述系统的不确定性，并结合摄动理论和优化技术，对具有参数不确定性的振动控制系统的闭环特征值和响应值的上、下界进行了讨论。 <WP=67>在本论文中，作者主要作了以下一些工作：1. 给出了具有参数不确定性的振动控制系统的研究方法。将不确定性的振动控制系统方程分成了确定性和不确定性两部分，用确定性部分来研究振动控制问题，例如用极点配置方法给出反馈控制律。然后用不确定性部分来研究系统的不确定参数对振动控制的影响。用非概率的未知但有界的凸模型来描述系统参数的不确定因素，其最大的不确定性参数作为多维椭球的半轴，所允许的可能不确定性参数都包含在所给定的多维椭球之中。这种不确定性的凸模型与优化技术结合就可建立具有参数不确定性的振动控制系统闭环特征值的上下届的分析理论。因此，本文首先应用凸模型理论来描述系统参数的不确定性，然后结合摄动理论和优化技术给出了具有参数不确定性的振动控制系统闭环特征值的上下届的计算方法。最后还给出了一个算例来说明本文方法的实际应用。3. 一般的，已知不确定性激励和系统求响应的问题有三种情况：系统是确定的，激励是不确定的；系统的参数具有不确定性，而激励是确定的；系统的参数和激励都具有不确定性。当系统比较简单时，一般可以看作是确定系统，而系统在服役期间的激励有时很难准确预测，而且通常受外来事件如地震、火灾、洪灾、强风等的影响，所以往往具有很大的不确定性。当结构系统比较复杂时，所建立的系统模型的参数往往具有误差或不确定性，若激励本身可以确定或为使分析简化假定激励是确定的，这时的问题属于第二类。若激励也比较复杂时，可以看作第三种情况。本文在假定系统的参数含不确定性，但不是时间的函数，以及激励函数与系统的参数无关的条件下，应用凸模型理论分析了具有参数不确定性的振动控制系统的响应问题。由振动控制方程的确定性部分给出了闭环系统响应的中值，并应用凸模型理论来描述系统参数的不确定性，最后由振动控制方程的不确定性部分给出了实际的闭环系统的响应与中值的偏差，从而得到了具有参数不确定性的振动控制系统<WP=68>闭环响应值的上下届的表达式。最后给出数值算例来说明本文方法的应用。更多还原

【Abstract】 The vibration control theory for the systems with deterministic parameters has been well developed. However, in actual situations, the structural parameters are often uncertain, such as the inaccuracy of the measurement, errors in the manufacturing process, invalidity of some components, etc. Therefore, the concept of uncertainty plays an important role in the control problem of the vibration structures. Many studies have been done about the problems only from the view point of mathematics, which are difficult applied to solve the actual engineering problems. Generally speaking, during structural analysis and design, these uncertainties need be quantified by some uncertain methods. And also there are some researches made on the relations among these uncertainties. Nowadays, according to the mathematical models with uncertainties, there are some models as follows, probability models, where uncertainties are described as random or probability variables; interval models, which use the interval variables to represent uncertainties and get interval analysis to solve them; fuzzy models, which use the fuzzy statistic to describe uncertainties by fuzzy variables or functions. Fuzzy optimizations can draw conclusions in fuzzy field in design dimensions; convex models, which can, with certain sets (such as ellipsoid), describe those uncertainties by many convex optimizations. The most common methods for solving uncertainty problems are to model the structural parameters as a random vector or fuzzy set. Unfortunately, the probabilistic approaches can not give reliable results unless sufficient experimental data are available to validate the assumptions about the joint probability densities of the random variables or functions involved. Similarly, for the fuzzy model, uncertainties in the fuzzy statistic still exist such as the fuzzy statistical errors or uncertainties in the fuzzy statistics, and the choice of subjection degree functions has the artificial uncertainties. Therefore, it is necessary to develop a new model, non-probability and non-fuzzy model, which needs less information and can give more reliable results, to describe the uncertainty. Recently, the convex model was used to deal with the uncertain problems in robust analysis of control system and structural failure. For example, Ben-Haim, Elishakoff and Lindberg used the convex model to study the dynamic response and failure of structures with pulse loads. Shi and Gao used the <WP=70>convex model to solve the robustness of control system. At present dissertation, the convex model is used to deal with the control problems of systems with uncertain parameters. The uncertainties of the structural parameters are described by an ellipsoid. And by combining the perturbation theory and optimization technique, the upper and lower bounds of the eigenvalues and the responds of the closed-loop system with uncertain parameters are discussed in this paper. There are some details as follows:The method to research the vibration control problems of the systems with uncertain parameters is developed at present dissertation. The equations of the system with uncertain parameters are divided into two parts, the deterministic part and the uncertain part. The deterministic part is used to investigate the vibration control problem, for example, it is can be used to design the feedback control law. then the uncertain part is used to determine the effects of the uncertain parameters on the actual uncertain system.A non-probability unknown-but-bounded convex model is used to describe the uncertain parameters of the actual uncertain system. First the uncertain parameters are all included in a given N-dimension ellipsoid and the uncertainties of the system are described by convex model theory. Then by combining the perturbation theory and optimization technique, the method for estimating the upper and lower bounds of the eigenvalues of the closed-loop system with uncertain parameters is derived at present dissertation. A numerical example is given to illustrate更多还原