不确定性振动控制系统的随机方法
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摘要
在工程实际中，结构分析和设计一般都是建立在确定的数学模型基础上的，然而由于结构的复杂性，制造误差和测量的不精确等等因素的影响，结构参数往往是不确定的。在大型复杂结构中，诸多不确定因素的综合作用将给结构分析带来很大的影响。因此，以不确定的参数为基础建立不确定数学模型，直接进行不确定性分析在工程中具有重要的意义。
不确定性的分类及其定量化方法就工程中不确定性的存在方式而言，大体上可分为下列三类：物理不确定性。它取决于荷载，材料性能，几何尺寸等实际物理量的不稳定性。一般来说，物理量的不确定性是由制造误差，安装误差或工作条件变化引起的。统计不确定性。处理工程中的误差或不确定性问题，目前大多采用概率统计方法。一方面，样本的大小受到实际情况和经济上的限制，另一方面，背景噪声的存在必然使统计存在某些误差或不确定性。这种不确定性是由于缺乏统计信息而产生的。模型的不确定性。结构分析和设计所利用的是把输出量（结构的位移，应力，应变）同一组输入量（荷载，材料的几何尺寸，弹性模量）联系起来的数学模型。典型的结构和构件的响应除了由基本的物理不确定性引起的不确定性外，本身也含有不确定性的成分。这种不确定性是由理论简化和未知的边界条件产生的，是理论分析模型与工程实际的偏差。
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。
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。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。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。
本文应用随机模型来描述系统的不确定性，用摄动理论给出了参数不确定性振动控制系统特征值和响应标准差的表达式。在计算中只用到随机参数标准差和相关系数矩阵，而不用随机参数概率密度函数，因而使问题得到简化。并对系统稳定性和响应的鲁棒性，以及系统可控性等问题进行了讨论。
在本论文中，作者主要作了以下一些工作：
1．把不确定性系统的控制问题近似的转化为确定性系统的控制问题来处理，将不确定性参数描述为随机变量，应用摄动法和概率方法计算出闭环系统特征值的实部和虚部的标准差及误差，并估计其上、下界。讨论了闭环系统稳定性的鲁棒性。
2．讨论了不确定参数对闭环系统的动力响应的影响，给出了控制系统的动力响应的标准差计算公式及其误差, 并讨论了响应的鲁棒性。
3．讨论了不确定性系统的可控性，通过重频模态控制矩阵奇异值来度量系统重频模态的控制能力，并根据摄动理论给出了奇异值的一阶摄动解。
In practical engineering problems, the theories of the design and analysis of structures are always established on the basis of the definite mathematics models. However, there are always uncertain factors in the structural engineering practices, such as the inaccuracy of the measurement, the complexity of the structures or errors in manufacture, etc. When the structures are large and complex, the combination of the uncertainty can have some effect on the systems. Therefore, it is necessary to design and analyze the structures with uncertain models directly.
The uncertainty can be described as following three kinds: 1.physical uncertainty. It is correlated to load, material and geometrical size. Generally speaking, the change of working conditions and the errors in manufacturing or installation can bring this kind of uncertainty. 2.statistical uncertainty. Nowadays, probability statistical methods are used in solving uncertainty. But because there is no enough statistic information, samples cannot represent all the system information. 3.model uncertainty. During structural analysis and design, the models are constructed between the inputs, including load, geometric size and plastic modules, and the outputs, the displacement, stress and strain. Even beside the physical uncertainty, there are uncertainties in modeling such as the theoretical simplifying and unknown boundary conditions.
Nowadays, during structural analysis and design, these uncertainties need be quantified by some uncertain methods. Generally speaking, according to the
mathematical models with uncertainties, there are some models as follows: probability models, where uncertainties are described as random or probability variables; 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; interval models, which use the interval variables to represent uncertainties and get interval analysis to solve them. The most common methods for solving uncertainty problems are to model the structural parameters as a random vector.
At present dissertation, the random model is used to deal with the control problems of systems with uncertain parameters. The uncertainties of structural parameters are described by a random vector. And using the perturbation theory, the standard deviations of eigenvalues and responses of closed-loop systems with uncertain parameters are discussed. The method presented in this dissertation will not require the distribution function of the random parameters of the systems other than their means and variances. Similarly, the distribution function of random eigenvalues and responses will not be computed other than their means and variances. The robustness of stability and response, and the controllability of repeated mode are discussed.
There are some details as follows:
The vibration control problems of systems with uncertain parameters are discussed, which is approximated with the corresponding deterministic one. The uncertain parameters are modeled to be random variables. The formulas for calculating the standard deviations of eigenvalues of closed-loop systems are derived with the random model and the perturbation. The upper and lower bounds of eigenvalues of closed-loop systems with uncertain parameters is estimated. The robustness of stability for closed-loop systems is discussed.
The affected response due to uncertain parameters of systems are discussed. The formulas for calculating the standard deviations of response of closed-loop systems are derived with the random model and the perturbation. The standard deviations of response can be used to estimate errors of that. The response robustness is discussed..
The controllability of uncertain systems with repeated eigenvalues is discussed. The singular values of repeated modal control matrix was used deal with th
不确定性的分类及其定量化方法就工程中不确定性的存在方式而言，大体上可分为下列三类：物理不确定性。它取决于荷载，材料性能，几何尺寸等实际物理量的不稳定性。一般来说，物理量的不确定性是由制造误差，安装误差或工作条件变化引起的。统计不确定性。处理工程中的误差或不确定性问题，目前大多采用概率统计方法。一方面，样本的大小受到实际情况和经济上的限制，另一方面，背景噪声的存在必然使统计存在某些误差或不确定性。这种不确定性是由于缺乏统计信息而产生的。模型的不确定性。结构分析和设计所利用的是把输出量（结构的位移，应力，应变）同一组输入量（荷载，材料的几何尺寸，弹性模量）联系起来的数学模型。典型的结构和构件的响应除了由基本的物理不确定性引起的不确定性外，本身也含有不确定性的成分。这种不确定性是由理论简化和未知的边界条件产生的，是理论分析模型与工程实际的偏差。
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。
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。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。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。
本文应用随机模型来描述系统的不确定性，用摄动理论给出了参数不确定性振动控制系统特征值和响应标准差的表达式。在计算中只用到随机参数标准差和相关系数矩阵，而不用随机参数概率密度函数，因而使问题得到简化。并对系统稳定性和响应的鲁棒性，以及系统可控性等问题进行了讨论。
在本论文中，作者主要作了以下一些工作：
1．把不确定性系统的控制问题近似的转化为确定性系统的控制问题来处理，将不确定性参数描述为随机变量，应用摄动法和概率方法计算出闭环系统特征值的实部和虚部的标准差及误差，并估计其上、下界。讨论了闭环系统稳定性的鲁棒性。
2．讨论了不确定参数对闭环系统的动力响应的影响，给出了控制系统的动力响应的标准差计算公式及其误差, 并讨论了响应的鲁棒性。
3．讨论了不确定性系统的可控性，通过重频模态控制矩阵奇异值来度量系统重频模态的控制能力，并根据摄动理论给出了奇异值的一阶摄动解。
In practical engineering problems, the theories of the design and analysis of structures are always established on the basis of the definite mathematics models. However, there are always uncertain factors in the structural engineering practices, such as the inaccuracy of the measurement, the complexity of the structures or errors in manufacture, etc. When the structures are large and complex, the combination of the uncertainty can have some effect on the systems. Therefore, it is necessary to design and analyze the structures with uncertain models directly.
The uncertainty can be described as following three kinds: 1.physical uncertainty. It is correlated to load, material and geometrical size. Generally speaking, the change of working conditions and the errors in manufacturing or installation can bring this kind of uncertainty. 2.statistical uncertainty. Nowadays, probability statistical methods are used in solving uncertainty. But because there is no enough statistic information, samples cannot represent all the system information. 3.model uncertainty. During structural analysis and design, the models are constructed between the inputs, including load, geometric size and plastic modules, and the outputs, the displacement, stress and strain. Even beside the physical uncertainty, there are uncertainties in modeling such as the theoretical simplifying and unknown boundary conditions.
Nowadays, during structural analysis and design, these uncertainties need be quantified by some uncertain methods. Generally speaking, according to the
mathematical models with uncertainties, there are some models as follows: probability models, where uncertainties are described as random or probability variables; 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; interval models, which use the interval variables to represent uncertainties and get interval analysis to solve them. The most common methods for solving uncertainty problems are to model the structural parameters as a random vector.
At present dissertation, the random model is used to deal with the control problems of systems with uncertain parameters. The uncertainties of structural parameters are described by a random vector. And using the perturbation theory, the standard deviations of eigenvalues and responses of closed-loop systems with uncertain parameters are discussed. The method presented in this dissertation will not require the distribution function of the random parameters of the systems other than their means and variances. Similarly, the distribution function of random eigenvalues and responses will not be computed other than their means and variances. The robustness of stability and response, and the controllability of repeated mode are discussed.
There are some details as follows:
The vibration control problems of systems with uncertain parameters are discussed, which is approximated with the corresponding deterministic one. The uncertain parameters are modeled to be random variables. The formulas for calculating the standard deviations of eigenvalues of closed-loop systems are derived with the random model and the perturbation. The upper and lower bounds of eigenvalues of closed-loop systems with uncertain parameters is estimated. The robustness of stability for closed-loop systems is discussed.
The affected response due to uncertain parameters of systems are discussed. The formulas for calculating the standard deviations of response of closed-loop systems are derived with the random model and the perturbation. The standard deviations of response can be used to estimate errors of that. The response robustness is discussed..
The controllability of uncertain systems with repeated eigenvalues is discussed. The singular values of repeated modal control matrix was used deal with th
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