不确定闭环振动控制系统特征值及响应的区间有限元法
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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要求而言，大体上可分为下列三类：物理不确定性。当系统承受载荷而运行时，系统的各种固有特性和响应中的误差或不确定性部分取决于控制其强度的有关材料性能和几何尺寸的实际值。因此，研究人员必须要关心物理量（荷载，材料性能，几何尺寸）的实际的不稳定性。一般来说，物理量的不确定性是有制造误差，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。与概率相反，统计和推断有关。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。这种不确定性只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。
现在结构分析中应用的不确定性模型主要有随机模型，模糊模型，凸模型以及区间模型。由于结构和结构不确定性参数的复杂性和多样性，很难用一种模型来解决所有的不确定性结构问题，必须根据结构和参数的特点来决定适当的数学模型。比如现在应用比较多的随机模型，发展比较完善，但是随机模型一般需要经过大量试验得到参数的联合概率密度函数或分布函数，至少在以下两种情况下，随机模型就不能很好地满足要求：（1）在没有足够的试验数据得出足够精度的联合概率密度（联合概率密度函数）时；（2）在知道结构不服从随机机制时。因此随机模型不是唯一的不确定分析的方法。另外，模糊模型和凸模型在使用时也受到一定的主观信息的限制。
区间模型只需要参数的上下界，或者说是误差，而这在工程上是很容易得到的，因此自从区间数学出现就引起众学者的关注。
振动控制问题是工程上的一个非常重要的问题，具有确定参数系统的振动控制问题已经得到很好的研究。振动闭环控制即根据受控对象的振动状态进行实时的外加控制，使其振动满足人们的预定要求。由于结构不确定性的存在，鲁棒性的研究在控制工程上具有重要地位，鲁棒性问题的广泛工程背景已得到充分认可。
本文运用区间有限元方法，讨论了具有区间参数的结构振动控制问题，将其转化为一个确定系统来研究。给出了求解闭环系统区间特征值和响应的一种方法。基于区间参数导出了区间刚度矩阵和质量矩阵，然后利用矩阵摄动理论和区间扩张理论，推导了复区间特征值和响应上下界估计的算法。这些结果是从二阶系统的左右特征向量出发得到的。
首先简单介绍一下区间数学和控制理论，作为解决闭环系统控制问题的基础。接着从二阶系统出发给出了问题的一般形式；为了系统的稳定，必须给系统以一定的反馈，用极点配置法给出了反馈矩阵的算法；给出了相关区
间矩阵的表达式；由复盘扩张和复矩阵摄动理论推导出特征值和响应的上下界估计的算法。数值例子的分析结果证明，本文所提出的理论和方法是有效的并且能够用来检测系统的鲁棒稳定性。
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.
There are several mathematical models in present researches or applications about uncertain structural analysis, such as stochastic model, fuzzy model, convex model and interval one etc. Due to the diversity and complexity of the uncertain parameters, it is impossible to apply one uncertain model to all kinds of problems. We have to select the better ones for each specific problem. For instance, although
the stochastic one has been developed well, it is difficult to use in the following two cases: (1) no more data can be given to obtain the statistical character; (2) the parameters disagree the random mechanism. For the fuzzy and convex models, there also is certain subjective information limited.
More attention has been given to the interval model since the interval mathematics appears, because it needs only the bounds of the structural parameters, which can be always obtained in the engineering.
The vibration control problems are very important in engineering and the deterministic one has been well developed. The closed-loop control systems are defined as following: the real-time control is applied to the controlled objects according the state of the objects to obtain the requirement in advance. Because of the uncertainties of the structures, the research of the robustness has important positions in controlling engineering. The extensive engineering background of the robustness question is already fully approved.
Using the interval analysis, the vibration control problem of structures with interval parameters is discussed，which is approximated by a deterministic one. The methods to solve the interval eigenvalues and dynamics responses of the closed-loop system are presented respectively. The expressions of the interval stiffness and interval mass matrix are developed directly from the interval parameters. With matrix perturbation and interval extension theory, the algorithm for estimating the upper and lower bounds of the interval complex eigenvalue and response are developed. The results are derived in terms of eigenvalues and left and right eigenvectors of the second-order systems. The present method is applied to a vibration system to illustrate the application. The numerical results show that the present methods are valid.
First, we briefly introduce the theory of interval mathematics and control
question as the base of solving questions of closed-loop systems. Then we give the general form of the vibration control problem based on the 2nd systems. Using the pole allocation method, the modal gains are obtained to guarantee the asymmetric stability for the approximate deterministic system. The expressions o
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要求而言，大体上可分为下列三类：物理不确定性。当系统承受载荷而运行时，系统的各种固有特性和响应中的误差或不确定性部分取决于控制其强度的有关材料性能和几何尺寸的实际值。因此，研究人员必须要关心物理量（荷载，材料性能，几何尺寸）的实际的不稳定性。一般来说，物理量的不确定性是有制造误差，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。与概率相反，统计和推断有关。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。这种不确定性只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。
现在结构分析中应用的不确定性模型主要有随机模型，模糊模型，凸模型以及区间模型。由于结构和结构不确定性参数的复杂性和多样性，很难用一种模型来解决所有的不确定性结构问题，必须根据结构和参数的特点来决定适当的数学模型。比如现在应用比较多的随机模型，发展比较完善，但是随机模型一般需要经过大量试验得到参数的联合概率密度函数或分布函数，至少在以下两种情况下，随机模型就不能很好地满足要求：（1）在没有足够的试验数据得出足够精度的联合概率密度（联合概率密度函数）时；（2）在知道结构不服从随机机制时。因此随机模型不是唯一的不确定分析的方法。另外，模糊模型和凸模型在使用时也受到一定的主观信息的限制。
区间模型只需要参数的上下界，或者说是误差，而这在工程上是很容易得到的，因此自从区间数学出现就引起众学者的关注。
振动控制问题是工程上的一个非常重要的问题，具有确定参数系统的振动控制问题已经得到很好的研究。振动闭环控制即根据受控对象的振动状态进行实时的外加控制，使其振动满足人们的预定要求。由于结构不确定性的存在，鲁棒性的研究在控制工程上具有重要地位，鲁棒性问题的广泛工程背景已得到充分认可。
本文运用区间有限元方法，讨论了具有区间参数的结构振动控制问题，将其转化为一个确定系统来研究。给出了求解闭环系统区间特征值和响应的一种方法。基于区间参数导出了区间刚度矩阵和质量矩阵，然后利用矩阵摄动理论和区间扩张理论，推导了复区间特征值和响应上下界估计的算法。这些结果是从二阶系统的左右特征向量出发得到的。
首先简单介绍一下区间数学和控制理论，作为解决闭环系统控制问题的基础。接着从二阶系统出发给出了问题的一般形式；为了系统的稳定，必须给系统以一定的反馈，用极点配置法给出了反馈矩阵的算法；给出了相关区
间矩阵的表达式；由复盘扩张和复矩阵摄动理论推导出特征值和响应的上下界估计的算法。数值例子的分析结果证明，本文所提出的理论和方法是有效的并且能够用来检测系统的鲁棒稳定性。
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.
There are several mathematical models in present researches or applications about uncertain structural analysis, such as stochastic model, fuzzy model, convex model and interval one etc. Due to the diversity and complexity of the uncertain parameters, it is impossible to apply one uncertain model to all kinds of problems. We have to select the better ones for each specific problem. For instance, although
the stochastic one has been developed well, it is difficult to use in the following two cases: (1) no more data can be given to obtain the statistical character; (2) the parameters disagree the random mechanism. For the fuzzy and convex models, there also is certain subjective information limited.
More attention has been given to the interval model since the interval mathematics appears, because it needs only the bounds of the structural parameters, which can be always obtained in the engineering.
The vibration control problems are very important in engineering and the deterministic one has been well developed. The closed-loop control systems are defined as following: the real-time control is applied to the controlled objects according the state of the objects to obtain the requirement in advance. Because of the uncertainties of the structures, the research of the robustness has important positions in controlling engineering. The extensive engineering background of the robustness question is already fully approved.
Using the interval analysis, the vibration control problem of structures with interval parameters is discussed，which is approximated by a deterministic one. The methods to solve the interval eigenvalues and dynamics responses of the closed-loop system are presented respectively. The expressions of the interval stiffness and interval mass matrix are developed directly from the interval parameters. With matrix perturbation and interval extension theory, the algorithm for estimating the upper and lower bounds of the interval complex eigenvalue and response are developed. The results are derived in terms of eigenvalues and left and right eigenvectors of the second-order systems. The present method is applied to a vibration system to illustrate the application. The numerical results show that the present methods are valid.
First, we briefly introduce the theory of interval mathematics and control
question as the base of solving questions of closed-loop systems. Then we give the general form of the vibration control problem based on the 2nd systems. Using the pole allocation method, the modal gains are obtained to guarantee the asymmetric stability for the approximate deterministic system. The expressions o
引文
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张景绘，动力学系统建模，国防工业出版社，2000。
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Qiu, Z.P., Chen, S.H., and Elishakoff I., Natural Frequencies of Structures with Uncertain-but-non-random Parameters, Journal of Optimization Theory and Applications, 1995,86(3):669-683.
Qiu, Z.P., Chen, S.H., and Elishakoff I., Bounds of Eigenvalues for Structures with an Interval Description of Uncertain-but-non-random Parameters, Chaos, Soliton, and Fractral, 1996,7(3):425-434.
10．Chen, S.H., and Qiu, Z.P., Perturbation Method for Computing Eigenbalue Bounds in Vibration System with Interval Parameters, Communications in Numerical Methods in Engineering, 1994,10(2):121-134.
11．Chen, S.H., and yang, X. W., Interval Finite Element Method of the Beam Structures, Finite Elements in Analysis and Design, Vol.34,pp.75-88,2000.
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Thoft-Christensen, P., and Bark, M.J., Structural Reliability Theory and Its Applications, Springer-Verlag, 1982.
张景绘，动力学系统建模，国防工业出版社，2000。
Ben-Haim, Y., and Elishakoff, I., Convex Models of Uncertainty in Applied Mechanics, Elsevier Science Publishers, Amsterdam, 1990.
Elishakoff, I., Essay On Uncertainties in Elastic and Viscoelastic Structures: From A.M. Frendenthal’s Criticisms to Modern Convex Modeling, Techanical Report, Center for Applied Stochastic Research, Florida Atlantic University, Dec., 1993.
Chen, S.H., and Qiu, Z.P., A New Method for Computing The Upper and Lower Bounds on Frequencies of Structures with Interval Parameters, Mechanics Research Communication, 1994,2:583-592.
Qiu, Z.P., Chen, S.H., and Elishakoff I., Natural Frequencies of Structures with Uncertain-but-non-random Parameters, Journal of Optimization Theory and Applications, 1995,86(3):669-683.
Qiu, Z.P., Chen, S.H., and Elishakoff I., Bounds of Eigenvalues for Structures with an Interval Description of Uncertain-but-non-random Parameters, Chaos, Soliton, and Fractral, 1996,7(3):425-434.
10．Chen, S.H., and Qiu, Z.P., Perturbation Method for Computing Eigenbalue Bounds in Vibration System with Interval Parameters, Communications in Numerical Methods in Engineering, 1994,10(2):121-134.
11．Chen, S.H., and yang, X. W., Interval Finite Element Method of the Beam Structures, Finite Elements in Analysis and Design, Vol.34,pp.75-88,2000.
12. 徐世杰,徐肖毫,黄文虎,动力学系统稳定性分析的区间矩阵法,振动工程学报,1989年,第2卷,第3期。
13. Hisda, T. & Nakagiri, S., A note on stochastic finite element method(part3)-An extension of methodology to nonlinear problems, Seisan-Kenkyu., 1980,32(12):572-575.
14. Hisda, T. & Nakagiri, S., A note on stochastic finite element method(part2)-Variation of stress and strain caused by fluctuations of material properties and geometrical boundary condition, Seisan-Kenkyu.,
1980,35(2):262-265.
15. Hisda, T. & Nakagiri, S., Role of the stochastic finite element method in struct. Safety and reliability. Proc. 4th Int. Conf. on Struct. Safety and Reliability Kobe, Japan,1985:213-219.
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