基于矩阵摄动理论的鲁棒极点配置的参数化方法

发布时间：2018-02-04 00:15

本文关键词： 极点配置参数化方法矩阵摄动理论鲁棒性　出处：《东北电力大学》2017年硕士论文　论文类型：学位论文

【摘要】：线性系统的鲁棒控制问题一直是控制领域研究的热点内容。通过设计反馈控制器使系统按照期望的规律运动是控制系统设计的基本方法。在实际的控制系统设计中,必须保证系统满足期望的稳态性能和暂态性能,利用特征结构配置的参数化方法设计状态或输出反馈控制器可以方便地实现这一目标。但是,实际系统都工作在不断变化的环境中,不可避免地会受到外界扰动和参数变化的影响,给系统建模带来不确定性,从而严重影响系统设计的稳态特性和动态响应特性。解决这个问题最好的方法就是对系统进行鲁棒性设计,对控制律和鲁棒性能指标进行不断地探究和创新。基于上述问题,本文针对一阶线性系统,进行的主要研究内容及取得的主要成果总结如下:首先介绍了通过矩阵的Jordan分解与系统初值的选取来获得期望的状态与输出控制的方法。研究了状态反馈与输出反馈极点配置的参数化方法,通过对Sylvester矩阵方程的求解,得到控制器的完全参数化表达式,通过适当地选取自由参数来获得满足控制要求的控制器。但是受限于自由参数的数量,系统有时会出现无解的情况,针对这种情况,本文给出了一种状态反馈下近似求解的优化指标,从而求得近似解。由此得到启发,可以通过增加自由参数的数量来避免这种近似求解情况。本文在特征向量矩阵参数化结果的基础上,进一步研究了如何通过引入新的可调参数来增加系统的自由度,以便更有利于实现期望的系统特性。然后在参数化方法的基础上,利用参数化结果为系统设计提供的自由度,对系统进行鲁棒性设计。本文通过矩阵摄动理论的分析,推导出了一种新的鲁棒指标。该鲁棒指标的原理不同于以往常用的谱条件数,它并不是关于特征向量的函数,所以其计算过程简单且效率高,因此更适合大系统。考虑到该指标的优点,进一步研究了大系统的鲁棒分散控制问题,利用该鲁棒指标可以很容易地将闭环极点配置在特定的区域。同时使得闭环特征值对外界扰动和参数变化具有最小的灵敏度。本文通过对参数化方法的研究得出了闭环特征值、特征向量与反馈增益矩阵的参数化表达式之间的相互关系。总结了相关算法,并通过数值算例表明了方法的有效性。在参数化方法的基础上,利用所提供的的自由度,研究基于矩阵摄动理论的新型鲁棒指标,解决了鲁棒分散控制问题。通过数值算例与仿真与以往方法进行对比,表明了新指标优越性。
[Abstract]:The robust control problem of linear systems is always a hot topic in the field of control. It is the basic method to design the control system by designing feedback controller to make the system move according to the expected law. Middle. The desired steady-state and transient performance of the system must be guaranteed. It is convenient to design the state or output feedback controller using the parameterized method of eigenstructure configuration. The actual system is working in the changing environment, which will inevitably be affected by the external disturbance and parameter change, which brings uncertainty to the system modeling. Therefore, the steady-state and dynamic response characteristics of the system design are seriously affected. The best way to solve this problem is to design the system robustness. The control law and robust performance index are constantly explored and innovated. Based on the above problems, this paper aims at the first order linear system. The main research contents and main results obtained are summarized as follows:. Firstly, the method of obtaining desired state and output control by Jordan decomposition of matrix and the selection of system initial value is introduced, and the parameterization method of state feedback and output feedback pole assignment is studied. The complete parameterized expression of the controller is obtained by solving the Sylvester matrix equation. By properly selecting the free parameters to obtain the controller which meets the control requirements, but limited by the number of free parameters, the system will sometimes have no solution, in view of this situation. In this paper, an optimization index for approximate solution under state feedback is given, and the approximate solution is obtained. This approximate solution can be avoided by increasing the number of free parameters. This paper is based on the parameterized results of eigenvector matrix. In this paper, we further study how to increase the degree of freedom of the system by introducing new adjustable parameters in order to achieve the desired system characteristics. Then, based on the parameterization method. Using the degree of freedom provided by the parameterized results, the robust design of the system is carried out. In this paper, the matrix perturbation theory is used to analyze the robustness of the system. A new robust index is derived. The principle of the robust index is different from the usual spectral condition number. It is not a function of the eigenvector, so the calculation process is simple and efficient. Therefore, it is more suitable for large scale systems. Considering the advantages of this index, the robust decentralized control problem of large scale systems is further studied. By using the robust index, the closed-loop poles can be easily disposed in a specific region. At the same time, the closed-loop eigenvalues have the minimum sensitivity to external disturbances and parameter changes. In this paper, the parameterization method is studied. The closed-loop eigenvalues are given. The correlation between the eigenvector and the parameterized expression of the feedback gain matrix. The correlation algorithm is summarized, and the effectiveness of the method is demonstrated by a numerical example. Using the degree of freedom provided, a new robust index based on matrix perturbation theory is studied, and the problem of robust decentralized control is solved. The numerical example is compared with the simulation method, and the superiority of the new index is shown.
【学位授予单位】：东北电力大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TP13

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