切换线性系统的谱坐标估计
发布时间：2019-02-13 01:05
【摘要】：切换系统是一类典型的混杂系统,具有广泛的实际应用背景和重要的理论研究价值,从而引起大量学者的广泛关注。一般地,切换系统是由若干个子系统和作用在这些子系统上的切换信号构成。切换信号的引入使得切换系统的动力学行为变得更加复杂,系统可能产生各个子系统所不具有的动态行为。因此,切换系统可以精确描述复杂的非线性过程。对于连续时间切换线性系统,虽然谱坐标是刻画切换线性系统性能的一个重要指标,但是谱坐标的计算或估计是非常困难的问题。为了能够对其进行估计,需要谱坐标的等价代数表征。Barabanov和Sun分别证明系统的谱坐标等于所有子系统的极小公共矩阵集测度。所以如何求出系统的谱坐标可以转换为求系统的极小公共矩阵集测度问题。另一方面,Blanchini指出,广义坐标变换后矩阵集的极小μ1测度可以任意精度逼近原系统的极小测度。换言之,广义坐标变换后的矩阵集极小μ1测度可以用于估计系统的谱坐标。然而由于变换矩阵的维数未知,上述结果仅具有理论意义。如何找到合适的广义坐标变换是目前尚未解决的具有挑战性的问题。本学位论文的主要研究工作是根据这一思路研究一般的连续时间切换线性系统的谱坐标估计问题,具体如下：1.坐标变换矩阵为可逆方阵。因为任一满秩方阵都可以表示成有限个初等矩阵的乘积,所以考虑坐标变换矩阵为一系列初等矩阵。即分别考虑第Ⅰ,Ⅱ,Ⅲ类坐标变换后矩阵集极小μ1测度的性质,从而寻找使得μ1测度下降的递归变换。由第Ⅰ类坐标变换的性质和矩阵集测度的定义可知,第Ⅰ类坐标变换不改变系统的极小μ1测度。逐行迭代实施第Ⅱ类坐标变换,得到单调下降的矩阵集μ1测度最小值序列,且该序列收敛到极小μ1测度。从而将这个极小μ1测度作为谱坐标的上界估计。类似的,递归的实施第Ⅲ类坐标变换,得到变换后的极小μ1测度,用于估计系统的谱坐标。2.值得注意的是,基于第Ⅱ类坐标变换得到极小μ1测度的充要条件是：坐标变换后系统矩阵集{P1*,P2*}的列和满足下列条件：由此,对矩阵集列和最小值列进行坐标变换,得到用于估计谱坐标的极小μ1测度,从而减少计算量。由于第Ⅲ类坐标变换后矩阵集μ1测度是连续函数,重新设计算法用于搜索每次坐标变换后的矩阵集μ1测度的最小值。迭代实施第Ⅲ类坐标变换,得到变换后的极小μ1测度用于估计系统的谱坐标。但是关于第Ⅱ类和第Ⅲ类坐标变换之间的关系问题,目前尚不能从理论上得以解决。给出一些数值算例用以说明,对于某些具有一定特征或特殊结构的系统,采用第Ⅱ类或第Ⅲ类坐标变换,得到的变换后矩阵集的极小μ1测度能够更好的逼近原系统的谱坐标。最后初步讨论基于一般形式的方坐标变换,得到系统谱坐标的估计方法。3.一般情况下,可逆方变换不足以实现变换后的极小μ1测度能够精确的逼近系统的谱坐标。因此需要寻找合适的行满秩的广义坐标变换。该方法的主要思路是：首先确定变换矩阵中的待定参数及自由未知量的值,得到每次变换后矩阵集μ1测度的最小值。逐行迭代的实施广义坐标变换可以得到单调下降的矩阵集μ1测度的最小值数列。进一步分析可得,该数列是收敛的,且其极限值可以用于估计系统的谱坐标。
[Abstract]:The switching system is a kind of typical hybrid system, which has a wide range of practical application background and important theoretical research value, thus causing extensive attention of a great number of scholars. In general, the switching system is composed of a number of subsystems and a switching signal acting on the subsystems. the introduction of the switching signal makes the dynamic behavior of the switching system more complex and the system may produce a dynamic behavior that the various subsystems do not have. Thus, the switching system can accurately describe a complex non-linear process. For continuous time-switched linear systems, although the spectral coordinates are an important index for describing the performance of switching linear systems, the calculation or estimation of the spectral coordinates is a very difficult problem. In order to be able to estimate it, the equivalent algebraic representation of the spectral coordinates is required. Barbatov and Sun show that the spectral coordinates of the system are equal to the minimum common matrix set measure of all the subsystems. So, how to find out the spectral coordinates of the system can be converted into the minimum common matrix set measure problem of the system. On the other hand, Blanchini points out that the small & mu; 1 measure of the matrix set after the generalized coordinate transformation can approximate the minimum measure of the original system at any precision. In other words, the matrix set of small. mu. 1 after the generalized coordinate transformation can be used to estimate the spectral coordinates of the system. However, since the number of dimensions of the transformation matrix is unknown, the above results are of theoretical significance only. How to find a suitable generalized coordinate transformation is a challenging problem that has not yet been solved. The main research work of this dissertation is to study the spectral coordinate estimation of a general continuous time-switching linear system based on this thought, which is as follows: 1. The coordinate transformation matrix is a reversible matrix. Since any full-rank matrix can be expressed as the product of a finite unitary matrix, it is considered that the coordinate transformation matrix is a series of unitary matrices. In other words, the properties of the matrix set with minimal. mu. 1 measure after the coordinate transformation of the first, the second and the third class are considered, so that a recursive transformation that makes the mu 1 measure fall is found. As can be seen from the definition of the property of the class I coordinate transformation and the measure of the matrix set, the class I coordinate transformation does not change the minimum. mu. 1 measure of the system. The second-class coordinate transformation is carried out by the line-by-line iteration to obtain a monotone-reduced matrix set. mu. 1 measure minimum sequence, and the sequence converges to a minimal. mu. 1 measure. so that this very small. mu. 1 measure is estimated as the upper boundary of the spectral coordinates. Similarly, a recursive implementation of the third class coordinate transformation yields a transformed minimum. mu. 1 measure for estimating the spectral coordinates of the system. It is worth noting that the necessary and sufficient condition for obtaining a minimal. mu. 1 measure based on the second class coordinate transformation is that the column of the system matrix set {P1 *, P2 *} after the coordinate transformation and the following conditions are satisfied: a very small. mu. 1 measure for estimating the spectral coordinates is obtained, thereby reducing the amount of computation. Since the matrix set. mu. 1 measure is a continuous function after the class III coordinate transformation, the re-design algorithm is used to search for the minimum value of the matrix set. mu. 1 measure after each coordinate transformation. The class III coordinate transformation is carried out iteratively to obtain the transformed small. mu. 1 measure for estimating the spectral coordinates of the system. However, the relationship between the second class and the third class coordinate transformation can not be solved theoretically. Some numerical examples are given to illustrate that, for some systems with certain features or special structures, using the second class or the class 鈪，
本文编号：2421000
[Abstract]:The switching system is a kind of typical hybrid system, which has a wide range of practical application background and important theoretical research value, thus causing extensive attention of a great number of scholars. In general, the switching system is composed of a number of subsystems and a switching signal acting on the subsystems. the introduction of the switching signal makes the dynamic behavior of the switching system more complex and the system may produce a dynamic behavior that the various subsystems do not have. Thus, the switching system can accurately describe a complex non-linear process. For continuous time-switched linear systems, although the spectral coordinates are an important index for describing the performance of switching linear systems, the calculation or estimation of the spectral coordinates is a very difficult problem. In order to be able to estimate it, the equivalent algebraic representation of the spectral coordinates is required. Barbatov and Sun show that the spectral coordinates of the system are equal to the minimum common matrix set measure of all the subsystems. So, how to find out the spectral coordinates of the system can be converted into the minimum common matrix set measure problem of the system. On the other hand, Blanchini points out that the small & mu; 1 measure of the matrix set after the generalized coordinate transformation can approximate the minimum measure of the original system at any precision. In other words, the matrix set of small. mu. 1 after the generalized coordinate transformation can be used to estimate the spectral coordinates of the system. However, since the number of dimensions of the transformation matrix is unknown, the above results are of theoretical significance only. How to find a suitable generalized coordinate transformation is a challenging problem that has not yet been solved. The main research work of this dissertation is to study the spectral coordinate estimation of a general continuous time-switching linear system based on this thought, which is as follows: 1. The coordinate transformation matrix is a reversible matrix. Since any full-rank matrix can be expressed as the product of a finite unitary matrix, it is considered that the coordinate transformation matrix is a series of unitary matrices. In other words, the properties of the matrix set with minimal. mu. 1 measure after the coordinate transformation of the first, the second and the third class are considered, so that a recursive transformation that makes the mu 1 measure fall is found. As can be seen from the definition of the property of the class I coordinate transformation and the measure of the matrix set, the class I coordinate transformation does not change the minimum. mu. 1 measure of the system. The second-class coordinate transformation is carried out by the line-by-line iteration to obtain a monotone-reduced matrix set. mu. 1 measure minimum sequence, and the sequence converges to a minimal. mu. 1 measure. so that this very small. mu. 1 measure is estimated as the upper boundary of the spectral coordinates. Similarly, a recursive implementation of the third class coordinate transformation yields a transformed minimum. mu. 1 measure for estimating the spectral coordinates of the system. It is worth noting that the necessary and sufficient condition for obtaining a minimal. mu. 1 measure based on the second class coordinate transformation is that the column of the system matrix set {P1 *, P2 *} after the coordinate transformation and the following conditions are satisfied: a very small. mu. 1 measure for estimating the spectral coordinates is obtained, thereby reducing the amount of computation. Since the matrix set. mu. 1 measure is a continuous function after the class III coordinate transformation, the re-design algorithm is used to search for the minimum value of the matrix set. mu. 1 measure after each coordinate transformation. The class III coordinate transformation is carried out iteratively to obtain the transformed small. mu. 1 measure for estimating the spectral coordinates of the system. However, the relationship between the second class and the third class coordinate transformation can not be solved theoretically. Some numerical examples are given to illustrate that, for some systems with certain features or special structures, using the second class or the class 鈪，
本文编号：2421000
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