切換線性系統(tǒng)的譜坐標(biāo)估計
發(fā)布時間：2019-02-13 01:05
【摘要】：切換系統(tǒng)是一類典型的混雜系統(tǒng),具有廣泛的實際應(yīng)用背景和重要的理論研究價值,從而引起大量學(xué)者的廣泛關(guān)注。一般地,切換系統(tǒng)是由若干個子系統(tǒng)和作用在這些子系統(tǒng)上的切換信號構(gòu)成。切換信號的引入使得切換系統(tǒng)的動力學(xué)行為變得更加復(fù)雜,系統(tǒng)可能產(chǎn)生各個子系統(tǒng)所不具有的動態(tài)行為。因此,切換系統(tǒng)可以精確描述復(fù)雜的非線性過程。對于連續(xù)時間切換線性系統(tǒng),雖然譜坐標(biāo)是刻畫切換線性系統(tǒng)性能的一個重要指標(biāo),但是譜坐標(biāo)的計算或估計是非常困難的問題。為了能夠?qū)ζ溥M(jìn)行估計,需要譜坐標(biāo)的等價代數(shù)表征。Barabanov和Sun分別證明系統(tǒng)的譜坐標(biāo)等于所有子系統(tǒng)的極小公共矩陣集測度。所以如何求出系統(tǒng)的譜坐標(biāo)可以轉(zhuǎn)換為求系統(tǒng)的極小公共矩陣集測度問題。另一方面,Blanchini指出,廣義坐標(biāo)變換后矩陣集的極小μ1測度可以任意精度逼近原系統(tǒng)的極小測度。換言之,廣義坐標(biāo)變換后的矩陣集極小μ1測度可以用于估計系統(tǒng)的譜坐標(biāo)。然而由于變換矩陣的維數(shù)未知,上述結(jié)果僅具有理論意義。如何找到合適的廣義坐標(biāo)變換是目前尚未解決的具有挑戰(zhàn)性的問題。本學(xué)位論文的主要研究工作是根據(jù)這一思路研究一般的連續(xù)時間切換線性系統(tǒng)的譜坐標(biāo)估計問題,具體如下：1.坐標(biāo)變換矩陣為可逆方陣。因為任一滿秩方陣都可以表示成有限個初等矩陣的乘積,所以考慮坐標(biāo)變換矩陣為一系列初等矩陣。即分別考慮第Ⅰ,Ⅱ,Ⅲ類坐標(biāo)變換后矩陣集極小μ1測度的性質(zhì),從而尋找使得μ1測度下降的遞歸變換。由第Ⅰ類坐標(biāo)變換的性質(zhì)和矩陣集測度的定義可知,第Ⅰ類坐標(biāo)變換不改變系統(tǒng)的極小μ1測度。逐行迭代實施第Ⅱ類坐標(biāo)變換,得到單調(diào)下降的矩陣集μ1測度最小值序列,且該序列收斂到極小μ1測度。從而將這個極小μ1測度作為譜坐標(biāo)的上界估計。類似的,遞歸的實施第Ⅲ類坐標(biāo)變換,得到變換后的極小μ1測度,用于估計系統(tǒng)的譜坐標(biāo)。2.值得注意的是,基于第Ⅱ類坐標(biāo)變換得到極小μ1測度的充要條件是：坐標(biāo)變換后系統(tǒng)矩陣集{P1*,P2*}的列和滿足下列條件：由此,對矩陣集列和最小值列進(jìn)行坐標(biāo)變換,得到用于估計譜坐標(biāo)的極小μ1測度,從而減少計算量。由于第Ⅲ類坐標(biāo)變換后矩陣集μ1測度是連續(xù)函數(shù),重新設(shè)計算法用于搜索每次坐標(biāo)變換后的矩陣集μ1測度的最小值。迭代實施第Ⅲ類坐標(biāo)變換,得到變換后的極小μ1測度用于估計系統(tǒng)的譜坐標(biāo)。但是關(guān)于第Ⅱ類和第Ⅲ類坐標(biāo)變換之間的關(guān)系問題,目前尚不能從理論上得以解決。給出一些數(shù)值算例用以說明,對于某些具有一定特征或特殊結(jié)構(gòu)的系統(tǒng),采用第Ⅱ類或第Ⅲ類坐標(biāo)變換,得到的變換后矩陣集的極小μ1測度能夠更好的逼近原系統(tǒng)的譜坐標(biāo)。最后初步討論基于一般形式的方坐標(biāo)變換,得到系統(tǒng)譜坐標(biāo)的估計方法。3.一般情況下,可逆方變換不足以實現(xiàn)變換后的極小μ1測度能夠精確的逼近系統(tǒng)的譜坐標(biāo)。因此需要尋找合適的行滿秩的廣義坐標(biāo)變換。該方法的主要思路是：首先確定變換矩陣中的待定參數(shù)及自由未知量的值,得到每次變換后矩陣集μ1測度的最小值。逐行迭代的實施廣義坐標(biāo)變換可以得到單調(diào)下降的矩陣集μ1測度的最小值數(shù)列。進(jìn)一步分析可得,該數(shù)列是收斂的,且其極限值可以用于估計系統(tǒng)的譜坐標(biāo)。
[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，
本文編號：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，
本文編號：2421000
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