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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 容志輝(Chee-Fai Yung) | |
| dc.contributor.author | Guan-Yu Chen | en |
| dc.contributor.author | 陳冠羽 | zh_TW |
| dc.date.accessioned | 2021-06-16T08:07:12Z | - |
| dc.date.available | 2022-07-01 | |
| dc.date.copyright | 2020-07-27 | |
| dc.date.issued | 2020 | |
| dc.date.submitted | 2020-07-21 | |
| dc.identifier.citation | [Barot, 2015] Barot, M. (2015). Introduction to the representation theory of Algebras. [Berger et al., 2012] Berger, T., Ilchmann, A., and Trenn, S. (2012). The quasiWeierstraß form for regular matrix pencils. Linear Algebra and Its Applications, 436(10):4052–4069. [Berger and Trenn, 2012] Berger, T. and Trenn, S. (2012). The QuasiKronecker Form For Matrix Pencils. SIAM Journal on Matrix Analysis and Applications, 33(2):336–368. [Dai, 1989] Dai, L. (1989). Singular Control Systems. Lecture Notes in Control and Information Sciences, 118, SpringerVerlag, Berlin, Heidelberg. [Dmytryshyn et al., 2016] Dmytryshyn, A., da Fonseca, C. M., and Rybalkina, T. (2016). Classification of pairs of linear mappings between two vector spaces and between their quotient space and subspace. Linear Algebra and Its Applications, 509:228–246. [Gantmacher, 1959] Gantmacher, F. R. (1959). The Theory of Matrices, volume I. and II, Chelsea, New York. [Kronecker, 1890] Kronecker, L. (1890). Algebraische reduction der schaaren bilinearer formen. [Kuijper, 1994] Kuijper, M. (1994). FirstOrder Representations of Linear Systems. Birkhauser, Boston. [Kunkel et al., 2006] Kunkel, M. P., Mehrmann, V., Kunkel, P., and Mehrmann, V. (2006). DifferentialAlgebraic Equations Analysis and Numerical Solution. European Mathematical Society. [Lewis, 1986] Lewis, F. L. (1986). A Survey of Linear Singular Systems. Circuits Systems Signal Process, 5(1):3–36. [Lewis, 1992] Lewis, F. L. (1992). A Tutorial on the Geometric Analysis of Linear Timeinvariant Implicit Systems. Automatica, 28(1):119–137. [Luenberger, 1977] Luenberger, D. G. (1977). Singular Dynamic Leontieff Systems. Econometrica, 45:991–995. [Newcomb and Dziurla, 1989] Newcomb, R. W. and Dziurla, B. (1989). Some Circuits and Systems Applications of Semistate Theory. Circuits, Systems, and Signal Processing, 8(3):235–260. [Noferini and Poloni, 2015] Noferini, V. and Poloni, F. (2015). Duality of matrix pencils, Wong chains and linearizations. Linear Algebra and Its Applications, 471:730–767. [Sandovici et al., 2005] Sandovici, A., De Snoo, H., and Winkler, H. (2005). The structure of linear relations in Euclidean spaces. Linear Algebra and Its Applications, 397(13): 141–169. [Verghese et al., 1981] Verghese, G. C., Lévy, B. C., and Kailath, T. (1981). A Generalized StateSpace for Singular Systems. IEEE Transactions on Automatic Control, 26(4):811–831. [Wang et al., 2006] Wang, H. S., Yung, C. F., and Chang, F. R. (2006). H∞ control for nonlinear descriptor systems. SpringerVerlag London Limited. [Weierstrass, 1858] Weierstrass, K. (1858). Uber ein die homogenen Funcktionen zweiten Grades betreffendes Theorem, nebst Anwendung desselben auf die Theorie der kleinen Schwingungen, Monatsh Akad. Berlin. [Wong, 1974] Wong, K.T. (1974). The eigenvalue problem λTx + Sx. Journal of differential Equations, 16(2):270–280. [Yung, 2007] Yung, C. F. (2007). Linear Algebra (in Chinese) 線性代數. WuNan 五南圖書, 1st edition. [Yung, 2011] Yung, C. F. (2011). An Algebraic Geometry Approach to Control Theory and Related Topics (以代數幾何方法研究控制理論及相關問題). Research Report, Ministry of Science and Technology of the Republic of China (992221E019003). [Yung, 2013] Yung, C. F. (2013). Controlled Solution Spaces and Related Control Problems for Linear Discrete Time Descriptor Systems (線性離散時間描述子系統可控解空間及相關控制問題之研究). Research Report, Ministry of Science and Technology of the ROC (1012221E019039). [Yung, 2018] Yung, C. F. (2018). A Maximal Strong Solution Space Geometric Approach to Strict Causality, Controllability and Related Control Problems for Linear Nonregular DiscreteTime Descriptor Systems (以最大強解空間幾何觀點研究Nonregular線性離散時間描述子系統嚴格因果性及可控性及相關控制問題). Research Report, Ministry of Science and Technology of the ROC (1062221E019001). | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/58161 | - |
| dc.description.abstract | 本論文研究非正規矩陣束$(E,A)$的幾何結構以及相應的自治離散時間描述子系統$Ex_{i+1}=Ax_i$。我們展示了Wong氏序列 [Wong, 1974] 的維度結構與Kronecker典型形式之間的聯繫。我們證明了狀態空間可以分解為奇異子空間、非平凡譜子空間和multishift子空間的直和。我們進一步根據相應的商空間結構找到各個子空間的Kronecker基底。此外，我們證明了各個子空間的Kronecker基底恰由奇異鏈、Jordan鏈和multishift鏈構成。這為非正規矩陣束的Kronecker典型形式提供了一種新的幾何證明方法，並且提供更多的幾何直觀。最後，離散時間描述子系統$Ex_{i+1}=Ax_i$的解行為將被完整刻劃。我們也給出數值範例用於說明。 | zh_TW |
| dc.description.abstract | Geometric structure of non-regular matrix pencils $(E,A)$ and the corresponding autonomous discrete-time descriptor systems $Ex_{i+1}=Ax_i$ are studied. We show the connection between the dimension structure of Wong's sequences [Wong, 1974] and the Kronecker canonical form. We also show that the state space can be decomposed as the sum of singularity subspace, nontrivial spectral subspace and the multishift subspace. We further find the Kronecker basis for each subspace in terms of pertinent quotient spaces. Moreover, we prove that the Kronecker basis for each subspace consists of singular chains, Jordan chains and the multishift chains. This provides a new geometric approach to the Kronecker canonical form of a non-regular matrix pencil, and gives more geometric insight. Finally, the solution behavior of the discrete-time descriptor systems $Ex_{i+1}=Ax_i$ is given. Some numerical examples are given for illustration. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-16T08:07:12Z (GMT). No. of bitstreams: 1 U0001-1407202021090800.pdf: 1246323 bytes, checksum: 315797210bc91f93e95d3ee8edd4ffc1 (MD5) Previous issue date: 2020 | en |
| dc.description.tableofcontents | 口試委員審定書i 誌謝iii 中文摘要v Abstract vii Contents viii List of Tables x Nomenclature xi 1 Introduction 1 1.1 Descriptor systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Matrix pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Preliminaries 7 2.1 Some facts in Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Sequence space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Matrix pencils and solvability 13 3.1 Pencil of maps and strict equivalence . . . . . . . . . . . . . . . . . . . . 13 3.2 Wong’s sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Solvability of linear descriptor system (E,A) . . . . . . . . . . . . . . . . 17 4 Singularity subspace X∞ 23 4.1 Singularity subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Singular chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2.1 Restriction of a matrix pencil . . . . . . . . . . . . . . . . . . . . . 28 4.2.2 Singular block representation . . . . . . . . . . . . . . . . . . . . . 29 4.3 Decomposition of singularity subspace . . . . . . . . . . . . . . . . . . . 30 4.4 Fundamental sequences for x0 = 0 containing finite nonzero states . . . . 40 4.5 Nonsingularity and quotient pencils . . . . . . . . . . . . . . . . . . . . 42 5 Multishift subspaceM∞ 49 5.1 Multishift subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.2 Multishift chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.2.1 multishift block representation . . . . . . . . . . . . . . . . . . . . 55 5.3 Decomposition of multishift subspace . . . . . . . . . . . . . . . . . . . . 57 5.3.1 Main theorem for multishift decomposition . . . . . . . . . . . . . 60 6 Eigenstructure 71 6.1 Eigenvalue and eigenspace . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.2 Complete chains and Jordan chains . . . . . . . . . . . . . . . . . . . . . 76 6.2.1 Jordan block representation . . . . . . . . . . . . . . . . . . . . . . 77 6.3 Regular pencils . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 79 6.3.1 Decomposition of reachability subspace . . . . . . . . . . . . . . . 79 6.3.2 The Weierstraß theorem . . . . . . . . . . . . . . . . . . . . . . . . 83 6.4 Nonsingular pencils and regularization . . . . . . . . . . . . . . . . . . . 90 6.5 Singular pencils and nontrivial spectral subspace . . . . . . . . . . . . . . 93 7 Applications 109 7.1 Maximal fundamental chains . . . . . . . . . . . . . . . . . . . . . . . . 109 7.2 Wong’s indices of matrix pencil . . . . . . . . . . . . . . . . . . . . . . . 110 7.3 Reverse matrix pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8 Conclusion 113 Appendix: Matlab Scripts 115 Bibliography 121 | |
| dc.language.iso | en | |
| dc.subject | 商空間幾何方法 | zh_TW |
| dc.subject | 差分代數方程組 (DAEs) | zh_TW |
| dc.subject | 奇異描述子系統 | zh_TW |
| dc.subject | Kronecker典型形式 | zh_TW |
| dc.subject | Wong氏序列 | zh_TW |
| dc.subject | 非正規矩陣束 | zh_TW |
| dc.subject | Kronecker canonical form | en |
| dc.subject | Wong's sequences | en |
| dc.subject | geometric approach using quotient space | en |
| dc.subject | difference algebraic equations (DAEs) | en |
| dc.subject | singular descriptor systems | en |
| dc.subject | non-regular matrix pencil | en |
| dc.title | 矩陣束之幾何結構 | zh_TW |
| dc.title | On Geometric Structure of Matrix Pencils | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 108-2 | |
| dc.description.degree | 碩士 | |
| dc.contributor.author-orcid | 0000-0003-3463-774X | |
| dc.contributor.oralexamcommittee | 陳榮凱(Jung-Kai Chen),蔡炎龍(Yen-Lung Tsai) | |
| dc.subject.keyword | 非正規矩陣束,Wong氏序列,Kronecker典型形式,奇異描述子系統,差分代數方程組 (DAEs),商空間幾何方法, | zh_TW |
| dc.subject.keyword | non-regular matrix pencil,Wong's sequences,Kronecker canonical form,singular descriptor systems,difference algebraic equations (DAEs),geometric approach using quotient space, | en |
| dc.relation.page | 123 | |
| dc.identifier.doi | 10.6342/NTU202001519 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2020-07-21 | |
| dc.contributor.author-college | 理學院 | zh_TW |
| dc.contributor.author-dept | 應用數學科學研究所 | zh_TW |
| 顯示於系所單位： | 應用數學科學研究所 | |
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