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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/34457

Title: 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
Controlled Solution Spaces and Related Control Problems for Linear Discrete-Time Descriptor Systems
Authors: 容志輝
Contributors: NTOU:Department of Electrical Engineering
Keywords: 線性離散時間描述子系統;幾何控制;(E;A;B)之可解性;(E;A;B) 可控解空間;干 擾解耦控制問題
linear discrete-time descriptor system;geometric control;solvability of (E;A;B);controlled solution spaces of (E;A;B);distur- bance decoupling control problem
Date: 2012
Issue Date: 2013-10-07T02:32:39Z
Publisher: 行政院國家科學委員會
Abstract: 令X為一佈於代數封閉體F之n維狀態向量空間, U為佈於F之m維輸入空間。設A及E表X上之線性映射, 且B : U ! X亦為線性映射。假設rankE = r n。考慮由(E;A;B)所定義出之線性離散時間描述子系統 Exk+1 = Axk + Buk; (1) 其中xk及uk分別為在時間k時之n維狀態以及m維輸入。許多實際的系統可表示成此類差分代數方程, 文獻證明了描述子系統表示法遠比傳統之狀態系統表示法更能有效達到控制之目的。線性離散時間描述子系統表示法雖具優點, 卻仍有其基本問題未解決。文獻指出: 並非對任意給定的初始狀態x0以及輸入數列u := (uk)1 k=0, 均存在狀態數列(xk)1 k=1滿足(1) 式。若對一給定之初始狀態x0以及一給定的輸入數列u := (uk)1 k=0存在狀態數列(xk)1 k=1滿足(1) 式, 則稱(x0; u) 為solvable。為達控制之目的, 一個非常重要的課題就是要能夠完全刻畫所有的solvable 初始狀態x0及輸入數列u = (uk)1 k=1所形成之幾何空間。本計畫主要目的就是要完整刻畫出所有的solvable 之(x0; u)所形成之空間及其他衍生之重要相關控制問題。本計畫首先研究無輸入之線性離散時間描述子系統Exk+1 = Axk, 此可解釋成系統(1) 加入了狀態回授後之閉迴路系統, 屬閉迴路系統之幾何分析。我們將進行下列研究主題: (一) 研究(E;A)之可解性, 並完整刻畫所有的(E;A)之solvable狀態所形成之幾何空間,也就是最大的(E;A) solution space。(二)證明所有的solvable狀態之fundamental sequence 之唯一性, 並求出其計算公式。(三) 求出所有可在有限時間內從零初始狀態到達之狀態所形成之空間, 以及求出所有可在有限時間內到達零狀態之所有初始狀態所形成之空間。之後, 我們將研究推廣到具輸入之線性離散時間描述子系統(1), 主要研究主題包含: (一)研究(E;A;B)之可解性, 並完整刻畫所有的(E;A;B) solvable 初始狀態x0及輸入數列u = (uk)1 k=1所成之幾何空間, 也就是最大的(E;A;B) controlled solution space X1。(二) 證明所有的solv- able 狀態之solution sequence 之唯一性, 並求出其數學計算公式。(三) 研究(E;A;B) controlled solution space 與狀態回授之間的關係。(四) 建立一套遞迴式的演算法計算X1, 並求出其close form。(五) 最後, 將上述所有結果應用在實際的干擾解耦控制問題上, 並求出能達成干擾解耦之所有控制器。
Let X be an n-dimensional state vector space over an algebraically closed eld F, and U be an m-dimensional input space over F. Let A and E be linear mappings on X, and B : U ! X be a linear mapping. Consider the linear discrete-time descriptor system de ned by Exk+1 = Axk + Buk; (1) where xk and uk represent the n-dimensional state and m-dimensional input at time k, respectively. Assume rankE = r n. It has been proven in the literature that descriptor systems have higher capability in describing and controlling physical systems that cannot be easily handled by using the conventional state-space representation. Although the discrete-time descriptor representation has many ad- vantages, one basic unsolved problem with it is as follows: Literature shows that it is not always possible to nd state sequence (xk)1 k=1 sat- isfying (1) for any arbitrarily given initial state x0 and input sequence u := (uk)1 k=0. (x0; u) will be called solvable if there exists a state se- quence (xk)1 k=1 satisfying (1). For control purpose, it is important and essential to characterize the space consisting of all solvable (x0; u). The main aim of the project is to characterize the space consisting of all solvable (x0; u) and other related control problems. To this end, we rst consider the autonomous discrete-time descrip- tor system Exk+1 = Axk, which can be regarded as the closed-loop system of (1) after applying a state feedback. The topics we will be addressing include the following: (a) Study the solvability of (E;A), and characterize the space consisting of all solvable x0, that is, the maximal (E;A) solution space. (b) Prove the uniqueness of funda- mental sequence for a solvable state and nd a formula to evaluate the fundamental sequence. (c) Construct the space of all states reachable from zero, and the space of all initial states reachable to zero. Then, we return to system (1), and the topics we will study include: (a) Address the solvability of (E;A;B), and characterize the space con- sisting of all solvable (x0; u), that is, the maximal (E;A) controlled solution space. (b) Prove the uniqueness of solution sequence for a solvable state of (E;A;B) and nd a formula to evaluate the solution sequence. (c) Study the connection of (E;A;B) controlled solution spaces with state feedbacks. (d) Determine a recursive algorithm to evaluate X1, and nd the close form of X1. (e) Finally, apply the above results to the disturbance decoupling control problem, and nd a parameterization of all controllers solving the problem.
Relation: NSC101-2221-E019-039
URI: http://ntour.ntou.edu.tw/handle/987654321/34457
Appears in Collections:[電機工程學系] 研究計畫

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