套代数框架下线性时变系统的若干问题
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摘要
控制理论是关于控制系统建模、分析与综合的一般理论.该理论在经济、生物、工程、医学等众多领域具有广泛的应用,对人类产生着重要而深远的影响.因此,对控制理论的研究具有不可估量的价值与意义.数学理论和方法,如：泛函分析、代数、拓扑、几何等应用于控制理论的研究中取得了深刻的成果,并使得控制理论极大地依赖于数学.Hilbert空间上的控制理论就是利用算子理论方法对控制理论进行研究.
本文主要利用算子理论方法对套代数框架下线性时变系统的若干问题进行研究.
第二章回顾Hilbert空间上控制理论的基本知识,首次对系统强左(右)表示相应的所有关联有界右(左)逆进行完整的刻画,并采用算子方程的方式给出闭环系统稳定性的新判据.
第三章研究线性时变系统之间间隙度量与时变间隙度量的关系.借助算子极小模的性质,得出控制系统逆图的直交补与控制器的图之间的时变间隙度量与间隙度量的等价性,并利用此结论将最优极小角的计算同时变Hankel算子的范数联系起来.
第四章首次利用对偶理论对时变4-block问题进行完整的求解.首先,通过一类一般的最优反馈控制问题向时变4-block问题的等价转化,说明4-block问题在最优控制领域中的重要地位.其次,借助套代数的性质以及算子理论中M-理想的概念,求出4-block问题中相应子空间的预零化子与零化子的形式,并对适用于一般控制系统与紧控制系统的对偶理论进行了完整的描述.最后,通过例举所得结论在时变度量反馈控制问题中的应用,证明利用对偶理论得到的系统最优解满足时变全通条件.
第五章利用离散套代数的完全有限性,证明一个系统可被镇定当且仅当它具有强左或者强右表示,并仅利用一种强表示给出系统所有镇定控制器的参数化形式.进一步地,对两种类型强表示存在的等价性加以说明以及给出系统可被强镇定的一个充分条件.这些结论是对经典Youla参数化定理的本质改进.
Control theory is the general theory about modeling,analysis and synthesis for control systems. It is widely used in many different fields such as economics, biology,engineering, medicine and so on. And great effects have been made for the human beings. It is clear that the research on control theory is of invaluable importance. Mathematical theories and methods, such as functional analysis, algebra, topology, geometry and so on, have been introduced into the study of control theory. Lots of powerful, beautiful and elegant techniques and results have been established. This makes that control theory greatly relies on mathematics. The control theory in Hilbert space is right a link between operator theoretic approaches and the study of control theory.
In this paper, several problems of linear time-varying systems by using the operator theoretic approaches in the framework of nest algebras are studied.
In Chapter2, some known results about the control theory in the Hilbert space are recalled. Moreover, all the causal bounded right (left) inverses of a strong left (right) representation for a plant are fully characterized for the first time. A new criterion for judging the stability of a closed loop system is presented in terms of operator equations.
Chapter3is concerned with the relationship between the gap metric and the time-varying gap metric for linear time-varying systems. With the help of the properties of the minimum modulus of a linear operator, we get that when measuring the distance between the orthogonal complement of the inverse graph of a plant and the graph of a controller in a feedback configuration, the gap metric and the time-varying gap metric are in fact identical. The developed criteria are also applied to compute the optimal minimal angles of stabilizable linear time-varying systems. And it is shown that the value of the cosine of the optimal minimal angle is equal to the norm of a time-varying Hankel operator.
In Chapter4, the duality theory for the time-varying4-block problem is completely studied for the first time. We express the importance of this block problem for the optimal control theory by converting a general feedback control problem equivalently to the time-varying4-block problem. By using the properties of nest algebra and the notion of M-ideal in the operator theory, we compute the appropriate preannihilator and annihilator. Specific duality theories are established for the general plants and the compact plants, respectively.Furthermore,an example on the measurement feedback control problem is given to show that the optimum obtained by duality theory is allpass.
In Chapter5,by using the complete finiteness of a certain discrete nest algebra, we show that a system is stabilizable if and only if it has one kind of strong representation and we also give a parametrization for all the stabilizing controllers in terms of this strong representation. Moreover, a fact is obtained that the other strong representation of a plant can always be derived from the given one and a sufficient condition for the strong stabilization problem is given. These results provide an essential extension to the classical Youla parametrization theorem.
本文主要利用算子理论方法对套代数框架下线性时变系统的若干问题进行研究.
第二章回顾Hilbert空间上控制理论的基本知识,首次对系统强左(右)表示相应的所有关联有界右(左)逆进行完整的刻画,并采用算子方程的方式给出闭环系统稳定性的新判据.
第三章研究线性时变系统之间间隙度量与时变间隙度量的关系.借助算子极小模的性质,得出控制系统逆图的直交补与控制器的图之间的时变间隙度量与间隙度量的等价性,并利用此结论将最优极小角的计算同时变Hankel算子的范数联系起来.
第四章首次利用对偶理论对时变4-block问题进行完整的求解.首先,通过一类一般的最优反馈控制问题向时变4-block问题的等价转化,说明4-block问题在最优控制领域中的重要地位.其次,借助套代数的性质以及算子理论中M-理想的概念,求出4-block问题中相应子空间的预零化子与零化子的形式,并对适用于一般控制系统与紧控制系统的对偶理论进行了完整的描述.最后,通过例举所得结论在时变度量反馈控制问题中的应用,证明利用对偶理论得到的系统最优解满足时变全通条件.
第五章利用离散套代数的完全有限性,证明一个系统可被镇定当且仅当它具有强左或者强右表示,并仅利用一种强表示给出系统所有镇定控制器的参数化形式.进一步地,对两种类型强表示存在的等价性加以说明以及给出系统可被强镇定的一个充分条件.这些结论是对经典Youla参数化定理的本质改进.
Control theory is the general theory about modeling,analysis and synthesis for control systems. It is widely used in many different fields such as economics, biology,engineering, medicine and so on. And great effects have been made for the human beings. It is clear that the research on control theory is of invaluable importance. Mathematical theories and methods, such as functional analysis, algebra, topology, geometry and so on, have been introduced into the study of control theory. Lots of powerful, beautiful and elegant techniques and results have been established. This makes that control theory greatly relies on mathematics. The control theory in Hilbert space is right a link between operator theoretic approaches and the study of control theory.
In this paper, several problems of linear time-varying systems by using the operator theoretic approaches in the framework of nest algebras are studied.
In Chapter2, some known results about the control theory in the Hilbert space are recalled. Moreover, all the causal bounded right (left) inverses of a strong left (right) representation for a plant are fully characterized for the first time. A new criterion for judging the stability of a closed loop system is presented in terms of operator equations.
Chapter3is concerned with the relationship between the gap metric and the time-varying gap metric for linear time-varying systems. With the help of the properties of the minimum modulus of a linear operator, we get that when measuring the distance between the orthogonal complement of the inverse graph of a plant and the graph of a controller in a feedback configuration, the gap metric and the time-varying gap metric are in fact identical. The developed criteria are also applied to compute the optimal minimal angles of stabilizable linear time-varying systems. And it is shown that the value of the cosine of the optimal minimal angle is equal to the norm of a time-varying Hankel operator.
In Chapter4, the duality theory for the time-varying4-block problem is completely studied for the first time. We express the importance of this block problem for the optimal control theory by converting a general feedback control problem equivalently to the time-varying4-block problem. By using the properties of nest algebra and the notion of M-ideal in the operator theory, we compute the appropriate preannihilator and annihilator. Specific duality theories are established for the general plants and the compact plants, respectively.Furthermore,an example on the measurement feedback control problem is given to show that the optimum obtained by duality theory is allpass.
In Chapter5,by using the complete finiteness of a certain discrete nest algebra, we show that a system is stabilizable if and only if it has one kind of strong representation and we also give a parametrization for all the stabilizing controllers in terms of this strong representation. Moreover, a fact is obtained that the other strong representation of a plant can always be derived from the given one and a sufficient condition for the strong stabilization problem is given. These results provide an essential extension to the classical Youla parametrization theorem.
引文
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[3]钱学森.工程控制论[M].北京：科学出版社,1980.
[4]孙德宝,王永骥,王金城.自动控制原理[M].化学工业出版社,2002.
[5]郑大钟.线性系统理论[M].第2版.北京：清华大学出版社,2002.
[6]ANTSAKLIS P J, MICHEL A N. Linear systems [M].2nd corrected printing.Birkhauser, 2006.
[7]KHALIL H K. Nonlinear systems [M].3rd ed.Prentice Hall,2002.
[8]李训经,雍炯敏,周渊.控制理论基础[M].高等教育出版社,2002.
[9]吴敏,桂卫华,何勇.现代鲁棒控制[M].第2版.中南大学出版社,2006.
[10]张国会,窦永丰.H∞控制理论及其航天应用[J].航天控制,1995,3：17-23.
[11]DULLERUD G E, PAGANINI F. A course in robust control theory:a convex approach [M]. New York:Springer,2000.
[12]FRANCIS B A. Notes on introductory state space systems [R]. Personal communication, 1997.
[13]BOLTYANSKII V G, GAMKRELIDZE R V, PONTRYAGIN L S. Theory of optimal processes: I.The maximum principle [J]. Izv. Akad. Nauk SSSR Ser. Mat.,1960,24(1):3-42.
[14]BELLMAN R. Dynamic programming and Lagrange multipliers [J].Proc. Natl Acad Sci USA,1956,42(10):767-769.
[15]Kalman R E. On the general theory of control systems [J].IFAC World Congress, 1960.
[16]高黛陵,吴麒.多变量频域控制理论[M].清华大学出版社,1998.
[17]WONHAM W M. Linear multivariable control:a geometric approach [M].New York: Springer-Verlag,1979.
[18]QUADRAT A. The fractional representation approach to synthesis problems:an algebraic analysis viewpoint part II:internal stabilization [J].SIAM J. Control Optim.,2003,42(1):300-320.
[19]QUADRAT A. A lattice approach to analysis and synthesis problems[J]. Math. Control Signals Systems,2006,18:147-186.
[20]QUADRAT A. On a general structure of the stabilizing controllers based on stable range [J]. SIAM J. Control Optim,2004,42(6):2264-2285.
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