线性系统状态空间模结构与可控性研究
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摘要
在线性系统理论的研究领域中,随着所采用的数学工具和所采用的系统描述的不同,已经形成了四个平行的分支。这些分支是:状态空间法、几何方法、代数理论以及多变量频域方法。本文将线性系统研究的状态空间方法和代数理论(模理论)结合起来,研究了线性系统状态空间的模论方法,将状态空间作为多项式环上的有限生成模处理,利用状态空间模结构理论,研究了状态空间的分解、可控的充要条件、可控规范型及状态反馈极点配置等问题。主要结果如下:
(1)给出了定常多变量线性系统状态空间的模结构分析方法。
讨论了状态空间(向量空间)的模结构分解,特别讨论了复数域和实数域上向量空间的分解情形,引入了变换(或矩阵)的特征值对应的生成根向量的定义,得到了循环模的生成元与变换的生成根向量之间的关系。运用实数域上向量空间的模结构分解,得到了实数域上矩阵的几种相似标准形并应用于线性系统的模态结构分解;利用主理想环上有限生成模的性质给出了状态空间的一种直和分解方法,并得到了系统矩阵准对角化方法,应用此方法可以研究可控子空间的分解,为系统矩阵的化简提供了完善的理论方法。
(2)讨论了线性系统的可控性问题,得到了一类新的块对角规范型。
利用系统传递函数(矩阵)推广了一类多输入线性定常系统完全可控的多项式判据;讨论了定常线性系统的可控性矩阵秩的性质,指出对输入矩阵施行列初等变换后不改变系统的可控性,提出了判断定常线性多输入系统可控性的一种快速迭代算法及其改进算法,得到了判断其可控性的最多迭代步数,而当迭代矩阵的秩没有增加时便可断定其不可控;提出了一种新的改进型求解线性定常系统能控规范型的迭代算法;利用主理想环上有限生成模的性质讨论了该模及其子模的零化子的性质,并应用于定常线性系统的可控性研究,得到了定常线性系统完全可控的充要条件并得到了基于向量空间的模结构分解和矩阵的有理标准形的一类新的块对角可控规范型。
(3)利用线性系统的可控性研究了某些特殊矩阵。
将Hankel矩阵和r-循环矩阵视为某单输入线性系统的可控性矩阵,通过可控性分析得到了Hankel矩阵和r-循环矩阵的可逆条件及求逆的方法;讨论了两个多项式矩阵右互质时其广义Sylvester矩阵的性质,指出了广义Sylvester矩阵与R-循环分块矩阵的联系,得到了R-循环分块矩阵可逆时的充要条件,为这一类分块矩阵及其相关矩阵的研究提供了一种新的方法。
(4)研究了定常线性系统的状态反馈极点配置问题。
提出了几种状态反馈极点配置问题的新算法,有基于第一能控规范型的状态反馈极点配置问题算法;基于广义Wonham可控规范型的状态反馈极点配置问题算法;基于循环矩阵的状态反馈极点配置问题的改进算法;这些算法的优点是不需要计算系统的特征多项式;并将利用主理想环上有限生成模的性质得到的一类新的块对角可控规范型应用于研究定常多输入线性系统的极点配置问题,此方法将多输入线性系统极点配置问题转化为个数为系统矩阵循环指数的单输入系统的极点配置问题,进而推导出确定一个反馈增益矩阵的最少元素个数即为系统的阶数。和通常的方法相比较,这些算法减小了计算量,并得到了反馈增益矩阵的一个含有任意参数的一般表达式。
There are four major branches for studying linear system theory according to being used mathematic methods and being described system. They are state space method, geometry method, algebra theory and multi-variable frequency-domain method. This dissertation combines state space method with algebra method, presents a module structure in state space for linear system. The state space can be associated with a module over the polynomial ring. By means of the properties of finitely generated module over a principal ideal domain, the decomposition in state space, the necessary and sufficient condition for controllability, controllability canonical forms and pole placement problem for state feedback are discussed. The main results are following.
(1) Module structure analysis for state space of linear time-invariant system is presented. A non-zero finite dimensional vector space over a field F can be associated with a module over the ring of polynomials with coefficients in F. Those over the fields of complex numbers and real numbers are especially studied. The definition of the generated generalized eigenvectors of a linear transformation is introduced. The relations between the generators of the cycle modules and the generated generalized eigenvectors are obtained. The decomposition theorems of a non-zero vector space on module structure can be used to discuss the matrices over the field of real numbers and some canonical forms are obtained, which are used to modal structure analysis of multivariable. By means of the properties of finitely generated module over a principal ideal domain, a method of direct decomposition of space is presented. The quasi-diagonal algorithm on system matrix is deduced. A perfect theory and method to predigest system matrix is provided.
(2) Controllability problem of linear system is studied and new block diagonal controllability canonical forms are obtained. By use of the transform function (or transform function matrix), the polynomial criterion on the controllability of a kind of time-invariant linear systems is discussed. The properties of the rank of the controllability matrix in time-invariant linear system is discussed and it is inferred that the controllability is not changed when perform elementary column operation on the input-matrix. A fast algorithm and its improvement for judging the controllability in the system are deduced by using the elementary column transformation on the controllability matrix. The most iteration steps judging the controllability is proved and the system is not controllable when the rank of the iteration matrix is not added. A new modified algorithm for getting controllable canonical forms of the multi-input system has been put forward. The annihilator of a vector space is discussed and applied to study the controllability in constant linear systems. A sufficient and necessary condition judging the controllability is easily obtained. Based on the decomposition theorem for vector space with module structure and rational canonical form of matrix, a kind of new block diagonal controllability canonical forms are inferred when the time-invariant linear multivariable system is completely controllable.
(3) Some special matrices are studied by using controllability. A kind of circulant matrices, example of Hankel matrices and r-circulant matrices, are viewed as controllability matrices on linear system. The invertible condition and the algorithms for finding inverse matrices are obtained. The properties of the generalized Sylvester matrix are discussed when two polynomial matrices are right coprime. The relations between an R-block circulant matrix and a suitable generalized Sylvester matrix are presented. The necessary and sufficient condition for R-block circulant matrix invertible is obtained. A new method to study the kind of circulant matrices is presented.
(4) Some New methods of pole placement are proposed. There are based on No.1 controllability canonical form, the generalized Wonham controllability canonical form, and circulant matrix. The advantage of this algorithm is no need to compute the characteristic polynomical of the system. A special new method of pole placement based on the new block diagonal controllability canonical forms in multi-input system is proposed. This method changes the problem of multi-input system to several problems of single-input. The number is the cycle index of system matrix. Compare with some known results, this method put forward that the least number of unknown entries of the feedback gain matrix is the order of the system matrix. The general expression containing arbitrary parameter is obtained for the feedback gain matrix.
(1)给出了定常多变量线性系统状态空间的模结构分析方法。
讨论了状态空间(向量空间)的模结构分解,特别讨论了复数域和实数域上向量空间的分解情形,引入了变换(或矩阵)的特征值对应的生成根向量的定义,得到了循环模的生成元与变换的生成根向量之间的关系。运用实数域上向量空间的模结构分解,得到了实数域上矩阵的几种相似标准形并应用于线性系统的模态结构分解;利用主理想环上有限生成模的性质给出了状态空间的一种直和分解方法,并得到了系统矩阵准对角化方法,应用此方法可以研究可控子空间的分解,为系统矩阵的化简提供了完善的理论方法。
(2)讨论了线性系统的可控性问题,得到了一类新的块对角规范型。
利用系统传递函数(矩阵)推广了一类多输入线性定常系统完全可控的多项式判据;讨论了定常线性系统的可控性矩阵秩的性质,指出对输入矩阵施行列初等变换后不改变系统的可控性,提出了判断定常线性多输入系统可控性的一种快速迭代算法及其改进算法,得到了判断其可控性的最多迭代步数,而当迭代矩阵的秩没有增加时便可断定其不可控;提出了一种新的改进型求解线性定常系统能控规范型的迭代算法;利用主理想环上有限生成模的性质讨论了该模及其子模的零化子的性质,并应用于定常线性系统的可控性研究,得到了定常线性系统完全可控的充要条件并得到了基于向量空间的模结构分解和矩阵的有理标准形的一类新的块对角可控规范型。
(3)利用线性系统的可控性研究了某些特殊矩阵。
将Hankel矩阵和r-循环矩阵视为某单输入线性系统的可控性矩阵,通过可控性分析得到了Hankel矩阵和r-循环矩阵的可逆条件及求逆的方法;讨论了两个多项式矩阵右互质时其广义Sylvester矩阵的性质,指出了广义Sylvester矩阵与R-循环分块矩阵的联系,得到了R-循环分块矩阵可逆时的充要条件,为这一类分块矩阵及其相关矩阵的研究提供了一种新的方法。
(4)研究了定常线性系统的状态反馈极点配置问题。
提出了几种状态反馈极点配置问题的新算法,有基于第一能控规范型的状态反馈极点配置问题算法;基于广义Wonham可控规范型的状态反馈极点配置问题算法;基于循环矩阵的状态反馈极点配置问题的改进算法;这些算法的优点是不需要计算系统的特征多项式;并将利用主理想环上有限生成模的性质得到的一类新的块对角可控规范型应用于研究定常多输入线性系统的极点配置问题,此方法将多输入线性系统极点配置问题转化为个数为系统矩阵循环指数的单输入系统的极点配置问题,进而推导出确定一个反馈增益矩阵的最少元素个数即为系统的阶数。和通常的方法相比较,这些算法减小了计算量,并得到了反馈增益矩阵的一个含有任意参数的一般表达式。
There are four major branches for studying linear system theory according to being used mathematic methods and being described system. They are state space method, geometry method, algebra theory and multi-variable frequency-domain method. This dissertation combines state space method with algebra method, presents a module structure in state space for linear system. The state space can be associated with a module over the polynomial ring. By means of the properties of finitely generated module over a principal ideal domain, the decomposition in state space, the necessary and sufficient condition for controllability, controllability canonical forms and pole placement problem for state feedback are discussed. The main results are following.
(1) Module structure analysis for state space of linear time-invariant system is presented. A non-zero finite dimensional vector space over a field F can be associated with a module over the ring of polynomials with coefficients in F. Those over the fields of complex numbers and real numbers are especially studied. The definition of the generated generalized eigenvectors of a linear transformation is introduced. The relations between the generators of the cycle modules and the generated generalized eigenvectors are obtained. The decomposition theorems of a non-zero vector space on module structure can be used to discuss the matrices over the field of real numbers and some canonical forms are obtained, which are used to modal structure analysis of multivariable. By means of the properties of finitely generated module over a principal ideal domain, a method of direct decomposition of space is presented. The quasi-diagonal algorithm on system matrix is deduced. A perfect theory and method to predigest system matrix is provided.
(2) Controllability problem of linear system is studied and new block diagonal controllability canonical forms are obtained. By use of the transform function (or transform function matrix), the polynomial criterion on the controllability of a kind of time-invariant linear systems is discussed. The properties of the rank of the controllability matrix in time-invariant linear system is discussed and it is inferred that the controllability is not changed when perform elementary column operation on the input-matrix. A fast algorithm and its improvement for judging the controllability in the system are deduced by using the elementary column transformation on the controllability matrix. The most iteration steps judging the controllability is proved and the system is not controllable when the rank of the iteration matrix is not added. A new modified algorithm for getting controllable canonical forms of the multi-input system has been put forward. The annihilator of a vector space is discussed and applied to study the controllability in constant linear systems. A sufficient and necessary condition judging the controllability is easily obtained. Based on the decomposition theorem for vector space with module structure and rational canonical form of matrix, a kind of new block diagonal controllability canonical forms are inferred when the time-invariant linear multivariable system is completely controllable.
(3) Some special matrices are studied by using controllability. A kind of circulant matrices, example of Hankel matrices and r-circulant matrices, are viewed as controllability matrices on linear system. The invertible condition and the algorithms for finding inverse matrices are obtained. The properties of the generalized Sylvester matrix are discussed when two polynomial matrices are right coprime. The relations between an R-block circulant matrix and a suitable generalized Sylvester matrix are presented. The necessary and sufficient condition for R-block circulant matrix invertible is obtained. A new method to study the kind of circulant matrices is presented.
(4) Some New methods of pole placement are proposed. There are based on No.1 controllability canonical form, the generalized Wonham controllability canonical form, and circulant matrix. The advantage of this algorithm is no need to compute the characteristic polynomical of the system. A special new method of pole placement based on the new block diagonal controllability canonical forms in multi-input system is proposed. This method changes the problem of multi-input system to several problems of single-input. The number is the cycle index of system matrix. Compare with some known results, this method put forward that the least number of unknown entries of the feedback gain matrix is the order of the system matrix. The general expression containing arbitrary parameter is obtained for the feedback gain matrix.
引文
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