系统解耦和极点配置问题与不定最小二乘问题
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摘要
本文主要研究了以下三个问题：
1．右可逆系统{C，A，B}的解耦和极点配置问题。
通过研究与系统相关的矩阵的秩的关系式，得到系统是右可逆的充要条件。给出右可逆系统的一种新的标准分解。给出数值例子，说明如何判定系统是右可逆的，以及构造性得到右可逆系统的标准分解。
利用该标准分解，研究右可逆系统的结构特性。得到与系统传递函数矩阵密切相关的矩阵多项式P(s)=(?)的史密斯形、有限零点、无穷极点零点以及秩的范围，并且得到右可逆系统可控的充分条件。
我们还利用该标准分解研究了右可逆系统的三角解耦、行行解耦问题和相关的极点配置问题，得到一些新的结论。右可逆系统在行置换的意义下总是可以三角解耦的，并且可以任意配置部分传递函数矩阵的极点。给出数值例子，说明结论的正确性。得到了正则行行解耦问题可解的充要条件，及在该条件下求出解和配置极点的方法。对于非正则的行行解耦问题，得到了在特殊情况下可解的充要条件。
2．不定最小二乘问题(ILS)和等式约束不定最小二乘问题(ILSE)．
定义了一种新的加权广义逆，得到ILS问题和ILSE问题解的形式，以及解的扰动界。
Krylov子空间方法是求解标准最小二乘问题(LS)的非常重要的迭代技巧。本文应用三种Krylov子空间迭代方法(共轭梯度法，上、下双对角化方法)求解不定最小二乘问题，得到的结果与它们求解最小二乘问题的结果是不同的。对于求解LS问题下双对角化方法是非常有效的，但是对于求解ILS问题却是不稳定的。通过数值实验发现，上双对角化方法效果最好。
对等式约束不定最小二乘问题，提出双曲Householder消元法。由于ILSE问题的解是一个无约束的加权不定最小二乘问题(WILS)的解的极限，基于此，可以用双曲QR方法来求解该WILS问题，再取极限。我们发现这个过程相当于对原ILSE消元的过程。该方法在合理的假设下是向前稳定的，且计算量约为文献中的GHQR算法的一半。误差分析和数值算例说明了我们的结论。
3．秩约束下矩阵方程AXA~H=B的Hermitian半正定最小二乘解
研究了矩阵方程AXA~H=B的秩约束Hermitian半正定最小二乘解，这里不要求B是Hermitian半正定的，也不要求该方程是相容的。得到了使该问题有解的秩p的范围，以及在此范围内，秩-p Hermitian半正定最小二乘解的一般形式。
The following problems are studied in this thesis:
1.The decoupling and pole assignment problems of the right invertiblesystem {C,A,B}.
We deduce several rank relations related to the matrices of the system {C,A,B}to obtain the necessary and sufficient condition for the system being right invertible,and propose a new canonical decomposition of the right invertible system.From thisdecomposition,we study the Smith form of the matrix pencil P(s)=(?)to find out the finite zeros and infinite zeros of P(s),the range of the ranks of P(s) fors∈C,and the controllability of the right invertible system.
We also apply this canonical decomposition to study decoupling and prescriptpole assignment problems of the right invertible system {C,A,B}.Some new resultsabout the triangular decoupling upon the row permutation,row by row decoupling andassociated pole assignment problems are deduced.
2.The indefinite least squares (ILS)problem and the equality con-strained indefinite least squares (ILSE) problem.
At first,perturbation analysis for indefinite least squares (ILS) problem and equal-ity constrained indefinite least squares (ILSE) problem are studied.By defining a newkind of weighted generalized inverse,the solutions and the perturbation bounds of thetwo problems are derived.
Krylov subspace methods are considered currently to be among the most im-portant iterative techniques available for solving linear equations and standard leastsquares (LS) problems.In this thesis,three kinds of Krylov subspace methods areapplied to solve the ILS problem.The results are different than they worked on theLS problem.The lower bidiagonalization method which is very efficient for solving theLS problem is quite unstable for solving the ILS problem.Several numerical experi-ments are shown to compare the performances of the three algorithms and illustrateour results.
The hyperbolic Householder elimination method is proposed to solve the ILSEproblem.The solution of ILSE is the limit of the solution of the corresponding uncon-strained weighted indefinite least squares problem (WILS).Based on this observation,we derive a type of elimination method by applying the hyperbolic QR method to aboveWILS problem and taking the limit analytically.Theoretical analysis show that the method obtained is forward stable under a reasonable assumption.We illustrate ourresults with numerical tests.
3.The rank-constrained Hermitian nonnegative-definite least squaressolutions to the matrix equation AXA~H=B.
We discuss rank-constrained Hermitian nonnegative-definite least squares solu-tions to the matrix equation AXA~H=B,in which conditions that B is Hermitian andnonnegative-definite and the matrix equation is consistent may not hold.We derivethe rank range and expression of these least squares solutions.
1．右可逆系统{C，A，B}的解耦和极点配置问题。
通过研究与系统相关的矩阵的秩的关系式，得到系统是右可逆的充要条件。给出右可逆系统的一种新的标准分解。给出数值例子，说明如何判定系统是右可逆的，以及构造性得到右可逆系统的标准分解。
利用该标准分解，研究右可逆系统的结构特性。得到与系统传递函数矩阵密切相关的矩阵多项式P(s)=(?)的史密斯形、有限零点、无穷极点零点以及秩的范围，并且得到右可逆系统可控的充分条件。
我们还利用该标准分解研究了右可逆系统的三角解耦、行行解耦问题和相关的极点配置问题，得到一些新的结论。右可逆系统在行置换的意义下总是可以三角解耦的，并且可以任意配置部分传递函数矩阵的极点。给出数值例子，说明结论的正确性。得到了正则行行解耦问题可解的充要条件，及在该条件下求出解和配置极点的方法。对于非正则的行行解耦问题，得到了在特殊情况下可解的充要条件。
2．不定最小二乘问题(ILS)和等式约束不定最小二乘问题(ILSE)．
定义了一种新的加权广义逆，得到ILS问题和ILSE问题解的形式，以及解的扰动界。
Krylov子空间方法是求解标准最小二乘问题(LS)的非常重要的迭代技巧。本文应用三种Krylov子空间迭代方法(共轭梯度法，上、下双对角化方法)求解不定最小二乘问题，得到的结果与它们求解最小二乘问题的结果是不同的。对于求解LS问题下双对角化方法是非常有效的，但是对于求解ILS问题却是不稳定的。通过数值实验发现，上双对角化方法效果最好。
对等式约束不定最小二乘问题，提出双曲Householder消元法。由于ILSE问题的解是一个无约束的加权不定最小二乘问题(WILS)的解的极限，基于此，可以用双曲QR方法来求解该WILS问题，再取极限。我们发现这个过程相当于对原ILSE消元的过程。该方法在合理的假设下是向前稳定的，且计算量约为文献中的GHQR算法的一半。误差分析和数值算例说明了我们的结论。
3．秩约束下矩阵方程AXA~H=B的Hermitian半正定最小二乘解
研究了矩阵方程AXA~H=B的秩约束Hermitian半正定最小二乘解，这里不要求B是Hermitian半正定的，也不要求该方程是相容的。得到了使该问题有解的秩p的范围，以及在此范围内，秩-p Hermitian半正定最小二乘解的一般形式。
The following problems are studied in this thesis:
1.The decoupling and pole assignment problems of the right invertiblesystem {C,A,B}.
We deduce several rank relations related to the matrices of the system {C,A,B}to obtain the necessary and sufficient condition for the system being right invertible,and propose a new canonical decomposition of the right invertible system.From thisdecomposition,we study the Smith form of the matrix pencil P(s)=(?)to find out the finite zeros and infinite zeros of P(s),the range of the ranks of P(s) fors∈C,and the controllability of the right invertible system.
We also apply this canonical decomposition to study decoupling and prescriptpole assignment problems of the right invertible system {C,A,B}.Some new resultsabout the triangular decoupling upon the row permutation,row by row decoupling andassociated pole assignment problems are deduced.
2.The indefinite least squares (ILS)problem and the equality con-strained indefinite least squares (ILSE) problem.
At first,perturbation analysis for indefinite least squares (ILS) problem and equal-ity constrained indefinite least squares (ILSE) problem are studied.By defining a newkind of weighted generalized inverse,the solutions and the perturbation bounds of thetwo problems are derived.
Krylov subspace methods are considered currently to be among the most im-portant iterative techniques available for solving linear equations and standard leastsquares (LS) problems.In this thesis,three kinds of Krylov subspace methods areapplied to solve the ILS problem.The results are different than they worked on theLS problem.The lower bidiagonalization method which is very efficient for solving theLS problem is quite unstable for solving the ILS problem.Several numerical experi-ments are shown to compare the performances of the three algorithms and illustrateour results.
The hyperbolic Householder elimination method is proposed to solve the ILSEproblem.The solution of ILSE is the limit of the solution of the corresponding uncon-strained weighted indefinite least squares problem (WILS).Based on this observation,we derive a type of elimination method by applying the hyperbolic QR method to aboveWILS problem and taking the limit analytically.Theoretical analysis show that the method obtained is forward stable under a reasonable assumption.We illustrate ourresults with numerical tests.
3.The rank-constrained Hermitian nonnegative-definite least squaressolutions to the matrix equation AXA~H=B.
We discuss rank-constrained Hermitian nonnegative-definite least squares solu-tions to the matrix equation AXA~H=B,in which conditions that B is Hermitian andnonnegative-definite and the matrix equation is consistent may not hold.We derivethe rank range and expression of these least squares solutions.
引文
[1]魏木生，广义最小二乘问题的理论和计算，科学出版社，2006．
[2]郑大钟，线性系统理论，第2版，清华大学出版社，2002．
[3]韩正之，陈树中，Morgan问题始末，控制与决策，5：4(1990)，52-58．
[4]曹敬，陈树中，The model of transfer line and an adaptive congestionoriented routing algorithm，中国科学：F辑—信息科学(英文版)，44：4(2001)，270-277．
[5]陈树中，基本阶、无穷零点和绝对能控能观系统，华东师范大学学报：科学版，2(1997)，24-33．
[6]陈树中，陈迁，有理函数矩阵的右零空间和解耦，控制理论与应用，5：15(1998)，663-667．
[7]曹敬，陈树中，极大代数意义下向量线性相关性研究，华东师范大学学报：自然科学版，3(2000)，43-49．
[8]许可康，非方系统的Morgan问题，控制理论与应用，12：6(1995)，694-703．
[9]王晓哲，李界家，吴成东，顾树生，多变量系统解耦方法综述，沈阳建筑工程学院学报，16：2(2000)，143-145．
[10]孙振东，关于Morgan问题的一个充分条件，控制理论与应用，16：1(1999)．
[11]孙振东，夏小华，一个结构分解算法及其在解耦问题中的应用，自动化学报，23：5(1997)，710-713．
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[15]D.Chu and Y.S.Hung,A matrix pencil approach to the row by row decoupling problem for descriptor systems,SIAM J.Matrix Anal.Appl,28 (2006),682-702.
[16]D.Chu and V.Mehrmann,Disturbance decoupling for linear time-invariant systems:A matrix pencil approach,IEEE Trans.Automat.Control,48 (2001),802-808.
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[25]N.Ito and H.Inaba,Block triangular decoupling for linear systems orer principal ideal Domains,SIAM J.Contrl.Optimiz.35:3 (1997),744-765.
[26]F.N.Koumboulis,Input-output triangular decoupling and data sensitivity,Automatica,32 (1996),569-573.
[27]T.Mora,Solving Polynomial Equation Systems I,Cambridge University Press,Cambridge,UK,2003
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[2]郑大钟，线性系统理论，第2版，清华大学出版社，2002．
[3]韩正之，陈树中，Morgan问题始末，控制与决策，5：4(1990)，52-58．
[4]曹敬，陈树中，The model of transfer line and an adaptive congestionoriented routing algorithm，中国科学：F辑—信息科学(英文版)，44：4(2001)，270-277．
[5]陈树中，基本阶、无穷零点和绝对能控能观系统，华东师范大学学报：科学版，2(1997)，24-33．
[6]陈树中，陈迁，有理函数矩阵的右零空间和解耦，控制理论与应用，5：15(1998)，663-667．
[7]曹敬，陈树中，极大代数意义下向量线性相关性研究，华东师范大学学报：自然科学版，3(2000)，43-49．
[8]许可康，非方系统的Morgan问题，控制理论与应用，12：6(1995)，694-703．
[9]王晓哲，李界家，吴成东，顾树生，多变量系统解耦方法综述，沈阳建筑工程学院学报，16：2(2000)，143-145．
[10]孙振东，关于Morgan问题的一个充分条件，控制理论与应用，16：1(1999)．
[11]孙振东，夏小华，一个结构分解算法及其在解耦问题中的应用，自动化学报，23：5(1997)，710-713．
[12]A,Ben-Israel and T.N.E.Greville,Generalized Inverses:Theory and Applications,John Wiley,New York,1974.
[13]C.-T.Chen,Linear System Theory and Design,Holt,Rinehart and Winston,New York,1984.
[14]D.Chu and Y.S.Hung,A numerical solution for the simultaneous disturbance decoupling and row-by-row decoupling problem,Linear Algebra Appl.,320 (2000),37-49.
[15]D.Chu and Y.S.Hung,A matrix pencil approach to the row by row decoupling problem for descriptor systems,SIAM J.Matrix Anal.Appl,28 (2006),682-702.
[16]D.Chu and V.Mehrmann,Disturbance decoupling for linear time-invariant systems:A matrix pencil approach,IEEE Trans.Automat.Control,48 (2001),802-808.
[17]D.Chu and Roger C.E.Tan,Solvability conditions and parameterization of all solutions for the triangular decoupling problem,SIAM J.Matrix Anal.Appl.,23 (2002),1171-1182.
[18]J.Descusse,J.F.Lafay,and M.Malabre,A necessary and sufficient condition for decoupling by state feedback linear systems with 2 outputs,IEEE Trans.Automat.Contr.,AC-30 (1985),914-918.
[19]J.Descusse,J.F.Lafay and M.Malabre,Solution to Morgan's Problem,IEEE Trans.Automat.Contr.,AC-33 (1988),732-739.
[20]P.L.Falb and W.A.Wolovich,Decoupling in the Design and Synthesis of Multivariable Control systems,IEEE Trans.Automat.Contr.,AC-12 (1967),651-659.
[21]J.C.M.Garcia and M.Malabre,The row by row decoupling problem with stability:A structural approach,IEEE Trans.Automat.Control,AC-39 (1994),2457-2460.
[22]I.Gohberg,P.Lancaster,and L.Rodman,Matrix Polynomials,Academic Press,New York,1982
[23]A.N.Herrera and J.F.Lafay,New results about the Morgan's problem,IEEE Trans.Automat.Contr.,AC-38 (1993),1834-1838.
[24]R.A.Horn and C.R.Johnson,Matrix Analysis,Cambridge University Press,Cambridge,UK,1985
[25]N.Ito and H.Inaba,Block triangular decoupling for linear systems orer principal ideal Domains,SIAM J.Contrl.Optimiz.35:3 (1997),744-765.
[26]F.N.Koumboulis,Input-output triangular decoupling and data sensitivity,Automatica,32 (1996),569-573.
[27]T.Mora,Solving Polynomial Equation Systems I,Cambridge University Press,Cambridge,UK,2003
[28]B.S.,Morgan,The Synthesis of Linear Multivariable Systems by State Variable Feedback,IEEE Trans.Automat.Contr.,AC-9 (1964),405-411.
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