线性系统Luenberger能观规范型的充要条件及特征
|
摘要
提出了一种完全能观线性MIMO(multi-input multi-output)系统的能观矩阵的线性无关行向量的搜索方案,并基于此搜索方案提出了一种将完全能观线性MIMO系统化为Luenberger能观规范型的变换矩阵的构造方法。通过对几个定理的证明,阐述了此非奇异变换矩阵与Luenberger能观规范型的关系,提出了将Luenberger能观规范型按照结构的差异划分为广义和狭义2种规范型的观点,并给出了完全能观线性MIMO系统的广义和狭义Luenberger能观规范型实现的充要条件,用3个实例验证了上述观点和方法的正确性和可行性。同时给出了一种方法使得一类不满足Luenberger能观规范型实现条件的线性MIMO系统能够在不改变系统物理结构的前提下变换为Luenberger能观规范型,并通过2个实例的分析和比较对此方法进行了阐述。
An approach is put forward to find the linealy independent vector of the observability matrix of completely observable linear multi-input multi-output(MIMO) system.Based on this approach,a method is advanced to obtain the transformation matrix which can be used for transforming the linear MIMO system to its Luenberger observable canonical form.Through the proving of several theormes,the relation between the transformation matrix and Luenberger observable canonical form is exposited and the Luenberger observal form is divided into two classes,the generalized Luenberger observable canonical form and the special Luenberger observable canonical form,according to its structure difference.The necessary and sufficient condition for the realization of generalized and special Luenberger observable canonical form of completed observable linear MIMO system are given,and three examples are used to verify the correctness and feasibility of the above viewpoint and method.Meanwhile,another method is put forward to transform a class of linear MIMO system,under the condition of unchanging its physical structure,which does not meet the above necessary and sufficient condition,to its Luenberger observable canonical form.Two examples are analyzed and compared to elaborate on the method.
An approach is put forward to find the linealy independent vector of the observability matrix of completely observable linear multi-input multi-output(MIMO) system.Based on this approach,a method is advanced to obtain the transformation matrix which can be used for transforming the linear MIMO system to its Luenberger observable canonical form.Through the proving of several theormes,the relation between the transformation matrix and Luenberger observable canonical form is exposited and the Luenberger observal form is divided into two classes,the generalized Luenberger observable canonical form and the special Luenberger observable canonical form,according to its structure difference.The necessary and sufficient condition for the realization of generalized and special Luenberger observable canonical form of completed observable linear MIMO system are given,and three examples are used to verify the correctness and feasibility of the above viewpoint and method.Meanwhile,another method is put forward to transform a class of linear MIMO system,under the condition of unchanging its physical structure,which does not meet the above necessary and sufficient condition,to its Luenberger observable canonical form.Two examples are analyzed and compared to elaborate on the method.
引文
[1]Luenberger D G.Canonical forms for linearmultivariable systems[J].IEEE Transactions onAutomatic Control,1967,12(3):290-293.
[2]Luenberger D G.Observers for multivariable systems[J].IEEE Transactions on Automatic Control,1966,11(2):190-197.
[3]Luenberger D G.An introduction to observers[J].IEEE Transactions on Automatic Control,1971,16(6):596-603.
[4]Kalman R E.Canonical structure of linear dynamicsystems[J].Proceedings of the National Academy ofSciences of United States of America,1962,48(4):596-600.
[5]Kalman R E,Ho Y C,Narendra K S.Controllability oflinear dynamical systems[J].Contributions toDifferential Equations,1961,1(3):189-213.
[6]Kalman R E.Mathematical descriptions of lineardynamical systems[J].Journal of the Society forIndustrial and Applied Mathematics,Series A:Control,1963,1(2):152-192.
[7]Gilbert E.Controllability and observability inmultivariable control systems[J].Journal of the Societyfor Industrial and Applied Mathematics,Series A:Control,1963,2(1):128-151.
[8]Morgan B S,Jr.The synthesis of linear multivariablesystems by state-variable feedback[J].IEEETransactions on Automatic Control,1964,9(4):405-411.
[9]Luo Z.Transformations between canonical forms formultivariable linear constant systems[J].IEEETransactions on Automatic Control,1977,22(2):252-256.
[10]江宁强,宋文忠.MIMO系统转化为Luenberger能控规范型的条件[J].控制理论与应用,2007,24(5):866-868.JIANG Ningqiang,SONG Wenzhong.Restraint onMIMO system in being transformed into theLuenberger’s canonical form[J].Control Theory andApplications,2007,24(5):866-868.
[11]郑大钟.线性系统理论[M].2版.北京:清华大学出版社,2005.
[12]刘豹.现代控制理论[M].2版.北京:机械工业出版社,2005.
[13]韩肖宁,董达生.线性系统能控性和能观性的几何判据:核空间、象空间、不变子空间的基本概念[J].电力学报,2008,23(1):21-25.HAN Xiaoning,DONG Dasheng.The geometriccriteria on controllability and observability of linearsystem:the basic concept of kernel space,image spaceand invariant space[J].Journal of Electric Power,2008,23(1):21-25.
[14]Petersen I R.A kalman decomposition for robustlyunobservable uncertain linear systems[J].Systems andControl Letters,2008,57(10):800-804.
[15]Pruneda R E,Solares C,Conejo A J,et al.An efficientalgebraic approach to observability analysis in stateestimation[J].Electric Power System Research,2010,80(3):277-286.
[2]Luenberger D G.Observers for multivariable systems[J].IEEE Transactions on Automatic Control,1966,11(2):190-197.
[3]Luenberger D G.An introduction to observers[J].IEEE Transactions on Automatic Control,1971,16(6):596-603.
[4]Kalman R E.Canonical structure of linear dynamicsystems[J].Proceedings of the National Academy ofSciences of United States of America,1962,48(4):596-600.
[5]Kalman R E,Ho Y C,Narendra K S.Controllability oflinear dynamical systems[J].Contributions toDifferential Equations,1961,1(3):189-213.
[6]Kalman R E.Mathematical descriptions of lineardynamical systems[J].Journal of the Society forIndustrial and Applied Mathematics,Series A:Control,1963,1(2):152-192.
[7]Gilbert E.Controllability and observability inmultivariable control systems[J].Journal of the Societyfor Industrial and Applied Mathematics,Series A:Control,1963,2(1):128-151.
[8]Morgan B S,Jr.The synthesis of linear multivariablesystems by state-variable feedback[J].IEEETransactions on Automatic Control,1964,9(4):405-411.
[9]Luo Z.Transformations between canonical forms formultivariable linear constant systems[J].IEEETransactions on Automatic Control,1977,22(2):252-256.
[10]江宁强,宋文忠.MIMO系统转化为Luenberger能控规范型的条件[J].控制理论与应用,2007,24(5):866-868.JIANG Ningqiang,SONG Wenzhong.Restraint onMIMO system in being transformed into theLuenberger’s canonical form[J].Control Theory andApplications,2007,24(5):866-868.
[11]郑大钟.线性系统理论[M].2版.北京:清华大学出版社,2005.
[12]刘豹.现代控制理论[M].2版.北京:机械工业出版社,2005.
[13]韩肖宁,董达生.线性系统能控性和能观性的几何判据:核空间、象空间、不变子空间的基本概念[J].电力学报,2008,23(1):21-25.HAN Xiaoning,DONG Dasheng.The geometriccriteria on controllability and observability of linearsystem:the basic concept of kernel space,image spaceand invariant space[J].Journal of Electric Power,2008,23(1):21-25.
[14]Petersen I R.A kalman decomposition for robustlyunobservable uncertain linear systems[J].Systems andControl Letters,2008,57(10):800-804.
[15]Pruneda R E,Solares C,Conejo A J,et al.An efficientalgebraic approach to observability analysis in stateestimation[J].Electric Power System Research,2010,80(3):277-286.