动态反馈解耦规范型时域结构特征分析及变换矩阵的构造
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摘要
从可解耦线性多输入多输出(Multi-input multi-output,MIMO)系统的结构特性指数出发,根据此类系统解耦后系统的可观测矩阵与基本向量矩阵的秩的关系,提出了按照这种关系将解耦规范型划分为4大类的观点,同时给出了一种针对各类积分型解耦系统构造相应的非奇异变换矩阵的构造方法.分析了解耦规范型及其变换矩阵的时域结构形式,通过一系列定理的证明,从一般意义上解释了解耦规范型的结构与变换矩阵的关系,并通过一个数值实例验证了所提出方法的正确性及可行性.
In this paper, based on the structure characteristic index of linear multi-input multi-output (MIMO) system which can be decoupled, we propose that the decoupling canonical form of the linear MIMO system can be divided into 4 classes according to the correlation between the rank of observable matrix of the decoupling system and the rank of basic vector matrix. Meanwhile, we put forward an approach for obtaining different nonsingular transformation matrixes for different integral decoupled systems. Then, we analyze the time domain structures of the decoupling canonical form and the transformation matrixes, and give an elaborate proof of a series of theorems in order to interpret the correlation between the decoupling canonical form and the transformation matrix. Finally, the validity and feasibility of the method are verified by a numerical example.
In this paper, based on the structure characteristic index of linear multi-input multi-output (MIMO) system which can be decoupled, we propose that the decoupling canonical form of the linear MIMO system can be divided into 4 classes according to the correlation between the rank of observable matrix of the decoupling system and the rank of basic vector matrix. Meanwhile, we put forward an approach for obtaining different nonsingular transformation matrixes for different integral decoupled systems. Then, we analyze the time domain structures of the decoupling canonical form and the transformation matrixes, and give an elaborate proof of a series of theorems in order to interpret the correlation between the decoupling canonical form and the transformation matrix. Finally, the validity and feasibility of the method are verified by a numerical example.
引文
1 Gilbert E G.The decoupling of multivariable systems bystate feedback.SIAM Journal on Control,1969,7(1):5063
2 Wang Yong-Chu.Decoupling Control System.Chengdu:Sichuan Publishing House of Science&Technology,1985.251292(王永初.解耦控制系统.成都:四川科学技术出版社,1985.251292)
3 Marino R,Cilini F.Input-output decoupling control bymeasurement feedback in four-wheel-steering vehicles.IEEETransactions on Control Systems Technology,2009,17(5):11631172
4 Fang J,Ren Y.Decoupling control of magnetically sus-pended rotor system in control moment gyros based onan inverse system method.IEEE/ASME Transactions onMechatronics,2011,17(6):11331144
5 Jung J,Nam K.A dynamic decoupling control scheme forhigh-speed operation of induction motors.IEEE Transac-tions on Industrial Electronics,1999,46(1):100110
6 Wonham W M.Linear Multivariable Control:A GeometricApproach.Berlin:Springer-Verlag,1985.221323
7 Falb P L,Wolovich W A.Decoupling in the design and syn-thesis of multivariable control systems.IEEE Transactionson Automatic Control,1967,12(6):651659
8 Liu Bao.Modern Control Theory(Second edition).Beijing:China Machine Press,2003.182189(刘豹.现代控制理论(第2版).北京:机械工业出版社,2003.182189)
9 Morgan B S Jr.The synthesis of linear multivariable systemsby state variable feedback.IEEE Transactions on AutomaticControl,1964,9(4):405411
10 Rekasius Z V.Decoupling of multivariable systems by meansof state feedback.In:Proceedings of the3rd Annual Aller-ton Conference on Circuit and System Theory,Monticello.Illinois.Urbana,IL:University of Illinois,1965.439448
11 Gilbert E G,Pivnichny J R.A computer program forthe synthesis of decoupled multivariable feedback systems.IEEE Transactions on Automatic Control,1969,14(6):652659
12 Wonham W M,Morse A S.Decoupling and pole assign-ment in linear multi-variable systems:a geometric approach.SIAM Journal on Control,1970,8(1):118
13 Zheng Da-Zhong.Linear System Theory(Second Edition).Beijing:Tsinghua University Press,2005.285301(郑大钟.线性系统理论(第二版).北京:清华大学出版社,2005.285301)
14 Zhang Liu,Wang Zi-Hua.A simple and efficient solutionto the decoupling and eigenstructure assignment problem inlinear multivariable systems.Journal of Zhengzhou Univer-sity(Engineering Science),2006,27(1):7578(张刘,王子华.MIMO系统解耦规范型实现及特征结构配置.郑州大学学报(工学版),2006,27(1):7578)
15 Tian Guo-Hui,Li Xiao-Lei,Yang Xi-Xia.A simple and effi-cient solution to the decoupling and pole assignment prob-lem in linear multivariable systems.Journal of ShandongUniversity of Technology,2000,30(4):301305(田国会,李晓磊,杨西侠.多变量线性系统中极点配置问题的一种简便解法.山东工业大学学报,2000,30(4):301305)
16 Horn R A,Johnson C R.Matrix Analysis.New York:Cam-bridge University Press,1990.130
2 Wang Yong-Chu.Decoupling Control System.Chengdu:Sichuan Publishing House of Science&Technology,1985.251292(王永初.解耦控制系统.成都:四川科学技术出版社,1985.251292)
3 Marino R,Cilini F.Input-output decoupling control bymeasurement feedback in four-wheel-steering vehicles.IEEETransactions on Control Systems Technology,2009,17(5):11631172
4 Fang J,Ren Y.Decoupling control of magnetically sus-pended rotor system in control moment gyros based onan inverse system method.IEEE/ASME Transactions onMechatronics,2011,17(6):11331144
5 Jung J,Nam K.A dynamic decoupling control scheme forhigh-speed operation of induction motors.IEEE Transac-tions on Industrial Electronics,1999,46(1):100110
6 Wonham W M.Linear Multivariable Control:A GeometricApproach.Berlin:Springer-Verlag,1985.221323
7 Falb P L,Wolovich W A.Decoupling in the design and syn-thesis of multivariable control systems.IEEE Transactionson Automatic Control,1967,12(6):651659
8 Liu Bao.Modern Control Theory(Second edition).Beijing:China Machine Press,2003.182189(刘豹.现代控制理论(第2版).北京:机械工业出版社,2003.182189)
9 Morgan B S Jr.The synthesis of linear multivariable systemsby state variable feedback.IEEE Transactions on AutomaticControl,1964,9(4):405411
10 Rekasius Z V.Decoupling of multivariable systems by meansof state feedback.In:Proceedings of the3rd Annual Aller-ton Conference on Circuit and System Theory,Monticello.Illinois.Urbana,IL:University of Illinois,1965.439448
11 Gilbert E G,Pivnichny J R.A computer program forthe synthesis of decoupled multivariable feedback systems.IEEE Transactions on Automatic Control,1969,14(6):652659
12 Wonham W M,Morse A S.Decoupling and pole assign-ment in linear multi-variable systems:a geometric approach.SIAM Journal on Control,1970,8(1):118
13 Zheng Da-Zhong.Linear System Theory(Second Edition).Beijing:Tsinghua University Press,2005.285301(郑大钟.线性系统理论(第二版).北京:清华大学出版社,2005.285301)
14 Zhang Liu,Wang Zi-Hua.A simple and efficient solutionto the decoupling and eigenstructure assignment problem inlinear multivariable systems.Journal of Zhengzhou Univer-sity(Engineering Science),2006,27(1):7578(张刘,王子华.MIMO系统解耦规范型实现及特征结构配置.郑州大学学报(工学版),2006,27(1):7578)
15 Tian Guo-Hui,Li Xiao-Lei,Yang Xi-Xia.A simple and effi-cient solution to the decoupling and pole assignment prob-lem in linear multivariable systems.Journal of ShandongUniversity of Technology,2000,30(4):301305(田国会,李晓磊,杨西侠.多变量线性系统中极点配置问题的一种简便解法.山东工业大学学报,2000,30(4):301305)
16 Horn R A,Johnson C R.Matrix Analysis.New York:Cam-bridge University Press,1990.130