线性控制系统模型转换方法的研究
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摘要
控制方法的设计一般必须基于某一特定的模型,如:状态方程、微分方程、传递函数阵、差分方程等。而模型之间又存在着一定的相互转换关系,本文的目的就是分析研究各种模型之间的相互转换,以期能够找到简单、有效、实用的转换方法。
首先讨论了控制系统中的各类模型的描述形式,对于连续与离散系统中常用的模型形式的定义、其建立的数学基础以及建立模型的方法都进行了一定的研究,为后面各模型之间的相互转换奠定基础。
在状态空间到传递函数阵的转换算法的研究中,介绍了一种基于迭代的实用转换算法,算法的核心是计算系统的特征多项式。详细分析对比了三种求系统特征多项式的算法,即莱弗勒算法、基于特征多项式相似变换的算法和待定系数法,并通过实例对这三种方法进行比较,总结出各种算法所适用的不同情况。
针对目前多变量系统传递函数阵的状态空间实现主要是基于各元素均为零极点模型的传递函数阵,并且不带时间滞后项这一问题,提出了一种能够处理带有时间滞后的有理分式矩阵形式传递函数阵的状态空间最小实现方法。根据系统的直接实现得到的状态空间能控能观特性与传递函数阵各通道元素是否具有公因式存在着相互关联的关系,提出了一种传递函数阵的分解方法。在对这一分解方法进行进一步的改进后,使原来的求解系统带有时间滞后的有理分式矩阵形式的传递函数的状态空间最小实现方法能够应用于非方系统。
最后研究和分析了系统的离散化方法。通过这些不同的离散化方法可以得到不同的离散系统模型(差分方程、脉冲传递函数、差分状态方程),并基于约当规范形的特性对传递函数离散化和连续状态空间离散化方法进行了改进,使计算过程更加简便,且不影响离散化的精度。
All control methods are besed on a special model such as state equation, transfer matrix, difference equation and so on. So the aim of this paper is mainly analyse and study the transfer approach of many different kinds of control models, in order to find a model transfer method which is simple, effective and useful.
Firstly, the paper discussed the description of each kind of models. It studied not only the definition of common used models in both continuous system and discrete system, also the mathematical theory the model based on and how to establish the model. This part is the basic for the study of the model transfer methods in the following chapters.
In the study of the approach transfer state equation into transfer matrix, the paper introduced a useful algorithm based on the iteration. The core of that algorithm is to compute the characteristic polynomial. Three different kinds of methods are discussed and compared. With the comparation of these algorithms, the most adaptive usage fields of each approach are known.
Presently, most algorithms of the transfer matrix realization are deal with the transfer matrix whose elements are zero-pole models and the models they deal with are the ones without time delay. To resolve these problems, a new algorithm which can deal with a MIMO system transfer matrix (rational fraction matrix form) with time delay is proposed in this paper. The controllability and observability of a state equation tansfor from a rational fraction matrix form transfer matrix, is connecting with the transfer matrix whose elements in each channal have a common divisor or not. Base on this relationship, a transfer matrix's decomposition algorithm is proposed in the paper. After developed this transfer matrix's decomposition algorithm, the minimal realization method proposed in this paper can be used to in the non-square processes.
Lastly, discretization algorithms are researched and compared. With these algorithms, different kinds of discrete system madels (difference equation, pulse transfer function, discrete state equation) can be got.
首先讨论了控制系统中的各类模型的描述形式,对于连续与离散系统中常用的模型形式的定义、其建立的数学基础以及建立模型的方法都进行了一定的研究,为后面各模型之间的相互转换奠定基础。
在状态空间到传递函数阵的转换算法的研究中,介绍了一种基于迭代的实用转换算法,算法的核心是计算系统的特征多项式。详细分析对比了三种求系统特征多项式的算法,即莱弗勒算法、基于特征多项式相似变换的算法和待定系数法,并通过实例对这三种方法进行比较,总结出各种算法所适用的不同情况。
针对目前多变量系统传递函数阵的状态空间实现主要是基于各元素均为零极点模型的传递函数阵,并且不带时间滞后项这一问题,提出了一种能够处理带有时间滞后的有理分式矩阵形式传递函数阵的状态空间最小实现方法。根据系统的直接实现得到的状态空间能控能观特性与传递函数阵各通道元素是否具有公因式存在着相互关联的关系,提出了一种传递函数阵的分解方法。在对这一分解方法进行进一步的改进后,使原来的求解系统带有时间滞后的有理分式矩阵形式的传递函数的状态空间最小实现方法能够应用于非方系统。
最后研究和分析了系统的离散化方法。通过这些不同的离散化方法可以得到不同的离散系统模型(差分方程、脉冲传递函数、差分状态方程),并基于约当规范形的特性对传递函数离散化和连续状态空间离散化方法进行了改进,使计算过程更加简便,且不影响离散化的精度。
All control methods are besed on a special model such as state equation, transfer matrix, difference equation and so on. So the aim of this paper is mainly analyse and study the transfer approach of many different kinds of control models, in order to find a model transfer method which is simple, effective and useful.
Firstly, the paper discussed the description of each kind of models. It studied not only the definition of common used models in both continuous system and discrete system, also the mathematical theory the model based on and how to establish the model. This part is the basic for the study of the model transfer methods in the following chapters.
In the study of the approach transfer state equation into transfer matrix, the paper introduced a useful algorithm based on the iteration. The core of that algorithm is to compute the characteristic polynomial. Three different kinds of methods are discussed and compared. With the comparation of these algorithms, the most adaptive usage fields of each approach are known.
Presently, most algorithms of the transfer matrix realization are deal with the transfer matrix whose elements are zero-pole models and the models they deal with are the ones without time delay. To resolve these problems, a new algorithm which can deal with a MIMO system transfer matrix (rational fraction matrix form) with time delay is proposed in this paper. The controllability and observability of a state equation tansfor from a rational fraction matrix form transfer matrix, is connecting with the transfer matrix whose elements in each channal have a common divisor or not. Base on this relationship, a transfer matrix's decomposition algorithm is proposed in the paper. After developed this transfer matrix's decomposition algorithm, the minimal realization method proposed in this paper can be used to in the non-square processes.
Lastly, discretization algorithms are researched and compared. With these algorithms, different kinds of discrete system madels (difference equation, pulse transfer function, discrete state equation) can be got.
引文
[1]周春辉.化工过程控制原理[M].北京:化学工业出版社,1998
[2]蒋慰孙,叶银忠.多变量控制理论分析与设计[M].北京:中国石化出版社,1997
[3]吴麒,王诗宓.自动控制原理[M].北京:清华大学出版社,2006
[4]王诗宓.多变量控制系统的分析和设计[M],北京:中国电力出版社,1996
[5]涂臍生,齐寅峰.多变量线性控制系统分析与设计:状态空间与多项式矩阵方法[M].北京:化学工业出版社,1989
[6]薛兴昌.工业自动化系统的技术发展[J].自动化博览,2003,8:48-50
[7]Katsuhiko Ogata.Discrete-Time Control System[M].Beijing:China Machine Press,2006
[8]K.J.Astrom,B.Wittenmark.Computer-Controlled Systems:Theory and Design,third ed.[M].Prentice-Hall,Englewood Cliffs,NJ,1997
[9]郑大中.线性系统理论[M].北京:清华大学出版社,2002
[10]张伯春.用相似变换计算矩阵特征多项式的方法[J].西南民族学院学报(自然科学版),1993,11:19(4).417-421
[11]Jiu Ding,Aihui Zhou.Characteristic polynomials of some perturbed matrices[J].Applied Mathematics and Computation,2008,199,631-636
[12]孙志和,窦在祥.特征多项式系数的矩阵表示[J].青岛理工大学学报,2006,27:3,112-115
[13]岳锡亭,张玉石,潘英剑.复合矩阵[J].长春工业大学学报,2003,24:2,51-53
[14]李巍,胡方景.关于矩阵的特征多项式的展开[J].青海师专学报(自然科学),2001,6,8-10
[15]钟宝东,张国政,高齐圣.关于方阵A的特征多项式及其逆矩阵的求法问题[J].郑州工学院学报,1991,3:12(1),126-130
[16]J.Ding,G.Yao.Eigenvalues of updated matrices[J].Appl.Math.Comput.,2007,185,415 - 420
[17]E.G.Gilbert.Controllability and observability in multi-variable control systems[J].SIAM J.Control,1963,1(2),128-151
[18]R.E.Kalman.Mathematical description of linear dynamical systems[J].SIAM J.Control,1963,1(2),152-192
[19]B.L.Ho,R.E.Kalman,Effective construction of linear state-variable models from input/output functions,in:M.E.Van Valkenburg(Ed.)[J].Proceedings of the Third Annual Allerton Conference on Circuit and System Theory,1965,10,449-459.
[20]B.L.Ho,R.E.Kalman,Effective construction of linear,state-variable models from input/output functions[J],Regelungstechnik,1966,14(12),545-548
[21]R.E.Kalman.On minimal partial realizations of a linear input/output map,in:R.E.Kalman,N.DeClaris(Eds.)[J].Aspects of Network and System Theory,Holt,Rinehart and Winston,New York,1971,19(5),385-407.
[22]B.De Sehutter.Minimal state-space realization in linear system theory:an overview[J].Journal of Computational and Applied Mathematics,2000,121,331-354
[23]薛定宇.控制系统计算机辅助设计[M].北京:清华大学出版社,2006
[24]郭文献.最大公因式的初等变换求法[J].安阳师范学院学报,2005,5,28-29
[25]胡少勇,易艺.多项式最大公因式求法探讨[J].九江学院学报(自然科学版),2005,3,29-31
[26]王新民,孙霞。最大公因式与最小公倍式的统一求法[J].数学通报,2001,12,41-42
[27]刘静,邱芳.最大公因式及其系数多项式的统一求法[J].滨州学院学报,2006,22:6,69-71
[28]王丹华.利用矩阵初等变换求多项式的最大公因式[J].高等数学研究,2005,8:1,15-17
[29]L.G.Van Willigenburg,W.L.De Koning.Linear systems theory revisited[J].Automatica,2008
[30]Patel R.,V.N.Munro.Multivariale System Theory and Design[M].Pergamon Press,1982
[31]A.J.Tether.Construction of minimal linear state-variable models from finite input-output data[C],IEEE Trans.Automat.Control,1970,17(4),427-436
[32]J.Rissanen.Recursive identification of linear systems[J].SIAM J.Control Optim.1971,9(3),420-430
[33]B.L.Ho.An effective construction of realizations from input/output descriptions[D].Stanford University,Stanford,CA,1966
[34]陈伦铭,陈潇涛.最小实现简洁法[J].兵工自动化,1998,1,7-10
[35]周洪.最小实现的直接方法[J].控制理论与应用,1993,10:4,484-488
[36]Franklin G F,Powell J D,Workman M.Digital Control of Dynamic System(Third Edition)[M].北京:清华大学出版社,2001
[37]Mitra S K.Digital Signal Processing--A Computer-based Approach(Second Edition)[M].北京:清华大学出版社,2001
[38]刘德贵,费景高.动力学系统数字仿真算法[M].北京:科学出版社,2000
[39]蒋心怡,吴汉松,陈少昌.计算机控制工程[M].长沙:国防科技大学出版社,2002
[40]肖田元,张燕云,陈家栋.系统仿真导论[M].北京:清华大学出版社,2000
[41]Dorf R C,Bishop R H.Modem Control System(Ninth Edition)[M].北京:科学出版社,2002
[42]吕志民,周茂林.使用Pade近似式处理数字控制系统中的纯滞后[J].中山大学学报(自然科学版),2001,1:40(1),114-115
[43]张民,黄凯,王田荣.连续控制系统非规范替换法离散化[J].海军工程大学学报,2003,6:15(3),9-13
[44]M.Haraguchi,H.Y.Hu.Using a new discretization approach to design a delayed LQG controlle[J].Journal of Sound and Vibration,2008,314,558-570
[45]孙连明,赵志魁,谢永斌,刘文江.连续系统离散化的一种简单方法[J].焦作工学院学报,1996,15(6),69-73
[46]宁璀,张泉灵,苏宏业.多变量一阶加纯滞后系统的预测函数控制[J].信息与控制,2007,36:6,702-714
[47]G.Cai,J.Huang.Optimal control method with time delay in control[J].Journal of Sound and Vibration,2002,251(3),383-394
[48]刘开培.基于Pade逼近的纯滞后系统增益自适应内模PID控制[J].武汉大学学报(工学版),2001,8:93-95
[49]龚晓峰,高袊畅,周春晖.时滞系统 PID控制器内模整定方法的扩展[J].控制与决策,1998,7:337-341.
[50]李少远,蔡文剑.工业过程辨识与控制[M].北京:化学工业出版社,2005.206-214
[51]Ogunnaike B A,Ray W H.Process dynamics,modeling and control[J].Oxford University Press,New York,1994
[52]尹先斌,周有训.基于Taylor和Pade能逼近的滞后系统IMC-PID研究[J].昆明理工大学学报:理工版,2006,31(2):76-79.
[53]李巍,王明明,潘立登.矩阵连接法在辨识与PID参数设计中的应用[J].石油化工高等学校学报,2007,20(增刊):8-11
[54]王福永.基于Pade逼近的纯滞后系统内模控制器的设计[J].苏州大学学报:工学版,2004,24(4),26-29.
[2]蒋慰孙,叶银忠.多变量控制理论分析与设计[M].北京:中国石化出版社,1997
[3]吴麒,王诗宓.自动控制原理[M].北京:清华大学出版社,2006
[4]王诗宓.多变量控制系统的分析和设计[M],北京:中国电力出版社,1996
[5]涂臍生,齐寅峰.多变量线性控制系统分析与设计:状态空间与多项式矩阵方法[M].北京:化学工业出版社,1989
[6]薛兴昌.工业自动化系统的技术发展[J].自动化博览,2003,8:48-50
[7]Katsuhiko Ogata.Discrete-Time Control System[M].Beijing:China Machine Press,2006
[8]K.J.Astrom,B.Wittenmark.Computer-Controlled Systems:Theory and Design,third ed.[M].Prentice-Hall,Englewood Cliffs,NJ,1997
[9]郑大中.线性系统理论[M].北京:清华大学出版社,2002
[10]张伯春.用相似变换计算矩阵特征多项式的方法[J].西南民族学院学报(自然科学版),1993,11:19(4).417-421
[11]Jiu Ding,Aihui Zhou.Characteristic polynomials of some perturbed matrices[J].Applied Mathematics and Computation,2008,199,631-636
[12]孙志和,窦在祥.特征多项式系数的矩阵表示[J].青岛理工大学学报,2006,27:3,112-115
[13]岳锡亭,张玉石,潘英剑.复合矩阵[J].长春工业大学学报,2003,24:2,51-53
[14]李巍,胡方景.关于矩阵的特征多项式的展开[J].青海师专学报(自然科学),2001,6,8-10
[15]钟宝东,张国政,高齐圣.关于方阵A的特征多项式及其逆矩阵的求法问题[J].郑州工学院学报,1991,3:12(1),126-130
[16]J.Ding,G.Yao.Eigenvalues of updated matrices[J].Appl.Math.Comput.,2007,185,415 - 420
[17]E.G.Gilbert.Controllability and observability in multi-variable control systems[J].SIAM J.Control,1963,1(2),128-151
[18]R.E.Kalman.Mathematical description of linear dynamical systems[J].SIAM J.Control,1963,1(2),152-192
[19]B.L.Ho,R.E.Kalman,Effective construction of linear state-variable models from input/output functions,in:M.E.Van Valkenburg(Ed.)[J].Proceedings of the Third Annual Allerton Conference on Circuit and System Theory,1965,10,449-459.
[20]B.L.Ho,R.E.Kalman,Effective construction of linear,state-variable models from input/output functions[J],Regelungstechnik,1966,14(12),545-548
[21]R.E.Kalman.On minimal partial realizations of a linear input/output map,in:R.E.Kalman,N.DeClaris(Eds.)[J].Aspects of Network and System Theory,Holt,Rinehart and Winston,New York,1971,19(5),385-407.
[22]B.De Sehutter.Minimal state-space realization in linear system theory:an overview[J].Journal of Computational and Applied Mathematics,2000,121,331-354
[23]薛定宇.控制系统计算机辅助设计[M].北京:清华大学出版社,2006
[24]郭文献.最大公因式的初等变换求法[J].安阳师范学院学报,2005,5,28-29
[25]胡少勇,易艺.多项式最大公因式求法探讨[J].九江学院学报(自然科学版),2005,3,29-31
[26]王新民,孙霞。最大公因式与最小公倍式的统一求法[J].数学通报,2001,12,41-42
[27]刘静,邱芳.最大公因式及其系数多项式的统一求法[J].滨州学院学报,2006,22:6,69-71
[28]王丹华.利用矩阵初等变换求多项式的最大公因式[J].高等数学研究,2005,8:1,15-17
[29]L.G.Van Willigenburg,W.L.De Koning.Linear systems theory revisited[J].Automatica,2008
[30]Patel R.,V.N.Munro.Multivariale System Theory and Design[M].Pergamon Press,1982
[31]A.J.Tether.Construction of minimal linear state-variable models from finite input-output data[C],IEEE Trans.Automat.Control,1970,17(4),427-436
[32]J.Rissanen.Recursive identification of linear systems[J].SIAM J.Control Optim.1971,9(3),420-430
[33]B.L.Ho.An effective construction of realizations from input/output descriptions[D].Stanford University,Stanford,CA,1966
[34]陈伦铭,陈潇涛.最小实现简洁法[J].兵工自动化,1998,1,7-10
[35]周洪.最小实现的直接方法[J].控制理论与应用,1993,10:4,484-488
[36]Franklin G F,Powell J D,Workman M.Digital Control of Dynamic System(Third Edition)[M].北京:清华大学出版社,2001
[37]Mitra S K.Digital Signal Processing--A Computer-based Approach(Second Edition)[M].北京:清华大学出版社,2001
[38]刘德贵,费景高.动力学系统数字仿真算法[M].北京:科学出版社,2000
[39]蒋心怡,吴汉松,陈少昌.计算机控制工程[M].长沙:国防科技大学出版社,2002
[40]肖田元,张燕云,陈家栋.系统仿真导论[M].北京:清华大学出版社,2000
[41]Dorf R C,Bishop R H.Modem Control System(Ninth Edition)[M].北京:科学出版社,2002
[42]吕志民,周茂林.使用Pade近似式处理数字控制系统中的纯滞后[J].中山大学学报(自然科学版),2001,1:40(1),114-115
[43]张民,黄凯,王田荣.连续控制系统非规范替换法离散化[J].海军工程大学学报,2003,6:15(3),9-13
[44]M.Haraguchi,H.Y.Hu.Using a new discretization approach to design a delayed LQG controlle[J].Journal of Sound and Vibration,2008,314,558-570
[45]孙连明,赵志魁,谢永斌,刘文江.连续系统离散化的一种简单方法[J].焦作工学院学报,1996,15(6),69-73
[46]宁璀,张泉灵,苏宏业.多变量一阶加纯滞后系统的预测函数控制[J].信息与控制,2007,36:6,702-714
[47]G.Cai,J.Huang.Optimal control method with time delay in control[J].Journal of Sound and Vibration,2002,251(3),383-394
[48]刘开培.基于Pade逼近的纯滞后系统增益自适应内模PID控制[J].武汉大学学报(工学版),2001,8:93-95
[49]龚晓峰,高袊畅,周春晖.时滞系统 PID控制器内模整定方法的扩展[J].控制与决策,1998,7:337-341.
[50]李少远,蔡文剑.工业过程辨识与控制[M].北京:化学工业出版社,2005.206-214
[51]Ogunnaike B A,Ray W H.Process dynamics,modeling and control[J].Oxford University Press,New York,1994
[52]尹先斌,周有训.基于Taylor和Pade能逼近的滞后系统IMC-PID研究[J].昆明理工大学学报:理工版,2006,31(2):76-79.
[53]李巍,王明明,潘立登.矩阵连接法在辨识与PID参数设计中的应用[J].石油化工高等学校学报,2007,20(增刊):8-11
[54]王福永.基于Pade逼近的纯滞后系统内模控制器的设计[J].苏州大学学报:工学版,2004,24(4),26-29.