由频率特性数据到传递函数的识辩研究
摘要
分析、设计系统的首要前提是建立其数学模型,当不清楚系统的结构、参数,而不能用基于理论推导的方法建立其数学模型时,用实验的方法建立其数学模型,就显得十分必要。频率特性就是一种基于实验方法的数学模型,但它是频率域的数学模型,如果能将其转化为传递函数,就可以利用时间域内的分析、设计方法,避免频率域分析、设计方法的不直观性。
本文就是用计算机编程的方法来实现频率特性到传递函数的识辨研究。首先用MATLAB软件来获得系统的实测数据,克服传统方法获得数据的缺点。然后本文针对用MATLAB和传统实验方法得到的数据分别给出了用MATLAB软件画伯德(Bode)图的方法,再次对实测数据采用扩展频率点数的多项式拟合并对拟合曲线求导,识别渐进线斜率、绘制渐近线并确定渐近线的交点。最后,探讨采用最小二乘曲线拟合技术,进一步提高识辨准确度。为实现频率域与时间域的分析研究架起了沟通的桥梁。
In the classical control theory, the control system is generally studied with mathematic model. The differential equation, transfer function, and frequency property can be used for the description of the mathematic model. At the respect of obtaining the control system mathematic model, the differential equation is obtained with theoretical derivation, under premise of known every part of the system and its operation principle. The study of the differential equation is mainly based time domain. While to obtain the frequency property is relied on the experiments under the premise of unknown each parts of the system and its operation principle. The study of frequency property is based on frequency domain. Transfer function is a bridge for interconversion of the differential equation and frequency property. Under the premise of known system differential equation and under the null initial condition, with Laplace transform the differential equation can be transformed to transfer function, and then with s=jωdirectly transform the transfer function to frequency property. This is a fully-fledged method; contrarily, with inverse Laplace transform the transfer function can be transformed to differential equation. But it is still a study subject of transforming directly the frequency property to transfer function.
At the daily life, the inner structure, mechanism and parameter of some systems are often unknown. Because of this, the differential equation of the system cannot be set up with the theory derivation. And then it makes the time domain analysis method invalid. Therefore the system study can be only done with the frequency property model from the experiment. As the machine and human being are located in the same time domain, the conclusion with the time domain analysis is respond with the human being time domain. While the conclusion obtained from frequency property is in the abstract frequency domain. Obviously the time domain analysis method is more directviewing than frequency domain analysis method. If the data with the frequency property can be distinguished as the transfer function, the frequency property can be transformed to the differential function. The system analysis can be more directviewing. The modern control theory describes the system with the mathematic model of state space expression. The mathematic model of state space expression has the correspondence with the differential equation and transfer function. If we can obtain distinguish of the data with frequency property to transfer function, the corresponding state space expression can be obtained and the essential premise and possibility can be provided for the study method with the modern control theory. This essay studied the distinguish for the type and parameter of data with frequency property to transfer function, and on this base the parameter is estimated.
The transformation of the frequency property to transfer function is realized manually. At first, the frequency property data of the control system is obtained with experiment, and expressed with decibel (dB), and then the Bobe diagram is drawn in the logarithm coordinate, again look for the asymptotic line manually. The intersection of the asymptotic line determines the corner frequency which is corresponding with its frequency property, therefore the data of transfer function is obtained. But it is slow and imprecision with this method. For overcoming this fault, the transformation of frequency property to transfer function must be done with computer. But till now, there is nobody who has done this. No body has studied the distinguish of frequency property to transfer function with asymptotic line by computer assemble language. This essay has realized the study of distinguish of frequency property to transfer function.
In this thesis, at first the mathematic model for automatic control system is introduced, mainly included differential equation, transfer function, dynamic structure diagram and frequency property. The mathematic model is a base for the research; next the relation between the mathematic models is deeply analyzed, included the relation between differential equation and transfer function, differential equation and dynamic structure diagram, transfer function and frequency property. Especially the relation of log magnitude-frequency characteristics asymptote intersection and corner frequency is discussed. It gives the theory foundation for frequency property determining the parameter of transfer function. And the third, the subject of data of frequency property distinguishing the data of transfer function.
The study of this thesis is based on the traditional theory and realized the computer automation. In the traditional method, the oscillograph, signal generator, circuit element are used for obtain the experiment data. Because this equipment has some weakness for research: the precision at lower and higher frequency is lower. Therefore as to grantee the data precision and not to break away from the reality, the MATLAB software is used at the first step of the research. At least the observation data is obtained with two methods. Then the decibel (dB) is calculated by computer. The Bobe diagram is drawn in the logarithm coordinate. The observation data from the electric equipment and MATLAB is given to Bobe to draw the diagram. One again multinomial fitting with extended frequency pointes for the observation data, and differentiating for the fitting. It provides the necessary grantee for distinguishing the slope of the asymptote line, for drawing the asymptote line and determining the intersection of the asymptote line. Besides the tolerance analysis for the method based on asymptote line is done. At last considering the distinguish accurate of the transfer function parameter, the least squares curve fitting technology is discussed. The distinguish accurate is improved. At the whole study, the corresponding program is programmed with the MATLAB’S functions and program environment. The corresponding diagram is drawn and the distinguish of the transfer function structure is realized.
In short, through the frequency property data of the observation system order connection– none parameter model, the corresponding transfer function– parameter model is distinguished. The system research method based on the engineering is explored from the aspect of principle, method, program and tolerance. Although it’s only a individual cases, it’s a good try to overcome the weakness at the theory research to experiment research method. In the study, the Laplace transform, least squares are used. They are relative classical mathematic subject. It indicates that the computer application field is extended with the combination of engineering mathematic and computer technology to solve the engineering system analysis.
At the research process of this thesis, it’s difficult to find relevant material and known achievement. Therefore the subject is only discussed at the first step. Besides, because of the limited time, there may be deficiency in this research work. If the transform from the frequency property to transfer function could be completely finished, it could give a great help for the future unknown system research.
本文就是用计算机编程的方法来实现频率特性到传递函数的识辨研究。首先用MATLAB软件来获得系统的实测数据,克服传统方法获得数据的缺点。然后本文针对用MATLAB和传统实验方法得到的数据分别给出了用MATLAB软件画伯德(Bode)图的方法,再次对实测数据采用扩展频率点数的多项式拟合并对拟合曲线求导,识别渐进线斜率、绘制渐近线并确定渐近线的交点。最后,探讨采用最小二乘曲线拟合技术,进一步提高识辨准确度。为实现频率域与时间域的分析研究架起了沟通的桥梁。
In the classical control theory, the control system is generally studied with mathematic model. The differential equation, transfer function, and frequency property can be used for the description of the mathematic model. At the respect of obtaining the control system mathematic model, the differential equation is obtained with theoretical derivation, under premise of known every part of the system and its operation principle. The study of the differential equation is mainly based time domain. While to obtain the frequency property is relied on the experiments under the premise of unknown each parts of the system and its operation principle. The study of frequency property is based on frequency domain. Transfer function is a bridge for interconversion of the differential equation and frequency property. Under the premise of known system differential equation and under the null initial condition, with Laplace transform the differential equation can be transformed to transfer function, and then with s=jωdirectly transform the transfer function to frequency property. This is a fully-fledged method; contrarily, with inverse Laplace transform the transfer function can be transformed to differential equation. But it is still a study subject of transforming directly the frequency property to transfer function.
At the daily life, the inner structure, mechanism and parameter of some systems are often unknown. Because of this, the differential equation of the system cannot be set up with the theory derivation. And then it makes the time domain analysis method invalid. Therefore the system study can be only done with the frequency property model from the experiment. As the machine and human being are located in the same time domain, the conclusion with the time domain analysis is respond with the human being time domain. While the conclusion obtained from frequency property is in the abstract frequency domain. Obviously the time domain analysis method is more directviewing than frequency domain analysis method. If the data with the frequency property can be distinguished as the transfer function, the frequency property can be transformed to the differential function. The system analysis can be more directviewing. The modern control theory describes the system with the mathematic model of state space expression. The mathematic model of state space expression has the correspondence with the differential equation and transfer function. If we can obtain distinguish of the data with frequency property to transfer function, the corresponding state space expression can be obtained and the essential premise and possibility can be provided for the study method with the modern control theory. This essay studied the distinguish for the type and parameter of data with frequency property to transfer function, and on this base the parameter is estimated.
The transformation of the frequency property to transfer function is realized manually. At first, the frequency property data of the control system is obtained with experiment, and expressed with decibel (dB), and then the Bobe diagram is drawn in the logarithm coordinate, again look for the asymptotic line manually. The intersection of the asymptotic line determines the corner frequency which is corresponding with its frequency property, therefore the data of transfer function is obtained. But it is slow and imprecision with this method. For overcoming this fault, the transformation of frequency property to transfer function must be done with computer. But till now, there is nobody who has done this. No body has studied the distinguish of frequency property to transfer function with asymptotic line by computer assemble language. This essay has realized the study of distinguish of frequency property to transfer function.
In this thesis, at first the mathematic model for automatic control system is introduced, mainly included differential equation, transfer function, dynamic structure diagram and frequency property. The mathematic model is a base for the research; next the relation between the mathematic models is deeply analyzed, included the relation between differential equation and transfer function, differential equation and dynamic structure diagram, transfer function and frequency property. Especially the relation of log magnitude-frequency characteristics asymptote intersection and corner frequency is discussed. It gives the theory foundation for frequency property determining the parameter of transfer function. And the third, the subject of data of frequency property distinguishing the data of transfer function.
The study of this thesis is based on the traditional theory and realized the computer automation. In the traditional method, the oscillograph, signal generator, circuit element are used for obtain the experiment data. Because this equipment has some weakness for research: the precision at lower and higher frequency is lower. Therefore as to grantee the data precision and not to break away from the reality, the MATLAB software is used at the first step of the research. At least the observation data is obtained with two methods. Then the decibel (dB) is calculated by computer. The Bobe diagram is drawn in the logarithm coordinate. The observation data from the electric equipment and MATLAB is given to Bobe to draw the diagram. One again multinomial fitting with extended frequency pointes for the observation data, and differentiating for the fitting. It provides the necessary grantee for distinguishing the slope of the asymptote line, for drawing the asymptote line and determining the intersection of the asymptote line. Besides the tolerance analysis for the method based on asymptote line is done. At last considering the distinguish accurate of the transfer function parameter, the least squares curve fitting technology is discussed. The distinguish accurate is improved. At the whole study, the corresponding program is programmed with the MATLAB’S functions and program environment. The corresponding diagram is drawn and the distinguish of the transfer function structure is realized.
In short, through the frequency property data of the observation system order connection– none parameter model, the corresponding transfer function– parameter model is distinguished. The system research method based on the engineering is explored from the aspect of principle, method, program and tolerance. Although it’s only a individual cases, it’s a good try to overcome the weakness at the theory research to experiment research method. In the study, the Laplace transform, least squares are used. They are relative classical mathematic subject. It indicates that the computer application field is extended with the combination of engineering mathematic and computer technology to solve the engineering system analysis.
At the research process of this thesis, it’s difficult to find relevant material and known achievement. Therefore the subject is only discussed at the first step. Besides, because of the limited time, there may be deficiency in this research work. If the transform from the frequency property to transfer function could be completely finished, it could give a great help for the future unknown system research.
引文
[1] 刘少军编著、现代控制方法及计算机辅助设计、中南大学出版社,2003 P20-P32
[2] 绪方胜彦著、现代控制工程、北京:科学出版社、1984 P52-P71 、 P285-P350
[3] 胡寿松编著、自动控制原理、国防工业出版社、1994 第 3 版、P7-64
[4] 孙梅、王彦良编著、自动控制系统及应用、北方交通大学、2007 P15-29,P78-112
[5] 薛定宇、陈阳泉著、基于 MATLAB/Simulink 的系统仿真技术与应用、北京:清华大学出版社、2002 P17-P62,P203-223
[6] 楼顺天、姚若玉、冶继民编著、基于 MATLAB 7.x 的系统分析与设计,西安电子科技大学出版社、1998 P1-P6,P139-P152
[7] 陈怀堔，吴大正，高西全编著、MATLAB 及在电子信息课程中的应用、北京:电子工业出版社、2002 P223-267
[8] 董景新、赵长德编著、控制工程基础、清华大学出版社、1992、P12-35
[9] 飞思科技产品研发中心编、MATLAB7 基础与提高、北京:电子工业出版社、2005 P115-134
[10] 颜庆津编著、数值分析、北京航空航天大学出版社、2004、P207-219
[11] 薛定宇 陈阳泉编著、高等应用数学问题的 MATLAB 求解、清华大学出版社、2004年 8 月 31 日、p84-214
[12] 高世臣编著、高等数学(二)、北京教育出版社、奥林匹克出版社、2001 年 6月第 3 版、p64-199
[13] 张忠平主编、李刚、夏德阳副主编、应用数学基础上册、辽宁人民出版社、2001年 8 月第上版、p62-88
[14]刘宝芹主编、沈雅芬副主编、计算机电路基础(1)、中央广播电视大学出版社、2000 年 7 月第一版、p8-14
[2] 绪方胜彦著、现代控制工程、北京:科学出版社、1984 P52-P71 、 P285-P350
[3] 胡寿松编著、自动控制原理、国防工业出版社、1994 第 3 版、P7-64
[4] 孙梅、王彦良编著、自动控制系统及应用、北方交通大学、2007 P15-29,P78-112
[5] 薛定宇、陈阳泉著、基于 MATLAB/Simulink 的系统仿真技术与应用、北京:清华大学出版社、2002 P17-P62,P203-223
[6] 楼顺天、姚若玉、冶继民编著、基于 MATLAB 7.x 的系统分析与设计,西安电子科技大学出版社、1998 P1-P6,P139-P152
[7] 陈怀堔，吴大正，高西全编著、MATLAB 及在电子信息课程中的应用、北京:电子工业出版社、2002 P223-267
[8] 董景新、赵长德编著、控制工程基础、清华大学出版社、1992、P12-35
[9] 飞思科技产品研发中心编、MATLAB7 基础与提高、北京:电子工业出版社、2005 P115-134
[10] 颜庆津编著、数值分析、北京航空航天大学出版社、2004、P207-219
[11] 薛定宇 陈阳泉编著、高等应用数学问题的 MATLAB 求解、清华大学出版社、2004年 8 月 31 日、p84-214
[12] 高世臣编著、高等数学(二)、北京教育出版社、奥林匹克出版社、2001 年 6月第 3 版、p64-199
[13] 张忠平主编、李刚、夏德阳副主编、应用数学基础上册、辽宁人民出版社、2001年 8 月第上版、p62-88
[14]刘宝芹主编、沈雅芬副主编、计算机电路基础(1)、中央广播电视大学出版社、2000 年 7 月第一版、p8-14