外文翻译---一个教学用的磁悬浮控制系统模型.docx

Modeling of a Didactic Magnetic Levitation System for Control EducationMilica B. NaumoviCAbsruo The magnetic levitation control system of a metallicsphere is an interesting and visual impressive device successfulfor demonstration many intricate problems for controlengineering research. The dynamics of magnetic levitation systemis characterized by its instability, nonlinearity and complexity. Inthis paper some approaches to the levitation sphere modeling areaddressed, that may he validate with experimentalmeasurements.Keywords - magnetic levitation system, control engineeringeducation, system modelingI. INTRODUCTIONMagnetic levitators not only present intricate problems forcontrol engineering research, but also have many relevantapplications. such as high-speed transportation systems andprecision bearings. From an educational viewpoint, thisprocess is highly motivating and suitable for laboratoryexperiments and classroom demonstrations, as reported in theengineering education literature 1-8.The classic magnetic levitation control experiment isprescnted in the form of laboratory equipment given in Fig.1.The complete purchase of the Feedback Instruments Ltd.Maglev System 33-006 9 is supported by WUS (WorldUniversity Service IO) - Austria under Grant CEP (Centerof Excellence Projects) No. 115/2002. This attraction-typelevitator system is a challenging plant because of its nonlinearand unstable nature. The suspended body is a hollow steel ballof 25 mm diameter and 20 g mass. This results in a visuallyappealing system with convenient time constants. Both analogueand digital control solutions are implemented. In addition, thesyslem is simple and relatively small, that is portable.This paper deals with the dynamics analysis of the consideredmagnetic levitation system. Although the gap between the realphysical systcm and the obtained nominal design model hascomplex structure, it should be robust stabilized in spite of modeluncertainties.II. SYSTEMD ESCRIPTIONThe Magnetic Levitation System (Maglev System 33-006given in Fig. I ) is a relatively new and effective laboratory setupvery helpful for control experiments. The basic control goalis to suspend a steel sphere by means of a magnetic fieldcounteracting the force of gravity. The Maglev Systemconsists of a magnetic levitation mechanical unit (an enclosedMilica B. Nauinovic is with the Faculty of Electronic Engineering,University of NE, Beogradska 14. 18000 Nil. Yugoslavia, E-mail:nmilicaelfak.ni.ac.yumagnet system, sensors and drivers) with a computer interfacecard, a signal conditioning unit, connecting cables and alaboratory manual.In the analogue mode, the equipment is self-contained withinbuilt power supply. Convenient sockets on the enclosurepanel allow for quick changes of analogue controller gain andstructure. The bandwidth of lead compensation may bechanged in order to investigate system stability and timeresponse. Moreover, user-defined analogue controllers may beeasily tested. Note, that the ,position of the sphere may beadjusted using the set-point control and the stability may bevaried using the gain control. In the digital mode, the Maglev System operates withMATLAB /SlMULrNK software. Feedback Software forSIMULLNiKs p rovided for the control models and interfacingbetween the PC and the Maglev system hardware. The Maglev System, both in analogue and digital mode,allows the study of various control strategies and other issuesfrom system theory, as follows:Analogue mode. Nonlinear modelingSystem stabilizationInfrared sensor characteristicsClosed-loop identificationLead-lag compensationPerturbation sensitivityPDcontrol;Linearization about an operating pointDigital mode Nonlinear modelingSystem stabilizationLinearization about an operating pointA D and DIA conversionClosed-loop identificationPerturbation sensitivityState space PD controlPosition regulation and tracking controla.S YSTEM MODELINAGN D IDENTIFICATIONA schematic diagram of the single-axis magnetic levitationsystem with principal components is depicted in Fig. 2. Theapplied control is voltage, which.is converted into a currentvia the driver within the mechanical unit. The current passesthrough an electromagnet which creates the correspondingmagnetic field in its vicinity. The sphere is placed along thevertical axis of the electromagnet. The measured position isdetermined from an array of infrared transmitters anddetectors, positioned such that the infrared beam is intersectedby the sphere. .Using the fundamental principle of dynamics, thebehaviour of the ferromagnetic ball is given by the followingelectromechanical equation where m is the mass of the levitated ball, g denotes theacceleration due to gravity, x is the distance of the ball fromthe electromagnet, i is the current across the electromagnet,and f ( x , i ) is the magnetic control force.A. Calculating the magnetic control force on the metallicsphereConsider a solenoid with an r radius, an 1 length, crossedby an I current. The sphere is located on the axis of the coilas shown in Fig. 3. The effect of the magnetic field from theelectromagnetic is to introduce a magnetic dipole in the spherewhich itself becomes magnetized. The force acting on thesphere is then composed of gravity and the magnetic forceacting on the induced dipole.The magnetic control force between the solenoid and thesphere can be determined by considering the magnetic field asa function of the balls distance x from the end of the coil.The magnetic field at some given point (see Fig. 3), maybe calculated according to the Biot-Savar-Laplace formula l l . Recall, that the magnetic field produced by a smallsegment of wire, dl , canying a current I (see Fig. 4a) isgiven byWhere u0 is the permeability of the free space and d l x r isthe vector product of vectors dl and r .Hence, the magnitude of the magnetic field becomesThe magnetic field of a circular contour with an a radius, asshown in Fig. 4b, is given byNote, that from considerations of symmetry, the fieldcomponent perpendicular to the coil axis dB, must be zero onthe axis.In order to evaluate the field due to the many turns ( N )along the axis of the coil, let n be the number of turns permetre. Also, consider the solenoid given in Fig. 3 as a series ofequidistant circular contours at the mutual distances dx,canying the current n l d x . The total axial field from all turnsof the coil becomesIntegrating Eq. (5) within the interval 1 2, giveswhich can be rewritten in the formHowever, the electromagnet is composed of many turnslayers with variable radius r1 r r2. Using the foregoing, inconjunction with Eq. (7), the magnetic field becomesAnd so, the total magnetic field is given byThe magnetic force on the metallic sphere can beexpressed as I I where S is the material surface crossed by the magnetic flux.Finally, the upwards force on the ball due to the field B isgiven byAnother approach to the determination of the force on theball due to the field is based on the fact that the force isproportional both to the induced dipole strength and field strength. Namely, the magnetic field induces a dipole in the sphere whose strength is proportional to the magnetic field and that means to the coil current. Of course, it is necessary to make the assumption, that the sphere is magnetized within the linear part of the magnetization curve and does not reach saturation. sphere whose strength is proportional to the magnetic field andThe location of thedipole within the sphere issuch that each pole is atthe center of mass of its respective hemisphere（see FIG.5）The forces on thesphere due to the magneticfield are an attractive forceon the North pole and arepulsive one on the South pole. Hence, the upwards force on the ball due to the magneticfield B in conjunction with Eq. (9) is given by and hencewhere G(X) denotes the derivative and X is the dipole separation. Using assumption that X may he taken as a constant,because it is sufficiently small compared to the electromagnet geometry (radius and length), the magnetic force becomesB. Some practicable approximations of magnetic forcebetween the electromagnet and the sphere Note that the analytical expression of the magnetic force/current/displacement relationships (Eqs. (1 1) and (14) is very complex for the experimental purpose. However, the magnetic force characteristics may be xperimentally calibrated as a function of the applied current I and the hall position X . Namely, the complex nonlinear function G ( X ) can be approximate by a polynomial function In the equilibrium position we havewhich can be rewritten in the formThe experiment consists of resting the levitation metallic sphere on an xyz- stage capable of Imm incremental positioning and determining the minimum current required topick up the ball at various heights 12-131, 4. Then the model of the foreeldistance relationship can he determined by means of least square fitting. Note, that the validity of so obtained curve is limited to some range Xmon, X Xmax , ,Also, the solenoid characteristics change with temperature,and the coefficients of the curve-fit change significantly when the system has been in operation for a while. This drift,combined with other nonlinear effects, results in a dispersion such that the calibration data can deviate from the curve-fit by up to 10% in qxtreme conditions.However, many studies dealing with the modeling of the magnetic levitation system based on model linearization using Taylors series I, 5, 14-151. Recall that this approach is really restrictive, because it is only available when the system variations are small.IV. THE COIL INDUCTANCEThe inductance L(x) is a nonlinear function of the balls position x l, 14. It has the largest value when the bearing ball is next to the coil, and decreases to a constant value as the bearing hall is enough removed. Some typical approximations for L(x) are given byNote that over the range of x considered in the experiments Xmon, X Xmax one can pick the parameters L1（），Li1, Xi0 ，i = l , 2 , 3 in Eqs. (18) to (20) for mutual approach purpose 7,8.V. CONCLUSIONUsing the magnetic field to levitate a steel ball, the Magnetic Levitation System as a teaching aid enables thetheoretical study and practical investigation of basic and advanced approaches to control of nonlinear unstable systems.The equipment may operate in stand alone mode without the PC.In the digital mode system operates with MAT LAB /SMULNKso ftware. The scope of experimentation includes dynamics modeling, identification, analysis and various conlroller design using classical and modern methods. For the modeling of the considered didactic magnetic levitation systein a set of formulas has been established.ACKNOWLEDGEMEThis work has been supported by the World University Service - Austria and Ministry of Science, echnologies and Development of the Republic of Serbia.REFERENCESI T. H. Wong, Design of a Magnetic Levitation Control System -An Undergraduate Project, IEEE Trans. on Educurion, Vol. E-29, No.4, pp. 196-200, November 1986.2 W.G. Herley and W.H. Wolfle, Electromagnetic Design of a Magnetic Suspension System, IEEE Trans. on Educarion., Vol. E-40, pp. 124-130, 1997. .3 V.A. Oliveira, E.F. Costa and J.B. Vargas, Digital Implementation of a Magnetic Suspension Control System for Laboratory Experiments, IEEE Trans. on Education., Vol. E-42, pp. 315-322, 1999.4 A.EI Hajjaji and M. Ouladsine, Modeling and Nonlinear Control of Magnetic Levitation Systems, IEEE Trans. Ind.Electron., Vol. 48, No.4. pp. 831-838, August 2001.5 R. K. H. Galvlo, T. Yoneyama, F. M. Ugulino de Aralijo, and R.G. Machado, A Simple Technique for Identifying a Linearized Model for a Didactic Magnetic Levitation System. IEEE Tronr on Education, Vol. 46, No., pp. 22-25, February 2003.6 M. B. NaumoviC, %e Magnetic Levitation System - A Laboratory Equipment, TELFOR2002, ConferenceProceedings, pp. 604-605, Beograd, Yugoslavia, 2002(in Serbian).7 M. B. NaumoviC, Magnetic Levitation System Analysis A Laboratory Approach, INFOTEH-JAHORINA 2003,Symposium CD Proceedings, Vol. 3, Ref. A-1, pp. 1-5,Jahorina, Republica Srpska-BA, 2003 (in Serbian).8 M. B. NaumoviC, Modeling of a Didactic Magnetic Levitation System, presented ut Conference ETRAN2003, June 8-13,Herceg Novi, Serbia and Montenegro, 2003 (in Serbian).9 Feedback Instruments Limited J IO I 11 I. Surutka, Elecrromagnerics, Beograd, NauEna knjiga, 1978 (in Serbian).I2 D. Cho, Z. Kato, and D. Spilman, Sliding Mode and Classical Control of Magnetic Levitation System, IEEE Conrrol Sysrerns,CS-M, pp. 42-48, 1993.I3 Jing-Chung Shen, H- control and sliding mode control of magnetic levitation system, Asian Journal of Control, Vol. 4,No.3, pp. 3331340, 2002.I4 W. Barie and J. Chiassan. Linear and nonlinear state-space controllers for magnetic levitation, Inrernarionul Journal of SystemsScience, Vol. 27, No.11, pp. 1153-1163, 1996.I5 Namerikawa, T., and Fujita, M., Uncertainty Structure and p-Synthesis of a Magnetic Suspension System, T.IEE Japan, 121-C. 1080-1087, 2001.一个教学用的磁悬浮控制系统模型 这是一个有趣并且令人印象深刻的能演示许多复杂现象的磁悬浮控制系统。磁悬浮系统的三大特点就是具有不稳定性，非线型和复杂性。这个实验是单人就能完成的简单实验。介绍磁悬浮到目前为止不仅仅在控制工程的研究上展现了许多的优点，还在许多方面有而且还有许多的实用的地方。比如，高速运输系统和精密的轨道设计。从教育的观点看，这个过程是相当有利于实验室和课堂上的演示的。 典型的磁悬浮控制实验的装置如图1所示。全部的仪器都是在反馈仪器有限公司购买的。悬磁列车系统 33-006 9是由WUS (世界大学服务 IO)提供的-奥地利倪属Grant CEP (优秀工程中心) No. 115/2002提供的资金支持。这种吸引力式的磁悬浮系统由于其非线形和不稳定性所以很有挑战性。被悬挂的物体是一个25毫米直径，20 克重的钢球。这个结果随时间有很明显的变化。计算机和模拟控制系统都可以实现这个实验。另外，这个系统相对来说比较简单和体形小，便于携带。这篇文章涉及考虑磁悬浮系统的力学分析。尽管在真正的物理系统和这个实验中的模型有着很大的区别，但是这个模型还是能反映很多的物理现象。系统描述 磁悬浮系统(磁悬浮系统 33-006 如图1所示)是控制实验非常对有帮助的一次相对新和有效的实验室安装。基本的实验目的就是将通过一个磁场使刚球的重力与磁力相互抵消。磁悬浮系统是由有电脑界面的机械悬浮装置和一个信号发生器有电缆连接组成。在这个模拟装置上，需要用一个独立的电源供电。考虑到模拟器的迅速的变化控和结构，在周围请确定有更多的电源接口。带宽变化是为了调查系统稳定性随时间的改变。而且，用户自定义的模拟控制系统是能被很简单的测试的。球的位置可以因为装置的不同参数来控制。在数字化的方面，悬磁系统运行用MATLAB / SlMULrNK 软件。在电脑和磁悬浮系统连接的反馈软件为控制模型和接口技术的SIMULLNiKs。悬磁系统，在模拟和数字模式里，都允许各种各样的控制参数并根据理论根据来产生不同的实验结果。例如：1.模拟方式：非线性的模型，系统的稳定性因素，红外传感器特性，封闭环鉴定，延时补偿，扰动敏感性，PD控制；2.数字模式:非线性的模型,系统的稳定性因素,零界点的线形化，数/摸的转变，封闭环鉴定，扰动敏感性，PD控制，位置确定和跟踪控制。系统模型和鉴定 一个单轴的磁悬浮系统的主要结构如图2所示。应用控制力是被机械设备转化成为电流的电压。电流通过电磁体，然后在周围产生一个磁场。纲球被放在这个磁场的垂轴上。球的位置是有2个相互垂直的红外线传感器所测量出来。红外探测器被假顶在需要操作的范围内。 电流A和调节入电压U有关。使用力学的基本原理，钢球满足下面的电磁学公式：其中M表示小球的质量，G表示重力加速度，X表示小球距离电磁铁的位置，I是通过电磁铁的电流，F（X，I）表示的是磁力大小。A。 钢球上的磁力计算：有一r半径，长度为1的螺旋管，中间流过电流强度为I的电流。钢球被放先线圈的轴线上，如图3所示。磁场穿过钢球，使得钢球，形成一个磁偶极子 ，钢球被磁化了。 钢球的受力为重力和磁力。钢球所受到的磁力由它距离线圈的距离X所控制。在一些给定的点上的磁场强度(参阅图 3),可以根据Biot-Savar-Laplace 公式计算出来。回忆一下有一短通电导体所产生的磁场，如：其中 u是空气的介电常数。dl X r是dl 和 r的矢积。因此，磁场强度变为：磁场就成为一个图4所示的圆形磁场了。 注意，考虑到对称性，垂直于线圈的磁场，肯定是0。为了各个线圈方向上的磁场强度，用n 表示各个方向上距离线圈的位置（单位米）。另外，考虑到图3中的螺旋管的原形对称磁场，dx就用电流表示出来：nldx，各个方向上的线圈成：结合公式5代定系数，就有写成另一种形式因此，电磁铁在不同的方向上就有个不同的层次，R1RR2.结合公式7就得到了磁场分布：所以总磁场就成所以，钢球上的磁力就写成S表示钢球穿过磁场的有效面积，钢球上所受到的力的总和就写成磁偶极子和磁场所产生的力也一样作用的球体上。也就是说，电流通过线圈产生了一个磁偶极子。当然，可以做出假设，磁体的磁化并没有达到饱和状态。球体内部的磁偶极子就在球内的顺磁的方向的重心方向2个极点上。球内部的磁偶极子产生的磁场，在北极产生一个吸引力，在南极产生一个排斥力。因此，作用在球上的力由磁力B和公式（9）一起产生：因此假设6X是一个常量，因为相对于电磁铁，它足够的小（半径和长度），电磁力就写成：注意到，分析磁力/电流/位移表达式在实验中十分的复杂，磁力大小在实验上可以由电流大小I和距离X描述，线形函数G（X）就是这样的一个多项式。在水平方向，我们有写成另一个形式 实验由金属的悬浮组成，在能Imm增加一个xyz面上的领域Xmon, X Xmax确定的位置和确定，最小电流要求在各种各样的位置的小球确定 12-13 4。然后力矩的关系式能通确定最小平方的方法适合。意到，那如此的有效性获得的曲线局限于一些X范围，5 X 5 X 。此外，螺线管特性随温度而变，系统在运行中以后曲线系数也随之改变。结合其他的非线形效果，结果可以有弯曲程度最高的10%左右。电磁感应：电感L(x)是球的一个关于球位置XL（14）的一个非线性的函数，当承重球紧贴线圈的时候，它有最大的价值，对于减少球的测移也有很大的作用。 一些典型的距离L（X）给出：Xmon, X Xmax注意到实验中反复的使球通过测出距离s L1（），Li1, Xi0 ，i = l , 2 , 3参考文献I T. H. Wong, Design of a Magnetic Levitation Control System -An Undergraduate Project, IEEE Trans. on Educurion, Vol. E-29, No.4, pp. 196-200, November 1986.2 W.G. Herley and W.H. Wolfle, Electromagnetic Design of a Magnetic Suspension System, IEEE Trans. on Educarion., Vol. E-40, pp. 124-130, 1997. .3 V.A. Oliveira, E.F. Costa and J.B. Vargas, Digital Implementation of a Magnetic Suspension Control System for Laboratory Experim