(MagLev) Student Handout
Contents
1. Kb Ky_b Asim Figure 9 Diagram used for the Real Time Implementation of the PIV Closed Loop with MultiQ PCI As mentioned in the pre lab assignments the ball position loop is based on the coil PI current controller as developed and tuned in the previous section The actual PI control scheme is depicted in Figure 6 above Similarly the position controller diagram also directly interfaces with your MAGLEV hardware as shown in Figure 7 above To familiarize yourself with the diagram it is suggested that you open the model subsystems to get a better idea of their composing blocks as well as take note of the I O connections You should also check that the signal generator block properties are properly set to output a square wave signal of amplitude 1 and of frequency 0 25 Hz It should be noted that a simple low pass filter of cut off frequency 80 Hz set by tau xb is added to the ball position sensor output signal in order to attenuate its high frequency noise content Moreover the ball vertical velocity is estimated by differentiating the ball position analog signal Therefore to get around potential noise problems a second order filter of cut off frequency 100 Hz set by wcf and damping ratio 0 9 set by etaf is also introduced after the differentiated velocity signal Although introducing a short delay in the signals low pass filtering allows for higher controller2. 8 6 6 Assignment 5 Ball Position Controller Design Pole Placement 10 TMP TEA Procedur sued baton toO RD Vin adr A veu UT aces NBN 13 7 1 Experimental Setup And WICIIS uestrae Eon iri e E eo uper EN E Nx Tea RU dU ILE REPRE 13 7 2 Real Time Implementation Tuning Of The PI Current Control Loop 13 ps WM MUNI CUNY EB TT 13 1 2 2 Experimental Procedure ue oes peo ven a RUP ep sux bM ubi URL P pesce vae ERa 13 7 3 Real Time Implementation Tuning Of The Feedforward Plus PIV Position Control Eis m 17 ENERO CLIVE M HN 17 1 3 2 Experimental Progedute sisipin aiea E UEM PRSE MR NERR SQ VEU K IO E 17 7 4 Resulting System s Actual Closed Loop Poles sssseeeenee 22 pu RC c M REE 22 7 4 2 Experimental PEOeSUUEossnoriteo Ye noo IO ue HERR b UNUM ens 22 Appendix A Nomenclature us cssee see vva ttes ex Nt phu Une PANE aE UON R US PER QN GE MPO ERA PUN YE pU ER UE VON Eu ais 23 Document Number 526 Revision 03 Page i Magnetic Levitation Control Laboratory Student Handout 1 Objectives The MAGLEV plant as illustrated in Figure 1 is an electromagnetic suspension system acting on a solid one inch steel ball It mainly consists of an electromagnet located at the upper part of the apparatus capable of lifting from its pedestal and sustaining in free space the st
3. ity PIV closed loop scheme with the addition of a feedforward action as illustrated in Fig ure 4 below As depicted in Figure 4 the current feedforward action is characterized by r gp Ke p X aes 9 and uc 10 As it can be seen in Figure 4 the feedforward action is necessary since the PIV control sys tem is designed to compensate for small variations a k a disturbances from the linearized operating point xw Lo In other words while the feedforward action compensates for the ball gravitational bias the PIV controller compensates for dynamic disturbances eK es J Xd F d x b des Q a cl c_des G s b Figure 4 Ball Position PIV plus Feedforward Control Loop The open loop transfer function G s takes into account the dynamics of the electromagnet current loop as characterized in Assignment 4 and is defined as shown below Document Number 526 Revision 03 Page 10 Magnetic Levitation Control Laboratory Student Handout KAS GA F s and GT Gs 11 c_des where x Gs TY 12 However for sake of simplicity we neglect in the following the dynamics of the electromagnet current loop In this analysis of the ball position PIV plus feedforward control loop as presented hereafter it is therefore assumed that D I r0 i e T s 1 13 Moreover due to the presence of the feedforwar
4. Revision 03 Page 7 Magnetic Levitation Control Laboratory Student Handout sa a and oe 5 Answer the following questions 1 Linearize the ball s EOM found in Assignment 2 about the quiescent operating point Xbo Lo Hint For a function f of two variables x and I a first order approximation for small variations at a point xp I Xs Lo is given by the following Taylor s series approximation fx 1 fo I as f x L x x are f x 12 ed ql 2 Determine from the previously obtained linear equation of motion the system s open loop transfer function in the Laplace domain as defined by the following relationship o iG G s I s 7 cl Express the open loop transfer function DC gain Kba and natural frequency as functions of X and Lo Is the system stable What are its order and its type As a remark it is obvious that linearized models such as the MAGLEV current to position transfer function are only approximate models Therefore they should be treated as such and used with appropriate caution that is to say within the valid operating range and or conditions However for the scope of this lab Equation 7 is assumed valid over the steel ball entire range of motion T 6 5 Assignment 4 Coil Current Controller Design Pole Placement Prior to control the steel ball position the current flowing through the electromagnet needs to be controlled The electromagnet current co
5. beforehand that the CONTROLLER TYPE flag is set to MANUAL This file initializes all the MAGLEV model parameters and user defined configuration variables needed by the Simulink diagram Given a set a closed loop pole locations pea and po you are now in a position to calculate the PI controller gains K and Ki from the equations you derived in pre lab Assignments 1 and 4 Step3 According to the PI controller design specifications the current response should be over damped i e no overshoot and relatively fast The zero steady state error re quirement should be satisfied by the controller Integral action Accordingly closed loop pole locations are chosen on the real axis such as shown below rad rad Pa 0 3 a and Po 188 4 Ed 16 Step4 From your results obtained in pre lab Assignments 1 and 4 calculate the PI con troller gains K and K satisfying Equation 16 Have your lab assistant check your values With his or her approval you can now enter your calculated values for K and K in the Matlab workspace by following the Matlab notations used for the controller gains as presented in Table A 2 of Appendix A You are now ready to go Document Number 526 Revision 03 Page 15 Magnetic Levitation Control Laboratory Student Handout ahead with running the actual PI current controller Step5 Build the real time code corresponding to your diagram by using the WinCon Build option from the Simulink
6. configure the experimental setup entirely yourself you should be at least completely familiar with it and understand it If in doubt refer to References 1 2 3 and or 4 The first task upon entering the lab is to ensure that the complete system is wired as fully described in Reference 1 You should be familiar with the complete wiring and connections of your MAGLEV system If you are still unsure of the wiring please ask for assistance from the Teaching Assistant assigned to the lab When you are confident with your connections you can power up the UPM The MAGLEV inside chamber should light up You are now ready to begin the lab 7 2 Real Time Implementation Tuning Of The PI Current Control Loop 7 2 1 Objectives El To tune through pole placement the PI controller for the actual electromagnet current El To implement in real time with WinCon the PI control loop for the actual MAGLEV coil current El To run the obtained PI current controller and compare the actual response against the controller design specifications 7 2 2 Experimental Procedure Please follow the steps described below Stepl If you have not done so yet you can start up Matlab now Depending on your sys tem configuration open the Simulink model file of name type q pi maglev ZZ mdl where ZZ stands for either for mq3 mqp q5 or nie Ask the TA assigned to this lab if you are unsure which Simulink model is to be used in the lab You should o
7. gains in the closed loop system and Document Number 526 Revision 03 Page 18 Magnetic Levitation Control Laboratory Student Handout therefore for higher performance Lastly your model sampling time should be set to 1 ms i e T 10 s and the solver type to ode4 Runge Kutta Step 2 Before being able to run the actual control loop the feedforward and PIV controller gains must be calculated and initialized in the Matlab workspace since they are to be used by the Simulink controller diagram Keep the PI current controller gain values as previously implemented As it has been seen in pre lab assignment 5 the quiescent feedforward term is added to L to compensate for the ball weight and to bring it to its operating position Evaluate the feedforward gain as determine in Assignment 5 so that the design requirements are satisfied Step 3 The controller integral action is included to eliminate any static steady state error However in the case of the MAGLEV it also compensates for the thermal drift in the coil force characteristics due to heating by electrical dissipation Furthermore it can be shown that the velocity or derivative action of the controller is crucial for the stability of the ball equilibrium position In other words damping appears to be critical in stabilizing the system This stability consideration can justify the design requirement of having an over damped position response Accordingly closed loop
8. menu bar After successful compilation and download to the WinCon Client you should see the green START button available on the Win Con Server window You can now start your PI control loop in real time by clicking on the START STOP button of the WinCon Server window Step6 In order to observe the system s real time responses from the actual system open the following WinCon Scope Jc Resp A You should now be able to monitor on the fly the actual coil current as it tracks the pre defined reference input as well as the corresponding control effort Hint 1 To open a WinCon Scope click on the Scope button of the WinCon Server window and choose the display that you want to open e g Ic Resp A from the selection list Hint 2 For a good visualization of the actual current response you should set the WinCon Scope buffer to 15 seconds Do so by using the Update Buffer menu item from the desired WinCon Scope Step7 Assess the actual performance of the current response and compare it to the design requirements Measure your response actual rise time Are the design specifications satisfied Explain If your current response does not meet the desired design specifications of Section Controller Design Specifications on page 3 review your PI gain calculations and or alter the closed loop pole locations until they do If you are still unable to achieve the required performance level ask your T A for advice Hint In order to acc
9. pole locations are chosen to lie on the real axis such as shown below rad rad an rad p 7 2 5 ER and pco a P43 51 6 ka 17 s s Step 4 Given the set of desired closed loop pole locations poi po and pis presented by Equation 17 calculate the PIV plus feedforward position controller gains Kg Ky v Ks and K from the equations you derived in pre lab Assignment 5 Have your lab assistant check your values With his or her approval you can now enter your calculated values for Kr Kp Ky and Ki in the Matlab workspace by following the Matlab notations used for the controller gains as presented in Table A 2 of Appendix A You are now ready to go ahead with running the actual PIV plus feedforward current controller Step 5 If absent place the steel ball on the pedestal inside the MAGLEV chamber Step 6 Build the real time code corresponding to your diagram by using the WinCon Build option from the Simulink menu bar After successful compilation and download to the WinCon Client you should see the green START button available on the WinCon Server window You can now start your feedforward plus PIV control loop in real time by clicking on the START STOP button of the WinCon Server window Step 7 Clicking on the START button should levitate the ball halfway between the post and the electromagnet core face with an airgap of xo If the ball is swaying left and right you can gently put your hand in and dampen
10. the motion Once the ball is stabilized at the operating point change the value of the Simulink constant block named 0 Tracking OFF 1 Tracking ON from 0 to 1 The ball should start tracking the desired 1 mm square wave setpoint around the operating position Document Number 526 Revision 03 Page 19 Magnetic Levitation Control Laboratory Student Handout Step 8 In order to observe the system s real time responses from the actual system open the three following WinCon Scopes xb Resp mm Ic Resp A and V Command V scope located for example in the following subsystem path MAGLEV System Actual Plant MAGLEV Actual Plant You should now be able to monitor on line as the ball moves the actual ball position coil current and command voltage proportional to the control effort sent to the power amplifier as the ball tracks the pre defined reference input Hint To open a WinCon Scope click on the Scope button of the WinCon Server window and choose the display that you want to open e g xb Resp mm from the selection list Step 9 Assess the actual performance of the ball position response and compare it to the design requirements Measure the response actual settling time Are the design specifications satisfied Explain If your position response does not meet the desired design specifications of Section Controller Design Specifications on page 3 review your feedforward and PIV gain calculations and or alter the
11. 526 Revision 03 Page 21 Magnetic Levitation Control Laboratory Student Handout Step 12 Include in your lab report your final values for Kg Kp Ki and Ky as well as the resulting response plot of x versus X as and the corresponding plots of I and Veommand Describe the ball position variations a k a jitter about the desired ball position as well as its instantaneous and static steady state errors Describe the corresponding current and voltage behaviours Is there any saturation in the system Ensure to properly document all your results and observations before moving on the the next section Step 13 You can now proceed to the next section 7 4 Resulting System s Actual Closed Loop Poles 7 4 1 Objectives m To numerically determine the system s actual closed loop poles by considering the coil current control system s dynamics previously neglected B To carry out block diagram reduction using some of the Matlab s Control System Toolbox functions 7 4 2 Experimental Procedure Please follow the steps described below Step 1 We now consider the dynamics of the electromagnet current loop contrary to our initial simplifying assumption of Equation 13 Using your expression for T s derived in Assignment 4 evaluate with Matlab G s as defined in Equation 11 Step 2 Let us define T s as the MAGLEV rig overall closed loop transfer function It consists of the feedforward plus PIV position control sch
12. Plant Quanser MAGLEV as represented in Figure 1 above El Real Time Control Software The WinCon Simulink RTX configuration as detailed in Reference 4 or equivalent For a complete and detailed description of the main components comprising this setup please refer to the manuals corresponding to your configuration 4 2 Wiring To wire up the system please follow the default wiring procedure for your MAGLEV as fully described in Reference 1 When you are confident with your connections you can power up the UPM The MAGLEV inside chamber should then light up 5 Controller Design Specifications In the present laboratory i e the pre lab and in lab sessions you will design and implement two control strategies in order to successfully regulate and track the ball position in mid air 5 1 Coil Current Controller Specifications The first closed loop system is to control the electromagnet coil current via the commanded coil voltage It is based on a Proportional plus Integral PI scheme In response to a 0 to 1 A square wave coil current setpoint tune the PI current controller in order to satisfy the following design performance requirements 1 The current response should be over damped i e no overshoot 2 Have no steady state error 3 Have a maximum rise time t o less than 0 35 seconds i e p E093 s 5 2 Ball Position Controller Specifications The second and last control strategy is to regulate and track in m
13. Specialty Plants Specialty Experiment PIV plus Feedforward C3 LJ ZX IM amp ERA INN OMVATTE ED UGA TE Control Magnetic Levitation MagLev Student Handout Magnetic Levitation Control Laboratory Student Handout Table of Contents IE C P 1 PM cT M 4 By toig Tc F P 2 4 Experimental SEDDEL os ox eate eso pM etu N a a REE A O E Na 2 4 1 Miti COmpoefils anos es rbenicexic t iian peti AER EEEE TEA S EE OE VERE 2 LUE Dl ETC E 3 a Controller Design Sp cificatiONSisssss asirai eiar ieaiai s iita UN 3 5 1 Coil Current Controller Specifications isssiiessisasaedsssssavnssssnsanctsienasvevsaancconsecasonnensdibansess 3 5 2 Ball Position Controller Specifications esses 3 b Pie Labo ASSIEUBIO S sses rsrsrsr X MEE EGOA DOR XP EEES EREE sugars tog eed 5 6 1 MagLev System Representation and Notations cccceccceesseeeeeeeeeeeceeseecseeeeeeeeees 3 6 2 Assignment 1 Electrical System Modelling senes 6 6 3 Assignment Z2 Electro Mechanical System Modelling Non Linear Equation Of Moton EOM CP 6 6 4 Assignment 3 Electro Mechanical System Modelling EOM Linearization and Transfer Fri Cit e 7 6 5 Assignment 4 Coil Current Controller Design Pole Placement
14. b tain a diagram similar to the one shown in Figure 5 below The model implements a Proportional plus Integral PI closed loop whose gains need to be calculated and ini tialized The actual PI control scheme is similar to the one described in Assignment 4 as illustrated in Figure 6 below To run your actual electromagnet current it di Document Number 526 Revision 03 Page 13 Magnetic Levitation Control Laboratory Student Handout rectly interfaces with your MAGLEV hardware as shown in Figure 7 below To fa miliarize yourself with the diagram it is suggested that you open the model subsys tems to get a better idea of their composing blocks as well as take note of the I O con nections You should also check that the signal generator block properties are properly set to output a square wave signal of amplitude 1 and of frequency 0 1 Hz The total current setpoint should result to be a square wave between 0 and 1 A zig xi File Edit View Simulation Format Tools Help WinCon Magnetic Levitation MAGLEV Experiment Coil Current Loop PI Controller Current Setpoint A lc Resp e Current Positive Setpoint Setpoint Coil Current Setpoint Amplitude A Limit A MAGLEV System Actual Plant Figure 5 Real Time Implementation of the PI Coil Current Control Loop And Setpoint Generation 2lq_pi_maglev_map MAGLE System Actual Plant a ci x File Edit View Simulation Format Tools Help WinCon MAGLEV Curr
15. closed loop pole locations until they do If you are still unable to achieve the required performance level ask your T A for advice Hint In order to accurately measure the position response settling time from your WinCon Scope plot you can first select Freeze Plot from the WinCon Scope Update menu and then reduce the window s time interval to for example 1 second by opening the Set Time Interval input box through the Scope s Axis Time menu item You should now be able to scroll through your plotted data Step 10 What are your final feedforward and PIV controller gain values Once your results are in agreement with the closed loop requirements your position response should look similar to the one displayed in Figure 10 below Note Fluctuations in the position signal can be due to actual fluctuations of the ball vertical position but also to swaying due to the hemispheric surface of the ball Document Number 526 Revision 03 Page 20 Magnetic Levitation Control Laboratory Student Handout ET q piv maglev mqp xb Resp mm 0 File Edit Update Axis Window Background Colour Text Colour Text Font Figure 10 Actual Ball Position Tracking Response Step 11 From the same run the corresponding coil current is displayed in Figure 11 LEJA q_piv_maglev_map Ic Resp A 1 Ele Edt Update Axis Window Background Colour Test Colour Text Font Figure 11 Actual Coil Current Response Document Number
16. d loop we have G s G s 14 where G s is as defined is Assignment 3 Answer the following questions 1 Analyze the ball position closed loop system at the static equilibrium point xo Ico and determine the current feedforward gain Ke 2 Using the MAGLEV current to position transfer function G s determined in Assignment 3 derive the ball position closed loop transfer function defined by Equation 15 below and illustrated by Figure 4 Ts x UN 15 b Xy des s l Hint 1 Block diagram reduction can be carried out Hint 2 T s should be a function of the feedforward gain Kr as well as of the PIV position controller gains K Ki and K 3 What is the normalized characteristic equation of the obtained electro mechanical system 4 We wish to place the closed loop poles of the position loop at the three following Document Number 526 Revision 03 Page 11 Magnetic Levitation Control Laboratory Student Handout locations psi po and po What would then be the system s desired characteristic equation as a function of pui po and po 5 Determine K Ki and Ky as functions of pui po and pis so that the desired closed loop pole locations for the position loop are achieved Document Number 526 Revision 03 Page 12 Magnetic Levitation Control Laboratory Student Handout 7 In Lab Procedure 7 1 Experimental Setup And Wiring Even if you do not
17. eel ball Two system variables are directly measured on the MAGLEV rig and available for feedback They are namely the coil current and the ball distance from the electromagnet face A more detailed description is provided in Reference 1 During the course of this experiment you will become familiar with the design and pole placement tuning of both PI current controller and PIV plus Feedforward ball position controller The challenge of the present laboratory is to levitate a one inch solid steel ball in air from the pedestal using an electromagnet The control system should maintain the ball stabilized in mid air and track the ball position to a desired trajectory Figure 1 The Magnetic Levitation Experiment At the end of the session you should know the following El How to mathematically model the MAGLEV plant from first principles in order to obtain the two open loop transfer functions characterizing the system in the Laplace domain El How to linearize the obtained non linear equation of motion about the quiescent point of operation Document Number 526 Revision 03 Page 1 Magnetic Levitation Control Laboratory Student Handout El How to design through pole placement a Proportional plus Integral PI controller for the MAGLEV electromagnet current in order for it to meet the required design specifications El How to design through pole placement a Proportional plus Integral plus Velocity PIV contr
18. eme applied to G s Using some of the Matlab s Control System Toolbox functions carry out block diagram reduction in order to evaluate T s which is the system s actual closed loop transfer function Hint The Matlab Control System Toolbox functions of interest include tf series parallel feedback Step 3 Use some of the Matlab s Control System Toolbox functions to calculate the system s actual closed loop zeros and closed loop poles Hint The Matlab Control System Toolbox functions of interest include zero pole Step 4 Compare the actual closed loop poles with their design value counterparts Explain Step 5 You can now move on to writing your lab report Ensure to properly document all your results and observations before leaving the laboratory session Document Number 526 Revision 03 Page 22 Magnetic Levitation Control Laboratory Student Handout Appendix A Nomenclature Table A 1 below provides a complete listing of the symbols and notations used in the MAGLEV and Ball mathematical modelling as presented in this laboratory The numerical values of the system parameters can be found in Reference 1 Symbol Description Units Matlab Notations Ls Coil Inductance mH Lc R Coil Resistance Q Re N Number Of Turns in the Coil Wire Nc Ic Coil Core Radius m rc Rs Current Sense Resistance Q Rs Tb Steel Ball Radius m rb Mb Steel Ball Mass kg Mb Tp Steel Ball Travel m Tb g Grav
19. ent Control Loop Coil Current PI Controller wots PI Current Controller 52 2 zetafwotstwot2 xb_dot m s Derivative Filter Ki c WALS MAX IC WDUP Current Setpoint A j 1 Air Sep em C ran m Coil Current Low P ass Filter le des A o des A K_o CWA UPM Voltage Limit le A MAGLEV Actual Plant Figure 6 Real Time Implementation of the PI Current Control Loop Document Number 526 Revision 03 Page 14 Magnetic Levitation Control Laboratory Student Handout q_pi_magley_mqp MAGLE System Actual Plant MAGLE Actual Plant k Jn ni x File Edit View Simulation Format Tools Help WinCon Interface to the Actual MAGLEV System V Command V Qua sul MultiQ PCI DAC DAC Limit Analog Output 0 To UPM Driving the Coil Air Gap Between Core and Ball xb m Quanser Consulting MultiQ PCl ADC Sensed Current Analog Inputs 80 2 s 7 1 Ball Position 2 3 Current Sensing lee Figure 7 Interface Subsystem to the Actual MAGLEV Plant Using the MultiQ PCI Card Step2 Before being able to run the actual control loop the PI controller gains must be calculated and initialized in the Matlab workspace since they are to be used by the Simulink controller diagram Start by running the Matlab script called setup lab ma glev_piv m However ensure
20. id air the ball position per se The closed loop scheme employed consists of a Proportional plus Integral plus Velocity PIV controller with a feedforward component Document Number 526 Revision 03 Page 3 Magnetic Levitation Control Laboratory Student Handout The first specification is to design the ball position controller for the following operating position a k a equilibrium position xw X 7 6 mm In response to a desired 1 mm square wave position setpoint from the ball equilibrium position in mid air the ball position behaviour should satisfy the following design performance requirements 1 Have an Percent Overshoot PO less than 15 i e PO lt 15 2 Have no static steady state error 3 Have a maximum settling time t less than 1 0 second 1 e t S lt 1 0 s 4 Minimize the control effort produced which is proportional to the coil input voltage Ve The power amplifier e g UPM should not go into saturation in any case Document Number 526 Revision 03 Page 4 Magnetic Levitation Control Laboratory Student Handout 6 Pre Lab Assignments 6 1 MagLev System Representation and Notations A schematic of the Magnetic Levitation MAGLEV plant is represented in Figure 2 below The MAGLEV system s nomenclature is provided in Appendix A As illustrated in Figure 2 the positive direction of vertical displacement is downwards with the origin of the global Cartesian frame of coo
21. ined Feedforward plus PIV position controller and compare the actual response against the controller design specifications B To investigate the effect of the nested PI current control loop on the system s closed loop poles 7 3 2 Experimental Procedure Please follow the steps described below Step 1 Depending on your system configuration open the Simulink model file of name Document Number 526 Revision 03 Page 17 Magnetic Levitation Control Laboratory Student Handout type g_piv_maglev_ZZ mdl where ZZ stands for either for mq3 mqp q8 or nie Ask the TA assigned to this lab if you are unsure which Simulink model is to be used in the lab You should obtain a diagram similar to the one shown in Figure 9 below The model implements a Feedforward plus Proportional plus Integral plus Velocity PIV closed loop as described in Assignment 5 The four ball position controller gains need to be calculated and initialized in Matlab as they are used by the Simulink diagram 73a piv maglev map Elle Edi view Simulation Format Tools Help WinCon Magnetic Levitation MAGLEV Experiment Ball Position Loop Feed Forward PIV Controller Coil Current Loop PI Controller Ball Position Setpoint m Ki b Almis MAX XB WDUP Kp_b Aim 1 Kit tf tau eset E
22. itational Constant on Earth m s g Ho Magnetic Permeability Constant H m mu0 Ks Ball Position Sensor Sensitivity m V K B Fe Electromagnet Force N Fe F Gravity Force N Fg Km Electromagnet Force Constant N m A Km I Actual Coil Current A Ic Ve Actual Coil Input Voltage V Ve Vs Current Sense Voltage V Vs Vo Ball Position Sensor Voltage V Vb Xb Actual Air Gap Between Core Face and Ball Surface m xb a k a Steel Ball Vertical Position d Steel Ball Vertical Velocity m s xb_dot ar Xo Steady State Air Gap m xb0 Document Number 526 Revision 03 Page 23 Magnetic Levitation Control Laboratory Student Handout Symbol Description Units Matlab Notations Steady State Coil Current Small Variation Around the Steady State Air Gap Small Variation Around the Steady State Coil Current Desired Coil Current Desired Air Gap Feedforward Coil Current Table A 1 MAGLEV System Model Nomenclature Table A 2 below provides a complete listing of the symbols and notations used in the design of both control loops ie PI and PIV plus Feedforward as presented in this laboratory Symbol Description Matlab Simulink Notation trc Coil Current Rise Time ts Ball Position Settling Time Ge Open Loop Coil Current Transfer Function Gc Keac Coil Open Loop DC Gain Kc dc Tb Coil Open Loop Time Constant tau c Te Closed Loop Coil Current Transfer Function Te Gb Open Loop Ball Position Transfer Function Gb Kba Ball Open Loop DC Gain Kb dc O
23. ntrol loop consists of a Proportional plus In tegral PI closed loop scheme as illustrated in Figure 3 below Document Number 526 Revision 03 Page 8 Magnetic Levitation Control Laboratory Student Handout K c r S L es d V L a K G s gt Figure 3 PI Current Control Loop Answer the following questions 1 Using the coil voltage to current transfer function G s determined in Assignment 1 derive the coil current closed loop transfer function defined by Equation 8 below and characterized by Figure 3 LG Ts 8 I a s Hint 1 Block diagram reduction can be carried out Hint 2 T s should be a function of the current PI controller gains Kp o and K e What is the normalized characteristic equation of the electrical system We wish to place the closed loop poles of the current loop at the two following locations pa and po What would then be the system s desired characteristic equation as a function of pa and po Determine K and Ki as functions of p and po so that the desired closed loop pole locations for the current loop are achieved Document Number 526 Revision 03 Page 9 Magnetic Levitation Control Laboratory Student Handout 6 6 Assignment 5 Ball Position Controller Design Pole Placement The steel ball position is controlled by means of a Proportional plus Integral plus Veloc
24. oller with feedforward action for the MAGLEV levitated ball position in order for it to meet the required design specifications m How to implement your two controllers in real time and evaluate their actual performances El How to numerically determine the system s actual closed loop poles by considering the coil current control system s dynamics 2 Prerequisites To successfully carry out this laboratory the prerequisites are i To be familiar with your MAGLEV plant main components e g mechanical design actuator sensors your power amplifier e g UPM and your data acquisition card e g MultiQ as described in References 1 2 and 3 ii To be familiar in using WinCon to control and monitor the plant in real time and in designing their controller through Simulink as detailed in Reference 4 iii To be familiar with the complete wiring of your MAGLEV specialty plant as per dictated in Reference 1 3 References 1 MAGLEV User Manual 2 Data Acquisition Card User Manual 3 Universal Power Module User Manual 4 WinCon User Manual 4 Experimental Setup 4 1 Main Components To setup this experiment the following hardware and software are required EJ Power Module Quanser UPM 2405 or equivalent El Data Acquisition Board Quanser MultiQ PCI MQ3 Q8 NI E Series or equivalent Document Number 526 Revision 03 Page 2 Magnetic Levitation Control Laboratory Student Handout m MAGLEV
25. p Ball Open Loop Natural Frequency wb To Closed Loop Ball Position Transfer Function Tb Gm MAGLEV Rig Open Loop Transfer Function Gm Ta MAGLEV Rig Overall Closed Loop Transfer Function Tm Kpc Current Proportional Gain Kp c Ki Current Integral Gain Ki c Document Number 526 Revision 03 Page 24 Magnetic Levitation Control Laboratory Student Handout Symbol Description Matlab Simulink Notation Current Dominant Closed Loop Pole Current Second Closed Loop Pole Ball Position Proportional Gain Ball Position Velocity Gain Ball Position Integral Gain Ball Position Feed Forward Gain Ball Position Dominant Closed Loop Pole Ball Position Second Closed Loop Pole Ball Position Third Closed Loop Pole Continuous Time Table A 2 Control Loops Nomenclature Document Number 526 Revision 03 Page 25
26. rdinates on the electromagnet core flat face as represented in Figure 2 Although the ball does have six Degrees Of Freedom DOF in free space only the vertical i e x axis is controlled It can also be seen that the MAGLEV consists of two main systems an electrical and an electro mechanical I Electromagnet j 1 Figure 2 Schematic of the MAGLEV Plant Document Number 526 Revision 03 Page 5 Magnetic Levitation Control Laboratory Student Handout 6 2 Assignment 1 Electrical System Modelling Assignment 1 derives the mathematical model of your MAGLEV electrical system The resulting model will provide you with the open loop transfer function G s which in turn will be used to design an appropriate controller Equation 1 defines the Laplace open loop transfer function G s of the MAGLEV coil voltage to the coil current G s is defined as shown below I s VG 1 G s Answer the following questions 1 As represented in Figure 2 above the MAGLEV coil has an inductance and a resistance Additionally the actual system is equipped with a current sense resistor in series with the coil and whose voltage Vs can be measured using the data acquisition board Analog To Digital converter The coil current can then be computed using the following relation ship V t lt 2 s Using the MAGLEV equivalent electrical circui
27. t depicted in Figure 2 derive the differ ential equation governing the current I flowing through the actual electromagnet coil as a result of the applied voltage V 2 From the previously obtained differential equation determine the electrical system trans fer function as defined in Equation 1 Express the open loop transfer function DC gain Kcae and time constant Te as functions of the basic electrical components Is the system stable What are its order and its type 6 3 Assignment 2 Electro Mechanical System Modelling Non Linear Equation Of Motion EOM Answer the following questions 1 Using the notations and conventions described in Figure 2 above derive the Equation Of Motion EOM of the MAGLEV electro mechanical system Is the MAGLEV electro mechanical system s EOM linear Document Number 526 Revision 03 Page 6 Magnetic Levitation Control Laboratory Student Handout Hint 1 The attractive force F generated by the electromagnet and acting on the steel ball is assumed to be expressed as 2 K I m Jom for 0 lt x 3 Xp Equation 3 shows that the pull of the electromagnet is proportional to the square of the current and inversely proportional to the air gap a k a ball position squared Hint 2 The Newton s second law of motion can be applied to the steel ball Hint 3 Express the resulting EOM under the following format 2 or where f denotes a function x f
28. urately measure the response rise time from your WinCon Scope plot you can first select Freeze Plot from the WinCon Scope Update menu and then reduce the window s time interval to for example 0 1 seconds by opening the Set Time Interval input box through the Scope s Axis Time menu item You should now be able to scroll through your plotted data Step8 What are your final PI controller gain values Once your results are in agreement with the closed loop requirements your current response should look similar to the one displayed in Figure 8 below Step9 Include in your lab report your final values for K and Ki as well as the resulting response plot of I versus lI des Step10 You can now proceed to the next section which deals with the implementation in real time of your PIV plus Feedforward position controller on the actual MAGLEV plus Ball system Document Number 526 Revision 03 Page 16 Magnetic Levitation Control Laboratory Student Handout g Scope q pi maglev mqp Ic Resp A 0 File Edit Update Axis Window Figure 8 Actual Ic Response PI Current Control Loop 7 3 Real Time Implementation Tuning Of The Feedforward Plus PIV Position Control Loop 7 3 1 Objectives El To tune through pole placement the Feedforward plus PIV controller for the actual MAGLEV ball position El To implement in real time with WinCon the Feedforward plus PIV control loop for the actual ball position B To run the obta
29. x D 4 2 The nominal coil current Lo for the electromagnet ball pair can be determined at the sys tem s static equilibrium By definition static equilibrium at a nominal operating point Xwo Lo is characterized by the ball being suspended in air at a constant position xy due to a constant electromagnetic force generated by Lo Express the static equilibrium cur rent I as a function of the system s desired equilibrium position xo and its electromagnet force constant Km Using the system s specifications given in Reference 1 and the de sired design requirements evaluate Io For reference also express the electromagnet force constant Km as a function of the system s desired equilibrium point Xo Ico 6 4 Assignment 3 Electro Mechanical System Modelling EOM Linearization and Transfer Function In order to design and implement a linear position controller for our system the Laplace open loop transfer function can be derived However by definition such a transfer function can only represent the system s dynamics from a linear differential equation Therefore the EOM found in Assignment Z2 should be linearized around a quiescent point of operation In the case of the levitated ball the operating range corresponds to small departure positions xy small departure currents I from the desired equilibrium point xyo Ico Therefore x and I can be expressed as the sum of two quantities as shown below Document Number 526
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