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1. Stefan B Williams amp Michael Jakuba March 2010 Updated April 2013 by Ian Manchester AMME3500 Systems Control Lab 1 System Identification Torsional Plant Note The objective of this lab is to familiarize you with techniques for identifying system parameters This lab contributes 25 towards your mark for assignment 2 This lab will be undertaken during Week 4 or 5 in the Control Systems lab in the EIE Building 522 At certain critical points these instructions ask you to discuss your results with your tutor Your tutor is invaluable for making sure you are on the right track To encourage you to use your tutor effectively your tutor has been provided with a set of questions designed to stimulate discussion and assess your grasp of the lab Your mark will be based on 1 your verbal answers to these questions and 2 a lab report prepared by your group and submitted with the second assignment This lab should take an average student 3 hours to complete plus 1 hour of preparation This document outlines experiments which identify the plant parameters using the ECP Systems Model 205a Torsional Plant This material is reproduced in large part from the appropriate lab manual which is available in the lab and on line for you to prepare yourself for the lab The safety portion of this manual Section 2 3 must be read and understood by any user prior to operating this equipment To become more familiar with these operations it is strongly recom
2. btained by double numerical differentiation is typically somewhat noisy however The student may want to verify this by observing the acceleration plot 6 1 3 6 1 4 6 1 5 16 Save any files or plots of interest Exit the executive program and power down the system Hardware Gain Calculation Software DAC Amplifier Motor Drive pulley Plant rotational position rad encoder counts Fig 6 1 2 Open loop system block diagram Consider the block diagram shown in Fig 6 1 2 The so called hardware gain Kh of the system is comprised of the product knw kckaktkpkeks 6 1 6 where kc the DAC gain 10V 32 768 DAC counts ka the Servo Amp gain to be determined approx 2 A V k the Servo Motor Torque constant to be determined approx 0 1 N m A kp the Drive Pulley ratio 3 N m disk N m Motor ke the Encoder gain 16 000 pulses 2x radians ks the Controller Software gain 32 controller counts encoder or ref input counts 8 In Step 15 we obtained the acceleration 6 counts s of a known inertia J Jm J41 with a known voltage applied at the DAC Thus by neglecting the relatively small friction Applied Torque J 6 J76 ke 6 1 7 and we have a direct measurement of the four term product kgkkekp i e 1 00V kakkp Applied Torque in Step 15 6 1 8 TIt contains software gain also This software gain kg is used to give higher controller internal numerical
3. double clicking on it 8 Remove the four masses from the third upper disk and repeat Steps 5 through 7 to obtain W for the unloaded disk If necessary repeat Step 3 to reduce the execution data sampling duration 9 Measure the reduction from the initial cycle amplitude X to the last cycle amplitude X for the n cycles measured in Step 8 Using relationships associated with the logarithmic decrement 6 1 2 a 32 1 nf Xo vl i Ca32 2am X find the damping ratio a32 and show that for this small value the approximations of Eq s 6 1 1 2 are valid Record the resulting plots and estimates for the damped frequency and natural frequency in your logbook and show your tutor 1 X gt ba f orsmall E3 10 Repeat Steps 5 through 9 for the lower disk disk 1 Here in Step 6 you will need to remove Encoder 3 position and add Encoder 1 position to the plot set up Hence obtain Ong gt ngj and E How does this damping ratio compare with that for the upper disk Record the results in your logbook and discuss any differences in the measured parameters with your tutor 11 Use the following information pertaining to each mass piece to calculate the portion of each disk s inertia attributable to the four masses for the d31 and d11 cases mass incl bolt amp nut 500g 5g dia 5 00 cm 0 02 cm Recall Moment of Inertia for a cylinder J 4 mr Parallel Axis Theorem J J md 12 Calling
4. en Loop Step and input 1 00 Volts 500 ms 2 reps Execute this open loop step via the Execute menu For higher open loop voltages this move may trip a software speed limit which disables the controller indicated by Limit Exceeded in the Controller Status box in the desk top Again to reset simply reselect Execute from the Execute menu Go to Set up in the Plot Data menu and select Encoder 1 velocity for plotting 15 Plot this data and observe four velocity profile segments with nominal shapes of linear increase constant acceleration constant zero acceleration linear decrease deceleration and constant Obtain the acceleration 0 counts s by carefully measuring the velocity difference and dividing by the time difference 500 ms through the positive sloped linear segment Repeat this for the negative sloped segment Calculate the average magnitude of the positive and negative accelerations for use in obtaining knw below Steps 11 and 12 may be done later away from the laboratory if necessary 3In plant configurations where a disk is used in the center location the inertia and damping parameters may be assumed to be the same as for the upper disk 4 Some small deceleration will exist due to friction SFor more precise measurement you may zoom in on this region of the plot using Axis Scaling in the Plotting menu 6 It is possible to read the accelerations directly by plotting Encoder 1 acceleration This data o
5. he approximations made in steps 7 and 9 to be valid D Explain the importance of accurately identifying the plant parameters and hardware gain in terms of how they will affect controller design
6. ile encoder data is collected to record this response Select OK after data is uploaded Select Set up Plot from the Plotting menu and choose Encoder 3 position then select Plot Data from the Plotting menu You will see the upper disk time response You can export data for later plotting or analysis with matlab via the Data Export Raw Data menu item note that this only saves the most recent recording Choose several consecutive cycles say 5 to 10 in the amplitude range between 100 and 1000 counts This is representative of oscillation amplitudes during later closed loop control maneuvers Much smaller amplitude responses become dominated by nonlinear friction effects and do not reflect the salient system dynamics Divide the number of cycles by the time taken to complete them being sure to take beginning and end times from the same phase of the respective harmonic cycles Convert the resulting frequency in Hz to radians sec This damped frequency wa approximates the natural frequency wn according to 6 1 1 w d Ora aa age f orsmall fas y Cas where the d31 subscript denotes disk 3 trial 1 Note that we will assume that the damping ratio j3 is small for this configuration We will estimate the damping ratio j32 in step 9 Close the graph window by clicking on the left button in the upper right hand corner of the graph This will collapse the graph to icon form where it may later be brought back up by
7. mended that the user read Chapter 2 in its entirety prior to undertaking the operations described here Remember here as always it is recommended to save data and control configuration files regularly to avoid undue work loss should a system fault occur This lab gives a procedure for identifying the plant parameters applicable to Eqs 5 1 1 through 5 1 8 in the user manual The approach will be to use certain fundamental properties of lightly damped second order systems to indirectly measure the inertia spring and damping constants of the plant by making measurements of the plant while set up in a pair of classical spring mass configurations Preparation a In your own words explain the purpose s of the experiment from an engineering point of view b Derive the equations of motion for the 1DOF torsional spring mass system with friction damping shown below and find the transfer function Simulate the unforced natural response using Matlab to an initial displacement of 20 Determine the natural frequency from the plot Use the following parameters i J 0 01 kgm ii k 2 5 Nm rad iii D 0 002 N rad s T t Ot c D k c Now draw a free body diagram of the various configurations shown in Fig 6 1 1 and used for the purposes of system identification Assume that the bearings in the system add viscous friction Based on your diagrams derive the equa
8. resolution and improves encoder pulse period measurement for very low rate estimates 8The controller counts are the counts that are actually operated on in the control algorithm i e The system input trajectory counts and encoder counts are multiplied by 32 prior to control law execution Use 6 1 6 through 6 1 8 to solve for knw using the specified values for k and ks Record this value in your log book and show your tutor This hardware gain quantity will be important in the design of controllers to be undertaken in future labs so do not lose this value Report Prepare a report outlining the findings of your lab This should be prepared as a group and submitted along with your first assignment Only one report needs to be submitted as long as you identify the apparatus you are using and all group members by name and SID The report should include the preparation work you did prior to the lab and describe the steps taken as outlined above showing the results from each step Include plots of the output where appropriate In addition address the following questions as part of your report A Report the measured properties and derived parameter values Parameters Second Order Model Values Jin Jai Ja3 Cal Ca3 kay ka3 khw B How do your values of Jq and cg compare with Jg3 and cq3 Can you explain why they may be different C What were your values for Gan and Ca Are these values sufficiently small for t
9. rom C L af shaft toC L af each mass Middle disk clam ped 4 masses om top and bottom disks Eecoder 1 Eecoder 1 4 masses om top and bottom disks Encoder 5 Eecede 1 a Set up To Begin Plant Identfi cation Procedure b Setup To Identify Disk Moment of Inertia c Set up for Khw gain Figure 6 1 1 Configurations For Plant Identification Model 205a shown Go to Set up Data Acquisition in the Data menu and select Encoder 1 and Encoder 3 as data to acquire and specify data sampling every 2 two servo cycles i e every 2 T s Select OK to exit Select Zero Position from the Utility menu to zero the encoder positions Select Execute from the Command menu Switch on the Normal Data Sampling checkbox Prepare to manually displace the upper disk approximately 20 deg Exercise caution in displacing the inertia disk displacements beyond 40 deg may damage and possibly break the flexible drive shaft Displacements beyond 25 deg will trip a software limit which disables the controller indicated by Limit Exceeded in the Controller Status box in the Background Screen To reset simply reselect Execute from the Command menu With the upper disk displaced approximately 20 deg lt 1000 encoder counts as read on the Background Screen display in either direction select Run from the Execute box and release the disk approximately 1 second later The disk will oscillate and slowly attenuate wh
10. this inertia Jm i e that associated with the four masses combined use the following relationships to solve for the unloaded disk inertia Jg3 and upper torsional shaft spring kg3 You may zoom the plot via Axis Scaling for more precise measurement in various areas For an even greater precision the data may be examined in tabular numerical form see Export Raw Data Section 2 1 7 3 ka3l Im Ja3 Onazi ka3 Ja3 Onaz2 Find the damping coefficient cg3 by equating the first order terms in the equation form s242 Zas2t c Is k J Repeat this for the lower unloaded disk inertia this includes the reflected inertias of the motor belt and pulleys spring and damping Jai Cai and kg respectively 2 Discuss your result with your tutor Now all dynamic parameters have been identified Values for J and J2 for any configuration of masses may be found by adding the calculated inertia contribution of the masses to that of the unloaded disk The following is necessary to establish the hardware gain for control modeling purposes Procedure 13 Remove the entire upper disk unfasten the mid shaft disk and clamp and replace the four masses on the lower disk only at the 9 0 center distance following the guidelines of Step 2 Verify that the masses are secure and that the disk rotates freely 14 In the Trajectory window deselect Unidirectional moves i e enabling bi directional inputs select Step Set up Choose Op
11. tions of motion for the system Do the upper and lower disks interact As part of this lab you will perturb the upper disk and observe the response Then you will remove the weights and observe the modified response What parameters can be identified from these two responses You will execute the same procedure on the lower disk Which parameters if any will be different Procedure l Obtain a set of masses springs and the lab manual from your tutor For Model 205a clamp the center disk to put the mechanism in the configuration shown in Figure 6 l la using the 1 4 bolt square nut and clamp spacer Only light torqueing on the bolt is necessary Secure four 500g masses on the upper and lower disks as shown in Fig 6 1 1 a Verify that the masses are secured per Section 2 2 2 and that each is at a center distance of 9 0 cm from the shaft center line Open the ECP program Start menu ECP ECP32 With the controller powered up enter the Control Algorithm box via the Set up menu and set T 0 00442 Enter the Command menu go to Trajectory and select Step Set up Select Open Loop Step and input a step size of O zero a duration of 4000 ms and I repetition Exit to the Background Screen by consecutively selecting OK This puts the controller board in a mode for acquiring 4 sec of data on command but without driving the actuator This procedure may be repeated and the duration adjusted to vary the data acquisition period 9 0m f

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