Rotary Motion Servo Plant: SRV02. Rotary Experiment #11: 1-DOF Torsion. 1-DOF Torsion Position Control using QuaRC. Student Manual

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1 Rotary Motion Servo Plant: SRV02 Rotary Experiment #11: 1-DOF Torsion 1-DOF Torsion Position Control using QuaRC Student Manual

2 Table of Contents 1. INTRODUCTION PREREQUISITES OVERVIEW OF FILES PRE-LAB ASSIGNMENTS Modeling System Representation Nomenclature Equations of Motion State-Space Representation Identifying Stiffness and Damping Parameters Finding the Natural Frequency and Damping Ratio Control Design Control Specifications LQR Design IN-LAB PROCEDURES Position Control Simulation Setup for Position Control Simulation Simulating Closed-Loop System Position Control Implementation Setup for 1-DOF Torsion Implementation Nominal LQR Controller Parameter Identification Tuned LQR Controller...35 Document Number 729 Revision 1.0 Page i

3 6. RESULTS SUMMARY REFERENCES...40 Document Number 729 Revision 1.0 Page ii

4 1. Introduction The challenge of this experiment is to design a feedback controller to position the output of a rotational system with one torsional flexible coupling as fast as possible with minimum vibration. Such a system emulates torsional compliance and joint flexibility, which are common characteristics in mechanical systems such as high-gear-ratio harmonic drives or lightweight transmission shafts. The laboratory objective is to design a state-feedback controller for the rotary torsion module which allows you to command a desired load angle position via a flexible coupling. The controller should eliminate the load's vibrations while maintaining a fast response. Full-state and partial-state feedback control strategies are compared. The Linear Quadratic Regulator (LQR) tuning algorithm is used. Frequency tests are carried out and the system's resonance observed. The following topics are covered in this laboratory: Model the rotary 1-DOF torsional system from first-principles. Using a nominal model parameters, design a state-feedback control using Linear Quadratic Regulator (LQR). Tune the LQR control in simulation. Implement the state-feedback control on the actual 1-DOF Torsion device using QuaRC. Evaluate the different between partial-state feedback and full-state feedback. Identify the stiffness and the damping of the system. Re-design the control based on the newly found parameters. Implement the re-designed controller. 2. Prerequisites In order to successfully carry out this laboratory, the user should be familiar with the following: Data acquisition card (e.g. Q8), the power amplifier (e.g. UPM), and the main components of the SRV02 (e.g. actuator, sensors), as described in References [1], [4], and [5], respectively. Document Number 729 Revision 1.0 Page 1

5 Wiring and operating procedure of the SRV02 plant with the UPM and DAC device, as discussed in Reference [5]. Transfer function fundamentals, e.g. obtaining a transfer function from a differential equation. Laboratory described in Reference [6] in order to be familiar using QuaRC with the SRV02. Designing a state-feedback controller using the Linear Quadratic Regulator technique. 3. Overview of Files Table 1 below lists and describes the various files supplied with the experiment. File Name SRV02 1-DOF Torsion Student Manual.pdf Description This laboratory guide contains pre-lab and in-lab exercises demonstrating how to design and implement a position controller on the Quanser SRV02 1-DOF Torsion plant using QuaRC. setup_srv02_exp11_torsion1d.m The main Matlab script that sets the SRV02 1-DOF Torsion motor and sensor parameters as well as its configuration-dependent model parameters. Run this file only to setup the laboratory. config_srv02.m config_1d_torsion.m SRV02_Torsion1DOF_ABCD_e qns calc_conversion_constants.m d_gui_lqr_tuning.m s_1d_torsion.mdl q_1d_torsion.mdl Returns the configuration-based SRV02 model specifications Rm, kt, km, Kg, eta_g, Beq, Jeq, and eta_m, the sensor calibration constants K_POT, K_ENC, and K_TACH, and the UPM limits VMAX_UPM and IMAX_UPM. Returns the 1-DOF Torsion stiffness, moment of inertia, and damping parameters: Ks, J2, and B2. Linear state-space matrices of 1-DOF Torsion. Returns various conversions factors. Used to generate the control gain K by tuning the Q and R LQR weighting matrices on-the-fly. Simulink file that simulates the partial and full-state feedback closed-loop controllers for the 1-DOF Torsion system. Simulink file that implements the partial and full-state feedback closed-loop controllers on the 1-DOF Torsion system using QuaRC. Document Number 729 Revision 1.0 Page 2

6 File Name Description Table 1: Files supplied with the SRV02 1-DOF Torsion experiment. 4. Pre-Lab Assignments 4.1. Modeling System Representation The SRV02 1-DOF Torsion schematic is depicted in Figure 1. Notice that the positive rotation is in counter-clockwise direction. Thus when a positive voltage is applied to the motor, a positive torque is generated and both the servo and the torsion load move counter-clockwise. Document Number 729 Revision 1.0 Page 3

7 Figure 1: Schematic of SRV02 1-DOF Torsion system Nomenclature Table 2, below, provides a complete listing of the symbols and notations used in the 1-DOF Torsion system mathematical modelling. Symbol Description Matlab / Simulink Notation V m SRV02 Motor Armature Voltage Vm R m Motor Armature Resistance Rm k t Motor Torque Constant kt η m Motor Efficiency eta_m k m Back-ElectroMotive-Force (EMF) Constant km K g Planetary Gearbox Gear Ratio Kg η g Planetary Gearbox Efficiency eta_g B 1 Equivalent viscous damping of SRV02 load. B1 B 2 Equivalent viscous damping of 1-DOF Torsion load. B2 J 1 Equivalent Moment of Inertia with torsion load as seen at the SRV02 Load Shaft. J1 Document Number 729 Revision 1.0 Page 4

8 Symbol Description Matlab / Simulink Notation J 2 Equivalent Moment of Inertial as seen at the 1-DOF Torsion load shaft. θ 1 SRV02 Load Shaft Angular Position theta_1 θ 2 Torsion Load Shaft Angular Position theta_2 τ 1 Load torque generated by SRV02 motor. tau_1 Ks Stiffness of flexible coupling. Ks k Control gain k Table 2: 1-DOF Torsion Model Nomenclature J Equations of Motion In this section, the equations that describe the angular position of the SRV02 and Torsion load shaft with respect to the torque applied will be developed. The 1-DOF Torsion equations of motion, or EOMs, can be represented by a set of linear equations of the form d 2 J + + = dt 2 q( t ) B d q( t ) K q( t ) τ [1] dt where J is the system inertia, B is the damping, K is the stiffness, τ is the applied torque, and q are generalized coordinates. For multi-axis systems, the terms J, B, and K would be matrices and q a vector. For the 1-DOF Torsion we would have two equations of the form: d 2 d = dt 2 θ 1 ( t ) g 1 θ ( ),,, 1 t θ 2 ( t ) θ ( ) dt 1 t τ 1 [2] and d 2 = dt 2 θ 2 ( t ) d g 2 θ ( ),, 1 t θ 2 ( t ) dt θ 2 ( t ), where g 1 and g 2 are functions that represent the angular acceleration of the SRV02 load shaft and the 1- DOF Torsion load shaft, respectively. The acceleration of the SRV02 depends on the torque being applied, τ 1, the angle of both the servo, θ 1, and the torsion module, θ 2, and the velocity of the SRV02 gear (due to damping). The 1-DOF Torsion and SRV02 systems are coupled through the flexible member. So in Equation [2], the angle of the torsion shaft will be introduced through the stiffness caused by the flexible coupling between the systems. Similarly in Equation [3], the SRV02 angle affects the acceleration of the torsion shaft through the flexible member. [3] Document Number 729 Revision 1.0 Page 5

9 Go through these exercises to obtain the EOMs of the 1-DOF Torsion plant: 1. Using Equation [1] and the schematic shown in Figure 1 develop an equation that describes the motions of SRV02 load angle θ 1 with respect to the applied torque τ 1. Make sure symbols defined in Table 2 are used. 2. Similarly, give the equation that describes the position of the 1-DOF Torsion shaft, θ 2. Make sure symbols defined in Table 2 are used State-Space Representation The linear state-space equations are given by = t x A x + B u [4] and Document Number 729 Revision 1.0 Page 6

10 y = C x + D u. [5] By expressing the dynamics of a device in this form, it makes it easy to use algorithm such as Linear- Quadratic Regulator or Ackermann's Pole Placement to design a control gain. Given the state vector x T = [ x 1, x 2, x 3, x 4 ] [6] we define the position states as { x 1 = θ 1 ( t ), x 2 = θ 2 ( t )} [7] and the velocities as d d { x 3 = θ ( ), } dt 1 t x 4 = θ ( ) dt 2 t. Recall that these variables are defined in Table 2. The (A,B) matrices are obtained from the EOMs developed in Section As for the output equation, only the position measurements of the SRV02 and Torsion load shaft angles are available and measurement noise will be neglected. Thus the output equation can be written x 1 y = [9] x 2. Go through these exercises to obtain the linear state-space representation of the 1-DOF Torsion plant: 1. Express the equations of motion obtained in Section in terms of the state and solve for the acceleration terms. [8] Document Number 729 Revision 1.0 Page 7

11 2. Enter the first two rows of the state-space matrix A elements in the table below. Document Number 729 Revision 1.0 Page 8

12 Matrix A Element A[1,1] Expression A[1,2] A[1,3] A[1,4] A[2,1] A[2,2] A[2,3] A[2,4] Table 3: State-space matrix A: first and second row elements. 3. Complete Table 4, below, with the third row state-space matrix A elements. Document Number 729 Revision 1.0 Page 9

13 Matrix Element A[3,1] Expression A[3,2] A[3,3] A[3,4] Table 4: State-space matrix A: third row elements. 4. Enter the fourth row elements of matrix A in Table 5. Matrix Element Expression A[4,1] A[4,2] A[4,3] A[4,4] Table 5: State-space matrix A: fourth row elements. 5. Enter matrix B in Table 6,below. Document Number 729 Revision 1.0 Page 10

14 Matrix Element B[1] Expression B[2] B[3] B[4] Table 6: State-space matrix B. 6. Fill the table below with the C matrix values. Matrix Element Expression C[1,1] C[1,2] C[1,3] C[1,4] C[2,1] C[2,2] C[2,3] C[2,4] Table 7: State-space matrix C. 7. Fill Table 8 with the matrix D elements. Document Number 729 Revision 1.0 Page 11

15 Matrix Element D[1] Expression D[4] Table 8: State-space matrix D Identifying Stiffness and Damping Parameters The theoretical results derived in this assignment will be used in your in-lab session to finely estimate your system parameter values K s and B 2 from your own experimental data. Answer the following questions: 1. Consider the Torsion module by itself without the actuated servo unit. This is a linear onedimensional spring-damper-mass system with stiffness parameter, K s, viscous damping parameter B 2, and moment of inertia J 2. Determine the equation of motion of the torsion module when in freeoscillation, i.e. when perturbed and non-actuated. 2. You should have found that the ordinary differential equation (ODE) describing the free-oscillatory motion of the torsion load is a second-order differential equation. Assuming the zero initial velocity, d θ ( ) = dt [10], show that the equation can be written in the format of a Laplace polynomial with its denominator in the form of the standard characteristic equation, s ζ ω n s + ω 2 n [10]. Document Number 729 Revision 1.0 Page 12

16 3. Assume that ω n and ζ can be determined from experimental measurements and J 2 is already known. Therefore, determine from your findings in the two relationships giving K s and B 2 as functions of the experimentally measured system parameters ζ, ω n, and J 2. Document Number 729 Revision 1.0 Page 13

17 Finding the Natural Frequency and Damping Ratio Consider the free response of a second-order system in Figure 2 after an impulse. Figure 2: Typical free response to an impulse Natural Frequency The period of the oscillations in a system can be found using t n + 1 t 1 T osc = [11] n where t n is the time of the n th oscillation, t 1 is the time of the first peak, and n is the number of oscillations considered. From this, the damped natural frequency (in radians per second) is 2 π ω d = T [12] osc and the undamped natural frequency is ω d ω n = 1 ζ 2. [13] Document Number 729 Revision 1.0 Page 14

18 Damping Ratio For a typical second-order underdamped system, the subsidence ratio (a.k.a. decrement), χ, is defined as O k + 1 χ = [14] O k. This gives an measure of how quickly the oscillations decay. In order to determine the overshoot of the k th oscillation, the equation k ζ π 1 ζ 2 [15] O k = x ss e. can be used. The x ss variable is the steady-state value of the response. Go through these exercises to find the damping ratio given the overshoots: 1. Using equations [14] and [15], express the subsidence ratio in terms of the damping ratio. 2. Solve for the damping ratio. Document Number 729 Revision 1.0 Page 15

19 Example Assume the following measurements are taken from the response shown in Figure 2, above. The first and second oscillation peaks are: O 1 = 21.0 [ deg ] [16] and O 2 = 16.0 [ deg ]. [17] The time of the first peak is t 1 = 1.12 [ s ] [18] And the time of the 5 th oscillation overshoot is t 5 = 1.71 [ s]. [19] 1. Find the damping ratio Document Number 729 Revision 1.0 Page 16

20 2. Find the period of this system using the measurements given. 3. Find the undamped natural frequency. Document Number 729 Revision 1.0 Page 17

21 4. Given the moment of inertia parameter for the load of the torsion module, J 2, in Reference [9], find the stiffness and damping of the device with that resulting response Control Design Control Specifications The time-domain specifications for controlling the position of the 1-DOF Torsion load shaft are: Document Number 729 Revision 1.0 Page 18

22 e ss 0.5 [ deg ], [7] t s [ s], and [8] PO 12.5 ["%"]. [9] Thus when tracking the load shaft reference, the transient response should have an overshoot of less than or equal to 12.5 %. The response should settle within 4% of its final value in less than or equal to seconds and its final value should be within ±0.5 degrees of the desired final position LQR Design A system is deemed as being controllable if its poles can be placed at any desired location via statefeedback. One method of determining if a system is controllable is called the rank test: rank( C o ) = n. [20] The matrix C o is called the controllability matrix. For a four-state system it is computed by C o = [ B, A B, A 2 B, A 3 B ] [21]. If the rank of the controllability matrix equals the amount of states, i.e. if n = 4, then the system is controllable and a state-feedback control can be designed. Assuming (A,B) is controllable, a control gain can be computed using the Linear-Quadratic Regular (LQR) optimization method. For the user-defined weighting matrices Q and R, LQR finds a signal u(t) that minimizes the cost function J = 0 x( t ) T Q x( t ) + u( t ) T R u( t ) dt. With the state-feedback control u = K x [23] LQR will compute a gain K that minimizes the J expression. For the 1-DOF Torsion system, the weighting matrices will be chosen as q q Q = [24] 0 0 q q 4. [22] Document Number 729 Revision 1.0 Page 19

23 and R = 200. [25] Generally speaking if R is kept constant and the diagonal elements in the Q matrix are increased then LQR will work harder to minimize J and the gains generated will be larger. Instead, matrix R can be varied. To generate a larger control gain, decrease the value of R while keeping Q constant. This way the algorithm must work harder against the smaller R value to minimize J and will yield a larger control gain. 5. In-Lab Procedures This section shows how to simulate the position control of the 1-DOF Torsion system in Simulink and implement the designed feedback system on the actual device using QuaRC Position Control Simulation Before going through the simulation procedure in Section 5.1.2, go through Section to configure the setup Matlab script properly Setup for Position Control Simulation Follow these steps to configure the lab properly: 1. Load the Matlab software. 2. Browse through the Current Directory window in Matlab and find the folder that contains the controller files, e.g. s_torsion_1d.mdl. 3. Double-click on the s_torsion_1d.mdl file to open the Simulink diagram shown in Figure Double-click on the setup_srv02_exp11_torsion_1d.m file to open the setup script for the position control Simulink models. 5. Configure setup script: The controllers will be ran on an SRV02 in the high-gear configuration with the 1-DOF Torsion module load. In order to simulate the system properly, make sure the script is setup to match this configuration, i.e. the EXT_GEAR_CONFIG should be set to 'HIGH' and the LOAD_TYPE should be set to 'TORSION_1DOF'. Also, ensure the ENCODER_TYPE, TACH_OPTION, K_CABLE, UPM_TYPE, and VMAX_DAC parameters are set according to the SRV02 system that is to be used in the laboratory. Next, set CONTROL_TYPE to 'LQR_TUNING'. 6. Run the script by selecting the Debug Run item from the menu bar or clicking on the Run button in the tool bar and the GUI Turning window shown in Figure 3 should load. Document Number 729 Revision 1.0 Page 20

24 Figure 3: GUI for tuning LQR matrices with initial parameters. 7. Select the Save parameters to a text file option and click on the Apply button. The messages shown in Text 1, below, should be generated in the Matlab Command Window. This is the initial control gain generated and it is based on the nominal model parameters of the system loaded. Trial # 1 : Q & R have been saved to the file "lqr_tuning_logfile.txt". Trial # 1 : K(1) = N.m/rad K(2) = N.m/rad K(3) = N.m.s/rad K(4) = N.m.s/rad Text 1: Display in Matlab prompt after running setup script and applying initial gain Simulating Closed-Loop System The s_torsion_1d Simulink diagram shown in Figure 4 is used to simulate the closed-loop position response of the 1-DOF Torsion when using a state-feedback gain. The Discretized State-Space block from the Simulink\QuaRC Targets library is used to simulate the system using the state-space matrices A, B, C, and D, that are loaded in Matlab. The Control Gain gain block contain the control gain parameter k, which can be generated using LQR. The System Timebase block forces the simulation to run in actual time. Document Number 729 Revision 1.0 Page 21

25 Figure 4: Simulink model used to simulate 1-DOF Torsion system. Follow these steps to simulate the 1-DOF Torsion response: 1. In the Setpoint subsystem, ensure the Signal Type field in the Signal Generator block is set to square in order to generate a step reference. 2. Also make sure the Frequency of the Signal Generator is set to 0.5 Hz 3. Set the Amplitude (deg) gain block to 10.0 degrees to generate a step with this amplitude. 4. Ensure the Manual Switch is set to Full-State Feedback mode, which is in the upward position. 5. Open the scope showing the angular position of the SRV02 load shaft, theta1 (deg), and the torsion load shaft, theta2 (deg), as well as the motor input voltage scope, Vm (V). 6. Start the simulation. By default, the simulation runs for 5 seconds. The scopes should be displaying responses similar to figures 5, 6, and 7. Note that in the theta1 (deg) and theta2 (deg) scopes, the yellow trace is the setpoint position while the purple trace is the simulated position. Document Number 729 Revision 1.0 Page 22

26 Figure 5: SRV02 load angle with initial control gain. Figure 6: Torsion load angle with initial control gain. Figure 7: Voltage when using initial control. 7. The task now is to tune the Q and R matrices in order to get a response that adheres to the specifications listed in Section without saturating the amplifier. To do this, change one of the Q elements, click on the Apply button, and re-run the simulation. Alternatively, you can set the simulation time of s_torsion_1d to inf so it continues to run continuously as you tune the gain through the GUI. Although not necessary, you can choose to tune the R matrix if it makes it easier to achieve the desired specifications. 8. Both the full-state feedback, i.e. FSF, and partial-state feedback, i.e. PSF, controllers will be ran on the actual device. Thus it's important to ensure the gains are reasonable and that the actuator is not saturated under both modes. Set the Manual Switch to Partial-State Feedback mode, which is the downward position, run the simulation and ensure the voltage is within the amplifier limits, i.e. ±15 V. 9. Once you have achieved a suitable response in both the FSF and PSF modes, generate a Matlab Document Number 729 Revision 1.0 Page 23

27 figure showing the full-state feedback position response and the input voltage. After each simulation run, each scope automatically saves their response to a variable in the Matlab workspace. For instance, the theta1 (deg) scope saves its response to the variable called data_theta1 and the Vm (V) scope saves its data to the data_u variable. The data_theta1 variable has the following structure: data_theta1(:,1) is the time vector, data_theta1(:,2) is the setpoint, and data_theta1(:,3) is the simulated angle. Similarly for the data_theta2 variable. For the data_vm variable, data_vm(:,1) is the time and data_vm(:,2) is the simulated input voltage. 10. Measure the steady-state error, the percentage overshoot, and the peak time of the simulated Document Number 729 Revision 1.0 Page 24

28 response. Does the response satisfy the specifications given in Section without saturating the servo motor? 11. Record the weighting matrices and the corresponding control gain generated that were used for the response. This will be called the nominal LQR control gain as it is based on the nominal model parameters. 12. Generate a Matlab figure showing the partial-state feedback position response and the input Document Number 729 Revision 1.0 Page 25

29 voltage. 13. If the specifications are satisfied without saturating the servo motor, proceed to the next section to implement the controller Position Control Implementation Section explains how to configure and setup Matlab properly before going through any of the Document Number 729 Revision 1.0 Page 26

30 proceeding sections. Follow the directions given in Section to run the nominal LQR controller, which was found in Section 5.1.2, on the actual SRV02 1-DOF Torsion system. To experimentally determine the stiffness of the flexible coupling and the damping of the torsion load, go through Section Once the actual parameters of the device are found, the LQR can be redesigned and the tuned controller implemented as dictated in Section Setup for 1-DOF Torsion Implementation Before beginning the in-lab exercises on the SRV02 device, the q_torsion_1d.mdl and the setup_srv02_exp11_torsion1d.m script must be configured. Follow these steps to get the system ready for this lab: 1. Setup the SRV02 in the high-gear configuration without the load as described in Reference [5]. 2. Load the Matlab software. 3. Browse through the Current Directory window in Matlab and find the folder that contains the torsion control files, e.g. q_torsion_1d.mdl. 4. Double-click on q_torsion_1d.mdl to open the 1-DOF Torsion Position Control Simulink diagram. 5. Configure DAQ: Double-click on the HIL Initialize block inside the SRV02 1-DOF Torsion subsystem and ensure it is configured for the DAQ device that is installed in your system. By default, the block is setup for the Quanser Q8 hardware-in-the-loop board. See Reference [6] for more information on configuring the HIL Initialize block. 6. Configure setup script: Set the parameters in the setup_srv02_exp11_torsion_1d.m script according to your system setup. See Section for more details Nominal LQR Controller The q_torsion_1d Simulink diagram shown in Figure 8 is used to perform the position control exercises in this laboratory. The SRV02 1-DOF Torsion subsystem contains QuaRC blocks that interface with the DC motor and sensors of the 1-DOF Torsion system. Document Number 729 Revision 1.0 Page 27

31 Figure 8: Simulink diagram used to run LQR controller on 1-DOF Torsion system using QuaRC. Follow the steps below: 1. Run the setup_srv02_exp11_torsion1d.m script. 2. Reload the nominal LQR control gain found in Section In the Setpoints subsystem, set Signal Type in the Signal Generator to square to generate a step reference. 4. Set the Amplitude (deg) gain block to 10.0 degrees to generate a step with this amplitude. 5. Set the Frequency of the Signal Generator to 0.5 Hz. 6. Set the Manual Switch to the Full-State Feedback mode, (i.e. the upward position). 7. Open the SRV02 and Torsion load shaft position scopes, i.e. theta1 (deg) and theta2 (deg), as well as the motor input voltage scope, Vm (V). 8. Click on QuaRC Build to compile the Simulink diagram. 9. Select QuaRC Start to begin running the controller. The scopes should be displaying responses similar to figures 9, 10, and 11. Note that in the position scopes, the yellow trace is the setpoint, the blue plot is the simulation, and the the purple trace is the measured position. Document Number 729 Revision 1.0 Page 28

32 Figure 9: SRV02 Angle. Figure 10: Torsion shaft angle. Figure 11: Input voltage. 10. When a suitable response is obtained, click on the Stop button in the Simulink diagram tool bar (or select QuaRC Stop from the menu) to stop running the code. Generate a Matlab figure showing the response and its input voltage. Attach it to your report. As in the simulation diagram, when the controller is stopped each scope automatically saves their response to a variable in the Matlab workspace. Document Number 729 Revision 1.0 Page 29

33 11. Measure the steady-state error, the settling time, and the percentage overshoot of the Torsion load gear. Does the response satisfy the specifications given in Section 4.2.1? Document Number 729 Revision 1.0 Page 30

34 12. Now set the Manual Switch to Partial-State Feedback (i.e. the downward position) and run the controller. 13. Similarly as in the FSF case, make a Matlab figure showing the angular position responses and the input voltage. Document Number 729 Revision 1.0 Page 31

35 14. Make sure QuaRC is stopped. 15. Shut off the power of the UPM if no more experiments will be performed on the SRV02 in this session Parameter Identification In order to identify the stiffness of the flexible coupling and the damping of the torsion load, the system is ran in partial-state feedback mode and the angular position of the torsion load is captured. Once that is done, the same procedure performed in Section can be used to find the actual stiffness and damping of the actual system. 1. Run the system in partial-state feedback mode, as was just performed in Section Plot the position of the torsion load angle and attach it to your report (no need for the SRV02 Document Number 729 Revision 1.0 Page 32

36 angle and voltage). 3. Calculate the damping ratio. Document Number 729 Revision 1.0 Page 33

37 5. Find the period of this system using the measurements given. 6. Find the undamped natural frequency. Document Number 729 Revision 1.0 Page 34

38 7. With the torsion load moment of inertia, J 2, given in Reference [9], find the stiffness and damping of the 1-DOF Torsion system. Compare these values with the nominal parameter given in Reference [9]. 8. Make sure QuaRC is stopped. 9. Shut off the power of the UPM if no more experiments will be performed on the SRV02 in this session Tuned LQR Controller By loading the newly found stiffness and damping parameters, the LQR control gain can be redesigned Document Number 729 Revision 1.0 Page 35

39 to get better performance. 1. Enter the stiffness and viscous damping parameters identified in Section as variables Ks and B2, respectively, in the setup_srv02_exp11_torsion1d.m script right before the SRV02_Torsion1DOF_ABCD_eqns.m script is ran, as shown in Text 2, below. % Load estimated parameters instead (uncomment following lines to do so) % Ks = 1.00; % B2 = 0.004; % Load 1-DOF Torsion Linear State-Space Model SRV02_Torsion1DOF_ABCD_eqns; Text 2: Loading estimated parameters for new control design. 2. Run the script with the same Q and R matrices used in Section This loads a new statespace model with your estimated parameters and, since the LQR is based on the (A,B) matrices, generates a new control gain. 3. Run q_torsion_1d in full-state feedback mode. Document Number 729 Revision 1.0 Page 36

40 4. If needed, you can tune the Q and R matrices to obtain a better response. Once complete, click on the Stop button in the Simulink diagram tool bar (or select QuaRC Stop from the menu) to stop running the code. Generate a Matlab figure showing the angular positions of the SRV02 and the Torsion load as well as the SRV02 input voltage using the tuned LQR controller. 5. Measure the steady-state error, the settling time, and the percentage overshoot of the Torsion load gear. Does the response satisfy the specifications given in Section 4.2.1? Document Number 729 Revision 1.0 Page 37

41 6. Record the weighting matrices and the corresponding control gain generated that were used for the response. This will be called the tuned LQR control gain as it is based on the experimentally found model parameters and, perhaps, some tuning of the weighting matrices Document Number 729 Revision 1.0 Page 38

42 6. Results Summary Fill out Table 9, below, with the pre-laboratory and in-laboratory results obtained. Section Description Symbol Value Unit Pre-Lab: Model Parameters 1. Damping ratio ζ 3. Undamped natural frequency ω n rad/s 4. Stiffness K s N.m/rad 4. Viscous damping B 2 N.m.s/rad In-Lab Simulation: Nominal LQR Controller 10 Steady-state error e ss deg 10 Settling time t s s 10 Percentage overshoot PO % 11 Q weighting matrix: Q(1,1) diagonal element q 1 11 Q weighting matrix: Q(2,2) diagonal element q 2 11 Q weighting matrix: Q(3,3) diagonal element q 3 11 Q weighting matrix: Q(4,4) diagonal element q 4 11 R weighting matrix R 11 SRV02 Proportional gain k 1 N.m/rad 11 Torsion Proportional gain k 2 N.m/rad 11 SRV02 Derivative gain k 3 N.m.s/rad 11 Torsion Derivative gain k 4 N.m.s/rad In-Lab Implementation: Nominal LQR Controller 11 Steady-state error e ss deg 11 Settling time t s s 11 Percentage overshoot PO % In-Lab Implementation: Parameter Identification 4. Damping ratio ζ 6. Undamped natural frequency ω n rad/s Document Number 729 Revision 1.0 Page 39

43 7. Stiffness K s N.m/rad 7. Viscous damping B 2 N.m.s/rad In-Lab Implementation: Nominal LQR Controller 16 Steady-state error e ss deg 16 Settling time t s s 16 Percentage overshoot PO % 17 Q weighting matrix: Q(1,1) diagonal element q 1 17 Q weighting matrix: Q(2,2) diagonal element q 2 17 Q weighting matrix: Q(3,3) diagonal element q 3 17 Q weighting matrix: Q(4,4) diagonal element q 4 17 R weighting matrix R 17 SRV02 Proportional gain k 1 N.m/rad 17 Torsion Proportional gain k 2 N.m/rad 17 SRV02 Derivative gain k 3 N.m.s/rad 17 Torsion Derivative gain k 4 N.m.s/rad Table 9: SRV02 Experiment #11: 1-DOF Torsion position control results summary. 7. References [1] Quanser. Q4/Q8 User Manual. [2] Quanser. QuaRC User Manual (type doc quarc in Matlab to access). [3] Quanser. QuaRC Installation Manual. [4] Quanser. UPM User Manual. [5] Quanser. SRV02 User Manual. [6] Quanser. SRV02 QuaRC Integration Instructor Manual. [7] Quanser. Rotary Experiment #1: SRV02 Modeling. [8] Quanser. Rotary Experiment #2: SRV02 Position Control. [9] Quanser. SRV02 Torsion User Manual. Document Number 729 Revision 1.0 Page 40

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