Rotary Experiment #04: BB01 Control Ball and Beam Position
Contents
1. Page 29 SRVO02 Ball and Beam Control Laboratory Student Manual 4 The third order characteristic equation is 2 2 0 5 t0 2 1 T 5 38 where T is the pole decay in seconds Where should the zero and pole lie and the gain be to meet the settling time and overshoot specifications First find an expressions for the pole location 7 that will satisfy the natural frequency and damping ratio requirements in Section 4 2 1 as well as the desired filter cutoff frequency given in 38 Document Number 711 Revision 1 1 Page 30 SRVO02 Ball and Beam Control Laborat ratory Student Manual 5 Find expressions for the zero location z and the compensator gain K that will satisfy and in Section 4 2 1 and the desired filter cutoff frequency in 38 Document Number 711 Revision 1 1 Page 31 nt Manual pr gt 6 Based on the expressions found evaluate numerically the pole time constant zero location and gain needed to satisfy the specifications SRV02 Ball and Beam Control Laboratory Student Manual 5 In Lab Procedures The closed loop response of the Ball and Beam is simulated in Section 5 1 and then implement on the actual BBO1 device in Section 5 2 5 1 Position Control Simulation Go through Section 5 1 1 to simulate the Ball and Beam system using the designed compensator and ensure it meets the specifications listed in Section 4 2 1 This section deals with only the outer loop c2. that is caused by gravity Fx 2 Find the force that is caused by rotational inertia of the ball in the x direction F Hint Use the sector formula to convert between linear and angular displacement e g or velocity and acceleration x 1 Yp 1 8 where y is the angle of the ball and r is the ball radius Document Number 711 Revision 1 1 Page 6 pr gt SRVO02 Ball and Beam Control Laborat ratory Student Manual 3 Give the nonlinear equation of motion of the ball and beam It should be in the form shown in 5 4 1 2 Adding SRV02 Dynamics In this section the equation of motion representing the position of the ball relative the angle of the Document Number 711 Revision 1 1 Page 7 SRV02 Ball and Beam Control Laboratory Student Manual SRV02 load gear is found The obtained equation is nonlinear includes a trigonometric term and it will have to be linearized in order for the model to be used for control design 1 Using the schematic given in Reference 9 find the relationship between the SRV02 load gear angle 0 and the beam angle a pr gt 2 Find the equation of motion that represent the ball s motion with respect to the SRV02 angle 4 Linearize the equation of motion about servo angle 0 t 0 pr gt 3 Simplify the expression by lumping the coefficient parameters of 0 t into parameter K This is the model gain of the Ball and Beam system Show the ne
3. 4 3 3 1 PD Compensator Analysis Consider the following PD compensator Cppls Ko s z 35 where K is the proportional gain and z is the location of the compensator zero 1 Plot the root locus of the BBO1 loop transfer function Lpp s Cyn Pap 36 when the compensator zero is placed somewhere between 0 2 and 0 5 along the negative real axis 008 SRV02 Ball and Beam Control Laboratory Student Manual 2 Could this compensator be used to satisfy the ball and beam settling time and overshoot specifications Explain with references to the location of the zero and the gain 3 Find the closed loop transfer function of the ball and beam X s X4 s when using the proportional velocity PV controller shown in Figure 9 This is a variation of the PD compensator that does not feed the setpoint velocity sX4 s and is similar to what is used to control the position of the SRV02 Figure 9 Ball and beam ideal PV controller Document Number 711 Revision 1 1 Page 24 SRV02 Ball and Beam Control Laboratory Student Manual ofie 4 3 3 2 Steady State Error In this section it is assessed whether the steady state error specification given in 13 can be met using the PD controller in 35 1 Find the BBO1 error transfer function when using PD compensator 35 Can the final value theorem be used on this system o pr gt Document Number 711 Revision 1 1 Page 25 SRV02 Ball and Beam
4. set CONTROL_TYPE to MANUAL 6 Run the script by selecting the Debug Run item from the menu bar or clicking on the Run button in the tool bar The messages shown in Text 1 below should be generated in the Matlab Command Window The model parameters and specifications are loaded but the SRV02 PV gains and compensator gain are all set to zero and the compensator pole and zero are set to 1 they n eed to be changed SRV SRV BBO BBO Cal 02 model parameters K 0 rad s V tau 0 s 02 Specifications tp 0 15 s PO 5 1 model parameter K bb 0 m s 2 rad 1 Specifications ts 3 5 s PO 10 culated SRVO2 PV control gains kp 0 V rad kv 0 V s rad Natural frequency and damping ratio wn 0 rad s zeta 0 BB01 PD compensator Ke 0 rad m Z 1 rad s wf 6 28 rad s Text 1 Display message shown in Matlab Command Window after running setup_srv02_exp04_bb01 m Document Number 711 Revision 1 1 Page 35 SRV02 Ball and Beam Control Laboratory Student Manual 5 1 1 2 Outer Loop Ideal PD Simulation In this section the root locus of the forward path is plotted using Matlab and the closed loop step position response of the BBO1 will be simulated to verify that the specifications are met As previously mentioned the simulation is performed using the nonlinear model of the Ball and Beam and the ideal PD controller that was designed Follow these st
5. time domain response requirements olaj Document Number 711 e Revision 1 1 e Page 16 SRVO02 Ball and Beam Control Laboratory Student Manual 4 3 2 Outer Loop Stability Analysis The inner loop that controls the position of the SRV02 load shaft is complete and it the servo dynamics are now considered negligible Thus it is assumed that the desired load angle equals the actual load angle IO 27 The outer loop shown in Figure 7 will be used to control the position of the ball on the beam BBO1 SRV02 Closed Compensator Loop System BBO1 Plant Xa s Figure 7 BBO1I closed loop system 4 3 2 1 Root Locus of Open Loop System The root locus shows how the poles of a closed loop system move when a proportional gain increases towards infinity 1 Using Figure 7 find the closed loop transfer function of the BB01 system with the proportional control A ke 28 Document Number 711 Revision 1 1 Page 17 SRV02 Ball and Beam Control Laboratory Student Manual pr gt 2 Plot the root locus of the BBO1 plant P s The closed loop transfer function found above can help describe how the poles behave as K goes to infinity Document Number 711 Revision 1 1 Page 18 SRVO02 Ball and Beam Control Laboratory Student Manual 4 3 2 2 Desired Location of Poles Consider the prototype second order system Y A R s 2 29 s 4 2 0 5t0 2 The location of the two poles of this system when th
6. 711 Revision 1 1 Page 44 11 Are the specifications in Section 4 2 1 still satisfied after adding the servo dynamics Also make sure the servo angle is within 56 0 degrees and the servo voltage is between 10 0 V Don t go back in the control design if some specifications are not met opr gt Document Number 711 Revision 1 1 Page 45 SRV02 Ball and Beam Control Laboratory Student Manual 5 1 2 2 Practical PD Cascade Simulation The practical PD controller developed in Section 4 3 3 4 is simulated in this section This is the compensator that will be used to control the actual BB01 device The control gain and zero may have to be fine tuned in order to compensate for the added dynamics of the filtering and the inner loop servo control Follow these steps to simulate the closed loop practical cascade PD response l 2 3 Enter the BB01 model gain found in Section 4 1 2 in Matlab as variable Kbb Enter the practical PD compensator gain Kc and zero z that were found in Section 4 3 3 4 The filter cutoff filter wf is already set by the script Follow steps 2 6 in Section 5 1 2 1 to setup the SRV02 model parameters and control gains and setup the Simulink diagram To simulate using the practical PD controller set the Manual Switch in the BB01 PD Position Control subsystem to the downward position Using Matlab plot the root locus of BB01 loop transfer function when using the practical PD compensator and attac
7. BO1 ball position response Figure 23 BBO1 servo angle response Ym Y 6H PAL Aa Sa gt Figure 24 BBO servo input voltage 6 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 ball position and servo angle response as well as the input voltage Attach it to your report As in the s bb01_ pos Simulink diagram each scope automatically saves their response to a variable in the Matlab workspace when the controller is stopped Document Number 711 Revision 1 1 Page 56 nt Manual 7 Measure the steady state error the settling time and the percentage overshoot Does the response satisfy the specifications given in Section 4 2 1 Give a reason why the designed gain and zero could fail to give a successful closed loop response on the actual system SRV02 Ball and Beam Control Laboratory Student Manual 8 Ifthe specification have not been satisfied tune the gains as described in Section 5 1 2 2 and run the experiment again until the a satisfactory response is obtained In the case where the steady state error is not satisfied integral action can be introduced in the outer loop controller To do this go into the BB01 Position Control subsystem and increase the Integral Gain block until the error is minimized Briefly explain the procedure to get those new control paramete
8. Control Laboratory Student Manual 2 Find the steady state error of the BB01 closed loop system with the PD controller Can the steady state error requirement in 13 be satisfied using a PD compensator 4 3 3 3 Ideal PD Control Design The proportional and velocity gains needed to meet the desired requirements in Section 4 2 1 are found in this section This is called the ideal PD controller 1 Where should the zero lie and the gain be to meet the settling time and overshoot specifications Find expressions for the zero location z and the compensator gain K that will satisfy a given natural frequency and damping ratio Document Number 711 Revision 1 1 Page 26 pr gt SRV02 Ball and Beam Control Laboratory Student Manual 2 Based on the expressions found evaluate numerically the zero location and gain needed to satisfy the specifications ofie 4 3 3 4 Practical PD Controller The designed ideal PD controller cannot be used to control the ball position on the actual BBO1 device because it takes a direct derivative to obtain the ball velocity The position of the ball is measured using an analog sensor and it has some inherent noise Taking the derivative of this type of signal would Document Number 711 Revision 1 1 Page 27 SRV02 Ball and Beam Control Laboratory Student Manual output results in an amplified high frequency signal that is eventually fed back into the motor and causes a grindin
9. Rotary Motion Servo Plant SRV02 Rotary Experiment 04 BB01 Control Ball and Beam Position Control using QuaRC Student Manual SRV02 Ball and Beam Laboratory Student Manual Table of Contents TST STRGODUCTION ii 1 A cagusacasasassedsasuavessaeexecseascenasaacess 1 3 OVERVIEW OF TIDES ccsesseceiededsavescsnsasiecscessusessneededadeespsuenscncadsss uid orae deos aes osoa ESE ioe SESA SEEE Aass eeoa Sises 2 A PRE LAB A O 4 4 1 Modeling from BrrstPENCIplES 2d A id A A E A A A E 4 4 1 1 Nonlinear Equation Of MoOtlOM ooonocccconnnionncionacooncnnononnoncnnnnnncnn no cnn rr nan nc nn n ron nn E nan rn ran rn nn nan nr rn nrnnnnnnnnnos 5 41 2 Adding SRV02 DOM inside 8 4 13 Obtaining Transfer Function sessirnar e ie sis dd dll 9 4 2 Desired Control Response neer aeons de ead a Wes 11 4 2 1 Time Domain Specifications sneis a e a i a e ae aae S 11 42 2 Settling TM a A oa 11 4 2 3 Percentage Overshoot and Peak TiMC oooocnnocccionccnonnnoonccnoncnconcncononconnncnonnn nn nonon orinar rr con nn carr rr naar nnnnnnnnns 13 4 24 Steady State FETO tc 13 4 3 Ball and Beam Cascade Control DesigW oooonoccnnoccnonoconccnonoconnconnnconccnonocan ccoo ncnn non n nro nconnncnnnrnnonnnss 14 4 3 1 Inner Loop Design SRV02 PV Position Controller ccccccsscccesscessseeeseeeesecesseeceseeeseeeeeseeeseeesteeenees 15 4 3 2 Outer Loop Stability Analysis nri ona a e no nonon cnn nn cnn nn rro nn ETa ran rn can E S 17 4 33 Outer Lo
10. e natural frequency is Document Number 711 Revision 1 1 Page 19 SRVO02 Ball and Beam Control Laboratory Student Manual rad S 0 1 5 30 n and the damping ratio 1s 0 6 31 is illustrated in Figure 8 Pole Zero Map 15 15 D8 24142 be eee i a ay 1 04 pe N po 8 05 E N sint 4 A ooi Ei gt Not gt m 1 T 0 eo ee E o o Eo 0S 4 1 al 2 15 4 0 5 0 0 5 1 15 2 Real Axis Figure 8 Desired pole locations As illustrated the natural frequency determines the radial length of the poles from the origin The damping ratio changes the angle where they are positioned from the imaginary axis according to the relationship 9 arcsir 32 The location of the poles along the imaginary axis is called the damped natural frequency 2 07 0 V 1 5 33 and the position of the poles along the real axis is described by the equation c 34 Document Number 711 Revision 1 1 Page 20 SRVO02 Ball and Beam Control Laboratory Student Manual 1 Find the natural frequency and damping ratio required to achieve the time domain specifications of the Ball and Beam plant given in Section 4 2 1 2 Similarly as shown in Figure 8 plot the region where the poles should lie to satisfy the specifications pr gt 3 Discuss the response if the poles lie beyond the radius circle along the diagonal lines i e away from the imaginary axis Als
11. econds then go back to your control design If the settling time does not satisfy the original specifications but is kept below the allowed tolerance explain any possible source for this discrepancy o pr gt Document Number 711 Revision 1 1 Page 40 SRVO02 Ball and Beam Control Laboratory Student Manual 5 1 2 Cascade Control Simulation The servo dynamics can now be added and the closed loop position response when using the cascade control system will be simulated using the Simulink diagram pictured in Figure 14 This Simulink model simulates the block diagram shown in Figure 5 EJ s_bb01_pos File Edit View Simulation Format Tools WinCon Quarc Help D F MeD gt m 25 Normal 1 Di le BEE Ball and Beam Experiment 4 Simulated Ball Position Control SRVO2 Signal Amplitude Generator fem A Setpoint m Constant Constant Setpoint om em BBO1 PD Position Control SRVO2 PV Position Control Mode Radians theta_I deg to Degrees Figure 14 Simulink diagram used to simulate cascade control system The SRV02 BB01 Model subsystem includes the nonlinear model of the BBO1 plant and the transfer function representing the SRV02 voltage to position relationship The proportional velocity position controller designed in Section 4 3 1 is implemented in the SRVO02 PV Position Control block The cascade controller is the algorithm that will be implemented on the actual SRV02 BBO01 device Before depl
12. ed in this laboratory Model the dynamics of the ball from first principles Obtain a transfer function representation of the system Design a proportional velocity PV compensator to control the position of the servo load shaft according to certain time domain requirements Using root locus assess whether or not the specifications can be met with a certain type of compensator Design a compensator that regulates the position of the ball on the beam and meets certain specifications This together with the servo control is the complete Ball and Beam cascade control system Simulate the Ball and Beam control using the model of the plant and ensure the specifications are met without any actuator saturation Implement the controllers on the Quanser BB01 device and evaluate its performance 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 Wiring and operating procedure of the SRV02 plant with the UPM and DAC device as discussed in Reference 5 Laboratory described in Reference 6 in order to be familiar using QuaRC with the SRV02 Designing a PV position control for the SRV02 as dictated in Reference 8 Document Number 711 Revision 1 1 Page 1 SRV02 Ball and Beam Con
13. eps to simulate the nonlinear BBO1 position response 1 Enter the BBO1 model gain found in Section 4 1 2 in Matlab as variable Kbb 2 Enter the BBO1 compensator gain Kc and the compensator zero z that were found in Section 4 3 3 3 3 Using Matlab plot the root locus of BBO1 loop transfer function when using the ideal PD controller and attach it to your report Similarly as in Figure 8 use the sgrid command to include the dashed lines that show the desired locations of the poles Ensure the poles go through the desired locations at the gain that was computed Document Number 711 Revision 1 1 Page 36 SRV02 Ball and Beam Control Laboratory Student Manual SU Select square in the Signal Type field of the SRV02 Signal Generator in order to generate a step reference Set the Amplitude cm slider gain block to 5 to generate a step with an amplitude of 5 0 centimeters Open the load shaft position scope theta_ deg and the ball position scope x m Start the simulation By default the simulation runs for 25 0 seconds The scopes should be displaying responses similar to figures 12 and 13 In the x m scope the yellow trace is the desired ball position and the purple trace is the simulated response Document Number 711 Revision 1 1 Page 37 SRVO02 Ball and Beam Control Laboratory Student Manual Hax theta_l deg SEs if Time offset 0 Time offset 0 Figure 12 Outer loop ideal PD ball position resp
14. ershoot over 10 After 3 5 seconds the ball should settled within 4 of its steady state value i e not the reference and the steady state should be within 5 mm of the desired position 4 2 2 Settling Time The response of a second order system y t when subjected to a unit step reference r t is shown in Figure 3 This response has a 5 settling time of 0 30 seconds Thus the response settles within 5 of its steady state value which is between 0 95 and 1 05 in 0 30 seconds Settling time is defined ll 10 17 where the initial step time is to and the time it takes to settle is t Document Number 711 Revision 1 1 Page 11 SRV02 Ball and Beam Control Laboratory Student Manual y t and r t 0 9 1 1 1 1 2 1 3 1 4 1 5 1 6 Figure 3 Settling time of unit step response An equation that expresses the settling time in terms of the natural frequency and the damping of a second order system C is required In order to find this an exponentially decreasing sinusoidal is used to approximate the upper bound of the response and is expressed ELO to e i Er 18 l 1 Given the settling time percentage equation Cts Yup l 19 and the upper bound formula show that the settling time equation is Ines y l 7 Le 20 CO Document Number 711 Revision 1 1 Page 12 SRVO02 Ball and Beam Control Laboratory Student Manual 4 2 3 Percentage Overshoot and Peak Time Recall from Reference 8 t
15. g noise As illustrated by H s in Figure 10 this is prevented by using a high pass filter The first order filter replaces the derivative in Figure 10 and has the form Q fs H s 37 For adequate filtering of the noise found in the BBO1 linear transducer the cutoff frequency will be set to 1 Hz or 38 Bae rad 0 f5 y Also added to the controller is the set point weight parameter b This varies the amount of setpoint that is used to compute the error velocity This compensator is called the practical PD controller Xa s 9 a O S F X s gt o Figure 10 BB01 PD controller with filtering Although filtering is often necessary when controlling actual systems to make it more robust against noise it does add dynamics to the system Thus the compensator gain and zero location have to be recomputed in order to meet the specifications listed in Section 4 2 1 1 Using the block diagram in Figure 10 find the closed loop equation of the BBO1 only the outer loop no servo dynamics Document Number 711 Revision 1 1 Page 28 SRV02 Ball and Beam Control Laboratory Student Manual pr gt 2 Find the BBO1 compensator Cpp s in Figure 7 when the set point weight of the practical PD controller is 1 i e bp 1 3 Find the locations of the zero and the pole of the compensator What type of compensator does the PD becomes when adding filtering e g lead or lag Document Number 711 Revision 1 1
16. h it to your report As in Figure 8 show the desired locations of the poles on the plots and ensure the poles go through the desired locations at the gain that was computed Document Number 711 Revision 1 1 Page 46 SRVO02 Ball and Beam Control Laborat ratory Student Manual 6 Open the ball position scope x m the load shaft position scope theta_I deg and the SRV02 motor input voltage scope Vm V 7 Start the simulation By default the simulation runs for 25 0 seconds The scopes should be displaying responses similar to figures 18 19 and 20 Document Number 711 Revision 1 1 Page 47 SRVO02 Ball and Beam Control Laboratory Student Manual y 9 ETT P theta_l deg 358 PSS ABB SAR 1865 OSX ABB OAR e ES Figure 19 Practical cascade control servo angle response response Fy m Y 6B PA ABB BGAR Figure 18 Practical cascade control ball position Figure 20 Practical cascade control input voltage 8 Generate a Matlab figure showing the practical cascade ball position servo angle and servo input voltage response and attach it to your report Document Number 711 Revision 1 1 Page 48 SRVO02 Ball and Beam Control Laborat ratory Student Manual 9 Measure the steady state error the settling time and the percentage overshoot of the simulated practical cascade PD control response Document Number 711 Revision 1 1 Page 49 SRV02 Ball and Beam Con
17. hat the peak time and percentage overshoot equations are P 2 21 and 2 22 PO 100 e 4 2 4 Steady state Error In this section the steady state error of the ball position is evaluated using a proportional compensator Recall the unity feedback system in Figure 4 Compensator Plant Figure 4 Unity feedback system Document Number 711 Revision 1 1 Page 13 SRV02 Ball and Beam Control Laboratory Student Manual 1 Find the steady state error the ball and beam Pi s with a unity compensator C s 1 23 and a reference step of Ro R s 24 s where Ro is the step amplitude Remark that in this calculation the SRV02 dynamics are ignored and only the BBO1 plant is being considered If there is no constant steady state error then describe the error of the system 4 3 Balland Beam Cascade Control Design The cascade control that will used for the SRV02 BBO1 system is illustrated by the block diagram given in Figure 5 Based on the measured ball position X s the ball and beam compensator Ci s in the outer loop computes the servo load angle needed O4 s to attain the desired ball position X s The inner loop is a servo position control system as described in Reference 8 Thus the servo compensator C s calculates the motor voltage required for the load angle to track the given desired load angle Document Number 711 Revision 1 1 Page 14 pr gt SRVO02 Ball and Beam Control Labo
18. he servo In the next few sections the time based motion equations are developed and from these equations of motion its transfer function is obtained Recall in Reference 7 that the SRV02 voltage to load angle plant transfer function was found to be P is 4 s 1 s This can be added to the system to get the full SRV02 BB01 model 4 1 1 Nonlinear Equation of Motion In this section the equation describing the motions of the ball x relative to the angle of the beam a is derived Thus the equation of motion or eom for short will be of the form 2 d AAC 5 dt where f a t is a nonlinear function The incomplete free body diagram of the ball on a beam is illustrated in Figure 2 Applying Newton s Law of Motion the sum of the forces acting on the ball alongside the beam equals gt d mo aX F 6 t where m is the mass of the ball 2 Figure 2 Free body diagram of Ball and Beam Document Number 711 Revision 1 1 Page 5 SRVO02 Ball and Beam Control Laboratory Student Manual Neglecting friction and viscous damping the ball forces can can be represented by d AE e F 7 dt where F is the force from the ball s inertia and F is the translational force generated by gravity For the ball to be stationary at a certain moment 1 e be in equilibrium the force from the ball s momentum must be equivalent to the force produced by gravity 1 Find the force in the x direction along the beam
19. ications can still be satisfied The B801 PD Position Control subsystem contains the ideal PD compensator designed in Section 4 3 3 3 Remark that it includes a Saturation block that limits the SRV02 angle between 56 degrees Go through the steps in Section 5 1 1 1 to setup the Matlab workspace The procedure to simulate the closed loop Ball and Beam outer loop response with the ideal PD compensator is detailed in Section 5 1 1 2 5 1 1 1 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 BBO1 controller files 3 Double click on the s_bb01_pos_outer_loop mdl file to open the Simulink diagram shown in Figure 11 4 Double click on the setup_srv02_exp04_bb01 m file to open the setup script for the BBO1 Simulink models Document Number 711 Revision 1 1 Page 34 SRV02 Ball and Beam Control Laboratory Student Manual 5 Configure setup script When used with the Ball and Beam the SRV02 must be in the high gear configuration and no load is to be specified Make sure the seript 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 NONE 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
20. imulation runs for 25 0 seconds The scopes should be displaying responses similar to figures 15 16 and 17 The yellow and purples plots in the x m scope is the ball position setpoint and the its simulated response Similarly in the theta_ deg scope the yellow trace is the desired servo angle position which is generated by the outer loop control and the the purple plot is the simulated servo response theta_l deg TE Time offset 0 Time offset 0 Figure 15 Ideal PD cascade control ball position Figure 16 Ideal PD cascade control servo angle response response Document Number 711 Revision 1 1 Page 42 SRVO02 Ball and Beam Control Laboratory Student Manual 38 2A ABB Ba Time offset 0 Figure 17 Ideal PD cascade input voltage Generate a Matlab figure showing the Ideal PD cascade ball position servo angle and servo input voltage response and attach it to your report As explained in the procedure of Section 5 1 1 2 the response from each scope is saved to a Matlab variable after each simulation run The SRV02 motor input voltage is saved in the data_vm variable data_vm 1 is the time and data_vm 2 is the voltage Document Number 711 Revision 1 1 Page 43 SRVO02 Ball and Beam Control Laborat ratory Student Manual opi gt 10 Measure the steady state error the settling time and the percentage overshoot of the ideal PD cascade control response Document Number
21. nse in a Matlab figure and attach it your report Document Number 711 Revision 1 1 Page 52 SRVO02 Ball and Beam Control Laborat ratory Student Manual ojij 14 Give the resulting steady state error settling time and percentage overshoot of the response Document Number 711 Revision 1 1 Page 53 SRVO02 Ball and Beam Control Laboratory Student Manual 5 2 Position Control Implementation The q_bb01_pos Simulink diagram shown in Figure 21 is used to perform the position control exercises in this laboratory The SRV02 ET BB01 subsystem contains QuaRC blocks that interface with the DC motor and sensors of the Ball and Beam system The BB01 PD Position Control subsystem implements the practical PD control detailed in Section 4 3 3 4 a q_bb01_pos File Edit view Simulation Format Tools WinCon Quarc Help D e Hg T E int Extemal Johe BEE BB01 Experiment 4 Ball Position Control Pot 1 Enc 2 SRVO2 Signal Amplitude Generator em Setpoint m Constant Constant Setpoint em Setpoint cm Source BBO1 PD Position Control Radians theta_ d SRVO2 PV to Degrees Position Control SRVO02 ET BB01 Figure 21 Simulink model used with QuaRC to run the practical PD controller on the Ball and Beam system Document Number 711 Revision 1 1 Page 54 pr gt SRV02 Ball and Beam Control Laboratory Student Manual Go through the steps in Section 5 2 1 to setu
22. nsor and observe the response obtained in the scopes Figures 25 26 and 27 show a sample response Document Number 711 Revision 1 1 Page 61 ona SRV02 Ball and Beam Control Laboratory Student Manual R cm theta_1 deg 6 PSA MMB SAR H SAN ABA Bar mona rea ae r A ili Time offset 25 027 Time offset 25 027 Figure 25 BBO1 ball position response with SS01 2 Vm Y 865 022 ABB DA Figure 26 BBO servo angle response with SSO1 y Figure 27 BBO1 servo input voltage with SSO1 5 When done click on the Stop button in the Simulink diagram tool bar or select QuaRC Stop from the menu to stop running the code 6 Shut off the power of the UPM if no more experiments will be performed on the SRV02 in this session Document Number 711 Revision 1 1 Page 62 Fill out Table 2 below with the pre laboratory and in laboratory results obtained such as the designed ideal and practical PD parameters along with the measured settling time percentage overshoot and steady state error obtained from the simulated and implemented step responses Section Description Symbol Value Unit 4 3 1 Pre Lab Model Parameters Open Loop Steady State Gain rad V s Open Loop Time Constant 4 3 1 Pre Lab PV Gain Design 3 Proportional gain V rad 3 Velocity gain V s rad 4 3 3 3 Pre Lab Ideal PD Control Design Ze Compensator Gain rad m Za Compensator Zero rad s 4 3 3 4 Pre Lab Practical PD Contr
23. o comment on what happens if the poles of the system lie inside the diagonal lines along the radius circle i e moving towards the real axis Make references to its effects on the settling time and overshoot of the response Document Number 711 Revision 1 1 Page 21 SRV02 Ball and Beam Control Laboratory Student Manual pr gt 4 Based on the root locus obtained previously can the specifications of the Ball and Beam system be satisfied using a proportional controller Discuss 4 3 3 Outer Loop Controller Design So far the analysis has been done using a proportional compensator Now the affects using a dynamic compensator in the loop path of Figure 7 is studied Generally speaking adding a zero in the forward path increases the bandwidth of the closed loop system Adding a pole increases the rise time and overshoot of the system but makes it overall less stable In our case the bandwidth must be increased and the overshoot has to be minimized The effects of adding a zero in the forward loop path are studied Section 4 3 3 1 and the obtained steady state error is assessed in Section 4 3 3 2 The first controller to be designed is a proportional derivative PD compensator This is done in Section 4 3 3 3 This design is then modified in Section Document Number 711 Revision 1 1 Page 22 SRV02 Ball and Beam Control Laboratory Student Manual 4 3 3 4 to handle some inherent practical issues
24. ol Design Compensator Gain Compensator Zero Compensator Pole Time Constant In Lab Simulation Outer loop Ideal PD Steady state error Settling time Percentage overshoot In Lab Simulation Cascade Ideal PD Steady state error Settling time Percentage overshoot In Lab Simulation Cascade Practical PD 9 Steady state error Es cm SRV02 Ball and Beam Control Laboratory Student Manual 9 Settling time ts S 9 Percentage overshoot PO In Lab Simulation Cascade Tuned Practical PD Compensator Gain Compensator Zero Steady state error Settling time Percentage overshoot In Lab Implementation Tuned Practical PD Steady state error Settling time Percentage overshoot In Lab Implementation Tuned 2 Practical PD Compensator Gain rad m Compensator Zero rad s Integral Gain i rad m s Steady state error E cm Settling time E S Percentage overshoot Table 2 SRV02 Experiment 4 Ball and Beam control results summary 7 References 1 Quanser 04 08 User Manual 2 Quanser QuaRC HTML Help Files 3 Quanser QuaRC Installation Manual 4 Quanser UPM User Manual 5 Quanser SRVO2 User Manual 6 Quanser SRV02 QuaRC Integration 7 Quanser Rotary Experiment 1 SRV02 Modeling 8 Quanser Rotary Experiment 2 SRV02 Position Control 9 Quanser Ball and Beam User Manual Document Number 711 Revision 1 1 Page 64
25. onse Figure 13 Outer loop ideal PD servo angle response 8 Generate a Matlab figure showing the nonlinear Outer Loop BB01 ball position response and the corresponding servo angle and attach it to your report After each simulation run each scope automatically saves their response to a variable in the Matlab workspace The x m scope saves its response to the variable called data_x and the theta_I deg scope saves its response to the variable data_theta_ The data_x variable has the following structure data_x 1 is the time vector data_x 2 is the setpoint and data_x 3 is the simulated ball position For the data theta lvariable data theta _ 1 is the time and data_theta_l 2 is the servo angle Document Number 711 Revision 1 1 Page 38 SRVO02 Ball and Beam Control Laborat ratory Student Manual opi gt 9 Measure the steady state error the settling time and the percentage overshoot of the simulated response Document Number 711 Revision 1 1 Page 39 SRV02 Ball and Beam Control Laboratory Student Manual 10 Does the outer loop ideal PD response satisfy the specifications given in Section 4 2 1 while keeping the servo angle between 56 degrees Some tolerance is allowed on the settling time specification it should not exceed 3 75 seconds rather then 3 50 seconds If the steady state error and percentage overshoot do no meet the desired specifications and the settling time goes over 3 75 s
26. ontrol In Section 5 1 2 the ball and beam is simulated using its full cascade control system That is the system that includes both the outer ball position control loop and the inner servo position control feedback loop 5 1 1 Outer Loop Simulation The s_bb01_pos_outer_loop Simulink diagram shown in Figure 11 is used to simulate the closed loop position response of the BB01 when using the outer loop control Thus the SRV02 dynamics are neglected i e Oa 0 The response is simulated using the developed nonlinear model of the Ball and Beam Document Number 711 Revision 1 1 Page 33 SRVO02 Ball and Beam Control Laboratory Student Manual E s_bb01_pos_outer_loop File Edit Yiew Simulation Format Tools WinCon Quarc Help DSHS ales T2 j Irom R aS Ball and Beam Experiment 4 Simulated Ball Position Control SRVO2 Signal Amplitude Generator cm Setpoint m Constant Constant Setpoint om cm BBO1 PD BB01 Nonlinear Model Position Control Radians theta_ deg to Degrees 100 Figure 11 Simulink diagram used to simulated the outer closed loop BB01 system The BB01 Nonlinear Model subsystem includes the Pi s transfer function that was derived in Section 4 1 3 Recall in Section 4 1 2 that the model had to be linearized in order to obtain the Pi s transfer function This nonlinearity is re introduced in the BBO Nonlinear Model subsystem in order to represent the plant more accurately and ensure the specif
27. op Controller Desi lia kn oeeteas 22 S AN LAB PROCEDURES S assicnsesnenssousisnercuedonvensenbasesanssanapeadtncsreunssseutastoancoeaiassetsdsaduoues savksteuvudnsaussnouseadsecineonies 33 Se POSITION Control MUA AAA alee A A 33 Sol Outer Loop Simulations eiii dada a ss id A tilda sd 33 5 1 2 Cascade Control SIMAO a the a AA a E see E EE 40 5 2 Position Control mpleMEITA ON AS A LAA 53 5 2 1 Setup for Position Control ImplementatiON oooocniocononnnconnnconnncnonccnonocnonocnoncnnon conan no can nn cnn nn cnn nrrnannnnnno 54 5 2 2 Running the Practical PD Controller ooooooononocionccnonccconcnoonncconnnnonnnonnrncnnnononn ocn nnnnon E corn r narra can nnnnos 54 Document Number 711 Revision 1 1 Page i SRV02 Ball and Beam Laboratory Student Manual 5 2 3 Controlling using the Remote Sensor Optional oooonnoccnionocionncononcnonncnonoconn ccoo cnnonnnnonnnnn nn rnnnnnnnnnnnnos 60 6 RESULTS SUMMARY ciscccccssiescaccescdecadessdcasesscsecssssedssdescdecstusvecsssssssscusssvstssdsscdccudescecsessscsactuseustsssescdascsssesesss OZ Ts REFERENCES OS Document Number 711 Revision 1 1 Page ii SRV02 Ball and Beam Control Laboratory Student Manual 1 Introduction The objective of the Ball and Beam experiment is to stabilize the ball to a desired position along the beam Using the proportional derivative PD family a cascade control system is designed to meet a set of specifications The following topics are cover
28. oyment we need to confirm that the specifications are still satisfied when the servo dynamics are added In addition the servo angle must be kept between 56 degrees and the servo voltage cannot exceed 10 V 5 1 2 1 Ideal PD Cascade Simulation The purpose of this simulation is to see how the settling time overshoot and steady state error of the response changes when adding the inner loop servo control that was designed in Section 4 3 1 Follow these steps to simulate the ideal PD closed loop SRV02 BB01 cascade control response 1 Follow steps 1 and 2 in Section 5 1 1 2 the to setup the Matlab workspace Document Number 711 Revision 1 1 Page 41 SRV02 Ball and Beam Control Laboratory Student Manual 2 Enter the low gear SRV02 model gain K and the model time constant tau in Matlab found in Section 4 3 1 3 Enter the SRVO02 PV gains called variables kp and kv in Matlab found in Section 4 3 1 4 Select square in the Signal Type field of the SRV02 Signal Generator in order to generate a step reference 5 Set the Amplitude cm slider gain block to 5 to generate a step with an amplitude of 5 0 centimeters 6 Open the ball position scope x m the load shaft position scope theta_I deg and the SRV02 motor input voltage scope Vm V 7 Place the Manual Switch in the BBOI PD Position Control subsystem to the upward position in order to use the ideal PD controller when simulating 8 Start the simulation By default the s
29. p the Matlab workspace The procedure to run the developed practical PD controller is outlined in Section 5 2 2 Section 5 2 3 shows how to run the same controller using the remote sensor module 1 e SSO1 5 2 1 Setup for Position Control Implementation Before beginning the in lab exercises on the Ball and Beam device the q_bb01_pos Simulink diagram and the setup _srv02 exp04 bb01 m script must be configured Follow these steps to get the system ready for this lab 1 Setup the SRVO02 with the BBO1 module as detailed in Reference 9 2 Load the Matlab software 3 Browse through the Current Directory window in Matlab and find the folder that contains the QuaRC BBO1 control file g bb01_pos madl 4 Double click on the g bb01_pos mal file to open the Ball and Beam Position Control Simulink diagram shown in Figure 21 5 Configure DAQ Ensure the HIL Initialize block in the SRV02 ET BB01 subsystem 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 Sensor The position of the load shaft can be measured using various sensors Set the Pos Src Source block in q bb01_pos as shown in Figure 21 as follows e to use the potentiometer e 2 to use to the encoder Note that when using the potentiometer there will be a discontinuity 7 Configure Setpoint The
30. ratory Student Manual BBO1 SRV02 Compensator Compensator Epp S SRVO02 Plant BB01 Plant Figure 5 Cascade control system used to control ball position in SRV02 BB01 plant In Section 4 3 1 the position controller for the SRV02 is designed similarly as explained in Reference 8 A compensator is introduced in Section 4 3 2 and it is assessed using root locus whether it can be used to meet the desired specifications Two different variations of a compensator are designed in Section 4 3 3 4 3 1 Inner Loop Design SRV02 PV Position Controller In this section the proportional velocity PV controller gains are computed for the SRV02 when it is in the high gear configuration and based on the specifications given in Section 4 2 1 The internal control loop is depicted in the block diagram shown in Figure 6 SRV02 SRVO02 Plant Compensator Es s Figure 6 SRV02 closed loop system 1 The nominal model parameters K and t when the SRV02 is in high gear configuration are Document Number 711 Revision 1 1 Page 15 SRVO02 Ball and Beam Control Laboratory Student Manual K 1 76 25 sV and t 0 0285 s 26 Given these parameters calculate the minimum damping ratio and natural frequency required to meet the SRV02 specifications given in Section 4 2 1 oa 2 Using the derivations in Experiment 2 SRV02 Position Control Reference 8 calculate the control gains needed to satisfy the
31. rs and give the gain and zero used to obtain the response including the integral gain if necessary This is called the Tuned Practical PD 2 control Document Number 711 Revision 1 1 Page 58 Laboratory Student Manual 9 Plot the response using the Tuned Practical PD 2 control in a Matlab figure and attach it to you report Document Number 711 Revision 1 1 Page 59 nt Manual 10 Give the measured steady state error the settling time and the percentage overshoot of the response Does the response satisfy the specifications given in Section 4 2 1 SRV02 Ball and Beam Control Laboratory Student Manual 11 Make sure QuaRC is stopped 12 Shut off the power of the UPM if no more experiments will be performed on the SRV02 in this session 5 2 3 Controlling using the Remote Sensor Optional In this lab the position of the ball on the BBO1 device will be controlled using the developed practical PD control but the setpoint is given with the remote sensor This procedure can only be undergone if with the remote sensor SS01 module detailed in Reference 9 Follow the steps below 1 Follow steps 1 4 in Section 5 2 2 to setup the Matlab workspace and build the q bb01_pos model 2 Place the Setpoint Source switch to the DOWN position in order to generate the setpoint using the SSO1 module 3 Select QuaRC Start to begin running the controller 4 Move the ball back and forth on the remote se
32. setpoint can be generated through the SRV02 Signal Generator Simulink block or via the SSO1 device see Reference 9 Place the Setpoint Source switch to the UP position in order to generate the setpoint through the Simulink model 8 Configure setup script Set the parameters in the setup_srv02_exp04_bb01 m script according to your system setup See Section 5 1 1 1 for more details 5 2 2 Running the Practical PD Controller In this lab the position of the ball on the BBO1 device will be controlled using the developed control Measurements will then be taken to ensure that the specifications are satisfied Follow the steps below 1 Enter the BBO1 model gain found in Section 4 1 2 in Matlab as variable Kbb 2 Enter the Tuned Practical PD 1 compensator gain Kc and zero z that were found in Section 5 1 2 2 or the original gains if tuning was not required 3 Follow steps 2 6 in Section 5 1 2 1 to setup the SRV02 model parameters and control gains and setup the Simulink diagram 4 Click on QuaRC Build to compile the Simulink diagram 5 Select QuaRC Start to begin running the controller The scopes should be displaying responses similar to figures 22 23 and 24 Document Number 711 Revision 1 1 Page 55 SRV02 Ball and Beam Control Laboratory Student Manual 3 cm al ineta_ deg A 8565 02 p2 BB SAR WEE SSX ABA Sa J AMY O AS i L sal Ate x 5 Time offset 25 023 Time offset 25 023 Figure 22 B
33. trol Laboratory Student Manual 3 Overview of Files Table 1 below lists and describes the various files supplied with the SRV02 Ball and Beam Position Control laboratory File Name Description 09 Ball and Beam User This manual describes the hardware of the Ball and Beam and Manual pdf explains how to setup and wire the system for the experiments 10 Ball and Beam Position This laboratory guide contains pre lab and in lab exercises Control Student Manual pdf demonstrating how to design and implement a position controller on the Quanser SRV02 Ball and Beam plant using QuaRC setup _srv02 exp04 bb0l m The main Matlab script that sets the SRV02 motor and sensor parameters the SRV02 configuration dependent model parameters and the BBO1 sensor parameters Run this file only to setup the laboratory config_srv02 m 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 config bb01 m Returns the configuration based BB01 model specifications L_beam r_arm r_b m_b J_b and g the servo offset THETA_OFF the min max servo limits THETA_MIN and THETA_MAX and the sensor calibration constant K_BS d_model param m Calculates the SRV02 model parameters K and tau based on the device specifications Rm kt km Kg eta_g Beq Jeq and eta_m calc conversion constants m Returns
34. trol Laboratory Student Manual oa 10 Does the simulated response satisfy the specifications given in Section 4 2 1 while keeping the servo angle between 56 0 degrees and the servo voltage between 10 0 V pr gt 11 Ifa specification is not satisfied then the control parameters need to be fine tuned One method is to redesign the compensator gain zero location and pole time constant according to more stringent restrictions For instance try simulating the system for a K z and 7 generated according to a percentage overshoot of 8 instead of 10 To do this write a short Matlab Document Number 711 Revision 1 1 Page 50 SRVO02 Ball and Beam Control Laboratory Student Manual script that computes the gains automatically according to a given set of percentage overshoot settling time and filter cutoff frequency specifications Then simulate the system and see if the specifications are satisfied Note that the cutoff filter frequency should remain as specified in Equation 38 or 1 Hz Attach the script to you report opi gt 12 Record the gain and zero have been fine tuned for the response to meet the specifications along with the new specifications used to generate those control parameters These control parameters will be called the Tuned Practical PD 1 Document Number 711 Revision 1 1 Page 51 SRVO02 Ball and Beam Control Laborat ratory Student Manual 13 Plot the simulated respo
35. various conversions factors s bb01_pos outer loop mdl Simulink file that simulates the closed loop system when using only the outer loop ball position controller with the BBO1 system i e no inner loop control of the servo position s bb01 pos mdl Simulink file that simulates the cascade ball position controller Both the outer loop ball position control and the inner loop servo position control are used in this file q _bb01 pos mdl Simulink file that implements a closed loop cascade position controller on the actual BBO1 system using QuaRC Document Number 711 Revision 1 1 Page 2 File Name Description SRV02 Ball and Beam Control Laboratory Student Manual File Name Description Table 1 Files supplied with the SRV02 Ball and Beam Position Control experiment 4 Pre Lab Assignments 4 1 Modeling from First Principles As illustrated in Figure 1 this system is comprised of two plants the SRV02 and the BBO1 SRV02 Plant BBO1 Plant Figure 1 Ball and Beam open loop block diagram The main objective in this section is to obtain the complete SRV02 BB01 transfer function P s Pps PS 1 where the BBO1 transfer function is X s Eb ys 2 and the SRV02 transfer function is 0 fs P s 3 OS Document Number 711 Revision 1 1 Page 4 SRV02 Ball and Beam Control Laboratory Student Manual The BBO1 transfer function describes the displacement of the ball with respect to the load angle of t
36. w simplified equation of motion Document Number 711 Revision 1 1 Page 8 SRV02 Ball and Beam Control Laboratory Student Manual Then evaluate the model gain numerically using the Ball and Beam parameters given in Reference 9 Hint Recall that the moment of inertia of a solid sphere is 2m oe J 9 5 where m is the mass of the ball and r is its radius 4 1 3 Obtaining Transfer Function In this section the transfer function describing the servo voltage to ball position displacement is found 1 Find the transfer function P s of the BB01 Assume all initial conditions are zero Document Number 711 Revision 1 1 Page 9 2 Give the complete SRV02 BBO1 process transfer function P s Document Number 711 Revision 1 1 Page 10 SRV02 Ball and Beam Control Laboratory Student Manual 4 2 Desired Control Response 4 2 1 Time Domain Specifications The time domain specifications for controlling the position of the SRV02 load shaft are es 10 t 0 15 s pe 11 PO 5 0 _ 12 Thus when tracking the load shaft step reference the transient response should have a peak time less than or equal to 0 15 seconds an overshoot less than or equal to 5 and no steady state error The specifications for controlling the position of the ball are e l lt 0 005 m 13 t 3 5 s 14 c 0 04 pe 15 PO 10 0 _ 16 Given a step reference the peak position of the ball should not ov
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