ENGS 26 CONTROL THEORY. DC Servomotor Laboratory

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1 the ENGS 26 CONTROL THEORY DC Servomotor Laboratory Equipment and Software Required: Thayer School DC Motor/Tachometer board DT2801-A Data Acquisition Board and PC Oscilloscope Connectors (1 BNC-BNC, 3 BNC-Banana, 5 pairs banana-banana, 2-3 pairs of alligator clips) + 10 V power amplifier (white box) Voltmeter Signal generator +15 V DC power supply DT VEE software Breadboard and analog control component kit Note: You should complete this experiment during the first week of the lab session, and write the report during the second week of the lab session. 1. Objective The objective of this lab is to design and implement a proportional-derivative, lead, OR proportional+tachometer compensator to control motor shaft position. 2. System modeling As in Lab 2, the DC motor is driven by a voltage through a power amp. An integrated tachometer provides a voltage proportional to shaft angular velocity, and a three turn potentiometer provides a voltage proportional to shaft position. In Lab 2, we identified an open-loop transfer function of the motor-tach-power amp based on the open-loop system response to step input voltages. With the potentiometer in the system, we cannot lump the tachometer transfer function in the system gain. We will represent the motor together with the power amp by θ V in = K1 sts ( + 1) (1) θ is the motor shaft position (rad), V in is the input voltage to the power amp (volts), T is the motor time constant in seconds identified in Lab 2, and K 1 is overall gain of the power amp and motor (rad/volt). K 1 can be determined based on the power amp gain (which was set to 1 in Lab 2 for open-loop transfer function identification), overall gain measured in Lab 2 (the variable we called K in Lab 2), and tachometer gain ( K t ). K t is given in manufacturer specs as 2.4 V/1000 RPM (or 2.4/104.2 V/rad/s). The potentiometer transfer function depends on its supply voltage. For example, if the supply voltage across the potentiometer is 15 V, the gain is approximately 15/3 = 5 V/rev or V/rad. The block diagram describing the open loop system is shown in Figure 1. It lumps the gains associated with the motor and power amp into K 1, assumes a system time constant T, and assumes gains K t and K pot for the tachometer and potentiometers, respectively. 1

2 D(s) V in + K1 θ (rad) (volts) + s(ts + 1) V out (volts) K t s + K pot Figure 1: Block diagram of the open-loop system Connect the tubing between the motor and potentiometer shafts. Apply +15V and 0V across the potentiometer s positive and negative terminals. Make sure you do not apply a voltage to the slide! Ask the lab TA for assistance, if necessary. Verify K pot. To do this, make sure that you read between 0 V and +15 V from ground to the slide when you turn the potentiometer 3 full turns. The slide voltage should increase by 5 V for each turn as you turn the pot by hand. From the open-loop system parameter K determined in Lab 2, compute the portion of the motor gain attributed to the tachometer ( K t ) and that attributed to the motor/power amp ( K 1 ). Assuming that you had the power amp gain set to 1 in Lab 2, K 1 represents the motor gain. 3. Position control of the DC motor using proportional control We will design a control system such that the following closed-loop specifications are met: ts < 0.15 sec (2% settling time) steady-state error = 0, for step input. (Note: We will define the steady-state error here as the reference input voltage - the potentiometer voltage) closed-loop damping ratio > 0.5 In addition you may consider the following (optional) specification closed-loop bandwidth > 5 Hz. We will consider three different control laws: a) Position feedback + proportional control b) Position feedback + PD control OR position feedback + lead compensation c) Position + tachometer feedback 2

3 3.1 Proportional control and position feedback Prelab: Draw a closed-loop system block diagram showing reference input θ d (voltage corresponding to desired shaft position), output θ (rad) and transfer functions for the motor, power amplifier, potentiometer, and control law Gc (). s Use motor parameters identified experimentally in Lab 2 for the motor transfer function. Using a proportional control law, Gc()= s Kp, and root locus analysis, show that the specifications cannot met. (On the outside chance that specs are met using proportional control for your motor, change the settling time spec to ts < 0.1 sec!) 3.2 Proportional-Derivative or Lead Compensation and Position Feedback Prelab: Design a proportional-derivative control law G s K K c() = d p( K s +1 ) OR design a lead p 1 s + compensator Gc() s = K τ c,. < < 1 01 α 1 such that the specifications are met. To do this, you s + ατ will need to define the region in the s-plane that constitutes acceptable locations for closed-loop roots, and place the zero of the PD compensator or the pole-zero pair of the lead compensator such that the root locus passes through this region. (There are multiple solutions for either method -- which is the best solution?) Sketch a compensator circuit to implement the control law, and choose resistor and capacitor values for the circuit. You will need a summing junction and PD or lead circuit. You may base your PD or lead circuit on Table 5.1 given in Ogata. Come to lab prepared to implement your circuit. 3.3 Position+Tachometer feedback and proportional control Prelab: Draw a block diagram showing position+tachometer feedback, with a proportional control gain Gc()= s Kp in the forward loop. Show that the form of the closed-loop characteristic equation for this system is identical to that of a PD compensator. What is the difference between the PD control law or lead control law and position + tach feedback? Design a position+tach feedback control law using root locus analysis. To do this, you will need to define the region in the s-plane which constitutes acceptable locations for closed-loop roots (as in section 3.2), and place the zero of the compensator such that the root locus passes through this region. You can either choose gains for the potentiometer and tachometer signals to place the zero, and fix the proportional control gain to select the closed-loop root along the root locus, or you can determine whether the existing gains for each transducer ( K pot and K t ) are sufficient to place the zero such that specifications are met. (Again, there are multiple solutions -- which is the best solution?) Design a position+tach feedback circuit to implement the control law. Choose values for resistors before you get to lab. An example circuit that uses a gain for each transducer is attached. If you do not implement potentiometer and tachometer gains, remember to make sure the sign of your motor voltage signal is correct for negative feedback. 3

4 3.4 Compensator implementation Lab: Choose one of your compensators (either PD, lead, or position+tach feedback) for implementation. PD or lead compensator: For implementation of the PD or lead circuit, construct a summing junction to sum the potentiometer voltage and the reference input. You may use a differential amplifier circuit as in Lab 2. The summing junction is followed by your compensator op amp circuit, and the output of your circuit is the input to the power amp. For the reference input signal, use the signal generator to provide a square wave at approx. 0.1 Hz. (See the note below for hints on how to adjust your reference signal amplitude.) You may, as usual, use the power amplifier to adjust the overall gain. To avoid loading affects, you must buffer the output of the potentiometer slide BEFORE it enters the summing junction. In addition, you may also want to buffer the +15 V supply across the potentiometer. See example circuits at the end of this handout. Implement the control law, and use DT-VEE to record the reference input signal (the square wave) and the output of the potentiometer on A/D channels zero and one. You may also want to record the output of the potentiometer and the input voltage to the motor to observe possible saturation, for one of your runs. Next, record the response for a smaller amplitude input signal and for a larger amplitude input signal. MAKE SURE THAT YOUR CIRCUIT WORKS BEFORE YOU CLOSE THE LOOP! Otherwise, you could break the pot. Note that if the motor shaft slips in the tubing when you operate the closed-loop system, you have done something wrong! Turn off the power amp immediately, and correct the problem in your circuit if this happens! In your report, show the closed-loop system block diagram, show your analysis, describe your control law design method, describe your circuit, and answer the following questions. Why did you choose the PD or lead compensator over the position+tach feedback compensator? What is the steady-state error for each input (experimental and MATLAB simulation)? What are the limitations, if any, in implementing the PD or lead control law? Can you implement any proportional and derivative control gains that you desire? How do experimental responses compare to simulated responses? Position+tach feedback: Construct a position+tach feedback circuit, implement the circuit, and test the circuit for reference input square wave (see notes below for hints on how to choose the reference input). An example circuit is given in this handout. To avoid loading affects, you must buffer the output of the potentiometer slide BEFORE it enters the summing junction. In addition, you may also want to buffer the +15 V supply across the potentiometer. See example circuits at the end of this handout. Use DT-VEE to record the reference input signal (the square wave) and the output of the potentiometer on A/D channels zero and one. You may also want to record the output of the potentiometer and the input voltage to the motor to observe possible saturation, for one of your runs. Next, look at the response for a smaller amplitude input signal and for a larger amplitude input signal. 4

5 Answer the following questions: Show the closed-loop system block diagram, show your analysis, describe your control law design, describe your circuit, and answer the following questions. Why did you choose position+tach feedback over PD or lead compensation? What is the steady-state error to each input (experimental and MATLAB simulation)? What are the limitations, if any, in implementing the position+tach feedback control law? Can you implement any position+tach feedback control law that you desire? How do experimental responses compare to simulated responses? Note on selecting reference input amplitude: Note that you cannot record position commands over the full range of the potentiometer (0 to 15V), since the A/D converter records voltages between +10V. When choosing your input amplitude, consider the angle that the motor will move through. For example, for a 5V input, you should expect 1 full turn, while for a 1V input, you will expect a fraction of a turn. Do not choose an input amplitude outside of the 0 to 10V range, and be aware that if your system overshoots, you will not see the overshoot recorded on the A/D channel, if it exceeds 10 V. For example, for a 5V input, you may want to choose the minimum voltage as 2 V and the maximum voltage as 7 V. Bandwidth determination (optional): Change the input signal to a sinusoidal voltage, and determine the closed-loop frequency response (amplitude only). To do this, record the magnitude ratio V pot for input frequencies ranging from θ d 0.1 Hz to 10 Hz. What is the closed-loop bandwidth? PLEASE TURN OFF INSTRUMENTS AND MAKE SURE ALL MATERIALS ARE RETURNED TO YOUR KIT BEFORE LEAVING THE LAB. PLEASE LEAVE YOUR STATION NEAT. 5

6 PD Compensation Circuit (One more capacitor is required to change this to a lead compensator see Ogata, Modern Control Engineering, Prentice Hall, 3 rd edition, p

7 Position + Tach feedback circuit 7