Exercise 7 Closed-Loop Speed Control, EXERCISE OBJECTIVE To describe the derivative control mode; To describe the advantages and disadvantages of derivative control; To describe the proportional-plus-integral-plus-derivative control mode; To define the terms derivative time, ideal and parallel configurations; To describe how a change in derivative time affects the controlled speed when using proportional-plus-integral control. DISCUSSION Closed-Loop Control of Actuator Speed In the previous exercise, you controlled the speed of a pneumatic motor with an open-loop system. The motor speed decreased when the load increased because the system was controlling the valve opening and not the actual motor speed. The addition of a controller and a feedback loop to a speed control system markedly reduces the variations in actuator speed. This type of system, called "closed-loop control system", is illustrated in Figure 7-1. The controller compares the setpoint (desired motor speed) to the measured speed and corrects for any difference between the two by modifying the opening of the valve until the system reaches a state of equilibrium. The feedback loop contains a speed transducer that measures the actual actuator speed and generates a proportional signal which is sent back to the controller. 7-1
Figure 7-1. Closed-loop control of actuator speed. The Derivative (D) Control mode Unlike the proportional control mode, which regards the "present" value of the error, or the integral control mode, which regards the "past" value of the error, the derivative control mode anticipates the "future" value of the error based on the rate at which it is changing. The derivative control mode opposes to changes in the actuator speed by producing immediate, significant correction of the error, thereby reducing overshoot of the actuator speed. Derivative action is achieved by using a derivative amplifier. This type of amplifier produces a signal proportional to the rate of change (slope, or derivative) of its input signal. Figure 7-2, for example, shows the signal produced at the output of a derivative amplifier when the signal at its input changes gradually. When the input signal increases, the output signal is positive and proportional to the rate of change of the input signal. When the input signal decreases, the output signal is negative and proportional to the rate of change of the input signal. When the input signal is constant, the output signal is zero. 7-2
Figure 7-2. Output signal of a derivative amplifier when the input signal changes gradually. The equation describing the action of a derivative amplifier is where D O (t) is the output signal at a specified time; K D is the derivative gain; di/dt is the rate of change (slope, or derivative) of the input signal. The equation shows that the derivative action is not only proportional to the rate of change of the input signal, but also to the derivative gain, K D. The derivative gain, in seconds, is the length of time during which the derivative mode anticipates the future value of the input signal. The higher the derivative gain is, the greater the derivative action will be, and the smaller the input rate of change required to produce any given output will be. Figure 7-3, for example, shows the output signal of a derivative amplifier for two different derivative gains. During the first segment of the input signal, which lasts 5 seconds, the signal increases by 1.5 V, which corresponds to a rate of change of 0.3 V/s. Therefore, the output signal during this interval is 0.6 V with a derivative gain of 2 seconds, while it is 1.2 V with a derivative gain of 4 seconds. 7-3
Figure 7-3. Output signal of a derivative amplifier for two different derivative gains K D. An important characteristic of the derivative control mode is that it cannot be used alone in a control system. The reason is that a null or constant error would produce a zero signal at the controller output and would stop the actuator. Instead the derivative mode is combined with the proportional and integral modes to reduce overshooting of the actuator speed and damp a tendency toward instability. 7-4
The Proportional-Plus-Integral-Plus-Derivative (P.I.D.) Control mode The proportional-plus-integral-plus-derivative control mode combines the advantages of each mode. Adding integral action to proportional action will eliminate the residual error, but will increase the overshooting and the tendency toward instability. However, by adding derivative action, the overshooting and tendency toward instability can be reduced. Figure 7-4 shows a diagram of a controller operating in the P.I.D. mode. The error signal produced by the error detector is first amplified by the proportional amplifier. The resulting signal is then passed through the integral and derivative amplifiers. Finally, the output signals of the three amplifiers are added at a summing point to produce the controller output signal. Figure 7-4. Simplified diagram of a controller operating in the proportional-plus-integral-plusderivative (P.I.D.) mode. The controller output at any specified time, t, is given by: where C O (t) is the controller output at a specified time; E P is the error at a specified time; K P is the proportional gain; K I is the integral gain; K D is the derivative gain; C O (t 0 ) is the controller output at the time the observation starts (t = 0). 7-5
In this equation, the first term corresponds to the proportional action, the second term corresponds to the integral action, and the third term corresponds to the derivative action. Thus, the controller looks at the current value of the error, the integral of the error over a recent time interval, and the current derivative of the error to determine not only how much of a correction to apply, but for how long. Figure 7-5 shows an example of what happens in a P.I.D. control system when the error changes suddenly. Both the proportional and derivative actions react immediately when the error changes by peaking to a high positive value. After this initial peak, however, the derivative action, which is directly related to the rate of change of the proportional action, becomes negative due to the decreasing proportional action. The derivative action acts in a way that opposes the proportional action, which in turn reduces overshooting of the actuator speed and causes it to stabilize quickly. Figure 7-5. Example of what happens in a proportional-plus-integral-plus-derivative system when the error changes suddenly. It is important to note that the P.I.D. mode must be used with care in systems where excessive noise is present. The derivative action tends to amplify noise, which may cause the controller output signal to become noisy and may result in system 7-6
instability. The derivative action may also overreact to sudden changes in error, resulting in increased overshooting of the actuator speed. Such problems can be overcome by adding a lowpass filter at the input of the derivative mode section of the controller. The purpose of the filter is to restore system stability by eliminating any change in error signal that is faster than the response of the system. The PID Controller supplied with your trainer is equipped with such a filter. Comparison of the Proportional, Proportional-Plus-Integral, and Proportional- Plus-Integral-Plus-Derivative Control modes Figure 7-6 is a comparison of the response of the actuator speed to a sudden change in setpoint with the proportional (P), proportional-plus-integral (P.I.), and proportional-plus-integral-plus-derivative (P.I.D.) control modes. Both the P.I. and P.I.D. modes eliminate the residual error inherent to the P. mode; With the P.I.D. mode, however, the derivative action reduces overshooting of the actuator speed and aids in stabilizing the system sooner than with the P.I. mode. The longer the derivative time is, the smaller the overshoot and the faster the stabilization time will be. However, the derivative time must not be too long. Indeed, there is a point beyond which increasing the derivative time will result in a very nervous, oscillatory system. Figure 7-6. Comparison of the proportional, proportional-plus-integral, and proportional-plusintegral-plus-derivative control modes. P.I.D. Controller Configuration P.I.D. controllers can be configured in different ways. Two common types of configurations are the "ideal" and "parallel" configurations. In the "ideal" configuration, the proportional amplifier is connected in series with the integral and derivative amplifiers, which are themselves connected in parallel, as Figure 7-4 shows. With this configuration, the proportional amplifier interacts with the integral and derivative amplifiers. As a result, increasing the proportional gain will at the same time increase the integral and derivative actions. 7-7
In the "parallel" configuration, the proportional, integral, and derivative amplifiers are connected in parallel. This results in a minimum interaction between these amplifiers, as Figure 7-7 shows. Hence, increasing the proportional gain will not affect the integral and derivative actions. Figure 7-7. Simplified diagram of the parallel configuration. Procedure summary In the first part of the exercise, Familiarization with the Proportional-Plus-Integral- Plus-Derivative Controller, you will familiarize yourself with the components and operation of a proportional-plus-integral-plus-derivative controller. 7-8
In the second part of the exercise, Proportional-Plus-Integral-Plus-Derivative Control of Motor Speed, you will study proportional-plus-integral-plus-derivative control of motor speed. You will observe the effect of increasing the integral and derivative gains on the response of the motor speed (feedback voltage) during a sudden change in setpoint. EQUIPMENT REQUIRED Refer to the Equipment Utilization Chart, in Appendix A of the manual, to obtain the list of equipment required to perform this exercise. PROCEDURE Familiarization with the Proportional-Plus-Integral-Plus-Derivative Controller G 1. Connect the circuit shown in Figure 7-8. In this system, an ideal P.I.D. controller is built with the ERROR DETECTOR, the PROPORTIONAL, INTEGRAL, and DERIVATIVE AMPLIFIERS, the SUMMING POINT, and the LIMITER section of the PID Controller. The RAMP GENERATOR is used to study the operation of the derivative amplifier. Note: Do not connect the 0-10 V output of the f/e converter to the negative input of the ERROR DETECTOR at this time. Also, do not connect the Bidirectional Motor to the Conditioning Unit at this time. These connections will be done later in the exercise. Note: To minimize the pressure drops, use a tube as short as possible between the outlet port of the motor and the inlet port of flow control valve FCV1, and between the outlet port of flow control valve FCV1 and the muffler module. G 2. Make the following settings on the PID Controller: RAMP 1........................................... MAX. PROPORTIONAL (P) GAIN range...................... LOW PROPORTIONAL (P) GAIN............................ MIN. INTEGRAL (I) GAIN............................. ½ of MAX. DERIVATIVE (D) GAIN............................... MAX. INTEGRATOR ANTI-RESET............................. O LOWER LIMIT....................................... MIN. UPPER LIMIT.................................. ½ of MAX. G 3. Turn on the DC Power Supply and PID Controller. G 4. On the PID Controller, set the SETPOINT potentiometers 1 and 2 to obtain 0.0 V and 0.25 V, respectively, at the proportional amplifier input. Select the SETPOINT potentiometer 2. 7-9
Figure 7-8. Proportional-plus-integral-plus-derivative (P.I.D.) control of motor speed. 7-10
G 5. On the PID Controller, set the PROPORTIONAL GAIN to obtain 0.25 V at the proportional amplifier output. This will set the proportional amplifier gain at about 1. Monitor the voltage at the output of the integral amplifier. Once this voltage has reached the positive saturation level of about 14.0 V, set the INTEGRATOR ANTI-RESET switch at I and set the UPPER LIMIT potentiometer to obtain 10.0 V at the integral amplifier output. Measure the voltage at the input of the derivative amplifier. This voltage corresponds to the proportional amplifier output voltage of 0.25 V. Measure the voltage at the derivative amplifier output. Is the voltage null (0.0 V)? Why? G 6. Readjust the SETPOINT potentiometer 2 to obtain 10.0 V at the derivative amplifier input. G 7. Select the SETPOINT potentiometer 1. This will cause the voltage at the input of the derivative amplifier to decrease from 10.0 V to 0.0 V at a rate of 3.3 V/s, as set by the RAMP GENERATOR potentiometer 1. While doing this, monitor the voltage at the derivative amplifier output. You should observe that the voltage stays at about -3.3 V for 3 s before returning to 0.0 V. The negative polarity indicates that the voltage decreases at the input of the derivative amplifier. Is this your observation? G Yes G No G 8. Select the SETPOINT potentiometer 2. This will cause the voltage at the input of the derivative amplifier to increase from 0.0 V to 10.0 V at a rate of 3.3 V/s, as set by RAMP GENERATOR potentiometer 1. While doing this, monitor the voltage at the derivative amplifier output. You should observe that the voltage stays at about 3.3 V for 3 s before returning to 0.0 V. The positive polarity indicates that the voltage increases at the input of the derivative amplifier. Is this your observation? G Yes G No G 9. Based on your observations, determine the current gain setting of the derivative amplifier, in seconds. 7-11
G 10. Select the SETPOINT potentiometer 1, and set the potentiometer to obtain 1.0 V at the proportional amplifier input. G 11. Measure the voltages at the SUMMING POINT inputs. These voltages correspond to the proportional, integral, and derivative amplifier output voltages, and should be about 1.0 V, 10.0 V, and 0.0 V, respectively. Measure the voltage at the SUMMING POINT output. Is this voltage about 11.0 V? Explain. G 12. On the PID Controller, measure the voltage at the LIMITER output. Is this voltage approximately 10.0 V? Explain. G 13. On the PID Controller, remove the RAMP GENERATOR from the controller circuit by disconnecting the leads at its terminals. Remove also the lead between the LIMITER output and the 0-10 V input of the Servo Control Valve. Proportional-Plus-Integral-Plus-Derivative Control of Motor Speed Preliminary settings Note: In order to obtain a motor speed which is stable, the motor should run at high speed during 1 minute. To do so, disconnect the motor from the circuit of Figure 7-8, put some pneumatic oil in the motor ports and connect the circuit shown in Figure 6-6. On the Conditioning Unit, open the main shutoff valve and the required branch shutoff valve at the manifold. Set the main pressure regulator to obtain 630 kpa (90 psi) on the regulated pressure gauge. After 1 minute approximately, close the shutoff valves and turn the regulator adjusting knob completely counterclockwise. Connect your motor in the circuit of Figure 7-8 and proceed with the rest of the exercise. 7-12
G 14. On the PID Controller, connect the SETPOINT output 1 to the 0-10 V input of the Servo Control Valve. Set the SETPOINT potentiometers 1 and 2 to obtain 0.0 V and 10.0 V, respectively, at the SETPOINT output 1. Then select the SETPOINT potentiometer 1. G 15. On the Conditioning Unit, open the main shutoff valve and the required branch shutoff valves at the manifold. Set the main pressure regulator to obtain 630 kpa (90 psi) on the regulated pressure gauge. G 16. Open the Flow Control Valve FCV1 completely (fully counterclockwise). G 17. Set the position of the pilot of the proportional control valve to obtain 1.0 V at the output N of the f/e converter. This corresponds to a speed of 1000 r/min. Refer to the section Positioning of the proportional control valve pilot in Exercise 6 if necessary. Span setting G 18. On the PID Controller, select the SETPOINT potentiometer 2. Set the flow control valve FCV1 to obtain 1.6 V at the output N of the f/e converter. This setting corresponds to a motor speed of 1600 r/min. G 19. Connect the DC voltmeter to the 0-10 V output of the f/e converter. Calibrate this output so that it provides 10.0 V using the potentiometer S (span). G 20. Once this setting is completed, open flow control valve FCV1 completely (fully counterclockwise). Note: Reduce the motor speed by decreasing the air flow with flow control valve FCV1 if the LED of the photoelectric switch skips or does not lite. 7-13
G 21. Turn off the PID Controller. Connect the SETPOINT output 1 to a positive input of the ERROR DETECTOR and reconnect the lead between the LIMITER output and the 0-10 V input of the Servo Control Valve. Place the system in the closed-loop mode by connecting the 0-10 V output of the f/e converter to the negative input of the ERROR DETECTOR. G 22. Make the following settings on the PID Controller: PROPORTIONAL (P) GAIN range....................... LOW PROPORTIONAL (P) GAIN....................... ¼ of MAX. INTEGRAL (I) GAIN range........................ ½ of MAX. DERIVATIVE (D) GAIN........................... ½ of MAX. System operation G 23. Turn on the PID Controller. Once the motor speed is stabilized, measure the voltage at the ERROR DETECTOR output. Is the error between the setpoint and feedback voltage (measured speed) null? Explain. G 24. Connect the DC voltmeter to the negative input of the ERROR DETECTOR. G 25. On the PID Controller, select the SETPOINT potentiometer 1 to reduce the motor speed, then set the DERIVATIVE GAIN to MIN. G 26. Create a 0-10 V step change in setpoint by selecting the SETPOINT potentiometer 2. You should observe that the feedback voltage stabilizes to the 10.0-V setpoint, but that it overshoots the setpoint before it stabilizes. Is this your observation? G Yes G No Note: If you answered "no" to the question, vary the PROPORTIONAL gain until the system operates as indicated. G 27. On the PID Controller, select the SETPOINT potentiometer 1 to reduce the motor speed. G 28. Now add derivative action to the controller by setting the DERIVATIVE GAIN at ½ of MAX. 7-14
G 29. Select the SETPOINT potentiometer 2 and observe what happens to the feedback voltage. You should observe that the addition of derivative action to the controller has reduced the overshooting of the feedback voltage and has damped the tendency toward instability. Is this your observation? G Yes G No G 30. Repeat the 0-10 V step change in setpoint with different DERIVATIVE GAINS. You will observe that the longer the derivative time, the smaller the overshoot and the faster the stabilization time. However, there is a point beyond which increasing the derivative time will result in a very nervous, oscillatory system. G 31. On the Conditioning Unit, close the shutoff valves, and turn the regulator adjusting knob completely counterclockwise. G 32. Turn off the PID Controller and the DC Power Supply. G 33. Disconnect and store all leads and components. CONCLUSION In this exercise, you studied proportional-plus-integral-plus-derivative control of motor speed. You saw that, similar to proportional-plus-integral control, proportionalplus-integral-plus-derivative control automatically eliminates the error between the setpoint and feedback voltage due to the integral action of the controller. You saw the effect of increasing the integral and derivative gains on the response of the motor speed (feedback voltage) during a sudden change in setpoint. REVIEW QUESTIONS 1. What does derivative gain mean? 2. What happens to the output signal of a derivative amplifier when the signal at its input increases gradually? 7-15
3. When is the output signal of a derivative amplifier equal to zero? 4. The higher the derivative gain is, the smaller is the input rate of change required to produce an output. G True G False 5. What is the advantage and disadvantage of derivative control? 6. What is the difference between ideal and parallel configurations? 7-16