UNIVERSITI KUALA LUMPUR BPB25103 CONTROL SYSTEM ASSESSMENT LABORATORY 1: TIME RESPONSE OF A CONTROL SYSTEM LECTURER: PROF. DR. BADRI ABU BAKAR GROUP: L02-B02
No 1. 2.
Student Name MUHAMMAD IKRAM BIN SHABRY MUHAMMAD SYAMIR BIN SUBRI Marking Scheme
ID 51212115124 51212115088 Marks
Student 1 ID /15 /15 /10 /10 /50
Student 2 ID /15 /15 /10 /10 /50
Task 2 CLO1
/15 /15 /10 /10
/15 /15 /10 /10
TOTAL CLO1 TOTAL
/50 /100
/50 /100
Task 1 CLO5
TOTAL CLO5
TABLE OF CONTENTS
Page 1.
PART A: Proportional Control of Servo Trainer Speed 1.1 Introduction 1.2 Procedure/ Setup 1.2.1 Part 1: Steady State Error 1.2.2. Part 2: Transient Response 1.3 Results 1.3.1 Part 1: Steady State Error 1.3.2. Part 2: Transient Response 1.4 Analysis / Discussion 1.5 Conclusion
2.
PART B: Proportional Plus Integral Control of Servo Trainer Speed 2.1 Introduction 2.2 Procedure/ Setup 2.2.1 Part 1: Effect of Integral Action on Steady State Errors 2.2.2. Part 2: Selection of Integral and Proportional Controller Gains 2.3 Results 2.3.1 Part 1: Effect of Integral Action on Steady State Errors 2.3.2. Part 2: Selection of Integral and Proportional Controller Gains 2.4 Analysis/ Discussion 2.5 Conclusion
1. PART A: PROPORTIONAL CONTROL of SERVO TRAINER SPEED
1.1 INTRODUCTION
Some of the general properties of a closed loop system can be illustrated by a simple example of speed control of a motor. The CE110 Servo Trainer (adjacent to CE120 Controller) is used in this experiment. The CE110 Servo Trainer is a device which are designed specifically for the theoretical study and practical investigation of basic and advanced control engineering principles. Page 1 of 39
Figure 1: The CE110 Servo Trainer used in Experiment 3.
The objectives of this experiment is to implement a proportional controller of the Servo Trainer speed and to investigate the closed transient response and the steady state errors.
Proportional Controllers
Proportional controller is also called as gain controller. It is where the output position is related to the deviation of the set point and the measured control process value. Proportional controller can be used where the processing time of equipment or process is very large or where error magnitude is not needed to minimize to zero. It is because the name specifies its output that will be the error deviation multiplied by a factor (electronic circuit gain). In other words, in the Page 2 of 39
proportional control algorithm, the controller output is proportional to the error signal, which is the difference between the set point and the process variable. In other words, the output of a proportional controller is the multiplication product of the error signal and the proportional gain.
Mathematically, it can be expressed as
Or
Removing the sign of proportionality we have,
Where, Kp is proportional constant also known as controller gain. It is recommended that Kp should be kept greater than unity. If the value of Kp is greater than unity, then it will amplify the error signal and thus the amplified error signal can be detected easily.
Proportional Offset Page 3 of 39
In proportional controller, the deviation will become zero if the control variables reaches the set point. Hence, the output valve (the multiplication of zero) will become zero. The proportional offset also can helps in bringing the controller variable near to the set-point which will helps to reduce the error in steady state.
Advantages of Proportional Controller The advantages of proportional controller are as follows: 1. Proportional controller helps in reducing the steady state error, which will make the system more stable. 2. With the help of proportional controller, the slow response of the over damped system can be faster. Disadvantages of Proportional Controller The disadvantages of proportional controller are as follows: 1. Due to presence of these controllers we some offsets in the system. 2. Proportional controllers also will increase the maximum overshoot of the system.
For Part 1 (Steady State Errors) of Experiment 3, it is to verify that the steady state error, e ss, for a constant reference signal, yr as follows:-
………………………equation 1
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Figure 2: Response characteristic
Figure 3: Block diagram of the closed-loop system in Experiment 3 (Part 1) While for Part 2 (Transient Response) of Experiment 3, it is to investigate on how the transient response of the Servo Trainer is affected by the proportional controller gain, kp. The closed-loop time constants, Tcl from the graph produced by the chart recorder are compared with the theoretical values obtained using the equation below:-
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…………………………. equation 2
Figure 4: Block diagram of the closed-loop system in Experiment 3 (Part 2)
1.2 PROCEDURE/SETUP
1.2.1 Part 1: Steady State Error
Initial Controller Settings:
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CE110 Clutch disengaged Large inertial load installed. Rear access door firmly closed. CE120 Potentiometer turned full anti-clockwise (i.e. set to 0V output) PID Controller: proportional gain set to 10 and switched in, derivative and integral blocks switched out.
………………………….. E3.1
PROCEDURE
1. The equipment is connected as shown in Figure 3.2. 2. The steady state error is being investigated whether it is proportional to reference signal, yr. Page 7 of 39
3. The steady state error signal is being recorded as we increasing the reference speed, in step by step according to the given potentiometer output which is in step of 1V from 2V until 10V. 4. The steady state error where the value of kp=10, G1=1 and yr, reference speed provided in the table are being calculated by using equation given E3.1. 5. The steady state error is being investigated whether it is inversely proportional to the controller gain, kp, by setting the potentiometer to 5V for the reference speed signal, yr. 6. The value of gain, kp, is being adjusted step by step from 1 until 10 and the corresponding error signal reading was recorded. 7. The theoretical values of the error for each k p value was calculated after recording the practical value, by using equation E3.1.
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Figure E3.1
Figure 5: Experimental setup of experiment 1.2.2 Part 2: Transient Response
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Initial Controller Settings: CE110 Clutch disengaged. Large inertial load installed. Rear access door firmly closed. CE120 Potentiometer set to 5V. Function generator set to square wave where frequency is 0.05Hz, offset 0V and level 1V. PID controller proportional kp=1, integral and derivative blocks switched out.
PROCEDURE
1. The equipment is connected as shown in Figure E3.3. 2. The transient response of the servo trainer is being investigated on how it is affected by the proportional controller 3. A series of step changes in reference speed was generated by using the square wave output. 4. The corresponding speed response using the chart recorder (suggested time base 10mm/sec) for proportional gains of kp=0.5, 1, 2, 4 was plotted. 5. The closed-loop time constants, Tcl1 was calculated with the result from the graph with the theoretical values obtain using the equation E3.2
…………………………. E3.2
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Figure E3.3
Figure 6: Experimental setup of experiment
1.3 RESULTS
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1.3.1 Part 1: Steady State Error Potentiometer Output (Reference Speed, yr) (V)
Measured Steady Speed Error Signal (V)
Theoretical Steady State Error Signal (V)
Percentage Error (%)
2 3 4 5 6 7 8 9 10
0.19 0.27 0.35 0.44 0.53 0.61 0.69 0.78 0.86
0.18 0.27 0.36 0.46 0.55 0.64 0.73 0.82 0.91
5.56 0 2.78 4.35 3.64 4.69 5.48 4.89 5.49
Table 1: Steady State Error for Various Reference Speeds
Proportional Controller Gain kp
Measured Steady State Error Signal (V)
Theoretical Steady State Error Signal (V)
Percentage Error (%)
1 2 3 4 5 6 7 8 9 10
2.34 1.56 1.17 0.94 0.79 0.68 0.60 0.53 0.48 0.44
2.50 1.67 1.25 1.00 0.83 0.71 0.62 0.55 0.50 0.45
6.40 6.59 6.40 6.00 4.82 4.23 3.23 3.64 4.00 2.22
Table 2: Steady State Error for Various Controller Gains 1.3.2 Part 2: Transient Response
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Gain, kp
Measured Closed Loop Time Constant (sec)
Theoretical Closed Loop Time Constant (sec)
Percentage Error (%)
0.5
0.78
1.00
22.00
1
0.71
0.75
5.33
2
0.52
0.50
4.00
4
0.36
0.30
20.00
Table 3: Comparison of Measured Closed Loop Time Constants with Theoretical Values
Table 3: Comparison of Measured Closed Loop Time Constants with Theoretical Values
The percentage error for each result was calculated using this formula:
% error =
Theoretical−Measured × 100 T heoretical
CALCULATION Part 1: Steady State Error Page 13 of 39
For Experiment 3, the formulae that will be used to calculate theoretical values of steady state error signal is: yr e = ss Steady state error, 1+ k p G1 Where; y r , reference speed k p , proportional gain G1 = 1 y r , reference speed ( y r ,=2,3,4,5,6,7,8,9,10 )( k p =10 )
Varying values of
yr 2
3
4 5
6
ess e ss =
yr 2 = =0.182 1+ k p G1 1+(10)(1)
e ss =
yr 3 = =0.272 1+ k p G1 1+(10)(1)
e ss =
yr 4 = =0.364 1+ k p G1 1+(10)(1)
e ss =
yr 5 = =0.455 1+ k p G1 1+(10)(1)
e ss =
yr 6 = =0.545 1+ k p G1 1+(10)(1)
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7
8
9
10
Varying values of
e ss =
yr 7 = =0.636 1+ k p G1 1+(10)(1)
e ss =
yr 8 = =0.727 1+ k p G1 1+(10)(1)
e ss =
yr 9 = =0.818 1+ k p G1 1+(10)(1)
e ss =
yr 10 = =0.91 1+ k p G1 1+(10)(1)
k p , proportional gain (
k p =1,2,3,4,5,6,7,8,9,10 ) y r ,=5 ¿ ¿
kp 1 2
3
4
5
ess e ss =
yr 5 = =2.5 1+k p G1 1+(1)(1)
e ss =
yr 5 = =1.67 1+ k p G1 1+( 2)(1)
e ss =
yr 5 = =1.25 1+ k p G1 1+(3)(1)
e ss =
e ss =
yr 5 = =1 1+ k p G1 1+( 4)(1)
yr 5 = =0.83 1+ k p G1 1+(5)(1)
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6
7
8
9
10
e ss =
yr 5 = =0.714 1+ k p G1 1+( 6)(1)
e ss =
yr 5 = =0.625 1+ k p G1 1+(7)(1)
e ss =
yr 5 = =0.56 1+ k p G1 1+( 8)(1)
e ss =
e ss =
yr 5 = =0.5 1+ k p G1 1+(9)( 1)
yr 5 = =0.45 1+ k p G1 1+(10)(1)
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Part 2: Transient Response Calculation For this part of the experiment, the formulae that will be used to calculate theoretical values of closed loop time constant is: Closed-loop time constants,
T cl = l
T 1+k p G1
Where: T , time=1.5 sec G1=1
Varying values of
k p =0.5,1,2,4
kp
Tcl1
0.5
T 1.5 T cl = = =1 1+k p G1 1+(0.5)(1)
1
2
4
l
T cl = l
T 1.5 = =0.75 1+k p G 1 1+(1)(1)
T cl =
T 1.5 = =0.5 1+k p G1 1+(2)(1)
T cl =
T 1.5 = =0.3 1+k p G1 1+(4)(1)
l
l
The Speed Response Plotted by Chart Recorder for Proportional Gains of kp = 0.5, 1, 2, 4 (Part 2)
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Figure 7: Speed response for kp=0.5, the measured closed loop time constant is 0.78s
Figure 8: Speed response for kp=1, the measured closed loop time constant is 0.71s
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Figure 9: Speed response for kp=2, the measured closed loop time constant is 0.52s
Figure 10: Speed response for kp=4, the measured closed loop time constant is 0.36s
1.4 ANALYSIS and DISCUSSION
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From the results obtained from Part 1 of Experiment 3 which has been recorded in Table 1 and Table 2, it shows that there are a slight difference between the readings obtained from practical and the one obtained theoretically. Based on the results obtained from both practical and theoretical as seen in Table 1, it can be seen that the greater the value of reference speed, y r, the greater the measured steady-state error signal will be. For the results shown in Table 2, it shows that as the value of potentiometer controller gain, k p used increase, the measured steady state error signal will decrease. We can simply said that the steady state error is directly proportional to the constant reference signal, yr and inversely proportional to the proportional controller gain, kp. These relationship can be clearly seen in equation below,
where ess = steady state error yr = reference speed/constant kp = potentiometer controller gain.
Supposedly, the difference between the measured and theoretical steady state errors decreases when the yr increases. This happens because errors were being introduced by the small dead-zone in the servo trainer. The errors introduced by the small dead-zone become smaller as the reference signal becomes much larger than the dead-zone width. But based on the results obtained from Part 1 of Experiment 3, the difference between the measured and theoretical steady state errors are not fixed (not decrease or increase as y r increases). This might be due to some errors done in the experiment. For Part 2 of Experiment 3, there are minor difference between the readings obtained from practical and the one obtained theoretically. The results has been recorded in Table 3. It can be noticed that the measured closed loop time constant will decrease as the gain, k p increase. Page 20 of 39
Measured closed loop time constant are inversely proportional with gain, k p. This is proved by the equation below:
Where Tcl = closed loop time constant kp = proportional controller gain.
This is because a proportional controller (k p) will have the effect of reducing the rise time and also will reduce the steady state error but never eliminate it. The difference between the actual and theoretical value obtained from Part 2 of Experiment 3 is not fixed. Allegedly, the difference should increase as the gain kp increases. This is because the actual and theoretical closed loop time constant will start to deviate far at higher values of k p. it is due to under high gain conditions, the drive amplifier saturates under transient conditions.
The minor difference of the readings may due to the error of the equipment itself. To avoid or minimize the error, the equipment must be calibrated accordingly. However, the results obtained from Experiment 3 still can be accepted as the average percentage error is only 20%. It can verify the theory of control system.
1.5 CONCLUSION
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As a conclusion for Experiment 3, the objectives of implementing a proportional controller of the Servo Trainer speed and to investigate the closed transient response and the steady state error has been achieved. For Part 1, it can be concluded that the steady state error is directly proportional to the reference speed yr but inversely proportional to the proportional controller gain, kp. For Part 2, the measured closed loop time constant are inversely proportional with gain, k p. This shows that a proportional controller of a Servo Trainer has been implemented. The closed transient response and the steady state error has been investigated. In this experiment, the specification of the transient response investigated is the rise time of a closed loop system. Rise time is the time required for the response to rise from 0% to 100% of the final value. The minor difference between the readings which may be caused by the equipment still can be accepted as the average percentage error is around 20%. The theory are proven.
2. PART B: PROPORTIONAL PLUS INTEGRAL CONTROL of SERVO TRAINER SPEED
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2.1 INTRODUCTION A proportional–integral–derivative controller (PID controller) is a control loop feedback mechanism (controller) commonly used in industrial control systems. PID stands for “Proportional, Integral, Derivative,” and it has to do with how the controller does what it does.
Figure 1: Response of a typical PID closed loop system
Proportional Response The proportional component depends only on the difference between the set point and the process variable. This difference is referred to as the Error term. The proportional gain (Kc) Page 23 of 39
determines the ratio of output response to the error signal. For instance, if the error term has a magnitude of 10, a proportional gain of 5 would produce a proportional response of 50. In general, increasing the proportional gain will increase the speed of the control system response. However, if the proportional gain is too large, the process variable will begin to oscillate. If Kc is increased further, the oscillations will become larger and the system will become unstable and may even oscillate out of control.
Figure 2: Block diagram of a basic PID control algorithm.
Integral Response The integral component sums the error term over time. The result is that even a small error term will cause the integral component to increase slowly. The integral response will continually increase over time unless the error is zero, so the effect is to drive the Steady-State error to zero. Steady-State error is the final difference between the process variable and set point. A phenomenon called integral windup results when integral action saturates a controller without the controller driving the error signal toward zero.
Derivative Response The derivative component causes the output to decrease if the process variable is increasing rapidly. The derivative response is proportional to the rate of change of the process variable. Page 24 of 39
Increasing the derivative time (Td) parameter will cause the control system to react more strongly to changes in the error term and will increase the speed of the overall control system response. Most practical control systems use very small derivative time (Td), because the Derivative Response is highly sensitive to noise in the process variable signal. If the sensor feedback signal is noisy or if the control loop rate is too slow, the derivative response can make the control system unstable.
Controller
Rise Time
Overshoot
Settling Time
S-S Error
Kp
Decrease
Increase
Small Change
Decrease
Ki
Decrease
Increase
Increase
Eliminate
Kd
Small change
Decrease
Decrease
Small change
Table 1: The characteristic of PID controller.
2.2 PROCEDURE/SETUP 2.2.1 Part 1: Effect of Integral Action on Steady State Errors Page 25 of 39
Initial Control Settings: CE110 Clutch disengaged Rear door firmly closed Largest inertial load installed CE120 Potentiometer turn fully anti-clockwise (i.e. set to 0V input). PID Controller: Proportional gain set to 1, integral gain set to 0.1 and switched out. Differential gain switched out. Function generator: select offset zero, level zero DC.
PROCEDURE
1. The equipment was connected as shown in Figure E4.1. 2. The potentiometer output voltage was slowly increased to 4V and the steady state error
was being observed. 3. The error signal was observed as integral action takes effect as follows: with k i=0.1, the integrator reset button is pressed and the integrator is switched into controller. 4. Observing the speed slowly increase and the error signal slowly decrease to zero as the
integrator output increase as to cancel the error. 5. The integrator was switched out of the circuit. 6. The above procedure for k1=0.5, 1, 2, 4, 6, and 10 was being repeated. 7. As k1 is increased the error is reduced to zero more rapidly until a points is reached when
the error overshoots zero and oscillates before setting to zero. 8. The k1 is increased as the oscillations pronounced.
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Figure E4.1
Figure 3: Experimental setup of experiment Page 27 of 39
2.2.2 Part 2: Selection of Integral and Proportional Controller Gains Initial Control Settings: CE110 Clutch disengaged Rear access door firmly closed Largest inertial load installed CE120 Potentiometer set to 5V PID Controller: proportional gain set to 1, integral gain set to3 Deferential gain switches out Function generator: select square wave, frequency 0.05Hz, offset zero and level 1V 1. The equipment has been connected as shown in Figure E4.1. 2. The square wave generator signal provides a series of step changes in the reference signal which can be used to investigate the step response of the servo-speed control system 3. The effect of proportional gain upon the control system step response was investigated by
plotting the response values of kp =1, 0.1 and 0.01. 4. The shape of the results in terms of speed of response and amount of overshoot was
commented. 5. The effect of the integral gain upon the control system step response by setting kp = 1 was investigated. 6. The step response for the value of k1 = 0.5, 1, 5, and 10 was being plotted. 7. The shape of the results in terms of speed of response and amount of overshoot was
commented.
2.3 RESULTS
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2.3.1 Part 1: Effect of Integral Action on Steady State Errors Value
Signal
of ki 0.5
1
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2
4
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6
10
Table 2: The Effect of Varying Integral Gain, Ki to the Steady State Error and the Signal
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2.3.2 Part 2: Selection of Integral and Proportional Controller Gains Kp Varies Value
Signal
of kp 0.01
0.1
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1
Table 3: The Effect of Proportional plus Integral Control Upon the Servo Motor Speed Control
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Ki Varies Value
Signal
of ki 0.5
1
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5
10
Table 4: The Effect of Proportional plus Integral Control Upon the Servo Motor Speed Control
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2.4 ANALYSIS and DISCUSSION
The results obtained from Part 1 of Experiment 4 has been tabulated in Table 2. From the results, we can see that as the integral control gain, ki increases, the rise time will become faster thus the steady state error was eliminated. At the rate of the integral control gain k i equals to 1, the steady state error has been removed and oscillations begins to appear. As the integral control gain ki, gets higher than 1, more oscillations appeared and higher overshoot occurs before it reach the steady state. The settling time also increase as the integral controller ki increase. This is because an integral control, ki will have the effect of eliminating the steady state error but it makes the transient response worse. It can be seen in this Part 1 of Experiment 4 where the steady state was eliminated when the integral control ki equals to 1 but there were more oscillations appeared and higher overshoot occurred as the integral control gets higher. For Part 2 of Experiment 4, the results has been tabulated in Table 3 and Table 4. For this part, it is about the selection of Integral and Proportional Controller gain, k i and kp. At constant integral controller ki = 3, the proportional controller kp being varied which is at 0.01, 0.1 and 1. As the results seen in Table 3, it shows that when the proportional gain k p was increased (altered) from 0.01 to 1, it shows that the damping of the response has been altered or reduced. The overshoot also has been reduced. From the results obtained in Table 3, it can be said that at constant integral controller ki=3, the most suitable value for proportional controller k p is 1 as the signal produced is the most stable with less overshoot and oscillation, less settling, short rise time and optimum disturbance rejection. Using the same equipment setup, the integral controller k i were varied at 0.5, 1, 5 and 10 at constant proportional controller kp=1. As the results seen in Table 4, it shows that at the value of ki=1, the steady state error was being eliminated. As the value of k i increases, overshoot and oscillations begins to occur and gets increasing. Damping of the response also increases as k i increases. But the rise time gets shorten at higher ki. From the results obtained in Table 4, it can be said that at constant proportional controller k p=1, the most suitable value for proportional controller ki is 5 as the signal produced is the most stable with less overshoot and oscillation, less settling time, short rise time and optimum disturbance rejection.
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2.5 CONCLUSION As a conclusion for Experiment 4, the effect of proportional plus integral control (k p and ki) upon the servo motor speed control loop in terms of steady state errors, disturbance rejection and transient response has been successfully investigated. The objective of this Experiment 4 is achieved. For Part 1 of experiment 4, as the integral control gain k i, gets higher than 1, more oscillations appeared and higher overshoot occurs before it reach the steady state. The settling time also increase as the integral controller ki increase and the steady state error was eliminated. For Part 2, at constant integral controller k i=3, the most suitable value for proportional controller kp is 1 as the signal produced is the most stable with less overshoot and oscillation, less settling, short rise time and optimum disturbance rejection. At constant proportional controller k p=1, the most suitable value for proportional controller ki is 5 as the signal produced is the most stable with less overshoot and oscillation, less settling time, short rise time and optimum disturbance rejection. The best selection of proportional and integral controller is the one which produce a stabilized signal with less overshoot, less rise time and settling time, and with optimum disturbance rejection. All the analysis and conclusion were explained using the table below which represent the characteristic of PID controller as a reference for the experiment.
Controller
Rise Time
Overshoot
Settling Time
S-S Error
Kp
Decrease
Increase
Small Change
Decrease
Ki
Decrease
Increase
Increase
Eliminate
Kd
Small change
Decrease
Decrease
Small change
Table 7: The characteristic of PID controller.
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(n.d.).
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from
https://www.mathworks.com/help/control/getstart/tune-pid-controller-to-balancetracking-and-disturbance-rejection.html 3. (n.d.).
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20,
2017,
from
http://www.sciencedirect.com/science/article/pii/S0019057809000585 4. P, PI and PID Controllers. (n.d.). Retrieved April 20, 2017, from https://gradeup.co/p-piand-pid-controllers-i-ba51cc88-c453-11e5-8e45-0f580d23b1d8
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