The tuning of PID is explained

1. PID Controller Tuning
Comparison of classical tuning methods
1

2. Content
June 16, 2015 2
▶ Introduction
▶ Objectives
▶ Closed-loop Methods
 Ziegler-Nichols Closed-loop
 Tyreus-Luyben
 Damped Oscillation
▶ Open-loop Methods
 Ziegler-Nichols Open-loop
 C-H-R
 Cohen-Coon
 Ciancone-Marlin
 Minimum Error Integral
▶ Simulation and Results
▶ GUI Description

3. Introduction
▶ PID tuning is to find the optimum Kp, Ki and Kd for the controller.
Control objective > Setpoint tracking, Disturbance rejection
Actions > Instantaneous proportional action, Reset integral action, Rate derivative
action
Optimum criteria > Depends on application and system requirements
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 3

4. Introduction
▶ Conceptual real-world example
Driver
(PID)
Car mechanism
(Process)
Crosswind
Front wheels
angle Car position
Driver’s eyes
(Feedback)
Desired position
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 4

5. Introduction
▶ PID configuration
𝐶𝑜𝑛𝑡𝑟𝑜𝑙𝑙𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡 = 𝐾𝑝 𝑒(𝑡) + 𝐾𝑖 ∫ 𝑒(𝑡)𝑑𝑡 + 𝐾𝑑
d𝑒(𝑡)
𝑑𝑡
= 𝐾𝑐 × (1 +
1
𝑟𝑖
∫ 𝑒(𝑡)𝑑𝑡 + 𝑟𝑑
d𝑒(𝑡)
𝑑𝑡
)
𝐾𝑝𝑒(𝑡)
𝐾𝑖 ∫ 𝑒(𝑡)𝑑𝑡
d𝑒(𝑡)
𝐾𝑑
𝑑𝑡
SP
PV
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 5
Controller output
e(t)

6. Introduction
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▶ Many tuning methods have been proposed for PID controllers each of which
has its advantages and disadvantages. So, no one can be considered the best
for all purposes.
▶ Closed-loop methods tune the PID while it is attached to the loop while in
open-loop methods the process is estimated using a FOPDT model
▶ A comparison of the most popular methods is to be done
▶ Simulation will be implemented for 1st, 2nd and 3rd-order processes, some of
which are lag-dominant and the others are dead-time dominant.
▶ IAE as criterion (which adds up the time and amplitude weight of the error)

7. Objectives
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 7
▶ Compare studied tuning methods for performance and robustness
▶ Develop a GUI to do the comparison automatically for a given process model

8. Closed-loop methods
▶ Ziegler-Nichols Closed-loop
▶ Tyreus-Luyben
▶ Damped Oscillation
PID Process
D
C PV
Feedback
SP
Tuning
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 8

9. Open-loop methods
▶ Ziegler-Nichols Open-loop
▶ C-H-R
▶ Cohen-Coon
▶ Ciancone-Marlin
▶ Minimum Error Integral
PID Process
D
PV
Tuning
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 9

10. Ziegler-Nichols Closed-loop
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 10
▶ ¼ decay ratio as design criterion (stability condition)
▶ Trial-and-error procedure to find 𝑲𝒖 and 𝑷𝒖
▶ Drives the process into marginal stability
▶ Performs well when 𝑟𝒎 ≥ 𝟐𝒕𝒅 (lag dominant)
▶ Performs very poorly for 𝒕𝒅 > 𝟐𝑟𝒎 (dead-time dominant)
▶ Fast recovery from disturbance but leads to oscillatory response
▶ Not applicable to open-loop-unstable processes
▶ Some processes do not have ultimate gain

11. Ziegler-Nichols Closed-loop
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 11
Controller 𝐾𝑐 𝑟𝑖 𝑟𝑑
P 0.5𝐾𝑢 - -
PI 0.45𝐾𝑢 0.83𝑃𝑢 -
PID 0.6𝐾𝑢 0.5𝑃𝑢 0.125𝑃𝑢
▶ Procedure:
 Set 𝐾𝑖 and 𝐾𝑑 to 0
 Increase 𝐾𝑝 till sustained oscillation and find 𝐾𝑢 and 𝑃𝑢
 Use the correlations in the table below

12. Tyreus-Luyben
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 12
▶ An improvement for Ziegler-Nichols closed-loop to make response less
oscillatory
▶ More robust to imprecise model
▶ Gives better disturbance response
▶ Procedure:
 Same procedure as Ziegler-Nichols closed-loop
Controller 𝐾𝑐 𝑟𝑖 𝑟𝑑
PI 0.45𝐾𝑢 2.2𝑃𝑢 -
PID 0.313𝐾𝑢 2.2𝑃𝑢 0.16𝑃𝑢

13. Damped Oscillation
▶ Another improvement for Ziegler-Nichols closed-loop
▶ Solves the problem of marginal stability
▶ Can be used with open-loop-unstable processes
0.8
0.6
0.4
0.2
0
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 13
1
1.2
0 10 20 30 40 50 60 70 80
4:1

14. Damped Oscillation
Controller 𝐾𝑐 𝑟𝑖 𝑟𝑑
PI 𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑃𝑑/6 -
PID 𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑃𝑑/6 𝑃𝑑/1.5
▶ Procedure:[1]
 Set 𝐾𝑖 and 𝐾𝑑 to 0
 Increase 𝐾𝑝 till ¼ damping ratio is maintained and find 𝑃𝑑 only
 Use the correlations in the table below to find 𝑟𝑖 and 𝑟𝑑
 Adjust 𝐾𝑝 till ¼ damping ratio is maintained again
[1] Lipták, Béla G., and Kriszta Venczel. Instrument Engineers' Handbook: Process Control 4thed, Volume T
w
o
.
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 14

15. Ziegler-Nichols Open-loop
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▶ ¼ decay ratio as design criterion
▶ Performs well when 𝑟𝑚 ≥ 2𝑡𝑑 (lag dominant)
▶ Performs very poorly for 𝑡𝑑 > 2𝑟𝑚 (dead-time dominant)
▶ Fast recovery from disturbance but leads to oscillatory response

16. Ziegler-Nichols Open-loop
𝐺𝑚 𝑠 =
▶ Procedure:
 The process dynamics is modeled by a first order plus dead time model
𝐾𝑚𝑒−𝑡𝑑𝑠
𝑟𝑚𝑠 + 1
2.5
2
1.5
1
0.5
0
-0.5
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 16

17. Ziegler-Nichols Open-loop
 PID parameters are calculated from the table below
Controller 𝐾𝑐 𝑟𝑖 𝑟𝑑
P 1 𝑟𝑚
𝐾𝑚 𝑡𝑑
- -
PI 0.9 𝑟𝑚
𝐾𝑚 𝑡𝑑
𝑡𝑑
0.3
-
PID 1.2 𝑟𝑚
𝐾𝑚 𝑡𝑑
2𝑡𝑑 0.5𝑡𝑑
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 17

18. C-H-R
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 18
▶ A modification of Ziegler-Nichols Open-loop
▶ Aims to find the “quickest response with 0% overshoot” or “quickest
response with 20% overshoot”
▶ Tuning for setpoint responses differs from load disturbance responses

19. C-H-R
Setpoint
Controller 𝐾𝑐 𝑟𝑖 𝑟𝑑 𝐾𝑐 𝑟𝑖 𝑟𝑑
0% overshoot 20% overshoot
P 0.3 𝑟𝑚
𝐾𝑚 𝑡𝑑
- -
0.7 𝑟𝑚
𝐾𝑚 𝑡𝑑
- -
PI 0.35𝑟𝑚
𝐾𝑚 𝑡𝑑
1.2𝑟𝑚 -
0.6 𝑟𝑚
𝐾𝑚 𝑡𝑑
𝑟𝑚 -
PID 0.6 𝑟𝑚
𝐾𝑚 𝑡𝑑
𝑟𝑚 0.5𝑡𝑑
0.95𝑟𝑚
𝐾𝑚 𝑡𝑑
1.4𝑟𝑚 0.47𝑡𝑑
Disturbance
P 0.3 𝑟𝑚
𝐾𝑚 𝑡𝑑
- -
0.7 𝑟𝑚
𝐾𝑚 𝑡𝑑
- -
PI 0.6 𝑟𝑚
𝐾𝑚 𝑡𝑑
4𝑡𝑑 -
0.7 𝑟𝑚
𝐾𝑚 𝑡𝑑
2.3𝑡𝑑 -
PID 0.95𝑟𝑚
𝐾𝑚 𝑡𝑑
2.4𝑡𝑑 0.42𝑡𝑑
1.2 𝑟𝑚
𝐾𝑚 𝑡𝑑
2𝑡𝑑 0.42𝑡𝑑
▶ Procedure:
 Same as Ziegler-Nichols Open-loop
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 19

20. Cohen-Coon
▶ Second in popularity after Ziegler-Nichols tuning rules
▶ ¼ decay ratio has considered as design criterion for this method
▶ More robust
𝒕
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𝒅
𝑟 𝑑
(i.e. 𝑡 > 2𝑟)
▶ Applicable to wider range of
▶ PD rules available

21. Cohen-Coon
▶ Procedure:[1]
 The process reaction curve is obtained by an open loop test and the FOPDT
model is estimated as follows:
3
𝑟𝑚 =
2
𝑡2 − 𝑡1
𝑡𝑑 = 𝑡2 − 𝑟𝑚 1.5
2
2.5
1
0.5
0
-0.5
[1] Smith,C.A.,A.B. Copripio; “Principles and Practice of Automatic Process Control”, John Wiley & Sons,1985
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 21

22. Cohen-Coon
Controller 𝐾𝑐 𝑟𝑖 𝑟𝑑
P
1 𝑟𝑚 𝑡𝑑
(1 + )
𝐾𝑚 𝑡𝑑 3𝑟𝑚
- -
PI
1 𝑟𝑚 𝑡𝑑
(0.9 + )
𝐾𝑚 𝑡𝑑 12𝑟𝑚
30 + 3𝑡𝑑
𝑟𝑚
𝑡𝑑 20𝑡𝑑
9 + 𝑟𝑚
-
PD
1 𝑟𝑚 𝑡𝑑
(1.25 + )
𝐾𝑚 𝑡𝑑 6𝑟𝑚
-
6 − 2𝑡𝑑
𝑟𝑚
𝑡𝑑 3𝑡𝑑
22 + 𝑟𝑚
PID
1 𝑟𝑚 𝑡𝑑
(1.33 + )
𝐾𝑚 𝑡𝑑 4𝑟𝑚
32 + 6𝑡𝑑
𝑟𝑚
𝑡𝑑 8𝑡𝑑
13 + 𝑟𝑚
4
𝑡𝑑 2𝑡𝑑
11 + 𝑟𝑚
 PID parameters are calculated from the table
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 22

23. Ciancone-Marlin
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 23
▶ Design criteria:
 Minimization of IAE
 Assumption of ±25% change in the process model parameters
▶ A set of graphs are used for the tuning
▶ Tuning for setpoint responses differs from load disturbance responses

24. Ciancone-Marlin
▶ Procedure:
 Estimate the process with FOPDT as for Cohen-Coon method
 Calculate the ratio
𝑡𝑑
𝑡𝑑+𝑐𝑚
𝑐 𝑚
 From the appropriate graph determine the values (𝐾 𝐾 , ,
𝑐𝑖 𝑐𝑑
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 24
𝑡𝑑+𝑐𝑚 𝑡𝑑+𝑐𝑚
)
 Do the calculation to find the PID parameters

25. Ciancone-Marlin
0
0.5
1
1.5
0 0.5 1
1.5
1.3
1.1
0.9
0.7
0.5
0 0.5 1
1
0.8
0.6
0.4
0.2
0
0 0.5 1
1
0.8
0.6
0.4
0.2
0
0 0.5 1
setpoint
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Disturbance

26. Ciancone-Marlin
0.5
0
1
1.5
2
0 0.5 1
0.5
0
1
1.5
2
0 0.5 1
0.2
0
0.4
0.6
0.8
0 0.5 1
1
0.8
0.6
0.4
0.2
0
0 0.5 1
0.25
0.2
0.15
0.1
0.05
0
0 0.5 1
0.25
0.2
0.15
0.1
0.05
0
0 0.5 1
setpoint
Disturbance
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 26

27. Minimum Error Integral
▶ Considers the entire closed loop response not like the ¼-decay tuning methods which
considers only the first two peaks
▶ Less oscillations in response than ¼-decay
▶ Performs well when 𝑟𝒎 ≥ 𝟐𝒕𝒅 (lag dominant)
▶ Performs very poorly for 𝒕𝒅 > 𝑟𝒎 (dead-time dominant)
▶ Tuning for setpoint responses differs from load disturbance responses
▶ Different error integrals can be used (IAE, ISE, ITAE, ITSE)
∞
𝐼𝐴𝐸 = ∫ 𝑒(𝑡) 𝑑𝑡 ,
0
∞
𝐼𝑆𝐸 = ∫ 𝑒(𝑡)2 𝑑𝑡 ,
0
∞
𝐼𝑇𝐴𝐸 = ∫ 𝑡 𝑒(𝑡) 𝑑𝑡 ,
0
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 27
∞
𝐼𝑇𝑆𝐸 = ∫ 𝑡𝑒(𝑡)2 𝑑𝑡
0

28. Minimum Error Integral
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 28
▶ Procedure:
 Estimate the process with FOPDT as for Cohen-Coon method
 Use the appropriate table to find the PID parameters

29. Minimum Error Integral
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 29
Error integral IAE ITAE
PI Controller
𝑎1 𝑡𝑑
𝐾𝑐 = ( )𝑏1
𝐾𝑚 𝑟𝑚
𝑎1 = 0.758
𝑏1 = −0.861
𝑎1 = 0.586
𝑏1 = −0.916
𝑟𝑚
𝑟𝑖 = 𝑡𝑑
𝑎2 + 𝑏2(𝑟𝑚
)
𝑎2 = 1.02
𝑏2 = −0.323
𝑎2 = 1.03
𝑏2 = −0.165
PID Controller
𝑎1 𝑡𝑑
𝐾𝑐 = ( )𝑏1
𝐾𝑚 𝑟𝑚
𝑎1 = 1.086
𝑏1 = −0.869
𝑎1 = 0.965
𝑏1 = −0.855
𝑟𝑚
𝑟𝑖 = 𝑡𝑑
𝑎2 + 𝑏2(𝑟𝑚
)
𝑎2 = 0.74
𝑏2 = −0.13
𝑎2 = 0.796
𝑏2 = 0.147
𝑡𝑑
𝑟𝑑 = 𝑎3𝑟𝑚( )𝑏3
𝑟𝑚
𝑎3 = 0.348
𝑏3 = 0.914
𝑎3 = 0.308
𝑏3 = 0.9292
▶ Setpoint tracking table

30. Minimum Error Integral
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 30
Error integral IST IAE ITAE
P Controller
𝑎1 𝑡𝑑
𝐾𝑐 = ( )𝑏1
𝐾𝑚 𝑟𝑚
𝑎1 = 1.411
𝑏1 = −0.917
𝑎1 = 0.902
𝑏1 = −0.985
𝑎1 = 0.49
𝑏1 = −1.084
PI Controller
𝑎1 𝑡𝑑
𝐾𝑐 = ( )𝑏1
𝐾𝑚 𝑟𝑚
𝑎1 = 1.305
𝑏1 = −0.959
𝑎1 = 0.984
𝑏1 = −0.986
𝑎1 = 0.859
𝑏1 = 0.977
𝑟𝑚 𝑡𝑑
𝑟𝑖 = ( )𝑏2
𝑎2 𝑟𝑚
𝑎2 = 0.492
𝑏2 = 0.739
𝑎2 = 0.608
𝑏2 = 0.707
𝑎2 = 0.674
𝑏2 = 0.68
PID Controller
𝑎1 𝑡𝑑
𝐾𝑐 = ( )𝑏1
𝐾𝑚 𝑟𝑚
𝑎1 = 1.495
𝑏1 = 0.945
𝑎1 = 1.435
𝑏1 = −0.921
𝑎1 = 1.357
𝑏1 = −0.947
𝑟𝑚 𝑡𝑑
𝑟𝑖 = ( )𝑏2
𝑎2 𝑟𝑚
𝑎2 = 1.101
𝑏2 = 0.771
𝑎2 = 0.878
𝑏2 = 0.749
𝑎2 = 0.842
𝑏2 = 0.738
𝑡𝑑
𝑟𝑑 = 𝑎3𝑟𝑚( )𝑏3
𝑟𝑚
𝑎3 = 0.56
𝑏3 = 1.006
𝑎3 = 0.482
𝑏3 = 1.137
𝑎3 = 0.381
𝑏3 = 0.995
▶ Disturbance rejection table

31. Simulation and Results
▶ Simulation performed for two purposes:
 Performance Assessment
 Robustness Assessment
▶ Simulation for two response objectives:
 Set point tracking
 Disturbance rejection
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32. Simulation and Results
▶ Test cases include processes of:
 Dead-time dominant (𝑡𝑑 > 2𝑟𝑚)
 Lag dominant (𝑟𝑚≥ 2𝑡𝑑)
 In-between cases
 Complex poles
 Unstable process
1
1. 𝐺 𝑠 =
𝑠+1
2. 𝐺 𝑠
0.5𝑠+1
= 1
𝑒−0.2𝑠
3. 𝐺 𝑠 =
1
0.5+1
𝑒−1.2𝑠
4. 𝐺 𝑠
1
=
30𝑠2+13𝑠+1
5. 𝐺 𝑠 =
1
𝑠2+3𝑠+1
𝑒−0.2𝑠
6. 𝐺 𝑠 =
1
𝑠2+1.8𝑠+1
𝑒−3𝑠
7. 𝐺 𝑠 =
1
25𝑠+1 20𝑠+1 30𝑠+1
8. 𝐺 𝑠
150𝑠3+95𝑠2+18𝑠+1
= 2
𝑒−0.5𝑠
9. 𝐺 𝑠 =
2
2𝑠3+5𝑠2+4𝑠+1
𝑒−4.2𝑠
10. 𝐺 𝑠
250
=
𝑠2+4𝑠+50
11. 𝐺 𝑠
7𝑠2+28𝑠+28
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 32
=
10𝑠3−10𝑠2−50𝑠−30

33. Simulation Example (Closed-loop)
▶ 𝐺 𝑠 =
1
0.5+1
𝑒−1.2𝑠
Method 𝑲𝒑 𝑲𝒊 𝑲𝒅
Ziegler-Nichols Closed-
loop
0.63 0.24 0
Tyreus-Luyben 0.44 0.06 0
Damped Oscillation 0.76 0.28 0
Method Overshoot Rise time
Settling
time
Ziegler-Nichols Closed-
loop
0 9.41773 20.10063
Tyreus-Luyben 0 41.5833 78.08328
Damped Oscillation 0 1.14425 17.86827
Method IAE ITAE ISE
Ziegler-Nichols Closed-
loop
4.287635 21.66082 2.14574
Tyreus-Luyben 16.21587 326.4134 6.600629
Damped Oscillation 3.657051 16.38796 1.930914
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 33

34. Simulation Example (Open-loop)
▶ 𝐺 𝑠 =
1
0.5+1
𝑒−1.2𝑠
Method 𝑲𝒑 𝑲𝒊 𝑲𝒅
Ziegler-Nichols Open-loop 0.38 0.096 0
C-H-R 0.26 0.50 0
Cohen-Coon 0.46 0.59 0
Ciancone-Marlin 0.65 0.61 0
Minimum Error Integral 0.36 0.19 0
Method IAE ITAE ISE
Ziegler-Nichols Open-loop 10.62439 133.3877 4.672032
C-H-R 2.534889 4.215979 1.916891
Cohen-Coon 2.23463 3.378988 1.687213
Ciancone-Marlin 2.31806 4.337486 1.623838
Minimum Error Integral 5.443972 29.46653 2.827566
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 34

35. Robustness Assessment Example
▶ 𝐺 𝑠 =
1
𝑠2+3𝑠+1
𝑒−0.2𝑠
≫ 𝐺 𝑠 =
1
𝑠2+3.4𝑠+1
𝑒−0.4𝑠
Method 𝑲𝒑 𝑲𝒊 𝑲𝒅
Ziegler-Nichols
Closed-loop
7.38 5.13 0
Tyreus-Luyben 5.13 1.35 0
Damped Oscillation 8.26 4.36 0
Method ∆%Overshoot ∆%Rise time
∆%Settling
time
Ziegler-Nichols
Closed-loop
2.53E+46 0.005528
Tyreus-Luyben 0.780894 0.021236 0.222945
Damped Oscillation 7.51E+58 0.002601
Method ∆%IAE ∆%ITAE ∆%ISE
Ziegler-Nichols
Closed-loop
65535 65535 65535
Tyreus-Luyben 0.578426 1.141222 0.534852
Damped Oscillation 65535 65535 65535
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 35
---- After process parameters change
With original process parameters
▶ Only Tyreus Luyben method could preserve the
system stability in this example

36. Results
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 36
Method
Example 1 Example 2 Example 3 Example 4 Example 5 Example 6
Set. Dis. Set. Dis. Set. Dis. Set. Dis. Set. Dis. Set. Dis.
ZN-Closed - - 0.445789 0.283633 4.287635 4.173887 - - 2.220379 0.30278 13.41728 13.1761
Tyreus-Luyben - - 1.102981 1.070794 16.21587 15.8735 - - 1.180371 0.735662 50.61003 49.72932
Damped Oscillation - - 0.612071 0.236871 3.657051 3.591137 5.435811 0.227883 2.036804 0.273401 12.38092 12.11599
ZN-Open - - 0.477394 0.283206 10.62439 10.40774 6.652971 0.659678 2.429928 0.313117 16.09085 15.75623
C-H-R - - 0.421681 0.25155 2.534889 9.219109 4.185609 1.19549 1.174634 0.444315 6.268245 14.07367
Cohen-Coon - - 0.903723 0.290855 2.23463 2.054926 6.597632 1.828374 1.629527 0.386198 6.621596 6.228913
Ciancone-Marlin - - 0.595529 0.316686 2.31806 2.235919 10.79177 4.51365 2.417798 1.027116 7.183998 6.603842
Minimum Integral E. - - 0.426224 0.264112 5.443972 3.585999 5.563018 1.75844 1.204237 0.367181 14.60711 10.23431
Method
Example 7 Example 8 Example 9 Example 10 Example 11 Average
Set. Dis. Set. Dis. Set. Dis. Set. Dis. Set. Dis. Set. Dis.
ZN-Closed 121.105 33.93362 24.0696 13.75189 19.49302 38.61412 - - - - 26.434 14.8908
Tyreus-Luyben 82.82336 75.37933 19.84668 36.28508 74.32392 145.8678 - - - - 35.1576 46.42
Damped Oscillation 74.90803 33.03475 18.32106 13.56714 17.76392 34.84851 0.8825 4.247397 2.4965 0.5507 13.849 10.269
ZN-Open 203.0636 48.10066 41.21583 19.02999 20.80098 40.49981 - - - - 37.669 16.8813
C-H-R 71.53518 62.488 15.79547 23.05193 10.29351 35.97429 - - - - 14.026 18.337
Cohen-Coon 82.23544 40.9686 18.73435 17.27418 11.04538 19.81969 - - - - 16.25 11.106
Ciancone-Marlin 72.66559 54.42106 17.36664 24.75492 10.93768 21.3825 - - - - 15.5346 14.4069
Minimum Integral E. 61.47353 37.4164 14.01516 15.94768 17.36168 29.64329 - - - - 15.0118 12.402
▶ Performance assessment

37. Results
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 37
Method
Example 12 Example 13 Example 14 Average
Set. Dis. Set. Dis. Set. Dis. Set. Dis.
ZN-Closed 0.30377 0.000776 - - 0.485444 0.391874 0.3946 0.1963
Tyreus-Luyben 0.013379 0.003065 0.578426 0.008142 0.027758 0.000149 0.2065 0.003785
Damped Oscillation 0.325173 0.164803 - - 0.322041 0.132218 0.3236 0.1485
ZN-Open 0.283954 0.000466 - - - - 0.283954 0.00466
C-H-R - 0.128355 0.619157 - 0.220264 - 0.4197 0.128355
Cohen-Coon - - - 0.903723 - 0.148872 - 0.52629
Ciancone-Marlin 0.004346 0.012664 0.009255 0.595529 0.01106 0.001862 0.00822 0.20335
Minimum Integral E. 0.293021 - 0.295112 0.426224 0.165632 0.101298 0.2512 0.26376
▶ Robustness assessment

38. GUI Description
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 38

39. GUI Description
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 39

40. GUI Description
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 40

41. GUI Description
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 41

42. GUI Description
University of Jordan, Department of Mechatronics Engineering, 2014 June 16, 2015 42