pid controller tuning
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University of Jordan, Department of Mechatronics Engineering, 2014
PID Controller TuningComparison of classical tuning methods
By Ahmad Taan
1
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 2
Content
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
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 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
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 4
Introduction
Conceptual real-world example
Driver(PID)
Car mechanism(Process)
Crosswind
Front wheels angle Car position
Driver’s eyes
(Feedback)
Desired position
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 5
Introduction
PID configuration
𝐾 𝑝𝑒(𝑡)
𝐾 𝑖∫𝑒(𝑡)𝑑𝑡
𝐾 𝑑
d𝑒(𝑡)𝑑𝑡
SP
PV
Controller outpute(t)
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 6
Introduction
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)
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 7
Objectives
Compare studied tuning methods for performance and robustness
Develop a GUI to do the comparison automatically for a given process model
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 8
Closed-loop methods
Ziegler-Nichols Closed-loop
Tyreus-Luyben
Damped Oscillation
PID Process
D
C PV
Feedback
SP
Tuning
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 9
Open-loop methods
Ziegler-Nichols Open-loop
C-H-R
Cohen-Coon
Ciancone-Marlin
Minimum Error IntegralPID Process
D
PV
Tuning
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 10
Ziegler-Nichols Closed-loop
decay ratio as design criterion (stability condition)
Trial-and-error procedure to find and
Drives the process into marginal stability
Performs well when (lag dominant)
Performs very poorly for (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
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 11
Ziegler-Nichols Closed-loop
Controller
P - -
PI -
PID
Procedure:
Set and to 0
Increase till sustained oscillation and find and
Use the correlations in the table below
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 12
Tyreus-Luyben
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
PI -
PID
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 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 10 20 30 40 50 60 70 800
0.2
0.4
0.6
0.8
1
1.2
4:1
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 14
Damped Oscillation
Controller
PI -
PID
Procedure:[1]
Set and to 0
Increase till damping ratio is maintained and find only
Use the correlations in the table below to find and
Adjust till damping ratio is maintained again
[1] Liptak, Bela G., and Kriszta Venczel. Instrument Engineers' Handbook: Process Control 4 thed, Volume Two.
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 15
Ziegler-Nichols Open-loop
¼ decay ratio as design criterion
Performs well when (lag dominant)
Performs very poorly for (dead-time dominant)
Fast recovery from disturbance but leads to oscillatory response
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 16
Ziegler-Nichols Open-loop
Procedure:
The process dynamics is modeled by a first order plus dead time model
0
0.5
1
1.5
2
2.5
𝜏𝑚
𝑡𝑑
𝐾𝑚
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 17
Ziegler-Nichols Open-loop
PID parameters are calculated from the table below
Controller
P - -
PI -
PID
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 18
C-H-R
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
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 19
C-H-R
Setpoint
Controller
0% overshoot
P- -
PI-
PID
Disturbance
P - -PI
4 -
PID
20% overshoot
- -
-
- -2.3 -
Procedure:
Same as Ziegler-Nichols Open-loop
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 20
Cohen-Coon
Second in popularity after Ziegler-Nichols tuning rules
¼ decay ratio has considered as design criterion for this method
More robust
Applicable to wider range of (i.e. > 2)
PD rules available
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 21
Cohen-Coon
Procedure:[1]
The process reaction curve is obtained by an open loop test and the FOPDT model is estimated as follows:
0
0.5
1
1.5
2
2.5
𝑡1
0 .632 𝑦 𝑠𝑠
𝑡 20 .283 𝑦𝑠𝑠
[1] Smith,C.A., A.B. Copripio; “Principles and Practice of Automatic Process Control”, John Wiley & Sons,1985
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 22
Cohen-Coon
Controller
P
- -
PI
-
PD
-
PID
PID parameters are calculated from the table
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 23
Ciancone-Marlin
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
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 24
Ciancone-Marlin
Procedure:
Estimate the process with FOPDT as for Cohen-Coon method
Calculate the ratio
From the appropriate graph determine the values (, , )
Do the calculation to find the PID parameters
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 25
Ciancone-Marlin
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
0 0.2 0.4 0.6 0.8 10.5
0.7
0.9
1.1
1.3
1.5
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
setp
oin
tD
istu
rbance
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 26
Ciancone-Marlin
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
setp
oin
tD
istu
rbance
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 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 (lag dominant)
Performs very poorly for (dead-time dominant)
Tuning for setpoint responses differs from load disturbance responses
Different error integrals can be used (IAE, ISE, ITAE, ITSE)
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 28
Minimum Error Integral
Procedure:
Estimate the process with FOPDT as for Cohen-Coon method
Use the appropriate table to find the PID parameters
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 29
Minimum Error Integral
Error integral IAE ITAE
PI Controller
3PID Controller
Setpoint tracking table
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 30
Minimum Error Integral
Error integral
IST IAE ITAE
P Controller 49PI Controller
859
PID Controller
749
56
Disturbance rejection table
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 31
Simulation and Results
Simulation performed for two purposes:
Performance Assessment
Robustness Assessment
Simulation for two response objectives:
Set point tracking
Disturbance rejection
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 32
Simulation and Results
Test cases include processes of:
Dead-time dominant (
Lag dominant
In-between cases
Complex poles
Unstable process
1.
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 33
Simulation Example (Closed-loop)
Method
Ziegler-Nichols Closed-loop
0.63 0.24 0
Tyreus-Luyben 0.44 0.06 0
Damped Oscillation 0.76 0.28 0
Method IAE ITAE ISE
Ziegler-Nichols Closed-loop
4.287635 21.66082 2.14574
Tyreus-Luyben 16.21587 326.41346.60062
9
Damped Oscillation 3.657051 16.387961.93091
4
MethodOvershoo
tRise 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
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 34
Simulation Example (Open-loop)
Method
Ziegler-Nichols Open-loop
0.38 0.096 0
C-H-R 0.26 0.50 0Cohen-Coon 0.46 0.59 0
Ciancone-Marlin 0.65 0.61 0
Minimum Error Integral
0.36 0.19 0Method IAE ITAE ISE
Ziegler-Nichols Open-loop
10.62439 133.38774.67203
2
C-H-R 2.534889 4.2159791.91689
1
Cohen-Coon 2.234633.37898
81.6872
13
Ciancone-Marlin 2.31806 4.3374861.62383
8Minimum Error
Integral5.443972 29.46653
2.827566
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 35
Robustness Assessment Example
Method
Ziegler-Nichols Closed-loop
7.38 5.13 0
Tyreus-Luyben 5.13 1.35 0Damped
Oscillation8.26 4.36 0
Method∆
%Overshoot
∆%Rise time
∆%Settling
timeZiegler-Nichols
Closed-loop2.53E+46 0.005528
Tyreus-Luyben 0.780894 0.021236 0.222945Damped
Oscillation7.51E+58 0.002601
Method ∆%IAE ∆%ITAE ∆%ISE
Ziegler-Nichols Closed-loop
65535 65535 65535
Tyreus-Luyben 0.578426 1.141222 0.534852Damped
Oscillation65535 65535 65535
---- After process parameters change
___ With original process parameters
Only Tyreus Luyben method could preserve the system stability in this example
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 36
Results
MethodExample 1 Example 2 Example 3 Example 4 Example 5 Example 6Set. 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 10.23431Method
Example 7 Example 8 Example 9 Example 10 Example 11 AverageSet. 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
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 37
Results
MethodExample 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
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 38
GUI Description
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 39
GUI Description
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 40
GUI Description
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 41
GUI Description
April 15, 2023University of Jordan, Department of Mechatronics Engineering, 2014 42
GUI Description
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