1. PID Controller Tuning
Comparison of classical tuning methods
By Ahmad Taan
1
University 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
June 16, 2015 2University of Jordan, Department of Mechatronics Engineering, 2014
3. Introduction
PID tuning is to find the optimum Kp, Ki and Kd for the controller.
June 16, 2015 3University of Jordan, Department of Mechatronics Engineering, 2014
Control objective > Setpoint tracking, Disturbance rejection
Actions > Instantaneous proportional action, Reset integral action, Rate derivative
action
Optimum criteria > Depends on application and system requirements
4. Introduction
Conceptual real-world example
June 16, 2015 4University of Jordan, Department of Mechatronics Engineering, 2014
Driver
(PID)
Car mechanism
(Process)
Crosswind
Front wheels
angle Car position
Driver’s eyes
(Feedback)
Desired position
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)
June 16, 2015 6University 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
June 16, 2015 7University of Jordan, Department of Mechatronics Engineering, 2014
8. Closed-loop methods
Ziegler-Nichols Closed-loop
Tyreus-Luyben
Damped Oscillation
June 16, 2015 8University of Jordan, Department of Mechatronics Engineering, 2014
PID Process
D
C PV
Feedback
SP
Tuning
9. Open-loop methods
Ziegler-Nichols Open-loop
C-H-R
Cohen-Coon
Ciancone-Marlin
Minimum Error Integral
June 16, 2015 9University of Jordan, Department of Mechatronics Engineering, 2014
PID Process
D
PV
Tuning
10. Ziegler-Nichols Closed-loop
¼ decay ratio as design criterion (stability condition)
Trial-and-error procedure to find 𝑲 𝒖 and 𝑷 𝒖
Drives the process into marginal stability
Performs well when 𝝉 𝒎 ≥ 𝟐𝒕 𝒅 (lag dominant)
Performs very poorly for 𝒕 𝒅 > 𝟐𝝉 𝒎 (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
June 16, 2015 10University of Jordan, Department of Mechatronics Engineering, 2014
11. Ziegler-Nichols Closed-loop
June 16, 2015 11University of Jordan, Department of Mechatronics Engineering, 2014
Controller 𝐾𝑐 𝜏𝑖 𝜏 𝑑
P 0.5𝐾 𝑢 - -
PI 0.45𝐾 𝑢 0.83𝑃𝑢 -
PID 0.6𝐾 𝑢 0.5𝑃𝑢 0.125𝑃𝑢
Procedure:
Set 𝐾𝑖 and 𝐾 𝑑 to 0
Increase 𝐾 𝑝 till sustained oscillation and find 𝐾 𝑢 and 𝑃𝑢
Use the correlations in the table below
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
June 16, 2015 12University of Jordan, Department of Mechatronics Engineering, 2014
Controller 𝐾𝑐 𝜏𝑖 𝜏 𝑑
PI 0.45𝐾 𝑢 2.2𝑃𝑢 -
PID 0.313𝐾 𝑢 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
June 16, 2015 13University of Jordan, Department of Mechatronics Engineering, 2014
0
0.2
0.4
0.6
0.8
1
1.2
0 10 20 30 40 50 60 70 80
4:1
14. Damped Oscillation
June 16, 2015 14University of Jordan, Department of Mechatronics Engineering, 2014
Controller 𝐾𝑐 𝜏𝑖 𝜏 𝑑
PI 𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑃𝑑/6 -
PID 𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑃𝑑/6 𝑃𝑑/1.5
Procedure:[1]
Set 𝐾𝑖 and 𝐾 𝑑 to 0
Increase 𝐾 𝑝 till ¼ damping ratio is maintained and find 𝑃𝑑 only
Use the correlations in the table below to find 𝜏𝑖 and 𝜏 𝑑
Adjust 𝐾 𝑝 till ¼ damping ratio is maintained again
[1] Lipták, Béla G., and Kriszta Venczel. Instrument Engineers' Handbook: Process Control 4thed, Volume Two.
15. Ziegler-Nichols Open-loop
¼ decay ratio as design criterion
Performs well when 𝜏 𝑚 ≥ 2𝑡 𝑑 (lag dominant)
Performs very poorly for 𝑡 𝑑 > 2𝜏 𝑚 (dead-time dominant)
Fast recovery from disturbance but leads to oscillatory response
June 16, 2015 15University 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
𝐺 𝑚 𝑠 =
𝐾 𝑚 𝑒−𝑡 𝑑 𝑠
𝜏 𝑚 𝑠 + 1
June 16, 2015 16University of Jordan, Department of Mechatronics Engineering, 2014
-0.5
0
0.5
1
1.5
2
2.5
17. Ziegler-Nichols Open-loop
PID parameters are calculated from the table below
June 16, 2015 17University of Jordan, Department of Mechatronics Engineering, 2014
Controller 𝐾𝑐 𝜏𝑖 𝜏 𝑑
P 1
𝐾 𝑚
𝜏 𝑚
𝑡 𝑑
- -
PI 0.9
𝐾 𝑚
𝜏 𝑚
𝑡 𝑑
𝑡 𝑑
0.3
-
PID 1.2
𝐾 𝑚
𝜏 𝑚
𝑡 𝑑
2𝑡 𝑑 0.5𝑡 𝑑
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
June 16, 2015 18University 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
𝒕 𝒅
𝝉
(i.e. 𝑡 𝑑 > 2𝜏)
PD rules available
June 16, 2015 20University 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:
𝜏 𝑚 =
3
2
𝑡2 − 𝑡1
𝑡 𝑑 = 𝑡2 − 𝜏 𝑚
June 16, 2015 21University of Jordan, Department of Mechatronics Engineering, 2014
-0.5
0
0.5
1
1.5
2
2.5
[1] Smith,C.A., A.B. Copripio; “Principles and Practice of Automatic Process Control”, John Wiley & Sons,1985
23. Ciancone-Marlin
June 16, 2015 23University of Jordan, Department of Mechatronics Engineering, 2014
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
June 16, 2015 24University of Jordan, Department of Mechatronics Engineering, 2014
Procedure:
Estimate the process with FOPDT as for Cohen-Coon method
Calculate the ratio
𝑡 𝑑
𝑡 𝑑+𝜏 𝑚
From the appropriate graph determine the values (𝐾𝑐 𝐾 𝑚,
𝜏 𝑖
𝑡 𝑑+𝜏 𝑚
,
𝜏 𝑑
𝑡 𝑑+𝜏 𝑚
)
Do the calculation to find the PID parameters
27. Minimum Error Integral
June 16, 2015 27University of Jordan, Department of Mechatronics Engineering, 2014
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 𝝉 𝒎 ≥ 𝟐𝒕 𝒅 (lag dominant)
Performs very poorly for 𝒕 𝒅 > 𝝉 𝒎 (dead-time dominant)
Tuning for setpoint responses differs from load disturbance responses
Different error integrals can be used (IAE, ISE, ITAE, ITSE)
𝐼𝐴𝐸 =
0
∞
𝑒(𝑡) 𝑑𝑡 , 𝐼𝑆𝐸 =
0
∞
𝑒(𝑡)2
𝑑𝑡 , 𝐼𝑇𝐴𝐸 =
0
∞
𝑡 𝑒(𝑡) 𝑑𝑡 , 𝐼𝑇𝑆𝐸 =
0
∞
𝑡𝑒(𝑡)2
𝑑𝑡
28. Minimum Error Integral
June 16, 2015 28University of Jordan, Department of Mechatronics Engineering, 2014
Procedure:
Estimate the process with FOPDT as for Cohen-Coon method
Use the appropriate table to find the PID parameters
31. Simulation and Results
June 16, 2015 31University of Jordan, Department of Mechatronics Engineering, 2014
Simulation performed for two purposes:
Performance Assessment
Robustness Assessment
Simulation for two response objectives:
Set point tracking
Disturbance rejection