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PID Controller Design, Tuning and Troubleshooting
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Disturbance responses for a close loop system
Disturbance responses for nine combinations of the controller gain and integral time for a first order + time delay simple close loop system.
Aggressive response
Aggr
essiv
e re
spon
se
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Performance criteria for closed-loop systemsIdeal performance criteria
• The close loop system MUST be stable• Effect of disturbances are minimized,
providing good disturbance rejection.• Rapid and smooth response to the set
point changes• Steady state error (offset) is eliminated• No excessive control• Control is robust. Insensitive to changes in
process conditions and to inaccuracies in the process model.
Performance
Robustness
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Performance criteria for closed-loop systemsPerformanc
e Robust
Kc -> lowI -> large
Reasonable degree of model inaccuracy
Wide range of conditions
Smooth responses
Rapid response to changes
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PID Controller settings
Techniques
• Direct Synthesis method (DS).• Internal Model Control method
(IMC).• Controller tuning relations• Frequency response techniques• Computer simulation• On-line tuning after control system is
installed.
Computer simulation
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Model Based Design MethodsDirect Synthesis Method 𝑌
𝑌 𝑠𝑝=
𝐾𝑚𝐺𝑐𝐺𝑣𝐺𝑝
1+𝐺𝑚𝐺𝑐𝐺𝑣𝐺𝑝
𝐺≜𝐺𝑐𝐺𝑣𝐺𝑝 𝐺𝑚=𝐾𝑚
𝑌𝑌 𝑠𝑝
=𝐺𝑐𝐺1+𝐺𝑐𝐺
What is ?
𝐺𝑐=1𝐺 ( 𝑌 /𝑌 𝑠𝑝
1−𝑌 /𝑌 𝑠𝑝)
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Model Based Design Methods
Lets see some options for
𝐺𝑐=1𝐺 ( 𝑌 /𝑌 𝑠𝑝
1−𝑌 /𝑌 𝑠𝑝)
A desired transfer function is used to make the problem simpler.
𝐺𝑐=1~𝐺 ( (𝑌 /𝑌 𝑠𝑝 )𝑑
1− (𝑌 /𝑌 𝑠𝑝 )𝑑 )
( 𝑌𝑌 𝑠𝑝 )𝑑=
1𝜏𝑐 𝑠+1
𝐺𝑐=1~𝐺
1𝜏𝑐 𝑠 is the desired closed-loop time constant
Direct Synthesis Method
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Model Based Design Methods
Lest see some options for
( 𝑌𝑌 𝑠𝑝 )𝑑=
𝑒−𝜃𝑠𝜏𝑐 𝑠+1
𝐺𝑐=1~𝐺
𝑒−𝜃 𝑠
𝜏𝑐 𝑠+1−𝑒−𝜃 𝑠 is a term to include the time
delay
Direct Synthesis Method
𝐺𝑐=1~𝐺
𝑒−𝜃 𝑠
(𝜏𝑐+𝜃 ) 𝑠
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Model Based Design MethodsDirect Synthesis Method
Lets derive controllers for two important process models
First Order Plus Time Delay Model
𝐺=~𝐺=
𝐾𝑒−𝜃𝑠𝜏 𝑠+1
𝐺𝑐=1~𝐺
𝑒−𝜃 𝑠
(𝜏𝑐+𝜃 ) 𝑠
𝐺𝑐=1
𝐾 𝑒−𝜃 𝑠𝜏 𝑠+1
𝑒−𝜃 𝑠
(𝜏𝑐+𝜃 )𝑠
𝐺𝑐=𝐾 𝑐(1+ 1𝜏𝐼 𝑠 )
𝐾 𝑐=1𝐾
𝜏𝜃+𝜏𝑐
,𝜏𝑐=𝜏
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Model Based Design MethodsDirect Synthesis Method
Lets derive controllers for two important process models
Second Order Plus Time Delay Model
𝐺=~𝐺=
𝐾𝑒−𝜃 𝑠
(𝜏1𝑠+1 ) (𝜏2𝑠+1 )
𝐺𝑐=1~𝐺
𝑒−𝜃 𝑠
(𝜏𝑐+𝜃 ) 𝑠
𝐺𝑐=𝐾 𝑐 (1+ 1𝜏 𝐼 𝑠
+𝜏𝐷𝑠)
𝐾 𝑐=1𝐾𝜏1+𝜏2𝜃+𝜏𝑐
𝜏 𝐼=𝜏1+𝜏 2 𝜏𝐷=𝜏1𝜏2𝜏1+𝜏2
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Model Based Design MethodsDirect Synthesis Method
Example: Use DS method to calculate the PID controller settings for the process. Consider three values of the desired closed loop time constant
𝐺𝑐=𝐾 𝑐(1+ 1𝜏 𝐼 𝑠
+𝜏𝐷𝑠)
𝐾 𝑐=1𝐾𝜏1+𝜏2𝜃+𝜏𝑐
𝜏 𝐼=𝜏1+𝜏 2
𝜏𝐷=𝜏1𝜏2𝜏1+𝜏2
𝐺=2𝑒− 𝑠
(10𝑠+1 ) (5 𝑠+1 )𝐺≜𝐺𝑐𝐺𝑣𝐺𝑝
𝐺𝑐=1~𝐺
𝑒−𝜃 𝑠
(𝜏𝑐+𝜃 ) 𝑠
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Model Based Design MethodsDirect Synthesis Method
Example: Use DS method to calculate the PID controller settings for the process. Consider three values of the desired closed loop time constant
𝐺=2𝑒− 𝑠
(10𝑠+1 ) (5 𝑠+1 )
3.75 1.88 0.682
15 (1/15)* 15 (1/15)* 15 (1/15)*
3.33 3.33 3.33
=3.75=1.88
=0.682
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Model Based Design MethodsDirect Synthesis Method : The problem with this method is that we need to guess
If the system can be simulated by then
1. and
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Model Based Design Methods
𝐺𝑐=𝐾 𝑐 (1+ 1𝜏 𝐼 𝑠
+𝜏𝐷𝑠)
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ON-Line Controller TuningZiegler and Nichols (1942). Paper “Continuous cycling method” this is a trial and error method
1. After process reaches steady state, eliminate the integral and derivative control actions. and (large value).
2. Set to a small value. Then set the controller in automatic.
3. Introduce an small set point change. Gradually increase until a sustained oscillation with constant amplitude occurs. This value of is called Ultimate gain. The period of the oscillation is the ultimate period,
4. Use the table to calculate 5. Evaluate the system by introducing a small set point
change. You may need fine tunning. Lower
Zeigler-NicholsP 0.5 --- ---
PI 0.45 ---
PID 0.6
𝐺𝐶=𝐾 𝑃 (1+ 1𝑇 𝐼
1𝑠 +𝑇 𝐷𝑠 )
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ON-Line Controller TuningUse the Ziegler and Nichols method to tune the following system
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ON-Line Controller TuningUse the Ziegler and Nichols method to tune the following system
1. After process reaches steady state, eliminate the integral and derivative control actions.
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ON-Line Controller TuningUse the Ziegler and Nichols method to tune the following system
2. Set to a small value. Then set the controller in automatic.
3. Introduce an small set point change. Gradually increase until a sustained oscillation with constant amplitude occurs. This value of is called Ultimate gain. The period of the oscillation is the ultimate period,
.0
𝐾 𝑐𝑢=10
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ON-Line Controller TuningUse the Ziegler and Nichols method to tune the following system
4. Use the table to calculate
Zeigler-NicholsP 0.5 --- ---
PI 0.45 ---
PID 0.6
𝜏 𝐼=0.5𝜏𝐷=1/8
𝐺𝐶=𝐾 𝑃 (1+ 1𝑇 𝐼
1𝑠 +𝑇 𝐷𝑠 )
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ON-Line Controller TuningUse the Ziegler and Nichols method to tune the following system
5. Tune the controller trial and error if it is necessary
𝐺𝐶=𝐾 𝑃 (1+ 1𝑇 𝐼
1𝑠 +𝑇 𝐷𝑠 )
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ON-Line Controller TuningDisadvantages
1. It is time consuming.2. There are many applications where this system do not
work as expected. In case of a chemical reaction it may cause a “runaway”.
3. It is not applicable to unstable systems.4. For first and second order models the ultimate gain do
not exist.
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ON-Line Controller TuningProcess Reaction Curve Method
1. After process reaches steady state2. Place the system in manual mode3. Adjust the controller out put signal to the values that
were working in automatic mode.4. Wait for steady state5. Introduce an step change in set point.6. Record the response of the variable as shown in the
plot.7. Return the controller to the initial values.8. Calculate the new set up of the parameters according
to the table.
Zeigler-NicholsP --- ---
PI ---
PID 0.5
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Model Based Design MethodsTuning with Matlab
Example:
𝐺=2𝑒− 𝑠
(10𝑠+1 ) (5 𝑠+1 )
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Model Based Design MethodsTuning with Matlab
Example:
𝐺=2𝑒− 𝑠
(10𝑠+1 ) (5 𝑠+1 )
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Model Based Design MethodsTuning with Matlab
Example:
𝐺=2𝑒− 𝑠
(10𝑠+1 ) (5 𝑠+1 )
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WorkshopNow you have time to solve the following question
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WorkshopUse the Ziegler and Nichols method and the curve response method to tune the proportional, proportional integral and PID controlled systems indicated below. Compare your results with the tuning results calculated by using Matlab.
𝐺=2
(𝑠+2 ) (0.18 𝑠2+0.6 𝑠+1 )
𝐺=2𝑒− 2𝑠
(10𝑠+1 ) (5 𝑠+1 )