Chemical Process Controls:
PID control, part II
Tuning
By Peter Woolf (pwoolf@umich.edu)
University of Michigan
Michigan Chemical Process
Dynamics and Controls
Open Textbook
version 1.0
Creative commons
Heater Example from
last class
F, Tin
Goal: Heat the output stream to
a desired set point temperature,
Tset
Assumptions:
• All liquid in lines and tank,
thus Fin=Fout=F
• Flow is constant
• Fluid does not boil
• No reactions
• Tank is well stirred
• Heater has no lag
• Heater has finite range
Question: How do we choose
PID control parameters?
F, T
idealized behav ior
Case: Heated CSTR
Starting at 120, and
Tset=130. Tfeed=100.
135
(see PID.example.xls)
130
temperature
125
120
idealized res pons e
T feed
limited ac tion
115
110
105
100
0
0 .5
1
1 .5
2
t ime
2 .5
3
3 .5
idealized behav ior
Case: Heated CSTR
Starting at 120, and
Tset=130. Tfeed=100.
140
135
(see PID.example.xls)
130
temperature
125
idealized res pons e
T feed
limited ac tion
120
115
Here we use a
smaller
 to show
integrator
windup
110
105
100
0
1
2
3
4
t ime
5
6
7
idealized behav ior
140
Case: Heated CSTR
Starting at 120, and
Tset=130. Tfeed=100.
135
(see PID.example.xls)
145
temperature
130
125
idealized res pons e
T feed
limited ac tion
120
115
110
Here  and Kc are
smaller.
105
What happened??
100
0
1
2
3
4
t ime
5
6
7
idealized behav ior
140
Case: Heated CSTR
Starting at 120, and
Tset=130. Tfeed=100.
135
(see PID.example.xls)
145
temperature
130
125
idealized res pons e
T feed
limited ac tion
120
115
110
Here  and Kc are
even smaller.
105
What happened??
100
0
1
2
3
4
t ime
5
6
7
Possible Tuning Strategies
1. Perturb system, see what happens
and use this response to choose PID
parameters
2. Adjust PID parameters until something
bad happens and then back off
3. Numerical optimization based on data
Reaction Curve Tuning
(=Open Loop)
• Based on a First
Order Plus Dead
Time (FOPDT)
process model
assumption
dT(t)
 k1T(t)  k2vt  
dt
First order process
delayed response
to signal v
time
• Change set point
from 39 to 42% CO
•Observe delay (0.8)
• Observe max slope
of response at T=27
Max slope
Slope=
(140 139)
 0.77
(26.2  27.5)
Kmax= output change/
Input change=k1/k2
0.77
 0.26
3


Units? %?
Relative to what?
Example from http://www.controlguru.com
i
d
Aside: intuition
•If slope is high (Kmax big) then want a small gain (Kc), as
the system is sensitive
•If large dead time, then want a small gain because response
is delayed, thus aggressive control could be dangerous.
•Large dead time also reduces the effect of integration, but
increases derivative. Integration can cause oscillations, and
with a large delay could be a problem. Derivative can still
work with time delay, in most cases.
i
d
Advantages of open loop tuning:
• fast: the experiment takes just one run
• does not introduce oscillations: Oscillations can be could
be dangerous in a large plant, so best avoided.
• Can be done before controller is installed
Disadvantages of open loop tuning:
• can be inaccurate: does not take into account control
dynamics or dynamics of other processes
• can be difficult to implement: max slope is not always
easy to find.
• Terms can be ambiguous
Closed Loop Tuning
Type of controller
P
PI
PID
Kc
0.5 Ku
0.45 Ku
0.6 Ku


Ti i
T dd
Pu/1.2
Pu/2
Pu/8
Zeigler-Nichols (Z-N) Tuning parameters for closed loop
Closed Loop Tuning
Advantages of closed loop tuning
1)Easy experiment
2)Incorporates in closed loop dynamics
Disadvantages of closed loop tuning
1)this experiment can be slow
2)Oscillations could be dangerous in some
cases, or if not at least wasteful
Model Based Tuning
• FOPDT is okay for a first approximation,
but we know what the process is doing.
• Given a model and normal operating data,
we can create a good model of the process.
• PID parameters can then be optimally
selected based on this model using
regression!
Model Based Tuning
Set points
Predicted model response
for a given Kc, i, and d.
temp
time
Goal: Use solver to find optimal values
of Kc, i, and d that minimize
t
T
set
i 0
(i)  Tmodel (i)
2
Model Based Tuning
Advantages of model based tuning
1) Incorporates in knowledge about the physical system
2)Incorporates in closed loop dynamics
3) Incorporates in physical limitations in valves and
sensors
4) Includes inherent noise in system
Disadvantages of model based tuning
1) Requires a good model that takes time to produce
2) Requires significant data describing a range of
operating behaviors
3) Optimization for large systems can be difficult.
4) Overkill for simple systems that are FOPD like
Light bulb control system
a little bit of real data…
Light bulb
(=heater)
Fan
(=pump)
inlet
Temperature
sensors
FLOW
Purge
valve
Note: valves don’t always look like
valves!
open
closed
Thermocouple
RTD
Sample Response Curve
(closed loop)
42
40
temp
38
Thermocouple
RTD
Set Point
36
34
32
30
0
50
100
150
200
250
300
Time (sec)
350
400
450
500
Closed loop
tuning?
41
Difficult to define as we have
(1) Limited control action, thus Ku tops out quickly.
(2) The oscillation frequency is only somewhat stable.
40.8
40.6
40.4
temp
40.2
Thermocouple
RTD
Set Point
40
39.8
39.6
39.4
39.2
39
0
50
100
150
200
250
300
Time (sec)
350
400
450
500
Different PID tuning
parameters
42
Small i, large Kc
Thermocouple
RTD
Set Point
40
38
All derivative
control
temp
Open recycle
36
34
Change set point
32
30
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Model based tuning?
42
1. Create a model
2. Parameterize the model based on
historical data
3. If fit is poor, adjust model in step 1
and repeat.
4. Fit PID tuning parameters to
optimize performance.
Thermocouple
RTD
Set Point
40
temp
38
36
34
32
30
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Model based tuning?
F, Tin
1. Create a model
2. Parameterize the model based on
historical data
3. If fit is poor, adjust model in step 1
and repeat.
4. Fit PID tuning parameters to
optimize performance.
What if this did not fit?
What might be a better model?
F, T
One idea… 4 CSTRs, each with different functions.
TC2 thermocouple
heater
TC1
Flow in due
to pressure
balance
RTD
Flow due to
recycle
(Look to your reactors text
for many more examples of
such lumped models of
multiple CSTRs)
Take Home Messages
• PID tuning parameters can be
estimated from data using a variety of
methods
• PID tuning can be difficult and time
consuming
• Complex physical processes can often
be broken down into smaller, more
familiar systems