Transcript 01_1 - Ferdowsi University of Mashhad
ميحرلا نمحرلا الله مسب
PID Controllers
1- Action, types and Modifications 2- Offline and online tuning
Ref 1: Smith & Corripio “Principles and Practice of Automatic Process Control”, 3 rd Ed., Wiley, 2006, Chapter 5 & 7.
Ref 2: C. C. Yu, Autotuning of PID controllers, 2 nd ed., springer, 2006, Chapters 2 &3.
Ref 3: K. J. Astrom and T. Hagglund, Advanced PID Control, ISA, 2006, Chapter 3.
Lecturer: M. A. Fanaei Ferdowsi University of Mashhad
1- Action of PID Controllers If the action is not correctly selected, the controller will not control
• Reverse action (increase/decrease)
In feedback control loop, the multiplication of Process gain (K
p
), Control valve gain (K
v
), Sensor gain (K
m
) and Controller gain (K
c
) must be positive.
Reverse action : If K
p K v K m
> 0 →
K c
> 0
2
1- Action of PID Controllers
• Direct action (increase/increase)
Direct action : If K
p K v K m
< 0 →
K c
< 0 To determine the action of a controller, the engineer must know: 1. The process characteristics 2. The fail-safe action of the control valve
3
2- Types of PID Controllers
• • •
Classic PID:
m
(
t
)
m
K c e
(
t
)
K c
I
e
(
t
)
dt
K c
D de
(
t
)
dt
Parallel PID:
G c
(
s
)
M
(
s
)
E
(
s
)
K c
1 1
I s
D s
Range : 0.05 to 0.2
(0.1)
G c
(
s
)
M
(
s
)
E
(
s
)
K c
1 1
I s
D D s s
1
Series PID:
G c
(
s
)
M
(
s
)
E
(
s
)
K
c
1 1
I s
D
D s s
1 1 4
2- Types of PID Controllers
5
3- Problems and Modifications of PID Controllers
Reset Windup
m
(
t
)
m
K c e
(
t
)
K c
I
e
(
t
)
dt
K c
D de
(
t
)
dt
6
3- Problems and Modifications of PID Controllers
Reset Feedback (RFB) M max
M
m
m
M M
min max
m
100
m
M min M max
M
(
s
)
E
(
s
)
K c
1
I s
1 1
K c
I
s I
1
s
M min
7
Internal Reset Feedback
3- Problems and Modifications of PID Controllers
Reset Feedback (RFB)
8
3- Problems and Modifications of PID Controllers
Reset Feedback (RFB)
External Reset Feedback
9
3- Problems and Modifications of PID Controllers
Proportional and Derivative Kick
m
(
t
)
m
K c e
(
t
)
K c
I
e
(
t
)
dt
K c
D de
(
t
)
dt
Proportional Kick Derivative Kick
Two Degrees of Freedom or ISA - PID
m
(
t
)
Where
m e p
K c e p
(
t
)
by sp
y
,
K c
I e d
e
(
t
)
dt
cy sp
K c
D y
,
e
de d
(
t
)
dt y sp
y
Range: 0-1 Range: 0-1, Commonly zero 10
3- Problems and Modifications of PID Controllers
11
4- Off-Line Tuning of PID Controllers More than 250 tuning rules are exist for PI and PID Controllers What is the suitable tuning rule? It really depends on your process (Type, Order, Parameters, Nonlinearity, Uncertainty, etc)
Ds K
p e
Ziegler-Nichols(1942):
Recommended for 0.1< D/
<0.5 ( )
s
1 12
4- Off-Line Tuning of PID Controllers
Tyreus-Luyben(1992): Recommended for time-constant dominant
processes ( D/
<0. 1 )
Ciancone-Marlin(1992): Recommended for dead-time dominant
processes ( D/
> 2
.0 )
13
4- Off-Line Tuning of PID Controllers
PID tuning based on IMC (Rivera et al., 1986)
14
4- Off-Line Tuning of PID Controllers Step Change m(t), % Final Control Element Process Record Sensor/ Transmitter c(t) , %
first order plus dead time
:
C
(
s
)
M
(
s
)
K
s e
Ds
1
Process Gain:
K
c s
m
15
4- Off-Line Tuning of PID Controllers
Fit 3 :
3 2 (
t
2
t
1 ) ,
D
t
2 16
5- On-Line Tuning of PID Controllers
Ziegler-Nichols Test (1942)
1. Set the controller gain
Kc
at a low value, perhaps 0.2
.
2. Put the controller in the automatic mode.
3. Make a small change in the set point or load variable and observe the response. If the gain is low, then the response will be sluggish.
4. Increase the gain by a factor of two and make another set point or load change.
5. Repeat step 4 until the loop becomes oscillatory and continuous cycling is observed. The gain at which this occurs is the ultimate gain
Ku ,
and the period of oscillation is the ultimate period
Pu.
17
5- On-Line Tuning of PID Controllers
Relay Feedback Test (Astrom & Hagglund, 1984)
Luyben popularized relay feedback method and called this method “ATV” (autotune variation).
18
5- On-Line Tuning of PID Controllers
1.
2.
3.
4.
5.
Relay Feedback Test
Bring the system to steady state.
Make a small (
e.g. 5%)
increase in the manipulated input. The magnitude of change depends on the process sensitivities and allowable deviations in the controlled output. Typical values are between 3 and 10%.
As soon as the output crosses the SP, the manipulated input is switched to the opposite position (
e.g. –5% change from the original value).
Repeat step 3 until sustained oscillation is observed .
Read off ultimate period the following Equation:
Pu
from the cycling and compute
K u
= 4h/(πa) , ω u = 2π/P
u
Ku
from 19
5- On-Line Tuning of PID Controllers
Advantages of Relay Feedback Test
1.
2.
3.
4.
5.
It identifies process information around the important frequency, the ultimate frequency (where the phase angle is
-π
).
It is a closed-loop test; therefore, the process will not drift away from the nominal operating point.
The amplitude of oscillation is under control (by adjusting
h ).
The time required for a relay feedback test is roughly equal to two to four times the ultimate period.
If the normalized dead time
D /
is less than 0.28
,
the ultimate period is smaller than the process time constant. Therefore the relay feedback test is more time efficient than the step test. Since the dead time can not be too large, the temperature and composition loops in process industries seem to fall into this category.
20
5- On-Line Tuning of PID Controllers
Advantages of Relay Feedback Test
K p
s e
Ds
1 21
5- On-Line Tuning of PID Controllers
Relay feedback responses of FOPDT processes
Assume an integrator plus dead time (Time constant dominant processes) Assume a FOPDT (Most slow processes) Assume a pure dead time (Dead time dominant processes) 22
6- Discrete Form of PID Controllers
Position Form
u
(
t
)
u s
K c e
(
t
)
K c
I
0
t e
(
t
)
dt
K c
D de
(
t
)
dt
Sampling Time :
T s
, Number of Sampling :
k
, Time :
t
=
kT s
Upper rectangula r approximat iom : 0
t e
(
t
)
dt
i k
1
e
(
iT s
)
T s
Backward Finite Difference :
de
(
t k dt
)
e
(
kT s
)
e
((
k
1 )
T s
)
T s
6- Discrete Form of PID Controllers
Position Form
u
(
t
)
u s
K c e
(
t
)
K c
I
0
t e
(
t
)
dt
K c
D de
(
t
)
dt u
(
k
)
u s
K c e
(
k
)
K
c T s I i K
1
e
(
i
)
K c
D
e
(
k
)
e
(
k
1 )
T s
Velocity Form
u
(
k
)
u
(
k
1 )
K c
e
(
k
)
e
(
k
1 )
K
c T s I e
(
k
)
K c
D T s
e
(
k
) 2
e
(
k
1 )
e
(
k
2 )
6- Discrete Form of PID Controllers
Velocity Form
u
(
k
)
u
(
k
1 )
g
0
e
(
k
)
g
1
e
(
k
1 )
g
2
e
(
k
2 )
Where:
g
0
K c
1
T s I
D T s
g
1
K c
1 2
D T s
g
2
K c
D T s
6- Discrete Form of PID Controllers
Backward Shift Operator (q -1 ) : y(k-n)=q -n y(k)
u
(
k
)
e
(
k
)
g
0
g
1
q
1 1
q
1
g
2
q
2
Tuning of Digital PID : Moore et al. (1969) Use the continuous tuning formula of PID controller with corrected dead time
D c
D
T s
2