بسم الله الرحمن الرحيم
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, 2nd 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
(Kp), Control valve gain (Kv),
Sensor gain (Km) and Controller
gain (Kc) must be positive.
Reverse action : If Kp Kv Km > 0 → Kc > 0 2
1 -Action of PID Controllers
• Direct action (increase/increase)
To determine the action of a controller, the engineer must know:
1. The process characteristics
2. The fail-safe action of the control valve
Direct action : If Kp Kv Km < 0 → Kc < 0
3
2 -Types of PID Controllers
• Classic PID:
• Parallel PID:
• Series PID:
dt
tdeKdtte
KteKmtm Dc
I
cc
)()()()(
s
sK
sE
sMsG D
Icc
1
1)(
)()(
1
11
)(
)()(
s
s
sK
sE
sMsG
D
D
Icc
1
111
)(
)()(
s
s
sK
sE
sMsG
D
D
Icc
Range :0.05 to 0.2(0.1)
4
2 -Types of PID Controllers
5
3 -Problems and Modifications of PID Controllers
Reset Windup
dt
tdeKdtte
KteKmtm Dc
I
cc
)()()()(
6
3 -Problems and Modifications of PID Controllers
Reset Feedback (RFB)
Internal Reset Feedback
Mmin
Mmax
Mmax
Mmin
mM
mMmmM
100max
min
s
sK
s
K
sE
sM
I
Ic
I
c
1
11
1)(
)(
7
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
dt
tdeKdtte
KteKmtm Dc
I
cc
)()()()(
yyeycyeybyeWheredt
tdeKdtte
KteKmtm
spspdspp
dDc
I
cpc
,,
)()()()(
Proportional Kick Derivative Kick
Range: 0-1 Range: 0-1, Commonly zero
Two Degrees of Freedom or ISA - PID
10
3 -Problems and Modifications of PID Controllers
11
4 -Off-Line Tuning of PID Controllers
Ziegler-Nichols(1942): Recommended for 0.1< D/ t <0.5 ( )1
s
eK Dsp
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)
12
4 -Off-Line Tuning of PID Controllers
Tyreus-Luyben(1992): Recommended for time-constant dominant
processes ( D/ t <0. 1 )
Ciancone-Marlin(1992): Recommended for dead-time dominant
processes ( D/ > t 2.0 )
13
4 -Off-Line Tuning of PID Controllers
PID tuning based on IMC (Rivera et al., 1986)
14
Final Control Element
ProcessSensor/
Transmitter
Step Change
Record
m(t), % c(t) , %
1)(
)(
:
s
eK
sM
sC
timedeadplusorderfirst
Ds
Process Gain:
m
cK s
15
4 -Off-Line Tuning of PID Controllers
Fit 3: 212 ,)(2
3tDtt
16
4 -Off-Line Tuning of PID Controllers
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
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
Relay Feedback Test
1. Bring the system to steady state.
2. 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%.
3. 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).
4. Repeat step 3 until sustained oscillation is observed .
5. Read off ultimate period Pu from the cycling and compute Ku from the following Equation: Ku = 4h/(πa) , ωu = 2π/Pu
19
5 -On-Line Tuning of PID Controllers
Advantages of Relay Feedback Test
1. It identifies process information around the important frequency, the ultimate frequency (where the phase angle is -π).
2. It is a closed-loop test; therefore, the process will not drift away from the nominal operating point.
3. The amplitude of oscillation is under control (by adjusting h ).
4. The time required for a relay feedback test is roughly equal to two to four times the ultimate period.
5. If the normalized dead time D /t 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
1
s
eK Dsp
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
5 -On-Line Tuning of PID Controllers
6 -Discrete Form of PID Controllers
Position Form
dt
tdeKdtte
KteKutu Dc
t
I
ccs
)()()()(
0
Sampling Time : Ts , Number of Sampling : k , Time : t = kTs
k
iss
tTiTedtte
10
)()(:iomapproximatr rectangulaUpper
s
ssk
T
TkekTe
dt
tde ))1(()()( :Difference Finite Backward
Position Form
Velocity Form
dt
tdeKdtte
KteKutu Dc
t
I
ccs
)()()()(
0
)1()()()()(1
kekeT
Kie
TKkeKuku
s
DcK
iI
sccs
)2()1(2)()()1()()1()( kekekeT
Kke
TKkekeKkuku
s
Dc
I
scc
6 -Discrete Form of PID Controllers
Velocity Form
Where:
)2()1()()1()( 210 kegkegkegkuku
s
Dc
s
Dc
s
D
I
sc
T
Kg
TKg
T
TKg
2
1
0
21
1
6 -Discrete Form of PID Controllers
Backward Shift Operator (q -1) : y(k-n)=q-ny(k)
Tuning of Digital PID : Moore et al. (1969)
Use the continuous tuning formula of PID controller with corrected dead time
1
22
110
1)(
)(
q
qgqgg
ke
ku
2s
c
TDD
6 -Discrete Form of PID Controllers