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Optimization-based PI/PID control for SOPDT process
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Summary on optimization-based PI/PID control
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Best achievable IAE performance by PI/PID control of FOPDT process
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Optimal rise-time vs, IAE in PI/PID control of SOPDT process
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Optimal rise-time vs, IAE in PI/PID control of SOPDT process
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FOPDT
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SOPDT
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According to the IMC theory, nominal loop transfer function of a control system that has an inverse-based controller will be of the following:
( )( ) ( )p
lp lp
G sG s F s
s
P
( ) ( ) ( )
where, ( ) serves as a loop filter in a control system,
and the ( ) represents the non-invertible part of G .
plp lp
lp
p
G sG s F s
sF s
G s
Loop transfer functions of IMC-PID Controllers
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IMC-PID for FOPDT process
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( )1
sp
p
k eG s
s
The resulting ( ) becomes (Chien and Fruehauf, 1990)lpG s
1
s
lp
eG
s
1 0 5
( )( 1)
s
lpf
e sG
s s
Loop transfer functions of IMC-PID Controllers
2( )
2 1
sp
p
k eG s
s s
the resulting loop transfer function becomes:
1s
lp
eG
s
FOPDT processes:
SOPDT processes:
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(1 )( )
1
sp
p
k s eG s
s
( 2 )
(1 ) (1 )( )
1 (1 )
(1 ) (1 ) =
1 (1 )
(1 ) =
1
sp
p
sp
sp
k s e sG s
s s
k s e s
s s
k s e
s
* 1.38 ( 2 )IAE
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(1 )( )
(1 )
so
loopk s e
G ss s
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(1 )( )
(1 )
so
LP n
k s eG s
s s
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• We should learn what happens to the Z-N tuned controllers?
• How inverse-based controllers are synthesized?
Inverse-based Controller Design
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• Inverse-based synthesis approach is used– Target loop transfer function (LTF)
– This LTF has satisfactory control performance as well as reasonable stability robustness
• ko and a are selected to meet desired control specification
Defaulted value: ko=0.65 a=0.4 GM = 2.7 PM = 60 o
(1 )( ) ( ) ( )
(1 )
so
loop c pf
k a s eG s G s G s
s s
Loop transfer functions of Inverse-based Controllers
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PI/PID Controllers Based on FOPDT Model
• A direct synthesis approach is used– PI controller
ko=0.5
• Controller parameters (actual PID)
– PID controller
ko=0.65 , a=0.4
( )s
oloop
k eG s
s
(1 )
( )(1 )
so
loopf
k a s eG s
s s
'
'
'
oc
p
R
D
kk
k
a
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PID Controller Based on SOPDT Model
• Controller parameters (ideal PID)
( )s
oloop
k eG s
s
ko=0.5
(2 )
2
2
oc
p
R
D
kk
k
2( )
2 1
sp
p
k eG s
s s
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0 0ˆ(1 ) (1 )s s
lp
k s e k s eG
s s
0 0ˆ andk k s s
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.41.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
^
p
ˆp
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1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
Am
m
Gain margin vs. phase margin at a=0.4
Phasemargin
Gainmargin
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Auto-tune
• Autotuning via relay feedback: Astrom and Hagglund (1984)
Referred as autotune variation (ATV): Luyben (1987)
Main advantage: under closed-loop
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4 2cu u
u
hk
A P
Apply Z-N or T-L tuning
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MODEL-BASED CONTROLLERS DESIGN
• Reduced order models– FOPDT Monotonic step response
• For zero offset, PI or PID controller is considered
• Usage of PI or PID controller depend on:– The application occasions– The dynamic characteristics of given process
• Processes are classified into two groups for controller tuning
- Underdamped SOPDT
Oscillatory step response
( )1
sp
p
k eG s
s
2 2( )2 1
sp
p
k eG s
s s
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Criterion for Classifying model order
• In general, processes with overdamped or slightly underdamped SOPDT dynamics can be identified with FOPDT models for controller tuning
Q: When an SOPDT process could be reduced to an
FOPDT parameterization?
A: Ku > 1
1 22 tan 11
; ; whereu
uu p cu
u u
KKK k k
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10.707 1
2UK
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PI/PID Controllers Based on FOPDT Model
• A direct synthesis approach is used– PI controller
ko=0.5
• Controller parameters (actual PID)
– PID controller
ko=0.65 , a=0.4
( )s
oloop
k eG s
s
(1 )
( )(1 )
so
loopf
k a s eG s
s s
'
'
'
oc
p
R
D
kk
k
a
In terms ofultimate data
(Ku = kp kcu)
2'
1 2
2'
1 2
'
1
tan 1
1
tan 1
o uc
p u
uR
u
u
Du
k Kk
k K
K
Ka
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• PI controller
Defaulted value: ko=0.55 a=0.4
1( ) 1c c
R
G s ks
0.9
o Rc
p
R
kk
k
a
2
1 2
2 1 2
0.9 1
tan 1
10.9 1 tan 1
uoc
p u
R u uu
Kkk a
k K
K a K
In terms of
ultimate data
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PID Controller Based on SOPDT Model
• Only PID controller is used for significant underdamped SOPDT dynamics, i.e.
• Controller parameters (ideal PID)
• The values of kp and need to be estimated in advance
( )s
oloop
k eG s
s
ko=0.5
(2 )
2
2
oc
p
R
D
kk
k
In terms ofultimate data
sin( )
sin( )
1 cos( )
sin( )
o u uc
p u
u uR
u
u uD
u u u
k Kk
k
K
K
K
1 2
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DynamicProcess
FOPDT Model SOPDT Model
PIDController
Group I Group II
PIController
PIDController
( )s
oloop
k eG s
s
(1 )
( )(1 )
so
loopf
k a s eG s
s s
1 2 1 2
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• Estimation of process gain kp
– Start the ATV test with a temporal disturbance to setpoint or process input
– Define
– and have
cycling responses
0
0
( ) ( )
( ) ( )
I t
I t
y t y d
u t u d
Iav
p Iav
yk
u
Iy Iu
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• Estimation from is subject to error,
sometimes as high as 20%
• From Fourier series expansion
• Ultimate gain is computed exactly as:
4cuk h A
0
0
0
0
( )( )
( )
uu
uu
j tt Pt
p u j tt Pt
y t e dtG j
u t e dt
1
( )cup u
kG j
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Estimation of Apparent Deadtime
• In an ATV test, two measured quantities are used to characterize the effect of the apparent deadtime
–
–
• For SOPDT process, this two quantities are
functions of and
A
A
pT
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• Underdamped SOPDT processes
cos( 3) sin( 3)
sin( 3) cos( 3)
p
p
AX
T A
AY
T A
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• Algorithm for estimation of apparent deadtime– Starting from a guessed value of – Calculate and , and feed them into networks to compute
and– Check if the eq. holds
– If not, increase the value
of until the above eq.
holds. At that time,
is the estimated apparent
deadtime
o X
Y
12 2
2tan
1u
uu
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• In ATV test, it provides and which are functions of and
pA k h uP
Locate on this figure • Zone I: FOPDT parameterization • Zone II: SOPDT parameterization
1 2
1 2
,u pP A k h
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• Initiate ATV test by a short period of manual disturbance and record y(t) and u(t) until constant cycling is attained
– Compute kp and kcu
– Estimate the apparent deadtime
– Classify the process by the location of
– If the process belongs to Group I, tune PI or PID controller based on FOPDT model parameterization
– If the process belongs to Group II, tune PID controller based on SOPDT model parameterization
( , )u pP A k h
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• Examples
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0 20 40 60 80 100 1200
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
Outp
ut
ProposedZ-N
0 10 20 30 40 50 600
0.5
1
1.5
Time
Outp
ut
ProposedZ-N
0 20 40 60 80 100 120 140 160 180 2000
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
Outp
ut
ProposedZ-N
Ex. 1 Ex. 2 Ex. 3
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• Ideal PID controller with an extra filter
• The value of kp and need to be known in advance
(2 )
2
2
oc
p
R
D
kk
k
sin( )
sin( )
1 cos( )
sin( )
o u uc
p u
u uR
u
u uD
u u u
k Kk
k
K
K
K
'
1 1( ) 1
1c c DR f
G s k ss s
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Optimal IAE Value for Set-point Tracking
– PI control
– PID control
• These optimal systems have reasonable stability robustness– PI control gain margin = 2.6 For unit step set-point change (H
uang and Jeng, 2002 )– phase margin = 55o
– PID control gain margin = 2.1, phase margin = 60o
1.0695s*PI
2.104 0.6023 for 5
2.104 for 5
eJ
1.5541s*PID
1.37 0.1134 for 3
1.37 for 3
eJ
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• Control systems designed for optimal input disturbance response will give smaller gain margin and phase margin than those designed for optimal set-point response.
• The optimal IAE value occurs at a phase margin about 30o to 50o – trade-off between disturbance performance and phase margin is not
always needed
PI control PID control
Optimal System for Disturbance Rejection
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Optimal IAE Value for Disturbance Rejection• The smaller the gain margin is (i.e. less robust), the lower the optimal IAE
value can achieve. – trade-off between disturbance performance and gain margin is
needed
• PI control
• PID control
d* 0.8931PI 0.8035 exp 0.0984m p mJ A k A
d* 1.0527PID 0.4681 exp 0.1071m p mJ A k A
PI control PID control
1.5 5mA Gain margin