optimization-based pi/pid control for sopdt process
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Optimization-based PI/PID control for SOPDT process
Summary on optimization-based PI/PID control
Best achievable IAE performance by PI/PID control of FOPDT process
Optimal rise-time vs, IAE in PI/PID control of SOPDT process
Optimal rise-time vs, IAE in PI/PID control of SOPDT process
FOPDT
SOPDT
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
IMC-PID for FOPDT process
( )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:
(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
(1 )( )
(1 )
so
loopk s e
G ss s
(1 )( )
(1 )
so
LP n
k s eG s
s s
• We should learn what happens to the Z-N tuned controllers?
• How inverse-based controllers are synthesized?
Inverse-based Controller Design
• 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
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
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
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
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
Auto-tune
• Autotuning via relay feedback: Astrom and Hagglund (1984)
Referred as autotune variation (ATV): Luyben (1987)
Main advantage: under closed-loop
4 2cu u
u
hk
A P
Apply Z-N or T-L tuning
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
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
10.707 1
2UK
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
• 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
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
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
• 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
• 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
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
• Underdamped SOPDT processes
cos( 3) sin( 3)
sin( 3) cos( 3)
p
p
AX
T A
AY
T A
• 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
• 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
• 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
• Examples
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
• 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
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
• 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
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