Loading in 2 Seconds...
Control of a Solution Copolymerization Reactor using Piecewise Linear Models. Leyla Özkan APACT03 York, UK April 30 th , 2003. Presentation Outline. Motivation MultiModel Predictive Control Formulation Implementation on MMVA Solution Copolymerization Reactor Stability Analysis
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Control of a Solution Copolymerization Reactor using Piecewise Linear Models
Leyla Özkan
APACT03 York, UK
April 30th , 2003
APACT0330 April, 2003
Reduce OffSpecification Product
APACT0330 April, 2003
min
J
(
k
)
p
u
(
.
)
Past
Future
Set point
dx
Predicted y
=
f
(
x
,
u
,
t
)
dt
Closed loop y
Open loop u
£
£
u
u
u
min
max
£
£
y
y
y
min
max
Closed loop u
k
k+1
k+Hc
k+Hp
Hc
Hp
APACT0330 April, 2003
[
]
¥
T
=
+
+
+
J
x
(
k
m

k
)
Q
x
u
Ru
(
k
m

k
)
å
¥
I
T
+
+
(
k
)
(
k
m

k
)
(
k
m

k
)
finite horizon cost
infinite horizon cost
m
=
0
2
å
+
+
x
(
k
m

k
)
2
n
+
+
+
+
u
u
(
(
k
k
m
m


k
k
)
)
x
(
k
m

k
)
R
Q
2
2
=
m
n+1
I
å
+
Q
R
I
=
m
0
T
+
+
+
+
x
)
)
(
k
n
1
P x
(
k
n
1
APACT0330 April, 2003
x(k+1k), u(k+1k)
x(k+2k), u(k+2k)
x(kk), u(kk)
n+1 m
(x,u) = (0,0)
u(k+mk)=Kx(k+mk)
Terminal region
x(k+n+1k)
APACT0330 April, 2003
t
J
J
¥
n
OP2
x2
OP1
x1
Control Recipe
If x terminal region
quasiinfinite horizon
If x terminal region
infinite horizon
APACT0330 April, 2003
g
min
é
ù
1
*
*
*
*
*
L
ê
ú
0
.
5
g
Q
x
(
k

k
)
I
0
0
0
0
0
I
ê
ú
0
.
5
ê
ú
g
R
u
(
k

k
)
0
I
0
0
0
0
ê
ú
0
.
5
³
+
g
0
Q
x
(
k
1

k
)
0
0
I
0
0
0
ê
ú
I
0
.
5
ê
ú
+
g
R
u
(
k
1

k
)
0
0
0
I
0
0
ê
ú
M
O
ê
ú
ê
ú
+
+
x
(
k
n
1

k
)
0
0
0
0
0
Q
ë
û
é
ù
T
0
.
5
T
0
.
5
T
+
Q
(
A
Q
B
Y
)
QE
Y
R
t
t
t
t
I
t
ê
ú
T
T
+
+
x
x
(
A
Q
B
Y
)
Q
b
b
b
e
0
0
ê
ú
t
t
t
t
t
t
t
ê
ú
T
T
³
0
x

x

E
Q
e
b
(
I
e
e
)
0
0
t
t
t
t
t
ê
ú
0
.
5
ê
ú
g
Q
Q
0
0
I
0
I
ê
ú
0
.
5
g
R
Y
0
0
0
I
ë
û
t
Finite horizon
x(k) terminal region
Infinite horizon
APACT0330 April, 2003
g
min
x
Q
,
,
Y
i
s.t.
é
ù
T
1
x
(
k

k
)
³
0
ê
ú
ê
ú
x
(
k

k
)
Q
û
ë
and
é
ù
T
0
.
5
T
0
.
5
T
+
Q
(
A
Q
B
Y
)
QE
Y
R
i
i
i
i
I
i
ê
ú
T
T
+
+
x
x
(
A
Q
B
Y
)
Q
b
b
b
e
0
0
ê
ú
i
i
i
i
i
i
i
ê
ú
T
T
³
0
x

x

E
Q
e
b
(
I
e
e
)
0
0
i
i
i
i
i
ê
ú
0
.
5
ê
ú
g
Q
Q
0
0
I
0
I
ê
ú
0
.
5
g
R
Y
0
0
0
I
ë
û
i



1
1
1
=
g
x
=
l
=
P
Q
K
Y
Q
i
i
Özkan, L. et.al. , Control of a solution copolymerization reactor using multimodel predictive control, Chem. E. Science, 58:12071221, 2003
Feasible
ONLY IF
bi=0
(x(k) terminal region)
APACT0330 April, 2003
j
=
1
,
2
,
…
,
n
£
u
(
k
)
u
u
k
0
j
j
,
max
³
Finite horizon cost:
Infinite horizon cost:
Impose directly
(
u
(
k
+
m

k
)
k
+
m

k
)
=
Kt x
+
£
Exists s.t.
X
u
(
k
m

k
)
é
ù
X
Y
j
2
t
³
£
0
X
u
ê
ú
jj
j
,
max
u
Y
Q
ë
û
j
t
,
max
m
=
0
, 1, …, n
j
j
=
=
1
1
,
,
2
2
,
,
…
…
,
,
n
n
u
u
m
=
n+1 ,…,
APACT0330 April, 2003
Monomer (FA)
Monomer (FB)
Initiator (FI)
Coolant
Solvent (FS)
Coolant
Transfer Agent (FC)
Polymer
Solvent
Unreacted feed
Inhibitor (FZ)
Richards, J. R. et.al. , Feedforward and Feedback Control of a Solution Copolymerization Reactor, AIChE Journal, 35(6):891907, 1989
Challenge: Optimization problem involves 200–300 variables
APACT0330 April, 2003
4
OP 2
3.8
Mw/E4 (kg/kmol)
3.6
OP 1
3.4
0
10
20
30
40
50
time (h)
APACT0330 April, 2003
(x,u)  (0,0)
x3
x
x(kk)
x2
x1
t
APACT0330 April, 2003
Ap
Y
30
30
0
0
10
10
20
20
time (h)
time (h)
2 models
4
353.5
3 models
353.4
3.8
T (K)
353.3
4 models
Mw/E4 (kg/kmol)
353.2
3.6
353.1
3.4
Off–spec. (kg)
0.64
27
26
0.62
Open loop302.4
25
0.6
2 models134.8
Gpi (kg/h)
0.58
24
3 models110.1
23
0.56
4 models109.8
0.54
22
APACT0330 April, 2003
Asymptotically
stable origin
V(x)
Time
APACT0330 April, 2003
Discrete
components
Continuous
dynamical systems
Logic commands (switches,automata)
Hybrid systems
APACT0330 April, 2003
Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automatic Control, 43(4):475482, 1998.
If Viis monotonically nonincreasing on(T) then system is stable in the sense of Lyapunov
APACT0330 April, 2003
V1(x)
t0
t1
t2
t3
t4
t5
t6
V2(x)
APACT0330 April, 2003
Theorem
Ifx(k) terminal region, the quasi–infinite horizon optimization problem is solvedincluding the contractive constraint. If x(k) terminal regionthe infinite horizon optimization problem is solved. The closed loop system is stable if the feasible solutions of the control strategy defined are implemented in a receding horizon fashion
APACT0330 April, 2003
terminal region
Ï
g
ì
(
k
)
x
(
k
)
=
V
í
a
(
(
x
))
T
terminal region
Î
x
(
k
)
Px
(
k
)
x
(
k
)
î
APACT0330 April, 2003
1
10
0
10
(k)
xT(k)Px(k)
1
10
log (V)
2
10
3
10
4
10
0
5
10
15
t(h)
APACT0330 April, 2003
APACT0330 April, 2003
Thank you
for your attention
Leyla Özkan
APACT0330 April, 2003