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Predictive Controller Tuning &
Steady State Target Selection:
A Covariance Assignment Approach
Donald J Chmielewski
Illinois Institute of Technology
Annual Meeting of the AIChE
November 2000, Los Angeles, CA
Previous Work
• Tuning of Predictive Controllers
- Cutler (1983)
- Shridhar & Cooper (1998)
- Loeblein & Perkins (1999)
• Steady State Target Selection
- Muske & Rawlings (1994)
- de Hennin et. al. (1994)
- Rao & Rawlings (1999)
- Loeblein & Perkins (1999)
Process
Controller
State
Estimator
-------------------
-------------------
-------------------
z
t
Outputs
Measurements
v ---------------------
t
---------------------
--------------------
--------------------
u
Sensor
Noise
---------------------
w
t
Disturbance
Inputs
Output Envelope Prediction
t
CSTR
Vent
Position
O2 out
To , F TR TF
Fuel Feed
Furnace
Example: Pre-Heated Reactor
Manipulated Variables:
• Reactant Feed Rate (F)
• Fuel Feed Rate (Ff)
• Vent Position (V)
Control Variables:
• Reactor Temperature (TR)
• Furnace Temperature (TF)
• Furnace Oxygen (O2)
• Furnace CO (CO)
CO out
Disturbance Input:
• Feed Temperature (To)
Infinite Horizon Predictive Control
Min { xT(k)Qx(k) + uT(k)Ru(k) }
s.t. x(k+1) = A x(k) + B u(k)
| xi (k)| < xi
| ui (k)| < ui
Σ k= 0
8
_
Unconstrained
Solution
P = (A-BL) P(A-BL) + L RL + Q
L = (B PB + R) B PA
T T
T T -1
_
Open Loop Response
475 480 485 490 495 500 505 510 515 520 525300
350
400
450
Reactor Temperature (F)
Fu
rna
ce
Te
mp
era
ture
(F
)
Closed Loop Response
495 496 497 498 499 500 501 502 503 504 5050
1
2
3
4
5
6
7
8
Reactor Temperature (F)O
2 C
on
ce
ntr
atio
n (
%)
495 496 497 498 499 500 501 502 503 504 5050
20
40
60
80
100
120
140
160
180
200
Reactor Temperature (F)
CO
Co
nce
ntr
atio
n (
pp
m)
495 496 497 498 499 500 501 502 503 504 5050.7
0.8
0.9
1
1.1
1.2
1.3x 10
4
Reactor Temperature (F)
Re
acta
nt
Fe
ed
R
ate
(b
bl/d
ay)
495 496 497 498 499 500 501 502 503 504 505300
350
400
450
Reactor Temperature (F)
Fu
rna
ce
Te
mp
era
ture
(F
)
495 496 497 498 499 500 501 502 503 504 5050.075
0.08
0.085
0.09
0.095
0.1
0.105
0.11
0.115
0.12
0.125
Reactor Temperature (F)
Ve
nt
Po
sitio
n
495 496 497 498 499 500 501 502 503 504 5050
2
4
6
8
10
12
14
16
18
20
Reactor Temperature (F)
Fu
el F
ee
d R
ate
Closed Loop Response
495 496 497 498 499 500 501 502 503 504 505300
350
400
450
Reactor Temperature (F)
Fu
rna
ce
Te
mp
era
ture
(F
)
495 496 497 498 499 500 501 502 503 504 5050
1
2
3
4
5
6
7
8
Reactor Temperature (F)O
2 C
on
ce
ntr
atio
n (
%)
495 496 497 498 499 500 501 502 503 504 505-200
-100
0
100
200
300
400
Reactor Temperature (F)
CO
Co
nce
ntr
atio
n (
pp
m)
495 496 497 498 499 500 501 502 503 504 5050.7
0.8
0.9
1
1.1
1.2
1.3x 10
4
Reactor Temperature (F)
Re
acta
nt
Fe
ed
R
ate
(b
bl/d
ay)
495 496 497 498 499 500 501 502 503 504 505-30
-20
-10
0
10
20
30
40
50
Reactor Temperature (F)
Fu
el F
ee
d
Ra
te
495 496 497 498 499 500 501 502 503 504 505
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Reactor Temperature (F)
Ve
nt
Po
sitio
n
Covariance Analysis
• Closed Loop Process
x(k+1) = (A – BL)x(k) + Gw(k)
• Covariance of the State
Σx = (A-BL) Σx (A- BL) + G ΣwG T
• Covariance of the Control Input
Σu = L Σx L T
T
T
Steady State Covariance
of Scalar Signals
State Variables:
lim E[ ( xi (k) )2 ] = ei Σx ei
Input Variables:
lim E[ ( ui (k) )2 ] = ei Σu ei = ei LΣx Lei
where e i is the ith Row of the Identity Matrix
k 8
T
T
8
k
T T
Covariance Bounded Design
There Exists L s.t. ei x eiT < xi
2 and eiL x LTei
T < ui2
If and Only If
There Exits X > 0 and Y s.t.
ui2 eiY
YTeiT X
> 0 and
X-AXAT+BYAT BY
+AYTB-G wGT
BTYT X
> 0
eiXeiT < xi
2
One such L is given by YX -1
Selection of Covariance Bounds
xi = min{( xi - xi ), ( xi - xi )}
ui = min{( ui - ui ), ( ui - ui )}
max
max
min
min
SS SS
SS SS
xi
-----------------------------------------------------------------------------------------------------
Ui
Xi
s u
s x
xi max
max ui
min ui
xi min
Selection of Covariance Bounds
xi = min{( xi - xi ), ( xi - xi )}
ui = min{( ui - ui ), ( ui - ui )}
max
max
min
min
SS SS
SS SS
xi
-----------------------------------------------------------------------------------------------------
Ui
Xi
s u / 2
s x
xi max
max ui
min ui
xi min
2
Covariance Bounded Tuning
• Covariance Bounded Synthesis:
Given: Σw , Dxi ‘s & Dui ‘s L
• Tuning of Predictive Controller:
Given: Σw , Dxi ‘s & Dui ‘s Q & R
Covariance Bounded LQR Design
There Exists Q > 0 & R > 0 s.t.
ei Σx ei < xi ; i = 1………n
ei LΣx Lei < ui ; i = 1………m
where Σx = (A-BL) Σx (A- BL) + G Σw G
P = (A-BL) P(A-BL) + L RL + Q
L = (B PB + R) B PA
T
T
T
T
T
T
T
- 1
2
2
T
T
If . . .
There Exists X > 0 & Y > 0 s.t.
ei A X (A ) ei < xi ; i =1 ….. n
ui eiYB
BYei X
X - BYB > 0
X - AXA + ABYBA > 0
X - AXA - AGΣwG A + 2ABYB A ABYB
BYB A X
> 0 ; i = 1 ….. m
-1 - 1
T
T
T
T
T
T T T
T T T T T T
T T
2
2
> 0
• Give a pair (X, Y) that satisfy the previous
Linear Matrix Inequalities (LMI’s).
Then,
Q = (X - BYB ) - A X A
R = Y
will yield the Covariance Bounded LQR
T T - 1
-1
- 1
Minimum Covariance Design
ei A Y (A ) ei < sxi ; sx
i < Dxi2
su
i eiYB
sui < Dui
2
BYei X
X - BYB > 0 X - AXA + ABYBA > 0
X - AXA - AGΣwG A + 2ABYB A ABYB
BYB A X
> 0 ;
-1 - 1
T
T
T
T
T
T T T
T T T T T T
T T
> 0
min { } cxi s
xi + c
ui s
ui
sxi s
ui
Closed Loop Response
( Minimized Temperature Covariance )
495 496 497 498 499 500 501 502 503 504 5050
1
2
3
4
5
6
7
8
Reactor Temperature (F)
O2 C
oncentr
ation (
%)
495 496 497 498 499 500 501 502 503 504 505300
350
400
450
Reactor Temperature (F)
Furn
ace T
em
pera
ture
(F
)
495 496 497 498 499 500 501 502 503 504 5050
20
40
60
80
100
120
140
160
180
200
Reactor Temperature (F)
CO
Concentr
ation (
ppm
)495 496 497 498 499 500 501 502 503 504 505
0.7
0.8
0.9
1
1.1
1.2
1.3x 10
4
Reactor Temperature (F)
Reacta
nt
Feed
Rate
(bbl/day)
495 496 497 498 499 500 501 502 503 504 5050
2
4
6
8
10
12
14
16
18
20
Reactor Temperature (F)
Fuel F
eed
Rate
495 496 497 498 499 500 501 502 503 504 5050.075
0.08
0.085
0.09
0.095
0.1
0.105
0.11
0.115
0.12
0.125
Reactor Temperature (F)V
ent
Positio
n
Closed Loop Response
( Minimized Reactant Feed Covariance )
495 496 497 498 499 500 501 502 503 504 505300
350
400
450
Reactor Temperature (F)
Furn
ace T
em
pera
ture
(F
)
495 496 497 498 499 500 501 502 503 504 5050
1
2
3
4
5
6
7
8
Reactor Temperature (F)
O2 C
oncentr
ation (
%)
495 496 497 498 499 500 501 502 503 504 5050
20
40
60
80
100
120
140
160
180
200
Reactor Temperature (F)
CO
Concentr
ation (
ppm
)495 496 497 498 499 500 501 502 503 504 505
0.7
0.8
0.9
1
1.1
1.2
1.3x 10
4
Reactor Temperature (F)
Reacta
nt
Feed
Rate
(bbl/day)
495 496 497 498 499 500 501 502 503 504 5050
2
4
6
8
10
12
14
16
18
20
Reactor Temperature (F)
Fuel F
eed
Rate
495 496 497 498 499 500 501 502 503 504 5050.075
0.08
0.085
0.09
0.095
0.1
0.105
0.11
0.115
0.12
0.125
Reactor Temperature (F)V
ent
Positio
n
Steady State Target Selection
• Minimum Covariance Design suggests that
Profit α ( xi + ui )
• Real Time Optimization Employs:
Profit α ( xi + ui )
2 2
Covariance Based Target Selection
ei A Y (A ) ei < sxi ; sx
i < Dxi2
su
i eiYB
sui < Dui
2
BYei X
X - BYB > 0 X - AXA + ABYBA > 0
X - AXA - AGΣwG A + 2ABYB A ABYB
BYB A X
> 0 ;
-1 - 1
T
T
T
T
T
T T T
T T T T T T
T T > 0
min { } cxi (sx
i ) + cu
i ( sui )
1/2 1/2
sxi s
ui
Acknowledgments
• Michael J.K. Peng
• Armor College of Engineering, IIT
• Department of Chemical & Environmental
Engineering, IIT
Covariance Bounded LQR Design
There Exists Q > 0 & R > 0 s.t.
ei Σx ei < xi ; i = 1………n
ei LΣx Lei < ui ; i = 1………m
where Σx = (A-BL) Σx (A- BL) + G Σw G
P = (A-BL) P(A-BL) + L RL + Q
L = (B PB + R) B PA
T
T
T
T
T
T
T
- 1
2
2
T
T
If and Only If. . .
There Exists X > 0 , Y > 0 and Z s.t.
ei Xei xi ; i =1 ….. n
ui ei Z
Z ei X
X - GΣwG AX -BZ
(AX-BZ) X
X (AX - BZ) Z
AX - BZ X 0
Z 0 Y
> 0 ; i = 1 ….. m T T
T
T
T
2
2
> 0
> 0
T
> T
• Give a triple (X,Y, Z) that satisfy the previous
Linear Matrix Inequalities (LMI’s).
Then,
Q = X [(AX-BZ) X (AX-BZ) - X + ZYZ]X
R = Y
will yield the Covariance Bounded LQR
T T - 1
-1
- 1 -1 -1
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