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Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 1
Quantitative methods for
Regulatory layer selection
Ramprasad Yelchuru Sigurd Skogestad
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 2
Outline
• Motivation
• Problem formulation
• Distillation column Case study
• Results
4.1 CV2 (c = H2y) as individual measurements
4.2 CV2 (c = H2y) as measurement combinations
5. Conclusions
CV2 – controlled variable y – measurements vector
'yn'd
ydG
cs = 0 +
+
+ +
+ - K
H2
yG y
c
u
dW nW
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 3
Control system hierarchy for plantwide control
Self optimizing control
Scope of this work
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 4
Regulatory layer should (1) facilitate stable operation regulate the process keep the plant in a linear operating region (2) be simple (3) avoid control loop reconfiguration
1. Motivation
How to quantify ?
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 5
1. Motivation
(1) Minimize state drift by assuming perfect control c(jω)=0 and excluding stabilization (2) Simple: Close minimum number of loops
(3) Avoid control loop reconfiguration
2
2( ) ( ) : state weighting matrixJ Wx j Wω ω=
Quantified the regulatory layer objectives
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 6
1. Motivation Typical frequency dependancy plot
with single closed loop with proportional control gain ’k’ for a 2 x 2 MIMO system
Steady state based state drift is fairly good over a frequency bandwidth
k = 0 :open loop k = 10 :close to perfect control
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 7
Outline
• Motivation
• Problem formulation
• Distillation column Case study
• Results
4.1 CVs (c = H2y) as individual measurements
4.2 CVs (c = H2y) as measurement combinations
5. Conclusions
CV2 – controlled variable y – measurements vector
'yn'd
ydG
cs = 0 +
+
+ +
+ - K
H2
yG y
c
u
dW nW
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 8
22
2 2
( , )
( )
opt opt
opt
L J J u d
Wx Wx d
= −
= −
Ref: Halvorsen et al. I&ECR, 2003 Kariwala et al. I&ECR, 2008
2. Problem Formulation
21/2 12 2 2 2( )
uu
yavg F
L J H G H Y−=
Loss is due to (i) Varying disturbances (ii) Implementation error in controlling c at set point cs
12 2 2
2
[( ) ]
[ ]uu ud
y yd d n
d n
Y G J J G W W
F W W
−= −
= 2
optyFd
∂=
∂
u
dGy Gd
y
Gx Gdx
ymu0x
H2c
+
-
Cs=0K(s) y
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 9
2
1/2 12 2 2 2min ( )
uu
y
FHJ H G H Y− Non-convex
optimization problem
1 1
-1 -1 1 -1 -12 y 2 2 y 2 2 y 2 2 y 2 (H G ) H (H G ) D DH (DH G ) DH (H G ) H −= = =
12 2H DH=D : any non-singular matrix
We use D to make 1/22
yuuH G J=
22 2H
min H Y F
subject to 1/22 2uu
yH G J=
Convex optimization problem
Global solution
Problem is convex in decision matrix H2
2. Problem : Convex formulation
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 10
2. Problem formulation
Pick nc columns in Hu Pick 1 column in Hy and nc -1 columns in Hu Pick 2 columns in Hy and nc -2 columns in Hu Pick k columns in Hy and nc -k columns in Hu Pick nc columns in Hy and 0 columns in Hu
2 0[ ]mc H y u=
u
dGy Gd
y
Gx Gdx
ymu0x
H2c
+
-
Cs=0K(s) y
Example
1,1 1,2 1,4 1,5 1,6 1,82
2,1 2,2 2,4 2,5 2,6 2,8
y uH H
h h h h h hH
h h h h h h
=
nym number of ym nu0 number of physical valves
nc = number of CVs =nu
nym =4 nu0 =4
nc = 2=nu
⇒
⇒
⇒
⇒
⇒
P1. Close 0 loops : Select (nc variables from u0)
or (0 variables from ym)
P2. Close 1 loops : Select 1 variables from ym
P3. Close 2 loops : Select 2 variables from ym
P4. Close k loops : Select k variables from ym
P5. Close nc loops : Select nc variables from ym
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 11
2. Solution approach
{0,1}1,2, ,
i
yi nσ ∈∀ =
1 2
1 2
1 2 2*2
( 1)* 1 ( 1)* 2 *
ny
ny
ny ny ny
nc ny nc ny nc ny nu ny
x x xx x x
H
x x x
σ σ σ
+ +
− + − + ×
=
( )
{ }
,
( 1)*
min
.
0,11,2, ,
T
T
x
y
i
ny ii i
nu ny i
i
y
x Fx
st G x jn
xM MxM M
M Mx
i n
δ δδ δσ
δ δ
δσ
σ σ
σ
+
− +
==
− − ≤ ≤ −
∈∀ =
δ
P
1
2
1yy
n n
δ
σσ
σ
σ×
=
(1, )max( ) / min( )
T
y
F Y Yones ny
upper bound for M J G
δ δ
δ
==
=
P
We solve this MIQP for n = nu to ny
Big M approach
1
2
* ( * ) 1nu ny nu ny
xx
X
x
δ
×
=
P, n are modified to fit our requirement
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 12
Outline
• Motivation
• Problem formulation
• Distillation column Case study
• Results
4.1 CVs (c = H2y) as individual measurements
4.2 CVs (c = H2y) as measurement combinations
5. Conclusions
CV2 – controlled variable y – measurements vector
'yn'd
ydG
cs = 0 +
+
+ +
+ - K
H2
yG y
c
u
dW nW
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 13
3. Case Study : Distillation Column
T1, T2, T3,…, T41
Tray temperatures qF
Binary Distillation Column Shown : LV configuration 41 Trays Level loops closed with D,B
2 MVs – L,V 41 Measurements – T1,T2,T3,…,T41
3 DVs – F, ZF, qF *Compositions are indirectly controlled by controlling the tray temperatures
2
2J Wx=
u
dGy Gd
y
Gx Gdx
ymu0x
H2c
+
-
Cs=0K(s) y
0
LL V
u uV D
B
= =
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 14
3. Case Study : Distillation Column
10.83 -10.96 5.85 11.17 10.9015.36 -15.55 8.30 15.86 15.47
;13.01 -12.81 5.85 13.10 12.90 8.76 -8.62 3.94 8.82 8.68
0.2 0 00 0.1 00 0 0.1
y yd
d
G G
W
= =
=
; (0.5* (41,1))Wn diag ones
=
41 2 41 3 2 2 3 3 3 41 412 2; ; ; ; ;
uu ud
y yd d nG G J S J W W× × × × ×
+∈ ∈ ∈ ∈ ∈ ∈
Data
21/2 12 2 2 2( )
uu
yavg F
L J H G H Y−= 12 2 2[( ) ]
uu ud
y yd d nY G J J G W W−= −
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 15
Outline
• Motivation
• Problem formulation
• Distillation column Case study
• Results
4.1 CVs (c = H2y) as individual measurements
4.2 CVs (c = H2y) as measurement combinations
5. Conclusions
CV2 – controlled variable y – measurements vector
'yn'd
ydG
cs = 0 +
+
+ +
+ - K
H2
yG y
c
u
dW nW
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 16
4. Results 4.1 CVs (c = H2y) as individual measurements
Pick 2 columns in Hu Pick 1 column in Hy and 1 column in Hu Pick 2 columns in Hy and 0 column in Hu
2 0[ ]mc H y u=
u
dGy Gd
y
Gx Gdx
ymu0x
H2c
+
-
Cs=0K(s) y
1,1 1,2 1,41 1,42 1,43 1,44 1,452
2,1 2,2 2,41 2,42 2,43 2,44 2,45
y uH H
h h h h h h hH
h h h h h h h
=
nym number of ym nu0 number of physical valves
nc = number of CVs =nu
nym =41 nu0 =4
nc = 2=nu
⇒
⇒
⇒
P1. Close 0 loops : Select (2 variables from u0)
or (0 variables from ym)
P2. Close 1 loops : Select 1 variables from ym
P3. Close 2 loops : Select 2 variables from ym
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 17
4.1 CV2 (c = H2y) as individual measurements
*
* In addition to two level loops
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 18
Distillation column : 1 loop closed (T18) + L constant
5 10 15 20 25 30 35 400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stages
Com
posi
tions
(x)
xopt for disturbance Fxopt for disturbance zFxopt for disturbance qFx for disturbance Fx for disturbance zFx for disturbance qF
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 19
Distillation column : 1 loop closed (T18) + L constant
5 10 15 20 25 30 35 4041-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Stages
∆ x
∆xopt for disturbance F
∆xopt for disturbance zF∆xopt for disturbance qF
5 10 15 20 25 30 35 4041-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Stages
∆ x
∆x for disturbance F∆x for disturbance zF∆x for disturbance qF∆x for measurement noise in L∆x for measurement noise in T18
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 20
Distillation column : 2 loops closed (T15,T27)
1 5 10 15 20 25 30 35 40 410
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Stages
Com
posi
tions
(x)
xopt for disturbance Fxopt for disturbance zFxopt for disturbance qFx for disturbance Fx for disturbance zFx for disturbance qF
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 21
Distillation column : 2 loops closed (T15,T27)
1 5 10 15 20 25 30 35 4041-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Stages
∆ x
∆xopt for disturbance F
∆xopt for disturbance zF∆xopt for disturbance qF
1 5 10 15 20 25 30 35 4041-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Stages
∆ x
∆x for disturbance F∆x for disturbance zF∆x for disturbance qF∆x for measurement noise in T15∆x for measurement noise in T27
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 22
4.2 CV2 (c = H2y) as combinations of measurements
2 3 4 5 410.1
0.12
0.14
0.16
0.18
0.2
Loss
1/2
||M||2 F
No. of used meas. (n)
One loop closed
2 3 4 5 410
0.005
0.01
0.015
0.02
0.025
0.03
Loss
1/2
||M||2 F
No. of used meas. (n)
Two loop closed
Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 23
5. Conclusions Quantified the regulatory layer objectives using state drift criterion
Extended the economic self-optimizing control concepts to state drift minimization in regulatory
layer
We showed that state drift minimization at steady-state is sufficient for distillation case study
Regulatory layer control variables (CV2) as individual measurements or measurement
combinations are selected to minimize the state drift
Optimal regulatory layer with 1,2 or more closed loops is obtained, to arrive at minimum
regulatory layer with acceptable loss.