Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 1
Quantitative methods for
controlled variables selection
Ramprasad Yelchuru
Department of Chemical Engineering
Norwegian University of Science and Technology
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 2
Thesis outline
Ch. 1. Introduction
Ch. 2. Brief overview of control structure design and methods
Ch. 3. Convex formulations for optimal CV using MIQP
Ch. 4. Convex approximations for optimal CV with structured H
Ch. 5. Quantitative methods for regulatory layer selection
Ch. 6. Dynamic simulations with self-optimizing CV
Ch. 7. Conclusions and future work
Appendices A - E
CV – Controlled Variables
MIQP - Mixed Integer Quadratic Programming
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 3
Presentation outline
Plantwide control : Self optimizing control formulation for CV, c = Hy – Chapter 2
Convex formulation for CV with full H - Chapter 3
Convex formulation
Globally optimal MIQP formulations
Case studies
Convex approximation methods for CV with structured H – Chapter 4
Convex approximations
MIQP formulations for structured H with measurement subsets
Case studies
Regulatory control layer selection – Chapter 5
Problem definition
Regulatory control layer selection with state drift minimization
Case studies
Conclusions and Future work
CV – Controlled Variables
MIQP - Mixed Integer Quadratic Programming
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 4
Presentation outline
Plantwide control : Self optimizing control formulation for CV, c = Hy – Chapter 2
Convex formulation for CV with full H - Chapter 3
Convex formulation
Globally optimal MIQP formulations
Case studies
Convex approximation methods for CV with structured H – Chapter 4
Convex approximations
MIQP formulations for structured H with measurement subsets
Case studies
Regulatory control layer selection – Chapter 5
Problem definition
Regulatory control layer selection with state drift minimization
Case studies
Conclusions and Future work
CV – Controlled Variables
MIQP - Mixed Integer Quadratic Programming
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 5
Plantwide control: Hierarchical decomposition
Each layer operates at different time
scales
The decisions are cascaded from top to
bottom
Top layer provides set points to the
bottom layer
Scope of the thesis: Optimal operation
constituting optimization layer and
control layers
Assumption: Economics are primarily
decided by steady-state
Focus is on the selection of controlled
variables CV1 and CV2
MPC
PID
RTO
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 6
Optimal operation
Ref: Kassidas et al., 2000
Engell, 2007
d
cs
Plant
(Gy,Gdy)
Controller
K
Real Time
Optimization (RTO)
+
-
y
u
+ ny
d
H
c cs
Plant
(Gy,Gdy)
Controller
K
Real Time
Optimization (RTO)
+
-
y
u
+ ny
Real time optimization Closed loop implementation with
a separate control layer
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 7
Self optimizing control
Self-optimizing control is said to occur when we can achieve an acceptable loss (in comparison with truly optimal operation) with constant setpoint values for the controlled variables without the need to reoptimize when disturbances occur.
Ref: Skogestad, JPC, 2000.
Acceptable loss
self-optimizing control
Controller
Process d
u(d)
c = Hy
cs e
-
+
+ n
cm
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 8
Optimal steady-state operation
( , ) ( ( ), )opt optL J u d J u d d
Problem Formulation, c = Hy
31( , ) ( ( ), ) ( ( )) ( ( )) ( ( ))
2
1( ( )) ( ( ))
2
T
opt u opt opt uu opt
T
opt uu opt
J u d J u d d J u u d u u d J u u d
L u u d J u u d
min ( , )u
J u d
d
Assumptions:
(1) Active constraints are controlled
(2) Quadratic nature of J around uopt(d)
(3) Active constraints remain same throughout the analysis
( )opt ou d
( )opt oJ d( )optJ d
( )optu d
Loss
u
J(u,d)
do
Real time optimization
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 9
Ref: Halvorsen et al. I&ECR, 2003
Kariwala et al. I&ECR, 2008
Problem Formulation, c = Hy
21/2 1( )y
avg uu FL J HG HY
Loss
d´,ny´ as random variables
1[( ) ]y y
uu ud d d nY G J J G W W
( , , )yL f H d n
(0,1)y
d
n
y
'd
Controlled variables, c yH
y
dG
cs = constant +
+
+
+
+
- K
H
yG
'yn
c
u
dW nW
H
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 10
Presentation outline
Plantwide control : Self optimizing control formulation for CV, c = Hy – Chapter 2
Convex formulation for CV with full H - Chapter 3
Convex formulation
Globally optimal MIQP formulations
Case studies
Convex approximation methods for CV with structured H – Chapter 4
Convex approximations
MIQP formulations for structured H with measurement subsets
Case studies
Regulatory control layer selection – Chapter 5
Problem definition
Regulatory control layer selection with state drift minimization
Case studies
Conclusions and Future work
CV – Controlled Variables
MIQP - Mixed Integer Quadratic Programming
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 11
Convex formulation (full H) 1/2 1min ( )y
uu FHJ HG HY Seemingly
Non-convex
optimization problem
-1 -1 -1 1 -1
1 y 1 y y y (H G ) H = (DHG ) DH = (HG ) D DH = (HG ) H
1H DH
D : any non-singular matrix
Objective function unaffected by D.
So can choose freely.
H is made unique by adding a constraint as
yHG
1/2y
uuHG J
Hmin HY F
subject to 1/ 2y
uuHG J
Full H
Convex
optimization problem
Global solution Problem is convex in decision matrix H
Ref: Alstad 2009
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 12
Vectorization
h
11 12 1
21 22 2
1 2 *
ny
ny
nu nu nu ny nu ny
h h h
h h hH
h h h
Hmin HY F
subject to 1/ 2y
uuHG J
min
.
T
h
T
h F X
st G X J
Problem is convex QP in decision vector
11
12
* ( * ) 1nu ny nu ny
h
hh
h
TF Y Y
is vectorized along the rows of H to form
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 13
Controlled variable selection
Optimization problem :
Minimize the average loss by selecting H and CVs as
(i) best individual measurements
(ii) best combinations of all measurements
(iii) best combinations with few measurements
min
.
T
h
T
h F h
st G h J
H
min HY F
st. 1/ 2y
uuHG J
1/2 1min ( )y
uu FHJ HG HY
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 14
MIQP formulation (full H)
{0,1}
1,2, ,
i
i ny
11 12 1
21 22 2
1 *
1
2
2
ny
ny
nu nu nu ny n
ny
u ny
h h h
h h hH
h h h
11
12
* ( * )
1
2
11nu ny nu ny ny ny
h
hh
h
is vectorized along the rows of H to form
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 15
MIQP formulation
Big-m method Indicator constraint method
,
1
2
min
.
1,2, ,
T
i i
T
x
y
i
i
nui
h F h
st G h J
P n
hm m
hm m
m mh
i ny
δ
,min
.T
T
x
y
h F h
st G h J
P n
δ
Indicator
constraints
1
2
10 0
1,2, ,
u
i
i
i n
nui
h
h
h
i ny
11 12 1
21 22 2
1 *
1
2
2
ny
ny
nu nu nu ny n
ny
u ny
h h h
h h hH
h h h
Selection of appropriate m is an iterative method
and can increase the computational requirements
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 16
Case Study : Distillation Column
T1, T2, T3,…, T41
Tray temperatures qF
Binary Distillation Column
LV configuration
(methanol & n-propanol)
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 2
, ,
, ,
D D s B B s
D s B s
y y x xJ
y x
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 17
Distillation Column : Full H
c = Hy
1
2
cc
c
1
2
41
T
Ty
T
1 11 1 12 2 141 41
2 21 1 22 2 241 41
c h T h T h T
c h T h T h T
11 12 120 130 141
21 22 220 230 241
h h h h hH
h h h h h
1/2 1( )y
avg uu FL J HG HY
Find H that minimizes
T1, T2, T3,…, T41
Tray temperatures qF
Binary distillation column
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 18
Case Study : Distillation Column
21/2 11
( ( ) )2
y
avg uu FL J HG HY 1
[ ]d n
y y
uu ud d
Y FW W
F G J J G
10.83 -10.96 5.85 11.17 10.90
15.36 -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
3.88 3.88
3.89 3
y y
d
uu
G G
J
1.96 3.96 3.88; ;
.90 1.97 3.97 3.89
0.2 0 0
0 0.1 0 ; (0.5* (41,1))
0 0 0.1
ud
d
J
W Wn diag ones
41 2 41 3 2 2 3 3 3 41 41; ; ; ; ;y y
d uu ud d nG G J S J W W
Data
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 19
Distillation Column Full H : Result
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 20
Distillation Column Full H : Result
Comparison with customized Branch And Bound (BAB)*
MIQP is computationally more intensive than Branch And Bound (BAB) methods
(Note that computational time is not very important as control structure selection is an offline method)
MIQP formulations are intuitive and easy to solve
* Kariwala and Cao, 2010
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 21
Other case studies
• Toy example
– 4 measurements, 2 inputs, 1 disturbance
• Evaporator system
– 10 measurements, 2 inputs, 3 disturbances
• Kaibel distillation column
– 71 measurements, 4 inputs, 7 disturbances
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 22
Presentation outline
Plantwide control : Self optimizing control formulation for CV, c = Hy – Chapter 2
Convex formulation for CV with full H - Chapter 3
Convex formulation
Globally optimal MIQP formulations
Case studies
Convex approximation methods for CV with structured H – Chapter 4
Convex approximations
MIQP formulations for structured H with measurement subsets
Case studies
Regulatory control layer selection – Chapter 5
Problem definition
Regulatory control layer selection with state drift minimization
Case studies
Conclusions and Future work
CV – Controlled Variables
MIQP - Mixed Integer Quadratic Programming
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 23
Convex approximation methods for structured H
Structured H will have some zero elements in H
Example:
decentralized H
(block-diagonal H)
triangular H
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 24
Convex approximations for Structured H
For a structured H like
or
only a block diagonal or triangular
preserves the structure in H and and the degrees of freedom in D is used to arrive at convex approximation methods
1/2 1min ( )y
uu FHJ HG HY
1H DH
-1 -1 -1 1 -1
1 y 1 y y y (H G ) H = (DHG ) DH = (HG ) D DH = (HG ) H
1H DHD : any non-singular matrix
1
2
0 0
0 0
0 0iun
D
DD
D
11 12 1
22 20
0 0
iu
iu
iu iu
n
n
n n
D D D
D DD
D
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 25
11 12
23 24
0 0
0 0 hH
h
h
h
CVs with structural constraints (structured H) : Convex
upper bound (structured H)
1H DH
Examples 1 :
1/2y
uuHG J
Full H 21
11 12 13 1
2 3 24
4
2 2h h
h hH
h
h
h
h 21 2
11
2
12
d d
d dD
2
1
2
1 0
0D
d
d
Decentralized H
22 2
11 11 1
3
1 2
22 24
1
1 0 0
0 0H DH
d
d h d
h d
h
h
1H DH
21 22 3 4
1 12
2
1
2
0 0
h h h
hH
h
h
Traingular H 1 22
11
2
0
d dD
d
For structured H, less degrees of freedom in
D result in convex upper bound
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 26
Convex approximation methods for structured H
Convex approximation method 1: matching elements in HGy to Juu
1/2
Convex approximation method 2: Relaxing the equality constraint to
inequality constraint
{0,1}l
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 27
Controlled variable selection with structured H
Optimization problem :
Minimize the average loss by selecting a structured H and CVs as
(i) best individual measurements
(ii) best combinations of all measurements
(iii) best combinations with few measurements
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 28
structured H with optimal measurement subsets
Convex approximation method 1: matching elements of HGy to Juu
1/2
Convex approximation method 2: relaxing equality constraint to
inequality constraint
{0,1}l
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 29
T1,T2,…,T20
T21,T22,…,T41
Distillation column : Decentralized H
1 11 1 12 2 120 20
2 221 21 222 22 241 41
11 12 120
221 241
0 0 0
0 0 0
c h T h T h T
c h T h T h T
h h hH
h h
Decentralized structure
qF
Top section
T21, T22, T23,…, T41
Bottom section
T1, T2, T3,…, T20
Binary distillation column
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 30
Distillation Column : Results
*clearly not optimal as the solutions must be same with CVs as individual measurements
Ɨ small differences in the optimal solution in convex approximation methods 1 and 2 for triangular H and block diagonal H
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 31
Decentralized H: Result
The proposed methods are not exact (Loss should be same for H full and H disjoint for individual
measurements)
Proposed method provide good upper bounds for the distillation case
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 32
Distillation column : Triangular H
1 121 21 122 22 141 41
2 21 1 22 2 241 41
121 122 141
21 22 220 221 222 241
0 0 0
c h T h T h T
c h T h T h T
h h hH
h h h h h h
Traingular structure
qF
Top section
T21, T22, T23,…, T41
All temperatures
T1, T2, T3,…, T41
Binary distillation column
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 33
Distillation Column : Results
**clearly not optimal as triangular H must at least be as good as H disjoint
Ɨ small differences in the optimal solution in convex approximation methods 1 and 2 for triangular H and block diagonal H
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 34
The proposed methods are not exact (Loss should be same for full H, triangular H for individual
measurements)
Proposed method provide good upper bounds for the distillation case
In convex approximation methods we are minimizing and smaller for
n = 5 than n = 4, but the loss is higher for n = 5 than n = 4 and
causes irregular behavior
Triangular H: Result
1/2 1( )y
uu FJ HG HY
FHY
FHYFHY
1/2 1( )y
uu FJ HG HY
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 35
Presentation outline
Plantwide control : Self optimizing control formulation for CV, c = Hy – Chapter 2
Convex formulation for CV with full H - Chapter 3
Convex formulation
Globally optimal MIQP formulations
Case studies
Convex approximation methods for CV with structured H – Chapter 4
Convex approximations
MIQP formulations for structured H with measurement subsets
Case studies
Regulatory control layer selection – Chapter 5
Problem definition
Regulatory control layer selection with state drift minimization
Case studies
Conclusions and Future work
CV – Controlled Variables
MIQP - Mixed Integer Quadratic Programming
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 36
Control system hierarchy for plantwide control
Self optimizing control
Regulatory control
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 37
Regulatory layer should
(1) facilitate stable operation
regulate the process
operate the plant in a linear operating region
(2) be simple
(3) avoid control loop reconfiguration
Regulatory control layer: Objectives
How to quantify ?
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 38
Regulatory control layer: Objectives
(1) Minimize state drift
(2) Simple: Close minimum number of loops
(3) Avoid control loop reconfiguration
2
2( ) ( ) : stateweighting matrixJ Wx j W
Quantified the regulatory layer objectives
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 39
Regulatory control layer: Justification to use steady state analysis
Typical frequency dependancy plot
2 x 2 MIMO system with single closed loop with proportional control gain ’k’
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 controlled variables selection, 40
22
2 2
( , ) ( ( ), )
( )
opt opt
opt
L J u d J u d d
Wx Wx d
Ref: Halvorsen et al. I&ECR, 2003
Kariwala et al. I&ECR, 2008
Regulatory control layer: Problem Formulation
21/2 1
2 2 2 2( )uu
y
avgF
L J H G H Y
Loss is due to
(i) Varying disturbances
(ii) Implementation error in
controlling c at set point cs
1
2 2 2
2
[( ) ]
[ ]
uu ud
y y
d d n
d n
Y G J J G W W
F W W
2
optyF
d
u
d
Gy Gdy
Gx Gdx
ym
u0
x
H2
c+
-
Cs=0
K(s) y
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 41
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
d
Gy Gdy
Gx Gdx
ym
u0
x
H2
c+
-
Cs=0
K(s) y
Example
15 16 1811 12 14
2
21 22 24 25 26 28
y uH H
h h hh 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 controlled variables selection, 42
MIQP formulation
{0,1}
1,2, ,
i
i ny
11 12 1
21 22 2
2
1
1 2
2
*
ny
ny
nu nu nu ny nu ny
ny
h h h
h h hH
h h h
11
12
* ( * )
1
2
11nu ny nu ny ny ny
h
hh
h
is vectorized along the rows of H to form
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 43
Regulatory layer selection: Solution approach
MIQP formulation
,
1
2
min
.
1,2, ,
T
i i
T
x
y
i
i
nui
h F h
st G h J
P n
hm m
hm m
m mh
i ny
δ
11 12 1
21 22 2
1 *
1
2
2
ny
ny
nu nu nu ny n
ny
u ny
h h h
h h hH
h h h
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 44
Case Study : Distillation Column
T1, T2, T3,…, T41
Tray temperatures qF
Binary Distillation Column
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 W x
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 45
Case Study : Distillation Column
10.83 -10.96 5.85 11.17 10.90
15.36 -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 0
0 0.1 0
0 0 0.1
y y
d
d
G G
W
; (0.5* (41,1))Wn diag ones
41 2 41 3 2 2 3 3 3 41 41
2 2; ; ; ; ;uu ud
y y
d d nG G J S J W W
Data
21/2 1
2 2 2 2( )uu
y
avgF
L J H G H Y 1
2 2 2[( ) ]uu ud
y y
d d nY G J J G W W
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 46
Regulatory control layer
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
d
Gy Gdy
Gx Gdx
ym
u0
x
H2
c+
-
Cs=0
K(s) y
1,1 1,2 1,41 1,42 1,43 1,44 1,45
2
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
Total nu+1 = 3 MIQP problems
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 47
Regulatory control layer: Result
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 48
Regulatory control layer results
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 49
Regulatory control layer result
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 50
Presentation outline
Plantwide control : Self optimizing control formulation for CV, c = Hy – Chapter 2
Convex formulation for CV with full H - Chapter 3
Convex formulation
Globally optimal MIQP formulations
Case studies
Convex approximation methods for CV with structured H – Chapter 4
Convex approximations
MIQP formulations for structured H with measurement subsets
Case studies
Regulatory control layer selection – Chapter 5
Problem definition
Regulatory control layer selection with state drift minimization
Case studies
Conclusions and Future work
CV – Controlled Variables
MIQP - Mixed Integer Quadratic Programming
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 51
Conclusions and Future work
Concluding remakrs
Controlled variables selection formulation in the self-optimizing control framework is presented
Using steady state economics, the optimal controlled variables, c= Hy, are obtained as
optimal individual measurements
optimal combinations of ’n’ measurements
for full H using MIQP based formulations.
Controlled variables c= Hy, are obtained with a structured H. The proposed convex approximation methods
are not exact for structured H, but provide good upper bounds.
Extended the self-optimizing control concepts to find regulatory layer control variables (CV2) that minimize
the state drift.
Future work:
Robust optimal controlled varaible selection methods
Fixed CV for all active constraint regions
Economic optimal CV selection based on dynamics
Acknowledgements: GASSMAKS and Research Council of Norway
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 52
Publications Chapter 3
1. Yelchuru, R., Skogestad, S., Manum, H., 2010. MIQP formulation for controlled variable selection in self optimizing
control. In: DYCOPS, July 7-9, Brussels. pp. 61--66.
2. Yelchuru, R., Skogestad, S., 2010. MIQP formulation for optimal controlled variable selection in self optimizing
control. In: PSE Asia, July 25-28, Singapore. pp. 206--215.
3. Yelchuru, R., Skogestad, S., Dwivedi, D., 2011. Optimal measurement selection for controlled variables for Kaibel
distillation column, AIChE National Meeting, October 16-21, Minneapolis. Presentation 652e.
4. Yelchuru, R., Skogestad, S., 2012. Convex formulations for optimal selection of controlled variables and
measurements using Mixed Integer Quadratic Programming. Journal of Process Control, 22, 995-1007.
Chapter 4
5. Yelchuru, R., Skogestad, S., 2010. Optimal controlled variable selection for individual process units in self
optimizing control with MIQP formulations, In: Nordic Process Control Workshop, August 19 - 21, Lund, Sweden,
Poster presentation.
6. Yelchuru, R., Skogestad, S., 2011. Optimal controlled variable selection for individual process units in self
optimizing control with MIQP formulation. In: American Control Conference, June 29 - July 01, San Francisco, USA.
pp. 342--347.
7. Yelchuru, R., Skogestad, S., 2011. Optimal controlled variable selection with structural constraints using MIQP
formulations. In: IFAC World Congress, August 28 - September 2, Milano, Italy. pp. 4977--4982.
Chapter 5
8. Yelchuru, R., Skogestad, S., 2012. Regulatory layer selection through partial control. In: Nordic Process Control
Workshop, Jan 25 - 27, Technical University of Denmark, Kgs Lyngby, Denmark.
9. Yelchuru, R., Skogestad, S., 2012. Quantitative methods for optimal regulatory layer selection. Accepted for
ADCHEM 2012, Singapore.
10. Yelchuru, R., Skogestad, S., 2012. Quantitative methods for Regulatory control layer selection. Manuscript
submitted for publication in Journal of Process Control.
Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 53
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