Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 1/16
MIQP formulation for optimal controlled variable selection in Self
Optimizing Control
Ramprasad YelchuruProf. Sigurd Skogestad
MIQP - Mixed Integer Quadratic Programming
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 2/16
Outline
1. Motivation
2. Problem formulation
3. MIQP formulation
4. Evaporator Case study
5. Comparison of MIQP & customized BAB
6. Conclusions
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 3/16
1.Motivation
Want to minimize cost J
Which two - individual measurements or - measurement combinations
should be selected as controlled variables (CVs) to minimize the cost J?
y = candidate measurements; H = selection/combination matrix
c = H y, H=?
Combinatorial problem1. Exhaustive search (10C2,10C3,…)
2. customized BAB
3. MIQP
100 200 2 3
1 2
600 0.6 1.009( )
0.2 4800
J F F F F
F F
2 MVs – F200, F1 Steady-state degrees of freedom
10 candidate measurements – P2, T2, T3, F2, F100, T201, F3, F5, F200, F1
3 DVs – X1, T1, T200
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 4/16
Optimal steady-state operation
( , ) ( , )opt optL J u d J u d
Ref: Halvorsen et al. I&ECR, 2003 Kariwala et al. I&ECR, 2008
2. Problem Formulation
21/2 1( )yavg uu FL J HG HY
Loss is due to(i) Varying disturbances(ii) Implementation error in controlling c at set point cs
31( , ) ( , ) ( ) ( ) ( )
2T
opt u opt opt uu optJ u d J u d J u u u u J u u
1[( ) ]y yuu ud d d nY G J J G W W
u
J
( )opt ou d
Loss
min ( , )u
J u d'd
Controlled variables,c yH
ydG
cs = constant +
+
+
+
+
- K
H
yG y
'yn
c
u
dW nW
d
optu
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 5/16
1/2 1min ( )yuu FHJ HG HY Non-convex
optimization problem(Halvorsen et al., 2003)
-1 -1 -1 1 -11 y 1 y y y (H G ) H = (DHG ) DH = (HG ) D DH = (HG ) H
1H DH
D : any non-singular matrix
Hmin HY F
st 1/ 2yuuHG J
Convex optimization
problem
Global solution
- Do not need Juu - And Q is used as degrees of freedom for faster solution
st yHG Q
Improvement 1 (Alstad et al. 2009)
Improvement 2 (this work)
Hmin HY F
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 6/16
Vectorization
X
1 2
1 2 2*
( 1)* 1 ( 1)* 2 *
ny
ny ny ny
nc ny nc ny nc ny nu ny
x x x
x x xH
x x x
TX H
Hmin HY F
subject to yHG Q
min
.
T T
X
T
X Y Y X
st G X Q
Problem is convex QP in decision vector
1
2
* ( * ) 1nu ny nu ny
x
xX
x
1 1 ( 1)* 1
2 2 ( 1)* 2
2* *
ny nc ny
ny nc ny
ny ny nc ny ny nu
x x x
x x xX
x x x
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 7/16
Controlled variable selection
Optimization problem :
Minimize the average loss by selecting H to obtain CVs as
(i) best individual measurements
(ii) best combinations of all measurements
(iii) best combinations with few measurements
min
.
T T
X
T
X Y Y X
st G X Q
H
min HY F
st. yHG Q1/2 1min ( )yuu FH
J HG HY
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 8/16
3. MIQP Formulation
{0,1}
1,2, ,i
i ny
( 1)*
min
.
0 0 0 0
0 0 0 0
1,2, ,
0 0 0 0
0,1
aug
T
Taug aug
x
ynew aug
aug
i
ny i
i i
nu ny i
i
x Fx
st G x Q
x n
xM MxM M
for i ny
M Mx
δ
P
1
2
( * ) 1
aug
ny nu ny ny
X
x
[ ( , )]
[ ( * , )]
[ (1, * ) (1, )]
max( ) / min( )
T
T
y Tnew
y
F Y Y zeros ny ny
G G zeros nu ny ny
zeros nu ny ones ny
upper bound for M Q G
P
1 2
1 2
1 2 2*
( 1)* 1 ( 1)* 2 *
ny
ny
ny ny ny
nc ny nc ny nc ny nu ny
x x x
x x xH
x x x
We solve this MIQP for n = nu to ny
Big M approach
high value M => high cpu time
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 9/16
4. Case Study : Evaporator System
2 MVs – F200, F1
10 candidate measurements – P2,T2,T3,F2,F100,T201,F3,F5,F200,F1
3 DVs – X1, T1, T200
100 200 2 3
1 2
600 0.6 1.009( )
0.2 4800
J F F F F
F F
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 10/16
Case Study : Results21/2 11
( ( ) )2
yavg uu F
L J HG HY
-0.093 11.678 -3.626 0 1.972
-0.052 6.559 -2.036 0 1.108 0.006
; ;
1.000 0 0 0 0
0 1 0 0 0
y yd uuG G J
0.25 0 0
-0.133 0.023 0 -0.001; ; 0 8 0 ; ( (10,1))
-0.133 16.737 -158.373 -1.161 1.4840 0 5
ud dJ W Wn diag measnoise
1[( ) ]y yuu ud d d nY G J J G W W
10 2 10 3 2 2 3 3 3 10 10; ; ; ; ;y yd uu ud d nG G J S J W W
Results Controlled variables (c)
Optimal individual measurements1 3
2 200
c F
c F
1 2 201 3 200
2 2 201 3 200
2.3527 0.0317 0.0605 0.0025
4.9523 0.0882 0.1659 0.0046
c F T F F
c F T F F
Loss2 = 3.7351
Loss4 = 0.4515
Data
Optimal 4 measurement combinations
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 11/16
Case Study : Results
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 12/16
Case Study : Computational time
** Branch and bound (BAB): Kariwala and Cao, IEEE Trans. (2010)
No. Meas
Optimal Measurements
MIQPcpu time (sec)
Downwards BABcpu time
(sec)
Partial BAB
cpu time (sec)
Exhaustive cpu time
(sec)* Loss
2 [F3 F200] 0.0310 0.0781 0.0600 0.45 3.7351
3 [F2 F100 F200] 0.0160 0.0000 0.1406 1.2 0.6501
4 [F2 T201 F3 F200] 0.0470 0.0313 0.0313 2.1 0.4515
5 [F2 F100 T201 F3 F200] 0.0320 0.0000 0.0313 2.52 0.3373
6 [F2 F100 T201 F3 F5 F200] 0.0160 0.0000 0.0313 2.1 0.2857
7 [P2 F2 F100 T201 F3 F5 F200] 0.0160 0.0313 0.0000 1.2 0.2532
8 [P2 T2 F2 F100 T201 F3 F5 F200] 0.0000 0.0000 0.0781 0.45 0.2296
9 [P2 T2 F2 F100 T201 F3 F5 F200 F1] 0.0000 0.0000 0.0000 0.1 0.2100
10 [P2 T2 T3 F2 F100 T201 F3 F5 F200 F1] 0.0000 0.0313 0.0000 0.01 0.1936
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 13/16
5. Comparison of MIQP, Customized Branch And Bound (BAB) methods
MIQP formulations can accommodate wider class than monotonic functions (J)
MIQPs are solved using standard cplex routines
MIQPs are simple and are easy to incorporate few structural constraints
MIQPs are computationally intensive than BAB methods
Single MIQP formulation is sufficient for the described problems
Customized BAB methods can handle only monotonic cost functions (J)
Customized routines are required
BABs require a deeper understanding of the customized routines to solve problems with structural constraints
Computationally faster than MIQPs as they exploit the monotonic properties efficiently
Monotonicity of the measurement combinations needs to be checked before using PB3 for optimal measuement subset selections
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 14/16
MIQP formulation with structural constraints
1 ( * )
1 ( * )
1 ( * )
1
2 ;
2
1 0 00 0 0 0 0 0 0
0 1 1 0 0 1 0 0 0 0
0 0 0 11 0 11 1 1
aug
nu ny
nu ny
nu ny
P
xP
If the plant management decides to procure only 5 sensors (1 pressure, 2 temperature, 2 flow sensors)
2 MVs – F200, F1
3 DVs – X1, T1, T200
10 candidate measurements – P2,T2,T3,F2,F100,T201,F3,F5,F200,F1
Loss5-sc = 0.5379
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 15/16
6. Conclusions The self optimizing control non-convex problem is
reformulated as convex problem
MIQP based formulation is presented for Selection of CVs as optimal individual measurements Selection of CVs as combinations of all measurements Selection of CVs as combinations of optimal measurement
subsets
MIQPs are more simple, intuitive and are easy compared to customized Branch and Bound methods
MIQPs are computationally intensive than customized Branch and Bound methods
Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 16/16