Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing...

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


Outline

1. Motivation

2. Problem formulation

3. MIQP formulation

4. Evaporator Case study

5. Comparison of MIQP & customized BAB

6. Conclusions


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


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


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


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


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


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


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


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


Case Study : Results


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


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


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


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

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