Quantitative methods for controlled variables selection€¦ · Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 2 Thesis outline Ch. 1. Introduction Ch.

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


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


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







Convex formulation


Case studies




Case studies


Problem definition


Case studies





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


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


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


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


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





Convex formulation


Case studies




Case studies


Problem definition


Case studies





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


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


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


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



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


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


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


Binary 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


Distillation Column Full H : Result


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


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





Convex formulation


Case studies




Case studies


Problem definition


Case studies





Convex approximation methods for structured H

Structured H will have some zero elements in H

Example:

decentralized H

(block-diagonal H)

triangular H


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


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


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


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


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


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



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


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


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



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


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





Convex formulation


Case studies




Case studies


Problem definition


Case studies





Control system hierarchy for plantwide control

Self optimizing control

Regulatory control


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 ?


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


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


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


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


P4. Close k loops : Select k variables from ym

P5. Close nc loops : Select nc variables from ym


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



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



T1, T2, T3,…, T41


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



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


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)



Total nu+1 = 3 MIQP problems


Regulatory control layer: Result


Regulatory control layer results


Regulatory control layer result





Convex formulation


Case studies




Case studies


Problem definition


Case studies






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


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.


Thank You

Quantitative methods for controlled variables selection€¦ · Ramprasad Yelchuru, Quantitative methods for controlled variables selection, 2 Thesis outline Ch. 1. Introduction Ch.

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