Quantitative methods for Regulatory layer selectionnpcw17.imm.dtu.dk/Proceedings/Session 7 Control...

Ramprasad Yelchuru, Quantitative methods for regulatory layer selection, 1

Quantitative methods for

Regulatory layer selection

Ramprasad Yelchuru Sigurd Skogestad


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


Control system hierarchy for plantwide control

Self optimizing control

Scope of this work


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 ?


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


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


Outline

• Motivation



• Results

4.1 CVs (c = H2y) as individual measurements

4.2 CVs (c = H2y) as measurement combinations

5. Conclusions


'yn'd

ydG

cs = 0 +

+

+ +

+ - K

H2

yG y

c

u

dW nW


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


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


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


P4. Close k loops : Select k variables from ym

P5. Close nc loops : Select nc variables from ym


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


Outline

• Motivation



• Results



5. Conclusions


'yn'd

ydG

cs = 0 +

+

+ +

+ - K

H2

yG y

c

u

dW nW


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

= =


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−= −


Outline

• Motivation



• Results



5. Conclusions


'yn'd

ydG

cs = 0 +

+

+ +

+ - K

H2

yG y

c

u

dW nW


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)




4.1 CV2 (c = H2y) as individual measurements

*

* In addition to two level loops


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


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


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


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


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


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.

Quantitative methods for Regulatory layer selectionnpcw17.imm.dtu.dk/Proceedings/Session 7 Control...

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