Integration of Design and Control: A Robust Control Approach by Nongluk Chawankul Peter L. Douglas Hector M. Budman University of Waterloo Waterloo, Ontario,

Integration of Design and Control: A Robust Control Approach

by

Nongluk ChawankulPeter L. Douglas

Hector M. Budman

University of WaterlooWaterloo, Ontario, Canada

Process design

Background

Control performance depends on the controller and the design of the process.

Traditional design procedure: Step 1: Process design (sizing + nominal operating conditions) Step 2: Control design

Idea of integrating design and control:

Process control+ = Integrated approach

Background

Traditional design and control design

Integrated design and control design

Step 1: Process design

Step 2: Control design

Only one step design

Cost = capital cost(x) + operating cost(x) + cost related to closed loop system(x,y)

where x is design variable y is control tuning parameter

Objective Function (Cost)

Two steps design

Process constraints Equality constraints, h(x) = 0 Inequality constraints, g(x,y) 0

Min Cost(x,y) x,y

s.t. h(x) = 0g(x,y) 0

Cost = Capital cost(x) + operating cost(x)

where x is design variable

Min Cost(x) x

s.t. h(x) = 0g(x) 0

Optimum design

Optimum design

Design controller

Closed loop system

• Nonlinear Dynamic Model (difficult optimization problem)

• Variability cost not into cost function: Multi-objective optimization

• Decentralized Control : PI /PID

• Linear Nominal Model + Model Uncertainty (Simple optimization problem)

• Variability cost into cost function : One objective function

• extended to Centralized Control : MPC

Previous studies Our study

Integrated Design and Control Design

Case study

• Case study II: SISO MPC• Case study III: MIMO MPC

Feed

RR

Ethane Propane Isobutane N-Butane N-PentaneN-Hexane

A

XD*+

-

Q

Feed

RR


A1

MPCXD*

+-

Q

IMC or MPC

SISO system MIMO system

A2

-+

XB*

XD

XB

XD

XB

MIMO case study:

RadFrac model in ASPEN PLUS was used.

Different column designs, 19 – 59 stages were studied.

Product specifications Mole fraction of propane in distillate product = 0.783 Mole fraction of isobutane in bottom product = 0.1

Design variables are functions of nominal RR at specific product compositions.

Case Study III: MIMO MPC

U is a vector of design variables. C is a vector of control variables. Lm is a set of uncertainty.

Optimization

Minimize Cost(U,C) = CC(U) + OC(U) + max VC(U,C) U,C Lm

Such that h(U) = 0 (equality constraints)

g(U,C) 0 (inequality constraints)

Objective Function

Objective Function

• Capital Cost, CC

– Cost of sizing, e.g. number of stages N and column diameter D

– Capital cost for distillation column from Luyben and Floudas, 1994 ($/day)

• Operating Cost, OC

– Operating cost from Luyben and Floudas, 1994 ($/day)

)5.17.0(245))76.06(486324615(3.12 22 DNDNDCC

UCOPRRHDOC tax )(

where tax = tax factorHD = reboiler duty (GJ/hr)OP = operating period (hrs)UC = Utility cost ($/GJ)

Capital Cost (CC) and Operating Cost (OC)

Feed

RR


A1

MPCXD*

+-

Q

A2

-+

XB*

V1

V2

t

t

t

t

t

- Variability cost, VC = inventory cost

- sinusoid disturbance induces process variability

- consider holding tank to attenuate the product variation

Variability Cost, VC

Variability Cost (VC)

)()1()1()()()( ,

11, zzzzz MPCmnomnpMPCmnomnn

Y WKSMKSMIWGY

Assume, W is sinusoidal disturbance with specific d. (alternatively, superposition of sinusoids)

WGY )( jYTjez With phase lag

Re

Imtan 1

Consider worst case variability :

)2,1(maxty variabilimaximumm

YY

max,mm

Calculation of Variability Cost (VC) - 1

Related to maximum VC

Objective Function (-cont-)

1

1

sC

C

in

out

Apply Laplace transform

din

out

Q

VC

C

,

1

12

The product volume in the holding tank

inC

VQ in Q out

Cin Cout

dout

in Q

C

CV

Calculation of Variability Cost (VC) - 2

VC1 = W1P1V1(A/P,i,N) VC2 = W2P2V2(A/P,i,N)

VC = VC1+ VC2

Equality Constraints

Process Models: ASPEN PLUS simulations at specific product compositions

Equality Constraints: Process models -1

3800

3900

4000

4100

4200

4300

4400

4500

4600

1.6 1.8 2 2.2 2.4 2.6 2.8

RR

Boi

lup

rate

,Q (

lbm

ol/h

r)

ASPEN simulation

equation (6.1)

0

10

20

30

40

50

60

70

1.6 1.8 2 2.2 2.4 2.6 2.8

RR

Num

ber

of s

tage

s, N ASPEN simulation

equation (6.2)

3.35

3.4

3.45

3.5

3.55

3.6

3.65

1.6 1.8 2 2.2 2.4 2.6 2.8

RR

Col

umn

Dia

met

er, D

(m

)

ASPEN simulation

equation (6.3)

Q(RR) N(RR)

30

31

32

33

34

35

36

37

1.6 1.8 2 2.2 2.4 2.6 2.8

RR

Hea

t d

uty

in

reb

oil

er,

HD

(G

J/h

r)

ASPEN Simulation

equation (6.4)

HD(RR) D(RR)

Process Models: ASPEN PLUS simulations

960

965

970

975

980

985

990

1.6 1.8 2 2.2 2.4 2.6 2.8

RR

Dis

tilat

e ra

te (

lbm

ol/h

r)

ASPEN simulation (tol = 0.001)

equation (6.5)


2235

2240

2245

2250

2255

2260

2265

1.6 1.8 2 2.2 2.4 2.6 2.8

RR

Bot

tom

rat

e (l

bmol

/hr)


equation (6.6)


DF(RR)

BF(RR)

Equality Constraints: Process models - 2

Process Models: Input/Output Model for 22 system

First Order Model

ip

t

ipi eKy ,)(

, 1

yi

t

S1

S2

S3

Sn

1%

35%

y

0

y1

y2

time

-35%

+35%

+1%

-1%

2%1

2

%35

1

,

RRy

RRy

K RRp

2

%1

2

%40

1

,

Qy

Qy

K Qp

Process gains

In a similar fashion, time constants and dead time

p(RR) and p(Q)

(RR)

and

Kp1(RR) for paring xD-RR

Kp2(RR) for paring xB-RR

Kp3(Q) for paring xD-Q

Kp4(Q) for paring xB-Q


0.12

0.14

0.16

0.18

0.2

0.22

1.6 2.1 2.6 3.1

RR

Proc

ess

gain

of

x D-R

R, K

p1

ASPEN PLUS simulation

equation (6.10)

0.027

0.028

0.029

0.03

0.031

0.032

0.033

1.6 2.1 2.6 3.1

RR

Proc

ess

gain

of

x B-R

R, K

p2


equation (6.11)

0.45

0.5

0.55

0.6

0.65

0.7

1.9 2.1 2.3 2.5

Q' (lbmole/hr)

Proc

ess

gain

of

x D-Q

', K

p3


equation (6.12)

0.11

0.12

0.13

0.14

0.15

0.16

1.9 2.1 2.3 2.5

Q' (lbmole/hr)

Pro

cess

gai

n o

f x B

-Q',

Kp

4


equation (6.13)

Process gains for 2 2 system


0

5

10

15

20

25

1.5 2 2.5 3 3.5

RR

Proc

ess

time

cons

tant

, p

1

(min

)

ASpen simulation

equation (6.16)

0

5

10

15

20

25

1.2 1.3 1.4 1.5 1.6 1.7

Q' (lbmole/hr)

Proc

ess

time

cons

tant

, p

2

(min

)

ASPEN simulation

equation (6.17)

4

4.5

5

5.5

6

6.5

1.6 1.8 2 2.2 2.4 2.6 2.8 3

RR

Dea

d t

ime,

(m

in)

ASPEN simulation

equation (6.18)

Process time constants: p(RR) and p(Q)

Process dead time: (RR)


max,m

Model uncertainty

lowernuppernnn

m SorSSS

SS,,

nomn,

nomn,max, ,max

Time

y

Sn,upper

Sn,lower

Sn,nom

)(

)(

)(

)(

4max,

3max,

2max,

1max,

Q

Q

RR

RR

m

m

m

m

xD-RR

xB-RR

xD-Q

xB-Q

max,m


00.0005

0.0010.0015

0.0020.0025

0.0030.0035

0.004

1.6 2.1 2.6 3.1

RR

Unc

erta

inty

(x

D-R

R)


equation (6.22)0

0.001

0.002

0.003

0.004

0.005

1.6 2.1 2.6 3.1

RR

Unc

erta

inty

(x

B-R

R)


equation (6.23)

0.11

0.16

0.21

0.26

0.31

0.36

1.9 2.1 2.3 2.5

Q' (lbmole/hr)

Unc

erta

inty

(x

D-Q

') ASPEN PLUS simulation

equation (6.24)

0.025

0.03

0.035

0.04

0.045

0.05

1.9 2 2.1 2.2 2.3 2.4 2.5

Q' (lbmole/hr)

Unc

erta

inty

(x

B-Q

')


equation (6.25)

Model Uncertainty for 22 system


Inequality Constraints

1. Manipulated variable constraint

Inequality Constraints- 1

εKu ).(MPC

)()()( zz MPCU WKGu

is a tuning parameter. Large less aggressive control

WezGRRp

TjUj

1

1max )(

maxmax uuu nom

WezGQp

TjUj

1

2max )(

Two manipulated variables Calculate RR and Q

and

2. Robust stability constraint (Zanovello and Budman, 1999)

Li

Mp

Kmpc T1T2 H N1

W1 W2

N2

Z-1I

+ + ++

++

+

-+N1

-

M

(k+1/k)u(k)

U(k)

U(k-1)

Z(k) w(k)

H H

Block diagram of the MPC and the connection matrix M

Z-1I

U(k) U(k+1)

M

w z

1))(( jM

Inequality Constraints- 2

Two different approaches

Integrated Method Traditional Method

0)(

0)(uch that

min

RRg

RRhs

OCCCRR

1))((..

min,

Mts

Robust Performance (Morari, 1989)

max

such that

UΔUU nom

1)( M

VCOCCCmLQRR

maxmin,,

Where U is manipulated variables

maxUΔUU nom

Results

Results - 1

Results from Integrated design and control design approach

w1 w2 RR* * 11 N D (m)

1 1 1.913 0.2350 3.65 26 3.392

5 1 1.911 0.2341 3.63 26 3.392

10 1 1.908 0.2338 3.62 27 3.391

15 1 1.753 0.2331 3.03 38 3.370

20 1 1.753 0.2332 3.03 38 3.370

1 5 1.912 0.1886 3.64 26 3.392

1 10 1.909 0.1848 3.62 26 3.391

1 15 1.906 0.1836 3.61 27 3.391

1 20 1.904 0.1830 3.60 27 3.390

w1 or w2 increases;

-RR* decreases smaller dead time

11 decreases interaction decreases as RR decreases

* decreases RS constraint is easy to satisfy as 11 decreases

Compare Results from Traditional and Integrated design and control design approaches.

Results - 2

600

700

800

900

1000

1100

1200

1300

0 5 10 15 20 25

w 1

Tot

al c

ost (

$/da

y)

Traditional approach

Integrated approach

600

800

1000

1200

1400

1600

1800

0 5 10 15 20 25

w 2

Tot

al c

ost (

$/da

y)

Traditional approach

Integrated approach

Results - 3

RR max RR* * N D (m) CC ($/day) OC ($/day) VC ($/day) TC ($/day)

2.633 1.913 0.2350 26 3.392 195.98 586.21 23.34 805.53

2 1.913 0.2349 26 3.392 195.98 586.21 23.34 805.53

1.9 1.854 0.2024 38 3.370 259.76 570.48 72.62 902.86

1.8 1.761 0.2021 39 3.370 262.60 570.20 93.04 925.84

Effect of RRmax on Total Cost (TC)

Conclusions

1- For the case ≠ 0, using the integrated method, the optimization tends to select smaller RR values which correspond to smaller dead time and smaller interaction.

2- The optimal design obtained using the integrated method resulted in a lower total cost as compared to the traditional method.

3- Limit on manipulated variable affects the closed loop performance and leads to more cost.

ProcessMPC

W (Sinusoid unmeasured disturbance)

yr=0-

+++u

)1()1()( kkk nn uSYMY

)/1()( kkk MPC εKu )1/()1( kkk MPCεKu

)/1()()1()/1( kkkkkk p WYMRε )1/()1()1/( kkkkk p WYMε

)()(11 zzz MPCnpMPCnn WKSMKSMIY

Substitute (k), u(k-1) into the first equation and apply z-transform

Calculation of Variability Cost (VC) -1

Process variability

Results - 1

Results from Integrated design and control design approach for = 0

w1 w2 RR* * 11 N D (m)

1 1 1.921 0.2503 3.68 26 3.393

5 1 1.955 0.2509 3.83 26 3.399

10 1 1.980 0.2518 3.95 25 3.404

15 1 2.012 0.2521 4.11 25 3.410

20 1 2.103 0.2527 4.63 24 3.431

1 5 1.960 0.2511 3.86 26 3.400

1 10 1.992 0.2516 4.01 25 3.406

1 15 2.102 0.2522 4.63 24 3.431

1 20 2.234 0.2540 5.57 23 3.465

w1 or w2 increases;

-RR* increases uncertainty decreases as RR increases

11 increases interaction increases as RR increases

* increases RS constraint is more difficult to satisfy as 11 increases

0

5

10

1.6 1.8 2 2.2 2.4 2.6 2.8

RR

Results - 4

Compare savings when = 0 and 0

05

1015202530

0 5 10 15 20 25

w 1

Sav

ings

(%

)

Series1

Series2 0 = 0

0

20

40

60

80

100

0 5 10 15 20 25

w 2

Savin

gs (

%)

Series1

Series2

= 0 0

Documents

Integration of Design and Control: A Robust Control Approach by Nongluk Chawankul Peter L. Douglas Hector M. Budman University of Waterloo Waterloo, Ontario,