Integration of Design and Control : Robust approach using MPC and PI

controllers

N. Chawankul, H. M. Budman and P. L. Douglas

Department of Chemical Engineering University of Waterloo

OutlineIntroduction

Objectives

Methodology

Case study

Results

Conclusions

Introduction

Traditional Approach

1- Design:

min (capital + operating costs)

2- Control for designed plant

-stability -actuator constraints -performance specs: Small Overshoot Short Settling Time Large closed loop bandwidth

Integrated Approach

1- Design+Control

min (capital +operating +variability costs)

st stability actuator constraints

Variability Cost

• Variability Cost = Cost of Imperfect Control

For a disturbance d (green):

What is the cost due to off-spec product (blue)?

disturbance

output

Robust Control Approach

• To test stability and calculate variability cost we need a model.

Nonlinear model: stability (Lyapunov-difficult) variability (numerically- difficult).

“Robust” linear model: nonlinear model= family of linear models

family of linear models= nominal model +model uncertainty (error)

• Nonlinear Dynamic Model (difficult optimization problem)

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

• Decentralized Control : PI /PID

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

• Variability cost into cost function : One objective function

• Centralized Control : MPC

Previous Approaches Our approach

Introduction

Objectives of the current work

Variability using MPC based on a nominal model and model error.

Cost of variability in one objective function together with the design cost.

Model uncertainty (as a function of design variables) into the objective function.

The robust stability criteria as a process constraint.

Compare the traditional method to integrated method.

Preliminary study on SISO system (distillation column) with MPC.

Methodology

• Model Predictive Control (MPC)

• Nominal Step Response and Uncertainty

• Process Variability • Optimization

- Objective Function- Constraints

MPC Controller

Sd

ProcessMPC

d

W

yr +-

++ ++

u(k)

k k+1 k+nk-1

k+2 k+3

targetpast future

y(k)y(k+1/k)

Simplified MPC block diagram

u

Nominal Step Response and Uncertainty

,.....1,1,1u ,...,,,0 321 SSSy

t

y

1

tS1

S2

S3

S4

S5

S6 Sn

Step Response Model, Sn

Nominal step response model, Sn,nom

2

,,,

lowernuppernnomn

SSS

Uncertainty, m

nomn

nomnnm S

SS

,

,

Actual Sn

mnomnn SS 1,

Upper bound

Nominal step response

Lower bound

tSn-Sn,nom

u

t

u1

-1

n

n

S

SSS

S3

2

1

uy

0

0

Process variability-1y = f(W)

ProcessMPC

W (Sinusoid unmeasured disturbance)

yr=0-

+++u

)1()1()( kuSkYMkY nn

)/1()( kkKku MPC )1/()1( kkKku MPC

)/1()()1()/1( kkWkYMkRkk p )1/()1()1/( kkWkYMkk p

)()(11 ZWKSZMKSMIZY MPCnpMPCnn

MPCnpMPCnn KSZMKSMIZWZY 11

)()(

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

Process variability-2

MPCmnomnpMPCmnomnn KSZMKSMIZWZY

ZG )1()1()()(

)( ,11

,

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

ImRe)(ˆ

jjGAAAR TjeZ With phase lag

ReImtan 1

ImRe,

,1,

1,2

,12,11,1

jG

GG

GGGG

AR ij

pnpnn

p

pGGGA ,12,11,1 ...1 B output, theof Bound

Consider worst case variability :

1092

83

74

65

56

47

38

29

1

1092

83

74

65

56

47

38

29

10

1)*(1

bZbZbZbZbZbZbZbZcbZbaZaZaZaZaZaZaZaZaZaa

Gmm

mij

Bm

maxty variabilimaximum

disturbance

output

Amplitude of disturbance,W

Amplitude of output,y

max,mm

u is a vector of design variables. c is a vector of control variables.

MinimizeCost(u,c) = Capital Cost + Operating Cost + Variability Cost u,c

Such that h(u,c) = 0 (equality constraints)g(u,c) 0 (inequality constraints)

Optimization

Constraints

h(u,c) = 0 (equality constraints)- steady state empirical correlations

g(u,c) 0 (inequality constraints)- manipulated variable constraint- robust stability

1. Manipulated variable constraint

Inequality Constraints- 1

TT

pT

pTT

pMPC SSSK1

I.

ppMPCnpMPCnnnnpMPC IKSZMKSMIMK

ZWZu 11

)()(

Consider the MPC controller gain, KMPC:

where is a manipulated variable weight

TeWeu

AZWZu

Au j

j

,)()(

max)()(

max],0[The infinity norm of

)()(

ZWZu

maxuuunom

A is the amplitude of the disturbance.

Inequality Constraints- 2

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

Li

Mp

Kmpc T1 T2 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 interconnection M-

Z-1IU(k) U(k+1)

M

w z

1))(( jM

Case study- Distillation Column Preliminary study: SISO system

Feed = 0.783 (propane)

RR

Depropanizer column from Lee, 1994

adjust reflux ratio to control the mole fraction of propane in distillate

Ethane Propane Isobutane N-Butane N-PentaneN-Hexane

A

MPC XD*

+-

Q

Process Model

RadFrac model in ASPEN PLUS different column design, 19 – 59 stages design variables (number of stages and column diameter) are

functions of nominal RR

010203040506070

1 2 3 4 5 6 7

RR

Num

ber o

f sta

ges,

N

Number of stages VS. RR

432 ))(ln(006.1

))(ln(934.3

))(ln(554.0

)ln(23.247303.6

RRRRRRRRN

7523.22365.1 RRD

649.196777.6 RRQ

The mathematic expressions of the process variables (N, D, Q) as functions of RR

(Equality Constraints)

Input/Output Model

First Order Model

00.020.040.060.080.1

0.120.140.16

0 0.2 0.4 0.6 0.8t

Nominal step response

Upper bound

Lower bound

3452.0)log(0963.0 RRK 432 ))(ln(363.0

))(ln(747.4

))(ln(41.21

ln892.41322.24

RRRRRRRRp

Step change on RR by 10 % (19- 59 stages)

p

t

eKy

1

5432max, 0469.07913.09532.45541.141858.200143.11 RRRRRRRRRRm

y

t

S1

S2

S3

Sn

Dynamic simulation using ASPEN DYNAMICS

63.2 %

RRy

K

y

max,m

p

(Equality Constraints)

Cost = CC(u) + OC(u) + VC(u,c)

Annualized capital cost, CC, (Luyben and Floudas, 1994) ($/day)

Operating cost, OC ($/day)

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

UCOPRRQOC )(

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

Objective Function

Variability cost, VC ($/day)- assume sinusoid unmeasured disturbance, W

- disturbance induces process variability - consider a holding tank to attenuate the product variation- calculate the volume of the holding tank - calculate the loss due to the product held in the tank

Variability Cost (Inventory cost) - 1

),,/(),( NiPAVPcuVC

where P = product price, N = payoff period (10 years), i = interest rate (10%) and V = volume of the holding tank

V1

V2

dout

in FCCV

1

2

The required volume of the holding tank

inC

F in F out

Cin Cout

The worst case variability:

BCm

in max

Distillation Column

Feed disturbance

A simple mass balance

Holding V

Variability Cost (Inventory cost) - 2

spec

Two different approaches

Integrated Method Traditional Method

maxmin..

min

RRRRRRts

OCCCRR

1))((..

min,

Mts

Robust Performance (Morari, 1989)

max

..

uuu

ts

nom

1)( M

VCOCCCmRR

maxmin,

Results-1

Results from Integrated Method: W is a product price multiplier.

W RR D (m) N Capital cost

($/day)

Operating cost

($/day)

Variability cost

($/day)

Total cost ($/day)

1 2.4 0.1849 5.8 31 551 654 68 1273

3 2.8 0.1705 6.23 28 530 703 168 1401

5 3.0 0.1653 6.5 27 529 726 257 1512

7 3.4 0.1357 6.9 25 538 774 284 1596

Results-2

Comparison using Traditional and Integrated methods

W Total Cost of integrated method ($/day)

Total Cost of traditional method ($/day)

Saving ($/day) % Saving

1 1273 1297 24 1.8

3 1401 1481 80 5.4

5 1512 1665 153 9.2

7 1596 1849 253 13.7

IMC Control

1)(

seK

sGp

sp

p

1111)(

sssksc

FIDc

Internal Model Control, IMC

C(s) Gp(s)

F(s)

d++ +

+--

r

y’

y

IMC-based PID parameters for

(Morari and Zafiriou, 1989) is used.

Gd(s)

Gp(s)

Results

Comparison using both methods

P Integrated Method Traditional Method Saving ($/day)

% Saving

RR c (sec) Total cost ($/day)

RR c (sec) Total cost ($/day)

1 1.5711 166.44 2624.2 1.5711 214.832 2666.67 42.45 1.59

3 1.3481 136.702 3970.6 1.5711 214.832 4592.41 621.81 13.53

5 1.3358 121.163 4259.7 1.5711 214.832 6518.15 2258.45 34.65

6 1.3314 115.616 4367.4 1.5711 214.832 7481.02 3113.62 41.62

Conclusions

• single objective function

• linear dynamic model + model uncertainty

• MPC variability cost is explicitly incorporated in the objective function

• integrated approach results in lower costs - savings can be significant; >13% for high value products

On-going work: Formulate the MIMO problem with MPC

Acknowledgement

Funding was provided by The Natural Sciences and Engineering Research Council (NSERC)