rosendo d í az-mendoza and hector budman adchem 2009 july 12–15 2009

Robust Nonlinear Model Predictive Control using Volterra Models and the Structured Singular Value ()

Rosendo Díaz-Mendoza and Hector Budman

ADCHEM 2009July 12–15 2009

► Chemical processes are nonlinear

► Nonlinear Model Predictive Control (NMPC)► First principles or empirical models► Robustness issues

► Robustness of NMPC► Simulation studies for different parameter

values

► Develop a Robust-NMPC methodology that considers parameter uncertainty

Diaz-Mendoza R. and Budman H Robust NMPC using Volterra Models and the SSV

Background and Motivation

Background and Motivation

UY

wrtminJ

pkypky

pkypky

kyky

kyky

yy

yy

nn

nn

ˆ

ˆ

ˆ

ˆ

sp

1sp1

sp

1sp1

Y

mku

mku

ku

ku

u

u

n

n

1

1

U

A model is required to calculate ŷ


Introduction Model Predictive Control

Model Predictive Control MPC

Parameters:► p, prediction horizon► m, control horizon► p ≥ m► ny, number of outputs► nu, number of inputs

Volterra Models

Why Volterra Models?►Represent a wide variety of nonlinear behavior►Model structure: nominal model + uncertain model

1

0

1

0

1

00,0

1

0

1

0

1

0101,01

0

,,

1,1,

ˆ M

n

M

i

M

ijnnjinn

M

n

M

i

M

ijjin

x

jkuikuhnkuh

jkuikuhnkuhky

uuunxuunx

xx

►M, system memory►nu, number of inputs►x є [1,,ny]; ny, number of outputs

Schetzen, M., The Volterra and Wiener theories of nonlinear systems; Robert E. Krieger, 1989


Introduction Volterra Models

CSTRA→B

ca

a

xxxxxDBx

dtdx

xxxDx

dtdx

212

212

2

12

211

1

1exp1

1exp1

CA

CA+CB

coolingfluid

coolingfluid

►Truncation error (M = 3)►High order dynamics

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Comparison Real Process vs Nominal Model

Sampling Instant

Cont

rol V

aria

ble

Real processNominal Model



Volterra Models

Nowak, R. D., and Van Veen, B. D. (1994). Random and pseudorandom inputs for Volterra filter identification, IEEE Transactions on Signal Processing, 42 (8), 2124–2135.

►Multilevel pseudo random binary sequence (PRBS)

►Nominal value = mean (parameters)

►Uncertainty = 2 (parameters)



Volterra ModelsIdentification

Output equation with parameter uncertaintySISO System

1

0

1

0

1

00,,00ˆM

n

M

i

M

ijjijinn jkuikuhhnkuhhky

►hn, hi,j, nominal value►hn, hi,j, parameter uncertainty



Volterra Models

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3


Sampling Instant

Con

trol V

aria

ble

Real processNominal ModelModel + 2

Model - 2

0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3


Sampling Instant

Con

trol V

aria

ble

Real processNominal Model



Volterra Models

Nonlinear Model Predictive Control

pkypky

kyky

ˆ

ˆ

sp

sp

Y

U

Ywrt

min

J

SISO System

How to consider parameter uncertainty?

0 1 2 3 4 5 6-6

-4

-2

0

2

4

6

Sampling Instant

Man

ipul

ated

Var

iabl

e

Worst Case Interpretation

wc(1) wc(2)

wc(3)

wc(4)

wc(5)


Introduction Nonlinear Model Predictive Control

0 1 2 3 4 5 60

1

2

3

4

5

6

Sampling Instant

Man

ipul

ated

Var

iabl

e

Worst Case Interpretation for thewhole prediction horizon

wc( Y )

Y

jin

yy

jin hh

nn

hhkyky

kyky

,, ,wrtsp

1sp1

,wrtmax

ˆ

ˆmax


Introduction



Structured Singular Value ()

Calculation of the worst ŷ(k) to ŷ(k+p) when parameter uncertainty is taken in consideration, i. e., for ŷ(k)

kdkw

jkuikuhh

nkuhhM

i

M

ijjiji

M

nnn

hh jin

1

0

1

,,

1

0

,wrt ,

max

0 1 2 3 4 5 60

1

2

3

4

5

6

Sampling Instant

Man

ipul

ated

Var

iabl

e

Worst Case Interpretation

wc(1)

wc(2)

wc(3)

wc(4)

wc(5)




Doyle, J., (1982). Analysis of feedback systems with structured uncertainties, IEE Proceedings D Control Theory & Applications, 129 (6), 242–250

Y

jin

yy

jin hh

nn

hhkyky

kyky

,, ,wrtsp

1sp1

,wrtmax

ˆ

ˆmax

Structured Singular Value (SSV)

SSV Theorem

ssvssvhhkk

jin

MY

,,wrtmax

ssv

k

khhk

ssv

ssvjin

M

Y

st

wrt ,wrt maxmax

,

Skew problem (convex)




Braatz, R. D., Young, P. M., Doyle, J. C., and Morari, M. (1994). Computational complexity of calculation, IEEE Transactions on Automatic Control, 39 (5), 1000–10002.

YM

UMMMMM

M

,,

21

12

2221

1211

ssv

ssv

kfkf

M, interconnection matrixΔ, uncertainty block structure

ssv

k

kk

ssv

ssv

Mst

wrt max M




2,12,221,11,1112,11,112,22

1,111,11

2

2

2

1100002020

0000000000000101000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

210202010

0110

010

00

0

hhhhhhhhhhhhhhicchhhhicc

kk

kk

kk

kk

kuuk

ukukuk

ukuk

ukk

kk

k

yy

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

ssv

0

0



Interconnection Matrix Example

Nominal UncertainFeedback

NMPC Cost Function

ssv

k

khhk

tcuu

ssv

ssvjin

M

Y

st

wrt limits

,wrt maxmax

, tcuukfkf

ssv

ssv

,,,,,

limits21

12

2221

1211

YMUM

MMMM

M




Additional termsManipulated variables movement penalization

1

11

mkumkuW

kukuWu

um

u

ssvu

m

ssvu

kmkumkuW

kkukuW

1max

1max 1




mkumku

kuku

kmku

mkuk

kku

kuk

ssvssv

ssvssv

bound

bound

bound

bound

max

max

max

max

Additional termsManipulated variables constraints

mkumkuk

kukuk

u

ssv

ssv

bound

bound

limits




Additional termsTerminal Condition

1 2 3 4 5 6 7 8 9 10-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Terminal Condition Representation

Sampling Instant

Cont

rol V

aria

ble




kdkwmkuhhmkuhhkyM

i

M

ijjiji

M

iii

sp 2

1,,

1

max

NMPC Cost Function

ssv

k

khhk

tcuu

ssv

ssvjin

M

Y

st

wrt limits

,wrt maxmax

, tcuukfkf

ssv

ssv

,,,,,

limits21

12

2221

1211

YMUM

MMMM

M

ssv

ssvjin

k

kssvhhk

tcuu

M

UU

Y

st

wrt wrt limits

,wrt wrt maxminmaxmin

,

NMPC Algorithm at each sampling instant




CSTRA→B

ca

a

xxxxxDBx

dtdx

xxxDx

dtdx

212

212

2

12

211

1

1exp1

1exp1

Control Specifications

► CV: x1 (dimensionless reactant concentration)

► MV: xc (cooling jacket di-mensionless temperature)

► β: process disturbance

Parameter Calculation► Multilevel PRBS► Parameter uncertainty

Doyle III, F. J., Packard, A., and Morari, M. (1989). Robust controller design of a nonlinear CSTR, Chemical Engineering Science, 44 (9), 1929–1947.

CA

CA+CB

coolingfluid

coolingfluid


Case Studies SISO

Case Study SISO System

CSTR with first order exothermic reaction


Case Studies SISO

Disturbance Characteristics

0 5 10 15 20 250.2

0.22

0.24

0.26

0.28

0.3

0.32

Sampling Instant

Disturbance Value dimensionless cooling rate

0 5 10 15 20 25-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Control Variablep=10 , m=2

Sampling Instant

Con

trol V

aria

ble

Wiu = 2.5

Wiu = 2.0

Wiu = 1.5

Wiu = 1.0

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Manipulated Variablep=10 , m=2

Sampling Instant

Man

ipul

ated

Var

iabl

e

Wiu = 2.5

Wiu = 2.0

Wiu = 1.5

Wiu = 1.0


Case Studies SISO

0 5 10 15 20 25 30 35 40 45 50-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Controlled VariableCase with constraints, p=10 , m=2 , W

iu = 0.5

Sampling Instant

Con

trolle

d V

aria

ble

0 5 10 15 20 25 30 35 40 45 500

0.05

0.1

0.15

0.2

0.25

Manipulated VariableCase with constraints, p=10 , m=2 , W

iu = 0.5

Sampling Instant

Man

ipul

ated

Var

iabl

e

Constraint set at 0.2


Case Studies SISO

Sum absolute error Robust = 1.46Sum absolute error Non-Robust = 1.556% improvement

0 5 10 15 20 25-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Control Variablep=10 , m=2 , W

iu = 2.5

Sampling Instant

Con

trol V

aria

ble

Robust ControllerNon-Robust Controller

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Manipulated Variablep=10 , m=2 , W

iu = 2.5

Sampling Instant

Man

ipul

ated

Var

iabl

e

Robust ControllerNon-Robust Controller


Case Studies SISO


Case Studies SISO

WiΔu Robust Controller is better

than Non-Robust1.5 37%

1 41%

0.75 50%

0.50 66%

25 different disturbances for each weight

im

mmk

k

kSX

f

k

KSSKSPP

XDPtP

XY

SSDtS

XdXtX

//1

dd

1dddd

2

/

► X, biomass concentration► S, substrate concentration► P, product concentration► D, dilution rate► Sf, feed substrate concentration

Control Specifications► CV: X and P► MV: D and Sf

► YX/S: process disturbance

Parameter calculation► Multilevel PRBS► Parameter uncertainty

FermenterXSP

DSf

Saha, P., Hu, Q., and Rangaiah, G., P. (1999). Multi-input multi-output control of a continuous fermenter using nonlinear model based controllers, Bioprocess Engineering, 21, 533–542.


Case Studies MIMO

Case Study MIMO System

0 5 10 15 20 25-0.14

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Control Variablesp=10 , m=2 , W

1

u = 0.2 , W2

u = 0.5

Sampling Instant

Con

trol V

aria

bles

BiomassProduct


Case Studies MIMO

0 5 10 15 20 25-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Manipulated Variable (Dilution Rate)p=10 , m=2 , W

1

u = 0.2 , W2

u = 0.5

Sampling Instant

Man

ipul

ated

Var

iabl

e

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Subustrate Feed Concentrationp=10 , m=2 , W

1

u = 0.2 , W2

u = 0.5

Sampling Instant

Man

ipul

ated

Var

iabl

e

Conclusions

► A Robust-NMPC algorithm was developed

► The algorithm considers all the features of previous NMPC formulations

► In average the robust controller results in better performance as the input weight is decreased


Preliminary Conclusions Preliminary Conclusions

► Computational demand

► Multivariable control


Current challenges

Preliminary Conclusions Challenges

rosendo d í az-mendoza and hector budman adchem 2009 july 12–15 2009

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