PREHRAMBENO -BIOTEHNOLOŠKI FAKULTET

1PREHRAMBENO -BIOTEHNOLOŠKI FAKULTET

Poslijediplomski studij: PREHRAMBENE TEHNOLOGIJE

MODELIRANJE, OPTIMIRANJE I PROJEKTIRANJE PROCESA

Prof.dr.sc. Želimir Kurtanjek

PBF

tel: 4605 294 fax: 4836 083

E-mail: [email protected]

URL: http:/mapbf.pbf.hr/~zkurt

2

MODELIRANJE

3MULTIDISCIPLINARNOST MATEMATIČKOG MODELIRANJA PROCESA

BIOTEHNIČKE

ZNANOSTI

MATEMATIČKE

ZNANOSTI

RAČUNARSKE

ZNANOSTI

4 TEORIJA SUSTAVA I MATEMATIČKO MODELIRANJE

Osnovni pojmovi o sustavu:

SUSTAV

OKOLINAGRANICASUSTAVA

{masa

energija

informacija}

masa

energija

informacija

Prikaz odnosa sustava i okoline

5

POČETAK

SISTEMSKI PRISTUP MODELIRANJU

SVRHA MODELA

DEFINIRANJE ULAZNIH VELIČINA X

DEFINIRANJE IZLAZNIH VELIČINA Y

IZVODI BILANCI MASE, ENERGIJE, KOLIČINE GIBANJA

IZBOR NUMERIČKE METODE

IZBOR RAČUNALNOGJEZIKA

RJEŠENJE JEDNADŽBI MODELA

ODREĐIVANJE PARAMETARA

PROVJERA MODELA

2M < NE

PRIMJENA

DA

6Značajke sustava

Sustav je apstraktna tvorevina, najčešće definira matematičkim relacijama ( npr. skupom diferencijalnih jednadžbi, diskretnih jednadžbi, neuralnim mrežama, neizraženom “fuzzy “ logikom, ekspertnim sustavom itd.).

1) za analizu nekog procesa, 2) upravljanje,3) projektiranje,4) nadzor ( monitoring )5) osiguranje kakvoće proizvoda6) optimiranje7) razvoj novih proizvoda8) zaštitu okoliša

Sustav se definira s obzirom na određenu svrhu, na primjer:

7NAČELO IZVOĐENJA BILANCI

V

dio volumena

ulazni tokovi:

tvari, energije,

količine gibanja

izlazni tokovi:

tvari, energije,

količine gibanja

8U procesnom inženjerstvu ( kemijskom, biokemijskom, prehrambenom, farmaceutskom .. ) matematičke modele izvodimo na osnovi slijedećih bilanci: mase (tvari), energije i količine

gibanja.

Osnovni oblik bilance je:

reakcijeebiokemijsk

iliikemijskezbogVvolumenu

uSpromjena

Vvolumenaiz

Stokovaizlaznih

zbroj

Vvolumenu

Stokovaulaznih

zbroj

Vvolumenuu

Saakumulacij

t /

gdje S označava masu ( količinu tvari), energiju i količinu gibanja.

9Modeli se razlikuju zavisno od izbora volumena za koji se postavlja bilanca.

Kada volumen obuhvaća ukupan volumen u kojem se zbiva proces ( na primjer biokemijski reaktor ) onda su to modeli s usredotočenim ili koncentriranim veličinama stanja.

Ako se kao volumen za koji se postavljaju bilance odabere samo dio cijelog volumena onda se radi o modelu s raspodjeljenim ili distribuiranim veličinama stanja.

Modeli s usredotočenim parametrima postaju sistemi običnih diferencijalnih jednadžbi, a modeli s distrubuiranim stanjima određeni su sistemom parcijalnih diferencijalnih jednadžbi.

10Razliku u načinu izvođenja bilanci možemo prikazati pomoću slijedećeg grafičkog prikaza:

V

U

U

U

I

I

1

2

3

1

2

V

U

U

U

I

I

1

2

3

1

2

u

u

i

i

1

2

1

2

S

U

U

U

1

2

3

ukupanvolumen V

diferencijalvolumena

dV

I

I

1

2

u

u1

2

i

i12

U,I su ulazni i izlazni tokovi za ukupanvolumen

u , i su ulazni i izlazni tokoviza diferencijal volumena

S

11U bilanci mase sastojka predznak ( + ) dolazi u slučaju kada je tvar

produkt reakcije, a predznak ( - ) kada je tvar reaktant u reakciji. Kod bilance energije predznak ( + ) dolazi kada je reakcija

egzotermna, a predznak ( - ) kada je reakcija endotermna.

Oznaka označava malu ali konačnu promjenu određene veličine,t je oznaka za vrijeme, je oznaka za malu konačnu promjenut je mala konačna promjena vremena(akumulacija S) je mala konačna promjena akumulacije ( sadržaja S)

dt

dF

t

Ft

0lim

Bilance postaju diferencijalne jednadžbe kada se provede granični postupak u kojem konačne diferencije, , postaju infinitezimalne veličine ( odnosno diferencijali, d ).

12Na primjer, za model s usredotočnim veličinama bilance mase za pojedine supstrate ima ima oblik:

Ni

Ni

NijsV

dt

d

,1 j

j,1 i

j,1 i

V u volumenus eproizvodnj

ili potrosnje brzina

reaktoru u ssupstrata

ijakoncentrac

Vizqprotok

volumniizlazni

pritoku u ssupstrata

ijakoncentrac

Vuqprotok

volumniulazni

13Opći oblik modela s raspodjeljenim veličinama stanja je:

txyfr

rtyt

,,,

gdje je vektor položaja. r

uz zadano početno stanje: ryrty o

,0

rubne uvjete: tyrty SSr

, i/ili ygrtyr Sr

,

txi ulazne veličine:

14Klasifikacija modela

Analitički modeli Neanalitički modeli

Regresijski

Neuralne mreže

“Fuzzy logic”

neizražena logika

Ekspertni sustavi

oooo

izvedeni iz

fundamentalnih

zakona fizike, kemije

i biologije

15Klasifikacija analitičkih modela

Deterministički Stohastički

Distribuirani UsredotočeniPopulacijski

Usredotočeni

DistribuiraniLinearni

Nelinearni

Diskretni

Kontinuirani

Nelinearni Linearni

Dif. jednadžbe

Prijenosne

funkcije

16Kontinuirani - diskretni modeli

Kontinuirani model sustava 1 reda

txktytydt

d

x(t) y(t)Sustav 1. reda

Zadane veličine:

1) parametri , k

2) početno stanje y(t = 0) = y0

3) ulazna veličina x(t), t [ 0, tf ]

17

x@t_D:= If@2< t< 8, 1, 0DPlot@x@tD,8t, 0, 10<D

2 4 6 8 10

0.2

0.4

0.6

0.8

1

Model u programskom jeziku:

Wolfram Research “Mathematica”

18k= 1.0; t =1.2;

NDSolve@8t * y'@tD+y@tD==k * x@tD, y@0D== 0.01<, y,8t, 0, 10<DPlot@Evaluate@y@tD. %D,8t, 0, 10<D

2 4 6 8 10

0.2

0.4

0.6

0.8

1

kontinuirandiskretan

korak

19Matematički modeli procesa u biotehnologiji

Matematički modeli procesa u biotehnologiji imaju vrlo istaknuti značaj. Na osnovu matematičkih modela analiziraju se:

odzivi mjernih sustava u biotehnološkim procesima,

procjenjuju se parametri i direktno nemjerljiva stanja procesa,

prijenos rezultata iz modela za laboratorijsko mjerilo u poluindustrijsko i industrijsko mjerilo

optimiranje procesa

nadzor ( “ monitoring” ) procesa

očuvanje kakvoće proizvoda

upravljanje ( automatizacija ) procesa

projektiranje novih procesa

20CONTENTS

1. Systems approach

2. Knowledge and system models

3. Fuzzy logic models

4. Example: Fuzzy logic control of flow rate

5. Neural networks

6. Control structures

7. Neural network control of a chemostat

8. Adaptive neural network fuzzy inference system

9. Computer demo exercises

10. Conclusions

21

Surroundings

System

xP

xI

y

Process subsystemSP

Control subsystemSC

Systems view of an industrial bioprocess

22Schematic diagram of mathematical forward M and inverse M-1 models

X Y

M

M-1

23Graphical representation of "transparency" of mathematical models in relation to knowledge and perception of

complexity of a system.

Neural networks

Fuzzy models

Analytical models

System complexity

Knowledge

X Y

24Objectives in modeling

Analytical models

Process analysis: studies of reaction mechanisms, kinetics, parameter estimation

Process design

Process optimization

Process control

Process on-line monitoring

Input - output models

Process on-line monitoring

Process control

25Fuzzy logic models

In fuzzy logic models input and output spaces are covered or appro-ximated with discourses of fuzzy sets labeled as linguistic variables

For example, if Ai X is an i-th fuzzy set it is defined as an ordered pair:

fAi ttXtxtxtxA ,0,,

where x(t) is a scalar value of an input variable at time t, and A is called a membership function which is a measure of degree of mem-bership of x(t) to Ai expressed as a scalar value between 0 and 1.

Typical membership functions have a form of a bellshaped or Gaussian, triangular, square, truncated ramp and other forms

26Gaussian membership functions

27

X

Input space of physical variables

Input space of linguistic variables

AX

Output space of linguistic variables

AY

Output space of physical variables

Y

Logical rules with linguistic variables

Fuzzy Logic Inference Systems

( Mamdani Model )

28Input output relationships are modeled by fuzzy inference system, FIS.It is based on fuzzy logic reasoning which is a superset of classical Boolean logic rules for crisp sets. Elementary logic operations with fuzzy sets are:

fuzzy intersection or conjunction ( Boolean AND )

xxTxAxA AjAiji ,A typical choice of T-norm operator is a minimum function corresponding to Boolean AND, i.e.:

xAxAxAANDxA jiji ,minand standard choice to Boolean OR and NOT:

xAxAxAORxA jiji ,max

xAxANOT 1

29Process of mapping scalar between input and output sets by Fuzzy Inference System.

Fuzzification Fuzzy inference Defuzzification

x(t)y(t)

30Sugeno (1988) Fuzzy Inference System

X

Space of input variables (numbers)

AX

Space of input logic variables

Z

Space of singelton MF (numbers)

Y

Space of output variables (numbers)

Developed for process modeling and identification.

Application in adaptive neural fuzzy logic systems ANFIS

Logic relations

31

In Sugeno FIS for fuzzy inference polynomial Pn approximation is applied

Y = Pn ( Z ), usually a linear model is used

Y = C1 Z + Co , C1 and Co are constants

Mapping to scalar variables is obtained by averaging

y = WT Y

32Example: Fuzzy logic control of flow rate

For example, consider a fuzzy logic model of control of a flow rate ( position of a valve piston) based on input values of temperature T and pH

flow ratevalve position

T

pH Q

BIOPROCESS

FUZZY LOGIC MODEL

33FIS model Q=f(T,pH)

FUZZYRULES

INPUTSPACE OFLINGUISTICVARIABLES

FUZZIFICATION

OUTPUTSPACE OFLINGUISTICVARIABLES

DEFUZZIFI-CATION

FUZZY INFERENCE SYSTEM

INPUT DATA T(t) pH(t)

OUTPUT DATA Q(t)

AGGREGATION

34

T

T

T

LOW T

GOOD T

HIGH T

T(t)

pH

pH

pH

LOW pH

GOOD pH

HIGH pH

pH(t)

35List of the fuzzy rules for control of valve position

IF T is low AND pH is low OR good THEN valve is half open IF T is low AND pH is low THEN valve is open IF T is high AND pH is high THEN valve is closed IF T is high AND pH is low THEN valve half open IF T is good AND pH is good THEN valve half open

36Membership function of the fuzzy sets in the output space

CLOSED HALF CLOSED

VALVE

VALVE

VALVE

OPEN

37Aggregation of fuzzy consequents from fuzzy inference system FIS into a single fuzzy variable output

(t)

VALVE

y(t) = valve position

FIS rules

Aggregation to output

dxx

dxxxty

~

~)(

centroid

38Schematic representation of a neurone with a sigmoid activation function

O

x1

x3

x2

xi

xN

ACTIVATION

0

0,2

0,4

0,6

0,8

1

1,2

-6 -4 -2 0 2 4 6

INPUTOUTPUT

)exp(1

1)(

ssf

39Schematic diagram of a feedforward multilayer perceptron

Y3

Y2

Y1

X1

X2

X3

X4

I H O

40Model equations

k

TtytyE

2

1

)1(

1

)1()1(

1

)1(lN

i

lj

li

lijl

lN

i

lj

li

lij

lj koWnetkoWfko

jiji W

EW

,,

Methods of adaptation:

On-line back propagation of error with use of momentum term

Batch wise use of conjugate gradients ( Ribiere-Pollack, Leveberg-Marquard)

41NN models for process control

NNARX: Regressor vector:

Tkbka nntuntuntytyt 11

Predictor: ,,1 tNNttyty

NNOE: Regressor vector:

Tkbka nntuntuntytyt 11

Predictor:

,tNNty

42Inverse neural network control

PROCESSNN-1

XI Y

n

Input information on referencetransients of output variables

Compensation of process noise ?

43Inverse neural network control coupled with a PID feedback loop

NN-1 PROCESS

PID

XI

nY

+

-

-

44Internal model control structure

+-

NN -1PROCESS

NN

n1 n2

n3

xI

Y-

45Chemostat as a single input single output SISO system

CHEMOSTAT

NN

D S

XSM

SMSXSS

S ccK

cYccD

dt

dc

0

XSM

SMX

X ccK

ccD

dt

dc

XXSM

SMPX

P cDccK

cY

dt

dc

46CHEMOSTAT SISO MODELS

)1(),(,1,1 kDkDkckcNNkc SSS

)(),1(,,11 1 kDkckckcNNkD sSS

NN

NN-1

47Responses of concentration of substrate chemostat to a sine perturbation of reference concentration obtained with direct inverse control. Reference signal is plotted as a solid curve and response is dotted. Frequency of perturbations are A: 0,0125 min-1; B: 0,025 min-1; C: 0,2 min-1; D: 0,1 min-1

0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

0 20 40 60 80 100 120 140 160 180 200 0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100 120 140 160 180 200 0

1

2

3

4

5

6

7

8

9

10

A B

C D

48Responses of substrate (s), dilution rate (D), product (p), and biomass (x) under direct inverse neural network control. Reference signal is a series of square impulses of substrate. The chemostat responses are dotted lines and the reference is a solid line.

0 20 40 60 80 100 120 140 160 180 200 3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4

s D

p x

49Responses of substrate under direct inverse neural (….)

network control and internal model (….) control .

0 20 40 60 80 100 120 140 160 180 200 0

2

4

6

8

10

12

50Comparison of direct inverse neural network control and internal model neural network control with 7,5% relative standard noise in substrate measurement

S

Time (min)

0 200100

51NN from B. yeast production in deep jet bioreactor (Podravka)

1-run2-run3-run

15 h15 h15 h

Measured NN model

EtOH

52Adaptive neuro fuzzy inference system ANFIS

Integration of neural networks with fuzzy logic modeling.

ANFIS does not require prior selection of fuzzy logic variables

ANFIS does not require prior logic inference rules

ANFIS requires only sets of input and output training data ( like for NN modeling )

ANFIS has Sugeno structure of fuzzy logic systems

53ANFIS provides fuzzy logic clustering of data to artificial linguistic variables.

ANFIS provides adaptive membership functions for definition of association of data to linguistic variables (fuzzy variables).

ANFIS provides combinatorial generation of logical relations for mapping between input and output fuzzy sets.

ANFIS provides adaptation of parameters in Sugeno mapping.

ANFIS provides back propagation method for adaptation of model to training data.

54ANFIS model of chemostat D(k)=f [ Sref,S(k),S(k-1)]

and

or

not

input

Input MF

rules

output MF

Sugeno i/o mapping

output

Sref

S(k)

S(k-1)D(k)

55DemoDEMO PROGRAMS

56ConclusionsNeural networks NN and Fuzzy logic inference (FIS) systems are very practical methods for modelling and control of bioprocesses.

Advanced computer supported instrumentation for physical, chemical and biological variables provide large data banks applicable for training NN and FIS models.

NN and FIS are best suited for on-line monitoring, soft identification and nonlinear multivariable adaptive control.

Unlike analytical models, NN and FIS can be developed without “a priori” fundamental knowledge of a process.

Analytical models are “very expensive” to develop, but they are the most valuable engineering tool.

57NN and FIS can integrate knowledge in a very general form. Information from on-line instruments, image analysis and human experience can be easily incorporated.

Analytical models are excellent for extrapolation in the entire process space, while NN and FIS are the best at interpolation in the training set and need to be tested for extrapolation outside training.

Integration of NN and FIS into Adaptive Neural Fuzzy Inference Systems ANFIS leads to models which combine the best properties of NN and FIS.

ANFIS are highly adaptive like NN, they are transparent for logical rules like FIS, automatically generate linguistic variables and logical rules, and are trained to extensive process data.

58

Model verification of NN, FIS and ANFIS is the most important step before their application in laboratory and industrial practice.