PREHRAMBENO -BIOTEHNOLOŠKI FAKULTET

1. PREHRAMBENO -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: zkurt@mapbf.pbf.hr URL: http:/mapbf.pbf.hr/~zkurt

2. MODELIRANJE

3. MULTIDISCIPLINARNOST MATEMATIČKOG MODELIRANJA PROCESA BIOTEHNIČKE ZNANOSTI MATEMATIČKE ZNANOSTI RAČUNARSKE ZNANOSTI

4. TEORIJA SUSTAVA I MATEMATIČKO MODELIRANJE Osnovni pojmovi o sustavu: Prikaz odnosa sustava i okoline

5. SISTEMSKI PRISTUP MODELIRANJU POČETAK SVRHA MODELA ODREĐIVANJE PARAMETARA RJEŠENJE JEDNADŽBI MODELA DEFINIRANJE ULAZNIH VELIČINA X IZBOR NUMERIČKE METODE DEFINIRANJE IZLAZNIH VELIČINA Y PROVJERA MODELA 2M <  NE IZVODI BILANCI MASE, ENERGIJE, KOLIČINE GIBANJA DA IZBOR RAČUNALNOGJEZIKA PRIMJENA

6. Značajke sustava Sustav je apstraktna tvorevina, najčešće definira matematičkim relacijama ( npr. skupom diferencijalnih jednadžbi, diskretnih jednadžbi, neuralnim mrežama, neizraženom “fuzzy “ logikom, ekspertnim sustavom itd.). Sustav se definira s obzirom na određenu svrhu, na primjer: 1) za analizu nekog procesa, 2) upravljanje, 3) projektiranje, 4) nadzor ( monitoring ) 5) osiguranje kakvoće proizvoda 6) optimiranje 7) razvoj novih proizvoda 8) zaštitu okoliša

7. NAČELO IZVOĐENJA BILANCI dio volumena V ulazni tokovi: tvari,energije, količine gibanja izlazni tokovi: tvari,energije, količine gibanja

8. U procesnom inženjerstvu ( kemijskom, biokemijskom, prehrambenom, farmaceutskom .. ) matematičke modele izvodimo na osnovi slijedećih bilanci: mase (tvari), energije i količine gibanja. gdje S označava masu ( količinu tvari), energiju i količinu gibanja. Osnovni oblik bilance je:

9. Modeli 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.

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

11. U 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 reakcijaendotermna. Oznaka  označava malu ali konačnu promjenu određene veličine, t je oznaka za vrijeme,  je oznaka za malu konačnu promjenu t je mala konačna promjena vremena (akumulacija S) je mala konačna promjena akumulacije ( sadržaja S) Bilance postaju diferencijalne jednadžbe kada se provede granični postupak u kojem konačne diferencije,  , postaju infinitezimalne veličine ( odnosno diferencijali, d ).

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

13. Opći oblik modela s raspodjeljenim veličinama stanja je: gdje je vektor položaja. uz zadano početno stanje: rubne uvjete: i/ili i ulazne veličine:

14. Klasifikacija modela Analitički modeli Neanalitički modeli Regresijski oooo izvedeni iz fundamentalnih zakona fizike, kemije i biologije Neuralne mreže “Fuzzy logic” neizražena logika Ekspertni sustavi

15. Klasifikacija analitičkih modela Deterministički Stohastički Populacijski Distribuirani Usredotočeni Usredotočeni Diskretni Distribuirani Linearni Kontinuirani Nelinearni Dif. jednadžbe Linearni Nelinearni Prijenosne funkcije

16. Kontinuirani - diskretni modeli Sustav 1. reda y(t) x(t) Kontinuirani model sustava 1 reda Zadane veličine: 1) parametri  , k 2) početno stanje y(t = 0) = y0 3) ulazna veličina x(t), t  [ 0, tf ]

17. Model u programskom jeziku: Wolfram Research “Mathematica”

18. kontinuiran diskretan korak

19. Matematički 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 projektiranje novih procesa nadzor ( “ monitoring” ) procesa očuvanje kakvoće proizvoda upravljanje ( automatizacija ) procesa optimiranje procesa

20. CONTENTS 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 Process subsystem SP y xI Control subsystem SC Systems view of an industrial bioprocess

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

23. Graphical representation of "transparency" of mathematical models in relation to knowledge and perception of complexity of a system. Analytical models X Y Fuzzy models Neural networks System complexity Knowledge

24. Objectives in modeling Analytical models Process analysis: studies of reaction mechanisms, kinetics, parameter estimation Process design Process optimization Process on-line monitoring Processcontrol Input - output models Process on-line monitoring Processcontrol

25. Fuzzy 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: 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

26. Gaussian membership functions

27. Fuzzy Logic Inference Systems ( Mamdani Model ) Logical rules with linguistic variables AX AY Y X Input space of linguistic variables Input space of physical variables Output space of linguistic variables Output space of physical variables

28. Input 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 ) A typical choice of T-norm operator is a minimum function corresponding to Boolean AND, i.e.: and standard choice to Boolean OR and NOT:

29. Process of mapping scalar between input and output sets by Fuzzy Inference System. Fuzzy inference Fuzzification Defuzzification y(t) x(t)

30. Sugeno (1988) Fuzzy Inference System Logic relations X AX Z Y Spaceof input variables (numbers) Space of input logic variables Space of singelton MF (numbers) Space of output variables (numbers) Developed for process modeling and identification. Application in adaptive neural fuzzy logic systems ANFIS

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

32. Example: Fuzzy logic control of flow rate flow rate valve position T BIOPROCESS FUZZY LOGIC MODEL pH Q 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

33. FIS model Q=f(T,pH) FUZZY INFERENCE SYSTEM OUTPUT SPACE OF LINGUISTIC VARIABLES INPUT SPACE OF LINGUISTIC VARIABLES FUZZY RULES AGGREGATION FUZZIFICATION DEFUZZIFI- CATION OUTPUT DATA Q(t) INPUT DATA T(t) pH(t)

34. LOW pH LOW T   pH T GOOD pH GOOD T   pH T HIGH pH HIGH T   pH T pH(t) T(t)

35. List 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

36. Membership function of the fuzzy sets in the output space HALF CLOSED CLOSED   VALVE VALVE  OPEN VALVE

37. Aggregation of fuzzy consequents from fuzzy inference system FIS into a single fuzzy variable output (t) FIS rules VALVE centroid y(t) = valve position Aggregation to output

38. Schematic representation of a neurone with a sigmoid activation function x1 x2 x3 O xi xN

39. Schematic diagram of a feedforward multilayer perceptron X1 Y1 X2 Y2 X3 Y3 X4 I H O

40. Model equations Methods of adaptation: On-line back propagation of error with use of momentum term Batch wise use of conjugate gradients ( Ribiere-Pollack, Leveberg-Marquard)

41. NN models for process control NNARX: Regressor vector: Predictor: NNOE: Regressor vector: Predictor:

42. Inverse neural network control Compensation of process noise ? n Y XI NN-1 PROCESS Input information on reference transients of output variables

43. Inverse neural network control coupled with a PID feedback loop n Y XI NN-1 - + PID - PROCESS

44. Internal model control structure n3 n1 n2 NN -1 PROCESS Y xI - - + NN

45. Chemostat as a single input single output SISO system D S CHEMOSTAT NN

46. CHEMOSTAT SISO MODELS NN NN-1

47. Responses of concentration of substrate chemostat to asine 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 B A D C

48. Responses 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. s D p x

49. Responses of substrate under direct inverse neural (….) network control and internal model (….) control .

50. Comparison of direct inverse neural network control and internal model neural network control with 7,5% relative standard noise in substrate measurement S 0 100 200 Time (min)