1

PREFERENCES AND DETERMINATION OF THE NOMINAL GROWTH RATE OF FED-BATCH PROCESS: CONTROL DESIGN OF COMPLEX PROCESSES

Bulgarian Academy of SciencesBulgarian Academy of Sciences,,Institute of Biophysics and Biomedical Institute of Biophysics and Biomedical

EngineeringEngineering

SOFIA

 

   

 

   

Yuri P. Yuri P. Pavlov, Peter VassilevPavlov, Peter Vassilev

www.clbme.bas.bg

yupavlov@clbme.bas.bgyupavlov@clbme.bas.bg

ilywrin@clbme.bas.bg

204/18/2304/18/23

The objectiveThe objective of this of this investigationinvestigation is is development ofdevelopment of comfortable tools and comfortable tools and mathematical methodology that are useful for mathematical methodology that are useful for dealing with the uncertainty of human behavior dealing with the uncertainty of human behavior and judgments in complex control problemsand judgments in complex control problems. . This investigation is based on This investigation is based on 44 approaches approaches: : Equivalent Brunovsky formEquivalent Brunovsky form of the fed-batch of the fed-batch model, model, Pontrjagin’s maximum principlePontrjagin’s maximum principle, , Sliding mode controlSliding mode control, Utility theory , Utility theory andand Stochastic programmingStochastic programming. .

304/18/2304/18/23

Main purpose ( ЦЕЛ НА ИЗСЛЕДВАНЕТОЦЕЛ НА ИЗСЛЕДВАНЕТО))

Synchronized utilization of the 4 approaches to overcome difficulties arising from the biotechnological peculiarities, non linearity, singular optimal control and determination of good control solutions.

4

MOTIVATION OF THE INVESTIGATION

• Complex control problem -Complex control problem -Assessment of human

value for determination of the “best” technological conditions;

• Non-linearity of the Monod kinetic model;

• Singular optimal control;

• Non observability of the Monod kinetic models.

504/18/2304/18/23

CONTINUOUS PROCESS: Monod-Wang modelCONTINUOUS PROCESS: Monod-Wang model

I. PECULIARITIES 1. Description of the models; 2. The cultivation processes are different from the

classical physical systems; 3. Appearance of non Gaussian noise in the

system; 4. Measurements difficulties; 5. Complex systems need complex models.

II. MODEL -X -biomass concentration,

-S -substrate concentration,

- -specific growth rate,

-Ks - Mihaelis-Menten constant,

-ν -white noise ,

-So substrate concentration in the feed,

-m - coefficient,

-D input –dilution coefficient,

-μm (T, pH) maximal value of the specific growth rate (as function of temperature T and the acidity pH),

-y -coefficient.

νSK

Sμm

DSSXy

S

DXXX

)(

)(1

sm

0

.

.

.

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

X1

X2

X3

EVALUATIONS OF MODEL (1) without control

604/18/2304/18/23

FED-BATCH FERMENTATION PROCESS: Monod-WangFED-BATCH FERMENTATION PROCESS: Monod-Wang

data presentation (CLBME, Bulgarian Academy of Sciences, MOBPS) (от модел)

1. Substrate concentration S; 2. Specific growth rate ;

0 500 1000 1500 2000 2500 3000 3500 4000-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Време 8 h

2

1

FV

νSK

Sμm

V

FSSX

yS

XV

FXX

0

sm

00

0

)(

)(1

MODEL (fed-batch-полупериодичен):• X -biomass concentration,• S -substrate concentration, -specific growth rate,• Ks - Mihaelis-Menten constant,• ν -white noise ,

• So substrate concentration in the feed,

• m - coefficient,• F ”is the substrate-feed rate”, input ,

• μm (T, pH) maximal value of the specific growth rate (as function of temperature T and the acidity pH),

• y -coefficient.

704/18/2304/18/23

BEST GROWTH RATE: The inclusion of a value expert model as a part and criteria of a BEST GROWTH RATE: The inclusion of a value expert model as a part and criteria of a dynamical control system can be done with the expected Utility theory.dynamical control system can be done with the expected Utility theory.

• Data modelling

1.Substrate concentration S;

2.Specific growth rate ;

• Objective function - U():

0 500 1000 1500 2000 2500 3000 3500 4000-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Време 8 h

2

1

.)(6

0

i

iicU

8

Human Judgment and Assessment of Human Value (Utility) in Complex Systems

(Frontiers: Decision making, Subjective Preferences, Value, Expected Utility, Probability theory, Theory of Measurement )

The assessment bases on mathematically formulated The assessment bases on mathematically formulated axiomatic principles and stochastic procedures. The axiomatic principles and stochastic procedures. The evaluation is a preferences-oriented machine learning evaluation is a preferences-oriented machine learning procedure with restriction of the “certainty effect” and procedure with restriction of the “certainty effect” and ”probability distortion” identified by Kahneman and Tversky ”probability distortion” identified by Kahneman and Tversky (prospects theory). The uncertainty of the human preferences (prospects theory). The uncertainty of the human preferences is eliminated as typically for the stochastic programming is eliminated as typically for the stochastic programming procedures.procedures.

This evaluation is based on This evaluation is based on 3 approaches3 approaches: : Decision Decision making theory,making theory, Utility theory of von Neuman and Utility theory of von Neuman and Potential function method of Aizerman, Braverman and Potential function method of Aizerman, Braverman and RozonoerRozonoer. .

9

EXPECTED UTILITY AND VALUE FUNCTION (mathematical definitions)

(pq , (p,q)P2 ) (u(.)dp u(.)dq), pP, q P.

Von Neumann and Morgenstern’s axioms:

• I. The preference relation () is negatively transitive and asymmetric one (weak order);

• II. (QP, 0<<1) ((P+(1-)R)((Q+(1-)R)) (independence axiom);

• III.(PQ, QR) ((P+(1-)R)Q)((P+(1-)R)Q), for ,(0,1) (Arhimed’s axiom);

Let X is the set of alternatives (XRm). According to von Neumann & Morgenstern this formula means that the mathematical expectation of the expert utility function u(.) is a measure for the expert preferences. These preferences are defined over the set of the probability distributions P ( P is defined over the set of the alternatives X). With is denote the preference relation over P (X P). The indifference relation () is defined as: (x y) x y x y .

A “value” function is a function u*(.) for which is fulfilled:

(x, y)X2 , x y u*(x)>u*(y).

10

UTILITY ASSESSMENT PROCEDURE

• The expert compares the "lottery" x,y, with the alternative zX ("better- ", "worse - " or "can’t answer or equivalent - "):

• x,y, ( or or ) z.

• The expert compares the "lottery" <x,y,> with z (the “learning point” (x,y,z,)) and with the probability D1(x,y,z,) relates it to the set Au= (x,y,z, u(x)+(u(y))>u(z), or with the probability D2(x,y,z,) - to the set Bu = (x,y,z, u(x)+(u(y))<u(z). At each “learning point” (x,y,z,) a juxtaposition can be made: f(x,y,z, =1 for (, f(x,y,z,=-1 for () and f(x,y,z,=0 for ( (subjective characteristic of the expert which contain the uncertainty of expressing his/her preferences).

• Let the "learning points" (the learning sequence) ((x,y,z,1, (x,y,z,2,…,(x,y,z,n,..) has the probability distribution F(x,y,z,. Then the probabilities D1(x,y,z,) and D2(x,y,z,) are the mathematical expectation of f(.) over Au and Bu, respectively: D1(x,y,z,)= M(f/x,y,z,, if M(f/x,y,z,>0, D2(x,y,z,)=-M(f/x,y,z,, if M(f/x,y,z,<0. Let D'(x,y,z,) is the random value: D'(x,y,z,)= D1(x,y,z,), M(f/x,y,z,>0; D'(x,y,z,)=-D2(x,y,z,), M(f/x,y,z,<0; D'(x,y,z,)=0, M(f/x,y,z,=0.

• We approximate D'(x,y,z,) by the function G(x,y,z,)=(g(x)+(-g(y)-g(z)), where. the function G(x,y,z,) is positive over Au and negative over Bu depending on the degree of approximation of D'(x,y,z,). In such case g(x) is an approximation of the empirical expert utility u*(.).

(1-) 1

x, y, ( or or ) z

11

APPROXIMATION OF THE EXPECTED UTILITY U*(.)

i

ii xcxg )()(

The expert answers has the presentation f(.): f=D'+, , M(x,y,z,)=0, M(2/x,y,z,)<d, dR. Let the utility function u*(.) is a “square-integrable function”: (u*2(x)dFx< +), where Fx is the probability distribution over X. The distribution Fx is defined by the probability distribution F(x,y,z, of the appearance of the learning points. The expected utility u(.) fulfils:

We note as (i(x)) a family of polynomials and riR.

i

iiL

xrxu )()(*2

)(,' 11))1(()1(1

n

innnnnn ttctDcc nii

0,,, nn

nn

nn 2

APPROXIMATION ALGORITHM: t=(x,y,z, ;

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

n

innnnn ttctfcc nii

1204/18/2304/18/23

0 2 4 6 8 10 12 14 16 180

0.05

0.1

0.15

0.2

0.25

0.3

0.35OPTIMAL CONTROL AND SMC

Time [h]

Sp

ecif

ic G

row

th R

ate

0 2 4 6 8 10 12 14 16 180

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

TIME [h]

Sp

eci

fic G

row

th R

ate

Without control and fixation of the system on the equivalent control position.

)(

,)(1

]0[]0[]0[))(

32s

2m3

20132

1131

max

xxK

xμmx

DxSxxy

x

Dxxxx

, D, D,Tt,, μμ,μU m

.

.

.

max(

FIXATION OF THE SYSTEM IN A POSITION CONFORMING TO THE EQUIVALENT CONTROL

The fixation is based on a “ time optimization” control. “Time optimization” control is solution of the next optimal problem (continuous process):

max(U((T))), where the variable is the third coordinate in the state vector of the model of the continuous process, ([0,m], D[0,Dmax]. Here U() is an aggregation objective unimodal function. Possible choice is the expert utility function.

1304/18/2304/18/23

EQUIVALENT MODEL – CONTINUOUS PROCESSEQUIVALENT MODEL – CONTINUOUS PROCESS • With the use of the GS algorithm the non-linear Wang-Monod model is presented

in Brunovsky normal form (нормална форма на Бруновски):

• Diffeomorphic transformations: Transformation 2

))()(

()23(

)(

2113

2

231

2211

233

21132

11

kuuuusK

umukkuuuY

kuuuY

uY

m

3

2

20

1

321

3

2

1

),,(

x

xxS

x

xxx

u

u

u

νSK

Sμm

DSSXy

S

DXXX

)(

)(1

sm

0

.

.

.

WY

YY

YY

3

32

21

WY

YY

YY

3

32

21

WY

YY

YY

3

32

21

1404/18/2304/18/23

OPTIMAL CONTROL OPTIMAL CONTROL Equivalent BRUNOVSKY NORMAL FORMEquivalent BRUNOVSKY NORMAL FORM::

• Gardner, Robert B.; Shadwick, William F.The GS algorithm for exact linearization to Brunovsky normalform.1992, Text.Article, IEEE Trans. Autom. Control 37, No.2, 224-230 (1992).

• Elkin, V. Reduction of Non-linear Control Systems: A Differential Geometric Approach–Mathematics and its Applications, 472, Handbound, Kluwer, 1999.

y

YY

VY

YY

YY

2.

4

.

3

3

.

2

2

.

1

FV

νSK

Sμm

V

FSSX

yS

XV

FXX

0

sm

00

0

)(

)(1

WY

YY

YY

3

32

21

WY

YY

YY

3

32

21

1504/18/2304/18/23

OPTIMIZATION PROBLEM – continuous processOPTIMIZATION PROBLEM – continuous process

• The optimization problem is:

• Hamilton based on the Brunovsky normal form :

)(

,)(1

]0[]0[]0[))(

32s

2m3

20132

1131

maxmax

xxK

xμmx

DxSxxy

x

Dxxxx

, D, D,Tt,, μμ,μU

.

.

.

max(

VΨYΨYΨH 33221

1604/18/2304/18/23

OPTIMAL CONTROL “Time minimization”OPTIMAL CONTROL “Time minimization”

2211

126

1

)1(

10

)(

)21()(

))((

kYY

kYYic

Y

TUb

i

ii

1)(

)21(

)(

1)(

))((211

12

211

6

1

)1(0

21

kYY

kYTY

kYYicTb

Y

TUb

i

ii

2

20

12Tb

Tbb

1)(2

)21()(

)(

)()(

1)(2

)21()(

)(

)()(

2)(

211

12

211

6

1

)1(

211

12

211

21

20

3

kYY

kYYtT

kYY

tTic

kYY

kYYtT

kYY

tTUbtb

tbt

i

ii

DxK

xKμkYYmDtCONTR

22s

2-0sm

211

)(

)(S)()(

0,0)(0,1)(

12

)21()()(

12

)21()())((

)()(

,max1

6

1

)1(

max1

6

1

)1(

max3

rrsignrrsign

DkYtT

icsign

DkYtT

tTicsign

DttCONTRsignD

i

ii

i

ii

opt

Online journal: Bioautomation (2004, 2005 – Pavlov Yuri)Online journal: Bioautomation (2004, 2005 – Pavlov Yuri)

1704/18/2304/18/23

OPTIMAL CONTROL - graphicsOPTIMAL CONTROL - graphics

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Време h;

Спе

циф

ична

ско

рост

на

нара

ства

не x

3 h-

1; 1

2

0 1 2 3 4 5 60

2

4

6

8

10

12

Време h;

Кон

цент

раци

и g/

l;

1

2

1804/18/2304/18/23

Evaluation of the system-fed-batch Evaluation of the system-fed-batch profileprofile

0 2 4 6 8 10 12 14 16 180

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time [h]

Sp

ec

ific

Gro

wth

Ra

te Sliding Mode Control

Equivalent Control

Optimal Control

Specific Growth Rate

1904/18/2304/18/23

SECOND ORDER SLIDING MODE CONTROLSECOND ORDER SLIDING MODE CONTROL

Here is used the so cold “contraction” algorithm. After Emelyanov the SM control law is:

The scientists Emelyanov, Korovin and Levant evolve high-order sliding mode methods in control systems. Out to this approach the second order SM manifold becoms:

))59.0)()(

μ((3

1

)1(3

2(

85,0

15,1)15,115,1(

32

2

21

2

32

mxKx

xm

Sx

xxKmUcontr

s

mm

s

sign

sign

0 2 4 6 8 10 12 14 16 180

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time [h]

Sp

ecif

ic G

row

th R

ate

Equivalent Control

Second Order Sliding Mode Control Optimal Control

Growth Speed Rate

0 2 4 6 8 10 12 14 16 180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [h]

Su

bst

rate

Co

nce

ntr

atio

n [

g/l]

X2

Equivalent Control

VYYX

SSmanifoldSYSX

,

)0101(,1,1

)31.03(111 xSwhereSS

2004/18/2304/18/23

Optimal profile: fed-batch processOptimal profile: fed-batch process

,0))(exp(1)(

)(

)(ln

))((

)()(

)())(exp(),,,,(

2

1

2

121

1

11

11

m

tf

m

tmfm

SKs

tSKs

tSS

tKsftfm

ttfKsSXManifold

mee

m

ee

.

)(

)(ln)(

))(())(

)()(ln(

))((

)(1

1

11

111

1

eem

eee

SKs

tSKsKstSS

tSStX

StStX

m

t

tfwhere

.0)()(

))(

(

,31.0],,0[],,[],,0[),))((min(

2,2

max21max

2

1

tt

SsK

Sm

XkS

XX

FFtttt

tdtt

e

m

ee

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

TIME [h]

Gro

wth

Rat

e

t1

t2

CONTROL F

2104/18/2304/18/23

Optimal profile and Optimal Optimal profile and Optimal ControlControl

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

TIME [h]

Gro

wth

Rat

e

t1

t2

CONTROL F

0 2 4 6 8 10 120

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

TIME [h]

Gro

wth

Rat

e

OPTIMAL PROFILE

Parameter Noise

CONTROL

Pavlov Y. (2008). Equivalent Forms of Wang-Yerusalimsky Kinetic Model and Optimal Growth Rate Control of Fed-batch Cultivation Processes, online journalBioautomation, Vol. 11, Supplement, November, p.p. 1 – 12.

2204/18/2304/18/23

CONCLUSIONSCONCLUSIONS

• This investigation permits utilization of Control theory and Utility theory to design a flexible methodology, useful in complex biotechnological processes and descriptions of the complex system “Technologist-Fed batch process”.

• The possibilities of the second order SM are investigated.

• Both controls SM and “time-minimization” synchronized utilization permits to overcome difficulties arising from the biotechnological peculiarities in order to obtain good control solutions.

• The inclusion of a value model as objective function as part of a dynamical system could be done with the expected utility theory. Such a utility objective function allows the user to

correct iteratively the control law in agreement with his value judgments.

2304/18/2304/18/23

Applications

• Pavlov Y., K. Ljakova (2004). Equivalent Models and Exact Linearization by the Optimal Control of Monod Kinetics Models. Bioautomation, v.1, Sofia, 42 – 56.

• Pavlov Y. (2005). Equivalent Models, Maximum Principle and Optimal Control of Continuous Biotechnological Process: Peculiarities and Problems, Bioautomation, v.2, Sofia, 24–29.

• Pavlov Y. (2007). Brunovsky Normal Form of Monod Kinetics Models and Growth Rate Control of a Fed-batch Cultivation Process, Bioautomation, v.8, Sofia, 13 – 26.

• Pavlov Y. (2008). Equivalent Forms of Wang-Yerusalimsky Kinetic Model and Optimal Growth Rate Control of Fed-batch Cultivation Processes, Bioautomation, Vol. 11, Supplement, November, p.p. 1 – 12.

Thank you for your attention!

24

МОДЕЛ НА ПРОЦЕСА НА ПОЛУЧАВАНЕ НА АЦЕТАТ - Микробиология

ν)KSSsK

Sm(μ

S)D(Soμxy

S

Dxμ.xx

m

.

0

2

1

Daμxka

ν)KSSsK

Sm(μ

S)D(Soμxy

S

Dxμ.xx

m

.

2

02

1

Еквивалентен диференциален модел :

• We discuss yeast’s C.blankii 35 continuos cultivation process .

• The system parameters are as follows: μm=0.776 [h-1], Ks= 14.81 [g/l], Ko=1/1231 [–], m=3.51 [–], Se=0.2625 [g/l], S0=9 [g/l], y=0.5584 [–], De=0.01 [h-1].

25

ПРИЛОЖНИ РАЗРАБОТКИ: БИОТЕХНОЛОГИЧЕН ПРОЦЕС ЗА ПОЛУЧАВАНЕ НА АЦЕТАТ - дифеоморфни трансформации

• Нелинейните дифеоморфни трансформации:

Трансформация 1 (x1=x; x2=S; x3=):

2xx

'xx 11

02

12 S'x

'xx

,'33 xx • Трансформация 2

22123 xkxxy

32

21

2212

23

22122

23. xkxkxxxkxkSSk

Smz

oSm

yx zy

Vz

)xk2xk3x(xkSSk

Sx.m3 3

221

22123

0S

m3

2

302

2212

20132

02

202

212 ][ xkSSk

SμxkxmDSSxxk

kSSk

SkkmxkxV

S

m

S

S

26

ПРИЛОЖНИ РАЗРАБОТКИ: ПРИНЦИП НА МАКСИМУМА И ОПТИМАЛНО УПРАВЛЕНИЕ - непрекъснат процес за добиване на ацетат с реални данни от института по Микробиология

• Оптимално управление с реални данни от института по Микробиология:

min (x (t1)-x0)2, където x е първата координата във вектора на пространството на състоянията на Бруновски модела, t[0,t1], D[0,D0]. В случая x0 е избрана предварително константа.

2704/18/2304/18/23

Bulgarian Academy of Sciences,Bulgarian Academy of Sciences,CBME “Prof. Ivan Daskalov”CBME “Prof. Ivan Daskalov”

• In chapter (3) is presented a control design for control and stabilization of the specific growth rate of fed-batch cultivation processes. The control design is based on Wang-Monod kinetic and on Wang-Yerusalimsky kinetic models and their equivalent Brunovsky normal form. The control is written based on information of the growth rate. The criterion for determination of the “best” growth rate is a utility function evaluated by the utilization of human preferences. By this way is obtained mathematical description of the complex system “technologist-biotechnological model”. The evaluation of the utility function is based on a scientific investigations and mathematical and programming results developed in Bulgaria, BAS by the author In the last two decades.

28

This work is partially supported by the Bulgarian National Science Fund under grant No. DID-02-29 “Modelling Processes with Fixed Development Rules”

29

Thank You For Your Attention!