Adaptive Estimation and Control of the Specific Growth Rate of a Nonlinear Fermentation Process via MRAC Method

Copyright © IFAC Modeling and Control of Biotechnical Processes, Colorado, USA, 1992

ADAPTIVE ESTIMATION AND CONTROL OF THE SPECIFIC GROWTH RATE OF A NONLINEAR FERMENTATION PROCESS VIA MRAC METHOD F.Y. Zeng

*'**, B. Dahhou*'**, G. Goma** and M.T. Nihtilii***

*Laboratoire d'Automatique et d'Analyse des Systemes, Centre National de La Recherche Scientifique, 7 Avenue du Colonel Roche, F-310n Toulouse, France **Centre de Transfert en Biotechnologie-Microbiologie, INSAIUPS, Avenue de Rangueil, F-310n Toulouse, France ***Control Engineering Laboratory, Faculty of Information Technology, Helsinki University of Technology, (TKK-S) Otakaari 5A , SF-02150 Espoo, Finland

ABSTRACf The estimation and control of the specific growth rate is a fundamental task for optimizing the fermentation process. This paper presents a new approach b,lsed on MRAC (Model Reference Ad
1. Introduction

In fermentation technology for culture where met
dX e Idt = !-le .Xe - Xe.u + a ,(Se - Sp)

(2)

dSe Idt = -R·!-Ie.Xe + (Sin - Se)'u + j),(Se - Sp) with

Using the plant J.l model given a priori, the first order Taylor expansion of !-ICEl,S,X) allows to obtain the approximate linear relationship between !-I estimation error te!-l ' par
(3)

(I)

where

dSp/dt = -R.!-Ip'X p + (Sin - Sp).u(t) we assume the structure of the specific growth rate is known : !-Ip=!-I(Elp' Sp' Xp)' The objective of the estimation consists of

llER+:the positive common factor of the components of i)!-I(El,S,X)/ilElIElp,Sp,Xp

two levels : to estimate !-Ip as a time-varying parameter and to estimate one of the kin etic par
Kp = (J!-I(El ,S ,X) /(JX IElp,Sp,Xp

ER

biochem ists
Substituting (3) in (2), the estimation error equations can be ohtained by the substrat ion (2)-( I), that is

model. 2. Adaptive estimation

dt e Idt = Ae· Ee + B. teEl T.w

The
(4)

where

all components of parameter vector ElpERm are unknown , (ii) the estimation of !-Ip(t) and anyone component of parameter

Ae = [ !-Ip - u+Kp.Xe

vector ElpERm in the case that the other m-I components of Elp

-R.!-Ip - R.Kp.Xe

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(5)

BT =11-[ 1 , -R ] w

= Xe .H p

where, eT =[J..lm ,Ks], J..lm is maximum growth rate, Ks is bounded function of time

"Michaeolis-Menton" parameter, we have

hT =[0 , 1]

In (4) matrix Ae is a time-varying matrix composed of some complecated components, however we have two adaptive gains a, j3 allowing to change its dynamical feature. For dynamic From (8.3), (9) we can obtain the parameters adaptive adjustment law as follows

system described by (4), if we take

(6)

a = (llp+Kp.Xe)/R

(10)

dJ..lme Idt = q .Xe·( Se-Sp)

j3 = -(Ilp +Kp.Xe)

dK se Idt = - r2.Xdl(fle,sp).( Se-Sp)/Sp

, according to some biological know ledges and by some simple manipulations, we can prove that the matrix Ae is stable, and

where,

transfer function We(p)=h T .(pI-Aet IB is S.P.R..

adjustable coefficients may be determined by simulation.

THEOREM.

rI

and

r2

are

two

3. Adaptive control

For dynamic system (4) using (6) for a , j3

following parameter adjustment law

The objective of the adaptive control presented is to regulate and track the specific biomass growth rate of the plant IIp(t) at

(7) results: when

ll(fle,Sp)=llme.Sp/(Kse+Sp)'

the desired value Ilc(t) by the acting on the dilution rate u(t). The desired trajectory of Ilc(t) is generated by a control

t-~oc,

reference model a priori determined as follows

(i)Ees= Se-Sp"~O , Eex=Xe-Xp"~O, EeJ..l =Ile-Ilp- ~O ,

(11)

dX c Idt = Ilc.xc - Xc·r(t)

(ii)~i=l- m hi ' ( eei - flpi)-+- 0, where, Hp= rh l,h2, .. h m JT .

dS c Idt = - R.llc'Xc + (Sin - Sc)·r(t) with

Proof According to well known Narendra Theorem (Narendraet al., 1974), Considering above S.P.R. condition and the fact that the adjustment dynamic of the reference model parameters fle is much faster than the variation of the plant

Taking over the biological error e(t)=RX(t)+S(t)-Sin EL2

parameters Hp, i.e ., dEeH/dt '" dfle/dt, above conclusions can be

(Zeng et al. , 1991), the Il dynamic model for the plant and the

directly derived.

reference model can be obtained respectively: - the plant Il-dynamic model:

Commelll.

(ii) implies that if any m-I components of parameter vector 8 p are known, i.e., the corresponding error

(12.1)

components of tee are equal to zero , the rest parameter can be

where

estimated; if more than two components of Hp are unknown the

a p = (gp.R - Ip).Xp,

estimation of parameters can

Ip = iJllp/iJXp

not

be

realized,

however,

according to (3) the estimation of specific growth rate can be achieved, i.e. , J..le-Ilp-+-O .

gp= iJllp/vSp

- the reference Il-dynamic model (13.1 )

A practical adaptive estimation algorithm. Since, in (6)(7) IIp' where

Kp and Hp are inaccessible, an approximate method taking their estimate values J..le, Ke and He to replace is adopted. So,

a c = (gc· R - Icl·Xc'

combining with (2), following equations consist of a practical adaptive estimation algorithm

Ic = vllelaXe

a = (lle+Ke.Xe)/R

(8.1)

j3 = -(Ile +Ke·Xe)

(8.2)

r=rT >0

( 12.2)

gc= iJllcliJSc

( 13.2)

If the control input to the plant is choosed as u (t) = G(t).ECIl(t) + ljJ(t).llp(t)+ (t).r(t)

( 14)

(8.3) the control error equation can he obtained from (12.1), (13 . 1) and(14)as

where the formulas of Ke and He are identical to Kp and Hp, but the variables are replaced by X e , Sp and He.

dECll ld t=-( ac+G( t ).a p ) .ECIl -( ac-a p +1jJ( t ).a p ) .Ilp+( ae-Cl)( t ).a p )' r( t) ( 15) Then a Lyapunov design gives the following practical adaptive control laws

An exemple of application. If the expression of the plant Il

364

G(t) =

(lad - ac)/ae

parameters Bp and at least to estimate [.Ip as a time-varying

d'l!ldt = [A1 · E~,d.le + P1.d(EIl.lle)/dt ],sgn(a e )

parameter. The distinguishing feature of [.Ip control prohlem is

(16)

that the controled variable [.Ip is a time-varing parameter of the

dCJ>/dt = [ A2.EI.t.r + P2,d(E Wr)/dt ],sgn(a e)

plant. So a Il-dynamic model is established both for plant and

where EI.t( t)=Ilc( t)-I.te( t)

for control reference model. On the base of [.I-dynamic model

ae = (ge· R - le)·Xe ge= (J1.t(B,S,X)/(JSISe,Sp,Xe

an indirect MRAC method is used, Here, "indirect" is in the meaning that in fact the controller uses the estimate value of [.Ip as controled variable and the algorithm used to update the controller parameters depends on also the estimation results, The simulation results make shown, despite of the large range variation of the plant parameters, the adaptivities and performences of estimation and control are satisfactory,

le = iJl.t(S,S,X)/iJXIBe,Sp,Xe

The adjustable coefficients AI,

PI, 1...2 and

P2 can be

determined by the simulation, REFERENCES

4. Simulation results

Bastin, G. (1991). Nonlinear and Adaptive Control in Biotechnology:a Tutorial.ECC 91, Proc. ,3, 2002-20121. Narendra KS, and Kudva P. (1974), Stable Adaptive Schemes for System Identification and Control - Part 11. IEEE Trans, on sys" man, and cyb" vol. SMC-4, No, 6, Uribelarrea, 1. L., 1. Winter, G, Goma and A. Pareilleux( 1990), Determination of maintenance coeffients of SaCCharomyces cerevisiae cultures with cell recycle by cross-flow membrane filtration . Biotechnol, Bioeng, 35,217-223. Yamane. T., T. Kume, E. Sad a and T. Takamatsu (1977).1. Ferment. Techn ol., 55. 587. Zeng, F.Y. and B. Dahhou (1991). Model reference adaptive estimation and control applied to a continuous flow fermentation process. ADCHEM'91, ProG'. (to be published)

(i) Plant conditions : The I.t model of the plant is Monod law I.tp=l.tme.Sp/(Ksp+Sp), the yield coefficient R=2, the initial conditions are Xp(0)=0.02 gll, Sp(O)= 0,5 gll. The plant parameters vary in the time : I.tmp=0,5 I/h tE[0, 120hl, I.tmp=0,7 lIh tE(l20, .. ); Ksp=O, I g/l tEI 0,200h I, Ksp=0,3 g/l tE(200,.. ), (ii) The estimator conditions: Xe(()=O,OI gll , Se(0)=O,3 gll , I.tme(0)=0,3 I/h, Kse(0)=0,3 gll. The coefficients of (10) are q =0,03, r2=0,02, (iii) The controler conditions: I.tmc=0,7 I /h, Ksc=0,3 g/l. The desired tracking trajectory of I.tc(t) is illustrated in Fig, I.. The coefficients of (16) are AI =0,2, 1...2=0,5, PI =P2=0,OO1.

The simulation results, Fig, I illustrates the adaptive estimation and control algorithm can adapt well to the variation of the plant. Despite of the large range variation of the plant : at moment A, plant parameter [.Imp changes 40%, at moment B. Ksp changes 200%, and at moment C the intluent substrate concentration Sin changes 18%, the specific hiomass growth rate of the plant I.tp can well track the desired value [.Ic' In the respect of adaptive estimation, we can observe: (i) In the case that one of the plant parameters is known, ex" [.Ime= [.Imp' the other plant parameter Ksp can he estimated (see Fig,2,), (ii) In the case that the plant parameters [.Imp and Ksp are unknown, we can also obtain satisfactory X-estimation and

[.1-

estimation as Fig, 1. and Fig,3. illustrate,

5. Conclusion A new approch via MRAC technique for estimating biomass concentration Xp and the specific biomass growth rate [.Ip and furthermore controlling [.Ip for a non-linear continous now fermentation process is presented in this paper. For estimation problem, from the Strictly Positive Real condition of estimation error equations, using Narendra Theorem, we obtain adaptive estimation law employing the plant output Sp as a single measurable variable. We prove by theory and simulations results that one of the advantages of the algorithm presented is to allow, in certain cases, to estimate simultaneously the plant specific biomass growth rate [.Ip and corresponding hiological

365

0.7

-- ---- - ------ ----- --. -- ------.----~

0 .8

0.5

0.4

0 .2

0.'

o · ·0. 1

o

300

200

'00

400

Fig. 1. The evolutions of specific biomass growth rate. !-lc(t) : set-point, !-lp(t) : plant, !-le(t) : estimate

0 .8

. . , - - - - - -. - - - - - - - -- -

~

:: ________ j 04

- - - - - -- --

- - . - -- -- - -

~ ~mp' ~m.

J---- - - -.- --

t

K
r

0 .3

- - Kse

0 .2

0.'

O - ~--_r--_,---,_--_r---.---._--_r--~

o

.00

200

400

300

Fig.2. The estimation of biological parameter Ksp when lAmp is known .

....

-- 1

' .7 ' .8 ' .5 ' .4 '.3

... ' .2

0 ."

0 .8 0.7 0 .8 0.5 0 .4 0 .3 0 .2

0.' ·0 .1 0

'00

200

300

400

Fig.3. The estimation of the biomass concentration. Xc(t) : reference model , Xp(t) : plant, Xe(t) : estimate

366