- Email: [email protected]
Control of Distributed Parameter Bioreactors Via Orthogonal Collocation
C:oprright
©
11'.-\(: Control of J)i s tri""t ~ d
Parameter S~-S(l'IIlS. Pt"r pigll ~ lIl.
Fr;I II(T.
I ~IHq
CONTROL OF DISTRIBUTED PARAMETER BIOREACTORS VIA ORTHOGONAL COLLOCATION D. Dochain* t and
J. P. Babary**
*Lllliom /oiJ"" d'AII/OIIIII/iqlll', DYII{[lIIiqlll' 1'1 AII"I.\'sl' rll'S SYS/t'iIlI'S ,
L'lI iJ'eni"; CII //i O/iqlll' rll' L(lIflIlli ll , Btiti III I'll/ ,\lIIX'I'"I/. PIliCl' dll Ln'tll(/, 3, 13-18 LOIn'aill-I,,-S I'lIl/(', Btigilllll *':Llllwm/oi (p rI'AII/OIIIII/iqllf 1'/ rI'Allllly.I'" rlf'S Sys/t'lII I'S rill CNRS, 7, {/l'I' 1I1I1' rill Colollel ROC/il' , 3 107, TOlllolISl', Fmll cl'
t!1l\'ited Professor a t the Ecole Polyt ec hlliqu c de f\.lontreal. Canada
Abstract. This paper deals with the control of fluidized and packed bed bioreactors, The distributed parameter model of the process and its reduction to ordrn,ary dlfferentl~1 equations by using orthogonal collocation is presented. A nonlinear adaptive controller IS designed based on the reduced model , i.e. ordinary differential equalions deduced by orthogonal collocation . Keywords. Distributed parameter systems, non linear systems, orthogonal collocation, bioreactors. 1. INTRODUCTION
also Issacs et ai , 1986) for the case df an air-lift bioreactor. Maendenius et al (1987) also consider the control of a packed bed bioreactor (in cascade with a Continuous STR (CSTR)), but they treated it as a CSTR, which seems fair in that application with respect to the small size of the bioreactor. In this paper, we shall concentrate on fluidized and packed bed bioreactors, which presents the following characterisctics : the active biomass is kept within the vessel and the different substrate(s) and product(s) flow through it. Furthermore, diffusion phenomenons will be coqsidered as negligible. The distributed parameter model of the process will then be constituted of first-order hyperbolic PDE's.
Over the last decades , there has been a growing and widespread development of industrial biotechnological processes. The operation in Stirred Tank Reactors (STR) has been (and is still) a widely used technology in fermentation processes . But, more a~d more , other technologies are considered for bioprocesses operation , such as fluidized bed, packed bed or air lift reactors, which present several advantages over the "classical" STR's (e.g . Schugerl , 1989). For instance, higher cell mass, more efficient product extraction and higher flow rates are often mentioned as advantages of fluidized and packed bed bioreactors. Typical examples are fluidized or packed bed anaerobic digesters (e.g . Poncelet et ai, 1985 ; Poncelet, 1986) and ethanol production with alginate entrapped yeasts (e .g. Williams and Munnecke, 1981).
Basically , there are two different approaches when treating the control of a distributed parameter system (e.g. Ray, 1981). In the first one , the controller design is based on the distributed parameter model and then eventually reduced to a finite dimension control solution. However, the very large uncertainty on the knowledge about the kinetics of fermentation processes (because they are highly complex processes which involve living organisms), renders this approach very difficult. That's why we made up our mind for a second approach, the early lumping. In a first step, the PDE's are reduced to ODE's by using orthogonal collocation and in a second step, the finite dimension ODE system is considered for the controller's design.
In the context of the monitoring and control of fermentation processes, the use of fluidized or packed bed bioreactors introduces new methodological problems to be solved. In fact , the behaviour of these reactors are expected to be more likely in plug flow conditions than in completely mixed ones. Then , strictly speaking, their dynamics cannot be described by ordinary differential equations (ODE's) as in STR's, but by partial derivative equations (PDE's). Very few papers have been concerned in the literature with the control of distributed parameter bioreactors , with the notable exception of Luttman et al (1985)(see
The proposed controller's design is in line with recent works on the control of STR bioreactors (Dochain, 1986; 12!1
D. Dochain and
1:10
.J.
1'. Bail;tl'\
Dochain and Bastin, 1989} and presents the following basic features :
external substrates) ,
- it is based on the well-known nonlinearities of the process ; - it is adaptive in order to deal with the parameter uncertainty.
and z is the space variable (z
The paper is organized as follows. Section 2 introduces the distributed parameter dynamical model of fluidized and packed bed bioreactors. Its reduction to ODE's by orthogonal collocation is presented in section 3. Section 4 is dedicated to the design of an adaptive linearizing control law. We shall concentrate in this paper on the regulation of one output, i.e . of one reactant concentration at the output of the bioreactor.
(dim
10, H]) .
Remark: let us recall that the reaction rates
2. DYNAMICAL MODEL OF FLUIDIZED AND PACKED BED BIOREACTORS
2} a death reaction of the micro-organisms : X---tX d where Xd is the non-active biomass concentration . If we assume that the non active micro-organisms leave the bioreactor, the dynamics of the above process will clearly be described by the equations (1 )(2) with ~, = X,
~2 = [S Xd1T, ~2,in = [Sin 01 T,
,
K = [1 -11
Fig.1. Scheme of fluidized bed reactor Let us consider a fluidized or packed bioreactor (Fig .1) in which m biochemical reactions with n reactants take place . Among these reactants, n, are micro-organisms entrapped or fixed on some support or fluidised and which remain within the reactor, and n2 other reactants (essentially substrates and products) flow through the reactor. For simplicity, we also consider the crosssection of the bioreactor is constant and equal to A. From mass balance considerations, we can deduce , in line with the model formulation in Dochain et al (1987) and Dochain and Bastin (1989), the following dynamical model:
(1 )
(2)
with the following limit conditions: ~ (t, z=O) = ~2 (t) 2 ,In
In the above equations, vector
(dim
~,
~2
2
=[-k, 0
0] 1
k, : yield coefficient
3. REDUCTION OF THE DISTRIBUTED PARAMETER MODEL TO ORDINARY DIFFERENTIAL EQUATIONS It is clear from the above model (1 )(2) that the state variables ~, and ~2 are functions of time and space, i.e. ~,
(t , z) and ~2(t, z). We shall now see how to
approximate the PDE's of each component of ~, and ~2 by a finite number (equal to p+ 1) of ODE's at p+ 1 discrete spatial positions along the bioreactor. There exist several methods to reduce the above PDE's to ODE's (e .g. Ray, 1981). They consist of expanding the variables as a finite sum of products of time functions and space functions. In the orthogonal collocation method (e.g. Wysocki, 1983), the state variables will be written as follows :
(3) ~,
= n,),
is the biomass concentration ~2
is the other reactant
concentration vector (dim ~2 = n2) ' ~2 , in(t} is the influent concentration of
K
(which is different from zero only for
~k(t, z} =
P+'
2. ~i(z)~ki(t}
k=1,2
(4)
i= 0 where ~ki(t) is the value of ~k(t,z} at some discrete spatial
13 1
Control of Distributed Paramete r Bioreactors
positions (called bioreactor, i.e. :
collocation
points)
along
the
(S)
{b.} IJ J=1 to p;.1
0
0
0
{b .. }. IJ J=1 to p+1
0
B.1 =
and the residuals ~i(z) are chosen as orthogonal functions (e.g. Bessel or Lagrange functions), such that :
0
0
{b.} IJ J=1 to p;.1
where 0ij is the delta function (Oij = 1 if i = j; 0ij = 0 if i "# j). P in equation (4) corresponds to the number of interior collocation points. Z = Zo and Z = zp+1 corresponds to the input (z = 0) and the output (z = H) of the bioreactor. Now we can write the partial derivate of ~2 with respect to z appearing in equation (2) as follows :
(6)
Let us note :
b ij
=
d~.(z = J
Comments : 1) an important condition for the model reduction to be valid is that the influent concentration of the reactants ~2' ~2 , in(t) must be C1 (i.e. continuously differentiable) time functions (see Wysocki, 1983). 2) Wysocki (1983) indicates that 3 or 4 interior collocation points are enough to describe the dynamics of first hyperbolic equations like (1 )-(3) .
z.) (7)
I
dz
4. THE CONTROL ALGORITHM Statement of the Control Problem
By introducing (4)-(7) into equations (1 )-(3), each PDE is transformed into p+ 1 differential equations at the interior collocation points and at the output of the reactor and we obtain ODE's system of order nx(p+ 1) :
(8)
(9)
We shall concentrate in this paper on the control of the concentration of one reactant at the output of the bioreactor, under the following conditions: C1 . The flow rate F is the control input; C2 . The concentration of the controlled reactant is measured not only at the output of the bioreactor but also at each interior collocation point, and at the reactor input (in case of an external substrate) ; C3 . The yield coefficients are positiv~, constant and unknown ; :i C4. m1 ::; m reaction rates are unknown. CS . The dynamics of the nonlinear system (8)(9) is minimum phase (e.g . Kravaris , 1988). Let us call Y the controlled output . By definition, it can be written :
K.I =
K.I
0
0
R.
0
o I
o
0
0
R.I
T T Y_C [X1]_C l' x - 2 ':o2 ,p+1 2
(10)
The dynamical equation of the output Y can then be written as follows : dY = _f. CTB x + f. CT5 ~ dt A 2 p+1 2 A 2 p+1 2,in +
C:R2CP(~1'P+1 ' ~2'P+1)
(11 )
By conSidering condition C4 , the last term of the above equation can be rewritten as follows :
D. Doch ,lill alld
132
.I .
1'. Bahan'
e
=Y
1+1
1+1
TFI T _ -Y +C [B x -b ~ ] I A 2 p+1 2,1 p+1 2,inl
where
o < y<::' 1
The adaptive version of the linearizing control equation is then equal to : i.e. a contains all the unknown parameters.
,.J Therefore , the output dynamical equation takes the following form:
By using first order Euler approximation for dY/dt, equation (12) becomes in discrete-time:
Y
TFI
1+1
= YI + - A
T -
T
C [b ~ - B x ] + Ta
(13)
where the index t is the time index. This equation is the basis for the derivation of the controller.
(1 - A)(Y' - Y ) - Ta
I
TCT [5 ~ 2 p+1 2,inl
I
-B
(16)
p+1 2,1
Comment: the above adaptive linearizing control algorithm (16) is in line with similar control schemes (e.g. Bastin and Dochain, 1988) for which theoretical stability properties have been studied.
Example Let us consider the same process as above . Assume that we wish to control the substrate concentration S at the output of the bioreactor, i.e . Y = Sp+I' Its dynamics is equal to : dS
~=-..E..[b S dt A p+l.0 in
Adaptive Linearizing Control
I
x ]
~1
+
~b p+l,i S]L... i
k
X (17) l J.!P+l p+l
i= 1
Assume that the desired discrete-time closed loop dynamics is equal to the following linear first-order equation:
The Iinearizing command (15) is here written :
(14)
Y' - Y + = A(Y' - Y ) I 1 I
F=A I
where Y' is the desired value of YI .
(1 - A)(Y' - Y) + k J.! X T I 1 p+1,I p+I,1 p+1 -Tb S -T ~ b S p+I,O in,1 L... p+l ,i i,1
(18)
i =1
This may be achieved by implementing the following control input which is calculated by combining equations (13)(14) :
F =A I
T (1-A)(Y'-Y)-Ta
TC T [5 ~ 2 P+ 1 2,inl
I
I
(15)
-B
Assume also that the death parameter kd is known (e.g . from some batch experiments), and that the yield coefficient k, and the specific growth rate J.! are unknown. In this (simple) example, let us further introduce the following definitions :
x ] p+ 1 2,1
From condition C3 and C4 , the parameters a are assumed to be unknown : we shall then estimate them via some adaptation schemes , e .g . a least square
1) J.!p+1 = Up+ISp+1
estimation algorithm with a forgetting factor y :
accordance with the physical reality, that J.!P+l = 0 if SP+I = 0). 2) the auxiliary variable Z = SP+I + k,Xp+, The dynamical behaviour of Z is given by :
with Up+1 an unknown positive function of the bioreactor state (this definition only implies , in
COlltr()1
+"
or
Distributed Parallle te r Biorcart()rs
p+1
dZ __ E. [b S dt - A p+1,Q in
£...
should then be modified so as to take them into account. b
.S.] - k (Z - S ) p+1 ,1 I d p+1
(19) It is also worth noting that the approach proposed in this paper can be easily extended to air lift bioreactors (e.g. Luttman et ai, 1985), at least when the diffusion phenomenons can be neglected.
1=1
or, in discrete-time : TF Z - Z - _ I [b S 1+1 - I A p+1,0 in,1 - k T(Z - S d
I
+"
p+l.I
p+1
£...
i =1
b
.S] p+1,1 1,1
Acknowledgement.
)
The adaptive version of the control scheme (18) is then given by : Y (Z - Y )T (1 - ;\..)(Y' - Y ) + & p+1 ,1 I I I I FI=A p+1 -Tb
(211
S - T" bS p+1,0 in,1 £... p+1,1 1,1 i =1
where Z is calculated from (20) and tile estimate of O:p+1 is updated by the following least square estimation algorithm : TF
p+1
e1+1 = Y1+1 - YI + _A 1 [b p+1 ,0S in,1 + " £...b p+1,1.S.l I, i= 1
(25)
&p+1.I+1 = &p+1,1 + TfIYI (YI- ZI)e 1+1
(26)
f _
l 1
13 ,1
(27)
5. CONCLUSIONS In this paper, we have presented a non linear adaptive control algorithm for fluidized or packed bed bioreactors based on a reduced ODE model which has been deduced by orthogonal collocation methods from the distributed parameter model. The proposed results constitute a very first approach to the control of these bioreactors and many questions remain to be solved. Let us mention some of them . 1) One condition of the proposed alogorithm is that it requires the knowledge of the values of the controlled variable at p+ 1 positions along the bioreactor (i.e., following Wysocki's recommendations, 4 or 5 positions) . They might be replaced, if not measured on-line, by online obseNations. However, as far as we know, the state observation problem, particularly in this situation, is far from being an easy matter. 2) Gas measurements usually represent very valuable information about the state and the kinetics of the process in STR bioreactors and also most probably in diofributed parameter bioreactors. The dynamical model
This work has been carried out during the 4 month stay of D. Dochain as "chercheur associe" of the CNRS (France) at the LAAS from March to June 1989.
REFERENCES BASTIN G. and D. DOCHAIN (1988). "Non linear Adaptive Control Algorithms for Fermentation Processes". Proc. 1988 American Control Conference, Invited Session on Advances in Control of Biotechnical Processes, W.J . Book (Ed.), IEEE Publ. SeN., pp 11241128. DOCHAIN D. (1986). On-line parameter estimation, adaptive state estimation and adaptive control of fermentation processes . Ph. D. thesis, Univ. Cath . Louvain, Belgium . DOCHAIN D., E. DE BUYL and G. BASTIN (1988). "Experimental validation of a methodology.' for on-line state estimation in bioreactors" . 41h International Congress on Computer Applications in Fermentation Technology, N.M. Fish & R.I. Fox (Eds.), Elsevier. DOCHAIN D. and G. BASTIN (1989) . On-line Estimation and Adaptive Control of Bioreactors . Book in preparation. ISAACS S., A. MUNACK and M. THOMA (1986). UseQf orthogonal collocation approximation for parameter identification and control optimization of a distributed parameter biological reactor. AIChE meetrng , Miami Beach. ,i KRAVARIS C. (1988) . Input/output linearization : a non linear analog of placing poles at process zeroes . AIChE Journal, vo1.34, n011 , 1804-1812. LUTTMAN R., A. MUNACK and M. THOMA (1985). Mathematical modelling, parameter identification and adaptive control of single cell protein processes in tower loop bioreactors. Advances in Biochemical Eng ., Springer Verlag, vo1.32, 95-205. MAENDENIUS C.F., B. MATTIASSON , J.P. AXELSSON and P. HAGANDER (1987). Control of an ethanol fermentation carried out with alginate entrapped Saccharomyces cerevisiae. Biotech. & Bioeng. , XXIX, 941-949. PONCELET D., R. BINOT, H. NAVEAU and E.J . NYNS (1985). Biotechnologie des lits fluidises en reacteurs cylindriques et tronconiques. Trib. Cebedeau, 38 (494), 3-12 ; 38 (497), 33-48. PONCE LET D. (1986) . Analyse fondamentale des lits fluidises biologiques. Ph. D. Thesis, Univ. Liege, Belgium.
134
D. Dochain a nd
RAY W .H. (1981) . Adavanced Process Control. Mc Graw-Hill. SCHUGERL K. (1989) . Biofluidization : application of the fluidization technique in biotechnology . Can . J. Chem. Eng., 67,178-184. WILLlAMS D. and D.M. MUNNECKE (1981) . The production of ethanol by immobilized yeast cells. Biotech. & Bioeng ., XXIII, 1813-1825. WYSOCKI M. (1983) . Application of orthogonal collocation to simulation and control of first order hyperbolic systems . Math. and Comp . in Simulation XXV, 335-345.
J.
P. Babary
©
11'.-\(: Control of J)i s tri""t ~ d
Parameter S~-S(l'IIlS. Pt"r pigll ~ lIl.
Fr;I II(T.
I ~IHq
CONTROL OF DISTRIBUTED PARAMETER BIOREACTORS VIA ORTHOGONAL COLLOCATION D. Dochain* t and
J. P. Babary**
*Lllliom /oiJ"" d'AII/OIIIII/iqlll', DYII{[lIIiqlll' 1'1 AII"I.\'sl' rll'S SYS/t'iIlI'S ,
L'lI iJ'eni"; CII //i O/iqlll' rll' L(lIflIlli ll , Btiti III I'll/ ,\lIIX'I'"I/. PIliCl' dll Ln'tll(/, 3, 13-18 LOIn'aill-I,,-S I'lIl/(', Btigilllll *':Llllwm/oi (p rI'AII/OIIIII/iqllf 1'/ rI'Allllly.I'" rlf'S Sys/t'lII I'S rill CNRS, 7, {/l'I' 1I1I1' rill Colollel ROC/il' , 3 107, TOlllolISl', Fmll cl'
t!1l\'ited Professor a t the Ecole Polyt ec hlliqu c de f\.lontreal. Canada
Abstract. This paper deals with the control of fluidized and packed bed bioreactors, The distributed parameter model of the process and its reduction to ordrn,ary dlfferentl~1 equations by using orthogonal collocation is presented. A nonlinear adaptive controller IS designed based on the reduced model , i.e. ordinary differential equalions deduced by orthogonal collocation . Keywords. Distributed parameter systems, non linear systems, orthogonal collocation, bioreactors. 1. INTRODUCTION
also Issacs et ai , 1986) for the case df an air-lift bioreactor. Maendenius et al (1987) also consider the control of a packed bed bioreactor (in cascade with a Continuous STR (CSTR)), but they treated it as a CSTR, which seems fair in that application with respect to the small size of the bioreactor. In this paper, we shall concentrate on fluidized and packed bed bioreactors, which presents the following characterisctics : the active biomass is kept within the vessel and the different substrate(s) and product(s) flow through it. Furthermore, diffusion phenomenons will be coqsidered as negligible. The distributed parameter model of the process will then be constituted of first-order hyperbolic PDE's.
Over the last decades , there has been a growing and widespread development of industrial biotechnological processes. The operation in Stirred Tank Reactors (STR) has been (and is still) a widely used technology in fermentation processes . But, more a~d more , other technologies are considered for bioprocesses operation , such as fluidized bed, packed bed or air lift reactors, which present several advantages over the "classical" STR's (e.g . Schugerl , 1989). For instance, higher cell mass, more efficient product extraction and higher flow rates are often mentioned as advantages of fluidized and packed bed bioreactors. Typical examples are fluidized or packed bed anaerobic digesters (e.g . Poncelet et ai, 1985 ; Poncelet, 1986) and ethanol production with alginate entrapped yeasts (e .g. Williams and Munnecke, 1981).
Basically , there are two different approaches when treating the control of a distributed parameter system (e.g. Ray, 1981). In the first one , the controller design is based on the distributed parameter model and then eventually reduced to a finite dimension control solution. However, the very large uncertainty on the knowledge about the kinetics of fermentation processes (because they are highly complex processes which involve living organisms), renders this approach very difficult. That's why we made up our mind for a second approach, the early lumping. In a first step, the PDE's are reduced to ODE's by using orthogonal collocation and in a second step, the finite dimension ODE system is considered for the controller's design.
In the context of the monitoring and control of fermentation processes, the use of fluidized or packed bed bioreactors introduces new methodological problems to be solved. In fact , the behaviour of these reactors are expected to be more likely in plug flow conditions than in completely mixed ones. Then , strictly speaking, their dynamics cannot be described by ordinary differential equations (ODE's) as in STR's, but by partial derivative equations (PDE's). Very few papers have been concerned in the literature with the control of distributed parameter bioreactors , with the notable exception of Luttman et al (1985)(see
The proposed controller's design is in line with recent works on the control of STR bioreactors (Dochain, 1986; 12!1
D. Dochain and
1:10
.J.
1'. Bail;tl'\
Dochain and Bastin, 1989} and presents the following basic features :
external substrates) ,
- it is based on the well-known nonlinearities of the process ; - it is adaptive in order to deal with the parameter uncertainty.
and z is the space variable (z
The paper is organized as follows. Section 2 introduces the distributed parameter dynamical model of fluidized and packed bed bioreactors. Its reduction to ODE's by orthogonal collocation is presented in section 3. Section 4 is dedicated to the design of an adaptive linearizing control law. We shall concentrate in this paper on the regulation of one output, i.e . of one reactant concentration at the output of the bioreactor.
(dim
10, H]) .
Remark: let us recall that the reaction rates
2. DYNAMICAL MODEL OF FLUIDIZED AND PACKED BED BIOREACTORS
2} a death reaction of the micro-organisms : X---tX d where Xd is the non-active biomass concentration . If we assume that the non active micro-organisms leave the bioreactor, the dynamics of the above process will clearly be described by the equations (1 )(2) with ~, = X,
~2 = [S Xd1T, ~2,in = [Sin 01 T,
,
K = [1 -11
Fig.1. Scheme of fluidized bed reactor Let us consider a fluidized or packed bioreactor (Fig .1) in which m biochemical reactions with n reactants take place . Among these reactants, n, are micro-organisms entrapped or fixed on some support or fluidised and which remain within the reactor, and n2 other reactants (essentially substrates and products) flow through the reactor. For simplicity, we also consider the crosssection of the bioreactor is constant and equal to A. From mass balance considerations, we can deduce , in line with the model formulation in Dochain et al (1987) and Dochain and Bastin (1989), the following dynamical model:
(1 )
(2)
with the following limit conditions: ~ (t, z=O) = ~2 (t) 2 ,In
In the above equations, vector
(dim
~,
~2
2
=[-k, 0
0] 1
k, : yield coefficient
3. REDUCTION OF THE DISTRIBUTED PARAMETER MODEL TO ORDINARY DIFFERENTIAL EQUATIONS It is clear from the above model (1 )(2) that the state variables ~, and ~2 are functions of time and space, i.e. ~,
(t , z) and ~2(t, z). We shall now see how to
approximate the PDE's of each component of ~, and ~2 by a finite number (equal to p+ 1) of ODE's at p+ 1 discrete spatial positions along the bioreactor. There exist several methods to reduce the above PDE's to ODE's (e .g. Ray, 1981). They consist of expanding the variables as a finite sum of products of time functions and space functions. In the orthogonal collocation method (e.g. Wysocki, 1983), the state variables will be written as follows :
(3) ~,
= n,),
is the biomass concentration ~2
is the other reactant
concentration vector (dim ~2 = n2) ' ~2 , in(t} is the influent concentration of
K
(which is different from zero only for
~k(t, z} =
P+'
2. ~i(z)~ki(t}
k=1,2
(4)
i= 0 where ~ki(t) is the value of ~k(t,z} at some discrete spatial
13 1
Control of Distributed Paramete r Bioreactors
positions (called bioreactor, i.e. :
collocation
points)
along
the
(S)
{b.} IJ J=1 to p;.1
0
0
0
{b .. }. IJ J=1 to p+1
0
B.1 =
and the residuals ~i(z) are chosen as orthogonal functions (e.g. Bessel or Lagrange functions), such that :
0
0
{b.} IJ J=1 to p;.1
where 0ij is the delta function (Oij = 1 if i = j; 0ij = 0 if i "# j). P in equation (4) corresponds to the number of interior collocation points. Z = Zo and Z = zp+1 corresponds to the input (z = 0) and the output (z = H) of the bioreactor. Now we can write the partial derivate of ~2 with respect to z appearing in equation (2) as follows :
(6)
Let us note :
b ij
=
d~.(z = J
Comments : 1) an important condition for the model reduction to be valid is that the influent concentration of the reactants ~2' ~2 , in(t) must be C1 (i.e. continuously differentiable) time functions (see Wysocki, 1983). 2) Wysocki (1983) indicates that 3 or 4 interior collocation points are enough to describe the dynamics of first hyperbolic equations like (1 )-(3) .
z.) (7)
I
dz
4. THE CONTROL ALGORITHM Statement of the Control Problem
By introducing (4)-(7) into equations (1 )-(3), each PDE is transformed into p+ 1 differential equations at the interior collocation points and at the output of the reactor and we obtain ODE's system of order nx(p+ 1) :
(8)
(9)
We shall concentrate in this paper on the control of the concentration of one reactant at the output of the bioreactor, under the following conditions: C1 . The flow rate F is the control input; C2 . The concentration of the controlled reactant is measured not only at the output of the bioreactor but also at each interior collocation point, and at the reactor input (in case of an external substrate) ; C3 . The yield coefficients are positiv~, constant and unknown ; :i C4. m1 ::; m reaction rates are unknown. CS . The dynamics of the nonlinear system (8)(9) is minimum phase (e.g . Kravaris , 1988). Let us call Y the controlled output . By definition, it can be written :
K.I =
K.I
0
0
R.
0
o I
o
0
0
R.I
T T Y_C [X1]_C l' x - 2 ':o2 ,p+1 2
(10)
The dynamical equation of the output Y can then be written as follows : dY = _f. CTB x + f. CT5 ~ dt A 2 p+1 2 A 2 p+1 2,in +
C:R2CP(~1'P+1 ' ~2'P+1)
(11 )
By conSidering condition C4 , the last term of the above equation can be rewritten as follows :
D. Doch ,lill alld
132
.I .
1'. Bahan'
e
=Y
1+1
1+1
TFI T _ -Y +C [B x -b ~ ] I A 2 p+1 2,1 p+1 2,inl
where
o < y<::' 1
The adaptive version of the linearizing control equation is then equal to : i.e. a contains all the unknown parameters.
,.J Therefore , the output dynamical equation takes the following form:
By using first order Euler approximation for dY/dt, equation (12) becomes in discrete-time:
Y
TFI
1+1
= YI + - A
T -
T
C [b ~ - B x ] + Ta
(13)
where the index t is the time index. This equation is the basis for the derivation of the controller.
(1 - A)(Y' - Y ) - Ta
I
TCT [5 ~ 2 p+1 2,inl
I
-B
(16)
p+1 2,1
Comment: the above adaptive linearizing control algorithm (16) is in line with similar control schemes (e.g. Bastin and Dochain, 1988) for which theoretical stability properties have been studied.
Example Let us consider the same process as above . Assume that we wish to control the substrate concentration S at the output of the bioreactor, i.e . Y = Sp+I' Its dynamics is equal to : dS
~=-..E..[b S dt A p+l.0 in
Adaptive Linearizing Control
I
x ]
~1
+
~b p+l,i S]L... i
k
X (17) l J.!P+l p+l
i= 1
Assume that the desired discrete-time closed loop dynamics is equal to the following linear first-order equation:
The Iinearizing command (15) is here written :
(14)
Y' - Y + = A(Y' - Y ) I 1 I
F=A I
where Y' is the desired value of YI .
(1 - A)(Y' - Y) + k J.! X T I 1 p+1,I p+I,1 p+1 -Tb S -T ~ b S p+I,O in,1 L... p+l ,i i,1
(18)
i =1
This may be achieved by implementing the following control input which is calculated by combining equations (13)(14) :
F =A I
T (1-A)(Y'-Y)-Ta
TC T [5 ~ 2 P+ 1 2,inl
I
I
(15)
-B
Assume also that the death parameter kd is known (e.g . from some batch experiments), and that the yield coefficient k, and the specific growth rate J.! are unknown. In this (simple) example, let us further introduce the following definitions :
x ] p+ 1 2,1
From condition C3 and C4 , the parameters a are assumed to be unknown : we shall then estimate them via some adaptation schemes , e .g . a least square
1) J.!p+1 = Up+ISp+1
estimation algorithm with a forgetting factor y :
accordance with the physical reality, that J.!P+l = 0 if SP+I = 0). 2) the auxiliary variable Z = SP+I + k,Xp+, The dynamical behaviour of Z is given by :
with Up+1 an unknown positive function of the bioreactor state (this definition only implies , in
COlltr()1
+"
or
Distributed Parallle te r Biorcart()rs
p+1
dZ __ E. [b S dt - A p+1,Q in
£...
should then be modified so as to take them into account. b
.S.] - k (Z - S ) p+1 ,1 I d p+1
(19) It is also worth noting that the approach proposed in this paper can be easily extended to air lift bioreactors (e.g. Luttman et ai, 1985), at least when the diffusion phenomenons can be neglected.
1=1
or, in discrete-time : TF Z - Z - _ I [b S 1+1 - I A p+1,0 in,1 - k T(Z - S d
I
+"
p+l.I
p+1
£...
i =1
b
.S] p+1,1 1,1
Acknowledgement.
)
The adaptive version of the control scheme (18) is then given by : Y (Z - Y )T (1 - ;\..)(Y' - Y ) + & p+1 ,1 I I I I FI=A p+1 -Tb
(211
S - T" bS p+1,0 in,1 £... p+1,1 1,1 i =1
where Z is calculated from (20) and tile estimate of O:p+1 is updated by the following least square estimation algorithm : TF
p+1
e1+1 = Y1+1 - YI + _A 1 [b p+1 ,0S in,1 + " £...b p+1,1.S.l I, i= 1
(25)
&p+1.I+1 = &p+1,1 + TfIYI (YI- ZI)e 1+1
(26)
f _
l 1
13 ,1
(27)
5. CONCLUSIONS In this paper, we have presented a non linear adaptive control algorithm for fluidized or packed bed bioreactors based on a reduced ODE model which has been deduced by orthogonal collocation methods from the distributed parameter model. The proposed results constitute a very first approach to the control of these bioreactors and many questions remain to be solved. Let us mention some of them . 1) One condition of the proposed alogorithm is that it requires the knowledge of the values of the controlled variable at p+ 1 positions along the bioreactor (i.e., following Wysocki's recommendations, 4 or 5 positions) . They might be replaced, if not measured on-line, by online obseNations. However, as far as we know, the state observation problem, particularly in this situation, is far from being an easy matter. 2) Gas measurements usually represent very valuable information about the state and the kinetics of the process in STR bioreactors and also most probably in diofributed parameter bioreactors. The dynamical model
This work has been carried out during the 4 month stay of D. Dochain as "chercheur associe" of the CNRS (France) at the LAAS from March to June 1989.
REFERENCES BASTIN G. and D. DOCHAIN (1988). "Non linear Adaptive Control Algorithms for Fermentation Processes". Proc. 1988 American Control Conference, Invited Session on Advances in Control of Biotechnical Processes, W.J . Book (Ed.), IEEE Publ. SeN., pp 11241128. DOCHAIN D. (1986). On-line parameter estimation, adaptive state estimation and adaptive control of fermentation processes . Ph. D. thesis, Univ. Cath . Louvain, Belgium . DOCHAIN D., E. DE BUYL and G. BASTIN (1988). "Experimental validation of a methodology.' for on-line state estimation in bioreactors" . 41h International Congress on Computer Applications in Fermentation Technology, N.M. Fish & R.I. Fox (Eds.), Elsevier. DOCHAIN D. and G. BASTIN (1989) . On-line Estimation and Adaptive Control of Bioreactors . Book in preparation. ISAACS S., A. MUNACK and M. THOMA (1986). UseQf orthogonal collocation approximation for parameter identification and control optimization of a distributed parameter biological reactor. AIChE meetrng , Miami Beach. ,i KRAVARIS C. (1988) . Input/output linearization : a non linear analog of placing poles at process zeroes . AIChE Journal, vo1.34, n011 , 1804-1812. LUTTMAN R., A. MUNACK and M. THOMA (1985). Mathematical modelling, parameter identification and adaptive control of single cell protein processes in tower loop bioreactors. Advances in Biochemical Eng ., Springer Verlag, vo1.32, 95-205. MAENDENIUS C.F., B. MATTIASSON , J.P. AXELSSON and P. HAGANDER (1987). Control of an ethanol fermentation carried out with alginate entrapped Saccharomyces cerevisiae. Biotech. & Bioeng. , XXIX, 941-949. PONCELET D., R. BINOT, H. NAVEAU and E.J . NYNS (1985). Biotechnologie des lits fluidises en reacteurs cylindriques et tronconiques. Trib. Cebedeau, 38 (494), 3-12 ; 38 (497), 33-48. PONCE LET D. (1986) . Analyse fondamentale des lits fluidises biologiques. Ph. D. Thesis, Univ. Liege, Belgium.
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J.
P. Babary