- Email: [email protected]
Model-Based Adaptive Control of Fed-Batch Fermentation Process with the Substrate Consumption Rate Application
Copyright @ IFAC Adaptive Systems in Control and Signal Processing, Glasgow, Scotland, UK. 1998
MODEL-BASED ADAPTIVE CONTROL OF FED-BATCH FERMENT A nON PROCESS WITH THE SUBSTRATE CONSUMPTION RATE APPLlCA nON
Jacek Czeczot
Institute of Automatic Control. Technical University of Silesia u!. Akademicka 16. 44-100 Gliwice. Poland le!. (48) 32 371473,fax. (48) 32 372127. Email: [email protected]
Abstract: This paper deals with the model-based adaptive predictive control of the fed-batch fermentation process. For this system the theoretical approach to the substrate consumption rate is given and the estimation procedure with application of the recursive least-squares method is proposed. On the basis of this estimated value two nonlinear model-based adaptive predictive controllers for substrate control have been designed. Both estimation procedure and adaptive controllers have been derived only on the basis of the general form of the state equation written for the substrate concentration so there is no necessity to know the form of the nonlinear part of the model of the system. The estimation accuracy and the control performance have been demonstrated by means of computer simulation. Copyright @ 1998 IFAC
Keywords: Biotechnology, Fennentation process, Adaptive control, Model-based control, Least-squares estimation, Recursive least-squares.
1. INTRODUCTION
in this paper but the control strategy has been derived on the basis of the phenomenological model of the process (the general fonn of the state equation written for the quantity being to be controlled).
The idea of the model-based control of dynamic systems has been the object of growing interest for last several years. In opposite to the case when blackbox linear approximate models are used to implement the control law, in this case the nonlinear phenomenological model in the fonn of state equations is the basis for deriving the model-based control algorithm . This idea was experimentally investigated in (Joshi. et al. , 1997) and its benefits were pointed out in comparison with the other nonlinear control strategies.
The perfectly mixed bioreactor. in which the fedbatch process (bacterial production of lysine) takes place, is considered as the example of the "true" bacterial system to be controlled. Because in practical applications of such systems the yield-productivity conflict occurs, there is a need to manage this problem by means of properly designed control system. In (Bastin and Dochain. 1990) and (Van lmpre and Bastin. 1995) it was shown that effective control of such systems consists in regulating the outlet substrate concentration at the set-point value despite the disturbances changes for the time of production with application of the feedback control loop. The optimal trajectory of the set-point value S sp is assumed to be known from preliminary technological considerations and the problem consists
In the case of nonstationary systems the adaptation should be implemented in the control law. In (Richalet. 1993) it is shown how to derive the modelbased predictive controller on the basis of the simplified model of the process, given in the fonn of the transfer function , and how to improve the control performance by means of adaptation and prediction techniques. The same idea is taken into consideration
321
in deriving the adaptive control algorithm which ensures the best possible tracking performance.
v O~S(t ) ~~ VI
V
VI
where:
Let us consider the general form of the state equation (2) describing the dynamic of the outlet substrate concentration. dS(t) = D(t) (Sin (t) - S(t») - R(t) dt
dt
dS(t) dt
= D(t)(S in (t ) -S(t))- k J!l(t)X(t)-
In our example the value of R(t) can be expressed as follows .
(I)
As it can be seen, the value of R(t) depends on the biomass concentration X(t) and on the time-varying parameters !let) and vet).
(2)
-k 2 v(t)X(t)
dP(n
- -' = v(t)X(t) dt
D(t)P(t)
4. SUBSTRA TE CONTROL (3)
Let us consider the problem of regulating the substrate concentration Set) at the set-point Sw In this paper the model-based adaptive predictive controller is applied for this purpose. Although the phenomenological model of the system consists of three nonlinear ordinary differential equations, the control law is designed only on the basis of the general form of the state equation written for the substrate concentration - see equation (6).
with a Monod (1949) model for the specific growth rate !l [I ih]
!let)
=!lmax
Set) KM +S(t)
(6)
In this equation R(t) [gll h] denotes the time-varying parameter which stands for the nonlinear part of the equation (2), describing the biological reaction taking place inside the reactor. Let us call this parameter substrate consumption rate (according to the fact that it describes the consumption of the substrate due to the biological reaction) and note that its value characterizes the intensity of the reaction. This parameter can be applied for the monitoring of biotechnological processes - e.g. see the nitrificationdenitrification process (Isaacs, et al., 1992).
As an example of the 'true' bacterial system the fedbatch fermentation process (bacterial production of amino acids, e.g. lysine) is considered in this paper. Its mathematical model (Bastin and Dochain, 1990) consists of three non linear state equations describing respectively biomass concentration X [gill, substrate concentration S [gill and product concentration P [gill, outcoming from the reactor, dX(t)
D - dilution rate [l ih] Sm - inlet substrate concentration [gill kJ. k2 - yield coefficients [ - ] Ilmax - maximum specific growth rate [I Ih 1 KM - saturation constant [gll] vo, V I - constant parameters
3. THEORETICAL APPROACH TO THE SUBSTRA TE CONSUMPTION RATE
2. DESCRIPTION OF THE SYSTEM
- - = !l(t)X(t) - D(t)X(t)
(5)
S( t) > --...2..
This paper is organized in the following way. First the mathematical model of the system is given. Then the theoretical approach to the substrate consumption rate as to an important parameter characterizing the intensity of the biological reaction is presented. In the next section the model-based predictive controllers for substrate control are proposed. One of them applies the dilution rate and the other the inlet substrate concentration as control quantity and both are derived on the basis of the general form of the state equation written for the outlet substrate concentration. Since the value of the substrate consumption rate must be known to calculate the control quantities and it cannot be measured directly, the procedure for on-line estimation with application of the recursive least-squares method has been proposed by author (Czeczot, 1997, I 997a, 1998) and is presented in the next section. Then on the basis of the estimated value of the substrate consumption rate the adaptive form of the control laws is proposed. Finally, the simulation results show the high accuracy of the proposed estimation procedure and very good control performance of the adaptive controllers in comparison with a conventional PI controller. Concluding remarks complete the paper.
(4)
and a parabolic specific production rate v [I Ih] .
Since it is assumed that substrate concentration measurements are accessible only at discrete 322
In (Bastin and Dochain. 1990). (Bastin. 1991) and (Dochain. 1992) the following approach to this problem was presented. For the same example of a system as the one considered in this paper the fonn of the equation (7) was simplified assuming that the value of k2 '" O. Such an assumption can be made in cases when the inequality k, » k2 occurs. Then the value of k, was assumed to be known from previous off-line identification experiments and non linear observer for the biomass concentration X(t) was designed on the basis of the fonn of the mathematical model of the system - equations (I) - (3) . Finally, the value of the parameter l1(t) was estimated on-line as the time-varying parameter by means of the recursive least-squares method.
moments of time. the control law has to have the fonn of a discrete-time equation. Thus let us rewrite the equation (6) in the discrete fonn (the subscript i denotes a discrete-time index).
Basing on the equation presented above a one-step ahead prediction of the substrate concentration Si +' can be expressed as follows .
with T R [min] - sampling time
For that approach some drawbacks can be pointed out. That idea cannot be applied to a system with unknown fonn of the equation (7), describing the value of the substrate consumption rate. Such a problem frequently occurs in practice. Moreover, in some cases fonn of mathematical model of a system (especially its non linear part) as well as values of model parameters can be unknown and thus the equation (7) cannot be simplified in any way.
In order to obtain the control law there is a need to set a one-step ahead prediction of the substrate concentration equal to the set point (Dochain and Bastin, 1984). (10) The outlet substrate concentration Set) can be controlled by means of manipulating the value of the dilution rate D(t) or the value of the inlet substrate concentration Sin(t). Both cases are taken into consideration and therefore two nonlinear modelbased predictive controllers have been derived for this purpose by combining the equations (9) and (10).
There is also another approach to this problem (Dochain, 1991). For a class of biotechnological processes it is possible to propose the adaptive controllers for which no feedback from the value of the substrate consumption rate is needed but that idea consists in reducing of the order of a model and then in space transfonnation. In result there is no necessity to know the fonn of the equation describing the value of R(t) but there is a need to know the fonn of a model to reduce its order and to carry out the space transfonnation. Thus it is impossible to apply that idea in cases when complete mathematical model of a system remains unknown .
(11)
(12)
Therefore, in this paper different approach is presented. This idea was proposed by Czeczot (1997 , I997a, 1998) and it bases only on the fonn of the equation (8) which is simply a mass balance equation .
In these control laws the values of all parameters are measurable on-line (which provides the feedforward action) or are known and constant except the substrate consumption rate value Ri. This value cannot be measured directly and a special procedure for estimating it on-line at discrete moments of time must be proposed.
If the derivative of the substrate concentration with respect to time is approximated by means of the twopoint finite difference fonnula, as it is shown below. dS(t») ( dt i
5. ESTIMATION OF THE SUBSTRA TE CONSUMPTION RATE
(13)
the following equation can be proposed.
As it was said in the previous section. in order to implement the adaptation in the controllers (11) and (12) there is a need to know the value of the substrate consumption rate Ri at discrete moments of time. Because of its physical nature this value cannot be measured directly and therefore the estimation procedure must be proposed.
Let us define the auxiliary measurable on-line parameter.
323
time-varying
and
the control laws presented in this paper. In order to obtain appropriate equations. there is a necessity to substitute the estimated value of the substrate consumption rate for its "true" value in the equations (11) and (12) (Czeczot. 1997. 1997a). Thus the following adaptive predictive controllers can be obtained.
Finally, the equation (14) can be rearranged in the following way. (16 )
Since the equation presented above is linear with respect to the parameter Ri that is to be estimated it is possible to propose the estimation procedure on the basis of the recursive least-squares method.
( 19)
(17)
(20)
It can be seen that the control laws are calculated only on the basis of the measurements of the inlet and outlet substrate concentration and of the dilution rate. These quantities are easily measured on-line. The adaptive nature of both controllers consists in application of the estimated value of the substrate consumption rate.
with: Ri
a
E
- estimated value of the consumption rate (0, I) - forgetting factor
substrate
7. SIMULATION RESULTS
Because there is only one parameter to estimate, the well-known equations (17) and (18) can be written as scalar equations. Moreover, the only "measured" value correlated with the estimated parameter is sampling time T R and its value can be considered to be constant and known. The application of the recursive least-squares method with forgetting factor a allows the estimation procedure to be resistant to the measurement noise by adjusting the value of a (Czeczot, 1998). These features allow us to expect very good estimation accuracy without any assumptions dealing with excitation signals.
In order to investigate the approach to the adaptive substrate control presented in this paper the successful simulation experiments have been carried out and the most interesting results are presented in this section. The following values of the parameters have been chosen for simulation: Ilm.x = 0.35, KM = kl = 0.4, k2 = 0.05, vo = 15, v I = 5. The steady state of the system is characterized by the following values of the parameters: Sin = 60, D = 0.2. X(O) = 29.1, S(O) = 0.53, P(O) = 956,6. The sampling time for both the estimation of the substrate consumption rate and the substrate control has been chosen as T R = 5 [min) .
Let us also note that the estimation procedure, considered in this paper, bases only on the current and previous measurements of the inlet and outlet substrate concentration and thus there is no need to make any assumptions dealing with the form of the equation describing the substrate consumption rate. Moreover, this estimation procedure is based only on the general form of the state equation written for the substrate concentration - see equation (8). Therefore the form of the nonlinear part of the mathematical model is no longer needed to be known and the model calibration and validation are no longer needed to be carried out.
Figure I shows the high accuracy of the estimation of the substrate consumption rate for two different values of the forgetting factor a in the presence of the step changes of the values of the parameters. The initial value of the substrate consumption rate has been chosen as Ro = O. Although, as it can be expected, for a = 0. 1 we observe much faster convergence of the estimated value of this parameter to its "true" value than for a = 0.8, in both cases there is no bias in the steady state. As it was said before. increasing the value of a allows the estimation procedure to be more resistant to measurement noise. Since in this paper the noiseless case is considered. further simulation results have been obtained for a = 0.1.
6. APPLICATION OF THE SUBSTRA TE CONSUMPTION RATE TO THE ADAPTIVE SUBSTRATECONTROL
In the left column of the figure 2 the control performance of the adaptive controller ( 19) with the dilution rate as the control quantity is presented. Simulation starts with the process being in the steady
The procedure for on-line estimation of the substrate consumption rate, described in details in the previous section, allows us to propose the adaptive version of
324
accuracy of the suOstrate consumption rate estimation
accuracy of the substrate consumption rate estimation 30m---~----~----r_--~----~----~--~----~
30 mr--------~--------~----
__--__----r_--~ Cl ~
a~OI
R
25
25
! ~
20
6 [hI
~" ! ~
15
t
20
2 [hI
5 .. 60 ---> 50
~
/) [hI
Um.\
l 15~~
0 .35 ---> 0.7
0.8
R
i
035---tO-;
= : [hI
..__ 5._.• _6--,0,---> 50
10
10 t = 4 [hI D 0 .2 -> 0.3
5
o~--~--~----~--~----~--~----~--~
o
t = 4 [hI D 0 .2 -> 0 .3
5
O~--~--~~--~--~----~--~----~--~
23456780
345
2
time[h]
6
7
8
time[h]
Fig. I. Accuracy of the estimation of the substrate consumption rate for different values offorgetting factor a.
substrate concentration S(t)
5ubstrate concentration S(t)
1.2,-----r-----,.---..-----.-----,.-------,
4,----,-----,.---~--
adaptive controller
v
3.5
1.1
3
I t = 20 [h)
0.9
2.5
Son : 50 -> 60 o
= 40 [h)
o o o o o o o
t
[hI S.. : 0.53 -> l.l
t= 5
1.5
0.7
1U'
PI controller
adapltve controller
= 20 [h)
D ' 50 ->
2
Il- . 0.35 -> 0.5
o
0.8
t
__-----r_--__.
6.~,// _- ----t =
.'
PI controller
40 [hI 035 -> 0.5
__':~--------I._ ..""'" o
~- .. - .. ----.--~!
0.6
t
_
= 5 [hI
0.5
s..
S .. . 0.53 -> Ll
S'"
0 . 5L----~--~--~-----I.---~--.-.J
o
10
20
30
40
50
60
O~--~--~-------I.----~--~--~
o
10
20
30
time[h]
40
50
60
time[h] control quanttty Sin(t)
control quantity O(t)
O. 38 r-----.....,...------~----__..------~----~------..,
250,----,-----,.----..-----.-----,.----, adaptive controller
ad apltve controller
0.36 200
0.34
[hI Son ' 50 -> 60
t = 20
0.32
150 0.3 0.28
100
t = 40 [h) 1J.m" . 0.35 -> 0.5
0.26 0.24
PI controller
[hI S" 053 -> I I
10
20
0 35~05
PI controller
0 . 2L---'--~--~--~--~-------~---~
o
Ilma\
S,' 053 -> 1.1
t ~ 5
0.22
-'-"oC·'" t ~ 40 [hI
t~5[h]
50
30
40
50
60
time[h]
O~----'------'---~----~----'-----'
0
10
20
30
40
50
time[h]
Fig. 2. Control performance of the adaptive controllers in comparison with a conventional PI controller. Left column - dilution rate D(t) as the control quantity. Right column - inlet substrate concentration SIn(t) as the control quantity .
325
60
state and. since it was found that the best productivity can be obtained for Set) = 1.1. at time t = 5 [h] the feedback control loop is closed with the set-point chosen as S'P = 1.1. In order to examine the influence of the disturbances changes on the control performance. additional step changes are applied on the parameters Sm and ~.x. In similar way the right colwnn of the figure 2 shows the control performance of the adaptive controller (20) with the inlet substrate concentration as the control quantity . After time t = 5 rh], when the feedback control loop is closed with the set-point S,p = I. I, the step changes are applied on the parameters D and ~ax .
Acknowledgment: This work was supported Polish Committee of Scientific Research (KBN).
by
REFERENCES Bastin G .. Dochain D. (1990). On-line estimation and adaptive control of bioreaClOrs. Elsevier Science Publishers B.V. , ISBN 0-444-88430-0. Bastin G. (1991). Nonlinear and adaptive control in biotechnology. A tutorial. European Control Conference ECC'91, Grenoble , France. July 2-5 . 1991. Czeczot J. (1997). On possibility of the application of the substrate consumption rate to the monitoring and control of the water purification processes. Ph.D. Thesis, Technical University of Silesia, Gliwice, Poland. (in polish). Czeczot 1. (I 997a). Substrate consumption rate application to the minimal-cost model-based adaptive control of the activated sludge process. In: Preprints of 71h International Workshop On Instrumentation. Control and Automation of Water and Wastewater Treatment and Transport Systems, pp. 327-334,Brighton, United Kingdom. Czeczot J. (1998). Application of the recursive leastsquares method to the estimation of the substrate consumption rate in the activated sludge process. In: Proceedings of the 91h International Symposium on "System-Modelling-Control", ed. P.S. Szczepaniak, Zakopane, Poland. Dochain D., Bastin G. (1984). Adaptive identification and control algorithms for nonlinear bacterial growth systems. Automatica, 20 , No . 5, pp. 621634. Dochain D. (1991). Design of adaptive controllers for non-linear stirred tank bioreactors: extension to the MIMO situation. 1. Proc. Cont .. 1, 1991. Dochain D. (1992). Adaptive control algorithms for non-mInimum phase nonlinear bioreactors. Computers Chem. Engng., 16, No. 5, pp. 449469. Isaacs. S.H ., Soeberg H., Kummel M. (1992). Monitoring and control of a biological nutrient removal processes: rate data as a source of information. IF AC Modelling and Control of Biotechnological Processes, Colorado, USA, pp. 239-242. Joshi N.V. , Murugan P., Rhinehart R.R. (1997). Experimental comparison of control strategies. Control Eng. Practice. 5, No . 7, pp. 885-896. Monod J. (1949). The growth of bacterial cultures. Ann. Rev. Microbial, 3, pp. 371-394 Richalet J. (1993). Industrial applications of model based predictive control. Automatica, 29. No. 5. pp. i251-1274. Van Impre J.F., Bastin G. (1995). Optimal adaptive control of fed-batch fermentation processes. Control Eng. Practice. 3. No . 7. pp. 939-954 .
In both cases we observe very good control performance of the considered adaptive controllers. The controlled output converges much faster for these controllers being in use than for conventional PI controllers applied for the same purpose. The adaptive controllers manipulates the control quantities smoothly, which is very important from practical point of view, and provide very short regulation time without overshooting.
8. CONCLUSIONS In this paper the simple model-based adaptive controllers for the fed-batch fermentation process have been proposed and their effectiveness has been demonstrated by means of simulation experiments. The proposed control laws have been derived on the basis of the phenomenological model of the process. However, the application of the substrate consumption rate allows us to avoid problems with estimation of the parameters of the nonlinear part of the mathematical model. Since both the procedure for estimation of the substrate consumption rate and the adaptive controllers have been derived only on the basis of the general form of the state equation written for the substrate concentration, there is no need to know anything about the nonlinear part of the mathematical model describing the intensity of the reaction . Moreover, there is no necessity to know or to estimate the values of the parameters such as, for instant, yield coefficients, biomass concentration or specific growth and production rates. These features allow this idea to be easily applicable in practice. The form of the state equation that is a basis for the idea presented in this paper is very general and therefore it satisfies every biological and chemical reaction as well as heat transfer processes. This feature allows us to expect the idea of the substrate consumption rate and based on its value adaptive controllers to be easily applicable for control of chemical and thermal processes.
326
MODEL-BASED ADAPTIVE CONTROL OF FED-BATCH FERMENT A nON PROCESS WITH THE SUBSTRATE CONSUMPTION RATE APPLlCA nON
Jacek Czeczot
Institute of Automatic Control. Technical University of Silesia u!. Akademicka 16. 44-100 Gliwice. Poland le!. (48) 32 371473,fax. (48) 32 372127. Email: [email protected]
Abstract: This paper deals with the model-based adaptive predictive control of the fed-batch fermentation process. For this system the theoretical approach to the substrate consumption rate is given and the estimation procedure with application of the recursive least-squares method is proposed. On the basis of this estimated value two nonlinear model-based adaptive predictive controllers for substrate control have been designed. Both estimation procedure and adaptive controllers have been derived only on the basis of the general form of the state equation written for the substrate concentration so there is no necessity to know the form of the nonlinear part of the model of the system. The estimation accuracy and the control performance have been demonstrated by means of computer simulation. Copyright @ 1998 IFAC
Keywords: Biotechnology, Fennentation process, Adaptive control, Model-based control, Least-squares estimation, Recursive least-squares.
1. INTRODUCTION
in this paper but the control strategy has been derived on the basis of the phenomenological model of the process (the general fonn of the state equation written for the quantity being to be controlled).
The idea of the model-based control of dynamic systems has been the object of growing interest for last several years. In opposite to the case when blackbox linear approximate models are used to implement the control law, in this case the nonlinear phenomenological model in the fonn of state equations is the basis for deriving the model-based control algorithm . This idea was experimentally investigated in (Joshi. et al. , 1997) and its benefits were pointed out in comparison with the other nonlinear control strategies.
The perfectly mixed bioreactor. in which the fedbatch process (bacterial production of lysine) takes place, is considered as the example of the "true" bacterial system to be controlled. Because in practical applications of such systems the yield-productivity conflict occurs, there is a need to manage this problem by means of properly designed control system. In (Bastin and Dochain. 1990) and (Van lmpre and Bastin. 1995) it was shown that effective control of such systems consists in regulating the outlet substrate concentration at the set-point value despite the disturbances changes for the time of production with application of the feedback control loop. The optimal trajectory of the set-point value S sp is assumed to be known from preliminary technological considerations and the problem consists
In the case of nonstationary systems the adaptation should be implemented in the control law. In (Richalet. 1993) it is shown how to derive the modelbased predictive controller on the basis of the simplified model of the process, given in the fonn of the transfer function , and how to improve the control performance by means of adaptation and prediction techniques. The same idea is taken into consideration
321
in deriving the adaptive control algorithm which ensures the best possible tracking performance.
v O~S(t ) ~~ VI
V
VI
where:
Let us consider the general form of the state equation (2) describing the dynamic of the outlet substrate concentration. dS(t) = D(t) (Sin (t) - S(t») - R(t) dt
dt
dS(t) dt
= D(t)(S in (t ) -S(t))- k J!l(t)X(t)-
In our example the value of R(t) can be expressed as follows .
(I)
As it can be seen, the value of R(t) depends on the biomass concentration X(t) and on the time-varying parameters !let) and vet).
(2)
-k 2 v(t)X(t)
dP(n
- -' = v(t)X(t) dt
D(t)P(t)
4. SUBSTRA TE CONTROL (3)
Let us consider the problem of regulating the substrate concentration Set) at the set-point Sw In this paper the model-based adaptive predictive controller is applied for this purpose. Although the phenomenological model of the system consists of three nonlinear ordinary differential equations, the control law is designed only on the basis of the general form of the state equation written for the substrate concentration - see equation (6).
with a Monod (1949) model for the specific growth rate !l [I ih]
!let)
=!lmax
Set) KM +S(t)
(6)
In this equation R(t) [gll h] denotes the time-varying parameter which stands for the nonlinear part of the equation (2), describing the biological reaction taking place inside the reactor. Let us call this parameter substrate consumption rate (according to the fact that it describes the consumption of the substrate due to the biological reaction) and note that its value characterizes the intensity of the reaction. This parameter can be applied for the monitoring of biotechnological processes - e.g. see the nitrificationdenitrification process (Isaacs, et al., 1992).
As an example of the 'true' bacterial system the fedbatch fermentation process (bacterial production of amino acids, e.g. lysine) is considered in this paper. Its mathematical model (Bastin and Dochain, 1990) consists of three non linear state equations describing respectively biomass concentration X [gill, substrate concentration S [gill and product concentration P [gill, outcoming from the reactor, dX(t)
D - dilution rate [l ih] Sm - inlet substrate concentration [gill kJ. k2 - yield coefficients [ - ] Ilmax - maximum specific growth rate [I Ih 1 KM - saturation constant [gll] vo, V I - constant parameters
3. THEORETICAL APPROACH TO THE SUBSTRA TE CONSUMPTION RATE
2. DESCRIPTION OF THE SYSTEM
- - = !l(t)X(t) - D(t)X(t)
(5)
S( t) > --...2..
This paper is organized in the following way. First the mathematical model of the system is given. Then the theoretical approach to the substrate consumption rate as to an important parameter characterizing the intensity of the biological reaction is presented. In the next section the model-based predictive controllers for substrate control are proposed. One of them applies the dilution rate and the other the inlet substrate concentration as control quantity and both are derived on the basis of the general form of the state equation written for the outlet substrate concentration. Since the value of the substrate consumption rate must be known to calculate the control quantities and it cannot be measured directly, the procedure for on-line estimation with application of the recursive least-squares method has been proposed by author (Czeczot, 1997, I 997a, 1998) and is presented in the next section. Then on the basis of the estimated value of the substrate consumption rate the adaptive form of the control laws is proposed. Finally, the simulation results show the high accuracy of the proposed estimation procedure and very good control performance of the adaptive controllers in comparison with a conventional PI controller. Concluding remarks complete the paper.
(4)
and a parabolic specific production rate v [I Ih] .
Since it is assumed that substrate concentration measurements are accessible only at discrete 322
In (Bastin and Dochain. 1990). (Bastin. 1991) and (Dochain. 1992) the following approach to this problem was presented. For the same example of a system as the one considered in this paper the fonn of the equation (7) was simplified assuming that the value of k2 '" O. Such an assumption can be made in cases when the inequality k, » k2 occurs. Then the value of k, was assumed to be known from previous off-line identification experiments and non linear observer for the biomass concentration X(t) was designed on the basis of the fonn of the mathematical model of the system - equations (I) - (3) . Finally, the value of the parameter l1(t) was estimated on-line as the time-varying parameter by means of the recursive least-squares method.
moments of time. the control law has to have the fonn of a discrete-time equation. Thus let us rewrite the equation (6) in the discrete fonn (the subscript i denotes a discrete-time index).
Basing on the equation presented above a one-step ahead prediction of the substrate concentration Si +' can be expressed as follows .
with T R [min] - sampling time
For that approach some drawbacks can be pointed out. That idea cannot be applied to a system with unknown fonn of the equation (7), describing the value of the substrate consumption rate. Such a problem frequently occurs in practice. Moreover, in some cases fonn of mathematical model of a system (especially its non linear part) as well as values of model parameters can be unknown and thus the equation (7) cannot be simplified in any way.
In order to obtain the control law there is a need to set a one-step ahead prediction of the substrate concentration equal to the set point (Dochain and Bastin, 1984). (10) The outlet substrate concentration Set) can be controlled by means of manipulating the value of the dilution rate D(t) or the value of the inlet substrate concentration Sin(t). Both cases are taken into consideration and therefore two nonlinear modelbased predictive controllers have been derived for this purpose by combining the equations (9) and (10).
There is also another approach to this problem (Dochain, 1991). For a class of biotechnological processes it is possible to propose the adaptive controllers for which no feedback from the value of the substrate consumption rate is needed but that idea consists in reducing of the order of a model and then in space transfonnation. In result there is no necessity to know the fonn of the equation describing the value of R(t) but there is a need to know the fonn of a model to reduce its order and to carry out the space transfonnation. Thus it is impossible to apply that idea in cases when complete mathematical model of a system remains unknown .
(11)
(12)
Therefore, in this paper different approach is presented. This idea was proposed by Czeczot (1997 , I997a, 1998) and it bases only on the fonn of the equation (8) which is simply a mass balance equation .
In these control laws the values of all parameters are measurable on-line (which provides the feedforward action) or are known and constant except the substrate consumption rate value Ri. This value cannot be measured directly and a special procedure for estimating it on-line at discrete moments of time must be proposed.
If the derivative of the substrate concentration with respect to time is approximated by means of the twopoint finite difference fonnula, as it is shown below. dS(t») ( dt i
5. ESTIMATION OF THE SUBSTRA TE CONSUMPTION RATE
(13)
the following equation can be proposed.
As it was said in the previous section. in order to implement the adaptation in the controllers (11) and (12) there is a need to know the value of the substrate consumption rate Ri at discrete moments of time. Because of its physical nature this value cannot be measured directly and therefore the estimation procedure must be proposed.
Let us define the auxiliary measurable on-line parameter.
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time-varying
and
the control laws presented in this paper. In order to obtain appropriate equations. there is a necessity to substitute the estimated value of the substrate consumption rate for its "true" value in the equations (11) and (12) (Czeczot. 1997. 1997a). Thus the following adaptive predictive controllers can be obtained.
Finally, the equation (14) can be rearranged in the following way. (16 )
Since the equation presented above is linear with respect to the parameter Ri that is to be estimated it is possible to propose the estimation procedure on the basis of the recursive least-squares method.
( 19)
(17)
(20)
It can be seen that the control laws are calculated only on the basis of the measurements of the inlet and outlet substrate concentration and of the dilution rate. These quantities are easily measured on-line. The adaptive nature of both controllers consists in application of the estimated value of the substrate consumption rate.
with: Ri
a
E
- estimated value of the consumption rate (0, I) - forgetting factor
substrate
7. SIMULATION RESULTS
Because there is only one parameter to estimate, the well-known equations (17) and (18) can be written as scalar equations. Moreover, the only "measured" value correlated with the estimated parameter is sampling time T R and its value can be considered to be constant and known. The application of the recursive least-squares method with forgetting factor a allows the estimation procedure to be resistant to the measurement noise by adjusting the value of a (Czeczot, 1998). These features allow us to expect very good estimation accuracy without any assumptions dealing with excitation signals.
In order to investigate the approach to the adaptive substrate control presented in this paper the successful simulation experiments have been carried out and the most interesting results are presented in this section. The following values of the parameters have been chosen for simulation: Ilm.x = 0.35, KM = kl = 0.4, k2 = 0.05, vo = 15, v I = 5. The steady state of the system is characterized by the following values of the parameters: Sin = 60, D = 0.2. X(O) = 29.1, S(O) = 0.53, P(O) = 956,6. The sampling time for both the estimation of the substrate consumption rate and the substrate control has been chosen as T R = 5 [min) .
Let us also note that the estimation procedure, considered in this paper, bases only on the current and previous measurements of the inlet and outlet substrate concentration and thus there is no need to make any assumptions dealing with the form of the equation describing the substrate consumption rate. Moreover, this estimation procedure is based only on the general form of the state equation written for the substrate concentration - see equation (8). Therefore the form of the nonlinear part of the mathematical model is no longer needed to be known and the model calibration and validation are no longer needed to be carried out.
Figure I shows the high accuracy of the estimation of the substrate consumption rate for two different values of the forgetting factor a in the presence of the step changes of the values of the parameters. The initial value of the substrate consumption rate has been chosen as Ro = O. Although, as it can be expected, for a = 0. 1 we observe much faster convergence of the estimated value of this parameter to its "true" value than for a = 0.8, in both cases there is no bias in the steady state. As it was said before. increasing the value of a allows the estimation procedure to be more resistant to measurement noise. Since in this paper the noiseless case is considered. further simulation results have been obtained for a = 0.1.
6. APPLICATION OF THE SUBSTRA TE CONSUMPTION RATE TO THE ADAPTIVE SUBSTRATECONTROL
In the left column of the figure 2 the control performance of the adaptive controller ( 19) with the dilution rate as the control quantity is presented. Simulation starts with the process being in the steady
The procedure for on-line estimation of the substrate consumption rate, described in details in the previous section, allows us to propose the adaptive version of
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accuracy of the suOstrate consumption rate estimation
accuracy of the substrate consumption rate estimation 30m---~----~----r_--~----~----~--~----~
30 mr--------~--------~----
__--__----r_--~ Cl ~
a~OI
R
25
25
! ~
20
6 [hI
~" ! ~
15
t
20
2 [hI
5 .. 60 ---> 50
~
/) [hI
Um.\
l 15~~
0 .35 ---> 0.7
0.8
R
i
035---tO-;
= : [hI
..__ 5._.• _6--,0,---> 50
10
10 t = 4 [hI D 0 .2 -> 0.3
5
o~--~--~----~--~----~--~----~--~
o
t = 4 [hI D 0 .2 -> 0 .3
5
O~--~--~~--~--~----~--~----~--~
23456780
345
2
time[h]
6
7
8
time[h]
Fig. I. Accuracy of the estimation of the substrate consumption rate for different values offorgetting factor a.
substrate concentration S(t)
5ubstrate concentration S(t)
1.2,-----r-----,.---..-----.-----,.-------,
4,----,-----,.---~--
adaptive controller
v
3.5
1.1
3
I t = 20 [h)
0.9
2.5
Son : 50 -> 60 o
= 40 [h)
o o o o o o o
t
[hI S.. : 0.53 -> l.l
t= 5
1.5
0.7
1U'
PI controller
adapltve controller
= 20 [h)
D ' 50 ->
2
Il- . 0.35 -> 0.5
o
0.8
t
__-----r_--__.
6.~,// _- ----t =
.'
PI controller
40 [hI 035 -> 0.5
__':~--------I._ ..""'" o
~- .. - .. ----.--~!
0.6
t
_
= 5 [hI
0.5
s..
S .. . 0.53 -> Ll
S'"
0 . 5L----~--~--~-----I.---~--.-.J
o
10
20
30
40
50
60
O~--~--~-------I.----~--~--~
o
10
20
30
time[h]
40
50
60
time[h] control quanttty Sin(t)
control quantity O(t)
O. 38 r-----.....,...------~----__..------~----~------..,
250,----,-----,.----..-----.-----,.----, adaptive controller
ad apltve controller
0.36 200
0.34
[hI Son ' 50 -> 60
t = 20
0.32
150 0.3 0.28
100
t = 40 [h) 1J.m" . 0.35 -> 0.5
0.26 0.24
PI controller
[hI S" 053 -> I I
10
20
0 35~05
PI controller
0 . 2L---'--~--~--~--~-------~---~
o
Ilma\
S,' 053 -> 1.1
t ~ 5
0.22
-'-"oC·'" t ~ 40 [hI
t~5[h]
50
30
40
50
60
time[h]
O~----'------'---~----~----'-----'
0
10
20
30
40
50
time[h]
Fig. 2. Control performance of the adaptive controllers in comparison with a conventional PI controller. Left column - dilution rate D(t) as the control quantity. Right column - inlet substrate concentration SIn(t) as the control quantity .
325
60
state and. since it was found that the best productivity can be obtained for Set) = 1.1. at time t = 5 [h] the feedback control loop is closed with the set-point chosen as S'P = 1.1. In order to examine the influence of the disturbances changes on the control performance. additional step changes are applied on the parameters Sm and ~.x. In similar way the right colwnn of the figure 2 shows the control performance of the adaptive controller (20) with the inlet substrate concentration as the control quantity . After time t = 5 rh], when the feedback control loop is closed with the set-point S,p = I. I, the step changes are applied on the parameters D and ~ax .
Acknowledgment: This work was supported Polish Committee of Scientific Research (KBN).
by
REFERENCES Bastin G .. Dochain D. (1990). On-line estimation and adaptive control of bioreaClOrs. Elsevier Science Publishers B.V. , ISBN 0-444-88430-0. Bastin G. (1991). Nonlinear and adaptive control in biotechnology. A tutorial. European Control Conference ECC'91, Grenoble , France. July 2-5 . 1991. Czeczot J. (1997). On possibility of the application of the substrate consumption rate to the monitoring and control of the water purification processes. Ph.D. Thesis, Technical University of Silesia, Gliwice, Poland. (in polish). Czeczot 1. (I 997a). Substrate consumption rate application to the minimal-cost model-based adaptive control of the activated sludge process. In: Preprints of 71h International Workshop On Instrumentation. Control and Automation of Water and Wastewater Treatment and Transport Systems, pp. 327-334,Brighton, United Kingdom. Czeczot J. (1998). Application of the recursive leastsquares method to the estimation of the substrate consumption rate in the activated sludge process. In: Proceedings of the 91h International Symposium on "System-Modelling-Control", ed. P.S. Szczepaniak, Zakopane, Poland. Dochain D., Bastin G. (1984). Adaptive identification and control algorithms for nonlinear bacterial growth systems. Automatica, 20 , No . 5, pp. 621634. Dochain D. (1991). Design of adaptive controllers for non-linear stirred tank bioreactors: extension to the MIMO situation. 1. Proc. Cont .. 1, 1991. Dochain D. (1992). Adaptive control algorithms for non-mInimum phase nonlinear bioreactors. Computers Chem. Engng., 16, No. 5, pp. 449469. Isaacs. S.H ., Soeberg H., Kummel M. (1992). Monitoring and control of a biological nutrient removal processes: rate data as a source of information. IF AC Modelling and Control of Biotechnological Processes, Colorado, USA, pp. 239-242. Joshi N.V. , Murugan P., Rhinehart R.R. (1997). Experimental comparison of control strategies. Control Eng. Practice. 5, No . 7, pp. 885-896. Monod J. (1949). The growth of bacterial cultures. Ann. Rev. Microbial, 3, pp. 371-394 Richalet J. (1993). Industrial applications of model based predictive control. Automatica, 29. No. 5. pp. i251-1274. Van Impre J.F., Bastin G. (1995). Optimal adaptive control of fed-batch fermentation processes. Control Eng. Practice. 3. No . 7. pp. 939-954 .
In both cases we observe very good control performance of the considered adaptive controllers. The controlled output converges much faster for these controllers being in use than for conventional PI controllers applied for the same purpose. The adaptive controllers manipulates the control quantities smoothly, which is very important from practical point of view, and provide very short regulation time without overshooting.
8. CONCLUSIONS In this paper the simple model-based adaptive controllers for the fed-batch fermentation process have been proposed and their effectiveness has been demonstrated by means of simulation experiments. The proposed control laws have been derived on the basis of the phenomenological model of the process. However, the application of the substrate consumption rate allows us to avoid problems with estimation of the parameters of the nonlinear part of the mathematical model. Since both the procedure for estimation of the substrate consumption rate and the adaptive controllers have been derived only on the basis of the general form of the state equation written for the substrate concentration, there is no need to know anything about the nonlinear part of the mathematical model describing the intensity of the reaction . Moreover, there is no necessity to know or to estimate the values of the parameters such as, for instant, yield coefficients, biomass concentration or specific growth and production rates. These features allow this idea to be easily applicable in practice. The form of the state equation that is a basis for the idea presented in this paper is very general and therefore it satisfies every biological and chemical reaction as well as heat transfer processes. This feature allows us to expect the idea of the substrate consumption rate and based on its value adaptive controllers to be easily applicable for control of chemical and thermal processes.
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