Substrate Control in Fedbatch Cultivations Using a Model-based Modification of a PI-regulator

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

SUBSTRATE CONTROL IN FEDBATCH CULTIV ATIONS USING A MODEL-BASED MODIFICATION OF A PI-REGULATOR P. Hagander* and O. Holst** *Department 0/ AuJomatic Comrol , Lund instituJe o/Technology, p.a . Box 118, S·221 00 Lund, Sweden **Department o/Biotechnology, Chemical Center, Lund University , p.a. Box 124. S·221 00 Lund. Sweden

Abstract A fed batch process shows exponential growth under ideal conditions. To obtain good substrate concentration control it is necessary that the regulator can track an exponentially growing feed demand, and standard PI-control has to be supplemented with an estimated basic dosage to get reasonable control. However, an exponentially growing concentration error is impossible to avoid. An I-term could be interpreted as an observer of a constant demand, and we have proposed to replace it with a model-based observer for an exponentially growing demand. In the resulting controller the integrator is replaced by an unstable pole at s = Ji., the specific growth rate, and the initial condition of this term is equivalent to the basic dosage part. The regulator can now track the exponentially growing feed demand without error. P8eudomona8 cepacia was grown on salicylate as sole carbon and energy source. Salicylate is a toxic substrate, so it is important to have good substrate control. Online measurement of salicylate concentration was carried out using a filtration system from which cell-free permeate was passed to a flow-through spectrophotometer.

Introducing more instability into the controller requires attention to the anti-windup features. No such problems were found during the cultivations or in simulations of the effect of conceivable disturbances like pump-failure, air-bubbles in the spectrofotometer, and low oxygen concentration induced growth-rate reduction . tion, as they normally are when fed-batch is used instead of batch processing, it would be advantageous or even necessary to keep the substrate concentration at a desired set-point by control of the feed-flow . The controller design of course depends on the sensor available and on how critical it is to keep a constant concentration despite the variations in the process, but an on/off control is normally not sufficient. In a situation with a toxic substrate or when a side reaction dominates at high substrate concentrations it is motivated with a detailed investigation of the process and a possibly more complicated controller design.

Introduction The extra complication of using fed-batch instead of batch fermentations is normally introduced to obtain optimal growth conditions and substrate utilization. Precalculated dosage schemes are often used industrially, but they should ideally be adjusted to variations in inoculum properties and substrate quality, This calls for manual adjustments of the feed-flow based on observations of the process, One suitable measurement would of course be the substrate concentration. During a fermentation there are other variables that are kept under control, like temperature, pH, and dissolved oxygen. Automatic on/offcontrollers are often used, since the loop timedelays are small enough to give acceptable fluctuations in the controlled variables. The control is normally designed in single loops. The interaction between the loops is difficult to account for without detailed modeling of the specific process. The possibility for substrate control has evolved with the development of on-line sensors for the substrate concentration or closely related compounds, When the fermentation conditions are influenced substantially by the substrate concentra-

Background The main objective of a substrate controller is to track the growing feed demand. Since the growth in a fed-batch process is normally exponential the controller has to track an exponential load disturbance, A natural attempt would be to use a PIregulator, which would be able to "reset" constant load disturbances. In this situation with an exponential load such a controller would normally be

77

where V is the volume, X the cell concentration, S the substrate concentration, F the manipulated substrate feedrate, and Sin its concentration. The two parameters JL, the specific growth rate, and qs, the specific substrate uptake rate, may have a quite complicated dependence of mainly the substrate concentration. In the application we describe here we will choose the substrate concentration set-point S.. such that JL and q, are saturated at their maximal values. If we also assume that the volume change is relatively small, then (1) simplifies to

s ~ __~~ ~l! _____________ :

Figure 1.

Block diagram of the process

quite bad. The feed-back gain is limited by stability, and there would inevitably be an asymptotically exponential control error. Reasonable control could in many situations be obtained using a PIregulator if it is supplemented with a basic dosage scheme, (e.g. Axelsson et al., 1988, Hagander et al., 1990a). Figure 1 describes this situation in a block diagram. The calculation of the basic dosage involves an estimate of the cell mass at the start of the fed-batch together with the growth rate, the yield, and the feed substrate concentration. It is impossible to get perfect basic dosage, but the PIregulator would reduce the effect of the resulting load disturbance. If the batch time is prolonged to obtain a high cell density, there would still appear an exponentially growing control error, and in some cases this might be unacceptable. The basic dosage scheme is really a model of the demand as an unstable first order linear system. Much of its uncertainty is in its initial condition. The I-term of the PI-regulator could be interpreted as an observer for a constant load, and we have proposed to replace it with an observer for the exponentially growing load (Hagander et al., 1990b). The fact that regulators should contain a model of the disturbance they are supposed to eliminate is often called the internal model principle. See e.g. (Astrom and Wittenmark, 1990), or (Bengtsson, 1977). In this paper we apply the modified controller to a cultivation on a toxic substrate. The observer is developed using the load model, and it is demonstrated how it can be characterized as a minor modification of a PI-regulator. The controller is then evaluated on real cultivations and also using simulation of conceivable disturbances.

(2) where Fo is the feed-flow demand at the start of the fed-batch, i.e.

SinFO

= q, V(O)X(O)

(3)

Neglecting the very slow dilution rate F/V the process (2) is simply an integrator with the gain

Kp

= Sin/V

Kp Gp(s) = -

(4)

s The main requirement for a controller is to match the exponentially growing substrate feed demand FO, c.f. Figure 1.

Formulation of an observer

The model (2) for the process and the disturbance could also be written as

:t

{

is=Kp(F-r) dt FO(t) = JLFO(t)

(5)

Equation (5) is then the basis for an observer of its state-variables. Since S is directly measured we use a reduced order observer, a Luenberger observer, e.g. (Friedland, 1986), which simplifies to

d

-Fo=JLFO-K dt 0 A

A

(d-S-Kp(F-r) A

(6)

)

dt

If we now use the estimate fro as the basic dosage Fb in Figure 1, we can try a proportional controller

Controller with disturbance model F The cultivations are performed in a well-mixed fermentor, and mass balances give the following equations

(:t - JL) fro

-=F dt

- dt = JLVX

(7)

S)

Combining (6) and (7) gives

dV

dVX

= fro + KR(S.. -

= Ko (:t + KpKR) (S.. -

S)

and the resulting controller transferfunction is then

(1)

dVS

----;It = -qs V X + Sin F 78

_ .

feed

'2

~'Ol

i:rdW3~1 ~

~oo

q4

14.5

15

15.5

16

16.5

17

17.5

18

18.5

time of day (h)

Figure 3. Fed-batch growth of Pseudomonas cepacia on salicylate. Substrate concentration (S) is controlled by feed-rate (F) using a poorly tuned PI-regulator. The reference substrate concentration (- - - -) is varied as a square-wave. LP-m....

Figure 2.

PC

presentation of the data. Using a mouse the operator can open up different windows and enter commands. The digital PID-controller included in the system was implemented as in (Astrom and Wittenmark, 1990). The regulator includes bumpless transfer from manual to automatic mode This also provides anti-reset windup to avoid bad behavior in case the pump saturates. To implement the modified controller it was only necessary to change one line of code, and the modularity made it possible to recompile only one implementation module.

The experimental setup

Thus GR could be seen as a PI-regulator, in which the integrator is made slightly unstable. Apart from the choice of jJ, of the unstable pole to match the growth-rate, two parameters should be tuned for good transient properties, just like K and 11 of a PI-regulator. The initial condition of the unstable regulator mode, FO(O), could be chosen based on (3), just as when the basic dosage, Fb was designed .

Preliminary experiments

Laboratory experiments The new controller was used for cultivation of Pseudomonas cepacia. Pseudomonas species are known for their ability to use aromatic structures as their carbon source, and they are useful for degradation of toxic wastes. Grown on salicylate Pseudomonas cepacia produces the intracellular enzyme salicylate hydroxylase, that is of interest in clinical chemistry for determination of salicylate in serum samples. Salicylate concentrations of 2 g/L are inhibitory, while no growth occurs at 10 g/L, so it is important to use fed-batch with substrate control. The experimental setup is shown in Figure 2. Continuous measurements of the salicylate concentration are obtained using a UV-spectrophotometer. A cell-free permeate is obtained from a microfilter device. See (Tocaj et al., 1992).

Calibration was made before each cultivation. The feed-pump characteristic was linear, while the sensor was slightly nonlinear. The reproducibility was good between experiments. It was found that the response of the pump was almost immediate, while the sensor system introduced a considerable time-delay, TD = 3 min. The time delay is mainly due to the transport of the small cell-free flow from filter unit to spectrophotometer. Closed loop identification was used in Figure 3 to verify the simple process model (4). The reference value to a poorly tuned PI-regulator was varied as a square-wave. During the first part of the experiment the integration time, T/, was chosen 20% faster than normal, while the regulator gain, K R , was 50% higher than normal during the second part. The expression Kp = Sin/V was satisfied, and no dependence on the S-variations could be detected in q•.

Computer implementation of the controller

Results using PI

The controller was implemented in Modula-2 on an IBM-PC-AT computer using a realtime kernel. The system includes interpolation in calibration data, possibilities to change controller parameters on-line, data-storage on diskette, and graphical

A typical result of a cultivation performed using the PI-regulator is shown in Figure 4. The narrow peaks in the sensor signal are caused by airbubbles that enter the flow-cuvette of the spectrophotometer. They can also be seen in the feed-flow

79

1 . 0 , - - _ - _ - _ - _ - - _ - _ -_ __,

1.4r--_-_--_-~-~--_-__,

0.9

1.2

0 .8 ~ 0.7

~

-!:9 ~

'-' 0.8

~

'E-

0.6

~

0.4

i

>.. u

i

0.6

0.5 0.4

'"

0.2 %~-~-~2--~3-~4~-~5--7--~

00

2

3

4

5

7

fed-batch time (h)

fed-batch time (h) 40

~,---------~-~------,

35 20

....... 30 c:

'El

:5--

20

<:::

15

Jl""

10

~ 0

2

3

4

5

00'

6

4

5

6

7

with the measurement of salicylate at the chosen wavelength. We are studying the problem further to get a more specific sensor. Anyhow the inability to keep the set point clearly demonstrates the drawback with the PI-regulator. Some comments could also be made about the details of the curves. The feed-flow levels off after about two hours. The substrate uptake rate, must have been reduced . The controller manages in the beginning to keep the substrate level at the set point. If we compare with a recording of the levels of dissolved oxygen, Figure 6, we notice that the aeration was not enough, and the substrate consumtion was reduced already at a saturation level of 20%. When the oxygen goes down almost to zero, the controller no longer manages to keep the salicylate down . As soon as a flow of pure oxygen is started, the salicylate comes down to the set point, and the feed-flow increases again.

140

~1:W c

~100

>.

~

""'"> '0 '"

3

fed-batch time (h) Figure 6. Salicylate concentration (upper) and feed-flow (lower) for a cultivation using the modified controller

fed-batch time (h) Figure 4. Salicylate concentration (upper) and feed-flow (lower) for a cultivation using PI-control

:a'"

~

40 20 %~-~-~ 2--~ 3 -~4~-~5--7--~

fed-batch time (h) Figure 5. Dissolved oxygen concentrations of the cultivation in Figure 4. The increases of the signal result. from increases in the oxygen flow, while the decreases occur after antifoam addition .

signal. The bubbles started to appear when we increased the aeration, and we have later installed a degassing filter that eliminated the problem. After about four hours there is a steady increase of substrate concentration . Although the controller reduces the feed rate well below the basic dosage, there is an increasing control error. The intuitive explanation is that this demonstrates the previously discussed shortcoming of the PIregulator. There may however be further explanations. When the feed was turned off after 5.8 hours the salicylate signal started to drop, but soon it unexpectedly became constant. If some more salicylate were added, that would immediately be consumed. This could indicate that some feed components or maybe by products interfered

Results using modified controller The first thing to notice from the cultivation in Figure 7 with the modified controller is that the controller manages keep the salicylate almost at the setpoint . The dashed line shows what corresponds to the basic dosage, i.e. the feed that would have resulted from the estimated initial feed demand, the initial condition to the unstable controller mode. In that calculation we used the measured initial cell density, but we chose here about half the expected yield coefficient, so the dashed

80

Sand Srel [glL]

line would have given a feed supply about twice the actual feed demand. The modified controller manages to reduce that estimate and supply what seems to be the adequate feed amount. This could not have been done by a PI-regulator. When the airbubble at 4 h was removed, the sensor was turned off for a few minutes, while the controller was still on. This is clearly seen in the feed-flow signal. The fedbatch was started at a rather high initial salicylate concentration, and the feed-flow was low. The salicylate reached the setpoint after about half an hour but it continued to decrease even if the pumpsignal was quite high. It was found that a high pressure had built up in the fermentor causing problems for the pumps. When the pressure was released at 45 min. the salicylate raised immediately with a large overshoot. This disturbance caused a substantial excitation of the system. The damping seems to be less than expected, and it may be that the time delay in the sensor system was prolonged. The salicylate peak at 5 h was caused by low oxygen during the change to a new oxygen gascylinder, while antifoam was added just at 6 h.

0.24

0.22 0.2f-------------------o

2

4

6

2

4

6

Feed flow [mUmin)

o Sand Srel [glL] 0.24

0.2+---.....:::=-------------o

2

4

6

2

4

6

Feed Ilow [mUmin)

o

Summing up

Figure 7. Simulation where the assumed initial VX is Sg while actual VX is 4g. The upper two panels show PI-control and the lower panels the modified controller.

Despite the problems with the sensor system both PI-control and the modified controller could keep the salicylate concentration within reasonable levels. The cell mass increased from less than 1 g dw /L to 12 g dw /L within 6 hours corresponding to a growth rate J.L of about 0.5 h -1 . The yield coefficient was 0.4 g dw / g. The controllers managed the sensor and pump failures reasonably well. The disturbances during the first hours with the modified controller suggested that a more robust tuning should have been chosen. The volume of the fermentor increases from 8 L to 10 L approximately, implying a 20 % decrease of the process gain Kp, so a controller gain that is slightly increasing with time could be motivated .

Sand Srel [g/L)

""!C: U t-L-------r,------~)L-----_,i 0.185

o

. :1

2

","00' ••

o

6

~,

o

'J"'",..

4

i

i

2

6

[U"[

i

2

4

6

Figure 8. An airbubble appears in the spectrophotometer.

Simulations of disturbances measurement of the cell-mass V X at the time of the fed-batch start. On the other hand the two controllers give identical responses to different other types of problems that have occured during the cultivations. Figures 8-9 show that they handle airbubbles and a 6 min breakdown of the pump without any problem.

From the cultivations obtained using PI-control and the disturbance-model controller, it was rather difficult to compare their properties. Different disturbances appeared during the different experiments. A simulation model was formulated for the system including equations (1) for the fermentor, a model for the measurement system, and discretetime versions of the controllers, just as the ones implemented in the control-computer. Different disturbances were introduced one at a time, and from Figure 7 it is clearly seen how superior the disturbance-model controller is in case of a bad

As always in feed-back control a sensor failure might be difficult. A 12 min loss of signal from the sensor should have been diagnosed, (S - Sr) should have been assumed to be zero, so that the controller continued open-loop instead of almost doubling the feed-flow as in Figure 10. Notice 81

References

S and Srell!yL]

0.2+--------------,+--==----

ASTR(}M, K.J. and B. WITTENMARK (1990): Computer Controlled Systems, (2nd edition), Prentice Hall Inc, Englewood Cliffs, N.J.

0.1

o

2

4

AXELSSON, J. P., C. F. MANDENIUS, O. HOLST, P. HAG AND ER and B. MATTIASSON (1988): "Experience in using an ethanol sensor to control molasses feed-rates in baker's yeast production," Bioprocess Engineering, 3, 1-9.

6

Feed flow [mUminj

0

2

Figure 9.

4

BENGTSSON, G. (1977): "Output regulation and internal models - a frequency domain approach," Automatica, 13, 333-345.

6

The feed-pump is down during 6 min.

:{"'S~,"'I 0.2 0

"r~"'''' ,., i"''"' ,.. ""I

,

,

,

2

4

6

I~

6

,

0 0

2

0 0

2

, Figure 10.

A

FRIEDLAND, B. (1986): Control system design: An introduction to state-space methods, McGraw-Hill,Inc, New York.

4

HAG AND ER, P., A. HOF, and O. HOLST (1990a): "Cultivation of P8eudomona8 cepacia on salicylate. Substrate control of a feedbatch process," Preprints of the 5th European Congress of Biotechnology, Copenhagen, Denmark.

,

HAGANDER, P., J. P. AXELSSON, and O. HOLST (1990b): "Substrate control ofbiotechnical processes - Robustness and the role of adaptivity," Preprints of the 11th IFAC World Congress, Tallinn, Estonia, USSR.

==1c:: 4

6

A. HOF, P. HAGANDER, and TOCAJ, A., O. HOLST (1992): "Fed-batch cultivation of P8eudomona8 cepacia with on-line control of the toxic substrate salicylate," (submitted for publication).

The sensor signal is lost during

12 minutes.

that the actual salicylate concentration goes up to three times the set point in this short time. Still the controllers recover nicely from the disturbance.

Conclusions The main task for a fed-batch substrate controller is to track a partly known exponential feed demand. We have aimed for the simpliest possible modification of a standard PI-controller. Such a controller was derived using a disturbance model. The new properties of the controller were here evaluated in simulation and in the laboratory. In (Hagander et al., 1990b) we also discussed possible adaptation of the JL-parameter of the controller. That extra complication was not necessary here.

Acknowledgements Financial support for this study was provided by the Swedish National Board for Technical Development, contract 86-4404. Several students have participated in this project, and we would like to thank them for their important contributions.

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