VII CAIQ2015 PENICILLIN PRODUCTION CONTROL BY …

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AAIQ Asociación Argentina de Ingenieros Químicos - CSPQ

PENICILLIN PRODUCTION CONTROL BY MONITORING

CONCENTRATION PROFILES OF THE PROCESS

M. C. Fernández, M. N. Pantano, S. Rómoli, D. Patiño, O. Ortiz and G. Scaglia.

Instituto de Ingeniería Química - Facultad de Ingeniería

(Universidad Nacional de San Juan)

Av. Libertador San Martín 1109 (O). San Juan. Argentina

[email protected]

Abstract. The objective of this work is to design a controller for process

variables for the penicillin production, carried out in a fed-batch reactor,

following predefined profiles. The design technique is based on linear

algebra, which allows the design of multivariable controllers and highly

nonlinear systems.

To achieve this, it is necessary to possess a mathematical model that

adequately represents the process and the concentration profiles that system

should follow. Simulation results are shown for different initial conditions.

Key words: controller design, nonlinear systems, fed-batch process.

1. Introduction.

The Penicillin is an antibiotic discovered by Alexander Fleming in 1928, this

discovery is known as the biggest revolution in medicine and public health in human

history (Biomed, 1999), because it allowed and still allows saving millions of lifes

around the world.

Although penicillin can be chemically synthesized since 1957, it is commonly

produced by microorganisms. To be more specific, only a few fungal species can

produce it, such as Penicillum crysogenum and Aspergillus nidulans (A. Herr, R.

Fischer, 2014).

The penicillin production as many other processes, like those to obtain primary and

secondary metabolites, proteins, and biopolymers (J. Lee, et al., 1999), can be carried

out in a fed-batch way. This consists in change substrate feed rate along the process and

not remove product until it ends (A. Survey, A. Johnson, 1987). Fermentations like that

http://www.sciencedirect.com/science/article/pii/S1096717614000937



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have many difficulties to be controlled, this is because the complex dynamic behavior of

microorganisms, nonlinear and sometimes unstable dynamics, external disturbances,

strong modeling approximations, unusual on-line measurements of most representative

variables, etc. All these conditions avoid the possibility to use classic industrial

controllers, being necessary to implement control algorithms specifically developed for

bioprocesses (H. De Battistaa, et al., 2012).

The substrate feed rate control profile for penicillin biosynthesis has been obtained in

different ways: analytically (H. C. Lim, et al., 1986), dynamic programming (R. Luus,

1993), by an evolutionary approach (M. Ronen, et al., 2002) and using orthogonal

collocation (C. A. M. Riascos, José M. Pinto, 2004). In this article, we propose an easy

form of tracking the optimal concentration profiles, in order to obtain the substrate feed

rate policy of the process. To achieve this goal, is important to have the next

information: the mathematical model that represents the process properly and the

concentration profiles that we want the system to follow.

First of all, the technique requires knowing the reference profiles, so that is the first

step to take; ones profiles are determined, the next stage is to compute the control action

causing the system to follow the references. The controller structure in this

methodology arises from the mathematical model of the process; that means that this

procedure can be used in many nonlinear systems, making it a very useful technique for

any bioprocess.

The paper is organized in four sections. The second one describes the bioreactor and

the mathematical model of the process. The third part explains the controller design.

The simulations that prove the effectiveness of the controller are developed in the last

section. Finally, conclusions are exposed.

2. Characteristics of the system and process.

A fed-batch bioreactor is used to produce penicillin from glucose. The system was

explained for the first time in (J. E. Cuthrell, L. T. Biegler, 1989), and many works were

done from it, obtaining with different methods the optimal feed rate profile (H. C. Lim,

et al., 1986), (R. Luus, 1993), (M. Ronen, et al., 2002) and (C. A. M. Riascos, José M.

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Pinto, 2004). The mathematical model (C. A. M. Riascos, José M. Pinto, 2004) that

represents the process is:

(1)

(2)

The state variables in the equations shown are: volume (V), concentrations

of biomass (X), product (P), and substrate (S), substrate feed rate (U) (that will be the

control variable). In the dynamic model, SF is the feed concentration, ( ) is the

specific biomass growth rate, ( ) is the specific penicillin production rate. Initial

variable values are shown in Table 1, whereas parameter definitions and values are in

Table 2.

Table 1. Initial variable values for penicillin biosynthesis

Variable Initial value

X (g/L) 1.5

P (g/L) 0.0

S (g/L) 0.0

V (L) 7.0

Table 2. Parameters of penicillin biosynthesis model

Parameter Definition Value

µmax Maximum specific biomass growth rate (h-1

) 0.11

Ρmax Maximum specific production rate (g P/g X h) 0.0055

𝑋 (𝑡) = (𝑋 𝑆)𝑋 − 𝑋

𝑆𝐹𝑉 𝑈

𝑃 (𝑡) = 𝜌(𝑆)𝑋 − 𝐾𝑑𝑒𝑔𝑃 − 𝑃

𝑆𝐹𝑉 𝑈

𝑆 (𝑡) = (𝑋 𝑆) 𝑋

𝑌𝑋/𝑆 − 𝜌(𝑆)

𝑃

𝑌𝑃/𝑆 −

𝑚𝑠𝑆

𝐾𝑚 + 𝑆 𝑋 + 1 −

𝑆

𝑆𝐹 𝑈

𝑉

𝑉 (𝑡) =𝑈

𝑆𝐹

(𝑋 𝑆) = 𝑚𝑎𝑥 𝑆

𝐾𝑋𝑋 + 𝑆

𝜌(𝑆) = 𝜌𝑚𝑎𝑥 𝑆

𝐾𝑃 + 𝑆(1 + 𝑆/𝐾𝑖𝑛)

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KX Saturation parameter for biomass growth (g S/g X) 0.006

KP Saturation parameter for production (g S/L) 0.0001

Kin Inhibition parameter for production (g S/L) 0.1

Kdeg Product degradation rate (h-1

) 0.001

Km Saturation parameter for maintenance

consumption (g S/L) 0.0001

ms Maintenance consumption rate (g S/g X h) 0.029

YX/S Yield factor for substrate to biomass (g X/g S) 0.47

YP/S Yield factor for substrate to product (g P/g S) 1.2

SF Feed concentration (g S/L) 500

3. Controller design.

As it was mentioned before, the penicillin production process was studied by various

authors ((H. C. Lim, et al., 1986), (R. Luus, 1993), (M. Ronen, et al., 2002) and (C. A.

M. Riascos, J. M. Pinto, 2004)). In those works, were determined the substrate feed rate

that makes upper the final product concentration. For this paper, it was taken as

reference (R. Luus, 1993). Fig. 1 shows the optimal reference profiles.

Fig 1. Reference Profiles.

In order to follow the references shown in Fig. 1, the technique is explained and

developed below.

0 20 40 60 80 100 120 1400

5

10

15

20

25

30

35

Time[hs.]

Concentr

ation[g

/L]

///

Volu

me [

L]

Cell

Product

Substrate

Volume

Fig. 1 Reference Profiles.

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3.1. Problem definition.

Assuming the following first-order differential equation:

= = ( ) (3)

Considering that ( ) = , where x is the system output, t the time and u the

control action. From here on, ( ) is the value of x at a discrete time (t=nT0), being T0

the sample time, and x(n+1) is x value in the next instant of time (n = 1, 2, …). To

obtain ( + 1) is necessary to integrate the last equation over the time interval

( + 1) , as it is shown next:

( + 1) = ( ) + ∫ ( ) ( )

(4)

It is supposed that u does not change it value in the integration interval.

There are many numerical methods used for approximate integrals, in this paper it

will apply Euler, despite of it is not the most accurate way, because there are methods

like Runge Kutta where the results are exacter, however the mathematical complexity

also increases.

( + 1) ( ) + ( ( ) ( ) ( )) (5)

With the numerical methods it can be calculated the system state at the time n+1 as a

function of the state, the variables at the time n and the control action.

The technique consist in obtain the profiles of some state variables (sacrificed)

making the tracking error tends to zero. To achieve this, the systems equations are

analyzed under the condition that they have exact solution. In this work, the substrate

concentration profile inside the reactor is the sacrificed variable, and the substrate feed

rate (U) is the control action. It is important to highlight that this methodology has been

employed in many systems with excellent results, such as: (G. Scaglia, et al., 2009), (A.

Rosales, et al., 2009), (G. Scaglia, et al., 2010), (A. Rosales, et al., 2011), (M. Serrano,

et al., 2013), (F. A. Cheein, G. Scaglia, 2013) and (S. Rómoli, et al., 2015).

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3.2. Controller.

In previous paragraphs it was detailed the mathematical model needed to estimate the

control action used to obtain the state variable profiles of Fig. 1. Now it is developed the

design of a controller whose function is to generate that control action; here it is taken

into account that the bioreactor state variables (cell and product concentrations) follow

the references shown in Fig. 1.

Applying Euler to the equations system (1) saw in section 2.:

(6)

Where and can be obtained as follow:

(7)

The difference between the real and the reference values of the state variables is

called “tracking error”. And it is calculated as follow:

2 2

( ) ( ) ( )n refn n refn ne X X P P

(8)

𝑋𝑛 = 𝑋𝑛 + 𝑇 𝑚𝑎𝑥 𝑆𝑛

𝐾𝑋𝑋𝑛 + 𝑆𝑛 𝑋𝑛 −

𝑋𝑛𝑆𝐹𝑉𝑛

𝑈𝑛

𝑃𝑛 = 𝑃𝑛 + 𝑇 𝜌𝑚𝑎𝑥 𝑆𝑛

𝐾𝑃 + 𝑆𝑛(1 + 𝑆𝑛/𝐾𝑖𝑛) 𝑋𝑛 − 𝐾𝑑𝑒𝑔𝑃𝑛 −

𝑃𝑛𝑆𝐹𝑉𝑛

𝑈𝑛

𝑆𝑛 = 𝑆𝑛 + 𝑇 𝑚𝑎𝑥 𝑆𝑛

𝐾𝑋𝑋𝑛 + 𝑆𝑛

𝑋𝑛𝑌𝑋/𝑆

− 𝜌𝑚𝑎𝑥 𝑆𝑛

𝐾𝑃 + 𝑆𝑛(1 + 𝑆𝑛/𝐾𝑖𝑛)

𝑃𝑛𝑌𝑃/𝑆

− 𝑚𝑠𝑆𝑠

𝐾𝑚 + 𝑆𝑛 𝑋𝑛

+ 1 −𝑆𝑛𝑆𝐹 𝑈𝑛𝑉𝑛

𝑉𝑛 = 𝑉𝑛 + 𝑇 𝑈𝑛𝑆𝑓

𝑋𝑛 = 𝑋𝑟𝑒𝑓 𝑛 + 𝑘 𝑋𝑟𝑒𝑓 𝑛 − 𝑋𝑛 − 𝑋𝑛

𝑃𝑛 = 𝑝𝑟𝑒𝑓 𝑛 + 𝑘2 𝑃𝑟𝑒𝑓 𝑛 − 𝑃𝑛 − 𝑃𝑛

𝑆𝑛 = 𝑆𝑟𝑒𝑓 𝑛 + 𝑘3 𝑆𝑟𝑒𝑓 𝑛 − 𝑆𝑛 − 𝑆𝑛

𝑉𝑛 = 𝑉𝑟𝑒𝑓 𝑛 + 𝑘4 𝑉𝑟𝑒𝑓 𝑛 − 𝑉𝑛 − 𝑉𝑛

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The controller parameters ( 2 3 4) take values among zero and one

( 1 ) for i = 1; 2; 3; 4, that makes possible the tracking error to tend to zero

when n tends to infinity. Seeing equations (7) it can be conclude that when k = 0, the

real profile reaches the reference in one step, and when k is between 0 and 1, the error

approaches zero gradually.

Therefore, the state variable in the sample time n+1 can be calculated as a function

of the real value of the variable in the time n, the reference profiles and the controller

parameters.

Equations system (6) can be expressed as a matrix with the form of:

A U = b (9)

That is to say:

(10)

Taking into account what it was expressed in (7):

(11)

−𝑋𝑛−𝑃𝑛

𝑆𝑓 − 𝑆𝑛1

𝑈𝑛

=

(𝑋𝑛 − 𝑋𝑛)

𝑇 𝑆𝑓𝑉𝑛 − (𝑋𝑛 𝑆𝑛)𝑋𝑛𝑆𝑓𝑉𝑛

(𝑃𝑛 − 𝑃𝑛)

𝑇 𝑆𝑓𝑉𝑛 − 𝜌(𝑆𝑛)𝑋𝑛𝑆𝑓𝑉𝑛 + 𝐾𝑑𝑒𝑔𝑃𝑛𝑆𝑓𝑉𝑛

(𝑆𝑛 − 𝑆𝑛)

𝑇 𝑆𝑓𝑉𝑛 + (𝑋𝑛 𝑆𝑛)


𝑆𝑓𝑉𝑛 + 𝜌(𝑆𝑛) 𝑋𝑛𝑌𝑃/𝑆

𝑆𝑓𝑉𝑛 + 𝑚𝑠𝑆𝑛

𝐾𝑚 + 𝑆𝑛 𝑋𝑛𝑆𝑓𝑉𝑛

(𝑉𝑛 − 𝑉𝑛)

𝑇 𝑆𝑓


𝑆𝑓 − 𝑆𝑛1

𝑈𝑛

=

𝑋𝑟𝑒𝑓 𝑛 − 𝑘 𝑋𝑟𝑒𝑓 𝑛 − 𝑋𝑛 − 𝑋𝑛

𝑇 𝑆𝑓𝑉𝑛 − 𝑚𝑎𝑥

𝑆𝑛𝐾𝑋𝑋𝑛 + 𝑆𝑛

𝑋𝑛𝑆𝑓𝑉𝑛

𝑃𝑟𝑒𝑓 𝑛 − 𝑘2 𝑃𝑟𝑒𝑓 𝑛 − 𝑃𝑛 − 𝑃𝑛

𝑇 𝑆𝑓𝑉𝑛 − 𝜌𝑚𝑎𝑥

𝑆𝑛

𝐾𝑃 + 𝑆𝑛 1 +𝑆𝑛𝐾𝑖𝑛 𝑋𝑛𝑆𝑓𝑉𝑛 + 𝐾𝑑𝑒𝑔𝑃𝑛𝑆𝑓𝑉𝑛

𝑆𝑟𝑒𝑓 𝑛 − 𝑘3 𝑆𝑟𝑒𝑓 𝑛 − 𝑆𝑛 − 𝑆𝑛

𝑇 𝑆𝑓𝑉𝑛 + 𝑚𝑎𝑥



𝑆𝑓𝑉𝑛 + 𝜌𝑚𝑎𝑥 𝑆𝑛

𝐾𝑃 + 𝑆𝑛 1 +𝑆𝑛𝐾𝑖𝑛

𝑋𝑛𝑌𝑃/𝑆



𝑉𝑟𝑒𝑓 𝑛 − 𝑘4 𝑉𝑟𝑒𝑓 𝑛 − 𝑉𝑛 − 𝑉𝑛

𝑇 𝑆𝑓

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As the volume is a variable that is present in all the equations, the last components of

the matrices A and b are redundant, so they can be neglected.

(12)

From (12) it is possible to obtain the control action (Un) at any sample time, knowing

that this is the value found by following the optimal trajectories. For this equations

system the parameters of the process and the reference profiles are presented in section

2.

To begin with the resolution of (12), it is important to establish the conditions

necessary for the system to has exact solution. For this to be fulfilled, looking equation

(9) it can be deduced that b have to be a linear combination of A columns. There are

several ways to make the system satisfy that condition, one of them is that both A and b

have to be parallel. In this paper that condition of parallelism will be expressed in (13),

for more information see (G. Strang, 2006).

(13)

In (13) is shown an equations system where am and bm (m = 1, 2, 3) are the

components of matrixes A and b, respectively. To solve the system there were used the

next expressions:

(14)


𝑆𝑓 − 𝑆𝑛

𝑈𝑛

=

𝑋𝑟𝑒𝑓 𝑛 − 𝑘 𝑋𝑟𝑒𝑓 𝑛 − 𝑋𝑛 − 𝑋𝑛






𝑆𝑛

𝐾𝑃 + 𝑆𝑛 1 +𝑆𝑛𝐾𝑖𝑛 𝑋𝑛𝑆𝑓𝑉𝑛 + 𝐾𝑑𝑒𝑔𝑃𝑛𝑆𝑓𝑉𝑛





𝑆𝑓𝑉𝑛 + 𝜌𝑚𝑎𝑥 𝑆𝑛





𝑎3𝑎 =𝑏3𝑏

𝑎3𝑎2=𝑏3𝑏2

𝑎3𝑏 = 𝑏3𝑎

𝑎3𝑏2 = 𝑏3𝑎2

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Replacing by (12) in (14):

(15)

(15)

This is a system with two equations and one unknown, the unknown variable is

called “sacrificed variable”, and it takes any value just to ensure the other variables to

track the references. To select that sacrificed variable is necessary to study and interpret

the system. In this system, analyzing (6), was selected the substrate concentration as

sacrificed variable (Sez), that is because all the other variables depend on it. So, in (15) it

can be replaced S by Sez.

𝑆𝑓 − 𝑆𝑛 𝑋𝑟𝑒𝑓 𝑛 − 𝑘 𝑋𝑟𝑒𝑓 𝑛 − 𝑋𝑛 − 𝑋𝑛




=


𝑇 𝑆𝑓𝑉𝑛

+ 𝑚𝑎𝑥 𝑆𝑛

𝐾𝑋𝑋𝑛 + 𝑆𝑛


𝑆𝑓𝑉𝑛

+ 𝜌𝑚𝑎𝑥 𝑆𝑛





(−𝑋𝑛)




𝑆𝑛


𝑋𝑛𝑆𝑓𝑉𝑛 + 𝐾𝑑𝑒𝑔𝑃𝑛𝑆𝑓𝑉𝑛

=





𝑆𝑓𝑉𝑛






(−𝑃𝑛)

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(16)

(16)

In the last expression (16), the subscript n represents the value that comes from the

system to the controller in the generic sample time n; the subscript ref n+1 and ref n

refer to the reference value taken at the time instants n+1 and n, respectively; finally

subscript ez represents the sacrificed variable. As it can be seen, there are two values for

Sez: Sez n+1 and Sez n, where the unknown is Sez n+1, because Sez n was calculated in the

previous sample time. To start the simulation it was considered that Sez n = Sez n+1 as a

first approximation (Euler zero order).

Observing Eq. (9) and (10) it can be seen that A is formed by one column linearly

independent, then using minimal square (G. Strang, 2006), (9) is expressed as:

= ( ) (17)

The control action (U) that makes all the state variables follow the references is

calculated with Eq. (17).

𝑆𝑓 − 𝑆𝑛 𝑋𝑟𝑒𝑓 𝑛 − 𝑘 𝑋𝑟𝑒𝑓 𝑛 − 𝑋𝑛 − 𝑋𝑛


𝑆𝑒𝑧𝐾𝑋𝑋𝑛 + 𝑆𝑒𝑧


=

𝑆𝑒𝑧 𝑛 − 𝑘3(𝑆𝑒𝑧 𝑛 − 𝑆𝑛) − 𝑆𝑛




𝑆𝑓𝑉𝑛






(−𝑋𝑛)




𝑆𝑒𝑧

𝐾𝑃 + 𝑆𝑒𝑧 1 +𝑆𝑒𝑧𝐾𝑖𝑛

𝑋𝑛𝑆𝑓𝑉𝑛 + 𝐾𝑑𝑒𝑔𝑃𝑛𝑆𝑓𝑉𝑛

=

𝑆𝑒𝑧 𝑛 − 𝑘3(𝑆𝑒𝑧 𝑛 − 𝑆𝑛) − 𝑆𝑛




𝑆𝑓𝑉𝑛






(−𝑃𝑛)

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The system under study has a single input (U) and multiplies outputs (X, P, S), so is

called a SIMO system. However, this strategy of control can be used without problem

for multi-input and multi-output systems (MIMO). To see some examples, go to (G.

Scaglia, et al., 2009), (M. Serrano, et al., 2013), (F. A. Cheein, G. Scaglia, 2013), and

(G. Scaglia, et al., 2014).

As it was mentioned before, this control method assumes that variables are known,

that is to say that is possible measure them. Nevertheless, it is a current problem that the

instrumental needed to measure the different variables is not always available.

4. Simulation results.

In this section it will be shown the performance of the controller. Moreover, in order

to evaluate the operation of the controller, there were applied different initial conditions

to the cell concentration.

In section 2. were shown the initial variable values and the process parameters of

penicillin biosynthesis model (R. Luus, 1993). To test the controller behavior in normal

conditions, there were used Table 1 and Table 2. The controller parameters are shown in

Table 3.

Table 3. Controller parameters

ki Value

k1 0.85

k2 0.85

k3 0.8

Figure 2 shows the tracking of the real and the references variables.

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Figure 2: Real and Reference profiles.

Looking Fig. 2 the real profiles reach and follow the references with minimal error,

and this error tends to cero as the process moves forward.

Figure 3 shows the tracking of the cells concentration (X) along the time of the

simulation. In that figure can be seen four traces, the red one represent the reference

(XR), the blue one correspond to the obtained tracking with the same initial conditions

(XS) as the reference, while green and black refer to the tracking with 20% more in

initial cells concentration (XM) and 20% less (XL), respectively.

Fig. 3. Cell Concentration.

0 20 40 60 80 100 120 1400

5

10

15

20

25

30

35

Time[hs.]

Co

nce

ntr

atio

n[g

/L]

///

Vo

lum

e [L

]

Cell Reference

Cells

Product Reference

Product

Substrate Reference

Substrate

Volume Reference

Volume

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

Cells Concentration

Time[hs.]

Co

nce

ntr

atio

n[g

/L]

XR

XS

XM

XL

Fig. 2 Cells Concentration

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Looking Figure 3 it can be concluded that whatever the initial conditions are, the

controller reaches and remains the reference perfectly, making the errors tend to zero,

and proving the excellent performance that it has.

Conclusion.

This work presents a new control technique based on linear algebra. With this

procedure can be designed multivariable controllers and highly nonlinear systems, such

as those of biotechnological processes, between others. Moreover, the controller

designed has many advantages: this technique has less mathematically complexity than

many others; it can be applied in a grate variety of systems and is versatile against

different changes and disturbances in the process and the system.

As it is shown in the simulation section, giving different initial conditions to the cell

concentration, the controller manages to achieve the reference profile successfully, and

that proves its good performance.

Acknowledgements.

A gratefully recognize to the National Council of Scientific and Technological

Research (CONICET) for funding this project, and for the Institute of Chemical

Engineering (IIQ) of the National University of San Juan (UNSJ) for their continued

collaboration.

References.

A. Herr, R. Fischer, “Improvement of Aspergillus nidulans penicillin production by targeting AcvA to

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