Archives of Control Sciences Volume 13(XLIX), 2003 No. 1, pages 5-19

Archives of Control Sciences

Volume 13(XLIX), 2003

No. 1, pages 5�19

Robust compensator control of aerobic continuous

fermentation processes with state estimation

IVAN SIMEONOV, STOYAN STOYANOV and VELISLAVA LUBENOVA

The paper deals with robust compensator control of continuous aerobic fermentation pro-

cesses described by a set of non-linear differential equations. For design purposes the non-linear

model is transformed into a linear one with interval parameters. The robust state-space compen-

sator is designed according to the internal model principle. Biomass concentration is estimated

on-line by an observer. Substrate concentration is obtained by model-based or indirect mea-

surement. The theoretical results are veri�ed by simulations in different cases. Robustness and

simple realisation are important features of the proposed algorithms.

Key words: fermentation processes; non-linear model; biomass and substrate concentra-

tion estimation; robust compensator control; simulation

1. Introduction

Bioprocesses are characterised by non-linear dynamics, time-varying parameters,

lack of reproducibility of experimental results and lack of cheap and reliable sensors

capable of providing on-line measurements of main process variables and parameters

[1,2,10,11].

Continuous fermentation processes are very perspective ones with their effectiveness

and productivity. Because of the very restrictive on-line information, the control of these

processes is often reduced to the regulation of one or more variables at desired values in

the presence of some perturbations [1,2,10,11]. Investigations show that classical linear

control algorithms have no good performances in this case [2]. More sophisticated non-

linear, linearizing, with variable structure and adaptive control algorithms for continuous

fermentation processes have been studied [1,10,12], but due to some implementation

dif�culties they are not so popular in practice.

A theoretical idea of robust compensator control of continuous fermentation pro-

cesses is presented in [8]. The proposed algorithm is with good robustness and sim-

ple realisation but it is supposed that reliable sensors for on-line measurement of main

I. Simeonov is with Institute of Microbiology, Bulgarian Academy of Sciences, Acad.G.Bonchev str.,

bl.26, 1113 So�a, Bulgaria, Fax:359(2)700109, e-mail: [email protected]. S. Stoyanov is with Tech-

nical University, Department of Automatics, So�a, Bulgaria. V. Lubenova is with Institute of Control and

System Research, Bulgarian Academy of Sciences, So�a, Bulgaria.

This work was supported by the Bulgarian National Scienti�c Fund under contract TH-1004/00.

6 I. SIMEONOV, S. STOYANOV, V. LUBENOVA

process variables are available. Stable observers of biomass concentration for aerobic

fermentation processes that need only on-line measurements of oxygen uptake rate are

presented in [4,5,6].

The aim of this paper is to study the possibility of controlling continuous non-linear

fermentation processes by linear robust control with estimation of biomass and substrate

concentrations.

2. Problem statement

A wide class of continuous aerobic fermentation processes can be described by the

following dynamical model

_X = (��D)X = Rx �DX; (1)

_S = D(S0 � S)� Ys�X = D(S0 � S)� YsRx; (2)

where X and S are state variables (biomass and substrate concentrations); D is control

input (dilution rate); S0 is main external perturbation (in�uent substrate concentration);

Rx is biomass growth rate and Ys is yield coef�cient for substrate consumption. Speci�c

growth rate � is accepted to be presented by the widely used Monod kinetic relation

� =�maxS

Ks + S; (3)

where �max � maximum speci�c growth rate, Ks � saturation constant.

In practice state variables (X and S), external disturbance ( S0) and control input

(D) belong to known intervals, depending on technological and biological requirements

0 < Xmin ¬ X ¬ Xmax; (4)

0 < Smin ¬ S ¬ Smax; (5)

0 < S0min ¬ S0 ¬ S0max; (6)

0 < Dmin ¬ D ¬ Dmax: (7)

For model (1), (2) and (3) it is assumed that:

� biomass growth rate (Rx) is an unknown time-varying parameter which is non-

negative and bounded, with bounded time derivative;

ROBUST COMPENSATOR CONTROL OF AEROBIC CONTINUOUS FERMENTATION PROCESSES 7

� concentration of biomass (X) cannot be measured on-line, substrate concentration

(S) is dif�cult to be measured, while the oxygen uptake rate (G) can be measured

on-line;

� oxygen uptake rate (G) is related to X and Rx by

G = YcRx + gX (8)

where yield coef�cient for oxygen consumption (Yc) and maintenance rate (g) are

constant and known. The on-line measurements of oxygen uptake rate (G) can be

computed using the following relationship [1,3]

G = Qin �Qout (9)

where input and output gaseous oxygen �ow rates, Qin and Qout respectively, are

measured on-line. In practice, oxygen �ow rates are computed using the relation-

ships:

Qin = FinCin=V Qout = FoutCout=V

where input and output air �ows Fin and Fout, oxygen concentrations Cin and

Cout in the input and output air �ows and reactor volume V are directly measur-

able on-line. In fact, the above relationships are derived by considering the low

solubility of oxygen in the liquid phase and by neglecting the liquid-gas transfer

dynamics of this substrate.

In this case it is assumed that substrate concentration S(t) is the process output. Theproblem is to design a robust control algorithm that ensures good dynamical properties

of the transfer from one to another steady-state (set point changes of S) and to keep S

close to a prespeci�ed value S� when changes in in�uent substrate concentration S0 or

variations of process kinetics occur.

3. Robust compensator control design

3.1. Robust compensator

A possible approach to solve this problem is the linearization of the non-linear model

in selected points and the design of a linear control algorithm. However, in this case an

interval instead of a point exists. Therefore, it is necessary to take into account the whole

working interval of S(t), which leads to the necessity of robust control.

The robust compensator design must ensure:

1. Closed-loop stability of the non-linear process (1), (2) and (3) in the whole se-

lected area.


2. Asymptotic tracking of step-wise changes of substrate concentration set points.

3. External disturbances (S0 changes) rejection.

In this work a transformation of the non-linear model (1), (2) and (3) into a linear one

with time-varying parameters is suggested [2]. After the transformation the new model

description is:

_X = �11(X;S;D)X + �12(X;D; S)S + �1(X;D; S)L (10)

_S = �21(X;S;D)X + �22(X;D; S)S + �2(X;D; S)D (11)

where �ij and �i, (i; j = 1; 2) are non-linear continuous functions of X , S and D [10].

For the transformed model (10) and (11) they are given by the following expressions:

�11 =�maxS

Ks + S�D �12 =

Ks�maxX

(Ks + S)2

�21 =Ys�maxS

Ks + S�22 =

�YsKs�maxX

(Ks + S)2

�1 = �X �2 = S0 � S:

When the dilution rate belongs to the interval (0;Dmax) and since �ij(X;S;D) and

�j(X;S;D) are continuous functions, model (10) and (11) can be presented as a linear

one with time varying interval coef�cients in the following matrix form

d

dt

��11 �12�21 �22

�=

�X

S

�+

�b1b2

�D; (12)

where the coef�cients in equation (12) belong to the following intervals:

a�ij = inf �ij ¬ sup�ij = a+ij ; i; j = 1; 2;

b�i = inf �i ¬ sup�i = b+i ; i = 1; 2:(13)

In these intervals uncertainties or variations of kinetic parameters �max and Ks are pos-

sible to be included. Every coef�cient is a sum of its nominal value (calculated as a mean

of the admissible interval) and an uncertain part

aij = a0ij + fij; bi = b0 + vi; i; j = 1; 2:

Then model (12) is transformed into

dx

dt= [A0 +�A(f)]x(t) + [b0 +�b(v)]D(t);

(14)

S(t) = cx(t);


where

A0 =

�a011

a012

a021

a022

�; �A(f) =

�f11 f12f21 f22

�;

bo0 =

�b01

b02

�; �b(v) =

�v1v2

�; c = [0 1];

and xT = [X S] is state vector, f = [fij], v = [vi], i; j = 1; 2, f 2 F , v 2 V (f

and v � matrix and vector of parameter uncertainties), matrices �A(f) and �b(v) arecontinuous, a0ij , b

0

i are nominal coef�cients of model (14). When fij = 0, vi = 0, thedescription of the nominal system (A0, b0) is obtained.

The control design is suggested to be carried out in two stages [7,9]. A feedback

matrix K0 for the nominal system (A0 ,b0) of (14) is designed in the �rst stage, which

ensures its stability. In the second stage an additional feedback matrix �bmK is ob-

tained, which compensates the parameter uncertainty. Finally, the feedback matrix is

K = K0 + �K, which ensures interval stability of the uncertain system (14). The

following assumptions are taken as standard ones [7]:

A1: Uncertainty sets F and V are compact;

A2: Matrix functions A(f) and b(v) are continuous;A3: Pair (A0; b0) is controllable;A4: The following condition holds

rank

�A0 b0c 0

�= n+ 1: (15)

A5: Matching conditions are available � continuous matrix functionsM(:) andE(:)exist so that

�A(f) = boM(f); �b(v) = boE(v);

I + 0:5[E(v) +ET (v)] > 0; for all f 2 F; v 2 V:

According to the above standard assumptions and the principle of the internal model,

it is necessary the nominal model of (14) (�A = 0; �b = 0) to be augmented by the

following dynamical system:

dq(t)

dt= e(t) = S�(t)� S(t); q(t0) = q0;

(16)

up(t)� kmq(t);


where q(t) is internal model state, up � the output of system (16). The augmented nom-

inal system (Au; bu) can be stabilised in the �rst stage by the following state feedback

D =K0z = [kx ks km ]

24 X

S

q

35 : (17)

Then the following description of the nominal closed loop system is obtained:

d

dt

24 X

S

q

35 =

24 a0

11a012

0a021

a022

00 �1 0

3524 X

S

q

35+

24 b0

1

b02

0

35 [kx ks km]

24 X

S

q

35+

24 0

01

35S�;(18)

S = [0 1 0]z

where z = [X;S; q]T is the state vector of the nominal closed loop system. If the system

(18) behaviour is de�ned by the following desired characteristic polynomial

Hd(s) = s3 + �2s2 + �1s+ �0; (19)

we can calculate the coef�cients of the nominal feedback matrixK0 = [kx ks km].In order to ensure robust stability of the linear interval system (14), it is necessary to

design an additional state feedback �K so that closed-loop uncertain system (�A 6= 0,�b 6= 0) becomes intervally stable for all f 2 F , v 2 V . According to the approach in

[7], the additional state feedback matrix �K is

�K = ��bTuP ; (20)

where � is scalar, P is a positive de�nite matrix and symmetric solution of the Lyapunov

equation

(Au + buK0)TP + P (Au + buK0) = �Q: (21)

MatrixQ is chosen to be positive de�nite and symmetric one. If � > ��, the requirement

for closed-loop interval stability will be satis�ed (some additional information can be

found in [9]). The calculation of �� can be quite complex. The required matrices are

often ill-conditioned and/or nearly singular. This makes the computation of �� not only

dif�cult but also subject to numerical inaccuracy. An alternative method for determining

a suitable value of � is based on iterative searching and leads to smaller gains in the

robust compensator.

3.2. Biomass concentration observer

It is assumed that only a noisy measurement G(t) of oxygen uptake rate is available

on-line

Gm(t) = G(t) + "(t); (22)


where "(t) is measurement noise. A stable observer of biomass concentration is designed

according to [6]:

dRx

dt=

1

yc

dGm

dt�

�g

Yc+D

�Rx +D

Gm

Yc+ C1

�Gm � YcRx � gX

�

(23)

dX

dt=

Gm

Yc�

�g

Yc+D

�X + C2

�Gm � YcRx � gX

�

where Rx and X are estimated values for Rx and X; G = ycRx + gX is the predicted

value of oxygen uptake rate; C1 and C2 are observer parameters which must be chosen

according to stability conditions.

The design of the above-presented observer (23) is based on such dynamical models

of biomass growth rate and biomass concentration which depend only on this unknown

time-varying parameter and this state variable. The proposed observer with linear struc-

ture allows a simple tuning procedure and has the advantage of estimating biomass con-

centration when only oxygen uptake rate is measurable. The results in [6] show that

under wrong initial conditions of Rx and X , the convergence rates of both estimates Rx

and X to their true values depend on experimental conditions through the values of dilu-

tion rates and kinetic parameters. It is proved that the structure of the X observer allows

to improve the convergence rate of the estimate, Rx (or X ) in the case that an exact

initial condition of X (or Rx) is known. This is realised by setting C1 (C2 respectively)

equal to zero. In the paper the above-proposed simple tuning procedure is used since it

is connected with practically realisable requirements.

3.3. Substrate concentration estimation

In some cases sensors for substrate (e.g. glucose) concentration are available [10].

However, in general, the on-line measurement of substrate concentration is a very dif�-

cult problem in industry and another way of estimating this variable is needed.

Model-based estimation

For real-time substrate concentration estimation the second equation of model (1),

(2) is used. In our case the estimates of Rx are supplied by the above-described observer.

Assuming that So is known (constant or measurable), from the equation (2) the following

result is obtained

dS

dt= D(S0 � S)� YSRx (24)

This equation is used for the calculation of substrate concentration estimate (S).

Indirect estimation


If S0 is unknown, another way of indirect substrate estimation is proposed. From

equations (1) and (3) the following relation is obtained

� =Rx

X=

�maxS

Ks + S(25)

From (25) it is obvious that the following equation for indirect substrate concentration

estimation may be used

S =Ks�

�max + �(26)

If �max and Ks are known constants, S is calculated using the estimates of X and Rx

obtained from the biomass observer.

Finally the control algorithm may be presented as follows:

D = (K0 +�K)

24 X

S

q

35 =

24 kx +�kx

ks +�kskm +�km

35T 24 X

S

q

35 (27)

with q, X and S calculated by (16), (23) and (24) or (26) respectively (with S instead of

S in (16).

The closed-loop system structure in case of biomass observation and indirect sub-

strate estimation is shown on Fig.1.

Figure 1. System structure

In practice, it is necessary to take into account the technological bounds on �ow rate,

and control law is then implemented as follows:

�D =

8<:

D if 0 ¬ D ¬ Dmax

0 if D < 0Dmax if D Dmax

(28)


where D is the value of the control calculated by the compensator, and D is the in-

put, which is effectively applied to the process. Dmax is a constant calculated to ensure

closed-loop system stability in the whole work area of substrate concentration S(t) andcontrol D(t) must be not greater than Dmax. If this requirement is not met, then the

so-called "washout" of microorganisms exists and is not desirable [2,10].

4. Numerical example and simulation studies

A fermentation process with mathematical description (1)-(3) is considered. The co-

ef�cients of the model are: �max = 0:33 h�1; Ks = 5 g/l; S0 = 5 g/l, Ys = 2. Oxygenuptake rate equation (8) coef�cients have the following values: Yc = 1:8; g = 0:006h�1. The values of estimation algorithms parameters are: C1 = 11:11 and C2 = 0. Atransformation of non-linear model (1), (2) and (3), when D 2 (0; 0:32), into linear

interval model (14) is performed, where:

A0 =

�0 0:07644

�0:17 �0:2378

�; �A =

�0:075 0:073860:15 0:07273

�;

b0 =

��1:284472:568935

�; �b =

�1:137412:2748

�;

�0:075 ¬ a11(t) ¬ 0:075; 0:00258 ¬ a12(t) ¬ 0:1503;

�0:32 ¬ a21 ¬ �0:02; �0:38558 ¬ a22(t) ¬ �0:08985;

�2:24188 ¬ b1(t) ¬ �0:17706; 0:29412 ¬ b2(t) ¬ 4:84375:

The desired characteristic polynom is chosen as follows

Hd(s) = s3 + 1:3178s2 + 0:4005s + 0:0324;

so that nominal closed-loop system has all its eigenvalues in the strict left half plane.

The following coef�cients of nominal feedback matrix are obtained

K0 = [kx ks km] = [�0:0368 � 0:0495 0:0578]:

The obtained nominal closed loop system is asymptotically stable and has eigenvalues

�1;2 = �0:1164 � j0:3672, �3 = �0:085. If the process parameters belong to the

above-obtained intervals but are different from their nominal values, it is possible for the

closed-loop system to become unstable, i.e. the nominal closed loop system is asymp-

totically stable, but the uncertain system is not intervally stable. It is necessary to design


an additional feedback matrix �K which compensates parameter uncertainties and en-

sures closed-loop system stability when the process parameters belong to their intervals.

If � = 0:12, additional feedback matrix �K is

�K = [2:0853 � 1:3644 1:076]:

Then the design is completed by calculating feedback matrix

K =K0 +�K = [2:0486 � 1:4139 1:1344]:

The eigenvalues of the closed-loop system are: �1 = �5:9246, �2 = �0:085, �3 =�0:4919. It is not dif�cult to verify that the closed-loop interval system is asymptotically

stable for all f 2 F , v 2 V and has good behaviour.

Figure 2. Simulation results for step changes of S� in the case of X observation and S measurement

Figure 3. Simulation results for step changes of So in the case of X observation and S measurement


Figure 4. Simulation results for step changes of S� in the case of X observation and model-based S esti-

mation

The above-presented robust control algorithm is applied on the non-linear model (1),

(2) and (3) of the process. Simulations of the system are carried out with step changes

of set points (S�) and of in�uent substrate concentration (S0) in their admissible ranges

with and without measurement noise. Different ways of measurement or estimation of

the variables which are essential for the control algorithm have been studied:

1. Oxygen uptake rate (G) and substrate concentration (S) are measurable and

biomass concentration (X) is estimated by the observer. Evolution of biomass

(X), biomass estimate (X), control input (D) and substrate concentration (S) for

step-wise set point (S�) changes and S0 = 5 = const are shown on Fig.2. The

same but for step changes of S0 (for S� = 1:6 = const) is presented on Fig.3.

2. Oxygen uptake rate (G) is measurable, 0 is available (constant or measurable),

biomass concentration (X) is observed (X) and model � based substrate concen-

tration estimation by equation (24) is realised. Evolution of X , X , D, S and S

without (Fig.4) and with 5% noise on the measurements of G (Fig.5) for step-

wise set point (S�) changes and S0 = 5 = const are shown. The same but for

sinusoidal changes of S0 (for S� = 4 = const) is presented on Fig.6.

3. Oxygen uptake rate (G) is measurable, biomass concentration (X) is observed

(X) and substrate concentration (S) is estimated indirectly (S) by equation (25).

Evolution of X , X , D, S and S for step-wise set point S� changes and S0 =5 = const are shown on Fig.7. The same but for sinusoidal changes of S0 (for

S� = 4 = const) is presented in Fig.8.

In all of the three above-mentioned cases, the designed robust control algorithm

satis�es the requirements for step-wise set points asymptotic tracking and disturbance

rejection without control saturation.


Figure 5. Simulation results for step changes of S� in the case of X observation and model-based S esti-

mation with 5% noise on G

Figure 6. Simulation results for sinusoidal changes of So in the case of X observation and model-based S

estimation with 5% noise on G

5. Conclusion

Linear robust state compensator control of non-linear continuous fermentation pro-

cesses has been developed. For design purposes the non-linear process model has been

transformed into a linear one with interval parameters. The control algorithm has been

designed on the basis of the internal model principle and ensures robust asymptotic track-

ing of the set points and external disturbances rejection in the whole range taken. Dif-

ferent ways of state variables estimation have been analysed and studied . An observer

of biomass concentration has been presented for those aerobic processes where the mea-

surement of oxygen uptake rate is available on-line. The design is based on dynamical

models of biomass growth rate and biomass concentration which depend only on this

unknown time-varying parameter and on this state.

The obtained theoretical results have been veri�ed by simulation. Simulations show

that in most of the cases a very good dynamical behaviour of the system exists. If model-


Figure 7. Simulation results for step changes of S� in the case ofX observation and S indirect measurement

Figure 8. Simulation results for sinusoidal changes of So in the case of X observation and S indirect

measurement

based substrate concentration estimation is realised and the value of S0 is unknown (vari-

able and unmeasurable on-line), the estimate S is displaced and a permanent regulation

error occurs. This fact may be accepted as a good proof of the theoretical results � when

S0 is not constant and unmeasurable, wrong estimates (S) are calculated on the base of

equation (24). If indirect substrate concentration estimation is realised by equation (26),

a bad in�uence of measurement noise on oxygen uptake rate (G) is observed. The latter

is due to the derivative of G in the equation of the observer (23).

The above-presented results prove the possibility of successful application of the

considered approach to the control of non-linear and uncertain continuous fermentation

processes. The main advantages of the proposed algorithms are their robustness and

simple realisation.


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Documents

Archives of Control Sciences Volume 13(XLIX), 2003 No. 1, pages 5-19