Biomass Concentration Estimation using the Extended JordanObservable Form

D. Tingey, K. Busawon and M. Saif

Abstract– In this paper, we propose a new observer designfor the estimation of the biomass concentration in a bioreac-tor, using the measurements of the substrate concentration.We consider two cases: in the first, a full knowledge of thespecific growth rate is assumed and in the second, only apartial knowledge of the latter is assumed. We show thatthe bioreactor model is in a special canonical form calledthe Extended Jordan Observable Canonical (EJOC) form. Itis shown, in particular, that an observer design for systemsin the EJOC form is fairly simple and straightforward. Noparticular nonlinear transformation is needed in the designprocedure. In addition, the gain of the observer is adaptivesince it is a function of the input and the output of thesystem, and can be tuned in order to obtain a desirable rateof convergence.Keywords: Biological systems, Observers, Nonlinear sys-

tems.

I. INTRODUCTION

Bioreactors are largely used in pharmaceutical in-

dustries and modern wastewater treatment plants. In

short, a bioreactor is a tank reactor in which living

microorganisms (biomass) such as bacteria, consume a

nutrient (substrate) in order to grow. One of the main

tasks of bioreactor control is the monitoring and control

of the biomass growth or concentration. However, such

tasks are not easy due to the lack of reliable sensors for

the measurement of the biomass concentration. For this

reason, the problem of observer design for the estimation

of the biomass concentration in a bioreactor has been

the focus of a great deal of research during past two

decades (see e.g. [1], [7], [8], [10]). In addition, the

mathematical model describing the underlying reaction

is highly nonlinear, and, since the theory of nonlinear

estimation is far from complete, a variety of approaches

and methods have been proposed to tackle the above

problem.

In e ect, a large class of bioreactor models is described

by [1]:

S = D(Sin S) YsµX

X = µX DX(1)

where X denotes the biomass concentration; S denotesthe substrate concentration; µ is the biomass’ specificgrowth rate; D is the dilution rate; Sin is the influent

This research is funded and supported by the EPSRC.D. Tingey and K. Busawon are with Northumbria University,

School of Engineering and Technology, Ellison Building, NewcastleUpon Tyne, NE1 8ST, UK, krishna.busawon@unn.ac.ukM. Saif is with the School of Engineering Science, Simon

Fraser University, Burnaby, British Columbia V5A 1S6, Canada.saif@cs.sfu.ca

substrate concentration and Ys is the yield coe cient for

substrate concentration. In general, the specific growthrate µ is not well-known, but the most commonly usedmodel is the well-known Monod model:

µ =µmaxS

KS + S(2)

where µmax is the maximum specific growth rate andKS is the saturation constant. Such a model has beenemployed in [7], [10] for the estimation of kinetic rates

and biomass concentration using high gain observers. In

[8], a bounded error observer has been given to estimate

the biomass concentration with partial knowledge of

the kinetic growth rate. In [9], the concept of interval

observers has been proposed in order to deal with

the kinetic growth uncertainty to estimate the biomass

concentration within acceptable bounds.

In this paper, we propose a new observer design for the

estimation of the biomass concentration using the above

model (1). We assume that the substrate concentration

S is measured, and that the specific growth rate µ isgiven by (2). We consider two cases: in the first, weassume a full knowledge of the specific growth rate andin the second, only a partial knowledge of the latter

is assumed. The key idea of the observer design is to

show that the above bioreactor model is in a special

observable canonical form which we call the Extended

Jordan Observable Canonical (EJOC) form. The EJOC

form is an extension of the Jordan canonical observable

form described in [3]. We show that an observer design

for systems in the EJOC form is fairly simple and

straightforward. As a result, the main advantage of the

proposed observer over the ones given in [7], [8], [10]

lies in its simplicity of design. No particular nonlinear

transformation is needed in the design procedure. In

addition, the gain of the observer is adaptive since it

is a function of the input and the output of the system,

and can be tuned in order to obtain a desirable rate

of convergence. In particular, we show that the observer

performs very well in the presence of measurement noise.

This is a clear advantage over the constant high gain

observers proposed in [7]; since it is well-known that

high gain observers have a tendency to amplify noise.

An outline of the paper is as follows: In the next

section, an observer design methodology is proposed for

single-output systems in EJOC form. In Section 3, the

design is applied to the bioreactor model given by (1) for

the estimation of the biomass concentration. Simulation

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

MA5.4

0-7803-9354-6/05/$20.00 ©2005 IEEE 143

results are provided to show the performance of the

proposed observer. Finally, some conclusions are drawn.

II. MAIN RESULT

Consider the following class of single-output systems

described by:

x = F (u, y)x+ g(u, y)y = Cx

(3)

where x Rn, u Rm and y R. The matrices F (u, y)and C are of the following form:

F (u, y) = J (u, y) + L (u, y)C (4)

where J (u, y) is an n-dimensional Jordan block definedby:

J (u, y) =

(u, y) (u, y) 0

0 (u, y). . .

.... . .

. . . (u, y)0 · · · 0 (u, y)

, (5)

L (u, y) =

l1 (u, y)l2 (u, y)...

ln (u, y)

(6)

and

C = 1 0 · · · 0 . (7)

We assume that:

A1) The function g(u, y) and the entries of the

matrix F (u, y) are smooth and bounded.A2) The function (u, y) > 0 for all u and y.

Remark 1

1) Note that if the functions (u, y), (u, y) and thevector L (u, y) are constant then the system (3)

is said to be in the Jordan observable canonical

form [3].

2) If, in particular, (u, y) = 0, (u, y) = 1 and thevector L (u, y) is constant, then system (3) is in

the Brunovskii observable canonical form.

3) Finally, since (u, y) = 0 for all u and y, it can bechecked that the rank of the observability matrix

(u, y) =

CCF (u, y)

...

CFn 1(u, y)

is equal to n for all u and y. Hence, system (3) is

observable for all u, or uniformly observable.

Based on the above remarks, we shall say the above

class of systems is of the Extended Jordan Observable

Canonical (EJOC) form. In fact, many physical systems

are already in the above form. In particular, as it will

be shown in the next section, the class of bioprocesses

considered in this paper is in the above form.

Note that the matrix J (u, y) can be written as:

J (u, y) = (u, y) In + (u, y)A (8)

where In is the n-dimensional identity matrix and

A =

0 1 0 · · · 0

0 0 1. . .

......

. . .. . .

. . . 0...

. . . 0 10 · · · · · · 0 0

. (9)

Now consider the following system:

.

x= F (u, y)x+ g(u, y) + K(u, y) (y Cx) (10)

where

• K(u, y) = (u, y)D 1K + L (u, y)• D = diag 1 1

2 · · · 1n where

˙ (t) =1

n(u, y) (t) if (u, y) 0

(u, y) (t) if (u, y) > 0; (t0) > 0

• K is chosen such that the matrix A KC is stable.

We can now state the following:

Theorem 1. Assume that system (3) satisfies Assump-tions A1)-A2). Then, the system (10) is an exponential

observer for system (3).

Proof:

Defining the estimation error as e = x x, we have

e = Fe KCe

= Fe D 1K + L Ce

= (J + L C)e D 1KCe L Ce

= J D 1KC e.

The arguments of the various matrices have been

dropped for the sake of convenience.

On the other hand, since J (u, y) = (u, y) In +(u, y)A, we have

e = Ine+ A D 1KC e.

Now, let = D e. Then,

˙ = D e+ D e

= D e+ D A D 1KC e+ D e

= D D 1 + D A D 1KC D 1 + D D 1 .

It can be verified that D AD 1 = A, CD 1 = C and

D D 1 =˙

where = diag 1 2 · · · n .Consequently,

˙ = In + (A KC)˙

= In˙

+ (A KC) .

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Now, since the matrix (A KC) is stable there exists asymmetric positive definite matrix P such that:

(A KC)TP + P (A KC) = In. (11)

Consider the following candidate Lyapunov function

V ( ) = TP . We then obtain:

V = 2 TP ˙

= 2 TP In˙

+ (A KC)

= 2 TP In˙

+ 2 TP (A KC)

= 2 TP In˙

T .

We now have two cases to study: 0 and > 0.

Case 1: If (u, y) 0 for all u and y; that is (u, y) =

0(u, y) with 0(u, y) 0 for all u and y, then

V = 2 TP 0In +˙

T .

If we set˙=

0

n, then

V = 2 0TP In

1

nT .

It can easily be verified that 2 0TP In

1

n0,

so that V 0.

Case 2: If (u, y) > 0 for all u and y, then

V = 2 TP In˙

T .

If we set˙= , then

V = 2 TP ( In)T .

It can easily be verified that 2 TP ( In) 0, sothat V 0.

This completes the proof of Theorem 1.

Remark: Note that if (u, y) = 0 then ˙ = 0 which, inturn, implies that is constant. This amounts to the

observer used in [11].

III. APPLICATION TO A BIOREACTOR

In this section, we present an observer design for the

estimation of biomass concentration in a bioreactor in

two cases. In the first case, we assume full knowledgeof the specific growth rate, whereas in the second weassume only a partial knowledge of the latter.

A. The case where µ is known

Consider the biological system described by (1). As-

suming that the substrate concentration S is measured,and by setting the coordinates of the system as: x1 =S, x2 = X, and u = D we obtain the following state

space representation of system (1):

.x1= Ysµx2 u(Sin + x1).x2= (µ u)x2y = x1

(12)

where we use the well-known Monod law as given by (2),

and assume full knowledge of µ.The above system can be expressed as

.x= F (u, y)x+ g (u, y)y = Cx

where

F (u, y) = J (u, y) + L (u, y)C

with

J (u, y) =(µ u) Ysµ0 (µ u)

,

L (u, y) =(µ u)

0,

C = 1 0

and

g (u, y) =u (Sin + x1)

0.

Here, (u, y) = (µ u) and (u, y) = Ysµ.By practical considerations the functions

g (u, y) , (u, y) and (u, y) are all bounded. In

addition, (u, y) > 0 for all u and y. We can thereforedesign an observer of the form (10) as follows:

.

x1= Ysµx2 u(Sin + x1) + 1 (x1 x1).

x2= (µ u) x2 + 2 (x1 x1)(13)

where

K(u, y) = 1

2= (u, y)D 1K + L (u, y)

with K =21

. That is,

1 = 2Ysµ µ+ u

2 = Ysµ2

where

˙ (t) =1

n(µ u) (t) if (u, y) 0

(µ u) (t) if (u, y) > 0; (t0) > 0

Note that the above observer will also function for any

other model of the specific growth rate other than theMonod law.

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B. The case where µ is partially known

In practice, as previously mentioned, the parameter

µ = µmaxS

KS+Smay not be perfectly known. Assume that

some nominal values of µmax = µmax and KS = KS areknown so that we can decompose µ as follows:

µ =µmaxS

KS + S+

µmaxS

KS + S

µmaxS

KS + S= µ0 + µ

where µ0 is the known nominal part and µ is unknown.Then,

S = D(Sin S) Ys (µ0 + µ)X

X = (µ0 + µ)X DX. (14)

Using the same change of variable as before x1 =S, x2 = X, and u = D, we obtain the following state

space representation of system (14):

.x1= Ys (µ0 + µ)x2 u(Sin + x1).x2= (µ0 + µ u)x2y = x1

. (15)

The above system can be expressed as:

.x= (F (u, y) + F (u, y))x+ g (u, y)y = Cx

where

F (u, y) = J (u, y) + L (u, y)C

with

J (u, y) =(µ0 u) Ysµ00 (µ0 u)

,

L (u, y) =(µ0 u)

0,

and

F (u, y) =0 Ys µ0 µ

.

Here (u, y) = (µ0 u) and (u, y) = Ysµ0.In such a case, one can show that the following system:

.

x1= Ysµ0x2 u(Sin + x1) + 1 (x1 x1).

x2= (µ0 u) x2 + 2 (x1 x1)(16)

where

K(u, y) = 1

2=

2Ysµ0 µ0 + uYsµ0

2

with

˙ (t) =1

n(µ0 u) (t) if (u, y) 0

(µ0 u) (t) if (u, y) > 0; (t0) > 0,

is an asymptotic observer for system (14).

C. Simulation results

The values of the model parameters used are:

X(0) = 0.5g/l Sin = 5g/l

S(0) = S(0) = 10.0g/l Ks = 4g/l

D = 0.1h 1 Ks = 3.8g/lµmax = 0.3h

1 Ys = 10µmax = 0.285h

1 k1 = 2

X(0) = 0.7g/l k2 = 1

.

In order to show the good convergence performance

both in the absence and in the presence of measurement

noise, measurements of S were corrupted by an additivenoise with an amplitude equivalent to 10% of the

corresponding values. The initial value of the adaptive

tuning parameter (u, y) was chosen as 0 = 2.

The first set of simulations conducted demonstratesthe performance of the adaptive gain observer where it

is assumed that the specific growth rate is known. Thisis illustrated by Figures ?? and ??, which show the real

(in solid lines) and estimated (in dotted lines) biomass

concentrations in the absence and in the presence of

measurement noise, respectively. It can be seen that the

convergence is smooth in both cases. Figure ?? shows

the variation of the adaptive tuning parameter (u, y).It can be observed that the tuning parameter adapts to

the variations of the output.

A further set of simulations was carried out in order to

show the performance of the adaptive gain observer for

the case where the specific growth rate is only partiallyknown. The results can be seen in Figures ?? and ??,

which show the real (in solid lines) and estimated (in

dotted lines) values of the biomass concentrations in

the absence and in the presence of measurement noise,

respectively. It can be seen that the convergence is

smooth in both cases.

Biomass estimation when µ is known

146

Biomass estimation under noisy measurement when µis known

Adaptive tuning parameter (u, y)

Biomass estimation when µ is partially known

Biomass estimation under noisy measurement when µis partially known

IV. CONCLUSIONS

In this paper, we have presented a new observer design

strategy for a class of single-output state a ne systems

which is in the Extended Jordan Observable Canonical

(EJOC) form. It is shown that observer design for such

systems is fairly simple and straightforward. No nonlin-

ear transformation is needed in the design procedure.

In particular, we have proposed a new observer design

for the estimation of the biomass concentration in a

bioreactor, using the measurements of the substrate

concentration. The observer gain is adaptive since it is

a function of the input and output of the system.

Simulation results have shown the good convergence

performance of the proposed observer in two cases. In

the first case, full knowledge of the specific growth rateis assumed, and in the second, only a partial knowledge

of the latter is assumed. Good convergence performance

is shown both in the absence and in the presence of

measurement noise.

References

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[2] G. Bornard, N. Couenne, and F. Celle. “Regularly persistentobserver for bilinear systems”, Proc. Colloque Internationalen Automatique Non Lineaire, Nante. (1988).

[3] K. Busawon. “On Jordan observable and controllable canoni-cal forms”, 1st African control conference, Cape Town, SouthAfrica. (2003).

[4] K. Busawon. “Sur les observateurs nonlineaires et le principede separation”, PhD thesis, Claude Bernard University, Lyon,France. (1996).

[5] K. Busawon, M. Farza, and H. Hammouri. “Observer designfor a special class of nonlinear systems”, Int. Journal ofControl, Vol. 71, pp. 405-418. (1998).

[6] K. Busawon, M. Saif and D. Tingey. “Nonlinear control de-sign using the extended Jordan controllable canonical form”,Mediterranean Journal of Measurement and Control, Vol.1,No.1, pp. 44-49. (2005).

[7] M. Farza, K. Busawon and H. Hammouri. “Simple NonlinearObservers for On-line Estimation of Kinetic Rates in Bioreac-tors”, Automatica, Vol. 34, No. 3, pp. 301-318. (1998).

[8] J-L. Gouzé and V. Lemesle. “A bounded error observerwith adjustable rate for a class of bioreactor models”, InProceedings of the European Control Conference, ECC 2001.Porto, Portugal. (2001).

[9] J. L. Gouzé A. Rapaport, M.Z. Hadj-Sadok. “Interval ob-servers for uncertain biological systems”, Ecological ModellingVol. 133, pp. 45-56. (2000).

[10] R. Oliveira, E. Ferreira, S. Feyo de Azevedo. “A study on theconvergence of observer-based kinetics estimators in stirredtank reactors”, Journal of Process Control, Vol. 6 (6), pp.367-371. (1996).

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