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Journal of Process Control 18 (2008) 71–79

Design and study of a risk management criterion for anunstable anaerobic wastewater treatment process

Jonathan Hess *, Olivier Bernard

INRIA (French National Institute for Research in Computer Science and Control), COMORE Research Team, Sophia Antipolis 06902, France

Received 11 August 2006; received in revised form 2 May 2007; accepted 16 May 2007

Abstract

In this paper, we consider an unstable biological process used for wastewater treatment. This anaerobic digestion ecosystem can havetwo locally stable steady states and one unstable steady state. We first study the model and characterise the attraction basin associated tothe normal operating mode. In the second step, we estimate the size of this attraction basin by using a simplified criterion that turns outto be a good approximation. Finally, we apply the approach on a real anaerobic digestion plant, and we show that the proposed criterionallows to rapidly detect the conditions of a destabilisation.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Haldane model; Anaerobic digestion; Nonlinear systems diagnosis

1. Introduction and motivation

Control of biological systems is a very delicate problemsince one has to deal with highly nonlinear systemsdescribed by poor quality models. In some cases this con-trol issue can be really crucial when perturbations canmove the system from one steady-state to another. Thisis especially the case for the anaerobic digestion process:a more and more popular bioprocess [2] that treats waste-water and at the same time produces energy through meth-ane (CH4). This process can also produce hydrogen (H2)under specific conditions. This complex ecosystem involvesmore than 140 bacterial species [7] that progressivelydegrade the organic matter into carbon dioxide (CO2)and methane (CH4). However, this process is known tobe very delicate to manage since it is unstable [8]: an accu-mulation of intermediate compounds can lead to the acid-ification of the digester.

0959-1524/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jprocont.2007.05.005

* Corresponding author. Tel.: +33 4 92 38 76 35; fax: +33 4 92 38 78 58.E-mail addresses: jonathan.hess@inria.fr (J. Hess), olivier.bernard@

inria.fr (O. Bernard).

To solve this problem, many authors have proposedcontrollers [10,14,9] that were able to ensure the local oreven the global stability of the system using the dilutionrate of the bioreactor as input.

These control laws are however difficult to apply inpractice due to the lack of available sensors on this typeof process. Moreover, by essence they act on the influentflow rate, and they may therefore not accept all the incom-ing wastewater. It means that this type of controllersimplies a storage of the wastewater to be treated. In prac-tice storage tanks are very small and this solution istherefore difficult to setup on a long term basis. As a con-sequence, the controllers are often disconnected at theindustrial scale and the plant manager manually operatesthe process trying both to avoid process destabilisationand wastewater storage.

The approach that we propose has the objective to pro-vide the operator with a risk index associated to his man-agement strategy. The idea is therefore to determine fromthe global analysis of the nonlinear system whether the pro-cess has been triggered to a dangerous working mode. Thisrisk index can also be used in parallel to a controller thatonly guarantees local convergence.

72 J. Hess, O. Bernard / Journal of Process Control 18 (2008) 71–79

The paper is organized as follows: in Section 2 a dynam-ical model of an anaerobic digestion process is recalled. Sec-tion 3 puts the emphasis on the analysis of the modeldynamics. A simple criterion to assess the stability of theprocess is set in Section 4, and finally this criterion is appliedto a real process to determine its destabilisation risk.

2. Model presentation

There exists numerous dynamical models for anaerobicdigestion, from the basic ones considering only one bio-mass [1] to detailed models including several bacterial pop-ulations and several substrates. Among complex modelsthe IWA Anaerobic Digestion Model 1 [4] has imposeditself as a useful tool to simulate a digestion plant withmore insight into the process dynamics. However, its exces-sive complexity makes any advanced mathematical analysisof the model critical.

We thus consider a simplified macroscopic model of theanaerobic process based on 2 main reactions [6], where theorganic substrate (S1) is degraded into volatile fatty acids(VFA, denoted S2) by acidogenic bacteria (X1), and thenthe VFA are degraded into methane CH4 and CO2 bymethanogenic bacteria (X2):

• Acidogenesis:

k1S1 !l1ðS1ÞX 1X 1 þ k2S2

• Methanogenesis:

k3S2 !l2ðS2ÞX 2X 2 þ k4CH4

where l1(S1) and l2(S2) represent the bacterial growthrates associated to these 2 bioreactions.

The mass balance model in the continuous stirred tankreactor (CSTR) can then straightforwardly be derived [3]:

_X 1 ¼ l1ðS1ÞX 1 � aDX 1 ð1Þ_S1 ¼ �k1l1ðS1ÞX 1 þ DðS1in � S1Þ ð2Þ_X 2 ¼ l2ðS2ÞX 2 � aDX 2 ð3Þ_S2 ¼ �k3l2ðS2ÞX 2 þ k2l1ðS1ÞX 1 þ DðS2in � S2Þ ð4Þ

where D is the dilution rate, S1in and S2in are respectively, theconcentrations of influent organic substrate and of influentVFA. The kis are pseudo-stoichiometric coefficients associ-ated to the bioreactions. Parameter a 2 (01] represents thefraction of the biomass which is not retained in the digester.We denote by n = (X1,S1,X2,S2)T the state vector.

In the sequel, we will consider the rather generic map-pings l1 and l2, satisfying the following properties:

Hypothesis 1. l1 is an increasing function of S1, withl1(0) = 0.

Hypothesis 2. l2 is a function of S2 which increases until aconcentration SM

2 and then decreases, with l2ðSM2 Þ ¼ lM

and l2(0) = 0.

For the numerical application in the real example, wewill consider the following kinetics that verify Hypotheses1 and 2:

l1ðS1Þ ¼ �l1

S1

S1 þ KS1

ðMonodÞ

l2ðS2Þ ¼ �l2

S2

S2 þ KS2 þS2

2

KI2

ðHaldaneÞð5Þ

In the mathematical analysis of this system, assumption ismade that the environment of the bacteria remains con-stant and we will thus assume that D, S1in and S2in are po-sitive constants. All the initial conditions are assumed to bepositive.

3. Model analysis

3.1. Analysis of the acidogenic dynamics

The subsystem (1) and (2) is close to a classical modelwith a Monod kinetics but slightly modified by the terma. This makes the study of this system less straightforwardthan for Monod model (with a = 1), for which the globalstability has been demonstrated (see e.g. [12]). However,its behaviour remains simple as stated in the followingProperty:

Property 1. Systems (1) and (2) with positive initial condi-

tions admit a single globally stable equilibrium. If aD <l1(S1in) this equilibrium is in the strictly positive orthant.

Proof. A detailed proof is given in Appendix B. h

It follows that the useful working point ðX �1; S�1Þ of sys-tems (1) and (2) are globally asymptotically stable. As aconsequence, we have the following property:

Property 2. After a transient time T, systems (1) and (2)

satisfy the inequality k1l1(S1)X1 6 DS1in.

Proof. First, let us note that at steady-state k1l1ðS�1ÞX �1 ¼DðS1in � S1Þ < DS1in. So the inequality of Property 2 holdsfor the equilibrium.

We have shown with Property 1 that the normal work-ing point n�1 ¼ ðX �1; S�1Þ was globally asymptotically stable.Therefore after a transient period all the trajectories willreach any neighbourhood of the steady state n�1 and espe-cially by continuity a neighbourhood where the inequalityholds. h

Remark. In practice this condition is often already met atinitial time and in all the cases the transient time T is small.

3.2. Analysis of the methanogenic dynamics

In anaerobic digestion the accumulation of VFA mightresult from the imbalance between acidogenesis andmethanogenesis leading thus to acidification. Since the

Fig. 1. Possible solutions for l2(S) = a D.

J. Hess, O. Bernard / Journal of Process Control 18 (2008) 71–79 73

methanogenesis is slower and can be inhibited it turns outto be the limiting step.

We now consider the methanogenic system given byEqs. (3) and (4) after a period greater than T (cf. Property2). The total concentration of VFA available for the secondstep of the process is S2in þ k2

D l1ðS1ÞX 1 6 S2in þ k2

k1S1in ¼eS 2in.

In order to study the methanogenesis as a stand-aloneprocess we consider eS 2in as a worst-case upper bound ofthe total concentration of VFA in the reactor.

Thus the methanogenic system is reduced to a one-stageprocess independent of the acidogenic phase:

_X 2 ¼ l2ðS2ÞX 2 � aDX 2

_S2 ¼ DðeS 2in � S2Þ � k3l2ðS2ÞX 2

(ð6Þ

This system is close to a generic Haldane model but, as forthe acidogenic subsystem, it is modified by the term a.

Property 3. System (6) with initial conditions in

X ¼ R�þ � Rþ admits a globally exponentially stable equi-

librium in the interior domain for aD < l2ðeS2inÞ. If

l2ðeS2inÞ < aD < lM it becomes locally exponentially stable

(l.e.s) and the acidification equilibrium is also l.e.s. For

aD > lM the acidification equilibrium becomes globallyexponentially stable (g.e.s.) (see Table 1 for more details).

Proof. We first study the bounds of the variables X2 and S2

in the same way as for the acidogenic phase, consideringthe quantity Z2 = S2 + k3X2. It follows that S2 6

maxðS20; eS 2inÞ and X 2 6maxðZ20;

eS 2ina Þ

k3.

The trivial steady state ny2 corresponding to the biore-actor acidification is given by ðX y2; S

y2Þ ¼ ð0; eS2inÞ.

Now we are going to explore the other steady states.They are solutions of the following system:

l2ðSH

2 Þ ¼ aD

X H

2 ¼ 1ak3ðeS2in � SH

2 Þ

(ð7Þ

Note that the steady states must verify SH

2 6eS 2in to have

0 6 X H

2 .First remark that, if eS 2in 6 SM

2 then l2 is an increasingfunction on the admissible domain ½0; eS 2in�. As a conse-quence the study of system (6) is identical to the study ofEqs. (1) and (2). We will then focus on the other case whereeS 2in > SM

2 .As illustrated in Fig. 1, five cases are possible, depending

on the parameters values.Cases 1 and 2. aD 2 ð0; l2ðeS2inÞ�: then the equation

l2(S2) = aD has a single solution for S2 2 ½0; eS 2inÞ:

ðX H

2 ; SH

2 Þ ¼eS 2in � l�1

2 ðaDÞak3

; l�12 ðaDÞ

!Case 3. aD 2 ðl2ðeS 2inÞ; lMÞ: here the equation l2(S2) =

aD has two solutions for S2 2 ½0; eS2inÞ. Let us denote S1H

2

and S2H

2 such that l2ðS1H

2 Þ ¼ l2ðS2H

2 Þ ¼ aD:

0 < S1H

2 < SM2 < S2H

2 < eS 2in

Application to the specific example: considering l2 as ex-pressed in (5) we can compute analytically these steadystates:

SiH2 ¼

KI2

2

�l2

aD� 1

� �þ ð�1Þi

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKI2

2

�l2

aD� 1

� �� �2

� KI2KS2

sð8Þ

with i = 1 for the useful working point ðX 1H

2 ; S1H

2 Þ and i = 2for the unstable equilibrium ðX 2H

2 ; S2H

2 Þ (see Table 1 for thejustification of this nomenclature). Then the two possibleequilibria are:

S1H

2 < SM2

X 1H

2 ¼ 1ak3ðeS 2in � S1H

2 Þ

(and

S2H

2 > SM2

X 2H

2 ¼ 1ak3ðeS 2in � S2H

2 Þ

(

Case 4. aD = lM: there is a unique solution to equationl2(S2) = aD:

ðX H

2 ; SH

2 Þ ¼eS 2in � SM

2

ak3

; SM2

!

Case 5. aD > lM: here there is no solution to the equa-tion l2(S2) = aD. In this case there is no other equilibriumthan the acidification point. h

3.3. Study of equilibria stability

We compute the Jacobian matrix of system (6) at anypoint on the non-negative orthant:

IðX 2; S2Þ ¼l2ðS2Þ � aD X 2

ol2

oS2ðS2Þ

�k3l2ðS2Þ �D� k3X 2ol2

oS2ðS2Þ

!ð9Þ

Fig. 2. Possible orbits in the phase plan: (a) case 3, (b) case 4.

Table 1Possible equilibria together with operating modes when eS 2in > SM

2

Case # Conditions int. acidi.

1 aD < l2ðeS2inÞ g.e.s. un.2 aD ¼ l2ðeS2inÞ l.e.s. un.�

3 aD 2 ðl2ðeS2inÞ;lM ÞS1H

2 l:e:s:S2H

2 un:

���� l.e.s.

4 aD = lM un� l.e.s.5 aD > lM – g.e.s.

74 J. Hess, O. Bernard / Journal of Process Control 18 (2008) 71–79

The stability of system (6) is now easy to assess by comput-ing the trace and the determinant of matrix (9) for all theconsidered cases:

• For the interior steady states ðX iH2 ; S

iH2 Þ:

trace ðIÞ ¼ �D� k3X iH2

ol2

oS2

ðSiH2 Þ

detðIÞ ¼ k3aDX iH2

ol2

oS2

ðSiH2 Þ

• For the acidification steady state ðX y2; Sy2Þ:

trace ðIÞ ¼ l2ðeS2inÞ � ð1þ aÞDdetðIÞ ¼ �Dðl2ðeS 2inÞ � aDÞ

It straightforwardly leads to the classification proposedin Table 11.

Remark. The 2 cases denoted by ‘un�.’ corresponding tonon-hyperbolic equilibria are

• Case 2: ð0; eS 2inÞ for aD ¼ l2ðeS2inÞ. Let us remark thatthe region fS2 6

eS2in;X 2 P 0g is positively invariant(any trajectory initialized in this domain stays in thisdomain). Moreover X2 is increasing in the sub-domainfX 2 > 0; S1H

2 6 S2 6eS 2ing. The only way to reach the

acidification X y2 ¼ 0 from the region fS2 6eS 2ing is thus

to start with a zero initial condition. This proves thatð0; eS2inÞ is unstable.

• Case 4: ðX H

2 ; SH

2 Þ for aD = lM. It is clear that in this case_X 2 6 0, and therefore the point is unstable (there is how-ever a region above X 2 ¼ X H

2 converging toward thissteady-state).

1 l.e.s.: locally exp. stable, g.e.s.: globally exp. stable, un.: unstable, int.:interior, acidi.: acidification.

3.4. Concluding remarks on stability

This study highlighted a special case of interest, foreS 2in > SM2 and aD 2 ðl2ðeS 2inÞ; lMÞ. Here there are 2 steady

states in the interior domain, one of which together withthe acidification are stable. In this case, illustrated inFig. 2a, the asymptotic state of the system is a priori notpredictable, and depends on the initial conditions.

The next section will consist in characterising the size ofthe domain of initial conditions leading to the interiorsteady state n1H

2 .

4. Attraction basin of the normal operating mode and

stability criteria

In this section, we still focus on the methanogenic stepto establish a stability criterion, but we assume here specificforms for l1(S1) and l2(S2) given by Eq. (5).

4.1. Definition of the attraction basin and of the stability

criterion

We have shown in the previous section that the statevector n2 remains bounded. We thus consider the accept-able domain for (X2,S2) as follows:

K ¼ 0;eS 2in

ak3

#� 0; eS 2in

h ið10Þ

Fig. 3. Maximal set of initial conditions considered within the set K (case4).

J. Hess, O. Bernard / Journal of Process Control 18 (2008) 71–79 75

Definition 1. For n1H

2 ¼ ðX 1H

2 ; S1H

2 Þ, the interior steady stateof system (6), we define its basin of attraction K as the setof initial conditions in K converging asymptoticallytowards it.

KðD; eS2inÞ ¼ fn20 2 Kj limt!þ1

n2ðn20; tÞ ¼ n1H

2 g

The main idea of this paper is to characterise the stabil-ity of the system by the area of the attraction basin K. Theprocess stability can then be assessed by the relative surfaceof K in K (11).

However, from the previous study (see Table 1) it isworth noting that there still exists a non empty attractionbasin KH ¼ KðlM

a ;eS 2inÞ associated to case 4 (aD = lM)

where the interior equilibrium is unstable (see Fig. 2b).We have seen that although this steady state is unstable

a b

Fig. 4. Definition of (a) the overloading tolerance (b) t

any trajectory initiated in the region K* above it (abovethe separatrix) will converge towards it. However it is obvi-ous that after any perturbation it will eventually convergetowards ny2. The region K* was then excluded from the con-sidered domain.

Definition 2. We define the Index of Stability ðISÞ as therelative area of the attraction basin on the domain K n KH

(see Fig. 3):

ISðD; eS 2inÞ ¼SðKðD; eS 2inÞ n KHÞSðK n KHÞ

ð11Þ

where application S is the area of the considered domain.

4.2. Numerical computation of the stability index

The separatrix can be computed numerically by inte-grating System (6) in inverse time along the stable directionof the saddle point ðX 2H

2 ; S2H

2 Þ starting very close to it. Thecomputation of the attraction basin area followsstraightforwardly.

However, the numerical computation of IS does notprovide any analytical expression of the stability index thatwould base a management strategy. In the following sec-tion we seek a simpler criterion related to IS.

4.3. Overloading tolerance of the process: a simple criterion

If the dilution rate is increased from zero, the interior

equilibrium will remain g.e.s. until D ¼ l2ðeS 2inÞa (cases 1

and 2). Then the second (unstable) steady state appearsin the interior domain together with a separatrix associatedto the attraction basin KðD; eS2inÞ that does no longeroccupy all the domain (case 3). The size of KðD; eS 2inÞ willthen decrease and finally vanish for D P lM

a (case 4). It isworth noting that the distance between the two interiorsteady states follows a rather comparable scheme: it will

he critical overloading tolerance in the phase plan.

76 J. Hess, O. Bernard / Journal of Process Control 18 (2008) 71–79

decrease from a maximum distance when D ¼ l2ðeS 2inÞa to zero

for D ¼ lMa (see Fig. 4a and b). For D > lM

a the acidificationsteady state is the only possible equilibrium and the dis-tance between the two interior steady states is no longerdefined. From this consideration, we define the notion ofoverloading tolerance (OT).

Definition 3. We define for aD 2 ½l2ðeS2inÞ; lM � the over-loading tolerance (OT), M which is simply the distancebetween the two interior steady states (see Fig. 4a):

Fig. 5. Relation between the stability index ISðD; eS 2inÞ and the relativetolerance margin mðD; eS 2inÞ for various couples (S1in in gDCO L�1, S2in inmmol L�1): (3,30), (0,25), (15,20), (30,30).

Fig. 6. Dilution rate, measured VFA, pH and compute

MðDÞ ¼ kn2H

2 � n1H

2 k ð12ÞWe also define the critical overloading tolerance (COT) Mc,which is the maximum value of the overloading tolerance

obtained for D ¼ l2ðeS 2inÞa .

The practical stability criterion that we will consider(named relative overloading tolerance (ROT)) is thendefined as follows:

mðD; eS2inÞ ¼

0 for aD > lM

MðDÞMcðeS 2inÞ

for aD 2 ½l2ðeS 2inÞ; lM �

1 for aD < l2ðeS 2inÞ

8>><>>:The OT given by the distance M between the two inte-

rior steady states can be computed straightforwardly fromEqs. (7) and (8):

MðDÞ ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1

a2k23

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiKI2

2

�l2

aD� 1

� �� �2

� KI2KS2

s

From this relation, we can see that the OT is a strictlydecreasing function of the dilution rate and that it is inde-pendent from influent composition (S1in,S2in). The COTcan be computed as follows:

MceS 2in

� �¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1

a2k23

s eS 2in �KI2KS2eS 2in

� �

d risk for an experiment performed at INRA LBE.

J. Hess, O. Bernard / Journal of Process Control 18 (2008) 71–79 77

4.4. Comparison between stability index and relative

tolerance

Using model parameters presented in Bernard et al. [6],we have computed the stability index IS and the ROTassociated with several working conditions (D, S1in andS2in).

As it can be seen in Fig. 5 the ROT represents a goodapproximation of the stability index IS based on the realcomputation of the attraction basin size.

The relative tolerance appears then as a simple but rele-vant criterion to assess the stability of an anaerobic digester.

Fig. 7. Risk index and VFA (zoom from Fig. 6).

Fig. 8. Dilution rate, measured VFA, pH and computed risk

From this criterion we define now the ‘‘risk index’’which will on-line indicate to the operator the destabilisa-tion risk he is taking;

r ¼ 1� mðD; eS 2inÞIn the next section, we use this operational criterion to as-sess the management strategy of a real anaerobic digester.

5. Application to the on-line determination of the

destabilisation risk

In this section, we apply the proposed index to a realexperiment performed at the LBE-INRA in Narbonne,France. The process is an up-flow anaerobic fixed bed reac-tor with a useful volume of 0.948 m3. The reactor is highlyinstrumented and many variables were measured duringthe experiments [6]. More details about the process andevaluation of its on-line instrumentation are available inSteyer et al. [13]. The experiments were performed withraw industrial wine distillery vinasses.

The risk index has been computed with parameters ofBernard et al. [6]. Nevertheless, in order to favour aprudent strategy, and in the framework of a ‘‘worst caseanalysis’’ the parameter KI2 defining the inhibition levelhas been multiplied by a security constant d (we havechosen d = 0.7). For low values of d the seeming inhibitionconstant for the calculus of the risk index is lower;therefore the system would be more easily inhibited and

for an overload experiment performed at INRA LBE.

78 J. Hess, O. Bernard / Journal of Process Control 18 (2008) 71–79

the risk index reacts more rapidly. On the contrary highvalues of d lead to a higher seeming inhibition constantand the risk index would be less responsive to smallchanges in the substrate concentration. The addi-tional parameter d enables to tune the sensibility of the riskindex.

The estimation of the risk index is presented in Fig. 6. Itis worth noting that the regimes associated with acid accu-mulation (i.e. more than 1 g L�1) are all characterised by avery high risk. More surprising, some a priori less danger-ous working modes are indeed also associated to a non zerorisk. This is especially the case at day 5 in Fig. 8, where therisk index is maximal while the dilution rate and the VFAlevel do not foreshadow any specific risk. This amplifiedsensibility is due to the tuning parameter d. A very impor-tant point is that the risk index increases immediately whileit takes time for the VFA to accumulate and even moretime to observe a pH decrease which is the usual indicatorof the process destabilisation. As such it anticipates apotential process acidification before it can be detectedand it becomes too late to avoid a dramatic biomass inhi-bition (see Fig. 7).

6. Conclusion

From the analysis of the nonlinear system describing theanaerobic process we have proposed a criterion thatassesses the risk associated to an operating strategy. Thisindex is highly correlated to the relative size of the normalworking mode attraction basin.

The criterion turns out to be relevant to diagnose anoperation strategy since it can predict very early a futureaccumulation of acids. It can thus be run as an indicatorthat helps an operator or even diagnoses the strategy ofan automatic controller which would not ensure globalstability.

As mentioned at the beginning of this paper, severalcontrol laws have yet been built to assure the durabilityof the process stability. A possible application of this sta-bility criterion is the triggering of an automatic robust con-troller whenever the risk index reaches a predefined level[5].

In order to take into account the biological evolution ofthe system in the risk index computation the next stepswould consist in:

(1) studying the index risk sensitivity to the variousparameters,

(2) developing a strategy for the on-line estimation of themost pertinent ones.

Acknowledgements

J. Hess is funded both by ADEME and by the Provence-Alpes-Cotes-d’Azur Region. The authors thank Jean-Phi-lippe Steyer along with the Process Engineering research

team at the LBE for providing the data presented in thispaper. The author also thank the reviewers for their com-ments that lead to the manuscript improvement.

Appendix A

Let us define the generic differential system:

_x ¼ f ðx; tÞxðt0Þ ¼ x0

�ð13Þ

with xðtÞ 2 Rn and f : Rn � Rþ ! Rn a continuous func-tion. A steady-state x* of system (13) is solution of:

0 ¼ f ðx�; tÞ 8t P t0

Definition 4 (Local asymptotic stability). The steady-statex* is said to be locally asymptotically stable if for any � > 0,there exists d > 0 and there exists T such that

jx0 � x�j < d) jxðx0; tÞ � x�j < � 8t P T þ t0

Definition 5 (Global asymptotic stability). The steady-statex* is said to be globally asymptotically stable if for any� > 0 and for any d > 0, there exists T such that

jx0 � x�j < d) jxðx0; tÞ � x�j < � 8t P T þ t0

Definition 6 (Local exponential stability). The steady-statex* is said to be locally exponentially stable if there exists

d > 0 and there exist non-negative constants c1 and c2, suchthat for all t0 > 0

jx0 � x�j < d) jxðx0; tÞ � x�j < c1jx0je�c2ðt�t0Þ 8t P t0

Definition 7 (Global exponential stability). The steady-statex* is said to be globally exponentially stable if for any d > 0there exist non-negative constants c1 and c2, such that forall t0 > 0

jx0 � x�j < d) jxðx0; tÞ � x�j < c1jx0je�c2ðt�t0Þ 8t P t0

Appendix B

The positivity of the variables of this system is trivialsince l1(0) = 0. To demonstrate the boundedness in a com-pact set of R2

þwe consider the quantity Z = S1 + k1X1:Since a 2 (0,1]:

DðS1in � ZÞ 6 _Z 6 aD Sin

a � Z

_S1 6 D S1in � S1ð Þ

(

It follows that Z and S1 are asymptotically bounded; SinceS1 P 0 the upper bound for Z is also an upper bound fork1X1:

J. Hess, O. Bernard / Journal of Process Control 18 (2008) 71–79 79

min Z0; S1inð Þ 6 Z 6 max Z0;S1in

a

0 6 S1 6 max S10; S1inð Þ

0 6 X 1 6max Z0;

S1in

a

k1

8>>><>>>:The considered system (1) and (2) has two steady states: thetrivial washout steady state X y1 ¼ 0; Sy1 ¼ S1in which existsfor any D, and another steady state in the positive domainif and only if aD < l1(S1in), given by:

SH

1 ¼ l�11 ðaDÞ

X H

1 ¼ 1k1a

S1in � SH

1

(ð14Þ

As l1(Æ) is monotone system (14) has a unique solutionin the positive domain if and only if aD 6 l1(S1in) (ensur-ing SH

1 6 S1in and thus X H

1 P 0).

Application to the specific case of the real example: con-sidering the expression of l1 given in (5) we get:

ðX H

1 ; SH

1 Þ ¼1

ak1

S1in � KS1

aD�l1 � aD

� �;KS1

aD�l1 � aD

� �In order to assess the stability of ðX H

1 ; SH

1 Þ and ðX y1; Sy1Þ we

study the Jacobian matrix of System (1), (2) at these steadystates:

IðX 1; S1Þ ¼l1ðS1Þ � aD X 1

ol1

oS1ðS1Þ

�k1l1ðS1Þ �D� k1X 1ol1

oS1ðS1Þ

!ð15Þ

The computation of the trace and determinant of (15) forboth equilibria gives:

• For the useful interior working point ðX �1; S�1Þ:

trace ðIÞ ¼ �D� k1X H

1

ol1

oS1

ðSH

1 Þ < 0

detðIÞ ¼ k1aDX I

1

ol1

oS1

ðSH

1 Þ > 0

• For the washout steady state ðX y1; Sy1Þ ¼ ð0; S1inÞ:

trace ðIÞ ¼ l1ðS1inÞ � ð1þ aÞDdetðIÞ ¼ �Dðl1ðS1inÞ � aDÞ < 0

This shows that only the useful working point ðX H

1 ; SH

1 Þis an attractor, the washout steady-state being a saddlepoint.

To conclude the proof and determine the global behav-iour of (1) and (2) we change variables (X1,S1) to (X1,Z).With this reformulation the system becomes:

_X 1 ¼ l1ðZ � k1X 1ÞX 1 � aDX 1

_Z ¼ D Sin � Zð Þ þ 1� að ÞDk1X 1

(Then the Jacobian matrix is:

IðX 1; ZÞ ¼H l01ðZ � k1X 1ÞX 1

ð1� aÞDk1 H

� �ð16Þ

It follows directly that this system is cooperative (i.e. theoff-diagonal terms of the Jacobian matrix are non-nega-tive). Furthermore the system is asymptotically boundedin a compact included in R2

þ. Hence from Theorem 2.2 ofChapter 3 in Smith [11] for two-dimensional systems, thelimit can only be a stable equilibrium point. Since the acid-ification equilibrium is unstable the system cannot con-verge towards it.

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