Robust regulation of a class of partially observed nonlinear continuous bioreactors

Robust regulation of a class of partially observednonlinear continuous bioreactors

A. Rapaporta,*, J. Harmandb

aINRA Biometrie, 2 pl. Viala, 34060 Montpellier, FrancebINRA LBE, Rte des Etangs, 11100 Narbonne, France

Received 26 January 2000; received in revised form 26 March 2001; accepted 18 April 2001

Abstract

We propose a design of robust dynamic output feedback for the global partial stabilization of a class of continuous stirredbioreactors. Our approach is robust with respect to exogenous disturbances and functional uncertainties of the kinetics, considered

as bounded unknown inputs. The method is based on a design of dynamic guaranteed bounds on the unmeasured variables, con-ditioned to past measurements and given bounds on the initial condition. This approach is suitable to systems which are notdetectable for unknown inputs. The flexibility in the design of the parameters allows the consideration of saturation constraints onthe control variable. # 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Min–max nonlinear control; Imperfect information; Interval observers; Robust partial stabilization; Bioreactor

1. Introduction

Control of bioreactors is a typical domain of potentialapplications of robust nonlinear control methods,because practical controllers have to tackle with severaldifficulties:

. nonlinear dynamics,

. large excursions of the state space during initialstages (linearizations about nominal points areusually not acceptable),

. partial measurements of the state variables,

. input saturations,

. time-varying uncertainties of the model and of theexogenous inputs.

Quite often, few reliable statistics are available and theonly reasonable assumption is to consider the uncertain-ties as unknown inputs but bounded (with knownbounds).In this paper, we consider a class of nonlinear con-

tinuously stirred bioreactors and propose a design of anoutput feedback regulation law, which is robust against:

. un-modeled time-varying kinetic,

. input saturations,

. external disturbances of feed concentrations.

A typical instance of such problems appears in waste-water treatment processes, where well-selected bacteriainside a bioreactor are expected to make the pollutantconcentration decrease. The control problem consists offeeding a continuous stirred reactor with wastewater loa-ded with a high pollutant concentration. The objective isthat the pollutant concentration of the output of thereactor reaches (after a transient stage) a prescribed value,fixed by the environmental authorities, and is then regu-lated about it (regulation stage). Reactions inside bior-eactors are often highly nonlinear, not very well knownand subject to external disturbances, such as the unknownpresence of pollutants. The manipulated variable is thedilution rate of the feeding, which has to be non-negative(remark that there is no interest to allow the concentra-tion to become lower than the prescribed value because asmaller dilution rate would be required, permitting lesswastewater to be treated during the same amount of time).The feed concentrations are not usually measured and canfluctuate rapidly between extreme values (a priori known).With industrial wastes, there are rarely reliable statisticson the time-evolution of these concentrations: uncertain-ties have to be considered as bounded unknown functionsof time. The only available measurement is the outputconcentration, to be regulated. Similar problems occur inthe pharmaceutical industry, but with bacteria selectedfor producing a substrate, instead of consuming it.

0959-1524/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.

PI I : S0959-1524(01 )00029-4

Journal of Process Control 12 (2002) 291–302

www.elsevier.com/locate/jprocont

* Corresponding author.

Some of these robustness issues have already beentackled for similar problems, using different approaches:

. PI based controllers can stabilize such systems,without requiring the knowledge of the reactionrates or the reconstruction of the unmeasured vari-ables [6]. Adaptive gain techniques can also copewith some exogenous uncertainties and handle inputsaturations [3]. The price to pay for completelyignoring the reaction rates is that no speed ofconvergence is guaranteed.

. I/O linearization techniques have been also appliedto these problems [7,13]. They consist of designingan exponential stabilizing law (under perfect infor-mation) and coupling it with an estimation of theunmeasured variables [4]. Extensions using adaptivetechniques allow consideration of kinetic uncertain-ties [5]. These approaches do not take into accountsaturation of the control variable and disturbanceson feed concentrations.

. In contrast, controllers based on Lyapunov func-tions can handle simultaneously uncertain reactionrates and input saturations [11,18,19]. Here also,the structure of the controllers is based on robustobservers. Nevertheless, these techniques do notseem to be suited to deal with exogenous uncer-tainties (unknown inputs) such that:� the system is no longer detectable for unknown

inputs (then it is not possible to obtain theconvergence of the observer),

� unknown disturbances on feed concentrationsare present (which bring a multiplicative uncer-tainty on the control variable).

To our knowledge, none of these approaches canhandle time-varying uncertainties simultaneously on thekinetic and on the feed concentrations. We propose hereanother approach, in the spirit of the theory of min–maxnonlinear control with imperfect information (see [14]).We consider uncertainties both of the kinetic and theexogenous inputs, requiring that:

. each un-modeled kinetic function is functionallybounded by two known kinetic functions (possiblytime-varying),

. the disturbances or unknown inputs are boundedwith known bounds (possibly time-varying).

We build a robust set-valued observer, in terms ofguaranteed intervals for the unmeasured variables (see[12]) and then use these dynamic bounds, instead of asingle-valued estimate, to design our control law. We donot require the system to be detectable for unknowninputs, as we do not require the set-observer to convergetowards a singleton. The proposed controller guaranteesa quadratic function to be a robust partial Lyapunov

function, providing an exponential stabilization of theoutput in the worst case (other Lyapunov candidatesare possible, see [1]).The solution we propose here turns out to be simple

and effective for the class of systems we investigate andwe believe that it illustrates a possible new methodologyfor the design of robust output feedback controllers.The consideration of bounds on the un-modeled kineticsbrings some flexibility in the design of the regulation law: it allows consideration of any functional knowledge(possibly time-varying) of the kinetics, even partial, andthe consideration of the two extreme cases:

. when the dynamics is known perfectly, the func-tional upper and lower bounds coincide, and ourcontrol law turns out to be a single nonlinear statefeedback.

. when no information at all is available on thekinetics, upper and lower bounds for the kineticsare constant functions. Then, the informationprovided by the interval observer is irrelevant forthe computation of our control law, which turnsout to be a static output feedback law.

This paper is a generalization of a previous work (see[16]), to more general classes of kinetics.The paper is organized as follows. In the coming sec-

tion, we present the considered model and state the con-trol problem. In the two following sections, we propose arobust ‘‘interval’’ observer and then a robust controller.In the next section, we illustrate this approach for asimple application, similar to the one studied in [15] butwith more than one reaction.

2. Modelling and problem statements

We consider a system of reaction equations of a con-tinuous stirred bioreactor:

Sð Þ : �:tð Þ ¼ Cr t; � tð Þð Þ þD tð Þ �in tð Þ � � tð Þ

� �� 0ð Þ ¼ �0

�ð1Þ

where:

. � is the vector of (non-negative) concentrations (ofboth biomasses and substrates), of dimension n.

. r is the vector of reaction kinetics (non-negativefunctions). Its dimension m (4n) is equal to thenumber of reactions.

. C is the (constant) stoichiometric matrix, ofdimension n�m.

. �in is the vector of feed concentrations.

. D (the dilution rate) is the manipulated variable,taking bounded values and subject to the ‘‘satura-tion’’ constraint: D50.

292 A. Rapaport, J. Harmand / Journal of Process Control 12 (2002) 291–302

Note 1. The sign of the terms Cij of the matrix C informsthen if the species i is consumed (Cij < 0), produced (Cij

> 0) or unaffected (Cij=0) by the j-th reaction.

We require usual assumptions on the kinetics (seeAppendix A), which ensures the existence of a compactset � of concentrations �, which is invariant by thedynamics (1).In this work, we consider that the vector r() of reac-

tion kinetics is not well known but functionally boun-ded by r� and r+:

r��ð Þ4r� t; �ð Þ4r t; �ð Þ4rþ t; �ð Þ4r� �ð Þ; 8� 2 �; 8t50

ð2Þ

where (r�, r+) are known functions that fulfill the sameassumptions as r. We consider that the system (1) is alsosubject to bounded uncertainties on the feed concentra-tions �in:

�in4�in� tð Þ4�in tð Þ4�inþ tð Þ4�in; 8t50 ð3Þ

where (�in�, �in+) are known functions.

Note 2. The comparison operator 4 applied betweenvectors should be understood as a collection ofinequalities applied between their coordinates.

We assume that an online measurements vector y(t)gives the values of p (4 n) concentrations of the vector� (t). Then, (S) can be rewritten in the following way:

Sð Þ :x:1 tð Þ ¼ C1r t; x1 tð Þ; x2 tð Þð Þ þD tð Þ xin1 tð Þ � x1 tð Þ

� �x:2 tð Þ ¼ C2r t; x1 tð Þ; x2 tð Þð Þ þD tð Þ xin2 tð Þ � x2 tð Þ

� ��

with dim x1=n–p, dim x2=p, � =(x1,x2)t, �in=(xin1 ; x

in2 )

t

and the measurement y=x2.Consider then a scalar output y1=Hy. Without loss

of generality, we assume in the following that H=(1 0. . . 0). We can now state our main interest:

The robust regulation problem (P) : Given a set X0 � Oof unknown initial conditions �0, find an output feed-back law D( ) that exponentially stabilizes y1 about areference value yr1, for any unknown �in( ), r( ) that fulfill(2,3) and �0 2 X0, such that D(t) 5 0 for any t5 0.

Remark 1. This is a robust partial stabilization problem, inthe sense that we do not require the robust stabilization ofall the variables. Notice that due to the existence of acompact set O invariant by the dynamics (see the assump-tions in Appendix A), the other concentrations remainbounded. Nevertheless, under additional assumptions, wecan prove the semi-global stabilization of the whole pro-cess (see Proposition 17 in Appendix C).

3. Interval observer

In this section, we propose a robust version of a wellknown asymptotic observer, developed in [7]. As in [7], werequire a sufficient condition for the detectability of (S)with unknown kinetics but with known disturbances �in( ):

Hypothesis 2. dim x2 5 rankC and rankC2=rankC.

Note 3. This hypothesis means that the dimension of theobservation should be at least equal to the rank of thestoichiometric matrix (which seems to be a reasonableassumption) and that the number of independent reac-tions in which measured variables interfere should beequal to the rank of the stoichiometric matrix (noticethat for one reaction with several species, only one mea-sured concentration is required, which is quite a weakassumption).

Under this hypothesis, we consider a matrix N suchthat C1+NC2=0 (it is clear that such a matrix existsunder hypothesis 2) and define the vectors:

zin� tð Þ ¼ xin�1 tð Þ þ1

2N xinþ2 tð Þ þ xin�2 tð Þ� �

� Nj j xinþ2 tð Þ � xin�2 tð Þ� ��

zinþ tð Þ ¼ xinþ1 tð Þ þ1

2N xinþ2 tð Þ þ xin�2 tð Þ� �

þ Nj j xinþ2 tð Þ � xin�2 tð Þ� ��

8>><>>:(where Nj j stands for the matrix Nj jij= Nij

).Then, we replace the observer x1 proposed in [7] for

known feed concentrations �in by two observers: x�1 andxþ1 bounding the unmeasured vector x1.

Proposition 3. Given the initial measurement y(0) and twonon-negative vectors x�1 (0), x

þ1 (0) bounding the unknown

initial condition: x�1 (0)4x1(0)4xþ1 (0), the followingdynamic system:

z:� tð Þ ¼ D tð Þ zin� tð Þ � z� tð Þ

� �; z� 0ð Þ ¼ x�1 0ð Þ þNy 0ð Þ

z:þ tð Þ ¼ D tð Þ zinþ tð Þ � zþ tð Þ

� �; zþ 0ð Þ ¼ xþ1 0ð Þ þNy 0ð Þ

�ð4Þ

provides guaranteed bounded intervals for the unmeasuredvariables:

x�1 tð Þ4x1 tð Þ4xþ1 tð Þ; 8t50 ð5Þ

where:

x�1 tð Þ ¼ max z� tð Þ �Ny tð Þ; diag e��1t; . . . ; e��n�ptð Þe�D� tx�1 0ð Þ

� �xþ1 tð Þ ¼ zþ tð Þ �Ny tð Þ

(

D is any upper bound on D( ).

Note 4. The parameters �i are provided by assumptions10 and 11 (see Appendix A). The operator max appliedbetween vectors should be interpreted coordinate bycoordinate.

A. Rapaport, J. Harmand / Journal of Process Control 12 (2002) 291–302 293

Proof. It is based on a simple argumentation of com-parison of the solutions of o.d.e. See Appendix B. &

Remark 4. When D is a persisting input, i.e. such that

9�; � > 0

ðtþ�

t

D sð Þds > �; 8t50

and the bounds �in�, �in+ are constant equal respectivelyto �in and �in, the vector xþ1 tð Þ � x�1 tð Þ converges asymp-

totically towards xin1 � xin1

� �+ Nj j xin2 � xin2

� �. In the last

section, we give an example which is not detectable for theunknown inputs (r,�in) but the interval x�1 ; x

þ1

� �converges

towards the set of indistinguishable x1.This interval observer is a robust version of the single-

valued observer proposed in [7] when �in�(t)=�in+(t),8t50.

4. Robust output stabilization

A natural way to use the guaranteed interval observerx�1 ; x

þ1

� �proposed in the previous section is to look now

for robust control laws in terms of feedbacks of the newstate x�1 ; x

þ1 ; y

� �t(which is perfectly known) instead of

the unknown state �.

4.1. The control law

l�, l+ being positive parameters, we consider thefeedback:

eDD t;x�1 ;xþ1 ; y

� �¼

�þ t;x�1 ; x

þ1 ; y

� �þ l� y1 � yr1

� �Hxin�2 tð Þ � y1

if y1 < yr1

�� t;x�1 ; x

þ1 ; y

� �þ lþ y1 � yr1

� �Hxinþ2 tð Þ � y1

if y15yr1

ð6Þ

with

� t;x�1 ;xþ1 ; y

� �¼ HC2 min

x12 x�1;xþ

1½ �r� t;x1; yð Þ;

þ t;x�1 ;xþ1 ; y

� �¼ HC2 max

x12 x�1;xþ

1½ �rþ t;x1; yð Þ

8><>: ð7Þ

We shall also consider simpler expressions of eDD forparticular cases:

eDDE t; x�1 ;xþ1 ; y

� �¼

�HC2r

þ t;xþ1 ; y� �

þ l� y1 � yr1� �

Hxin�2 tð Þ � y1if y1 < yr1

�HC2r

� t;x�1 ; y� �

þ lþ y1 � yr1� �

Hxinþ2 tð Þ � y1if y15yr1

ð8Þ

eDDP t; x1; yð Þ ¼ �HC2r t; x1; yð Þ þ l y1 � yr1

� �Hxin2 tð Þ � y1

ð9Þ

eDDS y1ð Þ ¼�HC2r�þ l� y1 � yr1

� �Hxin2 � y1

if y1 < yr1

0 if y15yr1

ð10Þ

4.2. Convergence results

Hypothesis 5. The process (S) is such that:

. The output y1 is a reactant of any reaction where itis involved (which is equivalent to require that thevector (HC2)

t is non null and non positive).. The output y1 converges asymptotically towards 0

when D(t)=0, 8t 5 0 and �0 > 0.. The reference value yr1 is below Hxin2 .

Note 5. This hypothesis means that we consider processesthat always consume the reactant y1 when there is nodilution, which ensures that the constraintD50 is feasiblefor the stabilization of y1 about any yr1 2

�0;Hxin2

�. Simi-

lar hypotheses can be required for processes that alwaysproduce y1.Of course, this hypothesis implies that a simple

‘‘bang-bang’’ control law:

D y1ð Þ ¼D� if y1 < yr10 if y15yr1

might stabilize y1 about y

r1 for large enough D� , without

requiring the knowledge of the kinetics or the recon-struction of unmeasured variables.Here, we show how to use the guaranteed estimation

(5) of x1(t) and functional bounds on the kinetics r toimprove this control law, guaranteeing an exponentialconvergence even in the worst case. The proposed con-trol law is also discontinuous at y1=yr1 but the magni-tude of the jump is dynamic and depends on theinformation collected during the transient stage. Thesimulations in next section show that the more infor-mation is available, the less is the magnitude of thejump.The output feedback law eDD can be seen as a robust

version of the I/O linearization state feedback eDDP pro-posed in [7], but where the unknown value of thedynamics r(t, x1(t), y(t)) at time t is replaced by theworst one (see the formula (7)), according to the givenbounds on the uncertainties (2,3) and the informationprovided by the interval observer (Proposition 3). Then,this worst estimation is dynamic and no longer leads toan I/O linearization controller.D� being a positive number, we denote by sat 0;D�½ � the

saturation function:


sat 0;D�½ � dð Þ ¼

0 if d < 0d if d 2

�0;D�

�D� if d > D�

:

Our main result is the following.

Proposition 6. There exists positive parameters D� , l� andl+ such that, for any set of unknown initial conditionsX0=[x�1 0ð Þ; xþ1 0ð Þ]� {y(0)} � O such that

� > 0; 8� 2 X0; ð11Þ

the output feedback law

D� t; x�1 ; xþ1 ; y

� �¼ sat 0;D�½ �

eDD t; x�1 ; xþ1 ; y

� �� ð12Þ

solves the problem (P) when x�1 and xþ1 are provided byProposition 3.

Proof. It consists in proving that V(y1)=12 y1 � yr1� �2

is apartial Lyapunov function. See Appendix B. &

Note 6. The parameters D� , l� and l+ have to be chosento fulfill conditions that can be checked directly on thedata of the problem (see the conditions C1-C2-C3 in theproof of Proposition 6 in Appendix B).

When the unknown kinetics vector r is monotonic withrespect to each coordinate of x1, the arg min and arg maxin (7) are unique and D* is then a ‘‘certainty equivalentcontroller’’ (see [10]). It corresponds to the state feedbacklaw in perfect information, as if the unknown terms(x1, xin2 , r( )) were equal to the unique ‘‘worst case’’(x�1 ; x

in�2 ; r�( )), thus the following corollary:

Corollary 7.When the kinetics r�j ; rþj (j=1 . . . m) are non

decreasing with respect to each coordinate of x1, then theoutput feedback law

D�E t; x�1 ; x

þ1 ; y

� �¼ sat 0;D�½ �

eDDE t; x�1 ; xþ1 ; y

� �� solves the problem (P).

The control laws D*, D�E are discontinuous at y1=yr1

but the magnitude of the jump at y1=yr1 is dynamic anddepends on the information collected during the tran-sient stage. In the two extreme cases, we have:

. when the system is perfectly known, i.e. whenr�=r+, �in�=�in+ and x�1 0ð Þ ¼ xþ1 0ð Þ, we canchoose l+=l�=l and there is no jump at all:

D�P t; x1; yð Þ ¼ sat 0;D�½ �

eDDP t; x1; yð Þ

� �turns out to be exactly the formula proposed in [7].

. when no information at all on the dynamics isavailable, we take r�� 0 and r+� r� (where r� is aconstant vector). Then,

D�S y1ð Þ ¼ sat 0;D�½ �

eDDS y1ð Þ

� �turns out to be a static output feedback law, whichno longer requires any information on the unmea-sured variables x1. The magnitude of the jump aty1=yr1 is then constant. See additional commentsin Appendix B (remark 16).

Remark 8. The proposed output feedback laws D*, D�E

and D�S are not continuous at y1=yr1 but can be

smoothed using a boundary layer (see the Corollary 15in Appendix B).

It is well known that an open-loop bioreactor can beunstable (for instance when an inhibition effect is pre-sent, see [8]). So, in general, we do not expect the stabi-lization of the global process. The trajectories of theother variables are driven by the unknown disturbances,as this can be seen on the simulation graphs. Never-theless, when the feed concentrations are constant andthe process has a stable ‘‘zero-dynamics’’, we can showthat the closed-loop stabilization of the whole process isguaranteed (see Appendix C).

5. An example

In this section, we present a simple application forillustrating the benefit of such a control law. Notice thatit has been applied successfully for the regulation of ananaerobic digestion process using a mass balance modelidentified from experimental data [9,2]. Its implementa-tion for controlling real processes is now under investi-gation [1].

5.1. The model

Consider a system of two reactions r1 and r2 involvingtwo biomasses and two substrates, whose concentra-

tions are respectively b ¼b1b2

� �and s ¼

s1s2

� �:

b:1 ¼ r1 t; b1; s1ð Þ �D b1

b:2 ¼ r2 t; b2; s1; s2ð Þ �D b2

s:1 ¼ �c1r1 t; b1; s1ð Þ � c12r2 t; b2; s1; s2ð Þ þD sin1 tð Þ � s1

� �s:2 ¼ �c2r2 t; b2; s1; s2ð Þ þD sin2 tð Þ � s2

� �8>><>>:(with c1 > 1, c12 > 0 and c2 > 1) to be regulated abouts1=sr1. The available measurements are: y(t)=s(t), 8t5 0.The system is subject to unknown bounded dis-

turbances on the input concentrations:


sin tð Þ ¼sin1 tð Þ

sin2 tð Þ

� �2

sin�1 ; sinþ1� �sin�2 ; sinþ2� � !

with sin�1 > sr1; 8t50:

In industrial environments, it is well known that thekinetics modelling is a particularly critical task. Forinstance, it is not always easy to detect when an inhibi-tion effect appears, because it might be caused by exter-nal unknown factors (for example, the presence of a toxicsubstance). The maximum reaction rates of the reactioncan be also very sensitive to pH or temperature, whichcan fluctuate with time and are often not measured witha good accuracy.We consider here that the reaction rate of r1 fluctuates

with time in an unknown way between a Monod law þ1

(monotonic) and a Haldane law �1 (non monotonic):

�1 s1ð Þb14r1 t; b1; s1ð Þ4þ

1 s1ð Þb1

with

þ1 sð Þ ¼ max1

s

K1 þ s

þ1 sð Þ ¼ max1

s

K1 þ sþ s2=Ki

8><>:(see Fig. 1). For the kinetics r2, we assume that it fol-lows a Monod law but with an unknown fluctuatingmaximum reaction rate:

max�2 12 s1ð Þ2 s2ð Þb24r2 t; b2; s1; s2ð Þ4maxþ2 12 s1ð Þ2 s2ð Þb2

with 2 sð Þ ¼ sK2þs and 12 sð Þ ¼ s

K12þs (see Fig. 2). This is

equivalent to say that the kinetics vector can be repre-sented in the following way:

r t; b; sð Þ ¼r1 t; b1; s1ð Þ

r2 t; b2; s1; s2ð Þ

� �¼

max1 1 p1 tð Þ; s1ð Þb1

p2 tð Þ12 s1ð Þ2 s2ð Þb2

� �with 1 p; sð Þ ¼

s

K1 þ sþ ps2;

subject to unknown bounded disturbances:

p tð Þ ¼p1 tð Þp2 tð Þ

� �2

0;K�1i

� �max�2 ; maxþ2

� � !:

One can easily check that the assumptions 10, 11 and12 of Appendix A and the hypotheses 2 and 5 are ful-filled and that the compact set:

� ¼ b; sð Þ 2 R4þ c1 b1 þ c2 þ c12ð Þb2 þ s1 þ s24sinþ1 þ sinþ2 ��

is positively invariant for any non-negative control law.

5.2. The robust control law

The equations of the interval observer [b�, b+] andthe control law eDD are:

b�1 ¼ z�1 �1

c1y1 þ

c12c1c2

y2

b�2 ¼ z�2 �1

c2y2

z:�1 ¼ D

1

c1sin�1 �

c12c1c2

sin�2 � z�1

� �z:�2 ¼ D

1

c2sin�2 � z�2

� �z:�1 0ð Þ ¼ b�1 0ð Þ þ

1

c1y1 0ð Þ �

c12c1c2

y2 0ð Þ

z�2 0ð Þ ¼ b�2 0ð Þ þ1

c2y2 0ð Þ

8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>:eDD b�; bþ; y� �

¼

c1max1 �

1 y1ð Þb�1 þ c12max�2 12 y1ð Þ2 y2ð Þb�2 � l� y1 � sr1

� �sin�1 � y1

ð13Þ

withFig. 1. þ1 and �

1 as functions of s1.

Fig. 2. þ2 and �

2 as functions of s2.


�1 y1ð Þ; b�1;

max�2 ; b�2; s

in�1 ; l�

� �¼

þ1 y1ð Þ; bþ1 ;

maxþ2 ; bþ2 ; s

in�1 ; l�

� �if y1 < sr1

�1 y1ð Þ; b�1 ;

max�2 ; b�2 ; s

inþ1 ; lþ

� �if y15sr1

Remark 9. The system is not detectable for the unknowninputs (p,sin), see the remark 14 in Appendix B.

5.3. Simulations

With the same occurrence of disturbances (see Fig. 3),we have made two simulations for the control law D*,given by Proposition (6), with the parameters given inTable 1:

1. The controller has the perfect knowledge of b1(t),b2(t), r1( ), r2( ) and sin1 ðtÞ (cf formula (9)): see Fig. 4.Notice that the controller does not require theknowledge of sin2 tð Þ.

2. The controller has partial knowledge of b1(t), b2(t),r1( ),r2( ) and sin2 tð Þ [cf formulae (6,7) with knownbounds given in Table 2 (see Fig. 5).

For both simulations, we have used the same designparameters: l=l�=l+=0.1 and D�=0.03 and used theregularized expression (26) with "=5 mg l�1.

Appendix A. Assumptions

Assumption 10. The reaction kinetics of vector r arenon-negative functions, measurable w.r.t. the time, Lip-schitz w.r.t. � and such that:

. The j-th reaction stops (i.e. rj(t,�)=0) exactly whenat least one of the variables �i involved in rj isequal to zero.

Fig. 3. Unknown disturbances.

Table 1

Simulation parameters

c1 c2 c12 sr1 K1 Ki max1 K2 K12 max�2 maxþ2

20 10 10 15 mg l�1 20 mg l�1 10 mg l�1 0.04 h�1 10 mg l�1 0.1 mg l�1 0.02 h�1 0.06 h�1

Fig. 4. Stabilization with perfect knowledge.

Table 2

Known bounds

sin�1 sinþ1 sin�2 sinþ2 b�1 (0) bþ1 (0) b�2 (0) bþ2 (0)

800 mg l�1 1200 mg l�1 160 mg l�1 240 mg l�1 30 mg l�1 60 mg l�1 10 mg l�1 15 mg l�1


. If Cij < 0 then the kinetic rj necessarily involvesthe concentration �i.

We shall denote Ij the set of indexes i such that thevariable �i is involved in reaction rj.The feed concentrations �in are measurable w.r.t. the

time.

Note 7. Assumption 10 guarantees the invariance of Rnþ:

�i=0 implies that rj (t,�)=0 for any j such that Cij < 0,so �

:i 5 0, for any i=1 . . .n.

Assumption 11. For any constant vector �in 2 Rnþ, there

exists a vector ! >0 such that �={� 2 Rnþ

wt�� in

�40

} is compact and positively invariant bythe dynamics (1), whatever are the kinetics r( ), the input�in( ) bounded above by �in and the control law D( )50.

Note 8. Assumption 11 is a consequence of the principleof mass conservation (see [7]) which claims the existenceof a vector !>0 such that !tC=0. Then !t �

:4–D!t(�–

�in), thus the positive invariance of �.

Let K be a positive number such that the functions rjare Lipschitz w.r.t. � of rank K on �. We define then thefollowing constants:

�i ¼ KXj

max 0;�Cij

� �; i ¼ 1 . . . nð Þ:

Assumption 12. For any compact subset � of Rnþ, there

exist positive numbers , �, � with �51 such that:

8� 2 �; 8t50; 8j ¼ 1 . . .m;

rj t; �ð Þ5 Yi2Ij

min �; �ið Þð Þ�:

Note 9. Assumption 12 is fulfilled for the usual kineticslaws (see [7]), even the ones with inhibition effects (i.e.non-monotonic reaction rates such as the Haldane law).

Appendix B. Proofs

We have the following lower estimates for the trajec-tories:

Lemma 13. Under assumptions 10 and 11, whatever is thecontrol law D(t) 2

�0;D�

�; 8t 5 0, we have:

�i tð Þ5�i 0ð Þe� �iþD�ð Þt; 8t50; 8i ¼ 1 . . . n;

Proof. Fix a control law bounded by D� and consider avariable �i for some i=1. . . n. Its dynamics satisfies theinequality:

�:i ¼

Xj

Cijrj t; �ð Þ þD �ini tð Þ � �i� �

5�Xj

max 0;�Cij

� �rj t; �ð Þ �D� �i

When Cij < 0, �i is involved in rj (by Assumption 10).Then, consider �~ such that �~k=�k for k6¼ i and �~i=0. rjbeing K-Lipschitz w.r.t. � (by Assumption 11), we have:jrj t; �ð Þ � rj

�t; �~�j4K�i. But, due to Assumption 10,

rj(t,�~)=0, so rj(t,�)4 K �i and we have:

�:i5� K

Pj

max 0;Cij

� �þD�

!�i ) �i tð Þ5�i 0ð Þe� �iþD�ð Þt;

8t50; 8i ¼ 1 . . . n:

&

Proof of Proposition 3. Lower bounds are already pro-vided by Lemma 13.Fix �0, �in( ), r( ) and a control law D( )5 0. Let �( )

and y( ) be the trajectories, respectively of the state andthe output of the system (S).Take z(0)=x1(0)+Ny(0) and consider the solution of

z:=D(t)(xin1 tð Þ þNxin2 tð Þ � z). We have clearly: x1(t)=z(t)–Ny(t), 8t 5 0.Whatever is the unknown input �in( ), we have: zin�(t)

4 xin1 tð Þ þNxin2 tð Þ 4 zin+(t) and we deduce that:

Fig. 5. Stabilization with partial knowledge [b�(t), b+(t), D*�(t) and

D*+(t) in dashed lines].


z:� z

:�5�D tð Þ z� z�ð Þ

z:þ � z

:5�D tð Þ zþ � zð Þ

�By standard argumentation of comparison of solu-

tions of differential equations, we have:

z tð Þ � z� tð Þ5e� tð Þzþ tð Þ � z tð Þ5eþ tð Þ

8t50

�where (e�, e+) is the solution of the linear system:

e:� tð Þ ¼ �D tð Þe� tð Þ; e� 0ð Þ ¼ z 0ð Þ � z� 0ð Þ

e:þ tð Þ ¼ �D tð Þeþ tð Þ; eþ 0ð Þ ¼ zþ 0ð Þ � z 0ð Þ

�Then, whatever is the initial condition �(0) such that

z�(0) 4 z(0) 4 z+(0), we have e�(t) 5 0 and e+(t) 5 0for any time and obtain the guaranteed interval:

z� tð Þ4z tð Þ4zþ tð Þ; 8t50:

We then conclude that z�(t)–Ny(t) 4 x1(t) 4 z+(t)–Ny(t), 8t5 0. Lemma 13 provides also the lower esti-mates:

�i tð Þ5��i 0ð Þe� �iþD�ð Þt; 8t50; 8i ¼ 1 . . . n� p

thus the guaranteed intervals x�1 tð Þ; xþ1 tð Þ� �

for x1(t).Consider now l(t)=z+(t)–z�(t) which is solution of

l:=D(t)(zin+(t)–zin-(t)–l). As zin+(t)–zin�(t) is boundedand D is non-negative, l(t) stays bounded. The state �(t)of the system remaining inside the invariant domain �,we deduce that the intervals x�1 tð Þ; xþ1 tð Þ

� �are also

bounded. &

Remark 14. In the example of Section 5, the system is notdetectable for the unknown inputs (p,sin): fix any open-loop control D( ), a pair of disturbances (pa( ),sin,a( )) andan initial condition (ba(0),s(0)). Name (ba( ),s( )) the cor-responding trajectory and consider another pair of dis-turbances:

pb1 tð Þ ¼1

s1 tð Þ2max1 s1 tð Þ ba1 tð Þ þ 1

� �max1 1 pa1 tð Þ; s1 tð Þ

� �ba1 tð Þ þD tð Þ 1

� K1 � s1 tð Þ

" #sin;b1 tð Þ ¼ sin;a1 tð Þ þ c1 1 þ c12 2

pb2 tð Þ ¼ pa2 tð Þ þD tð Þ

12 s1 tð Þð Þ2 s2 tð Þð Þ� pa2 tð Þ

� � 2

ba2 tð Þ þ 2

sin;b2 tð Þ ¼ sin;a2 tð Þ þ c2 2

8>>>>>>>><>>>>>>>>:where =( 1, 2)

t>0, with the initial condition (bb(0),s(0))=(ba(0)+ , s(0)). It is then straightforward to seethat (bb(0), pb( ), sin,b( )) generates exactly the same out-put s( ) but with a biomass history: bb(t)=ba(t)+ forany time. Nevertheless, one can easily see from (13) thatthe interval [b�(t), b+(t)] converges exponentially towardsthe set Ind(y2(t)) of indistinguishable b:

Ind y2ð Þ ¼

1

c1sin�1 � sr1� �

�c12c2

sinþ2 � y2� �

; sinþ1 � sr1� �

�c12c2

sin2 � y2� ��

1

c2sin�2 � y2; s

inþ2 � y2

� �0BB@

1CCAas soon as there exist positive numbers � and T such thatD*(t)5 � for any t 5 T.

Proof. For the design of our robust exponential outputstabilization law, we consider positive parameters D� , l�

and l+ that fulfill the following conditions:

C1: D� >

max�2�

�HC2r� �ð Þð Þ

Hxin2 � yr1

(this is possible by Hypothesis 5 and the compactness of�).

C2: l�4D� Hxin2 � yr1

� ��max

�2��HC2r� �ð Þð Þ

yr1

(this is possible by condition C1).

C3: lþ5�Xni¼1

�i þD��

For simplicity of notation, we shall denote in the fol-lowing:

þ tð Þ ¼ þ t; x�1 tð Þ; xþ1 tð Þ; y tð Þ� �

� tð Þ ¼ � t; x�1 tð Þ; xþ1 tð Þ; y tð Þ� �(

D� tð Þ ¼ D� t; x�1 tð Þ; xþ1 tð Þ; y tð Þ� �

eDD tð Þ ¼ eDD t; x�1 tð Þ; xþ1 tð Þ; y tð Þ� �(

For any closed subset X of �, we define a lowerbound for the initial kinetics of y1:

� Xð Þ ¼ �HC2

..

.

Qi2Ij

min �;min�2X

�i

� ��

..

.

0BBBB@1CCCCA

Note that �(X)5 0 by Hypothesis 5.The compact set � being positively invariant for any

non-negative control law (Assumption 11), we considera candidate partial Lyapunov function to ensure thestabilization of the output y1:

V y1ð Þ ¼1

2y1 � yr1� �2

;


the other variables remaining bounded. We have:

V:¼

y1 � yr1� �

HC2r t; x1; yð Þ þD� t; x�1 ; xþ1 ; y

� �Hxin2 tð Þ � y1� ��

:

ð14Þ

Case 1: y1<yr1. Necessarily, y1<Hxin2 tð Þ (by Hypoth-esis 5) and we have:

eDD tð Þ4max�2O

�HC2r� �ð Þð Þ þ l�yr1

Hxin2 � yr1

So, we have: eDD tð Þ4D� by condition C2, then, D*(t)5eDD(t). As Hxin2 tð Þ–y15Hxin�2 tð Þ–y1, (14) provides:

V:tð Þ4 y1 � yr1

� �HC2r t; x1; yð Þ þ eDD tð Þ Hxin�2 tð Þ � y1

� �h i4 y1 � yr1� �

HC2r t; x1; yð Þ � þ tð Þ � l� y1 � yr1� ��

But, HC2r(t,x1,y)–+(t)5 0 by (7). Then,

V:tð Þ4� l� y1 � yr1

� �2¼ �2l�V tð Þ: ð15Þ

Case 2: y1(0) 5 yr1. Note that � (X0) > 0 due to (11).Sub-case 2a: X0 is such that:

y1 0ð Þ < min Hxin2 ; yr1 þ

� X0ð Þ

lþ

� �; ð16Þ

Note that due to Hypothesis 5, such sets X0 exist (withy1(0) 5 yr1).By Lemma 13, we have �i(t)5min (�,min

�i2X0

�i) e� �iþD�ð Þt,

8t50. Name �~ ¼ �Pn

i¼1

��i þD�

�. Then, by Assumption

12:

r�j t; x1; y tð Þð Þ5 Yi2Ij

min �;min�2X

�i

� ��

e��~t

8x15x�1 tð Þ; 8j ¼ 1 . . .m

) � tð Þ4� � X0ð Þe��~t; 8t50:

So, as long as y1(t)5 yr1 and y1(t) < Hxin2 , we have:

eDD tð Þ5� X0ð Þe��~t � lþ y1 tð Þ � yr1

� �Hxinþ2 tð Þ � y1 tð Þ

: ð17Þ

We show now that the numeratorN(t)=� (X0)e

��~t�l+(y1(t)�yr1) stays non-negative. Wehave:

N 0ð Þ ¼ � X0ð Þ � lþ y1 0ð Þ � yr1� �

50; by 16ð Þ: ð18Þ

N:tð Þ ¼ ��~� X0ð Þe��~t

� lþ HC2r t; � tð Þð Þ þD� tð Þ Hxin2 tð Þ � y1 tð Þ� ��

ð19Þ

As long as N(t) 50, we have eDD(t)50 thus D*(t)4eDD(t). As Hxin2 tð Þ–y1(t)4 Hxinþ2 tð Þ–y1(t), we have from(19):

N:tð Þ5� �~� X0ð Þe��~t � lþ HC2r t; � tð Þð Þ � � tð Þ½ �

þ lþ2 y1 tð Þ � yr1� �

ð20Þ

But HC2r(t,� (t))–�(t)40 by (7) and l+(y1–yr1)=�(X0) e

��~t–N(t) (by definition of N(t)). So, (20) gives:

N:tð Þ5� lþN tð Þ þ � X0ð Þe��~t lþ � �~

� �ð21Þ

But l+5 �~ by C3. Then, (21) becomes:

N:tð Þ5� lþN tð Þ ð22Þ

So, as long as y1(t)5yr1 and y1(t)<Hxin2 , we haveN(t)50, by (18) and (22). Then, eDD(t)50 from (17) (thusD*(t)4eDD(t)) and, as Hxin2 tð Þ4Hxinþ2 tð Þ, (14) becomes:

V:tð Þ4 y1 � yr1

� �HC2r t; x1; yð Þ þ eDD tð Þ Hxinþ2 tð Þ � y1

� �h i4 y1 � yr1� �

HC2r t; x1; yð Þ � � tð Þ � lþ y1 � yr1� ��

ð23Þ

But, HC2r(t, x1, y)–�(t, x�1 ; x

þ1 , y)4 0 by (7). Then,

V:tð Þ4� lþ y1 � yr1

� �2¼ �2lþV tð Þ: ð24Þ

So, as long as y1(t)5yr1, V is decreasing which impliesalso that Hxin2 �y1(t) stays positive.Sub-case 2b: X0 does not fulfill the property (16).

Necessarily, y1>yr1. Consider the set Xt=[x�1 tð Þ;

x�1 tð Þ]�{y(t)} and the condition (16) at time t, i.e. thecondition:

y1 tð Þ < min Hxin2 ; yr1 þ

� Xtð Þ

lþ

� �; ð25Þ

Note that �(Xt)>0 for any time, due to (11) andLemma 13. As long as Xt does not fulfill (25), we have:

. either eDD(t)50 and y1(t)<Hxinþ2 tð Þ, then, as in (23)and (24), we have: V

:tð Þ4� lþV tð Þ,

. either eDD(t)<0 or y1(t)5Hxinþ2 tð Þ, then D*(t)=0and so V

:tð Þ4 y1 tð Þ � yr1

� �HC2r(�(t))<0 (due to

Hypothesis 5 and � (Xt)> 0).


In any case, y1(t) is decreasing towards yr1 and then Xt

fulfills (25) in finite time, say at t0: we are then in sub-case 2a.So, after a possible transient stage of finite duration

[0,t0] (sub-case 2b), we have from (15) and (24):

V:tð Þ4�min l�; lþ

� �V tð Þ; 8t5t0

thus the exponential convergence of y1 about yr1. &

Corollary 15. Let

D�� t; x�1 ; xþ1 ; y

� �¼ D� t; x�1 ; x

þ1 ; y

� �when y1 < yr1

D�þ t; x�1 ; xþ1 ; y

� �¼ D� t; x�1 ; x

þ1 ; y

� �when y15yr1

�;

then for any " 2 0; yr1� �

, the feedback law:

D�" t; x�1 ; x

þ1 ; y

� �¼eDD�� t; x�1 ; x

þ1 ; y

� �if y1 < yr1 � "

D�� t; x�1 ; xþ1 ; y

� � yr1 þ "� y12"

þD�þ t; x�1 ; xþ1 ; y

� � y1 � yr1 � "

2"if y� yr1 4"

D�þ t; x�1 ; xþ1 ; y

� �if y1 > yr1 þ "

ð26Þ

is continuous and ensures an exponential stabilization ofy1 about the set [y

r1–", y

r1+"].

Remark 16. When no information at all on the dynamicsis available, we take r�� 0 and r+� r� (where r� is a con-stant vector), which leads to the formula (10). The mag-nitude of the jump at y1=yr1 is then constant, equal to�HC2r�=

�Hxin2 � yr1

�.

Notice that r�� 0 does not fulfill Assumption 12, butany r� such that r�j �ð Þ ¼ min

Q�i; �

� �; 8j ¼ 1 . . .m ful-

fills Assumption 12 with arbitrary small � > 0. For any "> 0 and � small enough, we have HC2k k1�<l+", theneDD(y1)<0 for y15 yr1+�, thus the control law (10).

Appendix C. Stable ‘‘zero-dynamics’’ processes

For constant �in with yin1 > yr1, consider the change ofvariable �(�)=(�,�)=(�1(�),�2(�)) defined on a neigh-borhood of �

T{y1=yr1}:

�i ¼�i � �iniy1 � yin1

; i < n� pþ 1

�i ¼�iþ1 � �iniþ1y1 � yin1

; i 2 n� pþ 1; n� 1½ �

� ¼ y1 � yin1

8>>>>><>>>>>:

Write C~ for the matrix C without its (n�p+1)-th line,r(t, �, �) for r(t, �) and �r=yr1�yin1 . Then, the equationsof the system (S) can be written in the coordinates (�, �):

Sð Þ :�:¼ C~ � �HC2

� � r t; �; �ð Þ

�

�:¼ HC2r t; �; �ð Þ þD�

8<:Proposition 17. Under the same hypotheses of Proposition6, with constant (but unknown) vector �in, if the ‘‘zero-dynamics’’:

�:¼ C~ � �HC2

� � r t; �; �rð Þ

�r

admits a unique globally asymptotically stable equili-brium (�e, �r) in �1(� \ {y1=yr1} \ {�>0}), then theoutput feedback law (12) provides a robust semi-globalasymptotically stabilization.

Proof. Consider the closed loop system (S) with thecontrol law D* given by (12). The system being writtenin normal form and being minimum phase, it is wellknown that the local asymptotic stability of its zerodynamics implies the local asymptotic stability (see forinstance [17]). We prove now that this stability is semi-global on the set � \ {�>0}.Consider an initial condition that fulfills (11). From

the partial stabilization of y1 given by Proposition 6,there exists T<1 such that y1(T)<

12 (y

r1–y

in1 ). Then, the

semi-trajectory of (S) remains inside the bounded set �(� \ {y1 2 [0, y1(T)]}) for t5T, thus converges towardsthe unique equilibrium (�e, �r) (see for instance [18,Appendix A]). &

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