Global Stabilization of a Class of Partially Known Positive Systems

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Positive Systems

Jean-Luc Gouzé, Olivier Bernard, Ludovic Mailleret

To cite this version:

Jean-Luc Gouzé, Olivier Bernard, Ludovic Mailleret. Global Stabilization of a Class of Partially

Known Positive Systems. [Research Report] RR-5952, INRIA. 2006, pp.24. �inria-00086788v2�

inria-00086788, version 2 - 28 Jul 2006

a p p o r t

d e r e c h e r c h e

9

-6

3

9

9

IS

R

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IN

R

IA

/R

R

--5

9

5

2

--F

R

+

E

N

G

Thème BIO

Global Stabilization of a Class of Partially Known

Positive Systems

Ludovic Mailleret — Jean-Luc Gouzé — Olivier Bernard

N° 5952

Ludovi Mailleret

∗

,Jean-Lu Gouzé

†

, OlivierBernard

†

ThèmeBIOSystèmesbiologiques

ProjetComore

Rapportdere her he n°5952Juillet200624pages

Abstra t: Inthisreportwedealwiththeproblemofglobaloutputfeedba kstabilizationof

a lass of

n

-dimensionalnonlinearpositivesystemspossessingaone-dimensionalunknown,

thoughmeasured,part. Werstpropose ourmain result,anoutputfeedba k ontrol

pro- edure,taking advantageof measurementsof theun ertain part,ableto globally stabilize

thesystemtowardsan adjustableequilibrium pointin the interiorofthe positiveorthant.

Thoughquitegeneral,thisresultisbasedonhypothesesthatmightbedi ultto he kin

pra ti e. Then in ase ondstep, throughaTheorem ona lass ofpositivesystemslinking

theexisten eofastronglypositiveequillibriumtoitsglobalasymptoti stability,wepropose

otherhypothesesforourmainresulttohold. Thesenewhypothesesaremorerestri tivebut

mu hsimplerto he k. Someillustrativeexamples,highlightingboththepotential omplex

open loopdynami s (multi-stability, limit y le, haos) of the onsidered systems andthe

interestofthe ontrol pro edure, on ludethis report.

Key-words: Positivesystems,Partiallyknowsystems,Globalstabilization.

∗

URIH,INRA,400routedesChappes,BP167,06903Sophia-AntipolisCedex,Fran e

†

Résumé : Nous onsidéronsi i un problèmede stabilisation globalepar retour desortie

d'une lassedesystèmesnon-linéairespositifsdedimensionn possédantunepartiedeforme

analytiquein onnue,mais ependantmesurée. Nousproposonstout d'abordnotrerésultat

prin ipal, un ontrle par retour de sortie, qui stabilise globallement le système vers un

équilibre réglable appartenant à l'intérieur de l'orthant positif de l'espa e d'état. Bien

qu'assez général, e résultat est basé sur deshypothèsesqui peuvent serévéler di iles à

vériersurunexemple on ret. Dansunse ondtemps,parl'intermédiaired'unthéorèmesur

une lassede systèmes positifs liant existen eet stabilité globaled'un équilibre fortement

positif, nous proposons d'autres hypothèses permettant d'appliquer notre résultat. Ces

nouvelleshypothèses, bien que plus restri tives, sont nettement plus fa iles àvérier. Le

rapport s'a hève par des exemples qui illustrent à la fois la potentielle omplexité des

omportements en bou le ouverte qui peuvent être produits par les systèmes onsidérés

(multi-stabilité, y lelimite, haos)etl'intérêtdelapro édurede ontrleproposée.

1 Introdu tion

Positivesystemsof ordinarydierentialequationsaresystemsprodu ingtraje tories,that,

ifinitiatedinthe positiveorthantof

R

n

, remaininthis orthantforallpositivetime. Su h

systemsarebeingpayedmu hattentionsin etheyaboundinmanyappliedareassu haslife

s ien es,so ials ien es, hemi als ien es,tele ommuni ations,tra owset ... [11,4,1℄.

Inthispaperwedealwiththeproblemofglobaloutputfeedba kstabilizationofa lass

of

n

-dimensional nonlinear positive systems possessing a 1-dimensional unknown, though

measured, part. Our main result generalizes an earlier one obtainedin lowerdimensions

andforsimplerstru turesin the ontextof biopro ess ontrol[13℄.

This paper is organized as follows. We rst introdu e somenotations and denitions

related to positivity for ve tors and dynami al systems. We then introdu e the lass of

un ertainpositivesystemsthatis on ernedwithour ontrolproblem. Wethenproposean

output feedba kpro edure, taking advantageof measurementsof the un ertainpart,able

to globally stabilize the systemtowardsan adjustable equilibrium point in theinterior of

thepositiveorthant. The hypotheses onwhi h ourmain result is builtmight howeverbe

di ultto he kin pra ti e. Inorder toderivesomehypothesesthat aremu hsimplerto

he k,weproposeatheoremona lassofpositivesystems(thestronglypositive on ave

o-operativesystems)thatlinkstheexisten eofapositiveequilibriumtoitsglobalasymptoti

stability (GAS). Some examples found in the litterature illustrate the potential omplex

open loop dynami s that might be produ ed by the systemsunder study (multi-stability,

y li behavioror even haos). Simulationsonperiodi and haoti behaviorsstabilization

illustratethe ontrole ien y.

2 Preliminaries

2.1 Notations

We onsiderthefollowingautonomousnonlineardynami alsystemsin

R

n

:

˙x = f (x).

(1)

For the sake of simpli ity, we assume that the fun tions en ountered throughout the

paperaresu ientlysmooth.

Inthesequel,wewillusethefollowingnotations:

ˆ

Df (x)

denotestheJa obianmatrixofsystem(1)at state

x

.

ˆ

L

f

denotestheLiederivativeoperatoralongtheve toreld denedbysystem(1).

ˆ

x

|x

i

=0

denotesastateve torwhose

i−

th omponentequalszero.

Moreover,amatrixissaidtobeHurwitzifallofitseigenvalueshavenegativerealparts.

2.2 Positivity for ve tors and dynami al systems

Aswedealwith positivity in

R

n

and somerelated on epts,werst learly statewhat we

referto aspositivity(resp. strongpositivity)forve torsand dynami alsystems.

Denition1(Positivity)

x ∈ R

n

is positive (resp. strongly positive) and denoted

x ≥ 0

(resp.

x ≫ 0

) i all its

omponentsare non-negative(resp. positive).

We denote

R

n

+

and int

(R

n

+

)

the sets of positive ve tors and strongly positive ve tors

belongingto

R

n

, respe tively. Same denitionsof therelations

≥

and

≫

(i.e. omponent

by omponent)willbeusedformatri estoo.

Wenowstatethedenition ofpositivesystemsofordinarydierentialequationsaswell

asthedenitionofaspe ial lassof positivesystems,thestronglypositiveones.

Denition2(Positive Systems)

System(1)isapositive system(resp. stronglypositive)i:

∀i ∈ [1..n], ∀x

|x

i

=0

≥ 0, ˙

x

i

(x

|x

i

=0

) = f

i

(x

|x

i

=0

) ≥ 0, (resp. f

i

(x

|x

i

=0

) > 0).

Itis straightforwardthatDenition 2impliesthefollowing:

Proposition 1

Considerapositive system(1)(resp. stronglypositive). Then:

∀(x

0

, t) ∈ R

n

+

× R

+

(resp. ∀(x

0

, t) ∈ R

n

+

× R

+

∗

),

x(t, x

0

) ≥ 0 (resp. ≫ 0).

Remark1 Though wehaveequivalen ebetweenDenition2andProposition1inthe ase

ofpositivesystems(thisistheoriginaldenitionofpositivesystemsby[11℄),weonlyhavean

impli ationfromDenition2toProposition1forstronglypositive systems. Indeedsystems

with

f

i

(x

|x

i

=0

) ≥ 0

andastrongly onne tedJa obianMatrix mayprodu estrongly positive

traje tories too [4 ℄. We however keep Denition 2 as the denition of strongly positive

3 Main result

3.1 The onsidered lass of systems

Inthesequelwefo usonthe(global)asymptoti stabilizationproblemofthefollowing lass

ofnonlinearpositiveun ertainsystemsin

R

n

:

˙x = uf (x) + cψ(x),

y = ψ(x).

(2)

ˆ

u ≥ 0

beingthes alarinputofsystem(2). ˆ

f : R

n

_{→ R}

n

ˆ

c ∈ R

n

ˆ

ψ : R

n

_{→ R}

isanoutputofsystem(2)

Fun tion

ψ(.)

featurestheun ertain/unknownpartofthesystem. For ontrolpurposes,we

suppose that,thoughanalyti allyunknown,

ψ(.)

mightbe onlinemeasured. We moreover

assumethefollowing:

Hypotheses 1 (H1)

1.

∀x ∈

int

(R

n

+

), ψ(x) > 0

and

∀i, c

i

ψ(x

|x

i

=0

) ≥ 0

2.

f (.)

issu hthat the system

˙x = f (x)

ispositive

3.

∃β

m

∈ R

+

su hthat

∀β > β

m

,the system:

˙x = βf (x) + c.

is a strongly positive systemand possesses an equilibrium

x

⋆

β

≫ 0

whi h is GAS on

R

n

+

.

These hypotheseswill be ommented after thestatementof ourmain result. Fornow,

just noti e that HypothesesH1-1 and H1-2 togetherwith

u ≥ 0

ensurethe positivity of

the onsidered lassofsystems(2).

3.2 Main result

Theorem1

ontrollaw:

u = γy = γψ(x).

(3)

globally stabilizes system (2) on int

(R

n

+

)

towards the strongly positive equilibrium

x

⋆

γ

su h that:

f (x

⋆

γ

) =

−1

γ

c.

ToproveTheorem 1,weneedthefollowingLemmaonLyapunovfun tions forpositive

systemspossessingastronglypositiveGASequilibrium.

Lemma1

Apositive system(1)possesses astronglypositive equilibriumpoint

x

⋆

GASon int

(R

n

+

)

i

thereexistsasmooth real valuedfun tion

V (x)

su hthat:

∀x ≫ 0, x 6= x

⋆

_{, V (x) > 0,}

V (x

⋆

_{) = 0,}

∀x ≫ 0, x 6= x

⋆

_{, ˙}

_{V (x) < 0,}

[

α>0

{x, V (x) ≤ α} =

int

(R

n

+

).

Proof (Lemma1)

Considerapositivesystem(1)and suppose itpossessesastronglypositiveequilibrium

x

⋆

GASonint

(R

n

+

)

.

Now onsiderthefollowingin reasing(inthe

≫

sense) hangeofvariables:

z = ln



x

x

⋆



=















ln

 x

1

x

⋆

1



. . .

ln

 x

n

x

⋆

n

















.

thatmapsthestronglypositiveorthanton

R

n

and

x

⋆

to0. Thenwehave:

˙z =

f (x

⋆

_e

z

₎

x

⋆

e

−z

_.

(4) Sin e

x

⋆

isaGASequilibriumonint

(R

n

+

)

for(1),sois0on

R

n

forsystem(4). Then,from

Kurzweil's onverseLyapunovtheoremonGAStability(seeTheorem7in[10℄),thereexists

asmoothradiallyunboundedLyapunovfun tion

W (z)

forsystem(4). Comingba ktothe

originalvariables, onsiderthefun tion:

Sin e

W (z)

isaLyapunovfun tionforsystem(4),wehave:

∀x ∈

int

(R

n

+

), x 6= x

⋆

, V (x) = W (z) > 0,

V (x

⋆

_{) = W (0) = 0,}

∀x ∈

int

(R

n

+

), x 6= x

⋆

, ˙

V (x) = ˙

W (z) < 0.

Moreover,sin e

W (z)

issmoothsodoes

V (x)

andsin e

W (z)

isradiallyunboundedon

R

n

, wehave:

[

β>0

{z, W (z) ≤ β} = R

n

.

Thus:

[

α>0

{x, V (x) ≤ α} =

int

(R

n

+

).

whi h on ludestherstpartoftheproof.

Thereverseimpli ationiseasilyobtainedthrough lassi alLyapunovtheory.



Proof (Theorem1)

Considerthepositivesystem(2)under ontrollaw(3). Weget:

˙x = ψ(x)(γf (x) + c)

, g(x).

(5)

Sin e

γ > β

m

weknowfromH1-3that thesystem:

˙x = γf (x) + c

, h(x).

(6)

is strongly positive and possessesa (strongly positive) GAS equilibrium

x

⋆

γ

. Then, from

Lemma1,forall

γ > β

m

,system(6)admits areal valuedfun tion

V

γ

(x)

verifyingLemma

1properties.

Nowwe onsidersystem(5). FromH1-1,itis learthat wehave:

∀x ≫ 0, x 6= x

⋆

γ

, V

γ

(x) > 0,

V

γ

(x

⋆

γ

) = 0.

Moreover:

L

g

V

γ

(x) = ψ(x)L

h

V

γ

(x).

sothat sin eH1-1holds:

∀x ≫ 0, x 6= x

⋆

γ

, L

g

V

γ

(x) < 0.

andthefollowingstillholds:

[

α>0

{x, V

γ

(x) ≤ α} =

int

(R

n

+

).

So that we anapply Lemma 1to the ontrolled system (5). Then the equilibrium

x

⋆

γ

is

GASforthe ontrolledsystem(5) onint

(R

n

+

)

.



Remark2 It is important to noti e that the proposed ontrol law does not, to beapplied,

require any a priori analyti al or quantitative knowledge of the fun tion

ψ(x)

. The only

two requirements are (i) its positivity (a qualitative property), and (ii) the possibility to

measurethisquantityon-line. Thisisparti ularlyimportantforinstan einbiologi alpro ess

ontrol,whensomepartsofthemodelareonlyqualitativelyknown(usuallyintermsofsign)

while someother parts are pre isely (analyti ally) known. The spe ial ase of biopro esses

presentedby [13 ℄ does illustrate this pointandshows the interestof su h ontrol pro edures

for real life biopro esses management. We ome ba k to this point later in the examples

se tion.

3.3 Comments on Hypotheses H1

As wehave previouslynoted, Hypotheses H1-1 and H1-2 guarantees the positivity of the

systems(2).

FirstpartofHypothesisH1-1denesaqualitativepropertyoftheunknownpartofthe

system, whi h, though one-dimensional, possibly a ts on all state variables dynami s via

theve tor

c

. Noti ethat this isareally loose hypothesis sin etheonly requiredproperty

is

ψ(x)

positivityas

x

isstrongly positive. Nospe i analyti alform isassumed on

ψ(.)

.

These ondpartofH1-1 isrequiredtoensuresystem(2)'spositivityastheinput

u

equals

0. H1-2isrequiredforthesystem'spositivityastheinput

u

ispositive.

HypothesisH1-3isastrongerone. Itreadsthatiftheun ertainpart

ψ(.)

werepositive

and onstant,thentheremustexistaninputinmum

β

m

abovewhi hsystem(2)wouldbe

stronglypositiveand wouldpossess a(stronglypositive) GAS equilibrium. This property

might re all minimum phase" onditions, frequenlty en ountered when dealing with the

stabilizationofnonlinearand/orun ertainsystems. Minimumphase onditionsensuresthe

GAStabilityofanequilibriumasthesystem'soutputsare onstant. Minimumphasebased

ontrolresultsusuallystabilisestheoutput(s)inorderto on ludeonthewholesystem,using

results on as adednonlinear systems' stability. However un ertainties for these systems

are in generalnot on the outputs' dynami s, but rather in the intrinsi stable" part. A

minimumphase"approa h anthus notbeusedin our asesin etheoutput

ψ(x)

andits

dynami sis,toalargeextent,unknown.

Fromthe analyti alexpressionof thesystem that isto bestabilized, both Hypotheses

H1-1andH1-2areeasyto he k. Thisishowevernotthe aseforH1-3. Theexisten eofa

uniqueandstronglypositiveequilibrium of:

˙x = βf (x) + c

mightindeed beeasilyproven, but thedemonstrationof itsglobal stabilityis in generala

mu h hardertask. Nevertheless, for somespe ial lasses of positive systems,one an link

for instan e the ase for positive linear systems, see e.g. [11, 4℄. It is lear that su h a

propertyisofgreat interest: itallowstoprovein onestepboththeexisten eofastrongly

positiveequilibrium and its GAStability i.e. Hypothesis H1-3. Someworks anbefound

in the litterature regardingthis problem for nonlinear systems: see [18, 2℄. These results

anindeed be used to he k H1-3. It is to be noti ed that these works are all related to

ooperativesystems(seeDenition4inthesequel).

In the next se tion we also base our approa h on ooperative systems. We propose a

lass ofpositivesystems forwhi h theexisten e of astronglypositiveequilibrium implies

itsGAStability. We then derivea set ofeasy to he k" hypothesesthat are su ientfor

our ontrolpro edure(3)tobeapplied.

4 Su ient onditions to verify H1

4.1 Useful known results

Wenowre alltheMetzlermatri esdenition introdu edby[15℄inmathemati ale onomy.

Denition3(Metzler Matrix)

AmatrixisMetzler iallitso-diagonal elements arenon-negative.

Wenowfo usontworesultsonMetzler matri esthat are onsequen esof theoriginal

theoremofPerron-Frobeniusonpositivematri es[5℄. Therst omesfrom[11℄:

Theorem2

ConsideraMetzler matrix

A

andave tor

b ∈

int

(R

n

+

)

;Then,

A

isHurwitzi:

∃x ≥ 0, Ax + b = 0.

These ond anbefoundin[20℄:

Theorem3

ConsideraMetzler matrix

A

. Then:

ˆ The dominant eigenvalue of

A

(i.e. of largest real part) is real; its asso iated

eigen-ve torispositive.

ˆ onsider a matrix

B ≥ A

. Then the real part of the dominant eigenvalue of

B

is

Letusre allthedenitionofthe ooperativesystems",introdu edby[8℄(seealso[21℄)

andthatwill bepayedattentioninthesequel.

Denition4(CooperativeSystems)

System(1)isa ooperativesystemiitsJa obian

Df (x)

isMetzlerfor all

x ∈ R

n

.

Oneofthemostinterestingresulton ooperativesystemsistobefoundin[21℄andreads:

Theorem4

Consider a ooperative system (1) and two initial onditions

x

0

and

y

0

in

R

n

su h that

x

0

≥ y

0

(resp.

≫

). Then:

∀t ≥ 0, x(t, x

0

) ≥ y(t, y

0

) (resp. ≫).

4.2 Preliminary result

Proposition 2(Positive Cooperative Systems)

Considera ooperativesystem(1). Then,the followingare equivalent:

ˆ system(1)ispositive (resp. stronglypositive)

ˆ

f (0) ≥ 0

(resp.

f (0) ≫ 0

)

Proof

Suppose

f (0) ≥ 0

(resp.

f (0) ≫ 0

). Letus denotethe

i

-throwof

Df (x)

by

Df

i

(x)

. Then

wehaveforall

x

|x

i

=0

≥ 0

:

f

i

(x

|x

i

=0

) = f

i

(0) +

 Z

1

0

Df

i

(sx

|x

i

=0

)ds



. x

|x

i

=0

.

(7)

Sin e

Df (x)

isMetzlerforall

x

,theonlypossiblenegativeterminthes alarprodu tin(7)

iszero: indeed,

Df

i,i

(thesolepossiblenegativetermof

Df

i

)ismultipliedby

x

i

= 0

. Then:

∀x

|x

i

=0

≥ 0, f

i

(x

|x

i

=0

) ≥ f

i

(0) ≥ 0 (resp. > 0).

Sin ethisholdsforall

i ∈ [1..n]

,we on lude,usingProposition2,thatthesystemispositive

(resp. stronglypositive).

Remark3 Noti e that the lass of positive ooperative systems is a generalization of the

lass oflinearpositive systems[11 , 4 ℄.

Wenowstatearesultonglobalasymptoti stabilityofanequilibriumpointforaspe ial

lassofstronglypositive ooperativesystems.

Theorem5

Consider a strongly positive on ave ooperative (SPCC) system (1), i.e. that veries the

following ondition:

∀(x, y) ∈ R

n

+

× R

n

+

, x ≤ y ⇒ Df (x) ≥ Df (y).

(8)

Then,if system(1)has apositive equilibrium

x

⋆

,itissingle,stronglypositive andGASon

R

n

+

.

Proof:

We rst show the exponential stability of a positive equilibrium

x

⋆

(that is ne essarily

stronglypositive) ofaSPCCsystem(1).

Weobviouslyhave:

f (x

⋆

_{) = 0 = f (0) +}

 Z

1

0

Df (sx

⋆

_)ds



x

⋆

_.

(9)

Sin e

Df (x)

is Metzler for allpositive

x

, so doesthe bra keted matrix(denoted

F (x

⋆

₎

in

thesequel) in equation(9). Sin e the onsideredsystem isastronglypositive ooperative

system, we have from Proposition 2:

f (0) ≫ 0

. Applying Theorem 2, we on lude that

matrix

F (x

⋆

₎

isHurwitz. Ontheotherhand,equation(8)implies:

∀s ∈ [0, 1], Df (sx

⋆

_{) ≥ Df (x}

⋆

_).

whi h yields:

F (x

⋆

_{) =}

 Z

1

0

Df (sx

⋆

_)ds



≥ Df (x

⋆

_).

Sin e

F (x

⋆

₎

is a Hurwitz Metzler matrix, we show using Theorem 3 that its dominant

eigenvalueis negativereal. From the aboveinequationand Theorem 3,we on lude that

Df (x

⋆

₎

's dominanteigenvalueisnegativerealtoo,provingthat

x

⋆

isexponentiallystable.

Wenowshowtheuni ityofastronglypositiveequilibrium

x

⋆

.

Supposethatthe onsideredsystempossessestwostronglypositiveequilibria

x

⋆

1

and

x

⋆

2

, su h that

x

⋆

1

≥ x

⋆

2

. Then:

0 = f (x

⋆

1

) − f (x

⋆

2

),

=

 Z

1

0

Df (sx

⋆

1

+ (1 − s)x

⋆

2

)ds



(x

⋆

1

− x

⋆

2

).

(10)

From equation(8) and sin e

x

⋆

1

≥ x

⋆

2

, weuse similarargumentsthanin the previouspart

oftheprooftoshowthat:

 Z

1

0

Df (sx

⋆

1

+ (1 − s)x

⋆

2

)ds



≤ Df (x

⋆

2

).

Sin e

x

⋆

2

isastronglypositiveequilibrium,

Df (x

⋆

2

)

isHurwitz. Then,usingTheorem3,the

bra keted matrix is Hurwitz too. Then it possessesan inverse in

M

n

(C)

whi h, together

withequation(10),impliesthat

x

⋆

1

= x

⋆

2

.

Supposenowthatthe onsideredsystempossessestwostronglypositiveequilibria

x

⋆

1

and

x

⋆

2

, thatare notlinkedbytherelation

≥

, i.e. (

x

⋆

1

− x

⋆

2

)

isneitherpositive,nornegative.

Letus denetheparallelotopes

B

z

⊂ R

n

, su hthat:

∀z ∈ R

n

+

, B

z

=

x ∈ R

n

+

, x ≤ z

UsingTheorem4togetherwiththe onsideredsystem's ooperativityandstrongpositivity,

itis easyto showthat

B

x

⋆

1

and

B

x

⋆

2

arepositivelyinvariant sets. We nowdene thestate

x

3

,su hthat:

x

3

,i

= min(x

⋆

1,i

, x

⋆

2,i

)

Sin e

B

x

3

= B

x

⋆

1

∩ B

x

⋆

2

, it is positively invariant bysystem (1) too. Then, using Brouwer

xed point Theorem (seee.g. [23℄), there mustexist a third equilibrium

x

⋆

3

belonging to

B

x

3

. Notethat,from

x

3

denition,wene essarilyhave:

x

⋆

3

≤ x

⋆

1

and

x

⋆

3

≤ x

⋆

2

This asehas beentreatedpreviously and we on ludethat

x

⋆

1

= x

⋆

3

= x

⋆

2

, what yields a

ontradi tion.

We on ludethat if aSPCC system (1) possessesa positive equilibrium, itis the sole

positiveequilibrium.

Wea hievetheproofofTheorem5showingtheglobalattra tivity(on

R

n

+

)ofthepositive

equilibrum

x

⋆

ofaSPCCsystem(1).

Considerthefollowingreal valuedfun tion:

∀i ∈ [1..n], g

i

: R

+

→ R

k

7→ f

i

(kx

⋆

)

Sin e

f (0) ≫ 0

and

f (x

⋆

_{) = 0}

,itis learthat

g

i

(0) > 0

and

g

i

(1) = 0

. Then,thereexists a

k

0

∈ (0, 1)

su hthat:

dg

i

Fromequation(8),weget:

0 ≤ k

0

≤ k ⇒ 0 >

dg

i

dk

(k

0

) ≥

dg

i

dk

(k).

showingthatfun tion

g

i

stri tlyde reasesfrom

k

0

to

+∞

. Sin e

g

i

(1) = 0

,

g

i

(k) < 0

forall

k > 1

. Thisholdsforall

i ∈ [1..n]

,then:

∀k > 1, f (kx

⋆

) ≪ 0

Untilltheendoftheproof,weassume

k > 1

. Noti ethat,fromthe ooperativityandstrong

positivityproperties,

B

kx

⋆

isapositivelyinvariantset. Now onsider

x

¨

,thetimederivative

of

˙x

. Wehave:

¨

x = Df (x) ˙x

Sin e

Df (x)

is a Metzler matrix for all

x

, it is straightforward that if the initial state

velo ity

˙x(t = 0)

ispositive(resp. negative)thenitwillremainpositive(resp. negative)for

allpositivetime.

Remindthat

˙x(x = 0) ≫ 0

and

˙x(x = kx

⋆

_{) ≪ 0}

. Then,theforwardtraje toryofsystem

(1)initiatedat

x

0

= 0

(resp.

x

0

= kx

⋆

)is,inthe

≥

 sense,anin reasing(resp. de reasing)

fun tion of thetime. Moreover,from

B(kx

⋆

₎

positiveinvarian e, it islower(resp. upper)

boundedby

0

(resp.

kx

⋆

),thusit onverges,ne essarilytowardanequilibrium.

x

⋆

isthesoleequilibriumbelongingto

R

n

+

,thetwo onsideredtraje tories onvergethus

toward

x

⋆

. Fromsystem(1) ooperativitytogetherwithTheorem4,we on ludethat ea h

traje toryinitiatedin

B

kx

⋆

onvergestoward

x

⋆

. Sin e thisholdsforany

B

kx

⋆

(

k > 1

),we

haveshownthat

x

⋆

isgloballyattra tiveon

R

n

+

,whi h on ludestheproof.



Remark4 Proof of theorem 5uses toa largeextent the ideasproposedby [20 ℄. However,

his proof wasdedi ated toanother lass of systems: Kolmogoro-typepopulation dynami s

models of ooperating spe ies,whi h arebasedon dierent hypotheses. One an alsohavea

look at[19 ℄ for someresultsrelatedtoTheorem 5.

Remark5 Itis learthat Theorem 5isquitesimilar toHirs h'sresults[9 ℄on ooperative

systems, rephrased by [17℄, and statingthat for a bounded strongly monotonesystems(e.g.

ooperative systemsverifying

x

0

≥ y

0

⇒ x(t, x

0

) ≫ y(t, y

0

)

, the traje tories, apart froma

set of zero measure (thus atmost

n − 1

dimensional) of initial onditions, onvergeto the

set of equilibrium points. Thus if su h a systemwere bounded andthe equilibrium unique

thenalmostalltraje tories onvergetoit. Noti ealsothatSmithprovidesarelatedTheorem

(see [21 ℄, theorem 3.1). Here for SPCC systems, both uniqueness of the equilibrium (if it

exists) and(real) global onvergen e are guaranteed, whi h is of parti ular interest for the

4.3 Su ient onditions to verify H1

OnthebasisofSPCCsystemsandTheorem5weprovidesu ient onditionsthataremu h

simplerto he kandthatguaranteesthatHypothesesH1aresatised.

Considerasystem(2)underthefollowinghypotheses:

Hypotheses 2 (H2) 1.

∀x ∈

int

(R

n

+

), ψ(x) > 0

and

∀i, c

i

ψ(x

|x

i

=0

) ≥ 0

2.

f (0) ≥ 0

3.

f (.)

issu hthat system

˙x = f (x)

is ooperative

4.

∀x

1

, x

2

∈ R

n

+

, x

1

≤ x

2

⇒ Df (x

1

) ≥ Df (x

2

)

5.

∃β

m

∈ R

+

su hthat

∀β > β

m

, βf (0) + c ≫ 0

6.

∀β > β

m

, ∃x

⋆

β

∈

int

(R

n

+

)

su hthat

βf (x

⋆

β

) + c = 0

One anshowthat H2 impliesH1, thus that asystem (2)under H2 isa andidatefor

Theorem1.

Indeed, H2-1and H1-1are thesame. H2-2 togetherwith H2-3impliesthrough

Propo-sition2 that

˙x = f (x)

isa positive systemi.e. H1-2 holds. H2-3 togetherwith H2-5 and

Proposition2impliesthat forall

β > β

m

,thesystem:

˙x = βf (x) + c.

(11)

isastronglypositive ooperativesystem. Thisfa ttogether withH2-4impliesthat(11)is

aSPCC system. H2-6 ensuressystem 11 has, for all

β > β

m

, an equilibrium

x

⋆

β

that is,

throughTheorem5,GASon

R

n

+

sothatH1-3holds.



In the nextse tion weprovidenumeri al simulations, based onexamples fullling

hy-pothesesH2,thatillustratesTheorem 1interestande ien y.

5 Illustrative Examples

Weshowonthreeexamplesthatthepossibleopenloopdynami sofsystems(2)underH2

might be very omplex. Proofs of the omplex dynami s are not detailed asthey do not

fall beyond thes opeof thispaper,but they aneasilybeperformedwith thehelp ofthe

Example1(Bi-stability)Thissystemmodelsasimplebiorea tiono uringina

ontin-uousstirred tankrea torwith substrateinhibitory ee ts (see e.g. [22℄). This isa simple

model of an anaerobi digester, an apparatus used for waste water treatment [13℄. Both

variables represent on entrationsinside the rea tor:

x

1

denotes thesubstrate (pollutant)

onsumedbythebiomass(anaerobi mi roorganisms)

x

2

togrowataper apitarate

µ(x

2

)

.

u

denotes the dilution rate (i.e. passingowper volume unit) feeding therea tor with

substrate at a on entration

x

1

,in

and withdrawingablend of

x

1

and

x

2

from it. Weget

thefollowingsystemthat willbereferedtoas

(S

1

)

inthesequel.

˙x

= u



−1

0

0

−1



x +



x

1

,in

0



+



−k

1



µ(x

1

)x

2

,

, uf

1

(x) + c

1

ψ

1

(x).

with:

µ(x

1

) =

µ

m

x

1

K

m

+ x

1

+ x

2

1

/K

i

x

1,in

, k, µ

m

, K

m

and

K

i

arepositive onstants.

Itis learthatHypothesesH2-1,H2-2,H2-3andH2-4hold. MoreoeverH2-5holdswith

β

m

= k/x

1

,in

. Nowpi ka

β > β

m

,then

βf

1

(x) + c

1

ψ

1

(x) = 0

possessesasolution

x

⋆

β

su h that:

∀β > β

m

, x

⋆

β

=







x

1

,in

−

k

β

1

β







≫ 0.

sothat H2-6holds. ThenwehavethroughTheorem1:

Proposition 3 For all

γ > k/x

1,in

the ontrollaw

u(.) = γψ

1

(x)

(12)

globallystabilizes,on int

(R

n

+

)

,system

(S

1

)

towards

x

⋆

γ

,the(sole)solutionof

γf

1

(x)+c

1

= 0

.

Now let us onsider the open loop behaviorthat might be produ ed by

S

1

. Suppose

that

x

1

,in

>

argmax

µ

and

u ∈ (µ(x

1

,in

), max(µ))

is onstant. Thenthe onsideredsystem

possessesthreeequilibria,twoofwhi h beingstableontheirrespe tivebasinsofattra tion

separated by the stable manifold of the third equilibrium that is a saddle point. This

behaviorisdepi tedongure1;seee.g. [12℄forarigorousanalysis.

Thetwostableequilibria orrespondforthestronglypositiveonetotheoperatingpoint"

(thebiomasssurvivesinsidetherea torandthepollutant on entrationisredu ed ompared

toitsin theinow)whiletheotherdonot(thebiomassdisappearsandthewastewateris

nomoretreated). It is thenne essaryto ensurethat no traje torymaybedriven to this

Bi−stability behavior in open loop

X1

X2

0

2

4

6

8

10

12

14

16

18

20

0

5

10

15

20

25

30

Figure1: Statespa erepresentationofvariousopenloopforwardtraje toriesof

(S

1

)

witho

representingequilibriaand

u

belongingto

(µ(x

1,in

), max(µ))

: existen eoftwolo allystable

equilibria.

it orrespondsto theoutowofbiogazthat isprodu edbytheanaerobi digester. Taking

adavantageonthismeasurement, ontrolpro edure(3) hasbeenapplied toareallifepilot

s aleanaerobi digester. Ithasshownitsinterestande ien yfortheglobalstabilization

ofsu hmultistabledevi es. Itthusavoidsthe rashofthesystem,see[13℄formoredetails.

Example2(Attra tivelimit y le)

system

(S

2

)

inthesequel.

˙x

= u





−lx

1

µ

1

x

1

k

1

+

x

1

− x

2

+ α

1

µ

2

x

2

k

2

+

x

2

− x

3

+ α

2





+





1

0

0





1

1 + x

n

3

,

, uf

2

(x) + c

2

ψ

2

(x).

with

l = 2.1

,

µ

1

= 2/2.1

,

µ

2

= 4 ∗ (0.01 + 1/2.1)

,

k

1

= 1/4.2

,

k

2

= 0.01 + 1/2.1

,

α

1

= 0.01

,

α

2

= 1 − 2(0.01 + 1/2.1)

and

n = 80

. ItiseasilyproventhatH2-1,H2-2,H2-3andH2-4hold

true. MoreoverH2-5 holds with

β

m

= 0

. Nowpi ka

β > β

m

, then onsider theequation

βf

2

(x) + c

2

ψ

2

(x) = 0

. Wegetthat

x

3

mustbeasolutionof:

x

3

= α

2

+

µ

2

(µ

1

+ α

1

(k

1

βl(1 + x

n

3

) + 1))

µ

1

+ (α

1

+ k

2

)(k

1

βl(1 + x

n

3

) + 1)

evaluatingbothsidesofthisequationat

x

3

= 0

and

+∞

,weshowthatthereexistasolution

x

∗

β,3

whi h is positive for all

β > β

m

.

x

∗

β,2

and

x

∗

β,1

are then omputed and shown to be

positivetoo,sothatH2-6isveried.

Supposethat the onsidered systemoperatesin openloopwith thenon-negativeinput

u = 1

. Then, asdepi ted ongure 2,

(S

2

)

possessesan attra tivelimit y le around an

unstableequilibriumpoint. Theexisten eofanontrivialperiodi orbit anbeprovenwith

Theorem1from [7℄.

AsillustratedonFigure3,wegetfromTheorem 1:

Proposition 4 For all

γ > 0

the ontrol law

u(.) = γψ

2

(x)

(13)

globallystabilizes,on int

(R

n

+

)

,system

(S

2

)

towards

x

⋆

γ

,the(sole)solutionof

γf

2

(x)+c

2

= 0

.

Example3(Chaos)Thefollowingsystemisastatespa etranslationofthemodelofa

hemi alsystemso alledauto- atalator,see[16℄. It reads:

˙x = u













−1

0

k

2

1

k

1

−1

k

1

0

0

1

−1







x+





k

2

(k

3

−k

4

)

0

k

4











+







−1

1

k

1

0







x

1

x

2

2

,

, uf

3

(x) + c

3

ψ

3

(x).

k

1

ispositive,

k

2

∈ (0, 1)

and

k

3

> k

4

> 0

.

0.96

0.99

1.02

1.05

0

0.15

0.30

0.45

0.40

0.45

0.50

0.55

X2

X1

X3

Cyclic behavior in open loop

Figure 2: Statespa e representationof aforwardtraje tory of

(S

2

)

with

u = 1

: existen e

ofanattra tivelimit y le.

It is lear that H2-1, H2-2, H2-3 and H2-4 hold. Moreover H2-5 holds with

β

m

=

1/(k

2

(k

3

− k

4

))

. Now pi ka

β > β

m

, then

βf

3

(x) + c

3

ψ

3

(x) = 0

possesses asolution

x

⋆

β

su h that:

∀β > β

m

, x

⋆

β

=

















βk

2

k

3

+ k

2

− 1

β(1 − k

2

)

k

2

k

3

1 − k

2

k

2

(k

3

− k

4

) + k

4

1 − k

2

















≫ 0.

0

10

20

30

40

50

60

0

0.15

0.30

0.45

0

10

20

30

40

50

60

0.40

0.45

0.50

0.55

0

10

20

30

40

50

60

0.96

0.99

1.02

1.05

0

10

20

30

40

50

60

0.5

0.8

1.1

1.4

Time

Stabilized periodic forward orbit

X1

X2

X3

U

open loop

closed loop

Figure 3: Time varying representation of a forward traje tory of

(S

2

)

with

u = 1

before

t = 40

andunder ontrol law(13)from

t = 40

totheend, with

γ = 2 > β

m

(= 0)

omputed

s.t. asymptoti ally

u(.) = γψ

2

(x)

equalsitsopenloopvalue.

[16℄ prove that Example 5 exhibits a haoti behavior for the parameter values

k

1

=

0.015, k

2

= 0.301, k

3

= 2.5

and

u = 1

forall

k

4

. Asaspe ial ase,we hoose

k

4

= 0.56

.

Anexampleofaforward haoti traje toryisshownongure4.

Asillustratedongure5,wegetthefollowingresultfromTheorem 1:

Proposition 5 For all

γ > 0

the ontrol law

u(.) = γψ

3

(x)

(14)

globallystabilizes,on int

(R

n

+

)

,system

(S

3

)

towards

x

⋆

4

8

12

X2

1.5

1.7

1.9

2.1

X3

0

0.2

0.4

0.6

X1

Chaotic behavior in open loop

Figure4: Statespa erepresentationofaforward haoti traje toryofthesystem

(S

3

)

with

u = 1

.

6 Dis ussion

Inthis paperwehave onsidered aratherbroad lass of dynami alsystemsen ompassing

several possible dynami al behaviours. These models an espe ially represent biologi al

systemsexhibiting multistability,periodi solutionsoreven haoti behaviour. Inpra ti e,

thekeypointforstabilisingthesesystemsisthattheadequates alarfun tion

ψ(x)

,whose

exa t analyti al expression does not need to be known, is measured. In the example of

theanaerobi digesterthat anhave3steadystates,thebiogazoutowrate

ψ(x)

,iseasily

on-line measuredthough itsexa t analyti alexpression depending onba terial a tivity is

ratherun ertain.

The proposed ontrol law, although very simple, has proven its e ien y to globally

Time

0

5

10

15

20

25

30

0

0.2

0.4

0.6

0

5

10

15

20

25

30

0

4

8

12

0

5

10

15

20

25

30

1.3

1.5

1.7

1.9

0

5

10

15

20

25

30

0

0.5

1

1.5

X1

X2

U

Stabilized chaotic forward orbit

X3

open loop

closed loop

Figure 5: Time varying representation of aforwardtraje tory of the haoti system

(S

3

)

with

u = 1

before

t = 20

andunder ontrollaw(14)from

t = 20

totheend,with

γ ≈ 1.73 >

β

m

(≈ 1.71)

omputedsothatasymptoti ally

u(.) = γψ

3

(x)

equalsitsopenloopvalue.

when a part of the model (i.e.

ψ(x)

) is not perfe tly known. Moreover the ontrol law

naturally fulls the input non negativity onstraints whi h is lassi al e.g. for biologi al

systems.

The ontroller relevan e was demonstrated in a real asefor an anaerobi wastewater

treatmentplant(Mailleretet al.,2004). Despiteits omplexity(thise osystemisbasedon

morethan140intera tingba terialspe ies[3℄)andthegreatun ertaintiesthat hara terize

was also applied to aphotobiorea torwhere mi roalgae onsuming nitrate and light were

growing[14℄.

Another importantissue when dealingwith omplexsystemis the di ulty to on-line

measurethevariable that mustbe regulated. Theproposed ontrolapproa hhasthusthe

advantageofproposingaregulations hemethatdoesnotrequirethismeasurement,whi h

an be very di ult to arry out for biologi al systems. Moreover, and as it has been

demonstrated with the real implementation, it is rather robust and stabilizes the system

even if the model parameters are not perfe tly known. The pri e to pay is howeverthe

possible la kof a ura y,sin ethe solutionof

γf (x) + c = 0

andierfrom the expe ted

set point in ase of parametri un ertainty. In su h a ase the idea onsists in adding an

integrator to orre t the bias. This adaptiveapproa h was proposed in (Mailleret et al.,

2004)forthespe i aseofanaerobi wastewatertreatmentplantandthe onvergen eof

the ontrollerwasdemonstrated. Of ourseitrequiresmoremeasurement apabilityonthe

system,butweassumethat theadaptation anbeslowerandthusneedsalowersampling

frequen y. The onvergen e proofof theadaptive ontrollerin thegeneral aseis nowthe

next hallengethatwewillta kle.

A knowledgements: Theauthorswouldliketo thankV.Lemesle(COM-University

ofMarseille,Fran e)andF.Grognard(COMORE-INRIA, Fran e)forfruitfuldis ussions

onthesubje t.

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