Diagnosis and control using multiple models. Application to a biological reactor

Diagnosis and control using multiple models.

Application to a biological reactor

Jose Ragot

Institut National Polytechnique de Lorraine (INPL)Centre de Recherche en Automatique de Nancy, France (CNRS)

International Symposium on Advanced Control of IndustrialProcesses, Hangzhou, May 23-26, 2011

Introduction. Study background

◮ Processes related to environmentalAir pollution, water pollution, watertreatment plants, water distributionsystems, runoff, rainfall . . .

◮ Diagnostic of systemsValidation of measurements, detectionand fault location, fault tolerantcontrol. . .

◮ Models of environmental processesPhenomena are difficult to model,perturbations with critical effects,concentration of species difficult tomeasure. . .

◮ Fundamental issuesEnvironmental protection, protection ofcitizen . . .

1. A short presentation about our lab

The ≪ Centre de Recherche enAutomatique deNancy ≫ (Automatic Control) isa Research Centre funded by the”Centre National de laRecherche Scientifique (CNRS)”and two universities in Nancy :UHP (Universite Henri Poincare)and INPL (Institut NationalPolytechnique de Lorraine).

The CRAN was set up in Nancy (France) in 1980.It totals 200 persons. The research activitiesconcentrate on 5 principal themes :- Systems Observation and Control- System Identification and Signal Processing- Dependability and System Diagnosis- Health Engineering- Ambient Manufacturing Systems.

Dependability and

System Diagnosis

Automatic Control:

System Observation

and Control

System

Identification and

Signal Processing

Health

Engineering

Ambiant

Manufacturing

Systems

1 Introduction

2 Theoretical backgroundStructure of a multi-modelSeparating fast and slow modesA multimodel based observerController design

3 Application to a biological reactorModel of a biological reactorSlow and fast modes of functionningObserver

4 Conclusion

1. Structure of a multimodel

1.1 - How to obtain a multi-model ?

Non linear system

Structure

x(t) = f (x(t),u(t))

y(t) = g(x(t),u(t))

Handling difficulties

Multi-model

Structure

∑i=1

µi(x ,u)(Aix+Biu )

∑i=1

µi (x ,u)(Cix+Diu )

∑i=1

µi (x ,u) = 1 and µi (x ,u)≥ 0,∀(x ,u) ∈ Rn×R

Handling facilities

Non linear system

Structure

x(t) = f (x(t),u(t))

y(t) = g(x(t),u(t))

Handling difficulties

Multi-model

Structure

∑i=1

µi(x ,u)(Aix+Biu )

∑i=1

Handling facilities

• Non linear system{

x(t) = f (x(t),u(t))y(t) = g(x(t),u(t))

• Quasi-LPV system{

x = A(x ,u)x +B(x ,u)uy = C (x ,u)x +D(x ,u)u

• Sector non linearity approach

A(x ,u) =r

∑i=1

µi(x ,u)Ai B(x ,u) =r

∑i=1

µi(x ,u)Bi

C (x ,u) = . . .

• Multi-model (with weighting functions and decision variables)

∑i=1

µi(x ,u) [Aix+Biu ]

∑i=1

µi (x ,u) [Cix+Diu ]

∑i=1

x(t) = f (x(t),u(t))y(t) = g(x(t),u(t))

A(x ,u) =r

∑i=1

µi(x ,u)Bi

C (x ,u) = . . .

∑i=1

x(t) = f (x(t),u(t))y(t) = g(x(t),u(t))

A(x ,u) =r

∑i=1

µi(x ,u)Bi

C (x ,u) = . . .

∑i=1

x(t) = f (x(t),u(t))y(t) = g(x(t),u(t))

A(x ,u) =r

∑i=1

µi(x ,u)Bi

C (x ,u) = . . .

∑i=1

1.2 - Exercise : get the structure of the MM system

◮ Non linear state equation

x1 = x1+x1.x2+(2−x1).ux2 = x21 +u

← x = f (x ,u)

◮ State vector

◮ The system is nonlinear, but . . .

◮ A quasi LPV form

1 x1x1 0

2−x11

x = A(x).x+B(x).u

A(x) =

1 x1x1 0

B(x) =

2−x11

x1 = x1+x1.x2+(2−x1).ux2 = x21 +u

← x = f (x ,u)

◮ State vector

1 x1x1 0

2−x11

x = A(x).x+B(x).u

A(x) =

1 x1x1 0

B(x) =

2−x11

x1 = x1+x1.x2+(2−x1).ux2 = x21 +u

← x = f (x ,u)

◮ State vector

1 x1x1 0

2−x11

x = A(x).x+B(x).u

A(x) =

1 x1x1 0

B(x) =

2−x11

x1 = x1+x1.x2+(2−x1).ux2 = x21 +u

← x = f (x ,u)

◮ State vector

1 x1x1 0

2−x11

x = A(x).x+B(x).u

A(x) =

1 x1x1 0

B(x) =

2−x11

1.2 - Exercise : get the structure of a MM system

◮ LPV form{

x1 = x1+x1.x2+(2−x1).ux2 = x21 +u

A(x) =

1 x1x1 0

B(x) =

2−x11

◮ A useful decomposition : w =w −wmin

wmax −wminwmax +

wmax −w

wmax −wminwmin

◮ If x1 ∈ [−a a], it is always possible to write :

x1 = µ1(x)(a)+µ2(x)(−a)

µ1(x) =x1+a

2a. µ2(x) =

a−x1

◮ Thus :

A(x) = µ1(x).A1+µ2(x).A2

1 −a

−a 0

◮ Same decomposition for the matrix B

◮ Finally, the MM form obtained without loss of information :

x = A(x).x+B(x).u → x =2

∑i=1

µi (x)(Aix+Biu)

◮ LPV form{

x1 = x1+x1.x2+(2−x1).ux2 = x21 +u

A(x) =

1 x1x1 0

B(x) =

2−x11

wmax −wminwmax +

wmax −w

wmax −wminwmin

x1 = µ1(x)(a)+µ2(x)(−a)

µ1(x) =x1+a

2a. µ2(x) =

a−x1

◮ Thus :

A(x) = µ1(x).A1+µ2(x).A2

1 −a

−a 0

x = A(x).x+B(x).u → x =2

∑i=1

µi (x)(Aix+Biu)

◮ LPV form{

x1 = x1+x1.x2+(2−x1).ux2 = x21 +u

A(x) =

1 x1x1 0

B(x) =

2−x11

wmax −wminwmax +

wmax −w

wmax −wminwmin

x1 = µ1(x)(a)+µ2(x)(−a)

µ1(x) =x1+a

2a. µ2(x) =

a−x1

◮ Thus :

A(x) = µ1(x).A1+µ2(x).A2

1 −a

−a 0

x = A(x).x+B(x).u → x =2

∑i=1

µi (x)(Aix+Biu)

◮ LPV form{

x1 = x1+x1.x2+(2−x1).ux2 = x21 +u

A(x) =

1 x1x1 0

B(x) =

2−x11

wmax −wminwmax +

wmax −w

wmax −wminwmin

x1 = µ1(x)(a)+µ2(x)(−a)

µ1(x) =x1+a

2a. µ2(x) =

a−x1

◮ Thus :

A(x) = µ1(x).A1+µ2(x).A2

1 −a

−a 0

x = A(x).x+B(x).u → x =2

∑i=1

µi (x)(Aix+Biu)

◮ LPV form{

x1 = x1+x1.x2+(2−x1).ux2 = x21 +u

A(x) =

1 x1x1 0

B(x) =

2−x11

wmax −wminwmax +

wmax −w

wmax −wminwmin

x1 = µ1(x)(a)+µ2(x)(−a)

µ1(x) =x1+a

2a. µ2(x) =

a−x1

◮ Thus :

A(x) = µ1(x).A1+µ2(x).A2

1 −a

−a 0

x = A(x).x+B(x).u → x =2

∑i=1

µi (x)(Aix+Biu)

1.3 - In what areas using the multi-model forms ?

∑i=1

µi (x ,u)(Aix +Biu )

∑i=1

µi(x ,u)(Cix +Diu )

∑i=1

µi(x ,u) = 1 and µi (x ,u)≥ 0,∀(x ,u) ∈Rn×R

◮ Stability analysis

◮ Control

◮ State Estimation

◮ Fault Detection

◮ Fault Tolerant Control

2. Separating fast and slow

2.1 - Singular systems

◮ Standard form of a singular system with two time scales :

x(t) = f (x(t),u(t))y(t) = g(x(t))

ε xf (t) = ff (xs(t),xf (t),u(t),ε)xs(t) = fs(xs(t),xf (t),u(t),ε)y(t) = g(xs(t),xf (t))

where xs ∈ Rns and xf ∈ R

nf are respectively the slow and the fast variables,ε is a positive parameter, known as parameter of singular perurbation.

◮ When ε → 0, the system reduces to :

0 = ff (xs(t),xf (t),u(t),0)

xs(t) = fs(xs(t),xf (t),u(t),0)

y(t) = g(xs(t),xf (t))

◮ Reduced system (descriptor form) :

xf (t) = ϕ(xs(t),u(t))xs(t) = fs (xs(t),ϕ (xs(t),u(t)) ,u(t))

y(t) = g (xs(t),ϕ (xs(t),u(t)))

x(t) = f (x(t),u(t))y(t) = g(x(t))

0 = ff (xs(t),xf (t),u(t),0)

y(t) = g (xs(t),ϕ (xs(t),u(t)))

x(t) = f (x(t),u(t))y(t) = g(x(t))

0 = ff (xs(t),xf (t),u(t),0)

y(t) = g (xs(t),ϕ (xs(t),u(t)))

2.2 - Multi-model with singular structure

• MM form with slow and fast state variable :

∑i=1

µi (x ,u)[

Aiff xf (t)+Ai

fsxs(t)+B if u(t)

xs(t) =r

∑i=1

µi (x ,u)[

Aisf xf (t)+Ai

ssxs(t)+B isu(t)

y(t) = Cf xf (t)+Cs xs(t)

• Augmented output vector :

ya(t) =

−Bf (u(t))u(t)y(t)

• MM form with slow state variables and disturbances (fast variable)

xs(t) =r

∑i=1

µi (x ,u)[

Aissxs(t)+B i

su(t)+Aisf xf (t)

ya(t) =r

∑i=1

µi (x ,u) [Cixs(t)+Gixf (t)]

avec :

∑i=1

µi (x ,u)[

Aiff xf (t)+Ai

fsxs(t)+B if u(t)

xs(t) =r

∑i=1

µi (x ,u)[

Aisf xf (t)+Ai

ssxs(t)+B isu(t)

ya(t) =

xs(t) =r

∑i=1

µi (x ,u)[

Aissxs(t)+B i

su(t)+Aisf xf (t)

ya(t) =r

∑i=1

avec :

∑i=1

µi (x ,u)[

Aiff xf (t)+Ai

fsxs(t)+B if u(t)

xs(t) =r

∑i=1

µi (x ,u)[

Aisf xf (t)+Ai

ssxs(t)+B isu(t)

ya(t) =

xs(t) =r

∑i=1

µi (x ,u)[

Aissxs(t)+B i

su(t)+Aisf xf (t)

ya(t) =r

∑i=1

avec :

2.3 - Orgranigram

Sector transformation

Mode separation

Singular perturbations

Perturbed

Multi-

Singular

Multi-

Quasi-

system

Non linear

system

Multi-

Factorisation

x = f(x, u)

x = A(x, u).x+B(x, u).u

xs = Σiµi(x, u)(Asf,i.xf +Ass,i.xs +Bi.u)

ya =∑

iµi(x, u)(Ci.xs +Gi.xf )

0 = Σiµi(x, u)(Aff,i.xf +Afs,i.xs +Bi.u)

y = g(x)

y = C(x, u).x

y = Σiµi(x, u)(Cf,ixf + Cs,ixs)

x = Σiµi(x, u)(Ai.x+Bi.u)

y = Σiµi(x, u)Ci.x

3. Observer and control

using a multimodel

3.1 - A last form of the two time scles MM

• Perturbed form MM (or MM with unknown input)

xs(t) =r

∑i=1

µi (x ,u)[

Aissxs(t)+B i

su(t)+Aisf xf (t)

ya(t) =r

∑i=1

• Augmented state vector

• Perturbed form MM with augmented stateUsing the previous notations and the assumption of slowly varying unknowninputs, the system is equivalent to the augmented system :

xa(t) =r

∑i=1

µi(xa(t),u(t))[

Ai xa(t)+ Bi u(t)]

ya(t) =r

∑i=1

µi(xa(t),u(t))Ci xa(t)

xs(t) =r

∑i=1

µi (x ,u)[

Aissxs(t)+B i

su(t)+Aisf xf (t)

ya(t) =r

∑i=1

xa(t) =r

∑i=1

µi(xa(t),u(t))[

Ai xa(t)+ Bi u(t)]

ya(t) =r

∑i=1

xs(t) =r

∑i=1

µi (x ,u)[

Aissxs(t)+B i

su(t)+Aisf xf (t)

ya(t) =r

∑i=1

xa(t) =r

∑i=1

µi(xa(t),u(t))[

Ai xa(t)+ Bi u(t)]

ya(t) =r

∑i=1

3.1 - Structure of the multi-observer

• Model of the system : ≪ augmented MM form ≫

xa(t) =r

∑i=1

µi(xa(t),u(t))[

Ai xa(t)+ Bi u(t)]

ya(t) =r

∑i=1

ya(t)xa(t)

xa(t)u(t) ya(t)

• Proportional observer

˙xa(t) =r

∑i=1

µi(xa,u)[

Ai xa(t)+ Bi u(t) + Ki(ya(t)− ya(t))]

ya(t) =r

∑i=1

µi(xa,u)Ci xa(t)

• Adjusting the parameters Ki of the observer17/33

xa(t) =r

∑i=1

µi(xa(t),u(t))[

Ai xa(t)+ Bi u(t)]

ya(t) =r

∑i=1

ya(t)xa(t)

xa(t)u(t) ya(t)

˙xa(t) =r

∑i=1

µi(xa,u)[

ya(t) =r

∑i=1

µi(xa,u)Ci xa(t)

xa(t) =r

∑i=1

µi(xa(t),u(t))[

Ai xa(t)+ Bi u(t)]

ya(t) =r

∑i=1

ya(t)xa(t)

xa(t)u(t) ya(t)

˙xa(t) =r

∑i=1

µi(xa,u)[

ya(t) =r

∑i=1

µi(xa,u)Ci xa(t)

3.2 - Controller design

• Structure of the multi-model

∑i=1

µi(x ,u)(Aix+Biu )

∑i=1

• Control lawu(t) =−K x(t)

• System under control

∑i=1

µi(x ,u)(Ai −BiK )x

• Adjusting the gain K of the controller

∑i=1

µi(x ,u)(Aix+Biu )

∑i=1

µi(x ,u)(Aix+Biu )

∑i=1

µi(x ,u)(Aix+Biu )

∑i=1

2. Application to a biological

reactor

4.1 - Flowsheet

Bioreactor

Clarifier

Effluent

Recycling sludge

Feed qin qout

qwqR Wasted sludge

Figure: Diagram of actived sludge wastewater treatment

SNOXA XHWaster water

Biomass

Ammoniumcompounds

Solublecarbone

Waster water

Biomass

NitritesNitrates

Solublecarbone

Nitrogen gas

Figure: Reactions in nitrogen removal20/33

4.2 - Characteristic variables

Reactor Reactor Recycled

input output product

heterotrophic biomass XBH XBH ,in XBH ,out XBH ,R

autotrophic biomass XBH XBA,in XBA,out XBA,R

substrate SS SS,in SS,out SS,Roxygen SO SO,in SO,out SO,R

ammoniacal nitrogen SNH SNH ,in SNH ,out SNH ,R

nitrates SNO SNO,in SNO,out SNO,R

soluble nitrogen SND SND,in SND,out SND,R

ammonium SNH SNH ,in SNH ,out SNH ,R

Feed q qin qout qRAir flowrate qaVolume V

4.3 - Model : mass balance equations

General mass balance equation for component x(t)

V∆x(t) = qin(t)∆t xin(t)−qout(t)∆t xout(t)+ r(x(t))∆t

SS (t) = −1

SS (t)

KS +SS(t)

KOH +SO(t)XBH(t)+ (1− fP)bHXBH(t)+

qin(t)

V[SS,in(t)−SS(t)]

SO(t) =YH − 1

SS (t)

KS +SS(t)

KOH +SO(t)XBH(t)+

qin(t)

V[SO,in(t)−SO(t)]+Kqa(t) [SO,sat −SO(t)]

XBH(t) = µH

SS (t)

KS +SS(t)

KOH +SO(t)XBH(t)−bHXBH(t)+

qin(t)

XBH ,in(t)−XBH(t)+ fR1− fW

fR + fWXBH(t)

4.3 - Model : mass balance equations

General mass balance equation for component x(t)

V∆x(t) = qin(t)∆t xin(t)−qout(t)∆t xout(t)+ r(x(t))∆t

SS (t) = −1

SS (t)

KS +SS(t)

KOH +SO(t)XBH(t)+ (1− fP)bHXBH(t)+

qin(t)

V[SS,in(t)−SS(t)]

SO(t) =YH − 1

SS (t)

KS +SS(t)

KOH +SO(t)XBH(t)+

qin(t)

V[SO,in(t)−SO(t)]+Kqa(t) [SO,sat −SO(t)]

XBH(t) = µH

SS (t)

KS +SS(t)

KOH +SO(t)XBH(t)−bHXBH(t)+

qin(t)

XBH ,in(t)−XBH(t)+ fR1− fW

fR + fWXBH(t)

4.4 - Hypotheses

Bioreactor

Clarifier

Effluent

Recycling sludge

Feed qin qout

qwqR Wasted sludge

No dissolved oxygen in reactor input SO,in(t) = 0

Reactor with constant volume qout(t) = qin(t)+qR (t)

Recycled and rejected flows qR (t) = fR qin(t), 1≤ fR ≤ 2qW (t) = fW qin(t), 0< fW < 1

Perfect clarifier qout(t)XBH (t) = [qR(t)+qW (t)]XBH ,R (t)SS,R (t) = SS(t)

4.5 - Slow and fast variables

• Linearization of the nonlinear model around several operating points (x0,u0) :

x(t) = A0x(t)+B0u(t)

A0 =∂ f (x ,u)

∂x∣

(x0,u0)

B0 =∂ f (x ,u)

∂u∣

(x0,u0)

• Let us consider λ1 ≤ λ2 ≤ ...≤ λn the eigenvalues of A0, the largest eigenvalue(respectively smallest) corresponds to the slowest dynamics (respectively fastet).

• This separation is achieved by setting a threshold τ separating the two timescales :

Re(λ1)≤ ...≤ Re(λnf )<< τ ≤ Re(λnf +1)≤ ...≤ Re(λn)

!!! Linearization is only used to select fast and slow modes,

A0 =∂ f (x ,u)

∂x∣

(x0,u0)

B0 =∂ f (x ,u)

∂u∣

(x0,u0)

A0 =∂ f (x ,u)

∂x∣

(x0,u0)

B0 =∂ f (x ,u)

∂u∣

(x0,u0)

A0 =∂ f (x ,u)

∂x∣

(x0,u0)

B0 =∂ f (x ,u)

∂u∣

(x0,u0)

• The separation of slow and fast dynamics is confirmed by the eigenvalues λi ofthe Jacobian A0, as can be seen in the figure below showing these eigenvalues forforty operating points..

• Two eigenvalues (λ2 and λ3) are between −50 and −0.4 and the other (λ1)arround −250. By setting a threshold at τ =−70, we can deduce that the systemhas a fast dynamic (xf = SS) and two slow dynamics (xs = [SO XBH ]

0 5 10 15 20 25 30 35 40−250

−200

−150

−100

−0.5

Points de fonctionnement

λ1 (S

λ2 (S

λ3 (X

Figure: Eigenvalues λi of the Jacobian A0 calculated around 40 operating points

• The separation of slow and fast dynamics is confirmed by the eigenvalues λi ofthe Jacobian A0, as can be seen in the figure below showing these eigenvalues forforty operating points..

• Two eigenvalues (λ2 and λ3) are between −50 and −0.4 and the other (λ1)arround −250. By setting a threshold at τ =−70, we can deduce that the systemhas a fast dynamic (xf = SS) and two slow dynamics (xs = [SO XBH ]

0 5 10 15 20 25 30 35 40−250

−200

−150

−100

−0.5

Points de fonctionnement

λ1 (S

λ2 (S

λ3 (X

Figure: Eigenvalues λi of the Jacobian A0 calculated around 40 operating points

4.6 - How to get the multi-model structure ?

SS (t) =−1

SS (t)

KS +SS (t)

KOH +SO(t)XBH (t)+(1− fP )bHXBH (t)+

qin(t)

SS,in(t)−SS (t)]

SO(t) =YH −1

SS (t)

KS +SS (t)

SO (t)

KOH +SO(t)XBH (t)+

qin(t)

SO,in(t)−SO (t)]

+Kqa(t)[

SO,sat −

XBH (t) = µH

SS (t)

KS +SS (t)

KOH +SO(t)XBH (t)−bHXBH (t)+

qin(t)

XBH,in(t)−XBH (t)+ fR1− fW

fR + f

• Decision variables

z1(u(t)) =qin(t)

z2(x(t)) =1

KS +SS(t)

KOH +SO(t)XBH (t)

z3(u(t)) = qa(t)

• Transformation of the decision variables

zi (t) =zi (t)−zi ,min

zi ,max −zi ,minzi ,max +

zi ,max −zi (t)

zi ,max −zi ,minzi ,min

4.6 - How to get the multi-model structure ?

SS (t) =−1

SS (t)

KS +SS (t)

KOH +SO(t)XBH (t)+(1− fP )bHXBH (t)+

qin(t)

SS,in(t)−SS (t)]

SO(t) =YH −1

SS (t)

KS +SS (t)

SO (t)

KOH +SO(t)XBH (t)+

qin(t)

SO,in(t)−SO (t)]

+Kqa(t)[

SO,sat −

XBH (t) = µH

SS (t)

KS +SS (t)

KOH +SO(t)XBH (t)−bHXBH (t)+

qin(t)

XBH,in(t)−XBH (t)+ fR1− fW

fR + f

• Decision variables

z1(u(t)) =qin(t)

z2(x(t)) =1

KS +SS(t)

KOH +SO(t)XBH (t)

z3(u(t)) = qa(t)

• Transformation of the decision variables

zi (t) =zi (t)−zi ,min

zi ,max −zi ,minzi ,max +

zi ,max −zi (t)

zi ,max −zi ,minzi ,min

4.7 - Decomposition into local functionning modes

Quasi-LPV form of the model : the matrices A(t) = A(x(t),u(t)) andB(t) = B(u(t)) are decomposed under the form :

A(t) =

Aff (t) Afs(t)Asf (t) Ass(t)

B(t) =

Bf (t)Bs(t)

Aff (t) =−z1(t)−1

µH z2(t) Afs(t) =[

0 (1− fP)bH]

Asf (t) =

YH−1YH

µH z2(t)

Ass(t) =

−K z3(t)− z1(t) 00 −αz1(t)−bH

Bf (t) =[

z1(t) 0 0]

Bs(t) =

0 K Sosat 00 0 z1(t)

• The constant matrices Ai and Bi representing the 8 sub-models are definedusing the matrices A and B and the scalar zi :

Aiff = Aff (z1,σ1

i,z2,σ2

Aifs = [0 (1− fP)bH ]

Aisf = Asf (z2,σ2

Aiss = Ass(z1,σ1

i,z3,σ3

B if = Bf (z1,σ1

B is = Bs(z1,σ1

i) i = 1, ...,8

• The model is written equivalently in the form MM using separation of variablesinto state slow and fast variables.

• Consider the output vector y = Cf xf +Cs xs with the matrices :

0 01 0

Aiff = Aff (z1,σ1

i,z2,σ2

Aifs = [0 (1− fP)bH ]

Aisf = Asf (z2,σ2

Aiss = Ass(z1,σ1

i,z3,σ3

B if = Bf (z1,σ1

B is = Bs(z1,σ1

i) i = 1, ...,8

0 01 0

Aiff = Aff (z1,σ1

i,z2,σ2

Aifs = [0 (1− fP)bH ]

Aisf = Asf (z2,σ2

Aiss = Ass(z1,σ1

i,z3,σ3

B if = Bf (z1,σ1

B is = Bs(z1,σ1

i) i = 1, ...,8

0 01 0

4.8 - Measured input and observer implementation

u(t) y(t)

observerMulti

Reduced

Figure: Implementation ofthe observer

0 2 4 6 80

0 2 4 6 860

100qin

0 2 4 6 85

XBH,in

0 2 4 6 8150

250SS, in

Figure: Inputs of the plant

4.9 - State observer

• We apply the observer onthe ASM1 model,represented under MMequivalent form, with dataobtained from the completeplant.

• Results obtained for stateestimation are quiterelevant. The error betweenthe reconstructed state andthe true state is acceptable.

0 1 2 3

time (day)

0 1 2 3

time (day)

0 1 2 30

time (day)

0 1 2 3

time (day)

0 1 2 3500

time (day)

0 1 2 30

time (day)

XDCO,estimated

SNO,estimated

XBH,estimated

SNH,estimated

SO,estimated

XBA,estimated

Conclusion

Contributions

◮ State observer of a non linear system

◮ Complexity reduction without loss of information

◮ Slow and fast modes separation in the MM framework

◮ Unknown input observer

Perspectives

◮ Optimising the transformation SNL → MM

◮ Reducing the effect of measurement noise

◮ Taking into account the uncertain parameters

Thank you for your attention

Place Stanislas in Nancy

Diagnosis and control using multiple models. Application to a biological reactor

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