Parameters and states estimation for Dengue epidemic model

Ramdane Tami * ,1 , Driss Boutat 1 , Gang Zheng2 and Frederic Kratz 1 Abstract-This paper focuses on the states estimation and parameter identification for nonlinear Dengue epidemic model. This model supplies a change of coordinates that transforms the studied model into an extended output depending nonlinear observer normal form. Then we can use the high gain observer strategy, which allows both the parameter identification and the state estimation.

I. INTRODUCTION

The Dengue fever is endemic throughout the tropics and no specific treatment has been developed yet. The disease is transmitted between humans throughout mosquito bites. This spreading process through mosquitoes is given by a nonlinear dynamical system by involving two species (mosquitoes and humans). Each species is partitioned into two populations; the susceptible and the infected one. In this process, the mosquitoes are considered as vectors of disease. In order to minimize its impact on society, numerous studies have been conducted to understand the behavior of dynamics of such diseases which resulted in the development of several models aimed to analyze and predict the evolution of such epidemic in the purpose of minimizing its impact on society.

The modeling of epidemic spread was initiated by [START_REF] Kermack | A contribution to the mathematical theory of epidemics[END_REF], that had led to the well-known SIR model (Susceptible, Infected and Removed). Afterwards, it has been improved and adapted according to the type of disease. Based on these models, several works are realized to study the volatility of diseases [START_REF] Fan | Global stability of an seis epidemic model with recruitment and a varying total population size[END_REF], and to develop vaccination strategy [START_REF] De La Sen | Vaccination strategies based on feedback control techniques for a general seir-epidemic model[END_REF]. However, as far as we know, there are few works dedicated to the observer design [START_REF] Bernard | Nonlinear observers for a class of biological systems: application to validation of a phytoplanktonic growth model[END_REF].

To achieve an observer design for the dengue epidemic model, we will use the approach based on nonlinear observer normal forms (NONF). This concept has been set up in [START_REF] Bestle | Canonical form observer design for nonlinear time varying systems[END_REF] for a varying time and in [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF] for an invariant-time single output nonlinear dynamical system. Then, it has been adopted for the multiple outputs case in [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF], [START_REF] Xia | Nonlinear observer with linearizable error dynamics[END_REF]. Afterwards, several algorithms have been developed in [START_REF] Keller | Nonlinear observer design by transformation into a generalized observer canonical form[END_REF], [START_REF] Phelps | On constructing nonlinear observers[END_REF], [START_REF] Rudolph | A block triangular nonlinear observer normal form[END_REF], [START_REF] Hou | Observer with linear error dynamics for nonlinear multi-output systems[END_REF], [START_REF] Marino | Nonlinear control design: geometric, adaptive and robust[END_REF], [START_REF] Boutat | [END_REF], [START_REF] Lynch | Nonlinear observers with approximately linear error dynamics: the multivariable case[END_REF] and [START_REF] Lopez | Linearization by completely generalized input ouput injection[END_REF]. However, the conditions for the existence of diffeomorphism reported in the above works are relatively too restrictive. Therefore, to enhance this concept and to expand the class of nonlinear systems that allow a change of coordinates, another interesting form is the so-called depending output observer normal form introduced by [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF], and then improved in [START_REF] Zheng | A single output dependent observability normal form[END_REF], [START_REF] Zheng | Synchronisation of chaotic systems via reduced observers[END_REF], [START_REF] Boutat | Observer error linearization multi-output depending[END_REF], [START_REF] Zheng | Multi-output dependent observability normal form[END_REF], [START_REF]Output dependent observability linear normal form[END_REF] and [START_REF] Wang | Observer design using a generalized timescaled block triangular observer form[END_REF]. The most recent normal form, called extended nonlinear observer normal form due to [START_REF] Jouan | Immersion of nonlinear systems into linear systems modulo output injection[END_REF], allows to enlarge the class of observer normal forms. It has been generalized by [START_REF] Noh | Nonlinear observer design by dynamic observer error linearization[END_REF], [START_REF] Back | Dynamic observer error linearization[END_REF], [START_REF] Boutat | Geometrical conditions for observer error linearization via ∫ 0[END_REF], [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF] and [START_REF] Yang | Reduced-order dynamic observer error linearization[END_REF]. The idea of the last form is to add an auxiliary dynamics into the original system such that the augmented system fulfills commutativity condition of Krener and Isidori frame [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], which guarantees a transformation of the studied system into an extended normal form [START_REF] Tami | Extended output depending normal form[END_REF], [START_REF] Boutat | Extended nonlinear observer normal forms for a class of nonlinear dynamical systems[END_REF]. On the other hand, this observer normal form provides an adaptive observer to resolve the simultaneous parameters identification and state estimation problem [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF]. Thus, we believe that its application to epidemics model can constitute a very powerful decision-making tool in the vaccination campaign and fighting against diseases. The objective of this paper is intended to show the effectiveness of the observer design to estimate the spreading disease and to identify the parameters in the epidemic model. This paper is organized as follows. Section II illustrates the formalization of extended output depending observer normal form. Section III presents the dengue epidemic model. Section IV supplies a change of coordinates allowing the transformation of the dengue epidemic model into an extended output depending normal form. Section V deals with simultaneous state estimation and parameter identification of the studied system.

II. BACKGROUND ON NONLINEAR EXTENDED OBSERVER NORMAL FORMS

We Consider a nonlinear system in the following form:

{ ẋ = f (x) y = h(x) (1)
where x ∈ U ⊆ R n represents the state and y ∈ R p denotes the output. We assume that the vector field f and the output function h are sufficiently smooth. In the following, we also assume that the pair (h, f ) satisfies the observability rank condition, thus the 1-forms θ i = dL i-1 f h for i = 1 : n are linearly independent where L i-1 f h is the (i -1) th Lie derivative of the output h in the direction of the vector field f , and d is the usual differential. Under this assumption, theoretically the state can be estimated as x = σ(y, ẏ, ..., y n-1 ). However, it has been established that the use of successive output derivatives amplifies the noise in the measurement. Therefore, the observer design is a powerful tool to address the state estimation problem. To do so, we firstly present a class of nonlinear systems for which the design of observer is relatively straightforward. Consider the nonlinear system (1), we seek an auxiliary dynamics ẇ = η(y, w) so that the following extended dynamical system:

ẋ = f (x) (2) ẇ = η(y, w) (3) y = h(x) (4)
can be transformed through a change of coordinates (ξ T , ζ) T = ϕ(x, w) into the following extended output depending observer normal form:

ξ = A(y, w)ξ + B(y, w) (5) ζ = B n+1 (y, w) (6) y = Cξ (7)
where

ξ ∈ R n , y ∈ R p , ζ ∈ R, w ∈ R is an extra-output, and
A(y, w) =       0 ... ... ... 0 α 2 (y, w) 0 ... ... 0 0 α 3 (y, w) ... ... ... 0 ... ... ... ... 0 ... ... α n (y, w) 0       (8)
where α i (y, w) ̸ = 0 for i = 2 : n is a function depending only on the output y and the extra-output w. The normal form (5-7) has a great interest because it supports Kalmanlike observer or the following high gain observer presented in [START_REF] Busawon | Asimple observer for a class of nonlinear systems[END_REF]:

ξ = A(y, w) ξ + B(w, y) -Γ -1 (y)R -1 ρ C T (C ξ -y)(9) 0 = ρR ρ + G T R ρ + R ρ G -C T C ( 10
)
where

G =      0 • • • 0 0 1 • • • 0 0 . . . . . . . . . . . . 0 • • • 1 0      Γ(y, w) = diag[ n Π i=2 α i (y, w), n Π i=3 α i (y, w), • • • , α n (y, w), 1] R ρ (n + 1 -i, n + 1 -j) = (-1) i+j C j-1 i+j-2 ρ i+j-1 for 1 ≤ i ≤ n and 1 ≤ j ≤ n.
If we set e = ξ -ξ to be the observation error, then we see that its error dynamics is governed by the following equation:

ė = ξ -ξ = (A(y, w) -Γ -1 (y, w)R -1 ρ C T C)e.
If y and w are bounded, then the observation error dynamics is exponentially stable by well choosing ρ.

III. DENGUE DISEASE MODEL

Dengue is a viral infection transmitted by the bite of mosquito Aedes aegypti to human. It is an endemic in many countries through the world, it has not a specifical treatment and the more adapted solution is the prevention.

Throughout this paper, we consider the dengue epidemic model extracted from [START_REF] Esteva | Analysis of a dengue disease transmission model[END_REF], [START_REF] Hsieh | Turning points, reproduction number, and impact of climatological events for multi-wave dengue outbreaks[END_REF], whose behavior is governed by a set of differential equations describing epidemic transmission between humans and mosquitoes:

         dS(t) dt = b 1 -λ 1 SV -µ 1 S dI(t) dt = λ 1 SV -γI -µ 1 I dM (t) dt = b 2 -λ 2 M I -µ 2 M dV (t) dt = λ 2 M I -µ 2 V ( 11
)
where S(t) and I(t) are respectively the susceptible and infected individuals in human population, M (t) and V (t) are respectively the susceptible and infected mosquitoes. The involving parameters are the following senses: b 1 is the rate of natural birth on the host population, µ 1 is the natural death rate related to host population, λ 1 is the rate daily biting (or the rate of transmission from V ), γ is the rate at which the infected individuals are recovered, b 2 is the birth rate of vector population (i-e mosquitoes), λ 2 is the transmission rate from I to vectors (or rate daily biting of I), and µ 2 is the natural death rate of mosquitoes (i-e vectors).

Throughout this paper, it is assumed that all information concerning humans data S(t) and I(t) will be available thanks to the health department. Therefore, they are taken as the outputs of the model:

{ h 1 = y 1 = S h 2 = y 1 = I (12)
Now, we will check the observability rank condition of the above dynamical system [START_REF] Rudolph | A block triangular nonlinear observer normal form[END_REF] with the output defined in [START_REF] Hou | Observer with linear error dynamics for nonlinear multi-output systems[END_REF]. For this, let us compute the Lie derivative of the outputs as follows:

θ 1 = dL 0 f h 1 = dS θ 2 = dL 1 f h 1 = (λ 1 V -µ 1 ) dS + λ 1 SdV θ 3 = dL 2 f h 1 = - ( λ 1 b 1 -λ 1 µ 2 S -2µ 1 λ 1 S -2λ 2 1 SV
) dV

+ ( µ 2 1 + λ 1 (2µ 1 + µ 2 ) V + λ 2 1 V 2 -λ 1 λ 2 M I ) dS -λ 1 λ 2 SIdM -λ 1 λ 2 SM dI θ 4 = dh 2 = dI
We see that θ i for 1 ≤ i ≤ 4 are linearly independent which implies the observability of system [START_REF] Rudolph | A block triangular nonlinear observer normal form[END_REF]. Thus, we can estimate the state V (t) and M (t) from the outputs and their derivatives i.e. (V, M ) = σ(y, ẏ, y (2) ).

In order to simplify the calculation, as I(t) and V (t) are measured, then we can remove the dynamics of I in [START_REF] Rudolph | A block triangular nonlinear observer normal form[END_REF]. Specifically, we will deal with the following reduced dynamical system:

     dS(t) dt = b 1 -λ 1 SV -µ 1 S dM (t) dt = b 2 -λ 2 M I -µ 2 M dV (t) dt = λ 2 M I -µ 2 V ( 13
)
IV. EXTENDED OUTPUT DEPENDING OBSERVER NORMAL

FORM

This section provides a change of coordinates that transforms the model ( 13) into [START_REF] Bestle | Canonical form observer design for nonlinear time varying systems[END_REF][START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF][START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF]. For this we will process in two steps.

A. First change of coordinates

This subsection deals with a first transformation that enables to transform the dynamical system (13) into another dynamical system with depending output linear part modulo nonlinear terms which are only function of outputs I and S which will allow, thereafter, an easy use of the extended transformation algorithm.

Proposition 1: The following change of coordinates

z 3 = - 1 λ 1 ln(S) (14
)

z 2 = 1 λ 2 V - 1 λ 1 ( µ 2 λ 2 + S + I ) ln(S) + S λ 1 ( 15
)
z 1 = M + V (16)
transforms ( 13) into the following dynamical system

       ż1 = b 2 -µ 2 z 1 ż2 = Iz 1 + β 2 (y) ż3 = λ 2 z 2 + β 3 (y) y = z 3 (17)
where 16) implies ż3 = -Ṡ λ1S . Now, from the model [START_REF] Rudolph | A block triangular nonlinear observer normal form[END_REF], we obtain:

β 3 (y) = -λ 2 ( - ln(S) λ 1 ( µ 2 λ 2 + S + I ) + S λ 1 ) - b 1 λ 1 S + µ 1 λ 1 β 2 (y) = ( - b 1 λ 1 S + µ 1 λ 1 ) ( S + I + µ 2 λ 2 ) - ln(S) λ 1 (b 1 -µ 1 S -(γ + µ 1 ) I) + (b 1 -µ 1 S) λ 1 Proof: A direct derivation of change of coordinates (
ż3 = λ 2 z 2 -λ 2 ( - ln(S) λ 1 ( µ 2 λ 2 + S + I ) + S λ 1 ) - b 1 λ 1 S + µ 1 λ 1
which can be rewritten into the following compact form:

ż3 = λ 2 z 2 + β 3 (y) .
In the same way, the derivative of the second component of change of coordinates [START_REF] Lynch | Nonlinear observers with approximately linear error dynamics: the multivariable case[END_REF] gives:

ż2 = 1 λ 2 V + µ 2 ż3 + ż3 (S + I) + z 3 ( Ṡ + İ) + 1 λ 1 Ṡ = Iz 1 + ( - b 1 λ 1 S + µ 1 λ 1 ) ( S + I + µ 2 λ 2 ) - ln(S) λ 1 (b 1 -µ 1 S -(γ + µ 1 ) I) + (b 1 -µ 1 S) λ 1
Therefore, we obtain the second differential equation in Proposition 1 as follows:

ż2 = Iz 1 + β 2 (y) .
For the third component of the change of coordinates ( 16), a straightforward calculation leads to

ż1 = Ṁ + V = b 2 -µ 2 z 1
However, due to the first dynamics in system [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF], one can see that it is different from the proposed observer normal form ( 5)- [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF].And this motivates us to introduce the concept of extended dynamics (or immersion) to overcome this limitation.

B. Extended Dynamics

In this subsection, we will use the concept of extended dynamics. For this, let us extend the dynamical system [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF] as follows

       ż1 = b 2 -µ 2 z 1 ż2 = Iz 1 + β 2 (y) ż3 = λ 2 z 2 + β 3 (y) ẇ = η(y, w) ( 18
)
where w ∈ R is an auxiliary variable considered as an extra output and η(y, w) is a function to be determined ( for more details see [START_REF] Guay | Observer linearization by output-dependent time-scale transformations[END_REF], [START_REF] Jouan | Immersion of nonlinear systems into linear systems modulo output injection[END_REF], [START_REF] Back | Dynamic observer error linearization[END_REF], [START_REF] Boutat | Geometrical conditions for observer error linearization via ∫ 0[END_REF] and [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF]). Proposition 2: The following change of coordinates

ξ 1 = l(w)z 1 (19) ξ 2 = z 2 , ξ 3 = z 3 , ξ 4 = w ( 20
)
transforms the dynamical system (18) into the following extended nonlinear output depending observer normal form

             ξ1 = B 1 (y, w) ξ2 = I l(w) ξ 1 + B 2 (y, w) ξ3 = λ 2 ξ 2 + B 3 (y, w) ξ4 = η(y, w) y = S (21)
where η(w) is only function of w and l(w) = e ∫ w 0 µ 2 η(s) . Proof: Firstly, for a given η(y, w) ̸ = 0, let us firstly seek the function l(w). A direct derivative of the change of coordinates [START_REF] Zheng | Synchronisation of chaotic systems via reduced observers[END_REF]

implies ξ1 = z 1 l ′ (w) ẇ + l(w) ż1 .
Now, from the first and the fourth differential equations of (18), we obtain

ξ1 = z 1 l ′ (w)η(y, w) -l(w)µ 2 z 1 + l(w)b 2 .
As intended by observer normal form [START_REF] Zheng | Multi-output dependent observability normal form[END_REF], the dynamics ξ1 should be a function of the output (S, I, w), therefore, we require that z 1 l ′ (w)η(y, w) -l(w)µ 2 z 1 = 0 which leads to the following differential equation

l ′ (w) l(w) = µ 2 η(y, w) . ( 22
)
As l(w) is chosen as only a function of w, then from the last equation ( 22), the function η(w, y) = η(w) must be also only a function of w.

Then, by integrating of equation ( 22) we obtain

l(w) = e ∫ w 0 µ 2 η(s) ds .
Therefore, the dynamics of ξ 1 becomes

ξ1 = B 1 (y, w) = l(w)b 2 .
Now, taking into account the dynamical system [START_REF] Zheng | A single output dependent observability normal form[END_REF], the derivative of the second variable [START_REF] Boutat | Observer error linearization multi-output depending[END_REF] implies:

ξ2 = I l(w) ξ 1 + ( - b 1 λ 1 S + µ 1 λ 1 ) ( S + I + µ 2 λ 2 ) - 1 λ 1 (b 1 -µ 1 S -(γ + µ 1 ) I) (ln(S)) + 1 λ 1 (b 1 -µ 1 S)
which can be written as follows:

ξ2 = α 2 (y, w) ξ 1 + B 2 (y, w).
In the same way, the derivative of ξ 3 leads to

ξ3 = λ 2 ξ 2 -λ 2 ( - 1 λ 1 ln(S) ( µ 2 λ 2 + S + I ) + 1 λ 1 S ) - b 1 λ 1 S + µ 1 λ 1
and this can be described into the following compact form ξ3 = α 3 (y, w) ξ 2 + B 2 (y, w)

and finally the dynamics of ξ 4 is given by ξ4 = η(y, w) = B 4 (y, w) .

C. Simulation results

For simulation, we use the following parameters: b1 = 0.01, λ 1 = 5E -4, µ 1 = 0.01, γ = 0.01, λ 2 = 3E -4 : 10, b 2 = 60; µ 2 = 0.5. The initial conditions are S (0) = 784, I (0) = 216, R (0) = 0; M (0) = 0, V (0) = 100. To ensure the boundedness of w, we choose V. ADAPTIVE OBSERVER Unlike the above section, this section deals with simultaneous parameter identification and state estimation for system [START_REF] Marino | Nonlinear control design: geometric, adaptive and robust[END_REF]. It assumes that the parameters b 2 , µ 2 , γ 2 of the second and the third equation in system (13) are unknown. Therefore, inspired by the works of [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF], [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF] and [START_REF] Besanc ¸on | On adaptive observers for state affine systems and application to synchronous machines[END_REF], we rewrite system (21) into the following affine unknown parameters observer normal form:

{ ξ = A(w, y)ξ + φ(y) + ϕ(w, y)θ y = Cξ ( 23
)
where

ξ = [ ξ 1 ξ 2 ξ 3 ] T , θ = [ b 2 µ2 λ2 µ 2 λ 2 ] T A(w, y) =   0 0 0 α 2 (w, y) 0 0 0 α 3 0   =   0 0 0 I l(w) 0 0 0 λ 2 0   (24) ϕ(w, y) =    l(w) 0 0 0 0 µ1S-b1 λ1S 0 0 0 0 ln(S) λ1 (S+I) ln(S)+S λ1    and φ(y) =    0 ( µ1S-b1 λ1S ) I -(b 1 -µ 1 S -(γ + µ 1 ) I) ln(S) λ1 µ1 λ1 -b1 λ1S   
Similarly to [START_REF] Besanc ¸on | On adaptive observers for state affine systems and application to synchronous machines[END_REF] and [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], it is assumed as well that there exist positive constants α, β, T such that

αI ≤ ∫ t+T t Λ T (τ ) C T Σ(τ )C (τ ) Λ (τ ) dr ≤ βI and αI ≤ ∫ t+T t Ψ (t, τ ) T C T Σ(τ )CΨ (t, τ ) dr ≤ βI ∀t ≥ t 0
where Ψ is the transition matrix for the system { .

x = A(y, w)x y = Cx and Σ is a positive definite bounded matrix. Then, based on the results in [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF], [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF] and [START_REF] Besanc ¸on | On adaptive observers for state affine systems and application to synchronous machines[END_REF], the following system:

. x = A(y, w) x + φ(y) + ϕ(y, w) θ (25) + { ΛS -1 θ Λ T C T + S -1 x C T } Σ (y -C x) . θ = S -1 θ Λ T C T Σ(y -C x) ( 26
) . Λ = { A (y, w) -S -1 x C T C } Λ + ϕ (y, w) (27)
.

S x = -ρ x S x -A(y, w) T S x -S x A(y, w) + C T ΣC (28)
.

S θ = -ρ θ S θ + Λ T C T ΣCΛ, S x (0), S θ (0) > 0 (29)
where ρ x and ρ θ are positive constants, is an exponential adaptive observer for the nonlinear system [START_REF] Wang | Observer design using a generalized timescaled block triangular observer form[END_REF]. Remark 3: According to [START_REF] Busawon | Asimple observer for a class of nonlinear systems[END_REF], as we have the form [START_REF] Jouan | Immersion of nonlinear systems into linear systems modulo output injection[END_REF], then [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF] can be simplified to algebraic equation [START_REF] Phelps | On constructing nonlinear observers[END_REF]. In this case, it implies the following equation:

R ρ =    6 ρ 5 x -3 ρ 4 x 1 ρ 3 x -3 ρ 4 x 2 ρ 3 x -1 ρ 2 x 1 ρ 3 x -1 ρ 2 x 1 ρx   

A. Simulation results

The figures (4)- [START_REF] Xia | Nonlinear observer with linearizable error dynamics[END_REF] present the simulation results obtained with adaptive observer where ρ x = 0.77, ρ θ = 0.05, Σ = 1 and the initial conditions are the same as before. The fast convergences show the efficiency of the used approach to estimate the states (M , V ) and to identify the parameters (b 2 , λ 2 , µ 2 ).

VI. CONCLUSIONS AND FUTURE WORKS

The observer design for simultaneous states estimation and parameter identification of dengue epidemic model is studied. The efficiency of the proposed approach mixing the change of coordinates and extended dynamics to construct extended observer normal form is illustrated by the application to practical example.

.wFig. 1 .Fig. 2 .Fig. 3 .

 123 Fig. 1. Evolution of population (M )

Fig. 4 .

 4 Fig. 4. Identification of parameter b 2

Fig. 5 .Fig. 6 .

 56 Fig. 5. Identification of parameter λ 2

G. Zheng (gang.zheng@inria.fr) is with INRIA Lille-Nord Europe, 40 Avenue Halley 59650, France.


Ramdane Tami

Driss Boutat

Gang Zheng

Nonlinear Observer for the PM Synchronous Motor

This paper deals with nonlinear states and parameters estimation of a permanent magnet synchronous motor (PMSM) without mechanical sensors. In the proposed method, a change of coordinates is established in order to transform the PMSM system into an observer form which enables us to apply an adaptive or a high gain observer. Then the load torque, rotor speed and some parameters can be estimated from both measurable currents and voltages. The efficiency of the proposed methods is illustrated by the simulation results.

I. INTRODUCTION

Synchronous motor is widely used in industry, particulary in energy and transport. It must not only be reliable, but also have an optimum performance which are strongly dependent on its control and the access to measurements and parameters.

Enhancing the performance of control and fault monitoring requires accurate estimation of state variables and identification of parameters. Numerous observer strategies are developed to estimate the states and the parameters of PMSM, such as adaptive observer [START_REF] Besancon | On adaptive observers for state affine systems[END_REF], [START_REF] Ghanes | A robust sensorless output feedback controller of the induction motor drives: new design and experimental validation[END_REF], sliding mode observer [START_REF] Cascella | Adaptive sliding-mode observer for field oriented sensorless control of spmsm[END_REF], [START_REF] Paponpen | An improved sliding mode observer for speed sensorless vector control drive of pmsm[END_REF], [START_REF] Ghanes | On sensorless induction motor drives: sliding-mode observer and output feedback controller[END_REF], [START_REF] Zaltni | Observability analysis and improved zerospeed position observer design of synchronous motor with experimental results[END_REF] and Extended Kalman filter [START_REF] Bolognani | Sensorless full-digital pmsm drive with ekf estimation of speed and rotor position[END_REF], [START_REF] Bolognani | Extended kalman filter tuning in sensorless pmsm drives[END_REF], [START_REF] Boussak | Implementation and experimental investigation of sensorless speed control with initial rotor position estimation for interior permanent magnet synchronous motor drive[END_REF], [START_REF] Janiszewski | Extended kalman filter based speed sensorless pmsm control with load reconstruction[END_REF]. Recently in [START_REF] Ezzat | Sensorless speed control of pmsm via adaptive interconnected observer[END_REF], [START_REF] Hamida | An adaptive interconnected observer for sensorless control of pm synchronous motors with online parameter identification[END_REF] an adaptive interconnected observer has been proposed. This paper presents a new method to design nonlinear observer for PMSM. The strategy is based on constructing an extended output depending observer normal form through a change of coordinates [START_REF] Chiasson | Nonlinear speed observer for the pm stepper motor[END_REF], which enables us to apply the existing observers in the literatures, such as those presented in [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF], [START_REF] Busawon | Asimple observer for a class of nonlinear systems[END_REF], [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], [START_REF] Besanc ¸on | On adaptive observers for state affine systems and application to synchronous machines[END_REF].

The approach of observer normal form has been set up in [START_REF] Bestle | Canonical form observer design for nonlinear time varying systems[END_REF] for a varying time and in [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF] for an invariant time single output nonlinear dynamical system. Then, it has been adopted for the multiple outputs case in [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF], [START_REF] Xia | Nonlinear observer with linearizable error dynamics[END_REF]. After that, several relevant algorithms have been developed in [START_REF] Keller | Nonlinear observer design by transformation into a generalized observer canonical form[END_REF], [START_REF] Phelps | On constructing nonlinear observers[END_REF], [START_REF] Rudolph | A block triangular nonlinear observer normal form[END_REF], [START_REF] Hou | Observer with linear error dynamics for nonlinear multi-output systems[END_REF], [START_REF] Marino | Nonlinear control design: geometric, adaptive and robust[END_REF], [START_REF] Boutat | New algorithm for observer error linearization with a diffeomorphism on the outputs[END_REF], [START_REF] Lynch | Nonlinear observers with approximately linear error dynamics: the multivariable case[END_REF], [START_REF] Lopez | Linearization by completely generalized input ouput injection[END_REF]. Besides, another interesting form, the so-called depending output observer normal form, was introduced by [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF], and then was improved in [START_REF] Zheng | A single output dependent observability normal form[END_REF] and [START_REF] Wang | Observer design using a generalized timescaled block triangular observer form[END_REF]. The most recent normal form, called extended nonlinear observer normal form due to [START_REF] Jouan | Immersion of nonlinear systems into linear systems modulo output injection[END_REF], allows us to enlarge the class of observer normal forms. It has been generalized by [START_REF] Noh | Nonlinear observer design by dynamic observer error linearization[END_REF], [START_REF] Back | Dynamic observer error linearization[END_REF], [START_REF] Boutat | Geometrical conditions for observer error linearization via 0[END_REF], [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF], [START_REF] Tami | Extended output depending normal form[END_REF]. The idea of the last form is to add an auxiliary dynamics into the original system such that the augmented system fulfills existing conditions which gang.zheng@inria.fr guarantee a transformation of the studied system into an extended normal form. On the other hand, this observer normal form constitutes a very powerful tool to solve the problem of simultaneous estimation of states and parameters [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF].

The paper is organized as follows: Firstly, the observer form is presented in Section II. In Section III, a change of coordinates which transforms the PMSM model into observer form is proposed. In Section IV, the load torque and rotor speed are estimated when parameters are considered as known. The case of simultaneous estimation of states variables and the unknown parameters is considered in Section V. Finally, simulation results and conclusion are discussed in the last section.

II. BACKGROUND ON NONLINEAR EXTENDED OBSERVER NORMAL FORMS

In this section, a brief overview on the extended observer normal form is provided. We consider a multi-output nonlinear system in the following form:

ẋ = f (x) y = h(x) (1)
where x ∈ U ⊆ R n represents the state and y ∈ R p denotes the outputs. We assume that the vectors field f and the output function h are sufficiently smooth. Also, it is assumed that the pair (h, f ) satisfies the observability rank conditions, thus the 1-forms θ i = dL i-1 f h for 1 ≤ i ≤ n are linearly independent where L i-1 f h is the (i -1) th Lie derivative of the output h in the direction of the vector field f , and d is the differential operator.

Under the above assumptions, theoretically the state can be estimated as x = σ(y, ẏ, ..., y n-1 ) where σ is a function which can be determined from the dynamics of system (1). However, it has been established that the use of successive output derivatives amplifies the noise in the measurement. Therefore, the observer design is a powerful tool to address the state estimation problem. To do so, in the case of nonlinear system, the transformation into observer form is used for which the observer design is relatively straightforward. Consider the nonlinear system (1), we seek a one dimensional auxiliary dynamics ẇ = η(y, w) so that the following extended dynamical system:

ẋ = f (x) (2) ẇ = η(y, w) (3) y = h(x) (4)
might be transformed through a change of coordinates (ξ T , ζ) T = φ(x, w) into the following extended output depending observer form:

ξ = A(y, w)ξ + B(y, w) (5) ζ = B n+1 (y, w) (6) y = Cξ (7) where ζ ∈ R, w ∈ R is an extra-output, C = [0, ..., 0, 1], and
A(y, w) =       0 ... ... ... 0 α 2 (y, w) 0 ... ... 0 0 α 3 (y, w) ... ... ... 0 ... ... ... ... 0 ... ... α n (y, w) 0       (8)
where α i (y, w) = 0, for 2 ≤ i ≤ n are functions depending only on the output y and the extra-output w. The normal form given by ( 5)-( 7) allows to apply a high gain observer [START_REF] Busawon | Asimple observer for a class of nonlinear systems[END_REF].

III. PMSM MODEL

The model of PMSM [START_REF] Chiasson | Modeling and High-performance Control of Electric Machines[END_REF] is described by a set of differential equations in (d -q) reference frame as follows:

dΩ dt = - f v J Ω + p J φ f i q - 1 J T l (9)
di d dt = - R s L s i d + pΩi q + 1 L s u d (10)
di q dt = - R s L s i q -pΩi d -p 1 L s φ f Ω + 1 L s u q (11)
where Ω, i d , i q are the rotor speed, the d and q-axis currents, respectively, u d and u q are the d and q-axis voltages, respectively, T l is the load torque, R s is the winding stator resistance, L s is the stator winding inductance, φ f is the magnet flux, p is the number of the pair poles, J is the rotor moment inertia, and f v is the viscous friction coefficient. In the rest of the of this paper, it is assumed that

[Ω, i d , i q ] T = [x 1 , x 2 , x 3 ] T . The currents in the (d -q)
reference frame are chosen as the measurement vectors such that :

y 1 = i d (12) y 2 = i q (13)

A. Change of coordinates

In the following, two reduced observer forms can be constructed from the outputs y 1 = i d and y 2 = i q . The first one is realized with the dynamics of Ω and i d , the second one is realized with the dynamics of Ω and i q .

1) Observer normal form by using a rotor mechanical speed dynamics (x 1 = Ω) and stator curent (x 2 = i d ): Let us consider dynamics given in ( 9), ( 10), ( 12) and ( 13) as a subsystem:

.

x 1 = - f v J x 1 + p J φ f x 3 - 1 J T l (14)
.

x 2 = - R s L s x 2 + px 1 x 3 + 1 L s u d (15)
y 1 = x 2 = i d y 2 = x 3 = i q
Proposition 1: The change of coordinates

z 1 = x 1 e -w 0 1 k(S) ds (16) z 2 = x 2 (17)
where .

w = η = -fv J κ(w) and κ(w) is a function to be determined, transforms the dynamical system given by ( 14)-( 15) into following observer form:

.

z 1 = p J φ f y 2 - 1 J T l e -w 0 1 k(S) ds (18)
.

z 2 = py 2 e w 0 1 k(S) ds z 1 - R s L s y 1 + 1 L s u d (19) Proof:
The differential of ( 16) with respect to t gives:

.

z 1 = . x 1 e -w 0 1 k(S) ds -x 1 1 k(w) . we -w 0 1 k(S) ds = - f v J x 1 + p J φ f x 3 - 1 J T l e -w 0 1 k(S) ds -x 1 1 k(w) - f v J κ(w) e w 0 1 k(S) ds = p J φ f x 3 - 1 J T l e -w 0 1 k(S) ds
Thus, we obtain:

.

z 1 = p J φ f y 2 - 1 J T l e -w 0 1 k(S) ds
Moreover, the differential of equation [START_REF] Besanc ¸on | On adaptive observers for state affine systems and application to synchronous machines[END_REF] gives

. x 2 = px 3 e w 0 1 k(S) ds z 1 - R s L s x 2 + 1 L s u d
Consequently:

.

z 2 = py 2 e w 0 1 k(S) ds z 1 - R s L s y 1 + 1 L s u d
2) Observer normal form by using a rotor mechanical speed dynamics (x 1 = Ω) and stator current (x 3 = i q ): Let us consider another subsystem constructed by the dynamics given by ( 9) and ( 11) such that:

dx 1 dt = - f v J x 1 + p J φ f x 3 - 1 J T l ( 20
)
dx 3 dt = - R s L s x 3 -px 1 x 2 -p 1 L s φ f x 1 + 1 L s u q (21)
y 1 = x 2 = i d y 2 = x 3 = i q Proposition 2:
The following change of coordinates

ξ 1 = x 1 e -w 0 1 k(S) ds (22) ξ 2 = x 3 (23)
where .

w = η = -fv J κ(w) and κ(w) is a function to be determined, transforms the system given by ( 20)-( 21) into following observer form:

.

ξ 1 = p J φ f y2 - 1 J T l e -w 0 1 k(S) ds (24)
.

ξ 2 = -p y1 + φ f Ls e w 0 1 k(S) ds ξ1 - Rs Ls y2 + 1 Ls uq (25)
Proof: The differential of the change of coordinates [START_REF] Keller | Nonlinear observer design by transformation into a generalized observer canonical form[END_REF] gives:

.

ξ 1 = . x 1 e -w 0 1 k(S) ds -x 1 1 k(w) . we -w 0 1 k(S) ds = - f v J x 1 + p J φ f x 3 - 1 J T l e -w 0 1 k(S) ds -x 1 1 k(w) - f v J κ(w) e -w 0 1 k(S) ds = p J φ f x 3 - 1 J T l e -w 0 1 k(S) ds Then . ξ 1 = p J φ f y 2 - 1 J T l e -w 0 1 k(S) ds
Also, the differential of change of coordinates [START_REF] Phelps | On constructing nonlinear observers[END_REF] gives

.

ξ 2 = -p x 2 + φ f L s x 1 - R s L s x 3 + 1 L s u q = -p x 2 + φ f L s e w 0 1 k(S) ds ξ 1 - R s L s x 3 + 1 L s u q Then . ξ 2 = -p y 1 + φ f L s e w 0 1 k(S) ds ξ 1 - R s L s y 2 + 1 L s u q
Remark 1: Both the systems given by the dynamical equations ( 18)-( 19) and ( 24)-( 25) are obviously observable with simple appropriate structure, thus we can easily use the existing observers in the literature to estimate the states. [START_REF] Busawon | Asimple observer for a class of nonlinear systems[END_REF].

IV. HIGH GAIN OBSERVER

In this section, the load torque T l and rotors speed Ω are estimated when the parameters φ f , L s , R s are assumed to be known and the load torque T l is considered as constant piecewise function. It is easy to see, with the dynamics . T l = 0, (24) and ( 25), one can obtain the following output depending observer form:

   . T l . ξ 1 . ξ 2    =   0 0 0 α 11 (y, w) 0 0 0 α 12 (y, w) 0     T l ξ 1 ξ 2   +    0 p J φ f y 2 e -w 0 1 k(S) ds -Rs Ls y 2 + 1 Ls u q   
where α 11 (y, w) = - . The above form allows us to apply the high gain observer presented in [START_REF] Busawon | Asimple observer for a class of nonlinear systems[END_REF] as follows:

ξ = A(y, w) ξ + B(w, y) -Γ -1 (y)R -1 ρ C T (C ξ -y) (26)
where

G =      0 • • • 0 0 1 • • • 0 0 . . . . . . . . . . . . 0 • • • 1 0      Γ(y, w) = diag[ n Π i=2 α i (y, w), n Π i=3 α i (y, w), • • • , α n (y, w), 1] R ρ (n + 1 -i, n + 1 -j) = (-1) i+j C j-1 i+j-2 ρ i+j-1 for 1 ≤ i ≤ n and 1 ≤ j ≤ n.
If we set e = ξ -ξ to be the observation error, then we see that its dynamics is governed by the following equation:

ė = ξ -ξ = (A(y, w) -Γ -1 (y, w)R -1 ρ C T C)e.
If y and w are bounded, then the observation error dynamics is exponentially stable by well choosing ρ.

Remark 2: The rotor position can be estimated by integrating the rotor speed with the knowledge of the initial rotor position [START_REF] Zaltni | Observability analysis and improved zerospeed position observer design of synchronous motor with experimental results[END_REF].

A. Simulation results with high gain observer

The efficiency of the previous approach to estimates the load torque and speed rotor is shown in Fig. 1 and Fig.

V. ADAPTIVE OBSERVER

In this section, the parameters φ f , L s , R s are assumed to be unknown and T l is considered as constant piecewise function. In what follows, we are going to estimate the load torque T l , the rotor speed Ω, magnet flux φ f , stator inductance L s , and stator resistance R s of PMSM, by applying the adaptive observer [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF], [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF] and [START_REF] Besanc ¸on | On adaptive observers for state affine systems and application to synchronous machines[END_REF]. For this, let us consider the system composed by the dynamics . T l = 0, (18) and ( 19), then one obtains the following observer normal form: The dynamical system given by ( 27) is in adaptive observer form (with affine states and parameters) which enables to estimate the states and the parameters by using the adaptive observer.

  . T l . z 1 . z 2   =   0 0 0 α 21 (w, y) 0 0 0 α 22 (w, y) 0     T l z 1 z 2   +    0 0 0 py2e -w 0 1 k(S) ds J 0 0 0 -y 1 u d      φ f Rs Ls 1
Similarly to [START_REF] Besanc ¸on | On adaptive observers for state affine systems and application to synchronous machines[END_REF] and [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], it is assumed as well that there exist positive constants α, β, T such that

αI ≤ t+T t Λ T (τ ) C T Σ(τ )C (τ ) Λ (τ ) dr ≤ βI and αI ≤ t+T t Ψ (t, τ ) T C T Σ(τ )CΨ (t, τ ) dr ≤ βI ∀t ≥ t 0
where Ψ is the transition matrix for the following system:

. x = A(y, w)x y = Cx and Σ is a positive definite symmetric matrix. Then, based on the results in [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF], [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear time-varying systems[END_REF], [START_REF] Besanc ¸on | On adaptive observers for state affine systems and application to synchronous machines[END_REF], the following system:

. x = A(y, w) x + ϕ(y) + φ(y, w) θ (29)

+ ΛS -1 θ Λ T C T + S -1 x C T Σ (y -C x) . θ = S -1 θ Λ T C T Σ(y -C x) (30)
.

Λ = A (y, w) -S -1 x C T C Λ + φ (y, w) (31)
.

S x = -ρ x S x -A(y, w) T S x -S x A(y, w) + C T ΣC (32)
.

S θ = -ρ θ S θ + Λ T C T ΣCΛ, S x (0), S θ (0) > 0 (33)
is an exponential adaptive observer for the nonlinear system [START_REF] Boutat | New algorithm for observer error linearization with a diffeomorphism on the outputs[END_REF], where ρ x and ρ θ are positive constants.

A. Simulation results with adaptive observer

In this subsection, the efficiency of the proposed approach to estimate both state variables and parameters of PMSM has been verified. with a = 0.0005 ∈ ]0, 1[. The simulation results are presented in Fig. 4 and Fig. 5, Fig.

2

 2

  .

Fig. 1 :Fig. 2 :

 12 Fig. 1: Estimation of the load torque T l .

Ls   where α 21 1 k

 211 (w, y) = -1 J e -w 0 (S) ds , α 22 (w, y) = pe w 0 1 k(S) ds . The above system can be written in the following form:. z = A(w, y)z + φ(w, y)θ y = Cz(27)where z = T l z 1 z 2 T , θ = φ

Fig. 3 :

 3 Fig. 3: Implementation of Adaptive observer.

Fig. 7 andFig. 4 :

 74 Fig. 4: Estimation of the load torque.

Fig. 5 :Fig. 6 :Fig. 7 :

 567 Fig. 5: Estimation of the rotor mechanical speed.

Fig. 8 :

 8 Fig. 8: Estimation the of winding inductance.

Table I

 I

	Parameter	Symbol	Value
	stator resistance	Rs	2.875 Ω
	stator inductance	Ls	8.5 mH
	Number of pole pairs	p	3
	Rotor magnetic flux Moment of inertia	φ f J	0.175 W b 0.00003 kgm 2
	Viscous friction coefficient	fv	0.0034 N m/s

TABLE I :

 I TABLE of MOTOR PARAMETERS

	We take κ(w) = sin 2 (aw) (aw) 2


Ramdane Tami
email: ramdane.tami@insa-cvl.fr

Driss Boutat
email: driss.boutat@insa-cvl.fr

Gang Zheng
email: gang.zheng@inria.fr

2013xy Observateur

Keywords: estimation d'états, forme normale d système nonlinéaire. states estimation, observer normal form, extended dynamics, nonlinear system. -/

Cet article étudie le problème de la recherche d'un changement de coordonnées permettant de transformer un système non-linéaire en forme normale d'observabilité, qui est une forme idéale pour l'estimation d'états. Des conditions géométriques suffisantes pour l'existence d'un changement de coordonnées d'un un système non linéaire mono sortie sont données.

ABSTRACT. This paper deals with the observer design for nonlinear dynamical system after transformation through change of coordinates into an observer normal form which supports a high gain observer. Moreover, the sufficient geometrical conditions that guarantee the existante of change of coordinates are given. The proposed approach is applied to estimate the states of SEIR model.

Le caractère imprévisible des maladies et la mutation des virus nécessite la mise en place doutils permettant lanalyse, la prédiction et le suivi du phénomène de propagation épidémique. Le système dynamique SEIR (Susceptible, Exposed, In-fected, Removed) modélise la propagation dune maladie contagieuse à travers une population. Plusieurs travaux ont été menés pour améliorer ce modèle, étudier sa stabilité et développer des stratégies de vaccination. Cependant, peu de travaux se sont intéressés à la conception dobservateurs pour ce modèle [START_REF] Iggidr | Global analysis of new malaria intrahost models with a competitive exclusion principle[END_REF]. Par conséquent, se basant sur le modelé SRIR, ce travail vise à construire un observateur pour estimer l'évolution d'une épidémie. Dans cette optique, nous proposons une approche basée sur le concept des formes normales dobservabilité et de la dynamique étendue.

Le concept de forme normale d'observabilité a été introduit par [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], (Bestle, Zeitz, 1983) pour une certaine classe de systèmes dynamiques non linéaires mono-sortie, ensuite il a été étendu par [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF], [START_REF] Xia | Nonlinear observer with linearizable error dynamics[END_REF] aux systèmes dynamiques multi-sorties. A partir de là, plusieurs algorithmes ont été proposés donnant lieu à divers approches: algébrique [START_REF] Keller | Nonlinear observer design by transformation into a generalized observer canonical form[END_REF], [START_REF] Phelps | On constructing nonlinear observers[END_REF], [START_REF] Rudolph | A block triangular nonlinear observer normal form[END_REF], géométrique [START_REF] Hou | Observer with linear error dynamics for nonlinear multi-output systems[END_REF], [START_REF] Marino | Nonlinear control design: geometric, adaptive and robust[END_REF], [START_REF] Boutat | Observer error linearization multi-output depending[END_REF], [START_REF] Lynch | Nonlinear observers with approximately linear error dynamics: the multivariable case[END_REF] et les transformations directes [START_REF] Lopez | Linearization by completely generalized input ouput injection[END_REF].

Afin d'élargir la classe des systèmes non-linéaires transformables par diffeomorphisme sous forme normale d'observabilité, [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF] proposa la forme normale dépendante de la sortie, consolidée par la suite dans [START_REF] Zheng | A single output dependent observability normal form[END_REF], [START_REF] Wang | Observer design using a generalized time-scaled block triangular observer form[END_REF]. Toujours, dans l'optique de pouvoir transformer la plus large classe possible de systèmes non-linéaires sous forme appropriée à l'estimation d'états, [START_REF] Jouan | Immersion of nonlinear systems into linear systems modulo output injection[END_REF] développa la forme normale étendue dont le principe consiste à ajouter une dynamique auxiliaire au système non-linéaire de sorte que le système augmenté resultant satisfasse les conditions de la mise sous forme normale d'observabilité étendue. Cette méthode a été affinée progressivement dans [START_REF] Noh | Nonlinear observer design by dynamic observer error linearization[END_REF]), (Back et al., 2006), [START_REF] Yang | Reduced-order dynamic observer error linearization[END_REF], [START_REF] Zheng | A single output dependent observability normal form[END_REF], (Boutat, Busawon, 2011).

Malgré la multitudes d'algorithmes proposées permettant la construction des formes normales, il est nécessaire de les améliorer et de les renforcer pour plus d'efficacité face aux non-linéarités d'ordre supérieur. Dans cette optique, la forme normale étendue dépendante de la sortie est mise en oeuvre Dans ce papier nous proposons une nouvelle forme normale d'observabilité. L'idée consiste à coupler les formes normales étendues avec les formes normales dépendantes de la sortie. L'efficacité de cette approche est démontrée sur un modèle non-linéaire de propagation épidémique SEIR (Susceptible, Exposed, Infected and Removed).

Le caractère imprévisible des maladies et la mutation des virus nécessitent la mise en place d'outils permettant l'analyse, la prédiction et le suivi du phénomène de propagation épidémique. Par conséquent, dans ce papier, un observateur est synthétisé afin d'estimer les différentes populations concernées par l'épidémie et de prédire son évolution. L'approche utilisée est basée sur le concept des formes normales d'observabilité et de la dynamique étendue (immersion).

Cet article est organisé comme suit : la section (II) présente l'exemple de motivation. Un rappel sur la mise sous forme normale d'observabilité est donné dans la section (III). La section (IV ) est dédiée à la forme normale étendue dépendante de la sortie et les conditions suffisantes pour son obtention. La mise en application de la méthode étudiée et les résultats de simulation sont présentés dans la section (V ).

Modèle épidémiologique et forme normale étendue

Le modèle épidémique SEIR décrit la propagation d'une maladie contagieuse à travers une population, il met en jeu 4 variables: S(t) la population susceptible à la maladie, E(t) la population exposée mais qui ne présente pas encore de symptômes, I(t) la population infectée et R(t) la population traitée. Plus précisément, ce modèle est décrit par le système non linéaire:

dS dt = bN -µS -β SI N -pbE -qbI (1)
dE dt = β SI N + pbE + qbI -(µ + ε)E (2)
dI dt = εE -(r + δ + µ)I (3) dR dt = rI -µR (4) dN dt = (b -µ)N -δI (5)
où N = S +E +I +R représente la population totale, b le taux de naissance naturelle, µ le taux de fécondité, β le taux de transmission de l'épidémie, δ le taux de mortalité épidémique, ε le taux passage des individus exposés au cas infectés, p le taux de descendance chez la population exposée, q le taux de descendance chez la population infectée et r le taux de guérison.

Afin de faciliter l'écriture des equations, dans la suite de l'article, des variables normalisées sont utilisés, telles que

x 1 = S N , x 2 = E N , x 3 = I N , x 4 = R N et x 1 + x 2 + x 3 + x 4 = 1.
Après dérivation de ces nouvelles variables, le modèle SEIR (1)-(4) devient: Dans la section (V ), il est montré que le système dynamique ( 6)-( 8) ne satisfait pas les conditions géométriques existantes dans la littérature [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF], [START_REF] Kazantzis | Nonlinear observer design using lyapunoves auxiliary theorem[END_REF], (Boutat et al., 2009). Pour remédier à ce problème, une dynamique étendue est ajoutée, qui sera déterminée par la suite, de telle sorte que le nouveau système augmenté satisfasse les conditions suffisantes grandissant l'existence du diffeomorphisme. En effet, si la dynamique auxiliaire

ẋ1 = b -bx 1 + γ 1 x 1 x 3 -pbx 2 -qbx 3 (6) ẋ2 = βx 1 x 3 + γ 2 x 2 + δx 2 x 3 + qbx 3 (7) ẋ3 = εx 2 + γ 3 x 3 + δx 2 3 (8) ẋ4 = rx 3 + bx 4 + δx 3 x 4 (9) y = x 3 (10) . - / où γ 1 = -(β -δ), γ 2 = -(b + ε -pb), γ 3 = -(r + δ + b).
ẇ = -b + γ 1 x 3 (11)
est ajoutée au système ( 6)-( 8), alors le changement de coordonnées z = φ(x, w) définit par :

z 1 = βεx 1 e -w + bpβx 3 e -w z 2 = e -w x 2 - b + γ 2 ε x 3 e -w - δ -γ 1 2ε x 2 3 e -w z 3 = 1 ε x 3 e -w ξ = w
transforme le système ( 6)-( 8) augmenté de la dynamique auxiliaire (11) sous la forme normale étendue dépendante de la sortie donnée par :

ż1 = B 1 (w, y) (12) ż2 = yz 1 + B 2 (w, y) (13) ż3 = z 2 + B 3 (w, y) (14) ẇ = B 4 (w, y) (15)
y = z 3 (16)
où la variable w est considérée comme une sortie auxiliaire et les expressions des B i seront données dans la section (5).

Ce papier établit des conditions suffisantes pour l'existence du changement de coordonnées transformant un système non linéaire sous forme normale étendue dépendante de la sortie:

ż = A(y)z + B(w, y) . w = B n+1 (w, y) y = Cz avec C = [0, ..., 0, 1] et A =      0 • • • 0 0 α 2 (y) • • • 0 0 . . . . . . . . . . . . 0 • • • α n (y) 0      (17)
Cette forme a une configuration adaptée pour l'utilisation de l'observateur à grand gain [START_REF] Busawon | Asimple observer for a class of nonlinear systems[END_REF]:

˙ z = A(y) z + B(w, y) -Γ -1 (y)R -1 ρ C T (C z -y)
avec G, Γ et R ρ des paramètres donnés par :

G n×n =      0 • • • 0 0 1 • • • 0 0 . . . . . . . . . . . . 0 • • • 1 0      Γ(y) = diag[ n Π i=2 α i (y), n Π i=3 α i (y), • • • , α n (y), 1] R ρ (n + 1 -i, n + 1 -j) = (-1) i+j C j-1 i+j-2 ρ i+j-1
On montre que la dynamique de l'erreur d'observation est donnée par:

ė = (A(y) -Γ -1 (y)R -1 ρ C T C)e
Si y et w sont bornées, la dynamique de l'erreur d'observation est exponentiellement stable avec un ρ bien choisi.

Remarque 1. -La dynamique ajoutée (11) ne depend que de la sortie et d'elle même.

Préliminaires

Considérons un système non linéaire mono sortie:

ẋ = f (x) (18) y = h(x) (19) où x ∈ U ⊆ R n sont
θ i = dL i-1 f h pour i = 2 : n
sont indépendantes, avec L k f est la k eme dérivée de Lie de h dans la direction du champ de vecteurs f . Cette hypothèse permet de construire le repère de Krener & Isidori [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF]) τ = [τ 1 , ..., τ n ] où le premier champ de vecteurs τ 1 est donné par l'équation algébrique:

θ i (τ 1 ) = 0 pour i = 1 : n -1 θ n (τ 1 ) = 1 (21)
et les autres champs de vecteurs sont obtenus par induction:

τ i = -ad f τ i-1 = [τ i-1 , f ] pour 2 ≤ i ≤ n,
où le symbole [, ] désigne le crochet de Lie. A présent, nous allons énumérer quelques formes normales les plus connues et les conditions garantissant leur exitance. D'après [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF]) si la condition de commutativité est satisfaite:

[τ i , τ j ] = 0 (22)
alors, il existe un changement de coordonnées z = φ(x) permettant de transformer le système ( 18)-( 19) sous la forme normale d'observabilité

ż = Az + B(y) y = Cz (23) avec C = [0, ..., 0, 1] et A =      0 • • • 0 0 1 • • • 0 0 . . . . . . . . . . . . 0 • • • 1 0      (24)
La condition ( 22) assurant l'existence d'un changement de coordonnées est relativement difficile à satisfaire.

Dans [START_REF] Zheng | A single output dependent observability normal form[END_REF] des conditions moins restrictives ont été proposées pour l'existence du changement de cordonnées. En effet, supposons que le repère τ vérifie les conditions suivantes:

   [τ 1 , τ n ] = λ 1 (y)τ 1 [τ k , τ n ] = λ k (y)τ k modulo Eng {τ 1 , • • • , τ k-1 } pour 2 ≤ k ≤ n -2 (25)
Dès lors, on considère une famille de fonctions non nulles dépendantes de la sortie

α 2 (y), ..., α n (y) et soit π = n Π i=2 α i (y) = α 2 (y) × ... × α n (y).
Puis, on construit un nouveau repère comme suit:

τ 1 = πτ 1 τ i = 1 αi [τ i-1 f ] (26)
Si le repère τ est commutatif, c'est-à-dire :

[τ i , τ j ] = 0 (27)
alors il est prouvé dans [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF] pour α 2 = .... = α n et pour des α i différents dans [START_REF] Zheng | A single output dependent observability normal form[END_REF] que le système dynamique ( 18)-( 19) peut être transformé grâce à un changement de coordonnées z = φ(x) sous la forme d'observabilité dépendante de la sortie:

ż = A(y)z + B(y) y = Cz où A(y) est définie dans (17).
Dans le cas où ni la condition (22), ni la condition (27) ne sont satisfaites, alors on passe à l'algorithme utilisant la dynamique étendue.

Forme normale non linéaire étendue dépendante de la sortie

Cette section met en évidence des conditions géométriques suffisantes qui garantissent l'existence d'une dynamique auxiliaire ẇ = η(y, w) telle que le système dynamique augmenté:

ẋ = f (x) (28) ẇ = η(y, w) (29)
y = h(x) (30)
peut être transformé par un changement de coordonnées (z, ξ) = φ(x, w) sous la forme normale étendue dépendante de la sortie:

ż = A(y)z + B(y, w) (31) ξ = B n+1 (y, w) (32) y = Cz (33)
où A(y) est défini dans (17). Le cas où la matrice A(y) est constante (24) est largement étudié dans (Back et al., 2006 ;[START_REF] Jouan | Immersion of nonlinear systems into linear systems modulo output injection[END_REF][START_REF] Zheng | A single output dependent observability normal form[END_REF]Boutat, Busawon, 2011). . -/ Pour ce qui suit, la condition (25) est supposé satisfaite, par conséquent on peut determiner les fonctions α 2 (y), ..., α n (y) ainsi que le repère τ défini dans (26).

Maintenant, définissons un nouveau repère comme suit:

σ 1 = l(w)τ 1 où la fonction l(w) = 0 est à déterminer σ k = 1 α k [σ k-1 , F ]
où F = f + η(y, w) ∂ ∂w est le champ de vecteurs régissant le système étendue. Théorème 2. -S'il existe une fonction l(w) = 0 de la variable auxiliaire w telle que [σ i , σ j ] = 0 pour 1 ≤ i, j ≤ n, alors il existe un changement de coordonnées (z, ξ) = φ(x, w) qui permet de transformer le système dynamique étendu ( 28)-( 30) sous la forme normale étendue dépendante de la sortie ( 31)-( 33).

Preuve 3. -Supposons qu'il existe l(w) = 0 tel que [σ i , σ j ] = 0 pour 1 ≤ i, j ≤ n. Soit σ n+1 un champ de vecteurs indépendants de σ i pour tout i = 1 : n, tel que

[σ i , σ n+1 ] = 0 pour 1 ≤ i ≤ n et dw(σ n+1 ) = 1. Considérons la matrice Λ = (Λ i,j ) 1≤i,j≤n+1 définie comme l'evaluation des 1-formes θ i = dL i-1 F h pour i = 1 : n et θ n+1 = dw sur le repère σ = {σ 1 , ..., σ n , σ n+1 }, c'est-à-dire Λ i,j = θ i (σ j ) pour 1 ≤ i, j ≤ n + 1. Par construction la matrice s'écrit comme suit Λ =          0 • • • 0 0 1 * . . . 0 0 lα n * * 0 0 lπ α1α2 * * * 0 lπ α1 * * * * lπ * * * * * 0 0 0 0 0 1         
Il est facile de voir qu'elle est inversible. Par consequent, on peut définir les multi-1 formes différentielles suivantes :

ω = Λ -1 σ =     ω 1 ω 2 ... ω n+1    
il est clair que : ω i σ j = δ j i , avec δ j i est le symbole de Kronecker. Soit X et Y deux champs de vecteurs {σ 1 , ..., σ n , σ n+1 }. On a :

dω i (X, Y ) = L Y ω i (X) -L X ω i (Y ) -ω i [X, Y ] comme ω i (X) et ω i (Y ) sont constantes alors dω i (X, Y ) = -ω i [X, Y ]
par conséquent [X, Y ] = 0 pour tous X, Y si et seulement si dω i = 0 pour tous 1 ≤ i ≤ n + 1. D'où, grâce au lemme de Poincaré, il existe φ = φ 1 , ..., φ n+1 tel que :

ω = Dφ := φ * avec Dφ la différentielle de φ. Nous considérons φ * (σ i ) = ∂ ∂zi pour i = 1 : n et φ * (σ i ) = ∂ ∂ξ i .
A présent, nous verrons comment φ * transforme le champ de vecteurs F . Pour i = 1 : n -1, on a:

∂ ∂z i , φ * (F ) = [φ * (σ i ) , φ * (F )] = φ * [σ i , F ] = α i φ * σ i = α i ∂ ∂z i+1
Alors par intégration, on a żi = α i (y)z i + β i (y, w).

Nous terminons cette section par quelques remarques.

Remarque 4. --le calcul des fonctions α i est détaillé dans [START_REF] Zheng | A single output dependent observability normal form[END_REF]; -pour le calcul des fonctions l(w) et η(w, y), les algorithmes de [START_REF] Zheng | A single output dependent observability normal form[END_REF]Boutat, Busawon, 2011) peuvent être utilisés

Application au modèle SEIR

Dans cette section, on utilisera l'algorithme présenté ci-dessus pour le calcul du changement de cordonnées transformant le modèle SEIR ( 6)-( 8) sous la forme normale (12)-( 16).

Les 1-formes (20) d'observabilité du système ( 6)-( 8) sont données par:

θ 1 = dx 3 θ 2 = εdx 2 + (γ 3 + 2δx 3 )dx 3 θ 3 = εβx 3 dx 1 + ε(γ 2 + γ 3 + 3δx 3 )dx 2 + Q 1 dx 3 avec : Q 1 = εβx 1 + 3εδx 2 + εqb + γ 2 3 + 6δγ 3 x 3 + 6δ 2 x 2 3
Un calcul direct donne le repère de Krener & Isidori (21):

τ 1 = 1 εβx3 ∂ ∂x1 τ 2 = uτ 1 + 1 ε ∂ ∂x2 où u = -b + γ 3 + (δ + γ 1 )x 3 + ε x2 x3 τ 3 = -pb ε ∂ ∂x1 -(L f u) τ 1 + uτ 2 + 1 ε (γ 2 + δx 3 ) ∂ ∂x2 + ∂ ∂x3 10 . -
/ puis, on vérifie si la condition de commutativité ( 22) est satisfaite:

[τ 1 , τ 2 ] = [τ 1 , τ 3 ] = 0 [τ 2 , τ 3 ] = Q 2 τ 1 + 1 x 3 τ 2 avec Q 2 = -(3δ + 2γ 1 + 2γ 2 -γ 3 x3 -3ε x2 x 2 3
).

Comme [τ 2 , τ 3 ] = 0, c-a-d la condition de commutativité n'est pas satisfaite, alors le système ( 6)-( 8) ne peut pas être transformé, mais on utilisera les τ i pour construire le nouveau repère τ donné dans (26). En se référant à [START_REF] Zheng | A single output dependent observability normal form[END_REF], on peut utiliser l'equation:

[τ 2 , τ 3 ] = λ(y)τ 2 modτ 1 avec λ(y) = dα2(y)
dy 1 α2(y) pour determiner α 2 (y). Alors, on a dα2(y) dy

1 α2(y) = 1 x3 = 1 y ce qui donne α 2 (y) = x 3 . On peut toujours supposer que α 3 (y) = 1. Par conséquent, on a α 2 = x 3 , α 1 = 1 et π = α 2 α 2 = x 3 .
Alors, à partir de (26) on obtient:

τ 1 = x 3 τ 1 = 1 εβ ∂ ∂x 1 τ 2 = 1 x 3 [τ 1 , f ] = 1 εβx 3 (-b + γ 1 x 3 ) ∂ ∂x 1 + 1 ε ∂ ∂x 2 τ 3 = [τ 2 , f ] = (-pbβ + (-b + γ 1 x 3 ) 2 x 3 -bε x 2 x 2 3 )τ 1 + 1 ε (γ 2 -b + (δ + γ 1 )x 3 ) ∂ ∂x 2 + ∂ ∂x 3
Un simple calcul nous donne

[τ 1 , τ 2 ] = [τ 1 , τ 3 ] = 0 et [τ 2 , τ 3 ] = -2 b x 2 3 τ 1
Comme [τ 2 , τ 3 ] = 0, alors la condition (27) de commutativité n'est pas satisfaite. Cependant, en se référant au Lemme 3.1 dans (Boutat, Busawon, 2011), les conditions pour l'application de l'algorithme de la dynamique étendue sont satisfaites. Alors, nous chercherons une dynamique auxiliaire ẇ = η(y, w) et une fonction l(w) satisfaisant les conditions du théorème (2) énoncé ci-dessus.

Remarque 5. -D'après le Lemme 3.1 de (Boutat, Busawon, 2011)

[τ i , τ n ] - ̺(y)τ i-1 ∈ span {τ 1 , • • • , τ i-1
} est une condition assurant l'existence de η(y, w) et l(w), avec ̺(y) est une fonction de y.

Par consequent, posons σ 1 = l(w)τ 1 , ce qui donne :

σ 1 = l εβ ∂ ∂x 1
Par induction, on a :

σ 2 = 1 x 3 [σ 1 , F ] = 1 x 3 (lH -ηl ′ )σ 1 + l ε ∂ ∂x 2 où H = (-b + γ 1 x 3 ).
Ensuite, on a:

σ 3 = - pbl ε ∂ ∂x 1 + (lH -ηl ′ ) ε σ 2 - G εl σ 1 + l(γ 2 + qx 3 ) -ηl ′ ε ∂ ∂x 2 + l ∂ ∂x 3 avec G = ε 2 bl x2 x3 -ε 2 l ′ x 2 ( η ′ x3-η x 2 3 ) + η( ε x3 (l ′ H -ηl ′′ ))σ 1 et : η ′ = ∂η ∂x3 . Finalement, on obtient : [σ 2 , σ 3 ] = 2(ε 2 l ′ l η ′ x 3 -η x 3 - bε 2 l 2 x 2 3 )σ 1 On impose que [σ 2 , σ 3 ] = 0 ⇐⇒ l ′ l(η ′ x 3 -η) -bl 2 = 0
La solution de l'équation précédente est donnée par :

l = e w et η = γ 1 x 3 -b (34)
Alors le système dynamique étendu devient :

F =            ẋ1 = b -bx 1 + γ 1 x 1 x 3 -pbx 2 -qbx 3 ẋ2 = βx 1 x 3 + γ 2 x 2 + δx 2 x 3 + qbx 3 ẋ3 = εx 2 + γ 3 x 3 + δx 2 3 • w = γ 1 x 3 -b y = x 3           
Par l'application de l'algorithme précédemment décrit, et en remplaçant l et η par leur valeur touvée dans (34) on obtient :

σ 1 = e w εβ ∂ ∂x 1 , σ 2 = e w ε ∂ ∂x 2 σ 3 = - e w ε pb ∂ ∂x 1 + e w ε (γ 2 + b + (δ -γ 1 )x 3 ) ∂ ∂x 2
+e w ∂ ∂x 3 . -/ Ce nouveau repère σ vérifie la condition de commutativité, tels que :

[σ 1 , σ 2 ] = [σ 1 , σ 3 ] = [σ 2 , σ 3 ] = 0
Étant donné que le nouveau système, augmenté, est de dimension 4, alors il faut trouver un quatrième champ de vecteurs σ 4 qui commute avec σ 1 , σ 2 et σ 3 :

σ 4 = ∂ ∂w + x 1 ∂ ∂x 1 + (x 2 + δ -γ 1 2ε x 2 3 ) ∂ ∂x 2 + x 3 ∂ ∂x 3
on a [σ 4 , σ i ] = 0 pour 1 ≤ i ≤ 3. A partir de là, il devient simple de calculer la matrice d'observabilité θ e , telle que :

θ e = [dh, dL F h, dL 2 F , dw] T
puis on obtient :

Λ = θ e σ =     0 0 e w x 3 0 e w Λ 23 Λ 24 x 3 e w e w (γ 2 + δx 3 ) Λ 33 Λ 34 0 0 0 1     avec: Λ 23 = e w (b + γ 2 + x 3 (δ -γ 1 )) Λ 24 = ε x 2 + 1 2ε x 2 3 (δ -γ 1 ) Λ 33 = e w (γ 2 + δx 3 ) (b + γ 2 + x 3 (δ -γ 1 )) + εe w (βx 1 + δx 2 + bq) -bpβx 3 e w Λ 34 = εx 3 (βx 1 + δx 2 + bq) + ε (γ 2 + δx 3 ) x 2 + 1 2ε x 2 3 (δ -γ 1 ) + βεx 1 x 3
Par un simple calcul, le changement de coordonnées est donné par sa différentielle comme suit :

ω = Λ -1 θ e = dz = d     εβx 1 e -w + bpβx 3 e -w εx 2 e -w -(b + γ 2 )x 3 e -w -1 2 (δ -γ 1 )x 2 3 e -w x 3 e -w w    
Finalement, la forme normale du système (1)-( 5) est donnée par :

ż1 = B 1 (w, y) ż2 = yz 1 + B 2 (w, y) ż3 = z 2 + B 3 (w, y) ż4 = γ 1 y -b y = z 3 = ye -w avec : B 1 (y, w) = -(εβqb + bpβγ 3 + b 2 pβ)e -w x 3 + (bpβd -bpβγ 1 )e -w x 2 3 + εβbe -w B 2 (y, w) = (εqb -(b + γ 2 )γ 3 -(b + γ 2 )b)e -w x 3 + (-βpb -(b + γ 2 )d + (b + γ 2 )γ 1 -(δ -γ 1 )γ 3 - δ -γ 1 2 b)e -w x 2 3 + (-(δ -γ 1 )d + δ -γ 1 2 γ 1 )e -w x 3 3 B 3 (y, w) = (2b + γ 2 + γ 3 )e -w x 3 + 1 2 (d + δ -2γ 1 )e -w x 2 3

Résultats de simulation

Les valeurs des paramètres utilisés dans le modèle épidémique sont : N = 141, b = 0.221176/N, d = 0.002, p = 0.8, q = 0.95, β = 0.05, ε = 0.05, r = 0.003. S(0)=140; E(0)=0.01; I(0)=0.02; N(0)=141. Pour rendre la simulation proche de la réalité, on a ajouté un bruit blanc à la sortie du modèle I(t) et on a pris des conditions initiales différentes entre le modèle (1)-( 5) et l'observateur. En se référant aux figures (Fig. 2, Fig. 4, Fig. 6) la convergence des estimations est rapide mais des oscillations apparaissent due au choix du gain. Pour la simulation, on a choisi ρ = 0.35, ce qui satisfait le compromis entre la stabilité et la convergence de l'estimateur.

Conclusion

Ce papier présente une nouvelle forme d'observabilité étendue et dépendante de la sortie. Des condition suffisantes à l'existence du changement de coordonnées sont établies pour une classe de systèmes non linéaires. Cette nouvelle forme permet l'utilisation de l'observateur à grand gain. International Journal of Control, vol. 38, p. 419-431. Boutat D. (2007). Geometrical conditions for observer error linearization via R 0, 1, ..., (N -2). In 7th ifac symposium on nonlinear control systems nolcos'07,. Boutat D., Benali A., Hammouri H., Busawon K. (2009). New algorithm for observer error linearization with a diffeomorphism on the outputs. Automatica, vol. 45, n o 10, p. 2187-

  Les états x1 et x2 seront estimés par un observateur, tandis que l'état x4 sera déduit à partir de l'equation algébrique x4 = 1 -x1 -x2 -x 3 .

Figure 2 .

 2 Figure 2. Estimation de la population susceptible.

  Figure 1.

Figure 4 .

 4 Figure 4. Estimation de la population exposée.

  Figure 3.

Figure 6 .

 6 Figure 6. Estimation de la population traitée.

  Figure 5.


Ramdane Tami
email: ramdane.tami@ensi-bourges.fr

Gang Zheng
email: gang.zheng@inria.fr

Driss Boutat
email: driss.boutat@ensi-bourges.fr

Didier Aubry
email: didier.aubry@bourges.univ-orleans.fr

Partial observability normal form for nonlinear functional observers design

In this paper, we investigate Partial Observability Normal Forms (PONF) of nonlinear dynamical systems. Necessary and sufficient conditions for the existence of a diffeomorphism bringing the original nonlinear system into a PONF are established. This enables us to estimate a part of state of a nonlinear dynamical system. A concrete example (SIR epidemic model) is provided to illustrate the feasibility of the proposed results.

Introduction

Since the last four decades, many research activities have been developed to deal with the problem of state estimation of nonlinear dynamical systems. Several nonlinear state estimation methods have been performed to improve accuracy and performances of the control system design. Generally, we distinguish two approaches for nonlinear observer design. The first one is to design observer directly for the nonlinear systems which however highly depends on the studied system and there does not exist a uniform way to study general nonlinear systems. The second approach is based on some nonlinear transformations, using Lie algebra, to bring the original system into canonical observability normal form, from which the design of state observers is performed by using existing observer techniques in the new coordinates. The literature is vast about this second approach since the pioneer works of ( [START_REF] Bestle | Canonical form observer design for non-linear time-variable systems[END_REF][START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF]) for single output systems and [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF] for the case of multi outputs (see also [START_REF] Xia | Nonlinear observer with linearizable error dynamics[END_REF][START_REF] Phelps | On constructing nonlinear observers[END_REF][START_REF] Keller | Entwurf nichtlinearer Beobachter mittels Normalformen[END_REF][START_REF] Lopez-M.V | Linearization by completely generalized input ouput injection[END_REF][START_REF] Boutat | New algorithm for observer error linearization with a diffeomor-phism on the outputs[END_REF][START_REF] Zheng | A single output dependent observability normal form[END_REF][START_REF] Boutat | Observer error linearization multi-output depending[END_REF][START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF][START_REF] Kazantzis | Nonlinear observer design using lyapunoves auxiliary theorem[END_REF][START_REF] Lynch | Nonlinear observers with approximately linear error dynamics: the multivariable case[END_REF][START_REF] Marino | Nonlinear control design: geometric, adaptive and robust[END_REF][START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF][START_REF] Rudolph | A block triangular nonlinear observer normal form[END_REF][START_REF] Noh | Nonlinear observer design by dynamic observer error linearization[END_REF][START_REF] Jo | Observer design for non-linear systems that are not uniformly observable[END_REF]).

All the above-mentioned papers are dedicated to the full order case (i.e. the observer and the original system have the same dimension) and few works have been dedicated to partial observation which makes sense , in practice, when only a part or a function of states are required. Among the papers dedicated to this issue, let us quote the work of [START_REF] Kang | On the observability of nonlinear and switched systems[END_REF] on Z-observability or in [START_REF] Robenack | Observer design using a partial nonlinear observer canonical form[END_REF][START_REF] Santoboni | Application of partial observability for analysis and design of synchronized systems[END_REF][START_REF] Trinh | Partial-state observers for nonlinear systems[END_REF] where the authors proposed nonlinear observer based on particular canonical forms.

This paper proposes a PONF for partially observable nonlinear systems. In the new coordinates, a simple Luenberger observer is used to estimate a part of state of the studied system. Necessary and sufficient conditions are established to transform the original nonlinear system into the PONF.

This paper is organized as follows. Section 2 recalls Zobservability. In Section 3, PONF is presented. Necessary and sufficient conditions are deduced in Section 4 to bring the original nonlinear system into the PONF. An extension to nonlinear systems with inputs is presented in section 5. Section 6 generates the results by applying another diffeomorphism on the output, in which a concrete example (SIR epidemic model) is presented in order to highlight our results.

Z-observability

Let us consider the following nonlinear dynamical system:

ẋ = f (x) y = h(x) (1)
where x ∈ R n is the state vector, y ∈ R is the output, f : R n → R n and h : R n → R are analytic. Contrary to the classical observability analysis as in [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], where the full state vector is estimated, this paper considers the observability of the following variables

z = l (x) (2)
where z ∈ R p . This problem was firstly studied in [START_REF] Kang | On the observability of nonlinear and switched systems[END_REF], and is named as Z-observability.

Definition 1 (Z-observability) z = l (x) is said to be Zobservable with respect to system [START_REF] Bestle | Canonical form observer design for non-linear time-variable systems[END_REF], if for any two trajectories,

x i (t), 1 ≤ i ≤ 2, in U ⊂ R n defined on a same interval [t 0 , t 1 ], the equality h(x 1 (t)) = h(x 2 (t)), a.e. in [t 0 , t 1 ] implies Z(x 1 (t)) = Z(x 2 (t)), a.e. in [t 0 , t 1 ]
If for any trajectory x(t) in U there always exists an open set

U 1 ⊂ U so that z is Z-observable in U 1 , then z is said to be locally Z-observable in U .
The above definition of Z-observability can be interpreted in an algebraic way, which is linked to the classical definition of algebraical observability in [START_REF] Diop | Nonlinear observability, identifiability, and persistent trajectories[END_REF]. In this work we will adopt the following definition.

Definition 2 z = l (x) is said to be Z-observable with respect to system [START_REF] Bestle | Canonical form observer design for non-linear time-variable systems[END_REF], if it can be expressed as functions of the output and its derivatives, i.e.

z = l (x) = l y, ẏ, • • • , y (i) , • • •
In the following, by assuming that z is Z-observable, we are going to propose a universal approach to estimate z for system [START_REF] Bestle | Canonical form observer design for non-linear time-variable systems[END_REF]. This method is based on transforming nonlinear system (1) into a so-called partial observability normal form, from which a reduced order observer can be easily designed.

Partial observability normal form

This paper considers only the non trivial case of Zobservability, i.e. it is assumed that for dynamical system (1) we have rank dh, dL f h,

• • • , dL k f h, • • • = r < n. Let consider the following partial observability normal form    ξ = Aξ + β(y) ζ = η(ξ, ζ) y = Cξ (3)
where

ξ ∈ R r , ζ ∈ R n-r , y ∈ R, A is the r × r Brunovsky matrix: A =      0 0 • • • 0 1 0 • • • 0 0 . . . . . . . . . 0 • • • 1 0      ∈ R r×r C = (0, • • • , 0, 1) ∈ R 1×r , β : R → R n and η : R r × R n-r → R n-r .
For the form (3), one can easily design a reduced order observer to estimate only the partial state ξ.

Lemma 1 The following dynamical system:

ξ = A ξ + β(y) + K(ŷ -y) ŷ = C ξ
is an observer for the proposed partial observability normal form [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF].

Proof 1 Set e = ξ -ξ, we have ė = (A -KC) e. Since A ∈ R r×r is in the Brunovsky form and C = (0, • • • , 0, 1) ∈ R 1×r , thus the pair (A, C) is observable. One can arbitrarily choose K such that (A -KC) is Hurwitz, and this implies the exponential convergence of ξ to ξ.

It is shown that once system (1) can be transformed via a diffeomorphism (ξ T , ζ T ) T = φ(x) into the partial observability normal form (3), then one can design the above simple observer to estimate ξ. Moreover, if z is a function such that:

∂z ∂ζ = 0
then z is Z-observable for (1), and we can use ξ to estimate z.

Therefore, the rest of paper deals with the deduction of necessary and sufficient conditions which guarantee a diffeomorphism to transform system (1) into the proposed partial observability normal form (3). [START_REF] Robenack | Observer design using a partial nonlinear observer canonical form[END_REF]. which is written as:

Remark 1 The partial observability normal form considered in this work is quite different from the normal form introduced in the work of Röbenack and Lynch

   ξ = Aξ + β(y, ζ) ζ = η(ξ, ζ) y = Cξ
where β depends also on the second variable ζ.

Nonlinear systems without inputs

In this paper, it is assumed that there exists r < n such that rank dh, dL f h,

• • • , dL k f h, • • • is r. Thus system (1) is not fully observable. For 1 ≤ i ≤ r, set θ i = dL i-1 f h and ∆ =span{θ 1 , θ 2 , • • • , θ r }. Denote ∆ ⊥ =ker ∆ the distribu- tion kernel of ∆.
Let τ 1 be a vector field modulo ∆ ⊥ which satisfies the following conditions:

dL k f h(τ 1 ) = 0 for 0 ≤ k ≤ r -2 dL r-1 f h(τ 1 ) = 1
and by induction define the following family of vector fields

τ i = [τ i-1 , f ] modulo ∆ ⊥ for 2 ≤ i ≤ r, which implies τ i -[τ i-1 , f ] ∈ ∆ ⊥ ,
where [, ] denotes the conventional Lie bracket. Thus, one can choose a complementary family of vector fields {τ r+1 , ...,

τ n } such that τ = [τ 1 , τ 2 , • • • , τ n ] forms a basis and θ k (τ j ) = 0 for 1 ≤ k ≤ r, r + 1 ≤ j ≤ n.
Note

Λ 1 =      θ 1 θ 2 . . . θ r      (τ 1 , τ 2 , • • • , τ r ) =      0 • • • 0 1 . . . • • • 1 * 0 • • • * * 1 • • • * *      With the chosen {τ r+1 , τ r+2 , • • • , τ n }, one can freely choose {θ r+1 , θ r+2 , • • • , θ n } such that Λ 2 =    θ r+1 . . . θ n    (τ r+1 , • • • , τ n ) is of rank n -r.
Property 1 By giving the vector fields (τ 1 , τ 2 , • • • , τ r ) and the codistribution (θ 1 , θ 2 , • • • , θ r ), the chosen complementary τ i and

θ i for r + 1 ≤ i ≤ n should satisfy the following properties 1) τ = [τ 1 , τ 2 , • • • , τ n ] forms a basis; 2) θ k (τ j ) = 0 for 1 ≤ k ≤ r and r + 1 ≤ j ≤ n. 3) rankΛ 2 = n -r
Since τ is a basis, then it can be viewed as an invertible matrix. Therefore, in this basis f can be decomposed as follows:

f (x) = n i=1 f i (x) ∂ ∂x i = F 1 + F 2 (4)
with

F 1 = r i=1 F 1,i (x)τ i and F 2 = n j=r+1 F 2,j (x)τ j .
One can state the following theorem.

Theorem 1 Given a family of vector fields τ and θ satisfied Property 1, there exists a diffeomorphism (ξ T , ζ T ) T = φ(x) which transforms the dynamical system (1) into the partial observability normal form [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF] if and only if

• [τ i , τ j ] = 0 for all 1 ≤ i ≤ n and 1 ≤ j ≤ n; • [τ i , F 1 ] = 0 for all r + 1 ≤ i ≤ n, where F 1 is defined in (4).
Proof 2 Necessity:

If there existes a diffeomorphism (ξ T , ζ T ) T = φ(x) which transforms (1) into the form (3), then one has φ * (τ i ) = ∂ ∂ξi for 1 ≤ i ≤ n, which implies [φ * (τ i ), φ * (τ j )] = φ * ([τ i , τ j ]) = 0 for 1 ≤ i ≤ n. Since ξ = φ(x) is a diffeomorphism, one has [τ i , τ j ] = 0 for 1 ≤ i ≤ n and 1 ≤ j ≤ n.
Moreover, it is easy to see that for the dynamical system [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF] we have

∆ ⊥ =span{ ∂ ∂ζi , r + 1 ≤ i ≤ n}, therefore φ * (τ i ) = ∂ ∂ξi modulo ∆ ⊥ . Thus it is easy to check that [φ * (τ i ) , φ * (F 1 )] = φ * ([τ i , F 1 ]) = 0 for r + 1 ≤ i ≤ n. Finally one obtains [τ i , F 1 ] = 0 and then we have φ * (f ) = Aξ + β(y) η(ξ, ζ)
which implies that

φ * (F 1 ) + φ * (F 2 ) = Aξ + β(y) η(ξ, ζ)

Sufficiency:

Since [τ i , τ j ] = 0 for 1 ≤ i ≤ n and 1 ≤ j ≤ n, then according to Poincaré's lemma there exists locally a diffeomorphism

(ξ T , ζ T ) T = φ(x) such that dφ = φ * with φ * (τ i ) = ∂ ∂ξi for 1 ≤ i ≤ r and φ * (τ j ) = ∂ ∂ζj for r + 1 ≤ j ≤ n. Thus, for 1 ≤ i ≤ r -1, one has ∂φ * (f ) ∂ξ i = φ * ([τ i , f ]) = φ * (τ i+1 modulo ∆ ⊥ ) = ∂ ∂ξ i+1 modulo span{ ∂ ∂ζ k r + 1 ≤ k ≤ n} Moreover, for r + 1 ≤ j ≤ n, since [τ j , F 1 ] = 0, hence ∂φ * (F 1 ) ∂ζ j = φ * ([τ j , F 1 ]) = 0 (5)
By integration one has

ξ = φ * (F 1 ) = r i=1 ξ i ∂ ∂ξ i+1 + β(y)
and this proved the sufficiency.

Note θ = (θ 1 , • • • , θ n )
T , one can define the following matrix:

Λ = θτ = Λ 1 0 r×(n-r) * Λ 2
It is clear that Λ is invertible, thus one can define the following multi 1-forms:

ω = Λ -1 θ (6)
Theorem 2 Suppose that Theorem 1 is fulfilled, then the diffeomorphism (ξ T , ζ T ) T = φ(x) which transforms (1) into the form ( 3) is determined by

φ (x) = ω + φ (0)
Proof 3 For any two vector fields X, Y one has

dω(X, Y ) = L X (ω(Y )) -L Y (ω(X)) -ω([X, Y ])
By setting X = τ i and Y = τ j , one obtains

dω(τ i , τ j ) = L τi ω(τ j ) -L τj ω(τ i ) -ω([τ i , τ j ])
As ω(τ j ) and ω(τ i ) are constant, then one has

dω(τ i , τ j ) = -ω([τ i , τ j ])
which implies the equivalence between dω = 0 and [τ i , τ j ] = 0. Since Theorem 1 is fulfilled, thus one always has dω = 0. According to Poincaré's lemma, this implies that there exists a diffeomorphism φ (x) such that ω = dφ. Finally the diffeomorphism can be determined just by integration of ω defined in [START_REF] Keller | Entwurf nichtlinearer Beobachter mittels Normalformen[END_REF].

Example 1 Let us consider the following dynamical system:

           ẋ1 = -x 2 3 + x 3 x 2 x 1 -1 2 x 3 3 ẋ2 = x 1 -1 2 x 2 3 ẋ3 = -x 3 + x 2 x 1 -1 2 x 2 3 y = x 2 z = x 2 + 2x 1 x 2 -x 2 x 2 3 ( 7
)
A simple calculation gives rank dh, dL f h, dL 2 f h = 2, thus r = 2. One has

θ 1 = dx 2 and θ 2 = dx 1 -x 3 dx 3 Let ∆ =span{θ 1 , θ 2 }, and ker ∆ = span l 1 = x 3 ∂ ∂x 1 + ∂ ∂x 3
The frame τ is given by

τ 1 = ∂ ∂x 1 τ 2 = ∂ ∂x 2 + x 2 x 3 ∂ ∂x 1 + x 2 ∂ ∂x 3 modulo l 1 = ∂ ∂x 2
In order to form a basis which satisfies θ 1 (τ 3 ) = θ 2 (τ 3 ) = 0, the third complementary vector field can be chosen as follows

τ 3 = ∂ ∂x 3 + x 3 ∂ ∂x 1
which makes the following equality be satisfied

[τ 1 , τ 2 ] = [τ 1 , τ 3 ] = [τ 2 , τ 3 ] = 0
According to 4), one has

f = F 1 + F 2 = F 12 τ 2 + F 23 τ 3
where

F 1 = x 1 -x 2 3 /2 τ 2 and F 2 = -x 3 + x 1 x 2 -x 2 3 /2 τ 3 . It can be checked that [τ 3 , F 1 ] = 0
Then the second item of Theorem 1 is satisfied. In order to have rank Λ 2 =rank {θ 3 (τ 3 )} = 1, one can choose

θ 3 = dx 3 -x 2 dx 2 thus Λ = θτ =   0 1 0 1 0 0 0 0 1   which yields ω = Λ -1   θ 1 θ 2 θ 3   =   dξ 1 dξ 2 dζ 1   = d   x 1 -1 2 x 2 3 x 2 x 3  
Finally, one obtains the following diffeomorphism

φ (x) =   x 1 -1 2 x 2 3 x 2 x 3  
which transforms the studied system into the following form

       ξ1 = 0 ξ2 = ξ 1 ζ1 = η(ξ 1 , ξ 2 , ζ) y = ξ 2
Moreover, one has

z = x 2 + 2x 1 x 2 -x 2 x 2 3 = ξ 2 + 2ξ 1 ξ 2
and ∂z ∂ζ1 = 0 which implies that z is Z-observable, and one can use the estimated ξ to recover z in [START_REF] Lopez-M.V | Linearization by completely generalized input ouput injection[END_REF].

Extension to systems with inputs

In this section, we extend our results to systems with inputs in the following form:

   ẋ = f (x) + m k=1 g k (x) u k y = h(x) (2) (8)
where x ∈ R n is the state, u = (u 1 , ..., u m ) T ∈ R m is the inputs, y ∈ R is the output, f : R n → R n , g k : R n → R n and h : R n → R are sufficiently smooth. For system (8), the partial observability normal form is as follows:

           ξ = Aξ + β(y) + m k=1 α 1 k (y)u k ζ = η(ξ, ζ) + m k=1 α 2 k (ξ, ζ)u k y = Cξ (9)
where A, C, β and η are the same as those defined in the form [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF].

Following the same procedure, let define the projection of g k on τ as follows:

G k = G 1 k + G 2 k with G 1 k = r i=1 G 1,i k (x) τ i and G 2 k = n j=r+1
G 2,j k (x) τ j . Then we have the following theorem.

Theorem 3 Suppose that Theorem 1 is satisfied. There exists a diffeomorphism(ξ T , ζ T ) T = φ(x) which transforms [START_REF] Boutat | New algorithm for observer error linearization with a diffeomor-phism on the outputs[END_REF] into the form ( 9) if and only if

τ i , G 1 k = 0 for 1 ≤ i ≤ n, i = r and 1 ≤ k ≤ m.

Proof 4

From Theorem 1, one can state that there exists a diffeomorphism such that

φ * (F 1 ) = A(y)z + β(y) Now, for r + 1 ≤ i ≤ n and 1 ≤ k ≤ m, one has ∂φ * G 1 k ∂ζ i = ∂ ∂ζ i , φ * G 1 k = φ * τ i , G 1 k = 0 It is the same for 1 ≤ i < r such that φ * τ i , G 1 k = 0 Therefore φ * (G 1 k ) = α 1 k (y)
, and finally we proved Theorem 3.

Diffeomorphism on the output

By giving a family of vector fields τ and θ satisfied Property 1, if the conditions of Theorem 1 cannot be fulfilled, i.e. the Lie brackets of vector fields do not commute, then one can modify those vector fields to construct a new family of commutative vector fields, by applying another diffeomorphism on the output (see [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF][START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF]).

For this, let τ 1 be the vector field modulo ∆ ⊥ defined in Section 4. Denote s(y) = 0 a function of the output of (1), and one can construct a new vector field σ 1 according to the following equation:

σ 1 = s(y)τ 1
and by induction define the following new family of vector fields

σ k = [σ k-1 , f ] modulo ∆ ⊥ for 2 ≤ k ≤ r
Thus, one can choose a complementary family of vector fields {σ r+1 , ...,

σ n } such that σ = [σ 1 , σ 2 , • • • , σ n ] forms a basis and θ k (σ j ) = 0 for 1 ≤ k ≤ r, r + 1 ≤ j ≤ n. Note Λ1 =      θ 1 θ 2 . . . θ r      (σ 1 , σ 2 , • • • , σ r ) =      0 • • • 0 s . . . • • • s * 0 • • • * * s • • • * *      With the chosen {σ r+1 , σ r+2 , • • • , σ n }, one can freely choose {θ r+1 , θ r+2 , • • • , θ n } such that Λ2 =      θ r+1 θ r+2 . . . θ n      (σ r+1 , σ r+2 , • • • , σ n ) is of rank n -r.
Property 2 By giving the vector fields (σ 1 , σ 2 , • • • , σ r ) and the codistribution (θ 1 , θ 2 , • • • , θ r ), the chosen complementary σ i and θ i for r + 1 ≤ i ≤ n should satisfy the following properties

1) σ = [σ 1 , τ 2 , • • • , σ n ] forms a basis; 2) θ k (σ j ) = 0 for 1 ≤ k ≤ r and r + 1 ≤ j ≤ n. 3) rank Λ2 = n -r
Then, based on the new basis σ, f can be decomposed as follows:

f = r i=1 F 1,i (x)σ i + n j=r+1 F 2,j (x)σ j (10)
with

F 1 = r i=1 F 1,i (
• [σ i , σ j ] = 0 for all 1 ≤ i ≤ n and 1 ≤ j ≤ n; • [σ i , F 1 ] = 0 for all r + 1 ≤ i ≤ n,
where F 1 is defined in [START_REF] Boutat | Observer error linearization multi-output depending[END_REF].

Proof 5

The proof of this theorem is similar with that of Theorem 1, thus is omitted.

Remark 2

The deduction of such an output function s(y) = 0 in Theorem 4 is exhaustively investigated in [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF].

Remark 3 Following the same arguments in Section 4, the diffeomorphism φ(x) can be calculated by using φ(x) = ω + φ(0) where ω = Λ-1 θ with Λ = θσ.

The following example highlights the proposed result.

Example 2 Let consider the well-known SIR epidemic model that undergoes the spread of a contagious disease as follows:

           Ṡ = -βSI İ = βSI -γI Ṙ = γI y = I z = l(S, I) = N -I -S
where S denotes the suspected population, I denotes the infected, R denotes the removed population and the total population N is assumed to be known. The objective is to apply the proposed result of this paper to estimate the function l(S, I) = N -I -S. By using the same notations as in Section 4, a simple calculation gives: θ 1 = dI and θ 2 = βIdS + (βS -γ) dI which yield the following vector fields:

τ 1 = 1 βI ∂ ∂S and τ 2 = ∂ ∂I + (βS -γ -βI) τ 1
Unfortunately, these two vector fields do not commute, since [τ 1 , τ 2 ] = 2 I τ 1 . In order to construct a new family of commutative vector fields by introducing a diffeomorphism on the output, let follow the method proposed in [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF] to deduce a non-zero output function s(y). For this, set

σ 1 = s(y)τ 1 and σ 2 := [σ 1 , f ] modulo ∆ ⊥ = s(y)τ 2 - s ′ (y) (βSI -γI) τ 1 . Now, a straightforward calculation gives: [σ 1 , σ 2 ] = ( 2s 2 (y) I -2s(y)s ′ (y))τ 1 , thus [σ 1 , σ 2 ] = 0 if
[σ 1 , σ 2 ] = [σ 1 , σ 3 ] = [σ 2 , σ 3 ] = 0
Based on the new basis σ, according to [START_REF] Boutat | Observer error linearization multi-output depending[END_REF], one obtains

f = -βγIτ 1 + (βS -γ) τ 2 + γIτ 3 then F 1 = -βγIτ 1 + (βS -γ) τ 1 . It can be checked that [τ 3 , F 1 ] = 0
and the second item of Theorem 4 is satisfied. Since

Λ = θσ =   0 I 0 I -βI 2 + (-γ + Sβ) I 0 0 0 1   which yields ω = Λ-1   θ 1 θ 2 θ 3   =   dξ 1 dξ 2 dζ 1   = d

Conclusion

This paper deals with partial observability problem, which is useful sometimes when for example only partial states are need to be estimated, or only functions of certain states are required to be estimated. For this, a partial observability normal form is presented, for which a simple Luenberger observer can be applied directly to estimate the partial states. Necessary and sufficient conditions are deduced which guarantee the existence of a diffeomorphism transforming nonlinear systems into the proposed partial observability normal form. The results are extended to nonlinear systems with inputs and the transformation on the output as well. A concrete example was provided to illustrate the feasibility of the proposed results.

F 2 Theorem 4 1 s

 241 x)σ i and F 2 = n j=r+1 ,j (x)σ j . And one can state the following theorem. Given an output function s(y) = 0 which construct a new family of vector fields σ and θ satisfied Property 2, there exists a diffeomorphism (ξ T , ζ T ) T = φ(x) which transforms the dynamical system (1) into the partial observability normal form (3) with ξ r = ȳ = ψ(y) where ψ(y) = y 0 (c) dc. if and only if

  and only if function s(y) fulfils the following differential equation: Thus one can choose s(y) = y = I which yields construct σ and θ satisfying Property 2, one chooses σ 3 = ∂ ∂R and θ 3 = dR which makes

ξ 1 -

 1 βe ȳ -γ ζ1 = γe ȳ ȳ = ξ 2 = ln I In the transformed form, the Z-function becomes z = l(S, I) = l (ξ) = N -ξ 1 β which is independent of ζ 1 , thus it is Z-observable, and one can use the estimated ξ 1 to recover l(S, I). By setting β = 0.001, γ = 0.1, the simulation results are depicted in Fig. 1-2 for the estimation of suspected and infected population.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Estimation of suspected population (S)