G Zheng

D Boutat

J P Barbot

Multi-Output Dependent Observability Normal Form

Keywords: normal form, observer design, linearization, left invertibility

This article gives the sufficient condition which guarantees the existence of a reference frame in which a multi-output nonlinear system is linearizable with a linear part depending on its outputs. Our method is based on the design of a reference frame associated with nonlinear observable systems. Moreover, we give the generalization of the result obtained in [5] and [19]. And some examples are given in order to highlight our thinking.

Introduction

Given a nonlinear system, it is clear that no general approach can be used to design an observer. However, at least there are two ways to try. The first way is to design an observer directly to the given nonlinear system, including Kalman-like observer [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF], adaptive observer [START_REF] Besancon | A viewpoint on observability and observer design for nonlinear systems[END_REF], sliding mode observer [START_REF] Perruquetti | Sliding mode control in engineering[END_REF] and so on. Nevertheless, some extra conditions should be imposed for these direct approaches, such as Lipschitz, persistent excitation. The second method is based on the conception of normal form. A normal form represents a class of equivalent systems which possess the same properties (we focus on the observability in this paper). In other words, the appearance differences among this class of equivalent systems are not intrinsic, but because of the wrong choice of the coordinates.

Obviously one of the keys of this category is to find out a coordinate transformation with which a given system could be converted into the target observ-able normal form. And then the existed nonlinear observer design techniques can be easily applied. In this category, [START_REF] Besancon | A viewpoint on observability and observer design for nonlinear systems[END_REF][START_REF] Besancon | State equivalence based observer design for nonlinear systems[END_REF][START_REF] Besancon | Observer synthesis for a class of nonlinear control systems[END_REF] deal with the problem of transform a nonlinear system into a class of state affine 'normal forms'. In fact, in their works, the sense of normal form is more general since no structure constraints were imposed for its linear part. However, it is also because of this merit, some existed techniques for linear systems cannot be easily applied. And this leads to the raise of linearization problem of nonlinear system.

The linearization problem with output injection was firstly treated in [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] for single output system. And it was generated in multi-output systems in [START_REF] Krener | Nonlinear observer with linearizable error dynamics[END_REF][START_REF] Xia | Nonlinear observer design by observer error linearization[END_REF]. Some other results about input-output injection for the linearization problem were stated in [START_REF] Glumineau | New algebro-geometric conditions for the linearization by inputoutput injection[END_REF][START_REF] Plestan | Linearization by generalized input-output injection[END_REF]. Moreover, [START_REF] Respondek | Time scaling for observer design with linearization erro dynamics[END_REF][START_REF] Sampei | On time scaling for nonlinear systems: application to linearization[END_REF] gave the sufficient and necessary geometrical conditions to transform a nonlinear system into the so-called output-dependent time scaling linear canonical form. In [START_REF] Guay | Observer linearization by output diffeomorphism and outputdependent time-scale transformations[END_REF], the author gave independently the dual geometrical conditions of [START_REF] Respondek | Time scaling for observer design with linearization erro dynamics[END_REF]. Furthermore, the geometrical conditions to guarantee the existence of a local diffeomorphism and an output injection to transform a nonlinear system in a 'canonical' normal form depending on its output was presented [START_REF] Zheng | Output Dependent Observability Linear Normal Form[END_REF] [START_REF] Zheng | Single Output Dependent Observability Normal Form[END_REF], called Single Output Dependent Observability normal form (SODO). Moreover, an extension for this normal form with quadratic terms was also studied in [START_REF] Zheng | Output Depending Quadratic Observability Normal Form[END_REF]. As another natural extension for our previous study, we will extend our result for multi-outputs nonlinear system. Motivated by this interest, this paper focuses on the analysis of nonlinear systems with multiple outputs as follows:

     ẋ = f (x, u) y = (h 1 (x), ..., h m (x)) T (1)
where x ∈ IR n , u ∈ IR p , f : IR n × IR p → IR n and h :IR n → IR m are analytic, and we deal with the following problem: Find sufficient condition for the existence of a local diffeomorphism φ(x) = z to transform system (1) into

     ż = A (y, u) z + β (y, u) y = Cz (2)
where

z =           z 1 z 2 . . . z m           , β (y) =           β 1 (y 1 , u) β 2 (y 1 , y 2 , u) . . . β m (y 1 , ..., y m , u)           , C =           C 1 C 2 . . . C m           and A (y, u) =           A 1 (y 1 , u) 0 • • • 0 0 A 2 (y 1 , y 2 , u) • • • 0 . . . . . . . . . . . . 0 0 • • • A m (y 1 , ..., y m , u)           with z i =           z i,1 z i,2 . . . z i,k i           , β i (y 1 , ..., y i , u) =           β i,1 (y 1 , ..., y i , u) β i,2 (y 1 , ..., y i , u) . . . β i,k i (y 1 , ..., y i , u)           , C i = 0 • • • 0 1 1×k i and A i (y 1 , ..., y i , u) =           0 • • • 0 0 α i,1 (y 1 , ..., y i , u) • • • 0 0 . . . . . . . . . . . . 0 • • • α i,k i -1 (y 1 , ..., y i , u) 0          
with α i,j = 0 for all y and u in a certain studied neighborhood. For dynamical systems in the form 2, the high gain observer proposed in [START_REF] Busawon | A simple observer for a class of nonlinear systems[END_REF] can be applied directly (refer to [START_REF] Busawon | A simple observer for a class of nonlinear systems[END_REF] for more details).

In this paper, we will give the geometrical condition which is sufficient to guarantee the existence of a local diffeomorphism and an output injection to transform system (1) into the normal form (2). This kind of linearization will be named Multi-Output Dependent Observability normal form (MODO normal form) and generalizes the result obtained in [START_REF] Boutat | Observer Error Linearization Multi-Output Depending[END_REF][START_REF] Zheng | Single Output Dependent Observability Normal Form[END_REF] for nonlinear systems with single output. In section 2, we give some basic notations and a preliminary result is presented in Section 3 in order to introduce our method. In section 4, we propose our main result as a generalization of the result given in section 3.

Notations

Consider system (1), with a possible reordering of h i , we assume that there

exist k 1 ≥ k 2 ≥ ... ≥ k m ≥ 1 and m i=1 k i = n such that θ = θ 1 1 , ..., θ k 1 1 , .., θ 1 m , .., θ km m T ( 3
)
where

θ j i = dL j-1 f h i
is a frame of the cotangent bundle T * U. Thus, system (1) is observable. Integers (k i ) 1≤i≤m are called observability indices of system [START_REF] Besancon | State equivalence based observer design for nonlinear systems[END_REF]. For a nice description and more details about this assumption, see [START_REF] Krener | Nonlinear observer with linearizable error dynamics[END_REF]. Obviously the list of these integers is generally not unique. For 1 ≤ l ≤ m and 1 ≤ j ≤ k l , let (τ i,1 ) 1≤i≤m be the family of vector fields defined by:

θ j l (τ i,1 ) =      1, l = i, j = k i 0, otherwise (4)
and construct by induction the following family of vector fields:

τ i,r = [τ i,r-1 , f ] = (-1) r-1 ad r-1 f τ i,r-1 , for 2 ≤ r ≤ k i . ( 5
)
The family τ = (τ i,j ) 1≤i≤m and 1≤j≤k i is a basis of the tangent bundle T U. The frame T was addressed firstly in [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] for p = 0 and m = 1 and it is well-known in [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] that system (1) can be transformed into the normal form (2) if and only if we have

[τ 1,i , τ 1,j ] = 0, for 1 ≤ i, j ≤ n. ( 6
)
In this case, τ is a frame with which system (1) is in the form (2).

Preliminary results

In this section, we will generalize the result stated in [START_REF] Boutat | Observer Error Linearization Multi-Output Depending[END_REF] and [START_REF] Zheng | Single Output Dependent Observability Normal Form[END_REF] to systems with multi-output.

Lemma 1 For a system in the form [START_REF] Besancon | Observer synthesis for a class of nonlinear control systems[END_REF] we have for

1 ≤ l ≤ m, 1 ≤ t ≤ k l , τ l,t = 1 π l,t ∂ ∂z l,t +     l r=1
A l,t r,t-1 (y 1 , ..., y l , u) z r,kr-1 +η l,t t-1 (y 1 , ..., y l , u)

    ∂ ∂z l,t-1 + t-2 i=1     l r=1     A l,t r,i (y 1 , ..., y l , u) z r,kr-t+i + kr-1 j=kr-t+i+1 kr-1 s=j T l,t r,j,s (y 1 , ..., y l , u) z r,j z r,s         ∂ ∂z l,i + t-2 i=1         l r=1 kr j=kr-t+i+1 η l,t r,i (y 1 , ..., y l , u) z r,j +O [3] (y 1 ,...,y l ,u)    z 1,k 1 -t+i+1 , .., z 1,k 1 -1 , ...., z l,k l -t+i+1 , .., z l,k l -1            ∂ ∂z l,i , ( 7
)
where π l,k l = 1 and π l,t-1 = π l,t α l,t-1 for 2 ≤ t ≤ k l , η l,t i and T l,t r,j,q are some smooth functions of (y 1 , ..., y l , u), O [START_REF] Besancon | A viewpoint on observability and observer design for nonlinear systems[END_REF] (y 1 ,...,y l ,u)

   z 1,k 1 -t+i+1 , .., z 1,k 1 -1 , ...., z l,k l -t+i+1 , .., z l,k l -1    repre-
sents the residue higher than order 2 with coefficient which is function of (y 1 , ..., y l , u) and

A l,t r,i (y 1 , ..., y l , u) = (-1) t-i+1 S l,t r,t-i,1 ∂y r π l,i π 2 l,i π r,kr-t+i + (-1) t-i+1 t-1 m=t-i+1 S l,t r,t-i,m-t+i+1 ∂y r π l,t-m π 2 l,t-m m j=t-i+1 α l,k-j π r,kr-t+i , ( 8
)
where

∂ yr π l,i represents ∂π l,i (y 1 ,...,y l ,u) ∂yr
, S l,t r,t-i,1 and S l,t r,t-i,m-t+i+1 are defined as follows

S l,t r,j,1 = 1, S l,t r,j,s = S l,t-1 r,j-1,s + S l,t-1 r,j,s-1 , ( 9
)
for 1 ≤ l ≤ m, 1 ≤ r ≤ l, 2 ≤ t ≤ k l , 1 ≤ j ≤ t -1 and 1 ≤ s ≤ t -j.
Proof 1 For a system in the (2) form, for

1 ≤ l ≤ m, τ l,1 = 1 π l,1 ∂ ∂z l,1
, then we use equation ( 5) to obtain

τ 1,t = 1 π 1,t ∂ ∂z 1,t + A 1,t 1,t-1 (y 1 , u) z 1,k 1 -1 + η 1,t t-1 (y 1 , u) ∂ ∂z 1,t-1 + t-2 i=1 A 1,t 1,i (y 1 , u) z 1,k 1 -t+i + k 1 -1 j=k 1 -t+i+1 k 1 -1 l=j T 1,t j,l (y 1 , u) z 1,j z 1,l ∂ ∂z 1,i + t-2 i=1 k 1 j=k 1 -t+i+1 η 1,t i (y 1 , u) z 1,j + O [3] (y 1 ,u) (z 1,k 1 -t+i+1 , .., z 1,k 1 -1 ) ∂ ∂z 1,i , for 1 ≤ t ≤ k 1 and τ 2,t = 1 π 2,t ∂ ∂z 2,t +    A 2,t 1,t-1 (y 1 , y 2 , u) z 1,k 1 -1 + A 2,t 2,t-1 (y 1 , y 2 , u) z 2,k 2 -1 +η 2,t t-1 (y 1 , y 2 , u)    ∂ ∂z 2,t-1 + t-2 i=1             A 2,t 1,i (y 1 , y 2 , u) z 1,k 1 -t+i + k 2 -1 j=k 2 -t+i+1 k 2 -1 l=j T 2,t 1,j,l (y 1 , y 2 , u) z 1,j z 1,l + A 2,t 2,i (y 1 , y 2 , u) z 2,k 2 -t+i + k 2 -1 j=k 2 -t+i+1 k 2 -1 l=j T 2,t 2,j,l (y 1 , y 2 , u) z 2,j z 2,l             ∂ ∂z 2,i + t-2 i=1     k 1 j=k 1 -t+i+1 η 2,t 1,i (y 1 , y 2 , u) z 1,j + k 2 j=k 2 -t+i+1 η 2,t 2,i (y 1 , y 2 , u) z 2,j +O [3] (y 1 ,y 2 ,u) (z 1,k 1 -t+i+1 , .., z 1,k 1 -1 , z 2,k 2 -t+i+1 , .., z 2,k 2 -1 )     ∂ ∂z 2,i , for 1 ≤ t ≤ k 2 .
Then by an induction, for 1 ≤ t ≤ k l , we get

τ l,t = 1 π l,t ∂ ∂z l,t +     l r=1 A l,t r,t-1 (y 1 , ..., y l , u) z r,kr-1 +η l,t t-1 (y 1 , ..., y l , u)     ∂ ∂z l,t-1 + t-2 i=1     l r=1     A l,t r,i (y 1 , ..., y l , u) z r,kr-t+i + kr-1 j=kr-t+i+1 kr-1 s=j T l,t r,j,s (y 1 , ..., y l , u) z r,j z r,s         ∂ ∂z l,i + t-2 i=1         l r=1 kr j=kr-t+i+1 η l,t r,i (y 1 , ..., y l , u) z r,j +O [3] (y 1 ,...,y l ,u)    z 1,k 1 -t+i+1 , .., z 1,k 1 -1 , ...., z l,k l -t+i+1 , .., z l,k l -1            ∂ ∂z l,i ,
where

A l,t r,i (y 1 , ..., y l , u) = (-1) t-i+1 S l,t r,t-i,1 ∂ yr π l,i π 2 l,i + t-1 m=t-i+1 S l,t r,t-i,m-t+i+1 ∂ yr π l,t-m π 2 l,t-m   m j=t-i+1 α l,k-j     π r,kr-t+i ,
with the coefficients S l,t r,j,s defined by 9.

In order to determine the α i,j (y 1 , ..., y i , u) for 1

≤ i ≤ m, 1 ≤ j ≤ k i -1, we also impose that for 1 ≤ i, l ≤ m, ∂ ∂z i,j h l • φ =      1, if l = i and j = k i , 0, otherwise.
Now, we are ready to state a set of partial differential equations which enables us to compute functions α i,j (y 1 , ..., y i , u) for 1

≤ i ≤ m, 1 ≤ j ≤ k i -1.
Proposition 1 If there exists a diffeomorphism which transforms system (1) into form ( 2) then for

1 ≤ l ≤ m, 1 ≤ t ≤ k l -1, 1 ≤ s ≤ l -1,      [τ l,t , τ s,ks ] = λ s l,t τ l,t mod span {τ l,1 , ..., τ l,t-1 } [τ l,t , τ l,k l ] = λ l l,t τ l,t + G [1] l,k l ,t + R l,t
where

G [1] l,k l ,t = t-1 i=1 1 π l,t T l,t l,t,k l -t+i z l,k l -t+i ∂ ∂z l,i + 1 π l,t T l,t l,t,t z l,t ∂ ∂z l,2t-k l ,
and

R l,t = t-1 i=1         k l j=k l -t+i+1 ηl,t i (y 1 , ..., y l , u) +O [2] (y 1 ,...,y l ,u)    z 1,k 1 -t+i+1 , .., z 1,k 1 -1 , ...., z l,k l -t+i+1 , .., z l,k l -1            ∂ ∂z l,i and      λ s l,t = - ∂y s π l,t π l,t , λ l l,t = diag{δ l,
δ l,t i = A l,k l l,i -A l,k l l,k l -t+i - ∂ y l A l,t l,i A l,t l,i
for 1 ≤ i ≤ t -1, and A l,t l,i is given in [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF].

Proof 2 According to equation [START_REF] Guay | Observer linearization by output diffeomorphism and outputdependent time-scale transformations[END_REF], for

1 ≤ l ≤ m, 1 ≤ t ≤ k l -1, we have τ l,t = 1 π l,t ∂ ∂z l,t +     l r=1 A l,t r,t-1 (y 1 , ..., y l , u) z r,kr-1 +η l,t t-1 (y 1 , ..., y l , u)     ∂ ∂z l,t-1 + t-2 i=1     l r=1     A l,t r,i (y 1 , ..., y l , u) z r,kr-t+i + kr-1 j=kr-t+i+1 kr-1 s=j T l,t r,j,s (y 1 , ..., y l , u) z r,j z r,s         ∂ ∂z l,i + t-2 i=1         l r=1 kr j=kr-t+i+1 η l,t r,i (y 1 , ..., y l , u) z r,j +O [3] (y 1 ,...,y l ,u)    z 1,k 1 -t+i+1 , .., z 1,k 1 -1 , ...., z l,k l -t+i+1 , .., z l,k l -1            ∂ ∂z l,i ,
[τ l,t , τ l,k l ] = A l,k l l,t + ∂ y l π l,t π l,t 1 π l,t ∂ ∂z l,t + t-1 i=1     A l,k l l,i -A l,k l l,k l -t+i - ∂y l (A l,t l,i ) A l,t l,i A l,t l,i z l,k l -t+i + 1 π l,t T l,t l,t,k l -t+i z l,k l -t+i     ∂ ∂z l,i + 1 π l,t T l,t l,t,t z l,t ∂ ∂z l,2t-k l + t-1 i=1         k l j=k l -t+i+1 ηl,t i (y 1 , ..., y l , u) +O [2] (y 1 ,...,y l ,u)    z 1,k 1 -t+i+1 , .., z 1,k 1 -1 , ...., z l,k l -t+i+1 , .., z l,k l -1            ∂ ∂z l,i .
Set λ l l,t (y 1 , ..., y l , u) = diag{δ l,t 1 (y 1 , ..., y l , u) , ..., δ l,t i (y 1 , ..., y l , u) , ..., δ l,t t (y 1 , ..., y l , u) , ..., 0, ..., 0}, where δ l,t t = A l,k l l,t + ∂y l π l,t π l,t and

δ l,t i = A l,k l l,i -A l,k l l,k l -t+i - ∂ y l A l,t l,i A l,t l,i for 1 ≤ i ≤ t -1, then [τ l,t , τ l,k l ] = λ l l,t τ l,t + G [1] l,k l ,t + R l,t . ( 11
)
Remark 1 In equation [START_REF] Krener | Nonlinear observer with linearizable error dynamics[END_REF], λ l l,t could be uniquely determined since

G [1] l,k l ,t
might be separated according to the coefficients of second-order terms in τ l,k l .

Finally, the following result enables us to determine all the functions α l,i (y 1 , ..., y l , u)

for 1 ≤ l ≤ m and 1 ≤ i ≤ k l -1.
Proposition 2 If there exists a diffeomorphism which transforms system (1) into form (2), then α l,i = π l,i π l,i+1 for 1 ≤ l ≤ m and 1 ≤ i ≤ k l -2, and α l,k l -1 = π l,k l -1 , where the π l,i for 1 ≤ i ≤ k l -1 is the solution of the following partial differential equations:

             ∂y s π l,i π l,i = -λ s l,i (y 1 , ..., y l ) , f or 1 ≤ s ≤ l -1 and 1 ≤ i ≤ k l -1 ∂y l π l,i π l,i = exp δ l,i i -δ l,k l -1 i -δ l,i+1 i+1 dy l -Bl,k l -1 l,i , f or 1 ≤ i ≤ k l -2 ∂y l π l,k l -1 π l,k l -1 = δ l,k l -1 k l -1 - Āl,k l l,k l -1 2 (12)
with Bl,0 l,1 = 0 and for

1 ≤ i, t ≤ k l -1 Bl,t l,i = t-1 m=t-i+1 S l,t t-i,m-t+i+1 ∂ y l π l,t-m π l,t-m . ( 13
)
Proof 3 Obviously, according to [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF], we have

∂ ys π l,i π l,i = -λ s l,i (y 1 , ..., y l , u) , f or 1 ≤ s ≤ l -1 and 1 ≤ i ≤ k l -1 Define B l,t l,i = ∂ y l π l,i π l,i + Bl,t l,i . ( 14
)
where 1 ≤ l ≤ m and 1 ≤ i, t ≤ k l -1.

According to [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF], for

1 ≤ i, k ≤ k l -1, ∂ y l A l,t l,i A l,t l,i = ∂ y l B l,t l,i B l,t l,i - ∂ y l π l,i π l,i + ∂ y l π l,k l -t+i π l,k l -t+i . As δ l,t t = A l,k l l,t + ∂y l π l,t π l,t , hence δ l,k l -1 i = A l,k l l,i -A l,k l l,1+i - ∂ y l B l,k l -1 l,i B l,k l -1 l,i + ∂ y l π l,i π l,i - ∂ y l π l,1+i π l,1+i = δ l,i i -δ l,1+i 1+i -∂ y l ∂ y l π l,i π l,i + Bl,k l -1 l,i / ∂ y l π l,i π l,i + Bl,k l -1 l,i
. which yields

∂ y l π l,i π l,i = exp δ l,i i -δ l,k l -1 i -δ l,i+1 i+1 dy l -Bl,k l -1 l,i , f or 1 ≤ i ≤ k l -2
where Bl,k l -1

l,i
is defined in [START_REF] Perruquetti | Sliding mode control in engineering[END_REF] and c l,i = 0. As δ l,k l -1 k l -1 = 2

∂y l π l,k l -1 π l,k l -1 + Āl,k l l,k l -1 , where Āl,k l l,k l -1 = k l -1 m=2 S l,k l 1,m ∂y l π l,k l -m
π l,k l -m , then we obtain

∂ y l π l,k l -1 π l,k l -1 = δ l,k l -1 k l -1 -Āl,k l l,k l -1 2 4 Main result
If there exists a diffeomorphism which transforms system (1) into form (2), then equation ( 12) of Proposition 2 gives all α l,i (y 1 , ..., y l , u) for 1 ≤ l ≤ m and 1 ≤ i ≤ k l -1. Therefore, let us consider a new family of vector fields defined as follows:

τ l,1 = π l,1 (y 1 , ..., y l , u) τ l,1 (15)
τ l,i = 1 α l,i-1 [τ l,i-1 , f ] for i = 2 : k l where π l,1 (y 1 , ..., y l , u) = k l -1 i=1
α l,i (y 1 , ..., y l , u) . Set τ = ( τ i,j ) 1≤i≤m and 1≤j≤k i and Λ = θ τ , we can define the following multi 1-form:

ω = Λ -1 θ. ( 16
)
It is clear that

ω τ = I n×n
Then we are ready to state our main result.

Theorem 1 There exists a diffeomorphism which transforms system (1) into a MODO normal form (2) if i) there exists a family of functions α l,i (y 1 , ..., y l , u) for 1 ≤ l ≤ m and 1 ≤ i ≤ k l -1 such that the family of vector fields τ l,i for 1 ≤ l ≤ m and 1 ≤ i ≤ k l -1 defined in [START_REF] Respondek | Time scaling for observer design with linearization erro dynamics[END_REF] satisfies the following commutativity conditions

[ τ i,j , τ s,l ] = 0, f or 1 ≤ i, s ≤ m, 1 ≤ j ≤ k i and 1 ≤ l ≤ k s ( 17
)
or ii) there exists a family of functions α l,i (y 1 , ..., y l , u) for 1 ≤ l ≤ m and 1 ≤ i ≤ k l -1 such that the R n -valued form ω defined in [START_REF] Sampei | On time scaling for nonlinear systems: application to linearization[END_REF] satisfies the following condition

dω = 0. ( 18
)
Proof 4 Assume that there exist α l,i (y 1 , ..., y l , u) > 0 for 1 ≤ l ≤ m and

1 ≤ i ≤ k l -1 such that [ τ i,j , τ s,l ] = 0 for 1 ≤ i, s ≤ m, 1 ≤ j ≤ k i and 1 ≤ l ≤ k s ,
then it is well-known ( [START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF], [START_REF] Isidori | Nonlinear control systems[END_REF]) that we can find a local diffeomorphism φ = z such that

φ * ( τ l,i ) = ∂ ∂z l,i . As φ * ( τ l,i ) = ∂ ∂z l,i is constant, hence ∂ ∂z l,i φ * (f ) = φ * ([ τ l,i , f ]) = α l,i φ * ( τ l,i+1 ) = α l,i ∂ ∂z l,i+1 , thus ∂ ∂z l,i φ * (f ) = α l,i ∂ ∂z l,i+1 for 1 ≤ l ≤ m and 1 ≤ i ≤ k l -1.
Consequently, by integration we obtain: φ * (f ) = A(y, u)z + β(y, u). Moreover, as dh • τ l,i = 0 for 1 ≤ l ≤ m and 1 ≤ i ≤ k l -1 and dh • τ l,k l -1 = 1, we obtain h • φ = Cz where C is defined in [START_REF] Besancon | State equivalence based observer design for nonlinear systems[END_REF].

Finally, in order to prove that in Theorem 1 Condition i) is equivalent to Condition ii), it is sufficient to prove that equation ( 17) is equivalent to equation [START_REF] Zheng | Output Dependent Observability Linear Normal Form[END_REF]. Recall that for any two vector fields X, Y, we have

dω(X, Y ) = L X (ω(Y )) -L Y (ω(X)) -ω([X, Y ]).
Setting X = τ i.j and Y = τ s.l , we obtain

dω( τ i,j , τ s,l ) = L τ i,j ω( τ s,l ) -L τ s,l ω( τ i,j ) -ω([ τ i,j , τ s,l ]).
As ω( τ s,l ) and ω( τ i,j ) are constant, then we have

dω( τ i,j , τ s,l ) = -ω([ τ i,j , τ s,l ]).
Because ω is an isomorphism and ( τ i,j ) 1≤i≤m and 1≤j≤k l -1 is a basis of T U, then equation ( 17) is equivalent to equation [START_REF] Zheng | Output Dependent Observability Linear Normal Form[END_REF].

Example 1 Let us consider the following system:

                                         ẋ1,1 = γ(y 1 ,u) 1+x 3 x 1,1 x 1,2 , ẋ1,2 = µ(y 1 ,u) 1+x 1,3 x 1,1 , ẋ1,3 = γ (y 1 , u) x 1,2 , ẋ2,1 = β (y 1 , y 2 , u) x 2 2,1 (1+x 2,2 ) 2 ẋ2,2 = β (y 1 , y 2 , u) x 2,1 (1+x 2,2 ) y 1 = x 1,3 , y 2 = x 2,2 , ( 19
)
with y 1 ∈ ]-1, 1[ , y 2 ∈ ]-1, 1[ and γµβ = 0 for any u, y 1 and y 2 , then we can obtain

                                         τ 1,1 = 1+x 1,3 γµ ∂ ∂x 1,1 , τ 1,2 = 1 γ(y 1 ,u) ∂ ∂x 1,2 + (1 + x 1,3 ) ∂y 1 (γ(y 1 ,u)µ(y 1 ,u)) γ(y 1 ,u)µ 2 (y 1 ,u) x 1,2 ∂ ∂x 1,1 , τ 1,3 = ∂ ∂x 1,3 + ∂y 1 (µ(y 1 ,u)γ(y 1 ,u)) (µ(y 1 ,u)γ(y 1 ,u)) + ∂y 1 γ(y 1 ,u) γ(y 1 ,u) x 1,2 ∂ ∂x 1,2 + 1 1+x 1,3 - ∂y 1 (γ(y 1 ,u)µ(y 1 ,u)) γ(y 1 ,u)µ(y 1 ,u) x 1,1 ∂ ∂x 1,1 τ 2,1 = (1+x 2,2 ) β(y 1 ,y 2 ,u) ∂ ∂x 2,1 τ 2,2 = ∂ ∂x 2,2 +    2 x 2,1 (1+x 2,2 ) -∂ y 1 (1+x 2,2 ) β(y 1 ,y 2 ,u) γ (y 1 , u) x 2,2 -∂ y 2 (1+x 2,2 ) β(y 1 ,y 2 ,u) β (y 1 , y 2 , u) x 2,1 (1+x 2,2 )    ∂ ∂x 2,1 which yields                                  δ 1 1,1 = 0, δ 1 2,2 = 2 ∂y 1 γ(y 1 ,u) γ(y 1 ,u) + ∂y 1 (µ(y 1 ,u)γ(y 1 ,u)) (µ(y 1 ,u)γ(y 1 ,u)) , δ 1 2,1 = - ∂y 1 (µ(y 1 ,u)γ(y 1 ,u)) (µ(y 1 ,u)γ(y 1 ,u)) -2 ∂y 1 γ(y 1 ,u) γ(y 1 ,u) -∂ y 1 ∂y 1 (µ(y 1 ,u)γ(y 1 ,u)) (µ(y 1 ,u)γ(y 1 ,u)) / ∂y 1 (µ(y 1 ,u)γ(y 1 ,u)) (µ(y 1 ,u)γ(y 1 ,u)) , λ 1 1,1 = - ∂y 1 β(y 1 ,y 2 ,u) β(y 1 ,y 2 ,u) δ 2 1,1 = 2 ∂y 2 β(y 1 ,y 2 ,u) β(y 1 ,y 2 ,u) ,
Then, according to [START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF], we have

                   ∂y 1 π 11 π 11 = exp δ 1 1,1 -δ 1 1,2 -δ 1 2,2 dy 1 , ∂y 2 π 12 π 12 = 1 2 δ 1 2,2 - ∂y 1 π 11 π 11 ∂y 1 π 21 π 21 = -λ 1 1,1 = ∂y 1 β(y 1 ,y 2 ,u) β(y 1 ,y 2 ,u) , ∂y 2 π 21 π 21 = 1 2 δ 2 2,1 = ∂y 2 β(y 1 ,y 2 ,u) β(y 1 ,y 2 ,u) which gives us π 11 = c 1 γ (y 1 , u) µ (y 1 , u), π 12 = c 2 γ (y 1 , u) and π 2,1 = α 2,1 = c 3 β (y 1 , y 2 , u) , then we can obtain α 1,1 (y 1 , u) = c 1 c 2 µ (y 1 , u) and α 1,2 = c 2 γ (y 1 , u). Then we have              τ 1,1 = c 1 (1 + x 1,3 ) ∂ ∂x 1,1 , τ 1,2 = c 2 ∂ ∂x 1,2 , τ 1,3 = ∂ ∂x 1,3 + x 1,1 1+x 1,3 ∂ ∂x 1,1 .
and

     τ2,1 = c 3 (1 + x 2,2 ) ∂ ∂x 2,1 τ2,2 = ∂ ∂x 2,2 + x 2,1 1+x 2,2 ∂ ∂x 2,1 It is obvious [τ ij , τst ] = 0 for 1 ≤ i, s ≤ 2, j ∈ [1, 3], t ∈ [1, 2] . So we have ω = Λ -1 τ =               1 c 1 (1+x 1,3 ) 0 - 1 c 1 (1+x 1,3 ) 2 x 1,1 0 0 0 1 c 2 0 0 0 0 0 1 0 0 0 0 0 1 c 3 (1+x 2,2 ) -x 2,1 c 3 (1+x 2,2 ) 2 0 0 0 0 1               which yields z = φ (x) = x 1,1 c 1 (1 + x 1,3 ) , 1 c 2 x 1,2 , x 1,3 , x 2,1 c 3 (1 + x 2,2 ) , x 2,2
T with this diffeomorphism, system ( 19) is transformed into:

                                         ż1,1 = 0, ż1,2 = c 1 c 2 µ (y 1 , u) z 1,1 , ż1,3 = c 2 γ (y 1 , u) z 1,2 , ż2,1 = 0 ż2,2 = c 3 β (y 1 , y 2 , u) z 2,1 y 1 = z 1,3 , y 2 = z 2,2 .

Extension to system with unknown inputs

In this section, we extend our results to a system with unknown inputs and which is in following form:

     ẋ = f (x, u) + g(x, u, w) y = (h 1 (x), ..., h m (x)) T ( 20
)
where

x ∈ R n , u ∈ R p , w ∈ R q , f : R n × R p → R n , g : R n × R p × R q → R n
and h : R n → R m are analytic functions. For system [START_REF] Zheng | Output Depending Quadratic Observability Normal Form[END_REF] we seek the MODO normal form along its output trajectory as follows:

     ż = A (y, u) z + β (y, u) + η (y, u, w) y = Cz (21)
where A (y, u) , β (y, u) and C are defined in [START_REF] Besancon | Observer synthesis for a class of nonlinear control systems[END_REF], and the matrix η (y, u, w) is defined as the matrix β (y, u).

Theorem 2 System ( 20) can be transformed into MODO normal form ( 21) by a diffeomorphism and an output injection if and only if there is a family functions (α i,j (y, u)) 1≤i≤m and 1≤j≤k i -1 , satisfies both commutativity conditions:

i) For 1 ≤ i, s ≤ m, 1 ≤ j ≤ k i and 1 ≤ l ≤ k s , [ τ i,j , τ s,l ] = 0. ( 22
)
ii)

[g, τ i,j ] = 0 for 1 ≤ i ≤ m, 1 ≤ j ≤ k i -1
Proof 5 From Theorem 1, we can state that there exists a diffeomorphism φ such that φ * (f ) = A(y, u)z + β(y, u).

For 1 ≤ i ≤ m and 1 ≤ j ≤ k l -1, because φ * ( τ i,j ) = ∂ ∂z i,j is constant, hence we have

∂ ∂z i,j φ * (g) = φ * ([g, τ i,j ]) = 0.
Therefore φ * (g) = η(y, u, w). Thus, we obtain the form (21).

Remark 2 If g(x, u, w) = g 1 (x, u)w 1 +..+g q (x, u)w q , and also both conditions i) and ii) of Theorem 2 are fulfilled, then:

η(y, u, w) = B 1 (y, u)w 1 + .. + B q (y, u)w q (23)
Let us now study some special cases of the output injection.

Corollary 1 Let us assume that conditions i) and ii) of theorem2 are fulfilled:

a) if [g, τ i,k i ] = 0, for 1 ≤ i ≤ m, then: η(y, u, w) = η(u, w) b) if g(x, u, w) = g 1 (x, u)u 1 + .. + g q (x, u)u q and [g k , τ i,k i ] = 0 for 1 ≤ i ≤ m, 1 ≤ k ≤ p then η(y, u, w) = B 1 (u)w 1 + .. + B q (u)w q .
where B i are functions of u. c) if g(x, w) = g 1 (x)w 1 + .. + g q (x)w q and

[g k , τ i,k i ] = 0 for 1 ≤ i ≤ m, 1 ≤ k ≤ q then η(y, u, w) = B 1 w 1 + .. + B q w p
where B i are constant vector fields.

Example 2 Let us consider the following system

                           ẋ1,1 = γ (y 1 , u) x 2 1,1 (1+x 1,2 ) 2 + x 1,1 (1+x 1,2 ) w 1 ẋ1,2 = γ (y 1 , u) x 1,1 1+x 1,2 + w 1 + x 2,1 w 2 ẋ2,1 = β(y 1 , y 2 , u)x 1,2 y 1 = x 1,2 y 2 = x 2,1 (24)
with y 1 ∈ ]-1, 1[ , γβ = 0 for any u, y 1 and y 2 , then we can obtain

             τ 1,1 = 1+x 1,2 γ(y 1 ,u) ∂ ∂x 1,1 τ 1,2 = ∂ ∂x 1,2 + 3γ 2 (y 1 , u) x 1,1 (1+x 1,2 ) 3 -γ (y 1 , u) ∂ y 1 (γ (y 1 , u)) x 1,1 (1+x 1,2 ) 2 ∂ ∂x 1,1 τ 2,1 = ∂ ∂x 2,1 which yields δ 1,1 1 = 2 ∂ y 1 γ (y 1 , u) γ (y 1 , u)
so according to equation [START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF], we obtain 24) can be putted into the following form:

∂ y 1 π 1,1 π 1,1 = δ 1,1 1 2 = ∂ y 1 γ (y 1 , u) γ (y 1 , u) which gives π 1,1 = α 1,1 = c 1 γ (y 1 , u) . Consequently, we have              τ 1,1 = c 1 (1 + x 1,2 ) ∂ ∂x 1,1 τ 1,2 = ∂ ∂x 1,2 + x 1,1 1+x 1,2 ∂ ∂x 2,1 τ 2,1 = ∂ ∂x 2,1 As g 1 = ∂ ∂x 1,2 + x 1,1 1+x 1,2 ∂ ∂x 2,1 = τ 1,2 , so [g 1 , τ 1,1 ] = [g 1 , τ 1,2 ] = [g 1 , τ 2,1 ] = 0. Moreover, due to g 2 = x 2,1 ∂ ∂x 1,2 , then we have [g 2 , τ 1,1 ] = 0, [g 2 , τ 1,2 ] = 0 and [g 2 , τ 2,1 ] = 0. Thus system (
                           ż1,1 = -x 1,1 c 1 (1+x 1,2 ) 2 x 2,1 w 2 ż1,2 = c 1 γ(y 1 , u)z 1,1 + w 1 + z 2,1 w 2 ż2,1 = β(y 1 , y 2 , u)z 1,2 y 1 = z 1,2 y 2 = z 2,1 (25)
by the following diffeomorphism

φ =        x 1,1 c 1 (1+x 1,2 ) x 1,2 x 2,1       
Corollary 2 Assume conditions i) and ii) of theorem 2 are fulfilled and m ≥ q, then the OMC (Observability Matching Condition) for system (21) is as follows:

rank ∂ ∂w η i,1 (y, u, w) = q and ∂ ∂w η i,j (y, u, w) = 0, for 1 ≤ i ≤ m, 2 ≤ j ≤ k i
Here we give another example to highlight the above corollary.

Example 3 Consider the following system:

                           ẋ1,1 = γ (y 1 , u) x 2 1,1 (1+x 1,2 ) 2 + γ (y 1 , u) w 1 + w 2 ẋ1,2 = γ (y 1 , u) x 1,1 1+x 1,2 ẋ2,1 = β(y 1 , y 2 , u)x 1,2 + w 1 + β(y 1 , y 2 , u)w 2 y 1 = x 1,2 y 2 = x 2,1 (26)
with y 1 ∈ ]-1, 1[ , βγ = 0 for any u, y 1 and y 2 . According to example (24) we can get π 1,1 = α 1,1 = c 1 γ (y 1 , u) which yields the following new vector fields:

             τ 1,1 = c 1 (1 + x 1,2 ) ∂ ∂x 1,1 τ 1,2 = ∂ ∂x 1,2 + x 1,1 1+x 1,2 ∂ ∂x 2,1 τ 2,1 = ∂ ∂x 2,1
which means the diffeomorphism is as follows:

φ =        x 1,1 c 1 (1+x 1,2 ) x 1,2 x 2,1       
then system (26) can be putted into the following form: This article focused on seeking the MODO normal form. Firstly, the partial differential equations were given to deduce all functions α i,j , with which we can design a new frame of vector fields from the natural one associated with a dynamical system with outputs. And then sufficient geometrical condition was given in order to determine whether a system (1) can be transformed into the normal form [START_REF] Besancon | Observer synthesis for a class of nonlinear control systems[END_REF]. Finally the obtained result was extended for systems with unknown inputs. Nevertheless, one open problem remained to be solved is to find the necessary condition.

                           ż1,1 =

1 c 1 2 y 1 = z 1, 2 y 2 = z 2 1 c 1 ( 1 c 1 ( 1 κ 1 1 c 1 (E 2 κz 1 , 2

 112122211111111212 (1+x 1,2 ) (γ (y 1 , u) w 1 + w 2 ) ż1,2 = c 1 γ(y 1 , u)z 1,1 ż2,1 = β(y 1 , y 2 , u)z 1,2 + w 1 + β(y 1 , y 2 , u)w 1+y 1 ) (γ (y 1 , u) w 1 + w 2 ) 0 w 1 + β(y 1 , y 2 , u)w 2 1+y 1 ) γ (y 1 , u) or 1 ≤ i ≤ 2 and ∂ ∂w η 1,2 (y, u, w) = 0, if γ (y 1 , u) β(y 1 , y 2 , u) = 1 and c 1 (1 + y 1 ) = 0, then Corollary 2 is fulfilled, therefore both the state and the unknown input of system (26) can be recovered by an observer. Now, let us design an observer for system (27). We assume that u is bounded and γ (y 1 , u) β(y 1 , y 2 , u) = 1 and c 1 (1 + y 1 ) = 0, then the left invertibility problem for (27) may be solved by a step by step sliding mode observer[START_REF] Perruquetti | Sliding mode control in engineering[END_REF] of the following form:ż1,1 = E 1 κ 2 sign (z 1,1 -ẑ1,1 ) ż1,2 = c 1 γ(y 1 , u)ẑ 1,1 + κ 1 sign (z 1,2 -ẑ1,2 ) ż2,1 = β(y 1 , y 2 , u)ẑ 1,2 + κ 3 sign (z 2,1 -ẑ2,1 )wherez 1,2 = y 1 , z 2,1 = y 2 and z1,1 = ẑ1,1 + E sign(z 1,2 -ẑ 1,2 ) c 1 γ(y 1 ,u) 1+y 1 ) γ (y 1 , u) 2 sign (z 1,1 -ẑ1,1 ) E 2 κ 3 sign (z 2,1 -ẑ2,1 ) = ẑ1,2 , E 1 = 1, else E 1 = 0 if z 2,1 = ẑ2,1 and z1,1 = ẑ1,1 , then E 2 = 1, else E 2 = 0 6 Conclusion


G Zheng
email: gang.zheng@inria.fr

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

L Boutat-Baddas
email: latifa.boutat-baddas@iut-longwy.uhp-nancy.fr

T Floquet

J P Barbot
email: barbot@ensea.fr

A Multiple Data Secure Communication Based on Synchronization of Chaotic Systems with Zero Dynamics

 for nonlinear case a new observability matching condition are given, which relaxed again the previously mentioned conditions. Moreover, in the last reference the authors give conditions in order to recover the unknown

I. INTRODUCTION

Since chaotic system is extremely sensitive to initial conditions and parameters variations, its application to secure communication has provoked a great deal of interest, especially after Carroll and Pecora's outstanding work on successfully synchronizing two chaotic systems [START_REF] Pecora | Synchronization in chaotic systems[END_REF].

Up to now, most chaos-based communication systems are based on chaos synchronization technique. Actually, this synchronization can be studied as an observer design problem, since the work of H. Nijmeijer and I. Mareels in [START_REF] Nijmeijer | An observer looks at synchronization[END_REF].

In secure communication system, we need to estimate not only the state for chaos synchronization but also the input (confidential message), which can be considered as to design an unknown inputs observer with unknown inputs estimation. For the linear systems with unknown inputs and without input estimation, in [START_REF] Hautus | Strong detectability and observers[END_REF], [START_REF] Hou | Disturbance decoupled observer design: A unified viewpoint[END_REF], the necessary and sufficient conditions for the existence of a classical observer are presented. Moreover, an algorithms of full order observers and conditions for their existence are stated in [START_REF] Darouach | Full-order Observers for linear systems with unknown inputs[END_REF]. In the nonlinear case, at our knowledge, one of the first design of unknown inputs observer was given in [START_REF] Xiong | Sliding mode observer for nonlinear uncertain systems[END_REF] inputs for systems without zero dynamics. This problem can be see as a Left Invertibility Problem [START_REF] Singh | A modified algorithm for invertibility in nonlinear system[END_REF].

In this paper, we propose to apply the results of [START_REF] Barbot | A new observation algorithm for nonlinear system with unknown inputs[END_REF] taking into account the stability of the zero dynamics into multi data secure communication scheme.

This paper is organized as follows: Notations and problem statement are presented in section 2. In section 3, a new scheme of transmitter based on the stability of zero dynamics is proposed. Moreover, A sliding mode observer design with unknown inputs estimation (receiver) is proposed in section 4. Finally, in order to highlight the well found of the proposed method, an example is given simulated and commented.

II. NOTATIONS AND PROBLEM STATEMENT

A multiple secure communication system can be represented in the following form:

   ẋ = f (x) + m i=1 g i (x) u i y = [h 1 (x) , ..., h p (x)] T (1)
where U is an open set of R n , x ∈ U is the state vector and f : R n → R n is analytic. y ∈ R p is the output vector and u ∈ R m represents the confidential information to be transmitted. The vector fields f = [f 1 , . . . , f n ] T , g i = [g i1 , . . . , g im ] T and h = [h 1 , . . . , h p ] T are assumed to be sufficiently smooth on U, where f i , h j ∈ R and

g k ∈ R n , i ∈ [1, n], j ∈ [1, p], k ∈ [1, m],
. Without loss of generality, it is assumed that the distribution span {g 1 , . . . , g m } and the codistribution span {dh 1 , . . . , dh p } are nonsingular on U .

The relative degree r of system (1) is defined by r = {r 1 , ..., r p }, where

r i = min{s such that L g k L s-1 f h i = 0 for k = 1 : m}, i = 1 : p.
Under the case r = p i=1 r i = n, a multiple secure communication system can be established easily. However, this method does not lead to an enough complexity of computation for decryption. Therefore in order to improve the security, the complexity of the scheme can be increased by choosing the outputs y and the input channel vectors {g 1 , . . . , g m } such that r < n. In [START_REF] Barbot | A new observation algorithm for nonlinear system with unknown inputs[END_REF], an algorithm to solve the observation problem for nonlinear systems with unknown inputs when r < n has been proposed.

It is assumed that p ≥ m, and that the system (1) has a relative degree r = {r 1 , ..., r p }. It should be noted that in this paper we do not consider the case of infinite relative degree. Let us define the following sets that will be used in the sequel:

• Ω = {dh 1 , ..., dL r 1 -1 f h 1 , ..., dh p , ..., dL rp-1 f h p } and £ is the related distribution:

£ = span{h 1 , ..., L r 1 -1 f h 1 , ..., h p , ..., L rp-1 f h p } where r = dim Ω = p i=1 r i . • G is the smallest involutive distribution that contains {g 1 (x), ..., g m (x)}. Note k = dimG, m ≤ k ≤ n. • G ⊥ is the annihilator of G: G ⊥ = span{α 1 , ..., α n-k },
where the α i are one-forms such that for all λ ∈ G, l λ α i = 0 for i = 1 : n -k, where l λ α = α(λ) is the inner product of the vector field λ and α. Assume r < n. There exists a transformation (ξ, η) = φ(x) such that the system (1) can be locally transformed into the following normal form:

                     ξi 1 = ξ i 2 . . . ξi ri-1 = ξ i ri ξi ri = L ri f h i (x) + m j=1 L gj L ri-1 f h i (x)u j η = Q(ξ, η, u) y i = ξ i 1 ( 2
)
where ξ = ξ1 • • • ξ p T and

ξ i =    ξ i 1 . . . ξ i ri    =    h i (x) . . . L ri-1 f h i (x)    , for i ∈ [1, p].
If r < n and if the distribution span {g 1 , . . . , g m } is not involutive, u can not be obtained using classical observation algorithm. [START_REF] Barbot | A new observation algorithm for nonlinear system with unknown inputs[END_REF] provided an algorithm, under sufficient conditions, to allow the recovery of both the state and the unknown inputs in finite time. The main idea of this algorithm is to find extra information through functions of the outputs and their time derivatives. Let us define:

V = L r1 f h 1 (x) • • • L rp f h p (x) T + Γ(x)u (3) = y (r1) 1 • • • y (rp) p T (4)
that can be known using the normal form (2). Assume there exist 1 × p vector functions

K(x) = [k 1 (x), ..., k p (x)] ∈ £(x), withK(x) = 0 such that KΓ = 0 (5)
and define a fictitious output as follows:

ȳ = h(x) = KV = p i=1 k i (x)L ri f h i (x).
If ȳ / ∈ £(x), it can be considered as a suitable fictitious output in order to estimate more states 1 . Set y = [y, ȳ] T . The system has a new relative degree with respect to this output and the algorithm iterates until the new relative degree with all new fictitious outputs is equal to n, and then it has been shown in [START_REF] Barbot | A new observation algorithm for nonlinear system with unknown inputs[END_REF] that both the state x and the unknown input u can be estimated in finite time. And a more general case: r < n (even with all new fictitious outputs y), will be discussed in this paper. In the next section, several hypothesis will be given for system [START_REF] Darouach | Full-order Observers for linear systems with unknown inputs[END_REF] in order to deduce a practical form with zero dynamics, which can be applied into multi data secure communication based on synchronization of chaotic system. III. ZERO DYNAMICS Suppose that after s th steps iteration, all new fictitious outputs construct the £(x) as follows:

£(x) = span{h 1 , ..., L r1-1 f h 1 , ..., h p , ..., L rp-1 f h p , ȳp+1 , ..., L rp+1-1 f ȳp+1 , ...ȳ p+s , ..., L rp+s-1 f ȳp+s } (6)
and if dim £ = n, then we can recover all the states and all the unknown inputs. However, for the last time iteration of this algorithm, if there is no new ȳ / ∈ £, or if there is a singularity, moreover if dim £ < n, can we still recover all the states and all the unknown inputs? The answer is based on the Proposition 1.3 in Chapter 5 of [START_REF] Isidori | Nonlinear control systems[END_REF]:

Proposition 1: [6]If dimG = k = m, which means that the distribution spanned by {g 1 (x), ..., g m (x)} is involutive, then dimG ⊥ = n -m. Moreover if dim(G ⊥ ∪ Ω) = n, where Ω = {dh 1 , ..., dL r 1 -1 f h 1 , ..., dh m , ..., dL r m -1 f h m }
then there exists a diffeomorphism which can transform the original system into the following form with zero dynamics:

                     ξi 1 = ξ i 2 . . . ξi ri-1 = ξ i ri ξi ri = a i (ξ, η) + m j=1 b i j (ξ, η)u j η = Q(ξ, η) y i = ξ i 1 (7)
where

a i (ξ, η) = L ri f h i|φ -1 (ξ,η) , and b i j (ξ, η) = L gj L ri-1 f h i|φ -1 (ξ,η) , for i ∈ [1, p] Remark 1: For system (1), if p = m = 1
, there is only one input g, which is obviously involutive. So in this case, it can be also transformed into the system [START_REF] Nijmeijer | An observer looks at synchronization[END_REF].

Moreover, the following lemma is an extension of the above proposition for the case where the inputs are not involutive.

Lemma 1: Suppose that the last iteration of the algorithm gives £(x) defined in [START_REF] Isidori | Nonlinear control systems[END_REF] where dim £ = q < n and rankΓ = m. Define

Ω = {dh 1 , ..., dL r 1 -1 f h 1 , ..., dh m , ..., dL r m -1 f h m , dȳ p+1 , ..., dL rp+1-1 f ȳp+1 , ..., dȳ p+s , ..., dL rp+s-1 f ȳp+s } If dim(G ⊥ ∪ Ω) = n (8)
then system (2) can be described into form [START_REF] Nijmeijer | An observer looks at synchronization[END_REF], where

a i (ξ, η) = L ri f h i|φ -1 (ξ,η) , and b i j (ξ, η) = L gj L ri-1 f h i|φ -1 (ξ,η) , for i ∈ [1, p + s]. Proof: Suppose that dim(G ⊥ ∪ Ω) = n
with dim £ = q < n and rankΓ = m, it is possible to find (n -q) functions from the set {λ 1 , ..., λ n-k }, without loss of generality, choosing {λ 1 , ..., λ n-q }, such that, the n differentials {dh 1 , ..., dL

r 1 -1 f h 1 , ..., dh p , ..., dL r p -1 f h p , dȳ p+1 , ..., dL rp+1-1 f ȳp+1 , ..., dȳ p+s , ..., dL rp+s-1 f
ȳp+s , dλ 1 , ..., dλ n-q } are linearly independent. Then there exists a diffeomorphism

φ = {h 1 , ..., L r 1 -1 f h 1 , ..., h p , ..., L r p -1 f h p , ȳp+1 , ..., L rp+1-1 f ȳp+1 , ..., ȳp+s , ..., L rp+s-1 f ȳp+s , λ 1 , ..., λ n-q } (9)
which transforms system (2) into the form [START_REF] Nijmeijer | An observer looks at synchronization[END_REF], whose zero dynamics part is given by

η = Q(ξ, η) |ξ=0 (10)
Remarks 1: i) If dim £ = r < n and dim(G ⊥ ∪ Ω) < n , then there does exist zero dynamics part in the transformed system, i.e., equation [START_REF] Singh | A modified algorithm for invertibility in nonlinear system[END_REF]

becomes η = Q(ξ, η, u). ii) If for all i ∈ [1, p + s], j ∈ [1, n -q], ∂ ξi r i ∂η1 = ... = ∂ ξi r i
∂ηn-q = 0, i.e., equation [START_REF] Singh | A modified algorithm for invertibility in nonlinear system[END_REF] becomes η = Q(η), then it is possible to recover the unknown input u.

iii) If for a certain i ∈ [1, p + s], j ∈ [1, n -q], we have ∂ ξi r i ∂ηj = 0, then the unknown input u depends on the detectability of η.

For solving the problem stated above, we need the following hypothesis:

Hypothesis 1: The zero dynamics of system ( 7) is uniformly, at least exponentially detectable, i.e., there exists a strictly definitive positive Lyapunov function V (η -η) which satisfies, for all η, η and ξ, the following relations:

V (η -η) = ∂V ∂(η-η) [Q(ξ, η) -Q(ξ, η)] -K 0 V
where K 0 is a positive constant. And moreover φ defined in ( 9) is a diffeomorphism on U . Hypothesis 2: The system (1) is Bounded Input Bounded State (BIBS).

In the next section, for the proposed system (7), a sliding mode observer is designed in order to recover both state and unknown inputs.

IV. OBSERVER DESIGN WITH UNKNOWN INPUTS

ESTIMATION

Under the Hypothesis 1 and 2, we can observe both state and unknown inputs of system (7) using the following sliding mode observer

                           . ξi 1 = ξi 2 + λ i 1 sign(ξ i 1 -ξi 1 )
.

ξi 2 = ξi 3 + E i 1 λ i 2 sign( ξi 2 -ξi 2 ) . . . . ξi ri-1 = ξi ri + E i ri-2 λ i ri-1 sign( ξi ri-1 -ξi ri-1 ) . ξi ri = a i ( ξ, η) + E i ri-1 λ i ri sign( ξi ri-1 -ξi ri-1 ) . η = Q( ξ, η)E n-q (11)
with the auxiliary states:

ξi t+1 = ξi t+1 + E i t λ i t+1 sign( ξi t+1 -ξi t+1 ) (12)
where

E i t = 1 if ξi t+1 = ξi t+1 and E i t-1 = 1 0 otherwise E n-q = 1 if all E i t = 1 0 otherwise (13) for i ∈ [1, q] (Noted: ξi 1 = ξ i 1 )
.

Lemma 2: Thanks to the observer given in [START_REF] Xiong | Sliding mode observer for nonlinear uncertain systems[END_REF], for any initial condition, there exists λ j i and t r > 0 such that ∀t > t r , ξ (t) = ξ (t) = ξ (t) and lim t→+∞ |η (t) -η (t)| = 0.

Proof: Set e i t = ξ i t -ξi t , for i ∈ [1, p + s], t ∈ [1, r i ], ( ξ1 = ξ 1 ).
Assume that after the first (j -1) th step, j ∈ [2, p + s],

we have obtained:

ξi j-1 = ξi j-1
and E i j-1 = 1. Then we have ξi j = ξi j + λ i j-1 sign( ξi j-1 -ξi j-1 ) = ξ i j .

So for the j th step, the dynamic of observation error can be described in the following form:

ėi j = ξ i j+1 -ξi j+1 -E i j-1 λ i j sign( ξi j -ξi j )
We can choose the following Lyapunov function:

V i j = 1 2 e i j 2
Thanks to E i j-1 = 1, and ξi j = ξ i j , we have:

V i j = e i j ξ i j+1 -ξi j+1 -λ i j sign(e i j ) So if λ i j > ξ i j+1 -ξi j+1 max (i.e., ∃k i j > 0, such that λ i j = k i j + ξ i j+1 -ξi j+1 max
), then

V i j = -k |e| = -kV 1/2
which means there exists t i j , such that if t > t i j > t i j-1 , e i j = ėi j = 0, and we obtain:

ξ i j+1 -ξi j+1 = λ i j sign(e i j ) (14)
According to the definitions of ( 12) and ( 13), we have:

ξi j+1 = ξi j+1
and E i j = 1. Therefore, step by step we can recover all the components of vector ξ . But due to the unknown input u i , we cannot recover η through ξi r . However, under the Hypothesis 1, we can reconstruct the remain states as follows.

Under the Hypothesis 2, as the state is bounded for all t > 0, then η is bounded. Moreover the estimation error of the zero dynamics (e n-q = η -η) is equal to:

ėn-q = Q(ξ, η) -Q( ξ, η)
Thus from Hypothesis 1, we deduce that

lim t→∞ |η (t) -η (t)| = 0.
λ i ki > a i (ξ, η) -a i ( ξ, η) + m j=1 b i j (ξ, η)u j max ,
such that e i ri converges to zero in finite time. Consequently we obtain ξ = ξ.

Set

Γ =    b 1 1 (ξ, η) • • • b 1 m (ξ, η) . . . . . . . . . b p+s 1 (ξ, η) • • • b p+s m (ξ, η)   
As rankΓ = m, without loss of generality, we can assume that the first m rows are independent, i.e., rankB(ξ, η) = m,

where B(ξ, η) =    b 1 1 (ξ, η) • • • b 1 m (ξ, η) . . . . . . . . . b m 1 (ξ, η) • • • b m m (ξ, η)    . So we have . ξr = A( ξ, η) + B( ξ, η)u, where . ξr =      . ξ1 r1 . . . . ξm rm      , A( ξ, η) =    a 1 ( ξ, η) . . . a m ( ξ, η)    and u =    u 1 . . . u m   .
Once all the states ξ are observed and η is estimated exponentially, there exists t r > t i j , for all i ∈ [1, p + s], j ∈ [1, r i ], when t > t r , such that e A : = A(ξ, η) -A( ξ, η) then we can obtain

e A + B ξ, η u = E r-1 λ r sign( ξr -ξr ) (15)
where

E r-1 = diag{E 1 r1-1 , E 2 r2-1 , ..., E m rm-1 }, λ r = diag{λ 1 r1 , λ 2 r2 , ..., λ m rm } and sign( ξr -ξr ) =    sign( ξ1 r1 -ξ1 r1 ) . . . sign( ξm rm -ξm rm )    .
So if we assume:

ũ = B -1 ( ξ, η)E r-1 λ r sign( ξr -ξr ) (16)
for t > t r , we have

u -ũ = B -1 (ξ, η) -B -1 (ξ, η) E r-1 λ r sign( ξr -ξr ) -B -1 (ξ, η)e A
and consequently, from the convergence of e A and (η -η) to zero, we have lim

t→∞ |u -ũ| = 0 V. EXAMPLE
In order to highlight the proposed method, we construct a simple multiple secure communication system based on the Qi's chaotic system in [START_REF] Qi | On a four-dimensional chaotic system[END_REF], which is described as follows:

       ẋ1 = a(x 2 -x 1 ) + x 2 x 3 x 4 ẋ2 = b(x 1 + x 2 ) -x 1 x 3 x 4 ẋ3 = -cx 3 + x 1 x 2 x 4 ẋ4 = -dx 4 + x 1 x 2 x 3 (17)
where x i (i = 1 : 5) are the state variables, and a, b, c, d are all positive real constant parameters. Consider the following transmitter which is based on the chaotic system (17):

           ẋ1 = a(x 2 -x 1 ) + x 2 x 3 x 4 + m 1 ẋ2 = b(x 1 + x 2 ) -x 1 x 3 x 4 ẋ3 = -cx 3 + x 1 x 2 x 4 + x 3 x 5 m 2 ẋ4 = -dx 4 + x 1 x 2 x 3 -x 4 x 5 m 2 ẋ5 = -10x 5 + x 3 x 4 (18)
where g 1 = [ 1 0 0 0 0 ] T and g 2 = [ 0 0 x 3 x 5 -x 4 x 5 0 ] T . It is obvious that g 1 and g 2 are involutive. Assume that m 1 and m 2 are small, that m 2 > 0, and that the following condition is satisfied

m 2 + d -c x 5 > 0. (19)
Suppose that the outputs are set as y = x 1 x 2 T . The input channel vector fields g 1 and g 2 have been chosen such that the relative degree of the system is r = 3. So, following the lines of the algorithm proposed in [START_REF] Barbot | A new observation algorithm for nonlinear system with unknown inputs[END_REF], set

£(x) = span{h 1 , h 2 , L f h 2 }. Since L f h 2 = b(x 1 + x 2 ) -x 1 x 3 x 4 = x 3 x 4 mod{x 1 , x 2 } one has £(x) = span{x 1 , x 2 , x 3 x 4 }. Then let us calculate Γ = L g1 h 1 L g2 h 1 L g1 L f h 2 L g2 L f h 2 = 1 0 b -x 3 x 4 0 .
One can choose K = b -x 3 x 4 , -1 such that KΓ = (0, 0).

Then the following fictitious output can be defined:

ȳ = K L f h 1 L 2 f h 2 = (b -x 3 x 4 ) ẏ1 -ÿ2 = x 2 3 + x 2 4 mod£(x)
because ȳ / ∈ £(x), and ȳ can be selected as a new fictitious output. Then, let us set

y x 1 , x 2 , x 2 3 + x 2 4 T .
With this new output y, we have £(x) = span{x 1 , x 2 , x 3 x 4 , x 2 3 + x 2 4 }, where dim £(x) = 4 < 5. But due to the fact that {g 1 , g 2 } is involutive, so according to Proposition 1, transmitter (18) can be transformed into the following form [START_REF] Nijmeijer | An observer looks at synchronization[END_REF]. Due to Proposition 2, we can recover both the state and the unknown messages in finite time. For this, let us design a sliding mode observer for system (18) as follows:

                       . x1 = a (x 2 -x 1 ) + x 2 x3 x4 + E 1 λ 1 sign(x 1 -x1 ) . x2 = b(x 1 + x 2 ) + λ 2 sign(x 2 -x2 ) d(x3 x4) dt = -(c + d) x3 x4 +E 2 λ 3 sign(x 3 x4 -x3 x4 ) d(x 2 3 +x 2 4 ) dt = -2cx 2 3 -2dx 2 4 + 4x 1 x 2 x3 x4 +2E 3 λ 4 sign x2 3 + x2 4 -x2 3 + x2 4 . x5 = E 3 (-10x 5 + x3 x4 ) (20) with λ i > 0, i = 1, ..., 4 E 1 = 1 if x 2 = x2 0 otherwise E 2 = 1 if E 1 = 1 and x 1 = x1 0 otherwise E 3 = 1 if E 2 = 1 and x3 x4 = x3 x4 0 otherwise E 4 = 1 if E 3 = 1 and x5 = x 5 0 otherwise
and with the auxiliary states:

x3 x4 = - λ 2 sign(x 2 -x2 ) x 1 (21) x2 3 + x2 4 = E 2 λ 3 sign(x 3 x4 -x3 x4 ) x5 x 1 x 2 . ( 22
)
2 , then V (e 5 ) = -10e 2 5 < 0, which implies that the Hypothesis 1 is satisfied with K 0 = 10, and we obtain lim

t→+∞ |x 5 -x5 | = 0.
Then we can define: x3 x4 = A and x2

3 + x2 4 = B, obviously there are two groups of solutions:

S 1 : x2 31 = B+ √ B 2 -4A 2 2 x2 41 = B- √ B 2 -4A 2 2
and

S 2 : x2 32 = B- √ B 2 -4A 2 2 x2 42 = B+ √ B 2 -4A 2 2 (26)
Suppose that S 1 is the correct solution. From (25), the confidential message can be recovered correctly as follows:

-cx

Using equations ( 27) and ( 28), one has:

m 22 = -cx 2
ė3 2 +4 2 = 2x 5 (x 2 3 -x 2 4 )m 2 -2E 3 λ 4 sign (e 3 2 +4 2 ) Thus, tuning λ 4 > x 5 (x 2 3 -x 2 4 )m 2 max ensures that lim t→+∞ x 5 (x 2 3 -x 2 4 )m 2 -λ 4 sign (e 3 2 +4 2 ) = 0.
Then the relation (24) leads to the estimation of the second confidential message: 2 illustrates the original messages and their estimation. Figure 1 shows that the states of the receiver converge fast to those of the transmitter. It can be seen in Figure 2 that, once the state is estimated, the confidential messages are well reconstructed. The topic of this paper is to apply the the algorithm presented by [START_REF] Barbot | A new observation algorithm for nonlinear system with unknown inputs[END_REF] into multi data secure communication. Based on this algorithm, we propose a new scheme, under the condition on zero dynamics, to solve the Left Invertibility Problem. Moreover, an illustrated example based on one Qi's chaotic system is established in order to highlight the proposed method.

Proposition 2 :

 2 Under Hypothesis 1 and 2, with rankΓ = m, thanks to the observer (11), for any given bounded initial condition, there exists t r > 0 such that ∀t > t r ξ (t) = ξ (t) = ξ (t) , lim t→+∞ |η (t) -η (t)| = 0 and lim t→+∞ |u (t) -ũ (t)| = 0 Proof: Reconstruction of the unknown input u As ξ and u are bounded, we can choose

Figure 1

 1 Figure1exhibits the states of the transmitter and those of the receiver. Figure2illustrates the original messages and their estimation. Figure1shows that the states of the receiver converge fast to those of the transmitter. It can be seen in Figure2that, once the state is estimated, the confidential messages are well reconstructed.

  Figure1exhibits the states of the transmitter and those of the receiver. Figure2illustrates the original messages and their estimation. Figure1shows that the states of the receiver converge fast to those of the transmitter. It can be seen in Figure2that, once the state is estimated, the confidential messages are well reconstructed.

Fig. 1 .Fig. 2 .

 12 Fig. 1. States observation for transmitter and the receiver

It should be noticed that there may exist a submanifold of singularity S = {x ∈ U such that h(x) ∈ £(x)}.

Obviously the submanifold of observability singularity S is equal to S = {x 1 = 0} ∪ {x 1 x 2 = 0}.

Let us also define

.

where

The observation errors are defined by:

From system (18), it can be computed that:

and

Thus the dynamics of the observation error is given by

The convergence of the sliding mode observer has been proved in the previous section.