G Zheng

L Yu

D Boutat

J-P Barbot

Algebraic observer for a class of switched systems with Zeno phenomenon

For a large class of switched systems with zeno phenomenon, classical observer cannot be applied directly since the terms leading to zeno phenomenon are not derivable. However in this paper, by assuming that these terms are integrable in the less restrictive way, we can define a new output, with which algebraic observer can then be adopted to estimate the states of the studied switched systems with zeno phenomenon. For simplicity, the main idea is explained via normal forms, while it can also be extended to generic switched systems.

I. INTRODUCTION

Switched systems are widely used for modeling complex systems, such as biological systems and engineering systems, since they can complicatedly describe the interactions of a family of continuous-time or discrete-time dynamical systems [START_REF] Liberzon | Switching in system and control[END_REF]. During the study of the switched system, most generally we exclude the case where zeno phenomenon occurs.

However, zeno phenomenon is well-known in physical domain, for example the bouncing ball case leading to many theoretical developments and simulations [START_REF] Ames | Is there life after zeno? taking executions past the breaking (zeno) point[END_REF], [START_REF] Heymann | Analysis of zeno behaviours in hybrid systems[END_REF], [START_REF] Lamperski | On the existence of zeno behavior in hybrid systems with non-isolated zeno equilibria[END_REF]. The bouncing ball is an impact system, which may be seen as a switched system with jump at the transient. In this paper we have not treated the case of switched systems with state jump, but the observation of such system which, in our point of view, is a very challenge problem. Moreover, in the widely used theory of variable structure for the purpose of control design or observer design, for example the first order sliding mode [START_REF] Utkin | Sliding Modes in Optimization and Control[END_REF] and high order sliding mode [START_REF] Emelyanov | Applying the principle of control by deviation to extend the set of possible feedback types[END_REF], [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF], there exists also zeno phenomenon. Let us remark that a first order sliding mode can generate a chattering zeno phenomenon and high order sliding mode will generally produce a genuinely zeno phenomena [START_REF] Ames | Is there life after zeno? taking executions past the breaking (zeno) point[END_REF]. Obviously, in practice, it is impossible to design a control law which commutes at an infinite frequency, and zeno phenomena is only approximated by hight frequency switching for systems driven by sliding mode control, nevertheless all methods developed in this paper also work for hight frequency switching. Some other links between zeno phenomenon and sliding mode can be found G. Zheng with respect to the homogeneous concept, see for example [START_REF] Goebel | Lyapunov characterization of zeno behavior in hybrid systems[END_REF] and [START_REF]Homogeneity approach to high-order sliding mode design[END_REF].

Since zeno phenomenon cannot be avoided in some cases, hence this paper focuses on the study of observer design problem for a class of switched systems with zeno phenomenon. As the existence of zeno phenomenon makes the trajectory of studied system non derivable since it is high frequency switching, traditional observer design techniques, such as sliding mode observer [START_REF] Perruquetti | Sliding mode control in engineering[END_REF], high-gain observer [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF], algebraic observer [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF], cannot be applied directly. Consequently this paper proposes a new approach to design an algebraic observer for a class of switched systems with zeno phenomenon, where basic idea is to re-define a new derivable output with which numerical differentiation algorithm ( [START_REF] Fliess | Reconstructeurs d'état[END_REF], [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]) can be used to calculate its successive derivative, and at the same time the built-in low-pass filters of this algorithm can remove the influence of the high frequency signals (terms containing/causing zeno phenomenon). Since normal form represents a class of equivalent systems with the same properties, for the sake of simplicity we will explain our main idea based on normal forms.

This paper is organized as follows. The notation, motivations and the problem statements are given in section 2. In section 3, we present our main idea which is to define a new output based on which an observer can be designed to estimate the states for a class of switched systems with zeno phenomenon. After that, we highlight the fact that an algebraic observer can be used in such situation. This paper ends up with an illustrative example for the purpose of highlighting the feasibility of the proposed method.

II. PROBLEM STATEMENT

Let us consider the following class of switched systems with zeno phenomenon:

   ẋ = f (x) + g (x) q = f (x) + n i=1 g i (x) q i y = h(x) (1)
where x ∈ U ⊂ R n is the set of admissible state, y ∈ R is the measured output and vector fields f : U → R n , g i : U → R n , q i : R + → Q = {0, 1} and h : U → R are sufficiently smooth.

In our study, q is not considered as an unknown perturbation but as a known input, which leads to zeno phenomenon. Hence we assume that q preserves the observability of system (1) since q is a known input. If (1) has a relative degree equal to its dimension, then the observer design problem becomes trivial and lots of nonlinear observers can be applied directly.

Hence this paper focuses only on one of the non trivial cases where system (1) has a relative degree smaller than n. Moreover, in this case if q is derivable, the regularly weakly locally observability of system (1) implies the following expression

x = Ψ(y, ẏ, • • • , y (n-1) , q, • • • , q (k) ) (2)
with k ∈ {1, ..., n -2}. From (2), many observers, including algebraic observer, can be employed in order to estimate the state x. However, for the case where zeno phenomenon exists in system (1), q is considered as a high frequent switched signal which leads to zeno phenomenon. Consequently q is not derivable. Hence it is no longer possible to apply most of existing observers for (1) with zeno phenomenon, since the successive derivatives of q in (2) do not exist. So, strongly motivated by this point, this paper deals with the problem: First to find a new equation similar to (2) which contains only derivable elements, and after that from this new equation to design an algebraic observer for a class of switched systems in the form (1) with zeno phenomenon in order to estimate its states.

III. MAIN RESULT

In this paper, we consider the Henstock-Kurzweil integral (HK integral) [START_REF] Henstock | The General Theory of Integration[END_REF][5], also known as the gauge integral, the generalized Riemann integral. Let us recall both the definitions related to Riemann integral and the HK integral in order to clarify their differences.

Definition 1: Let us consider a real function

f defined on an interval [a, b] ⊂ R. For any tagged partition P of [a, b] such as a = x 0 < x 1 < x 2 < • • • < x n = b, and xi ∈ [x i-1 , x i ] ⊂ [a, b], consider F (P ) := n i=1 f (x i )(x i -x i-1 ), for any given function δ : [a, b] → R * + , P is said to be a δ-fine tagged partition of [a, b] if h i := x i -x i-1 < δ(x i )
where δ is called a gauge for each i.

Definition 2:

A number I is called the Riemann integral (respectively the HK integral) of f : [a, b] → R if for each constant ε > 0, there exists a constant δ (respectively a function δ : [a, b] → R * + ), such that whenever P is a δ-fine tagged partition of [a, b], we have |I -F (P )| < ε.
Roughly speaking, the integral of f on [a, b] is obtained by approximating the region under the curve defined on [a, b] as a union of small rectangles. The Riemann integral requires that all those rectangles depend on a constant δ while the HK integral uses more sophisticated δ (δ depends on xi ). In fact, when a function f oscillates more quickly at some points of the subinterval [a, b], one has to tighten the step h i at these points in order to approximate more precisely the associated surface. It is possible by choosing h i < δ(x i ) such that δ(x i ) is a sufficiently small positive value that depends on the place from where the rectangle of height f (x i ) is considered.

According to the characteristics of zeno phenomenon, such as the Chattering Zeno (i.e. after some time the dwell time is exactly equal to zero) or the Genuinely Zeno (i.e. the dwell time is never equal to zero) [START_REF] Ames | Is there life after zeno? taking executions past the breaking (zeno) point[END_REF], we make the following assumption:

Assumption 1: q is HK integrable1 or HKP integrable2 . Moreover its averages q f is measured via a low-pass filter of sufficiently large bandwidth on time interval [0, c] where c ∈ R + .

Based on Assumption 1, our main idea is to construct a new output Y based on the knowledge of the old output y, q, the successive derivatives of y and successive integrals of q since y is derivable until q appears in its derivative, and y, q are integrable, i.e.

Y = ϕ y, (m) ρ (y) q, d k dt k y
where k ∈ {1, 2, ..., n -1} represents the kth derivative of y before that q appears, (m) ρ (y) q with m ∈ {1, 2, ..., n -1}

represents the mth HK integral of the variable ρ (y) q where ρ (y) a smooth function of y.

For the sake of clarity, we denote by ρ (y) qdt ′ the exact derivative of HK integral which is equal to ρ (y) q and ˙ ρ (y) qdt the practical derivative of HK integral which is equal to ρ (y) q f . As it is highlighted hereafter numerical differentiation introduced by Fliess and Sira-Ramirez [START_REF] Sira-Ramirez | An algebraic state estimation approach for the recovery of chaotically encrypted messages[END_REF] [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF] gives the second solution of practical differentiation.

With the new defined output Y , if we can express the state x of (1) as follows

x = Ψ(Y, Ẏ , • • • , Y (n-1) , χ, χ, • • • χ (k) ) (3)
where χ is a set of new known kth derivable variables defined as function of integrals of y and q since they are integrable. From (3) it is possible to design an observer to estimate the states of system (1). It is well-known that normal form represents a class of equivalent dynamical systems possessing the same properties, such as controllability [START_REF] Kang | Bifurcation and normal form of nonlinear control system: Part i and ii[END_REF] and observability [START_REF] Barbot | Observability normal form[END_REF]. Hence for the sake of simplicity, we will explain our main idea for certain normal forms.

A. Normal form of [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] Throughout this paper, L i-1 f h for 1 ≤ i ≤ n denotes the (i -1) th Lie derivative of output h in the direction of f, and set θ i = dL i-1 f h as its differential. For (1), assume that for all x ∈ U , the pair (y, f ) is regularly weakly locally observable, i.e., the codistribution span{dh, dL f h, • • • , dL n-1 f h} is of rank n. Let us consider the vector field τ 1 defined in [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] as

follows θ i (τ 1 ) = 0, f or 1 ≤ i ≤ n -1, θ n (τ 1 ) = 1, (4)
and by induction we define

τ k = (-1) k-1 ad k-1 f (τ 1 ) , f or 2 ≤ k ≤ n. (5)
Then if the following conditions are satisfied [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF]:

[τ i , τ j ] = 0 [τ k , g i ] = 0 for 1 ≤ i ≤ n, 1 ≤ j ≤ n and 1 ≤ k ≤ n -1, then (1)
can be transformed into the following normal form 3 :

             ż1 = z 2 + α 1 (y) + β 1 (y)q . . . = . . . żn-1 = z n + α n-1 (y) + β n-1 (y)q żn = α n (y) + β n (y)q y = z 1 (6)
where

z = (z 1 , • • • , z n ) T , α = (α 1 , • • • , α n ) T , q is n × 1 matrix and β i is 1 × n matrix for 1 ≤ i ≤ n.
As explained before, because of the existence of zeno phenomenon in the studied switched system, it is no longer possible to apply directly most of existing observers, including algebraic observer proposed in [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF], to systems in the form [START_REF] Emelyanov | Applying the principle of control by deviation to extend the set of possible feedback types[END_REF]. However, we will show that, by re-defining a new output for the form [START_REF] Emelyanov | Applying the principle of control by deviation to extend the set of possible feedback types[END_REF], it is still possible to apply the algebraic observer to estimate its states. And this possibility is based on the Assumption 1.

Due to Assumption 1, we can define the following measurable variables:

χ i (t) = t t-T • • • s s-T i times β i (y(ξ))q f dξ • • • ds i times := (i) β i (y(ξ))q f dξ (7)
where T represents the time windows used by algebraic observer when calculating the derivative of a signal. It is obvious that the jth derivative of χ i satisfies:

χ (j) i = (i-j) β i (y(ξ))q f dξ, if j < i β i (y)q f , if j = i (8)
where q f implies the average value of q, measured thought a low-pass filter. Let us remark that the superscript j should be smaller than or equal to the subscript i, i.e. j ≤ i, in [START_REF] Fliess | Reconstructeurs d'état[END_REF] since the term β i (y)q f is not derivable. It is clear that χ (j) i for 1 ≤ j ≤ i ≤ n is a measurable term. Consequently we can define a new output for the form (6) as follows:

Y = y - n i=1 χ i (9)
Proposition 1: If form ( 6) is algebraically observable for all q sufficiently derivable with the old output y, then it is also algebraically observable under the new output Y define in [START_REF] Fliess | Analyse non standard du bruit[END_REF]. Moreover its states can be expressed as follows:

z = Ψ(Y, Ẏ , • • • , Y (n-1) , χ i , χi , • • • χ (i-1) i ) where χ i is define in (7) for 1 ≤ i ≤ n -1.
Proof: For (6), let us consider the first exact derivative of the new output Y defined in (9), then we have 3 The equivalent relation considered here is by diffeomorphism and output injection of the form α (y) + β (y) q.

(Y ) ′ = y ′ -β 1 (y)q f - n i=2 χi = z 2 + β 1 (y)(q -q f ) - n i=2 χi
In this equation, since q f is the average of q, then the term (qq f ) can be seen as noise since it varies with a very high frequency (zeno phenomenon). Then a low-pass filter can be implemented in order to remove this noise 4 , which therefore yields the estimate of z 2 . So in this case we deal with practical derivative, i.e. replace (Y )

′ by Ẏ , and we obtain

z 2 = Ẏ + n i=2 χi
According to the above equation, the exact derivative of z 2 gives:

Ẏ ′ = (z 2 ) ′ -β 2 (y)q f - n i=3 χi = z 3 -β 2 (y)(q -q f ) - n i=3 χi
Analogously, because of the low-pass filter or practical derivative we obtain the estimate of z 3 :

z 3 = Ÿ + n i=3 χi
Iteratively, with the same procedure, for the estimate of z j we have:

z j = Y (j-1) + n i=j χ (j-1) i for 1 ≤ j ≤ n.
Finally, we can express the states of (6) as a function without q as follows:

z = Ψ(Y, Ẏ , • • • , Y (n-1) , χ i , χi , • • • χ (i-1) i
) which implies that for example an algebraic observer can be applied to observe the states of (6) even if it has zeno phenomenon.

Remark 1: Let us remark that in the form (6), the term β i (y)q is not derivable, but integrable, and it allows us to introduce a derivable term χ i which is certain integrals of the term β i (y)q f . The introduction of the term χ i is for the purpose of compensating the effect of the term β i (y) (qq f ). Consequently, the compensated residue of these two terms can be regarded as a noisy signal with high frequency, which allows us to use low-pass filters to remove its effect.

B. More generic normal form

Let us consider the following more generic normal form:

                           ż1 = z 2 + α 1 (y) + β 1 (y)q . . . = . . . żr-1 = z r + α r-1 (y) + β r-1 (y)q żr = ρ 1 (z) + β r (y)q żr+1 = ρ 2 (z) . . . = . . . żn = ρ n-r+1 (z) y = z 1 ( 10
)
where 6), [START_REF] Fliess | Real-time estimation for switched linear systems[END_REF] represents a larger class of switched system, including the case where (1) cannot be linearized with output injection into the form [START_REF] Emelyanov | Applying the principle of control by deviation to extend the set of possible feedback types[END_REF].

z = (z 1 , • • • , z n ) , q is n × 1 matrix and β i is 1 × n matrix for 1 ≤ i ≤ r.. Compared to (
The part that cannot be linearized with output injection is transformed to the part of zero dynamic in [START_REF] Fliess | Real-time estimation for switched linear systems[END_REF]. Remark 2: To seek the sufficient and necessary condition of transforming ( 1) into ( 6) is beyond the scope of this paper, however they can be deduced based on the same idea of [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF]. It should be noted that we can only constructively determine r ∈ [1, n[ vector fields from ( 4) and ( 5) whose Lie brackets are commutative, which implies the first r states of (10) are linearizable with output injection. The undetermined vector fields corresponding to zero dynamics of ( 10) might be determined through commutative condition of Lie bracket between g and these vector fields.

In order to design an observer for the form [START_REF] Fliess | Real-time estimation for switched linear systems[END_REF], we need firstly study its observability. As stated before, q in the form [START_REF] Fliess | Real-time estimation for switched linear systems[END_REF] is regarded as a known signal, but we can only measure its average value because of its high fluctuation. The first part of ( 10) is linear with output injection, which is exactly in the form [START_REF] Emelyanov | Applying the principle of control by deviation to extend the set of possible feedback types[END_REF]. As discussed in the last section, it is possible to define some measurable variables in the form [START_REF] Fliess | Non-linear estimation is easy[END_REF] based on which a new output can be defined in the form ( 9) such that observers can be used to estimate z i for 1 ≤ i ≤ r of the normal form [START_REF] Fliess | Real-time estimation for switched linear systems[END_REF]. In other words, it yields

z = Ψ(Y, Ẏ , • • • , Y (r) , χ i , χi , • • • χ (i-1) i ) (11)
where z = (z 1 , • • • , z r ) T , χ i and Y can be defined as the form ( 7) and ( 9) for a certain i ∈ {1, 2, ..., r}. Hence, let us consider the second part of the form [START_REF] Fliess | Real-time estimation for switched linear systems[END_REF], which can be written as follows:

.

z = ρ(z, z) ỹ = h(z, z) = ρ 1 (z, z) (12)
with z = (z 1 , • • • , z r ) T , z = (z r+1 , • • • , z n ) T , ρ(z, z) = [ρ 2 (z, z), • • • , ρ n-r+1 (z, z)]
T where z and ỹ are known with the help of the application of observer for the first part of [START_REF] Fliess | Real-time estimation for switched linear systems[END_REF]. Assumption 2: (12) is uniformally weakly locally observable with respect to z.

Remark 3: Assumption 2 implies that there exists a smooth function Ψz , such that the following equation satisfied:

z = Ψz ỹ, . ỹ, • • • , ỹ(n-r) (13)
It is evident that [START_REF] Fliess | Real-time estimation for switched linear systems[END_REF] is locally weakly observable if Assumption 2 is fulfilled. Moreover, it is possible to design an observer to estimate the states of (10) according to [START_REF] Fliess | Complex continuous nonlinear systems: their black box identification and their control[END_REF] and [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF].

Remark 4: Using the concept of normal form in order to explain our main idea is firstly for facilitating the explanation and secondly for establishing a uniform formula. The main idea presented in this section can, without any difficulty, be adapted to deal with other normal forms [START_REF] Xia | Nonlinear observer design by observer error linearization[END_REF], [START_REF] Zheng | Single output dependent observability normal form[END_REF], [START_REF] Yu | Observability normal forms for a class of switched systems with zeno phenomena[END_REF] with zeno phenomenon.

IV. ALGEBRAIC OBSERVER

Since the time derivatives will amplify noise, hence for the proposed method, a low-pass filter should be firstly applied for the purpose of removing the noisy residue, and a numerical differentiation algorithm can then be employed in order to calculate its derivative. Thus, each signal should be equipped with a pair of filter and differentiator. In order to simplify the implementation, we use algebraic observer proposed in [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF], which is based on the numerical differentiation technique proposed by Fliess et al in [25][22]. This algebraic approach has at least the following advantages:

• It provides explicit formulae, which can be directly implemented; • It is non-asymptotic, which is a significant advantage for real-time applications; • It does not require any assumption concerning the statistical distribution of the unstructured noise; • ... Mathematical foundation of this approach can be referred to [START_REF] Fliess | Reconstructeurs d'état[END_REF], [START_REF] Fliess | Analyse non standard du bruit[END_REF], [START_REF] Fliess | Non-linear estimation is easy[END_REF]. Some applications of such method can be cited in [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF], [START_REF] Zheng | Praivate communication based on hybrid chaotic system with delays[END_REF] for secure communication, [START_REF] Pereira | Adaptive input shaping for manoeuvring flexible structures using an algebraic identification technique[END_REF] for adaptive input shaper design, [START_REF] Fliess | Real-time estimation for switched linear systems[END_REF] for switched system, [START_REF] Fliess | Complex continuous nonlinear systems: their black box identification and their control[END_REF] for model-free control.

Let y(t) = x(t) + n(t) be a noisy observation on a finite time interval of a real-valued signal x(t), the derivatives of which we want to estimate. Then replace the convergent Taylor expansion

x(t) = ∞ n≥0 a n t n n!
by its truncated Taylor expansion

x N (t) = N n≥0 a n t n n!
It reads in the operational domain as:

s N +1 x N (s) = s N x N (0)+s N -1 ẋN (0)+• • •+x (N ) N (0) (14)
And then annihilate the remaining coefficients x j N (0), j = n by multiplying by linear differential operators of the form

Π N,n κ = d n+κ ds n+κ • 1 s • d N -n ds N -n κ ≥ 0
It yields the following estimator for x (n) (0):

x(n) N (0) s ν+n+κ+1 = (-1) n+κ (n + κ)!(N -n)! 1 s ν Π N,n κ (s N +1 x) (15)
which is strictly proper whenever ν is of the form ν = N + 1+µ with µ > 0. It should be noted that µ here represents the number of low-pass filter. After transferring both equations [START_REF] Heymann | Analysis of zeno behaviours in hybrid systems[END_REF] into the time domain, and several calculations, the estimate of the n th -order derivative of x(t) with the form as follows 5 .

x(n)

N (0) = A(ν, n, κ, N ) N -n i=0 B(i, n, N ) n+κ j=j0 C(i, j, n, κ) 1 0
(1τ ) ν+κ-i-j-1 (-τ ) i+j x(T τ )dτ [START_REF] Kang | Bifurcation and normal form of nonlinear control system: Part i and ii[END_REF] where j 0 = max (0, ki) and

A(ν, n, κ, N ) = (-1) n+κ (ν+n+κ)! (n+κ)!(N -n)! B(i, n, N ) = N -n i (N +1)! (n+i+1)! C(i, j, n, κ) = n + κ j (n+i)! (i+j-κ)!
Remark 5: An n th -order truncated Taylor expansion is appropriate for estimating the n th -order derivative. 6 Remark 6: The resulting derivative estimation is always subject to a time delay τ 0 [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF], defined as follows:

τ 0 = κ + 2 κ + ν + 4 T (17)
which is needed to be compensated.

Remark 7: This algebraic differentiator can also remove noise, since that it only contains integrators which perform slightly as a low-pass filter. Corrupting noises are viewed as highly fluctuating phenomena, which are attenuated via low-pass filters (see [START_REF] Fliess | Reconstructeurs d'état[END_REF] for more details). Such filters are realized by multiplying both sides of (15) by s -v , where v = N + 1 + µ with µ > 0.

V. ILLUSTRATIVE EXAMPLE

This section is devoted to highlighting the feasibility of the proposed method. For this, let us consider directly a simple example which is already in the form [START_REF] Fliess | Real-time estimation for switched linear systems[END_REF] as follows:

           ż1 = z 2 -z 1 ż2 = z 3 -2z 1 + β(y)q ż3 = z 2 + z 4 ż4 = -z 4 -z 3 y = z 1 (18)
with β(y) = y and q = sign(w), where w is a high frequency noise around 0 and this form of q leads to a zeno phenomenon or at least hight frequency switching (see [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems[END_REF] for Zeno phenomena simulation). It is easy to check that Assumptions 1 and 2 are satisfied, hence algebraic observer can be applied for system [START_REF] Lamperski | On the existence of zeno behavior in hybrid systems with non-isolated zeno equilibria[END_REF] to estimate its states.

Based on the principal explained before, let us define χ according to [START_REF] Fliess | Non-linear estimation is easy[END_REF] as follows:

χ(t) = t t-T s s-T y(ξ)q f (ξ)dξds
and a new output can be defined Y = yχ 5 See [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF] for details. 6 See Lemma 1 in [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF].

Thus, a straightforward calculation yields the estimations of z 1 and z 2 as follows:

   z 1 = Y + χ z 2 = Ẏ + χ + z 1 = Ẏ + χ + Y + χ
and we have

z 3 = Ÿ + χ + z 2 + z 1 + (Y + χ) (q f -q) = Ÿ + χ + Ẏ + χ + 2Y + 2χ + (Y + χ) (q f -q)
By using algebraic observer, the estimate of z 3 can be obtained as

z 3f = Ÿ + χ + Ẏ + χ + 2Y + 2χ
Since the 3rd time derivative of Y does not exist, hence we use the derivative of z 3f , the estimate of z 3 , to reconstruct z 4 , which yields:

z 4 = ż3f -z 2 = ż3f -Ẏ -χ -Y -χ
Based on the deduced equations, an algebraic observer can be applied to estimate Ẏ , Ÿ , χ, χ and ż3f . It is necessary to fix a appropriate window size T for the integrator and the algebraic differential operator, while we set this parameter as T = 0.8s corresponding to the fixed time step t δ = 0.001s.

For the other parameters in algebraic differential operator [START_REF] Kang | Bifurcation and normal form of nonlinear control system: Part i and ii[END_REF], ν = κ = 2, and N = 1 for the first order derivative (N = 2 for the second order derivative). Moreover, since the zeno phenomenon cannot be simulated, hence in this example we approximate it by q = sign(w) where w is a high frequency noise signal with the lowest frequency equal to 1KHz. In the simulation, we use the fourth-order Runge-Kutta method to approximate the solution of ODEs with a fixed time step equal to 0.001s. The simulation results are depicted in Fig. 1, where it is worthy mentioning that the incorrect estimation at the beginning is due to the time delay of numerical integrator and the numerical algebraic differential operator. For the inherent time delay [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] inside the algebraic differential operator, it can be calculated and has been compensated in the simulation results.

VI. CONCLUSION

In this paper, we demonstrate a novel scheme to construct an observer for a large class of switched systems with zeno phenomenon. Since that the zeno terms are not differentiable, algebraic observer cannot be adopted directly. However, assuming that these terms are integrable, we can reconstruct the output and renew the discrete states to make these terms integrable in the final observer, where algebraic observer can be possibly utilized. Meanwhile, we have proved the equivalence between the novel observer with new output and states and the observer with original ones. An academical example is given to illustrate our theorems.

C D

Fig. 1. A: z 1 and its estimate with z 1 (0) = 2; B: z 2 and its estimate with z 2 (0) = 2; C: z 3 and its estimate with z 3 (0) = 4; D: z 4 and its estimate with z 4 (0) = -2.

  is with Project ALIEN, INRIA Lille-Nord Europe, 50, avenue Halley, 59650 Villeneuve d'Ascq, France gang.zheng@inria.fr L.Yu is with Signal Processing Laboratory, Electronic and Information School, Wuhan University, China, and ECS ENSEA, 6 Avenue du Ponceau, 95014 Cergy-Pontoise, and Project ALIEN, INRIA, France.

	lei.yu@ensea.fr

D. Boutat is with ENSI de Bourges PRISME, 10 Boulevard de Lahitolle, 18020 Bourges, France driss.boutat@ensi-bourges.fr J-P.

Barbot

is with ECS ENSEA, 6 Avenue du Ponceau, 95014 Cergy-Pontoise, and Project ALIEN, INRIA, France barbot@ensea.fr

see also Denjoy-Khinchine integrable.

Henstock-Kurzweil-Pettis Integral is a generalization of Henstock-Kurzweil Integral where a Flippov solution may be taken into account[START_REF] Talvila | Estimates of the remainder in taylor's theorem using the henstock[END_REF]

The choice of filter bandwidth is easy where zeno phenomenon appears but may be difficult where the switching frequency is widely variable.


G Zheng
email: zheng@ensea.fr

W Aggoune
email: aggoune@ensea.fr

J.-P Barbot
email: barbot@ensea.fr

Analogue private communication based on hybrid chaotic systems with delays

Keywords: Chaotic systems, hybrid systems, weakly left invertibility, numerical differentiation

Since most of private communication schemes based on chaotic synchronization are not robust again plain-texts attacks, the introduction of delays in the schemes can be regarded as an efficient method to improve the security degree with respect to such attack. As an extension of our recent work, this paper proposes a new analogue private communication scheme based on hybrid chaotic systems with delays. The proposed scheme is based on the notation of weakly left invertibility of switched systems, and an illustrative example is given for the purpose of highlighting the feasibility of the proposed method.

INTRODUCTION

After [START_REF] Pecora | Synchronization in chaotic systems[END_REF] successfully synchronized two identical chaotic systems with different initial conditions, chaos synchronization has been intensively studied in various fields. Since the work of [START_REF] Nijmeijer | An observer looks at synchronization[END_REF], unidirectional synchronization can be viewed as a special case of observer design problem, i.e. the state reconstruction from measurements of an output variable under the assumption that the system structure and parameters are known. For the private communication system based on the synchronization of chaotic systems, a receiver (an observer from a control theory point of view) is designed in order to be synchronized with respect to the transmitter (a chaotic system with unknown inputs from a control theory point of view) and to reconstruct the confidential messages (the unknown inputs of the chaotic systems from a control theory point of view). Many techniques arising from observation theory have been applied to the problem of synchronization, such as observers with linearizable dynamics in [START_REF] Huijberts | Nonlinear discrete-time synchronization via extended observers[END_REF], adaptive in [START_REF] Fradkov | Adaptive observer-based synchronization for communication[END_REF], generalized hamiltonian form based observers in [START_REF] Sira-Ramirez | Synchronization of chaotic systems: a generalized hamiltonian approach[END_REF], algebraic method in Sira-Ramirez and Fliess (2006), [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF] or inverse system in [START_REF] Feldmann | Communication by chaotic signals: The inverse system approach[END_REF]. However, for most of private communication schemes based on chaotic synchronization and message inclusion, it is shown that they are not robust to known plain-texts attacks (see [START_REF] Anstett | Chaotic cryptosystems: Cryptanalysis and identiability[END_REF]). According to the famous [START_REF] Kerkhoff | La cryptographie militaire[END_REF] principe, it is assumed that hackers know all the details about the cryptosystem except the secret key. Roughly speaking, if the keys are only the parameters of chaotic systems, it can be proved that all the "useful" 1 parameters can be identifiable when trying the known plain-texts attacks.

1 It means the parameters which play a role in the message transmission.

Recently, in [START_REF] Zheng | Secure data transmission based-on multi-input multi-output delayed chaotic system[END_REF] delays were used in chaotic systems in order to improve the robustness of cryptosystems with respect to known plain-texts attacks, since delays are more difficult to be identified (see [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF]). As an extension of our previous work, we propose a new strategy in this paper for the purpose of improving the robustness of private communication, and this extension is based on hybrid systems with delays. From control theory point of view, the problem of recovering the message in the private communication scheme based on chaotic synchronization can be regarded as a left invertibility problem [START_REF] Hirschorn | Invertibility of nonlinear control systems[END_REF], [START_REF] Singh | Invertibility of observable multivariable nonlinear system[END_REF], [START_REF] Respondek | Right and left invertibility of nonlinear control systems[END_REF]), on which the new private communication scheme is also based. Some sufficient conditions are also given in order to solve the left invertibility problem for systems with delays. In addition, as there exist lost packets in the private communication in practice, a resynchronization technique will be proposed as well. This paper is organized as follows: Section 2 gives a presentation of the problem statement. In Section 3, a new scheme based on hybrid chaotic systems with delays is proposed. Then the robustness of the proposed scheme is analyzed in Section 4. And Section 5 is devoted to highlighting the feasibility of the proposed scheme by an illustrative example.

PROBLEM STATEMENT

This section is devoted to analyzing the robustness of most proposed schemes. Without loss of generality, we consider first a n-dimensional chaotic system in the following generic form: ẋ = f (x) (1) where x ∈ U is the state vector, U is an open set of R n , and f : R n → R n is analytic.

Hence, the private communication system based on (1) can be represented in the following form:

ẋ = f (x, K) + g (x, K) u y = h (x) (2)
where K ∈ R q is the key vector, y ∈ R is the output vector and u ∈ R represents the confidential information to be transmitted. The vector fields f : R n × R q → R n , g : R n × R q → R n and h : R n → R are assumed to be sufficiently smooth on U .

In this paper, we focus only on the analysis of the scheme based on inverse system, which implies, according to (2), one can express all states and unknown inputs as functions of the original outputs y, their time derivatives and the key vector as follows (see [START_REF] Diop | Nonlinear observability, identifiability and persistent trajectories[END_REF] for details):

   x = Ξ y, ẏ, ..., y (n-1) , K u = Ψ y, ẏ, ..., y (n-1) , K (3)
However, (3) cannot resist against known plain-texts attacks when all plain-texts are known. Indeed, consider the second equation of (3) at different instants t i for 1 ≤ i ≤ l, it is possible to obtain several independent equations with respect to K:

u(t i ) = Ψ y(t i ), ẏ(t i ), ..., y (n-1) (t i ), K (4)
Remark 1. If we can obtain q = l independent equations from (4), thus all useful parameters are identifiable. If l < q, which means that q -l parameters are not identifiable, then they can not play the role of the key. Thus, the knowledge of these parameters is not necessary for recovering the message and those parameters are of no interest in the transmitter design.

Consequently such a scheme is not robust against known plain-texts attacks. For the data transmission scheme based on discrete chaotic system, the similar expression between all states and unknown inputs could be deduced as that of (3), replacing time derivatives of outputs by time delays of outputs. Analogously, those schemes are not robust against known plain-texts attacks as well.

PROPOSED SCHEME

In order to overcome this drawback, in [START_REF] Zheng | Secure data transmission based-on multi-input multi-output delayed chaotic system[END_REF] we proposed a more robust scheme with respect to known plain-texts attacks by introducing delays in multiinput multi-output continuous chaotic system, which is also a part of the unknown parameters (part of the key).

Consequently, the introduction of the delay operator into the input-output relation equation seems to exhibit a robust characteristics with respect to known plain-texts attacks. Inspired by the work of [START_REF] Tan | Invertibility, flatness and identifiability of switched linear dynamical systems: an application to secure communications[END_REF], as an extension of our previous work, we propose a new analogue private communication scheme based on hybrid chaotic systems with delays, which is described in Fig. 1.

For the transmitter, the switching signal is generated by the discrete system with the message and the partition block. It was used to activate the corresponding chaotic subsystem for the purpose of encoding the message. For the receiver, the received signal is decoded according to activated subobserver, determined by the switching signal generated by the discrete system and partition block in the receiver.

Description of transmitter

The discrete system (Σ D ) of the transmitter in this scheme can be described as

Σ D : z(k + 1) = F (z(k), u(kη), K) y D = H(z(k)) (5)
where z ∈ R N D is the state vector, K ∈ R q is the key vector, y D ∈ O is the output vector where O is the output space such that O ⊆ R O D , u ∈ R represents the confidential information and η represents the sampled period. The vector fields F : R N D × R × R q → R N D and H : R N D → O are assumed to be sufficiently smooth. Since Σ D is discrete chaotic system, such that it is sensitive to its initial condition and parameters, hence those variables (initial condition and some critique parameters of Σ D ) can be served as part of keys.

The function of partition block in the scheme acts as a generator of switching signal from y D (the output of Σ D ).

Definition 1. Given the output space of Σ D , its associated partition S 1 , . . . , S m can be defined

m i=1 S i = O and S i S j = Ø, for 1 ≤ i, j ≤ m, i = j
Consequently, the switching signal generated by the discrete system and partition block can be defined as follows.

Definition 2. The switching signal is a piecewise constant function S : R + 0 × O × N × N → M where R + 0 represents the non negative real, N represents the natural and M = {1, . . . , m} ⊂ N . For a given y D ∈ O, we have

S(t, y D , I, I 0 ) = i, if y D (t) ⊂ S i ⊆ O and mod(t, Iη) = 0 i 0 , if mod(t, I 0 η) = 0
where I ∈ N is a predefined natural characterizing iteration times of the discrete system, I 0 ∈ N represents a predefined positive reinitialization period of the proposed scheme, i ∈ M is the value of the switching signal, i 0 ∈ M is the default value of the switching signal, determined by the initial conditions of z and u in (5) and mod(a, b) represents a modulo b.

The generated switching signal plays the role to determine how and when to activate a subsystem. A simple determi-nation rule can be defined as follows.

Rule 1. Activate the jth subsystem when S(t, y D , I, I 0 ) = j for t ∈ R + 0 , y D (t) ⊆ O, I ∈ N , I ∈ N and j ∈ M. Remarks 1. i) The reinitialization procedure will be proposed in Section 4 to handle loss packets during data transmission. A predefined period is given by I 0 η, and at the reinitialization instant the switching signal will be reset, the same as those of the discrete system and the switched systems in the scheme, which will be explained in the next section.

ii) It can be seen that the associated partition of the output space O of Σ D is not unique, and the activate rule of subsystem can be different as well. Hence the partition manner and the activate rule can be served as part of keys in order to improve the robustness of the proposed scheme.

Left invertibility of delayed system

In order to simplify the presentation of the description of transmitter, we introduce the following notations. Classically, δ i ∈ R + 0 for i ∈ N , denotes the delay operator defined for any function a(•), such as δ i a(t) = a(t -τ i ). As usual, δ 0 i a(t) = a(t), and recursively, one has 6) which is homogeneous with the delay τ i for state x (i.e. it contains only the delay τ i ), satisfying f δi = δ i f and f (0) = 0. Let us remark that the derivative and the delay operators are commutative, i.e. Let us now consider the switched system in the transmitter. In the following we will focus on the left invertibility problem of the jth subsystem, hence for the sake of simplicity we drop the superscript j, and then the jth singleinput single-output subsystem: Σ j for j ∈ M is described as follows

δ k i a(t) = δ i (δ k-1 i a(t)), for k ≥ 1. Moreover, the delay operator satisfies: δ i (a + b) = δ i a + δ i b and δ i (a • b) = δ i a • δ i b. For any function f , we note f δi (x(t)) = f (δ i x(t)) = f (x(t -τ i )) (
Σ S j : ẋ = f 0 (x, K) + f δ 1 (x, K) + g 2 (x, K)δ 2 u + g δ 3 (x, K)u x(t) = φ(t), u(t) = ψ(t) t ∈ [-τm, 0] y = h(x) (7)
where

x ∈ R Nj , K ∈ R q , u ∈ R, y ∈ R and δ i for 1 ≤ i ≤ 3 represents the time delays of x or u for the jth subsystem. The functions φ(t) ∈ C([-τ m , 0], R n ) and ψ(t) ∈ C([-τ m , 0], R n ) represent the initial condition of the jth subsystem, defined over the interval [-τ m , 0], where τ m is a positive constant such that τ m = max{τ 1 , τ 2 , τ 3 }, C([-τ m , 0], R n ) is the Banach space of continuous func- tion mapping [-τ m , 0] into R n , with the norm φ = sup t∈[-τm,0] |φ(t)| with the Euclidean norm of φ(t) ∈ R n denoted by |φ(t)|.
The functions f 0 and g 2 are smooth functions of x without delays, while f δ1 and g δ3 are smooth functions which are homogeneous with δ 1 and δ 3 , respectively. Moreover we note f δ1 = f 1 and g δ3 = g 3 for the sake of simplicity.

Inspired by the work of [START_REF] Vu | Invertibility of switched linear systems[END_REF] and [START_REF] Tanwani | Invertibility of nonlinear switched systems[END_REF], we define the weakly left invertibility for system (7) as follows. In order to prove our main result, we introduce the following technical definitions. Definition 4. An input-output relation for system (7) is defined as

L gi L fj k • • • L fj 1 h = 0 (8) with i ∈ {2, 3} and j l ∈ {0, 1} with 0 ≤ l ≤ k < n,
where j represents the number of input-output relation. Moreover for each input-output relation j, it is associated with a delay index d j = k l=1 τ j l + τ i for j l ∈ {0, 1} and i ∈ {2, 3}. The input-output relation set for system (7) is the set of all input-output relations for this system. Remark 2. Obviously, as the derivative of u is also unknown, the input-output relation with the derivative of the input is not considered. However, if we take into account the derivative of the input, then a derivative of Lie-Backlun must be considered. Now similarly to systems without delays, we define the relative degree for system (7) as follows. Definition 5. The relative degree r of system ( 7) is equal to k * + 1 where k * is the greatest value of k in the inputoutput relation set which satisfies the rank condition

Rank     dh dL f l h . . . dL fj k • • • L fj 1 h     = k * + 1
At the relative degree r, we also associate a delay index d r which is the smallest delay index of all input-output relations with k = k * which satisfies the rank condition. Remark 3. If L g2 L f0 h = 0 and L g2 L f1 L f0 h = 0, this implies that the delay in this Lie derivative between the input and the output is equal to τ 2 + τ 1 . This fact is the starting point of the previous definition and it is with the rank condition the origin of the next theorem.

From the previous definition, we are able to give the following theorem. Theorem 1. System ( 7) is weakly left invertible if:

• the relative degree r for this system is equal to n;

• all delay indices of input-output relations with k < n -1 are strictly greater than d r . Proof 1. From the definition of the relative degree, it is clear that from the knowledge of the previous state and input we can recover the input at time t -d r . Moreover from the same definition of relative degree, the system is 'observable' with respect to the knowledge of previous states and input. Consequently we can 'observe' or estimate the state at least at time t-d r . Consequently system (7) is weakly left invertible in the sense of Definition 3.

Description of receiver

The receiver part in the scheme is to decode the encoded messages receiving through the public channel. At the end of receiver, the discrete system ΣD is the same structure used in the transmitter, i.e.

ΣD :

ẑ

(k + 1) = F (ẑ(k), û(kη), K) ŷD = H(ẑ(k)) (9)
where ẑ, û and ŷD are of the same dimension as those defined in Σ D for the transmitter, representing their estimates. K represents the key shared by both the transmitter and receiver. Proposition 1. If all the subsystems are weakly left invertible and the key K of the proposed scheme contains the following elements:

• for discrete system Σ D : the initial conditions of messages and states, and some parameters of Σ D ; • the partition manner and activate rule;

• for each subsystem Σ Sj for j ∈ M: the initial conditions of states and messages, some parameters and time delays of x and u for Σ Sj ;

then we can design a receiver, which might successfully recover the messages encoded by the transmitter. Proof 2. It can be seen that if the initial conditions and some parameters of the discrete system are used as part of keys, and if one knows the initial condition of u, one can have ŷD → y D if one can prove û → u. Moreover, if the partition manner and activate rule are served as part of keys as well, one has Ŝ → S which implies the switching signal could be recovered and consequently the correct corresponding subsystem might be activated. According to Theorem 1, one can design an observer in order to retrieve the encoded messages (i.e. û → u) from the active subsystem, provided one knows the initial conditions, the parameters and time delays of x and u for the active subsystem. It should be noted that this recovery of message may be with delays, and that is the reason why in Fig. 1 a block of predefined delays is placed in front of the discrete system for the purpose of keeping the applied input synchronized. Consequently, if key K contains all the elements mentioned above, one can design a receiver, which might successfully recover the messages encoded by the transmitter.

Concerning the observer design for each subsystem, according to Theorem 1 one can deduce an input-output relation with delays for the purpose of reconstructing the message. Based on [START_REF] Barbot | An algebraic framework for the design of nonlinear observers with unknown inputs[END_REF], an algebraic observer can be applied for this situation. This algebraic approach is based on the numerical differentiation technique used in Sira-Ramirez and Fliess ( 2006), Fliess and Sira-Ramirez ( 2004), [START_REF] Fliess | Complex continuous nonlinear systems: their black box identification and their control[END_REF], [START_REF] Fliess | Nonlinear estimation is easy[END_REF] and [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]. Roughly speaking, for an analytic signal x(t), its Taylor expansion at t = 0 can be written as

x(t) = ∞ i=0 x (i) (0) t i i! . Then the corresponding truncated Taylor expansion is x N (t) = N i=0 x (i) (0) t i i! with d N +1
dt N +1 x N (t) = 0. Rewrite it in the well-known notation of operational calculus:

x N (s) = N i=0 x (i) (0) s i+1
where d ds corresponds to the multiplication by -t in the time domain. By multiplying both sides d j ds j s N +1 with 0 ≤ j ≤ N and s -v with v > N , it yields the following triangular linear equations

s -v d j (s N +1 x N ) ds j = s -v d j ds j N i=0 x (i) (0)s N -i (10)
and we can obtain the numerical differentiation of x(t) by applying the inverse Laplace transform to (10).

ANALYSIS OF ROBUSTNESS

For a new proposed private communication scheme, one of the main tasks is to analyze its robustness with respect to attacks.

Robustness to known plain-texts attacks

Due to the introduced delays for the state variables and the inputs, the proposed scheme becomes robustness against known plain-texts attacks. More precisely, one can obtain the input-output relation with delays, hence at different instants t it is possible to obtain independent equations with respect to K, such that u = Ψ(y, . . . , y (n-1) , δy, . . . , δy (n-1) , δu, . . . , δu (n-1) , K) (

where δ and δ represent the delays for the output and input, respectively. However since at each instant t, the output and the derivative of the output with delays are not known because the delays are part of key, it becomes more difficult to identify K according to (11). Consequently the proposed scheme seems more robust than schemes without delays to the known plain-texts attacks.

Another firewall against known plain-texts attacks of the proposed scheme is the combination effect of the discrete system affected by the unknown input and the partition block, which implicitly implies that one can find out a relation between the unknown input and the generated switching signal i(t) = Γ(t, z, u, K) (12) In the situation of known plain-texts attacks, according to the relation ( 12) it is possible to identify some part of keys of K with the knowledge of the generated switching signal i(t). However, since ( 12) is function of the unknown input u, for different u the output space of the discrete system is different, and after the partition block the generated switching signal i(t) becomes difficult to be predicted, which signifies that the combination technique seems to be robust to the known plain-texts attacks.

Robustness to lost packets

From practical point of view, there exists an eventuality of lost packets during the data transmission, which will lead absolutely bad estimate of the states and messages. Since the chaotic system is quite sensitive to the initial condition and its parameters, those bad estimates will make future estimates totally worst. In order to rectify the bad influence of the lost packet during data transmission, we impose reinitilization rule in the proposed scheme. Rule 2. At the instant t reset ∈ R + 0 , i.e. mod(t reset , I 0 η) = 0, then set S(t + reset , y D , I, I 0 ) = i 0 ∈ M, and reinitialize the active subsystem Σ Si 0 at the instant t reset by setting

x(t reset ) = φ(t) and u(t reset ) = ψ(t) for t ∈ [t reset - τ m , t reset ].
Since this rule is shared by both the transmitter and receiver, if there exists lost packets phenomenon, the bad estimates of messages will occur only until the next reset time. When the reinitialization operation is imposed for the transmitter and the receiver, a new synchronization is established, and consequently the influence of lost packets for future estimates of state variables and messages is avoided.

ILLUSTRATIVE EXAMPLE

This section is devoted to illustrating the feasibility of the proposed scheme. We choose the simple Logistic map as the discrete chaotic system described in the scheme:

Σ D : z(k + 1) = µz(k)(1 -z(k)) + 0.01u(kη) y D = z(k) ( 13
)
with µ = 1.38, u(t) = 1 + sin(t)cos(15t + 20), η = 0.0001s, and with the initial condition z(0) = 0.01 and u(0) = 0. The partition of the output space of ( 13) is set to be S 1 = [0, 0.6] and S 2 =]0.6, 1]. The predefined constant variable I is set to be 2000 and I 0 = 40000, i 0 = 1. Hence the switching signal is defined

S(t, z, I, I 0 ) = 1, if 0 ≤ z ≤ 0.6 and mod(t, 2) = 0 2, if 0.6 < z ≤ 1 and mod(t, 2) = 0 1, if mod(t, 20) = 0
The uniform distribution of output space is illustrated in Fig. 2. Concerning the switched systems described in the transmitter, we consider only two subsystems for the sake of simplicity. The first chaotic system is based on Lorenz system. After introducing delays into the state variable and input, it can be written in the following form:

Σ s1 :      ẋ1 ẋ2 ẋ3 = a(x 2 -x 1 ) x 1 (b -x 3 ) -x 2 x 1 x 2 -cx 3 + x 2 δ 1 u + δ 1 x 1 u y = x 1 (14)
with a = 10, b = 28, c = 8/3, δ 1 u = u(t -τ 1 ) and δ 1 x 1 = x 1 (t -τ 1 ) where τ 1 = 7ms. In the simulation, the initial conditions of x 1 (t) and u(t) for t ∈ [-τ 1 , 0] are randomly generated over the time interval

[-τ 1 , 0]. Note f 0 = a(x 2 -x 1 ) x 1 (b -x 3 ) -x 2 x 1 x 2 -cx 3 , g 2 = 0 0 x 2 and g 3 = 0 0 δ 1 x 1
, and one can find the following input-output relation L g3 L f0 L f0 h = 0 if -ax 1 δ 1 x 1 = 0. It is easy to check such input-output relation has a regular relative degree 3 since

Rank   dh dL f0 h dL 2 f0 h   = 3
and consequently Theorem 1 is satisfied. More precisely, the state variables and the unknown message can be represented by the output and its derivative as follows:

           x 2 = ẏ a + y x 3 = b - ẋ2 + x 2 x 1 u = ẋ3 + cx 3 -x 1 x 2 -x 2 δ 1 u δ 1
x 1 hence one can easily design a subobserver for Σ S1 . The second subsystem is based on Chen chaotic system. Analogously by introducing the delays, Σ S2 can be described as follows:

Σ s2 :      ẋ1 ẋ2 ẋ3 = β(x 2 -x 1 ) x 1 (γ -β) -x 1 x 3 + γx 2 x 1 x 2 -ρx 3 + x 2 δ 2 u + δ 1 x 2 u y = x 1 (15)
with β = 35, γ = 28, ρ = 3 and τ 2 = 6ms. It is easy to check that Theorem 1 is also fulfilled and the state variables and the unknown input can be recovered. By applying the algebraic approach to calculate the numerical differentiation (see [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF]), the simulation results are depicted in Fig. 3, where only confidential message and its recovery are given. It is clear that the encode message u is reconstructed when the active subsystem is correctly determined.

CONCLUSION

This paper proposed a new analogue private communication scheme based on hybrid chaotic systems with delays in order to improve the security degree. The proposed scheme is based on the weakly left invertibility problem and sufficient conditions are given in this paper in order to solve such a problem. Moreover, the robustness of the proposed scheme is discussed from two aspects: the robustness to known plain-texts attacks and robustness to lost packets. In addition, the algebraic derivative method is adopted to compute the successive derivatives of the output. Finally an example is studied in order to illustrative the proposed scheme.

Fig. 1 .

 1 Fig. 1. Scheme for private communication system based on hybrid chaotic systems with delays.

  ∂f δ i ∂x = δ i ∂f ∂x, based on which the derivative of Lie can be extended to systems with delays.

Fig. 2 .

 2 Fig. 2. Uniform distribution of the output state space of discrete system (13)

Fig. 3 .

 3 Fig. 3. Confidential message and its recovery.

  Definition 3. System (7) is weakly left invertible if unknown input u(t) and state x(t) can be recovered for any t from the knowledge of the output y(t) for t ∈ [0, b] with b positive, the initial condition of states φ(t) and initial condition of input ψ(t) for t ∈ [-τ m , 0].