M Ghanes

G Zheng

J De Leon-Morales

On simultaneous parameter identification and state estimation for cascade state affine systems

Keywords: Adaptive observer, synchronization, identification, Rossler's chaotic system

In this paper, an adaptive observer is proposed to solve the problem of simultaneous parameter identification and state estimation for a class of cascade state affine systems. Sufficient conditions are given in order to guarantee the exponential convergence of the proposed observer. Furthermore, simulation results are given illustrating the performance of the proposed observer when it is applied in the synchronization and identification problem of Rossler's chaotic system.

I. INTRODUCTION

Nonlinear observer design is an important problem in the theory of systems. It is clear that no general procedure exists to construct a nonlinear observer for a general nonlinear system. However, for a class of nonlinear systems, under some extra assumptions, such as Lipchitz, persistent exciting..., it is possible to construct the observer. For the purpose of state estimation, many works have been devoted [START_REF] Kreisselmeier | Adaptive observers with exponential rate of convergence[END_REF], [START_REF] Bastin | Stable adaptive observers for nonlinear time varying systems[END_REF], [START_REF] Hammouri | Observers synthesis for state affine systems[END_REF], [START_REF] Besancon | Remarks on nonlinear adaptive observer design[END_REF]. However, when dealing with the problem of parameter identification, it becomes more difficult. The simultaneous parameter identification and state estimation problem has been attracted the attention of various research groups, since it is very useful to treat many practical problems, such as fault detection, signal transmission or control, and recently for synchronization of chaotic systems. Motivated by this interest, several approaches have been proposed to simultaneously estimate the state and identify the parameters. In [START_REF] Zhang | Adaptive observers for MIMO linear time-varying systems[END_REF], author has proposed a novel adaptive observer with an appropriate adaptation law for the unknown parameters. And in [START_REF] Besancon | On adaptive observers for state affine systems[END_REF], the unknown parameters were treated as the extended state of the system, in such a way the existed 'classical' nonlinear observer design methods can be applied to the augmented system. All these methods can be used to design nonlinear observer for a large class of nonlinear system, such as linear time invariant/variant systems, and a class of state affine systems.

In this paper, our goal is firstly to design an adaptive observer in order to estimate the unmeasurable state variables of the system and to identify the unknown parameters simultaneously for a class of cascade state affine systems. Furthermore, some sufficient conditions in order to guarantee the exponential convergence of the proposed adaptive observer are given.

Besides, according to [START_REF] Nijmeijer | An observer looks at synchronization[END_REF], the problem of synchronization of chaotic systems has been related to the concept of observer design from theoretical point of view. Therefore, in order to highlight the feasibility of the proposed observer, we apply it to treat the problem of chaotic synchronization. In fact, the topic of synchronization of chaotic system has attracted many researchers' attention since the work of [START_REF] Pecora | Synchronization in chaotic systems[END_REF]. After that, many applications of chaotic synchronization have been developed, such as chaotic secure communication based on synchronization of chaotic system [START_REF] Kovarev | Experimental demonstration of secure communications via chaotic synchrionization[END_REF], [START_REF] Parlitz | Transmission of digital signals by chaotic synchronization[END_REF], [START_REF] Wu | A simple way to synchronize chaotic systems with applications to secure communications systems[END_REF], [START_REF] Feldmanne | Communication by chaotic signals: The inverse system approach[END_REF] and [START_REF] Parlitz | Transmission of digital signals by chaotic synchronization[END_REF].

Inspired by these works, in this paper, secondly, we also try to deal with the synchronization problem for Rossler's chaotic system based on the proposed observer in order to not only estimate the state of system, but also identify its unknown parameters at the same time.

The paper is organized as follows: In section 2, some basic notations about the considered nonlinear systems are introduced. And section 3 is devoted to adaptive observer design for such class of cascade state affine systems. In section 4, an illustrative example dealing with the synchronization and identification parameter problem of Rossler's chaotic systems is given. Simulation results are presented in order to emphasize the performance of the proposed adaptive observer.

II. NOTATIONS

In this paper, we are interested in designing an exponential observer for the following cascade state affine system:

ż = A (y, u, z, θ) z + β (y, u, z, θ) + ϕ (y, u, z, θ) θ y = Cz (1) where z ∈ R n , u ∈ R l , y ∈ R p , θ ∈ R q are
respectively the state, known input, output of the system and the parameter, function A, β, ϕ and C are the matrices of appropriate dimensions, and the components of matrix A, and vectors β and ϕ are continuous functions depending on u, y and z 1 , ..., z i-1 , θ 1 , ..., θ i-1 , for 1 ≤ i ≤ p and uniformly bounded, with

z =    z 1 . . . z p    , y =    y 1 . . . y p    , θ =    θ 1 . . . θ p    A (y, u, z, θ) = diag   A 1 (y, u) , • • • , A p (y, u, z 1 , ..., z p-1 , θ 1 , ..., θ p-1 )   , C = diag C 1 , • • • , C p , β (y, u) =    β 1 (y, u)
. . .

β p y, u, z 1 , ..., z p-1 , θ 1 , ..., θ p-1    , ϕ (y, u, z, θ) = diag   ϕ 1 (y, u) , • • • , ϕ p (y, u, z 1 , ..., z p-1 , θ 1 , ..., θ p-1 )   where z i ∈ R ni , θ i ∈ R qi , y i ∈ R, A i , β i , ϕ i , C i are the matrices of appropriate dimensions, for 1 ≤ i ≤ p, u ∈ R l . And p i=1 n i = n, p i=1 q i = q.
In the sequel, the dependence of u, y, z 1 , ..., z i-1 , θ 1 , ..., θ i-1 and ẑi-1 , θ1 , ..., θi-1 , with A i , ϕ i , β i and Âi , φi , βi will be omitted in order to lighten notations.

III. ADAPTIVE OBSERVER FOR CASCADE SYSTEMS

Before introducing our main result, we establish the following assumptions: Assumption A1. If the input is persistently exciting, in the sense that there exist α 2,i > α 1,i > 0, T 1,i > 0 and t 0 ≥ 0 such that for all initial condition x 0 , the following condition for all t ≥ t 0 is satisfied:

α 1,i I ≤ t+T1,i t Ψ T (u,x0,i) (s, t) C T i Σ i (s) C i Ψ (u,x0,i) (s, t) ds ≤ α 2,i I
where Ψ (u,x0,i) denotes the transition matrix for the system

żi = A i (y, u, z 1 , ..., z i-1 , θ i , ..., θ i-1 ) z i , y i = C i z i
and Σ i is some positive definite bounded matrix, for 1 ≤ i ≤ p.

Assumption A2. Considering matrix

Λ = diag {Λ 1 , ..., Λ p } ,
where Λ i is a matrix defined by

Λi = A i -S -1 i C T i Σ i C i Λ i + ϕ i
Assume that ϕ i is persistency exciting so that there exist γ 2,i > γ 1,i > 0, T 2,i > 0 and t 0 ≥ 0 and some positive definite matrix Σ, such that the following inequality holds

γ 1,i I ≤ t+T2,i t Λ T i (s) C T i Σ i (s) C i Λ i (s) ds ≤ γ 2,i I
for all t ≥ t 0 and for 1 ≤ i ≤ p.

Assumption A3. We assume that the components of z and θ are bounded i.e. there exist positive constants such that

z i ≤ δ z i , θ i ≤ δ θ i
and the following inequalities hold

Âi -A i ≤ i-1 j=1 δ A j e j + i-1 j=1 ̺ A j e θj βi -β i ≤ i-1 j=1 δ β j e j + i-1 j=1 ̺ β j e θj φi -ϕ i ≤ i-1 j=1 δ ϕ j e j + i-1 j=1 ̺ ϕ j e θj
where e i = z i -ẑi and e θi = θ i -θi , for 1 ≤ i ≤ p.

Theorem 1: Consider system (1), if assumptions A1, A2 and A3 are satisfied, then the following system ż = A y, u, ẑ, θ ẑ + β y, u, ẑ, θ + ϕ y, u, ẑ, θ θ

+ S -1 C T + ΛΓ -1 Λ T C T Σ (y -C ẑ)
where

                   żi = Âi ẑi + βi + φi θi + S -1 i C T i + Λ i Γ -1 i Λ T i C T i Σ i (y i -C i ẑi ) Ṡi = -ρ i S i -ÂT i S i -S i Âi + C T i Σ i C i Λi = Âi -S -1 i C T i Σ i C i Λ i + φi Γi = -λ i Γ i + Λ T i C T i Σ i C i Λ i θi = Γ -1 i Λ T i C T i Σ i (y i -C i ẑi ) (2)
is an exponential observer for system [START_REF] Qi | On a four-dimensional chaotic system[END_REF], where ρ i and λ i are sufficiently large positive constants and Σ i are some positive definite matrices for 1 ≤ i ≤ p.

Proof: Set e = (e 1 , ..., e p )

T and e θ = e θ1 , ..., e θp T , the state estimation error and the parameter estimation error respectively, where e i = z i -ẑi and e θi = θ i -θi , and θ i is a constant, for 1 ≤ i ≤ p.

For a general case of i, for 1 ≤ i ≤ p, the estimation error dynamics for e i = z i -ẑi and e θi = θ i -θi can be obtained as follows

ėi = A i -S -1 i C T i Σ i C i -Λ i Γ -1 i Λ T i C T i Σ i C i e i +ϕ i e θi + Âi -A i z i + βi -β i + ( φi -ϕ i ) θ i and ėθi = -Γ -1 i Λ T i C T i Σ i C i e i
Introducing the following change of variable

ǫ i = e i -Λ i e θi (3)
we get

ǫi = A i -S -1 i C T i Σ i C i ǫ i + Âi -A i z i + βi -β i + ( φi -ϕ i ) θ i
Because S i and Γ i are some positive definite matrices according to Assumptions A1 and A2, a candidate Lyapunov function can be chosen as follows

v i = ǫ T i S i ǫ i + e T θi Γ i e θi
Hence its time derivative is given by

vi = -ρ i ǫ T i S i ǫ i -λ i e T θi Γ i e θi -(ǫ i + Λ i e θi ) T C T i Σ i C i (ǫ i + Λ i e θi ) +2ǫ T i S i Âi -A i z i + βi -β i + ( φi -ϕ i ) θ i
According to Assumptions A1, A2 and A3, we obtain

vi ≤ -ρ i ǫ T i S i ǫ i -λ i e T θi Γ i e θi +2 ǫ i S i    δ z i   i-1 j=1 δ A j e j + i-1 j=1 ̺ A j e θj      +2 ǫ i S i    i-1 j=1 δ β j e j + i-1 j=1 ̺ β j e θj    +2 ǫ i S i    δ θ i   i-1 j=1 δ ϕ j e j + i-1 j=1 ̺ ϕ j e θj      which can be rearranged as follows vi ≤ -ρ i ǫ T i S i ǫ i -λ i e T θi Γ i e θi +2 ǫ i S i i-1 j=1 δ z i δ A j + δ β j + δ θ i δ ϕ j e j +2 ǫ i S i i-1 j=1 δ z i ̺ A j + ̺ β j + δ θ i ̺ ϕ j e θj
Hence, using the following inequality

x y ≤ 1 2 x 2 + y 2 we get vi ≤ -ρ i ǫ T i S i ǫ i -λ i e T θi Γ i e θi + i-1 j=1 L i j,1 + L i j,2 ǫ i 2 + i-1 j=1 L i j,1 e j 2 + i-1 j=1 L i j,2 e θj 2
where

L i j,1 = S i δ z i δ A j + δ β j + δ θ i δ ϕ j L i j,2 = S i δ z i ̺ A j + ̺ β j + δ θ i ̺ ϕ j for 1 ≤ i ≤ p, 1 ≤ j ≤ i -1, with L 1 j,1 = L 1 j,1 = 0.
By applying

x + y 2 ≤ 2 x 2 + y 2
and after straightforward computations, we obtain

vi ≤ -ρ i - i-1 j=1 L i j,1 + L i j,2 η i ǫ T i S i ǫ i -λ i e T θi Γ i e θi + i-1 j=1 2L i j,1 η j ǫ T j S j ǫ j + i-1 j=1 2L i j,1 Λ j 2 + L i j,2
χ j e T θj Γ j e θj which follows

vi ≤ -µ i v i + i-1 j=1 κ i,j v j where µ i = min ρ i - i-1 j=1 (L i j,1 +L i j,2 )
ηi , λ i and

κ i,j = max 2L i j,1 ηj , 2L i j,1 Λj 2 +L i j,2 χj
, with κ 1,j = 0.

As a result, the whole Lyapunov function can be chosen as v = p i=1 v i , whose time derivative is v = p i=1 vi , and we have

v ≤ p i=1    -µ i v i + i-1 j=1 κ i,j v j    = p i=1   -µ i + p j=i+1 κ j,i   v i Consequently, if µ = min µ i |µ i > p j=i+1 κ j,i , 1 ≤ i ≤ p , then we obtain v ≤ -µv < 0
and this ends the proof.

IV. APPLICATION INTO CHAOTIC SYNCHRONIZATION AND PARAMETER IDENTIFICATION

As an illustrative example of adaptive state affine observation, let us consider the synchronization and parameter identification problem of a chaotic system. It is well-known that a chaotic system is a nonlinear deterministic system having a complex and unpredictable behavior. The sensitive dependence on initial conditions and the parameter variations is a prominent feature of chaotic behavior. In the sequel, we apply the proposed observer to simultaneously estimate states and identify parameters for a given chaotic system.

V. CONCLUSION

In this paper, the problem of adaptive observer for a class of cascade state affine systems has been discussed. Sufficient conditions have been given in order to guarantee the exponential convergence of the adaptive observer. Furthermore, it has been shown that the adaptive observer has an arbitrarily tunable rate. One practical interest of such observer is to study the synchronization and parameter identification problem of chaotic systems. An example of synchronization and parameter identification of Rossler's chaotic system has been studied in order to illustrate the feasibility of the proposed observer.

2 Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .

 24567 Fig. 1. z 1,1 and its estimate

A. Chaotic system

Considering the following 3-dimensional autonomous Rossler's chaotic system described by [START_REF] Chang | Parameter identification of Rossler's chaotic system by an evolutionary algorithm[END_REF]

where x i (1 ≤ i ≤ 3) are the state variables, and a, b, c are all positive real constant parameters. The actual system parameters of ( 4) are set to a = 0.2, b = 0.2, and c = 5.7 for exhibiting the chaos phenomenon.

B. Change of coordinate and transformation into cascade system

By considering the following change of coordinate

where z 1 ∈ R 2 , z 2 ∈ R, the chaotic system (4) can be rewritten into cascade system (1) as follows:

where

From figure 4, the chaotic system [START_REF] Feldmanne | Communication by chaotic signals: The inverse system approach[END_REF] with the parameters a = 0.2, b = 0.2, c = 5.7 and initial conditions z 1,1 = 2, z 1,2 = 3, z 2 = 2, exhibits the chaotic dynamics. Now, for this cascade system (5), the proposed adaptive observer ( 2) is applied and designed as follows:

where

C. Simulation results

The aim here is to illustrate the simulation results obtained by the proposed cascade observer when it is applied in the synchronization and parameter identification problem of Rossler's chaotic system. The chosen numerical values (initial conditions and parameters) for Rossler's chaotic cascade system (5) are given in section (IV-B) while the values of its observer (6) are as follows. The initial conditions are ẑ1,1 = ẑ1,2 = ẑ2 = 0; θ1 = â = 0, θ2 = b ĉ = 0 1×2 . S 1 (0) = 2I 2×2 , Λ 1 (0) = 0 2×2 , Γ 1 (0) = 10I 2×2 ; S 2 (0) = 1, Λ 2 (0) = 0 1×2 , Γ 2 (0) = 10I 2×2 . The gain were chosen as ρ 1 = 20, λ 1 = 15; ρ 2 = 20, λ 2 = 15.

The simulation results obtained with the proposed observer are illustrated in Figs. 1 to 7.


S Solvar

V Le

M Ghanes

J P Barbot

G Santomenna

Sensorless second order sliding mode observer for induction motor

Keywords: Induction Motor, Sensorless, Observer, Sliding mode, Super Twisting

In this paper, a second order sliding mode observer for the induction motor without mechanical sensor is presented. This observer converges in finite time and is robust to the variation of parameters. Using Matlab/Simulink, the simulation results show the performance of the proposed observer. Furthermore, an industrial application is presented in order to highlight the technological interest of the proposed method and also show the difficulties due to real time computation constraints.

I. INTRODUCTION

Nowadays Induction Motors (IMs) have replaced DC motors in the industrial application. There are many different ways to drive an induction motor. The differences between them are the performance of motor and the cost of implementation. The voltage over frequency method is the simplest method. This is a basic scalar control which imposes a constant relation between stator voltage and frequency. It is normally used without speed sensor. However, this control has not a good accuracy in speed and torque response because the flux and the torque are not controlled directly [START_REF] Buja | Direct Torque Control of PWM Inverter-Fed AC Motor-A Survey[END_REF]. Contrarily, in the vector control which is based on relations valid for dynamic states, magnitude and instantaneous position of voltage, current and flux are controlled. This control allows to have a good motor's performance with respect to permanent state and dynamic state behavior. But, in the vector control the mechanical speed and the flux are needed. For this reason, in the recent years high performance sensorless induction motor drivers have attracted a great attention in industrial applications due to theirs advantages such as low cost, low maintenance, high reliability etc. The sensorless control helps to reduce the cost and resolves installing problem in many applications: lack of place, dangerous environment. Present effort are devoted to improve the performance of observer at low speed [START_REF] Ghanes | A robust output feedback controller for induction motor drivers: new design and experimental validation[END_REF] and to develop a robust observer against perturbations and variation of parameters. Several methods have been proposed to estimate speed and flux of IM such as: Luenberger observer [START_REF] Kubota | DSP-Based Speed Adaptative Flux Observer of Induction Motor[END_REF] and Kalman filter [START_REF] Hilairet | Speed and rotor flux estimation of induction machines using a two-stage extended Kalman filterstar[END_REF], high gain and adaptive observers [START_REF] Khalil | Sensorless Speed Control of Induction Motor[END_REF], [START_REF] Ghanes | A robust output feedback controller for induction motor drivers: new design and experimental validation[END_REF] neural networks and signal injection [START_REF] Wolbank | Combination of signal injection and neural networks for sensorless control of inverter fed induction machines[END_REF] and sliding mode observer [START_REF] Ghanes | On Sensorless Induction Motor Drives: Sliding-Mode Observer and Output Feedback Controller[END_REF] etc. Compared with other observers, sliding mode technics [START_REF] Perruquetti | Sliding mode control in engineering[END_REF] has attractive advantages of robustness to disturbances and insensitivity to variation of parameters when the sliding mode happens. However, the chattering behavior, that is inherent in standard sliding mode techniques, is often an obstacle for practical application if neglected. Higher-Order Sliding Mode [START_REF] Bartolini | Chattering Avoidance by Second-Order Sliding Mode Control[END_REF] is one of the solutions which does not compromise robustness and avoid filtering estimation considered by other methods. In this paper, a new sensorless second sliding mode observer is proposed to estimates flux and speed of IM under unknown load torque. Firstly the observer is designed in continuous time. Thereafter, the work consists to show how to get a sliding mode observer under sampling in order to implement it on an experimental set-up for an industrial application. Here the sliding mode observer is over sampled in order to by pass the difficulties due to slow data acquisition. This paper is organized as follows: the section II recalls the IM model. In section III the observabilty of the IM is presented. After that the section IV proposes a second order sliding mode observer which is applied for IM. In sections V and VI, the simulation and experimentation results are shown to confirm the good performance of the second order sliding mode observer. Some conclusions and remarks are drawn in section VII.

II. IM PER-UNIT MODEL

In order to construct the proposed observer for an industrial application, the IM model given by [START_REF] Chiasson | Modeling and High-Performance Control of Electric Machines[END_REF] in the fixed (α, β) frame is rewritten under the following Per-Unit (p.u.) equations:

           ẋ1 = γ • x 1 + θ • (b • x 3 + c • x 5 x 4 ) + ξ • v 1 ẋ2 = γ • x 2 + θ • (b • x 4 -c • x 5 x 3 ) + ξ • v 2 ẋ3 = a • x 1 -b • x 3 -c • x 5 • x 4 ẋ4 = a • x 2 -b • x 4 + c • x 5 • x 3 ẋ5 = h • (x 3 • x 2 -x 4 • x 1 ) -d • x 5 -e • T l (1)
with: inductance. p is the number pairs of pole. J is motor inertia and f v the viscous friction coefficient. T l is the load torque. Notice that for the sake of homogeneity, hereafter the simulation and experimental results will be given in per-unit (p.u.).

x 1 = isα I ref , x 2 = i sβ I ref , x 3 = ω sref •φrα V ref , x 4 = ω sref •φ rβ V ref , x 5 = p•Ω ω sref , σ = 1 - M 2 sr Ls•Lr , γ = Rs•L 2 r +Rr•M 2 sr σ•Ls•L 2 r ,T r = Lr Rr , K = Msr σ•Ls•Lr , a = Msr•I ref •ω ref Tr •V ref , b = 1 Tr , c = ω ref , d = fv J , e = p J•ω ref , h = p 2 •Msr•I ref •V ref J•ω 2 ref •Lr , θ = K•V ref I ref •ω ref , ξ = V ref σ•Ls•I ref

III. BRIEF RECALL OF IM OBSERVABILTY

It is shown in ( [START_REF] Ghanes | Observation et commande de la machine asynchrone sans capteur mcanique[END_REF], [START_REF] Ghanes | A robust output feedback controller for induction motor drivers: new design and experimental validation[END_REF]) that the IM observability cannot be established in the particular case when the fluxes φ rα , φ rβ and the speed Ω are constant even if we use the higher derivatives of currents. This is a sufficient and necessary condition for lost of observability. This operating case match to the following physically interpretation : 1) when the fluxes are constant ( φrα = φrβ = 0), or equivalently, the stator pulsation voltage is zero (ω s = 0), it implies that: pΩ

+ RrTe pφ 2 d = ω s = 0 or: T em = -KΩ where T em is the electromagnetic torque, K = p 2 φ 2 d Rr and φ 2 d = φ 2 rα + φ 2 rβ . 2) if the speed motor is constant; thus: T em = (f v Ω + T l ) = -KΩ. This last equation defines the unobservability curve in the map (T l , Ω) with M = p 2 φ 2 d Rr + f v (Figure 1). - 6 Ω T l @ @ @ @ @ @ @ @ -M Fig. 1. Unobservability curve in the map (T l , Ω) .
Obviously, the observability is lost gradually when we approach this unobservability curve.

IV. SECOND ORDER SLIDING MODE OBSERVER DESIGN

A. Recall on sliding mode observer

The sliding mode technique is used at first in control theory, but in the recent years sliding mode is presented such as a good observer for many applications [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF], [START_REF] Floquet | Super twisting algorithm based stepby-step sliding mode observers for nonlinear systems with unknown inputs[END_REF], [START_REF] Boiko | Super twisting algorithmbased step-by-step sliding mode observers for nonlinear systems with unknown inputs[END_REF]. Considering a system:

             ẋ1 = x 2 ẋ2 = x 3 . . . ẋn = f (x 1 , • • • , x 2 ) y = x 1 (2)
To this system, a second order sliding mode observer is designed in the following way:

                           ẋ1 = x2 + λ 1 • |x 1 -x1 | 1 2 • sign(x 1 -x1 ) ẋ2 = α 1 • sign(x 1 -x1 ) ẋ2 = E 1 • x3 + λ 2 • |x 2 -x2 | 1 2 • sign(x 2 -x2 ) . . . ẋn = E n-2 • α n-1 • sign(x n-1 -xn-1 ) ẋn = E n-1 • θ + λ n • |x n -xn | 1 2 • sign(x n -xn ) θ = E n-1 • α n • sign(x n -xn ) (3) with E i = 1 if xi -xi = 0 0 if not B. Application for IM 1)
Continuous observer: Firstly, considering a change of variable like that:

               z 1 = x 1 z 2 = x 2 z 3 = b • x 3 + c • x 5 x 4 z 4 = b • x 4 -c • x 5 x 3 z 5 = ż3 z 6 = ż4 (4)
From the IM model ( 1) and this change of variable (4), a dynamical system for (4) can be obtained as:

               ż1 = -γ • z 1 + θ • z 3 + ξ • v 1 ż2 = -γ • z 2 + θ • z 4 + ξ • v 2 ż3 = z 5 ż4 = z 6 ż5 = z 7 ż6 = z 8 (5)
Now by applying the second order sliding mode observer (3) recalled on the section (IV-A) for system [START_REF] Perruquetti | Sliding mode control in engineering[END_REF] we get an observer in the following way:

                                         ż1 = θ • z3 -γ • z 1 + ξ • v 1 + λ 1 • |e 1 | 1 2 • sign(e 1 ) ż3 = α 1 • sign(e 1 ) ż2 = θ • z4 -γ • z 2 + ξ • v 2 + λ 2 • |e 2 | 1 2 • sign(e 2 ) ż4 = α 2 • sign(e 2 ) ż3 = E 1 • E 2 • (z 5 + λ 3 • |e 3 | 1 2 • sign(e 3 )) ż5 = E 1 • E 2 • α 3 • sign(e 3 ) ż4 = E 1 • E 2 • (z 6 + λ 4 • |e 4 | 1 2 • sign(e 4 )) ż6 = E 1 • E 2 • α 4 • sign(e 4 ) ż5 = E 1 • E 2 • E 3 • E 4 • (z 7 + λ 5 • |e 5 | 1 2 • sign(e 5 )) ż7 = E 1 • E 2 • E 3 • E 4 • α 5 • sign(e 5 ) ż6 = E 1 • E 2 • E 3 • E 4 • (z 8 + λ 3 • |e 6 | 1 2 • sign(e 6 )) ż8 = E 1 • E 2 • E 3 • E 4 • α 6 • sign(e 6 ) (6)
with

E i 1 if e i = zi -ẑi = 0 0 if not
The functions E i ensure that the next steps errors do not escape too far before one has the convergent of the last step error. By choosing the gains α i , λ i which respect to the reachability condition of the Super Twisting algorithm [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF], [START_REF] Floquet | Super twisting algorithm based stepby-step sliding mode observers for nonlinear systems with unknown inputs[END_REF], [START_REF] Saadaoui | Contribution la synthse d'observateurs non linaires pour des classes de systmes dynamiques hybrids[END_REF] for example:

α 1 > z 5max , λ 1 > 4 • z 5max • α1+z5max α1-z5max
, we get e 1 = e 2 = ... = e 6 = 0, ie ẑ1 = z 1 ,ẑ 2 = z 2 ...ẑ 6 = z 6 after some finite time T. To find out the estimated speed and flux, equations (4) are solved as follows. One has

z 3 = b • x 3 + c • x 5 • x 4 , i.e., x 3 = 1 b • (z 3 -c • x 5 • x 4 ) (7)
Furthermore, one has

x 4 = 1 b • (z 4 + c • x 5 • x 3
). If one replaces x 3 by equation ( 7) one obtains:

x 4 = z 4 + c b • z 3 • x 5 b + c 2 •x 2 5 b (8)
In the similar way, we get:

x 3 = z 3 -c b • z 4 • x 5 b + c 2 •x 2 5 b (9)
If one derives the equation ( 7) we obtain:

z 5 = b•(-z 3 + a • x 1 )+c• ẋ5 •x 4 +c•x 5 •(-z 4 + a • x 2 ) (10)
In the similar way, we have:

z 6 = b•(-z 4 + a • x 2 )-c• ẋ5 •x 3 -c•x 5 •(-z 3 + a • x 1 ) (11)
So one has 4 equations (8,9,10,11) with 4 unknowns x 3 , x 4 , x 5 and ẋ5 . Solving these equations we can find out the estimated speed and rotor fluxes. To estimate the rotor speed we can add a following hypothesis: the speed is constant in comparison with the variation of stator current. From this hypothesis and equations (4) one can estimate the speed and the rotor flux.

x5 = ẑ5 + b • ẑ3 -b • a • ẑ1 -c • z 4 + c • a • ẑ2 (12) φrα = ẑ3 -c•x5•ẑ4 b b + c 2 •x 2 5 b (13)
φrβ = ẑ4 + c•x5•ẑ3 b b + c 2 •x 2 5 b (14)
The designed second order sliding mode has another advantage because one does not need to know the load torque to estimate the motor's speed.

2) Discrete Observer: For the industrial application in real time, the discrete observer is constructed. The explicit Euler's method is chosen to transform continuous observer to discrete observer. This is due to the simplicity of computation. Nevertheless in order to obtain the request precision this discretization is used n times, doing so the discretization error is in nO(( Te n ) 2 ) instead of O(( Te n ) 2 ) with T e is the data acquisition period. This method is performed in section [START_REF] Ghanes | A robust output feedback controller for induction motor drivers: new design and experimental validation[END_REF].

Applying the explicit Euler's method for the second order sliding mode observer, the discrete observer is obtained:

                                                                                 ẑ1 (k) = ẑ1 (k -1) + T e • (θ • z3 (k -1) -γ • z 1 (k -1) +ξ • v 1 (k -1) +λ 1 • |e 1 (k -1))| 1 2 • sign(e 1 (k -1))) z3 (k) = z3 (k -1) + T e • α 1 • sign(e 1 (k -1)) ẑ2 (k) = ẑ2 (k -1) + T e • (θ • z4 (k -1) -γ • z 2 (k -1) +ξ • v 2 (k -1) +λ 2 • |e 2 (k -1)| 1 2 • sign(e 2 (k -1))) z4 (k) = z4 (k -1) + T e • α 2 • sign(e 2 (k -1)) ẑ3 (k) = z3 (k -1) + T e • E 1 • E 2 • (z 5 (k -1) +λ 3 • |e 3 (k -1)| 1 2 • sign(e 3 (k -1))) z5 (k) = z5 (k -1) + T e • E 1 • E 2 • α 3 • sign(e 3 (k -1)) ẑ4 (k) = ẑ4 (k -1) + T e • E 1 • E 2 • (z 6 (k -1) +λ 4 • |e 4 (k -1)| 1 2 • sign(e 4 (k -1))) z6 (k) = z6 (k -1) + T e • E 1 • E 2 • α 4 • sign(e 4 (k -1)) ẑ5 (k) = ẑ5 (k -1) + T e • E 1 • E 2 • E 3 • E 4 • (z 7 (k -1) +λ 5 • |e 5 (k -1)| 1 2 • sign(e 5 (k -1))) z7 (k) = z7 (k -1) +T e • E 1 • E 2 • E 3 • E 4 • α 5 • sign(e 5 (k -1)) ẑ6 (k) = ẑ6 (k -1) + T e • E 1 • E 2 • E 3 • E 4 • (z 8 (k -1) +λ 3 • |e 6 (k -1)| 1 2 • sign(e 6 (k -1))) z8 (k) = z8 (k -1) +T e • E 1 • E 2 • E 3 • E 4 • α 6 • sign(e 6 (k -1)) (15)
with

E i 1 if e i = zi -ẑi = 0 0 if not
For a good accuracy of this discrete observer, one must use a sampling time small enough and it needs a fast DSP. In the industrial application at GS Maintenance, the clock frequency of DSP is only 150M Hz, and it does not allow to work on a small sampling time. So in experimentation an over-sample technique is proposed in the following way: T e-new = Te n with T e is the data acquisition time, T e-new is the new computation period, n is the number of over-sample. Considering a differential equation Ẋ = f (x):

The explicit Euler's method with a sampling time T e gives:

X(k) = X(k -1) + T e • f (X(k -1))
with T e is the data acquisition period and computation period too. With an over sampling time T e-new = Te n , the explicit Euler's gives:

                                   X(kT e + Te n ) = X(kT e ) + Te n • f (X(kT e )) + O( Te n ) 2 X(kT e + 2 Te n ) = X(kT e ) + Te n • f (X(kT e )) + Te n • f (X(kT e + Te n )) + 2O( Te n ) 2 ... X(kT e + T e ) = X(kT e ) + Te n • f (X(kT e )) + Te n • f (X(kT e ) + Te n • f (X(kT e + Te n )) + Te n • f (X(kT e + 2 Te n )) + ... + Te n • f (X(kT e + (n -1) Te n )) + nO( Te n ) 2 (16)
Hence it can be remarked that the error is O(T e ) 2 for a sample time explicit Euler's while for explicit Euler's with an over sampling time, the error is nO( Te n ) 2 , which reduce by n the discretisation error.

In order to reduce again the computation time, one used a table of square root pre-calculated.

V. SIMULATION RESULTS

To verify the effectiveness of the proposed observer, a simulation has been carried out using Matlab/Smulink. The used sampling period is 10 -4 s. The simulation results are divided in two part: in one hand the observer is tested under nominal condition (identified parameters), in the other hand the influence of parameters variation is considered (robustness tests). The figures (2,3) illustrate estimated speed and estimated rotor flux. This result present a good performance of the second order sliding mode observer. Error between real speed and estimated speed is very small (0.5%). An error occurs in the time of changed speed because in the simulation one does not respect the hypothesis constant speed, in experimentation this problem does not appear due to system inertia. The figures [START_REF] Canudas De Wit | Observability conditions of induction motors at low frequencies[END_REF][START_REF] Perruquetti | Sliding mode control in engineering[END_REF][START_REF] Khalil | Sensorless Speed Control of Induction Motor[END_REF][START_REF] Wolbank | Combination of signal injection and neural networks for sensorless control of inverter fed induction machines[END_REF] prove the robustness of proposed observed with respect to the variation of parameters. In this simulation test, the variation of ±50% of R s , ±50% of R r , +20% of L s , +20% of L r is considered. The results simulation show that the observer is insensible to R r , L s , L r variation. It exists only a small error (1%) in the case of ±50% of R s .

VI. EXPERIMENTAL RESULTS

The proposed second order super twisting sliding mode observer for IM has been implemented at GS Maintenance. The system consists to a 1.1kW/380V/50Hz induction motor fed by an inverter. The observer is implemented in a single fixed-point TMS320F2812. A mechanical sensor is mounted on the motor's shaft to compare measured speed and estimated speed. At first, the observer has been tested at nominal speed. The figures [START_REF] Buja | Direct Torque Control of PWM Inverter-Fed AC Motor-A Survey[END_REF][START_REF] Chiasson | Modeling and High-Performance Control of Electric Machines[END_REF] show the estimated speed and rotor flux. In comparison the estimated speed with the real speed which is measured by an encoder, one obtains a good performance of the observer. After that the performance of observer is tested at low speed (5Hz) (figure 10). This figure shows a good result of estimated speed. The figure 11 shows that the observer diverges under unobservable condition (very low speed) 1 . To ensure the performance of speed observer, at low speed a speed estimator is proposed and tested 2 . The figure 12 shows under unobservable conditions the observer diverges but the estimator give a satisfactory result. For sum up, our practical approach consists in a variable structure observer: under observability conditions a second order sliding mode observer is used and under unobservable 1 see section III for more details 2 the estimator is simply an integration of normalized motor model conditions an estimator is used.

VII. CONCLUSION

In this paper a second order sliding mode observer is designed in order to estimate the speed of IM without mechanical sensor. An industrial application using a usual DSP shows that it is possible to implement a second order sliding mode observer in a single fixed point DSP. The over-sample technique allows to have a very fast sampling with Euler's method. Furthermore, the speed is estimated without the knowledge of load torque. The simulation and experimentation results affirm a good performance of the proposed second order sliding mode observer such that: finite time convergence, robustness with respect to the variation of parameters. The bad behavior of the observer under unobservable conditions (at very low speed) is remedied by using an estimator of speed.

Annexe: Induction Motor data

  and i sα , i sβ , φ rα , φ rβ , Ω are respectively the stator currents, the rotor fluxes and the speed. I ref , V ref , are the maximal values of stator current and voltage. ω sref is the nominal stator pulsation. R s and R r are the stator and rotor resistance. L s and L r are the stator and rotor inductance. M sr is mutual

Fig. 2 .

 2 Fig. 2. Estimated speed (blue) and real speed (red).

Fig. 3 .

 3 Fig. 3. Estimated rotor flux in the fixed frame.

Fig. 4 .

 4 Fig. 4. Estimated speed (blue) and real speed (red) when Rs varies +50%.

Fig. 5 .

 5 Fig. 5. Estimated speed (blue) and real speed (red) when Rs varies -50%.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Estimated speed (blue) and real speed (red) when Rr varies +50%.

Fig. 8 .

 8 Fig. 8. Estimated speed (blue) and real speed (red) when Ls varies +20%.

Fig. 9 .

 9 Fig. 9. Estimated speed (blue) and real speed (red) when Lr varies +20%.

Fig. 10 .

 10 Fig. 10. Estimated speed (red) and real speed (black) at nominal speed (p.u.).

Fig. 11 .

 11 Fig. 11. Estimated rotor fluxes in the fixed frame.

Fig. 12 .

 12 Fig. 12. Estimated speed (blue) and real speed (red) in the dynamic regime (p.u.).

Fig. 13 .

 13 Fig. 13. Speed observer (blue), speed estimator (red) and real speed (black) (p.u.).


H Hamiche
email: hamid.hamiche@ensea.fr

M Ghanes
email: ghanes@ensea.fr

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

S Djennoune
email: djennoune@mail.ummto.dz

Secure Digital Communication based on Hybrid Dynamical Systems

In this work, a transmission scheme based on the hybrid and chaotic dynamics for private communications is proposed. The transmitter is composed of a continuoustime system and a discrete-time system in which the message is inserted by inclusion. The states of the continuous system are also included, after sampling, in the discrete system. The receiver is composed of a discrete-time delay observer and a continuous-time observer. The principle of the proposed hybrid method is to show that the reconstruction of discrete states of the receiver and the message passes at first by a synchronization of the two continuous-time chaotic systems. This new strategy makes the system of transmission robust, in particular against an attack known plaintext. Simulation results are presented to highlight the performances of the proposed method.

I. INTRODUCTION

In recent years, the theory of complex nonlinear systems and especially chaotic systems has been applied to cryptography in order to propose other methods of encryption. In 1990, T. Pecorra and L. Carroll [START_REF] Pecora | Synchronization in Chaotic Systems[END_REF] have shown the possibility of synchronizing chaotic systems. So, many encryption schemes based on chaos have been proposed in the literature [START_REF] Koracev | Chaos-based cryptography : a brief overview[END_REF]. However, very little work has really made the link between encryption algorithms standard and those based on the generation of chaotic sequences. In normal cryptography, from a variety of mechanisms encryption, there are symmetric encryption and public key encryption. The link between the standard symmetric encryption and the encryption by inclusion based on the chaos has been proposed by F. Anstett [START_REF] Anstett | Chaotic Cryptosystems: Cryptanalysis and Identifiability[END_REF]. Many methods of encryption using chaotic systems has been proposed in the literature, we can cite the addition method [START_REF] Wu | Simple way to synchronize chaotic systems with applications to secure communications systems[END_REF], the parametric modulation [START_REF] Parlitz | Transmission of digital signals by chaotic synchronization[END_REF], modulation chaotic [START_REF] Dedieu | Newblock Chaos shift keying: modulation and demodulation of a chaoticcarrier using self-synchronizing Chua's circuits[END_REF], etc. These methods are all based on synchronization receiver with the transmitter in order to extract the message originally embedded in the chaotic carrier. In 1997, H. Nijmeijer and I. Mareels [START_REF] Nijmeijer | On Synchronization of Chaotic Systems[END_REF], [START_REF] Nijmeijer | An observer Looks at Synchronization[END_REF] showed that the unidirectional synchronization of the chaotic systems can be considered as an observer design problem. Different observers has been proposed for chaotic systems (observers used only to reconstruct the transmitter states), [START_REF] Freeland | Nonlinear state observers for chaotic systems and their application to communications signal process[END_REF], [START_REF] Ghanes | On simultaneous parameter identification and state estimation for cascade state affine systems[END_REF] and for chaotic systems with unknown inputs (observers for reconstructing the states of the transmitter and to recover the information) [START_REF] Barbot | Observability Bifurcations: Application to Cryptography[END_REF]. The right work of these observers depends on several conditions: the observability condition for reconstruct the system states, the observability matching condition to recover the system states and information embedded in the system (left invertibility of system), the condition of identifiability of parameters which represents the encryption keys. In this work, we are interested to study a transmission scheme composed of a hybrid dynamical system. The transmitter is composed of a continuous-time chaotic system called Colpitts and a discrete-time chaotic system called Henon. The receiver is composed from continuous and discrete observers. On the level of the receiver, the states of the continuous-time system are introduced in the dynamic of the discrete-time system to make its structure more complex. Thus, the new obtained hybrid system makes the system hardly observable since it gives more unknown parameters than the equations, and this is valid also for attacks with a known-plaintext. The transmitted output to the receiver is composed of a synchronization signal that comes from the continuous-time system and of a useful signal that contain the message (added by the inclusion method) that comes from the discrete-time system. To have a good transmission, it is necessary to have the ration of the useful signal to the transmitted signal as close to one as possible. The reconstruction of the states and the message of the observer discrete passes through the synchronization of two continuous-time chaotic systems (transmitter and receiver) before the synchronization of the two discrete-time chaotic systems. This approach present the advantage that the synchronization of the two systems can be obtained even in the case where the synchronization is lost. The work is organized as follow: in the section 2, the principle of the proposed method is presented by studying the transmitter and the receiver of the transmission system. The section 3 gives the simulation results. Finally, we conclude with a conclusion.

II. PRESENTATION OF THE METHOD

In this work a communication system based on the synchronization of two chaotic hybrid systems is realized. The global scheme of the proposed system for private communications is given by the Figure 1. In order to simplify the analysis, the added delays between the continuous and the discrete parts are removed. These delays ensure the robustness against an attack to a known clearly text. The developed method is presented as follow:

A. Study of the transmitter

The transmitter is composed of three blocks: a continuous-time chaotic system, a discrete-time chaotic system and a multiplexing block. These blocks are detailed as follows:

1) Continuous-time chaotic system: This system has been widely studied in the literature [START_REF] Maggio | Nonlinear Analysis of the Colpitts Oscillator and Application to Design[END_REF] and [START_REF] Maggio | Experimental manifestations of chaos in the Colpitts oscillator[END_REF]. The state equations of the normalized Colpitts's oscillator in a continuous time is given as:

Σ C :    ż1 = a 1 (-exp(-z 2 ) + 1 + z 3 ) ż2 = a 2 z 3 ż3 = -a 3 (z 1 + z 2 ) -a 4 z 3 (1)
with

a 1 = g q(1-k) , a 2 = g qk , a 3 = qk(1-k) g , a 4 = 1 q .
To have a chaotic behavior, the parameters of system (1) are given as follows: g = 4.46; q = 1.38 and k = 0.5 with the initial conditions z 1 (0) = 1.6; z 2 (0) = 8 and z 3 (0) = 0.1.

First, we begin by the study of the observability system with output y 1 = z 1 . Using the results given in [START_REF] Hernault | Feasability of Analogue Realization of Sliding Mode Observer: Application to Data Transmission[END_REF], it is easy to show that the rank observability matrix of system ( 1) is equal to 3, so, the system is locally weakly observable. Consequently, it is possible to obtain all the state from the output y 1 = z 1 and its derivatives.

2) Discrete-time chaotic system: The discrete-time chaotic system used in this work is the modified Henon's system. This system has been widely studied in the literature; one can site for instance the work by Vesely [START_REF] Veselyand | Chaos in a modified Henon-Heiles system describing geodesics in gravitational waves[END_REF]. The system is given by the following equations :

Σ D :    x 1 (n + 1) = a -x 2 2 (n) -bx 3 (n) x 2 (n + 1) = x 1 (n) x 3 (n + 1) = x 2 (n) (2)
To have a chaotic behavior, the parameters of the system (2) are given as follows: a = 1.76 and b = 0.1 with initial conditions x 1 = 0.1, x 2 (0) = 0.1 and x 3 (0) = 0.1. At the emission level, our aim is to make the structure of the discrete system more complex. To do this, we introduce (without delay) in the dynamics of discrete-time system, the states z 1 , z 2 and z 3 of the continuous-time system. It is noted that the states of the continuous-time system are first sampled with a rate T 1 before their introduction into the discrete-time system. The period T1 (the value is given in simulation) is chosen in order to ensure the synchronization between the transmitter and the receiver of the two continuous-time systems (this is argued in the section (II-B.1)). Despite the fact that the system is a discrete time pure Henon's system, unconnected with the time here each iteration will be made after a fixed period of time T 1. This fixed choice of the frequency was done to simplify the implementation of the receiver. In this part, the system (2) will be used as the transmitter with y 2 (n) = x 2 (n) as its output. In our work, the two states (z 2 , z 3 ) and the message m are added to the third dynamic of the system (2) in order to make the letter observable and not left invertible. The details of the reconstruction of the states and the message m are given in Section (II-B.2). The new obtained hybrid system is given as follows:

Σ H :        x 1 (n + 1) = a -x 2 2 (n) -bx 3 (n) x 2 (n + 1) = x 1 (n) x 3 (n + 1) = x 2 (n) + Az 2 (n) + Bz 3 (n) + Cm(n) (3)
with: A, B and C, the new coefficients of the discretetime system and m the message to send. To preserve the chaotic behavior of the system defined by (3), these parameters are chosen with precaution. In our case, we must respect the following values: A ≤ 0.04, B ≤ 0.1 and C ≤ 1.

Based on the works of [START_REF] Belmouhoub | Observability quadtatic Normal Form for Discrete-Time systems[END_REF], [START_REF] Djemaï | Discrete-Time Normal Form for Left Invertibility Problem[END_REF], it is easy to show that the rank of the observability matrix of the system (3) is 3, so the system is locally observable. Moreover, the recovery condition of observability is guaranteed here, the message appears after 3 iterations. Consequently, it is possible to reconstruct all states and the message from the output and its delayed output. By using the method input-output relationship approach detailed in the work of [START_REF] Anstett | Chaotic Cryptosystems: Cryptanalysis and Identifiability[END_REF], it is easy to show that the system (3) is identifiable from the output and sufficient of these iterations and from the message and sufficient of these iterations if the continuous variables change frequently.

3) Multiplexing block: The output signal y 1 = z 1 comes from the continuous-time system will be first sampled with a period T 2 but only blocked during T 1. The signal y 1 = z 1 plays the role synchronization signal, it is sent into the channel during a time equal to T 1. The signal y 2 , is sent during 9T 1. The discrete-time system is not iterated for the tenth cycle (T1) which corresponds to the sending of the signal synchronization. Thus, the cycle is composed of 10 periods T 1 illustrated by the figure 2.

B. Study of the receiver

The receiver is composed of three blocks: a continuoustime chaotic observer, a discrete-time chaotic observer and a multiplexing block. In the following, we will be interested to study the synchronization between the transmitter and the receiver in the proposed transmission scheme. We will present two observers (continuous and discrete) that allow to synchronize the systems ( 1) and ( 3) with their receivers, respectively.

1) Continuous-time chaotic observer: Consider the system (1) with the output y 1 = z 1 . The aim is to design a reduced-order observer (see [START_REF] Parlitz | Transmission of digital signals by chaotic synchronization[END_REF]) that allows to reconstruct the states ẑ1 ,ẑ 2 and ẑ3 of the system (1) from the output y 1 = z 1 . The equations of the observer are given by ( 4):

Σ OC :        ż1 = a 1 (-exp(ẑ 2 ) + 1 + ẑ3 ) +K(y 1 -ẑ1 ) ż2 = a 2 ẑ3 ż3 = -a 3 (ẑ 1 + ẑ2 ) -a 4 ẑ3 (4)
where ŷ1 = ẑ1 and K a real constant defined hereafter. Define the observer errors:

e 1 = z 1 -ẑ1 , e 2 = z 2 -ẑ2 , e 3 = z 3 -ẑ3 (5)
The gain K is chosen so that the dynamic of the synchronization error e 1 = z 1 -ẑ1 will be stable. This allows to ensure the convergence of the synchronization error e 3 = z 3 -ẑ3 to zero, consequently the convergence of the synchronization error e 2 = z 2 -ẑ2 to zero. Subtracting equation (1) from equation ( 4) and using equation ( 5), we obtain the following errors system:

   ė1 = a 1 (exp(-z 2 ) exp(e 2 -1) + e 3 ) -Ke 1 ė2 = a 2 e 3 ė3 = -a 3 (e 1 + e 2 ) -a 4 e 3 (6)
Neglecting higher order terms in the Taylor series expression of exp(e 2 ), we obtain the simplified errors system:

   ė1 = a 1 (exp(-z 2 )e 2 + e 3 ) -Ke 1 ė2 = a 2 e 3 ė3 = -a 3 (e 1 + e 2 ) -a 4 e 3
which can be rewritten as follows:

Ė = AE (7)
where

A =   -K exp(-z 2 )a 1 a 1 0 0 a 2 -a 3 -a 3 -a 4   (8) (9)
and E = e 1 e 2 e 3 T .

In order to ensure the convergence of the estimated variables ẑ1 , ẑ2 and ẑ3 to z 1 , z 2 and z 3 respectively, we must guarantee the asymptotical stability of [START_REF] Dmitriev | Chaotic synchronization of henon mappings: The information approach[END_REF]. Our main result is given by the following proposition.

Proposition

The errors system ( 7) is asymptotically stable if there exist a gain K which satisfy the following condition:

K > a 2 3 a 2 1 -2a 3 a 1 a 2 4 + a 4 4 + 4a 4 a 3 a 1 a 2 M -4a 2 4 a 3 a 2 2a 4 - a 3 a 1 + a 2 4 2a 4 (10)
where M 1 is the bound of exp(-z 2 ) i.e.,

| exp(-z 2 )| ≤ M. (11)
Proof. We define a Lyapunov function for system [START_REF] Dmitriev | Chaotic synchronization of henon mappings: The information approach[END_REF] as follows:

V = E T P E (12)
where P is symmetric positive definite matrix, which is a solution of the following Lyapunov function

A T P + P A = -Q ( 13
)
where Q is chosen semi-positive definite due to the zero eigenvalue of the matrix A (8), i.e.

Q =   1 0 0 0 1 0 0 0 0   (14)
Taking account [START_REF] Koracev | Chaos-based cryptography : a brief overview[END_REF], the time derivative of V (12) along the trajectory of the system (7) leads to:

V = E T A T P E + E T P AE = E T (A T P + P A)E
which can be written, under the condition [START_REF] Koracev | Chaos-based cryptography : a brief overview[END_REF], in the following quadratic form:

V ≤ -E T QE.
According to the choice of Q (14), the system ( 7) is asymptotically stable if the matrix P defined by ( 13) is positive definite, which is guaranteed by the condition [START_REF] Hernault | Feasability of Analogue Realization of Sliding Mode Observer: Application to Data Transmission[END_REF]. Now let's compare this direct Lyapunov method to the Azerman conjuncture one. The Azerman conjuncture consists to compute the eigenvalues of A (8) by using Routh criterium. The obtained result showed us that the eigenvalues of A are negative if the condition ( 10) is satisfied. So we get the same condition of K (10) with both methods, which is not true in a general case. This result represents a particular case and is justified by the eigenvectors of A which not change the direction.

This ends the proof.

2) Discrete-time chaotic observer: In this part, the system (3) with the output y 2 (n) = x 2 (n) is considered. For the reception, based on the works of [START_REF] Belmouhoub | Observability quadtatic Normal Form for Discrete-Time systems[END_REF] and [START_REF] Djemaï | Discrete-Time Normal Form for Left Invertibility Problem[END_REF], we will design a delayed discrete observer, that works with a sampling time T 1, as follows: -Construction of the state x1 : From the system (3), we have:

x2 (n + 1) = x1 (n)
By applying one delay on the output, we deduce the state x1 as following:

x1 (n -1) = y 2 (n) (15)
-Construction of the state x3 : From the system (3), we have also:

x3 (n) = a -x1 (n + 1) -x2 2 (n) b
Now, let us apply two delays on the output and using equation [START_REF] Nijmeijer | An observer Looks at Synchronization[END_REF], we obtain the state x3 as following:

x3 (n -2) = a -y 2 (n) -y 2 2 (n -2) b (16)
-Construction of message m : From system (3), we have:

m(n) = x3 (n + 1) -x2 (n) -Aẑ 2 (n) -B ẑ3 (n))
C Using three delays on the output and equation ( 16), we get: 17) allows constructing the message but only after the synchronization of the two continuous-time systems (1) and (4).

m(n -3) = a -y 2 (n) -y 2 2 (n -2) b ( 17
) - y 2 (n -3) + Aẑ 2 (n -3) + B ẑ3 (n -3) C Equation (
3) Demultiplexing block: At the reception, the received signal is first demultiplexed on two signals y 1 and y 2 . The signal y 1 witch is accessible during the period T 1 is now memorized during a period T 2 = 10T 1 (see figure 2). Then, the signal is introduced in the continuous observer. It should be noted that the sampling time T 2 = 10T 1 is chosen according to the Shannon's theorem. Another signal y 2 is accessible during 9 cycles and changes all cycles. This signal is introduced in the discrete observer (see figure 2).

III. SIMULATION RESULTS

In this section, the performances of the proposed method are evaluated using the Mtlab/Simulink software. First we will present the simulation results concerning the synchronization of the two continuous-time systems. The numerical value of M is equal to 23.16 and the gain K is equal to 10 and is chosen to ensure the stability of the observer (condition 10 given in section II-B.1). The initial conditions of the observer (4) are: z 1 (0) = 1.8, z 2 (0) = 3 and z 3 (0) = 0. The initial conditions of the transmitter system are given in section II-A.1.

A. Synchronization results of the two continuous-time systems

Figures (3, 5 and 7) and (4, 6 and 8) give the states and their synchronization errors, respectively. From t = 15s, we can note that the amount percentage for errors e 1 , e 2 and e 3 (Figures (4, 6 et 8)) are about 4.2%, 0.8% et 0.08%. This report allows us to assert that the errors are negligible, and all the states are perfectly estimated by the continuous observer.

B. Synchronization results of the two hybrid systems

In the following, we will present the simulation results concerning the synchronization of the two hybrid systems, i.e. the system (3) and the observer given in section (II-B.2). The new parameters A, B, C of the observer are: A = 0.04, B = 0.1 and C = 1 and the message to sent is a square signal with amplitude equal to C (the choice of theses amplitudes are justified just after the observer (3)). In our case, the continuous-time system (1) and its observer (4) still operate with the period T 1 = 0.001s. The same period is used by discrete-time system (3) and its observer (15-16-17). The outputs (y 1 and y 2 ) are transmitted in the transmission channel at each period T 2 = 10T 1. The Figures [START_REF] Ghanes | On simultaneous parameter identification and state estimation for cascade state affine systems[END_REF][START_REF] Koracev | Chaos-based cryptography : a brief overview[END_REF] and [START_REF] Hernault | Feasability of Analogue Realization of Sliding Mode Observer: Application to Data Transmission[END_REF][START_REF] Maggio | Nonlinear Analysis of the Colpitts Oscillator and Application to Design[END_REF] give the states and the synchronization errors of the hybrid system with its corresponding observer, respectively. Note that the states x 1 et x 3 don't depend on the states z 1 , z 2 and z 3 of the continuous-time system. This allows to establish that the error e 1 = x 1 -x1 vanishes after the period T 1 = 0.001s which corresponds to the delay of one step on the output. (agree with equation ( 15)), and the error e 3 vanishes after a period T = 2T 1 = 0.002s which correspond to a delay of two steps on the output (agree with equation [START_REF] Parlitz | Transmission of digital signals by chaotic synchronization[END_REF]. On other hand, the reconstitution of the message m (see equation ( 17)) depends on the synchronization of the states z 1 , z 2 and z 3 of the continuous-time transmitter (1) and the continuous receiver(4). From the previous figures (3, 5 and 7) and (4, 6 and 8)), we can notice that the time of the synchronization of the two systems (1) et ( 4) is obtained at t = 15s, and from this time the message is reconstituted (see figures 13 and 14).

IV. CONCLUSION

In this paper, a transmission scheme based on dynamic hybrid systems is designed for private communications. In the transmitter part, the continuous chaotic system of type Colpitts is combined with a discrete chaotic system of type Henon in order to materialize a complex structure of the transmitter that is robust especially to a known-plaintext attacks. Also, the system is made robust by introducing delays [START_REF] Zheng | Secure data transmission based on multi-input multi-output delayed chaotic system[END_REF] and continuous states, that don't ensure the observability recovery condition (this will be developed in our future works). In the transmission channel, the sent outputs are chosen with agreement with the rank of observability condition of the continuous-time system (Colpitts) and with the rank observability condition of the continuous-time system and the observability recovery condition of the discrete-time system (Henon). The strategy adopted to send the output signal consists with composition of the two outputs y 1 = z 1 (Colpitts) and y 2 = z 2 (Henon) in the heterogeneous output signal which is transmitted in the channel transmission. In our study, we have demonstrated that the reconstruction of the discrete-time transmitter message depends on the synchronization of the transmitter and the receiver (continuoustime observer) and the exchange of the information is performed in discrete form. The choice of the sampling time of the continuous-time system must be done with more attention but this make the identification of its parameters complex. The study is validated in simulation, since the obtained results show that the message is reconstructed succefully on the level of the receiver (discrete observer) if and only if the two continuous systems (Colpitts and its observer) are synchronized.

Fig. 1 .

 1 Fig. 1. Transmission chain based on a hybrid dynamical system

Fig. 2 .

 2 Fig. 2. Transmission cycles of y 1 and y 2

1 Fig. 4 .Fig. 5 . 2 Fig. 6 . 3 Fig. 8 .

 1452638 Fig. 3. z 1 and ẑ1

1 Fig. 10 . 1 0 3 Fig. 12 .Fig. 14 .

 110131214 Fig. 9. x 1 and x1


Dalila Zaltni
email: dalila.zaltni@ensea.fr

Malek Ghanes
email: ghanes@ensea.fr

Jean Pierre Barbot
email: barbot@ensea.fr

Mohamed Naceur Abdelkrim
email: naceur.abdelkrim@enig.rnu.tn

J-P Barbot

J-P Barbot

Synchronous Motor Observability Study and an Improved Zero-speed Position Estimation Design

This paper deals with the Permanent Magnet Synchronous Motor (PMSM) observability analysis for sensorless control design. The problem of loss of observability at low frequency range is always recognized in experimental settings. Nevertheless, there are no sufficient theoretical observability analyses for the PMSM. In the literature, only the sufficient observability condition has been presented. Therefore, the current work is aimed especially to the necessary observability condition analysis. Furthermore, an Estimator/Observer Swapping system is designed here for the surface Permanent Magnet Synchronous Motor (PMSM) to overcome position observability problems at zero speed which is an unobservable state point.

I. INTRODUCTION

Industries concerned by PMSMs are continuously seeking for cost reductions in their products. These reductions often impose the minimization of number of sensors used for control purposes because they substantially contribute to increase the complexity and cost of the full installation (additional cables, maintenance, etc.), and the default probability. This imposes the use of observers in order to realize sensor-less control design. Since many observers for electrical machines with sensor-less control are available, as the extended Kalman filter [START_REF] Bolognani | Extented Kalman Filter tuning in sensor-less PMSM drives[END_REF], the full order and the reduced order observers [START_REF] Solsona | A non linear reduced order observer for permanent magnet synchronous motors[END_REF], the LMI based methods [START_REF] Lian | LMI-based sensor-less control of permanent magnet synchronous motors[END_REF], the high-frequency signal injection methods ( [START_REF] Piippo | Signal injection in sensor-less PMSM drives equipped with inverter output filter[END_REF], [START_REF] Jang | Sensor-less drive of surfacemounted permanent magnet motor by high frequency signal injection based on magnetic saliency[END_REF]) the sliding mode observers ( [START_REF] Paponpen | An improved sliding mode observer for speed sensor-less vector control drive of PMSM[END_REF], [START_REF] Ghanes | On Sensorless Induction Motor drives: Sliding-Mode Observer and output Feedback Controller[END_REF]), the adaptive observers [START_REF] Ghanes | A robust output Feedback Controller of the induction motor drives: New design and experimental validation[END_REF], and so on, the main research stream has been focused on searching for reliable speed and position estimation methods for PMSM with the aim to replace the mechanical sensors with the observer in the control system ([9]- [START_REF] Wu | A permanent magnet motor drive without a shaft sensor[END_REF]). However, the current problems to successfully apply sensor-less control for PMSM are the existence of operating regimes for which the observer performance is remarkably deteriorated due to the difficulties in estimating correctly the motor position. The failure of sensor-less schemes in some particular operating conditions has been always recognized in experimental setting. In the case of induction motors, the observability has been studied by many authors ( [START_REF] Ghanes | Observability Study and Observer-Based Interconnected Form for sensor-less Induction Motor[END_REF], [START_REF] Ibarra-Rojas | Global observability analysis of sensor-less induction motors[END_REF]). Nevertheless, there are no sufficient theoretical observability studies for the PMSM. Only the sufficient observability condition has been presented in literature. For instance, in [START_REF] Junfeng | A New Control method for Permanent Magnet Synchronous Machines with Observer[END_REF], observability is analyzed in the case of constant high speed operation. In [START_REF] Vaclavec | Synchronous Machine Drive Observability Analysis for sensor-less Control Design[END_REF] and [START_REF] Vaclavec | Synchronous Machine Drive Observability Analysis and sensor-less Control Design[END_REF], the author gives only the sufficient observability condition (not necessary) of the PMSM in the particular case of constant speed. The current work is aimed especially to the necessary observability condition analysis. In [START_REF] Zaltni | Super Twisting Observer for Sensor-less Control of PMSM[END_REF] and [START_REF] Zaltni | Observability Analysis of PMSM[END_REF], we have given the sufficient condition of loss of observability for the Surface PMSM (SPMSM). In this paper, observability of both the Interior PMSM (IPMSM) and the SPMSM is studied and discussed at different operating conditions then the necessary and sufficient observability condition is presented. Furthermore, an Estimator/Observer Swapping system is proposed, for the SPMSM, to overcome position observability problems at zero speed which is an unobservable state point. The designed observer based on Higher Order Sliding Mode (HOSM) is used in order to ensure the robustness against disturbances and to avoid the chattering phenomenon [START_REF] Bartolini | Chattering avoidance by second-order sliding mode control[END_REF]. This paper is organized as follows: In the second section, mathematical models of both IPMSM and SPMSM are presented. In section three, the non linear observability is recalled. The observability analysis of both IPMSM and SPMSM is given in section four. The proposed observer is designed in section five. Simulation results are illustrated in section six. Finally, some concluding remarks are drawn in the last section.

II. SYNCHRONOUS MOTOR MODELS

A. Interior Permanent Magnet Case

The dynamic model of the IPMSM in the (α, β ) fixed coordinate is given by equation ( 1) and (2) ( [START_REF] Chen | An Extended Electromotive Force Model for Sensor-less Control of Interior Permanent-Magnet Synchronous Motors[END_REF], [START_REF] Zaltni | A HOSM Observer with an improved zero speed position estimation for Surface PMSM Control[END_REF]):

iα iβ = Γ -1 u α u β - R -2L 1 ω e sin(2θ e ) 2L 1 ω e cos(2θ e ) 2L 1 ω e cos(2θ e ) R + 2L 1 ω e sin(2θ e ) i α i β -ω e K e -sin(θ e ) cos(θ e ) (1) ωe
= P J [2L 1 (cos(θ e )i α + sin(θ e )i β ) + φ m ](-sin(θ e )i α + cos(θ e )i β ) - f v J ω e - T l J ( 2
)
where

Γ -1 = 1 L 2 0 -L 2 1 × L 0 -L 1 cos(2θ e ) -L 1 sin(2θ e ) -L 1 sin(2θ e ) L 0 + L 1 cos(2θ e
) ω e is the electric rotor speed; R is the stator resistance; P is the pair pole number; J is the moment of inertia; φ m is the rotor flux; f v is the viscous friction; T l is the load torque; [i α i β ] T and [u α u β ] T are the (α -β ) stator current and voltage vector respectively.

L 1 = L d -L q 2
and L 0 =

L d +L q
2 , where L d , L q are the (d-q) stator inductance components;

B. Surface Mounted Permanent Magnet Case

In the SPMSM we have L d = L q then L 1 = 0. Thus, from equation ( 1) and (2), we can deduce the dynamic model of the SPMSM:

iα iβ = 1 L 0 u α u β -R i α i β -ω e K e -sin(θ e ) cos(θ e ) (3) ωe
= P J φ m (-sin(θ e )i α + cos(θ e )i β ) - f v J ω e - T l J (4)
Where K e is the BEMF constant.

III. NONLINEAR OBSERVABILITY

In this section, the nonlinear observability is recalled [START_REF] Hermann | Non linear Controllability and Observability[END_REF]. We consider systems of the form:

∑ : ẋ = f (x, u) y = h(x) (5)
Where x ∈ X ⊂ R n is the state vector, u ∈ U ⊂ R m is the control vector, y ∈ R p is the output vector, f and h are C ∞ functions.

Definition 3-1(Locally weak observability) Consider the system ∑ and let x 0 be a point of the state space X.

• ∑ is locally weakly observable at x 0 if there exist an open neighborhood V of x 0 such that for every open neighborhood v of x 0 contained in V , I v (x 0 ) = x 0 and is locally weakly observable if it is so at every x ∈ X. • ∑ is locally regularly weakly observable at x 0 if it is locally weakly observable at x 0 and the n -1 derivatives outputs are sufficient to locally observe the system.

Rank Criterion

A sufficient locally regularly weakly observable condition at x 0 of ( 5) is that there exists (u, u, ..., ) such that:

rank(J) | x 0 = rank           dh dL f h dL 2 f h . . . dL n-1 f h           | x 0 = n (6) Remark 3-1
1. The notion of locally regularly weakly observability is introduced in order to design an observer of dimension equal to n.

2. The condition (6) depends on (u, u, ...) and this is an implicit justification of the universal inputs introduced in [START_REF] Gauthier | Observability for any u(t) of a class of nonlinear systems[END_REF].

IV. OBSERVABILITY ANALYSIS OF THE PMSM

Let's consider the state vector x = [i α , i β , θ e , ω e ] T and the output vector y = [i α , i β ] T . Voltages and currents are assumed to be measurable. The order of the state vector of the PMSM is n = 4. Thus, according to the observability rank criterion mentioned earlier, the PMSM is locally regularly weakly observable at x 0 for (u, u, ...) if the following condition is fulfilled:

rank(J)| x 0 ,(u, u,...) = 4 (7)
A. Observability analysis of the IPMSM Consider the system ( 1) and ( 2) as:

    ẋ1 ẋ2 ẋ3 ẋ4     =     (Λ 11 γ 1 + Λ 12 γ 2 )/(L 2 0 -L 2 1 ) (Λ 21 γ 1 + Λ 22 γ 2 )/(L 2 0 -L 2 1 ) x 4 T e -mx 4 -τ     (8)
Where

Λ = L 0 -L 1 cos(2θ ) -L 1 sin(2θ ) -L 1 sin(2θ ) L 0 + L 1 cos(2θ ) Λ i j is the i th row of the j th column of the matrix Λ γ 1 = u α -(R -2L 1 x 4 sin(2x 3 ))x 1 + 2L 1 x 4 cos(2x 3 )x 2 + x 4 K e sin(x 3 ) γ 2 = u β -(R -2L 1 x 4 sin(2x 3 ))x 2 -2L 1 x 4 cos(2x 3 )x 1 - x 4 K e cos(x 3 ) T e = P J [2L 1 (cos(x 3 )x 1 + sin(x 3 )x 2 ) + φ m ](-sin(x 3 )x 1 + cos(x 3 )x 2 ) is the electromagnetic torque. Let f (x) =     (Λ 11 γ 1 + Λ 12 γ 2 )/(L 2 0 -L 2 1 ) (Λ 21 γ 1 + Λ 22 γ 2 )/(L 2 0 -L 2 1 ) x 4 T e -mx 4 -τ     and h(x) = y.
Then, look at the vector of information generated from the output and its only first derivatives:

O 1 =     h 1 h 2 L f h 1 L f h 2     (9)
The associated observability matrix is:

J 1 =       1 0 0 0 0 1 0 0 ∂ L 1 f h 1 ∂ x 1 ∂ L 1 f h 1 ∂ x 2 ∂ L 1 f h 1 ∂ x 3 ∂ L 1 f h 1 ∂ x 4 ∂ L 1 f h 2 ∂ x 1 ∂ L 1 f h 2 ∂ x 2 ∂ L 1 f h 2 ∂ x 3 ∂ L 1 f h 2 ∂ x 4       (10)
The computation of the corresponding determinant gives:

∆ 1 = [[2L 1 sin(2x 3 )γ 1 + (L 0 -L 1 cos(2x 3 )) ∂ γ 1 ∂ x 3 -2L 1 cos(2x 3 )γ 2 -L 1 sin(2x 3 ) ∂ γ 2 ∂ x 3 ].[-L 1 sin(2x 3 ) ∂ γ 1 ∂ x 4 (11)
+ (L 0 + L 1 cos(2x 3 )) ∂ γ 2 ∂ x 4 ] -[-2L 1 cos(2x 3 )γ 1 -L 1 sin(2x 3 ) ∂ γ 1 ∂ x 3 -2L 1 sin(2x 3 )γ 2 + (L 0 -L 1 cos(2x 3 )) ∂ γ 2 ∂ x 3 ][(L 0 -L 1 cos(2x 3 )) ∂ γ 1 ∂ x 4 -L 1 sin(2x 3 ) ∂ γ 2 ∂ x 4 ]]/(L 2 0 -L 2 1 )
Case 1: IPMSM at zero speed: It is important to note that for interior permanent magnet synchronous motor L 1 is always different from 0, consequently the J 1 determinant at zero speed (x 4 = 0) is:

∆ 1 = [[2L 1 sin(2x 3 )(u α -Rx 1 ) -2L 1 cos(2x 3 )(u β -Rx 2 )].[-L 1 sin(2x 3 )(2L 1 sin(2x 3 )x 1 (12) + 2L 1 cos(2x 3 )x 2 + K e sin(x 3 )) + (L 0 + L 1 cos(2x 3 ))(-2L 1 cos(2x 3 )x 1 -2L 1 sin(2x 3 )x 2 -K e cos(x 3 ))]]/(L 2 0 -L 2 1
) Remark 4-1: Looking at the previous expression [START_REF] Wu | A permanent magnet motor drive without a shaft sensor[END_REF], we remark that at zero speed operation ∆ 1 depends on current and voltage. Therefore, we have always the opportunity to find again the observability property by injection of a continue current. Thus, we can conclude that the IPMSM is always observable.

B. Observability analysis of the SPMSM

In this section, we present the observability analysis of the SPMSM and we give a sufficient condition of loss of the observability property. In this case, we have L d = L q then L 1 = 0. Therefore, the expression of the determinant of J 1 given in [START_REF] Nakashima | Sensor-less initial rotor position estimation of surface permanent magnet synchronous motor[END_REF] becomes:

∆ 1 = -K 2 e x 4 ( 13
)
Remark 4-2: The determinant ∆ 1 is dependant only on x 4 . Thus, for the considered output and only its first derivative, the SPMSM is locally weakly observable at x 0 if x 4 = 0. This condition is independent on the considered input u α,β . Now the question is to look if higher derivatives of output overcome the observability singularity at zero speed (x 4 = 0). For that, let's consider the model of the SPMSM given by equations ( 3) and (4) in the form of (5) where :

f (x, u) =     ax 1 + bx 4 sin(x 3 ) + cu α ax 2 -bx 4 cos(x 3 ) + cu β x 4 k t (-sin(x 3 )x 1 + cos(x 3 )x 2 ) -mx 4 -τ     and h(x) = [x 1 , x 2 ] T , with a = -R L 0 , b = K e L 0 , c = 1 L 0 , k t = pφ m J , m = f v J and τ = T l J .
Let's look to the following vector of information generated from:

O 2 =         h 1 h 2 L f h 1 L f h 2 L 2 f h 1 L 2 f h 2         (14)
The associated observability matrix is:

J 2 = ∂ ∂ x O 2 (15)
Condition ( 7) can be tested by searching for a regular matrix constructed from any four rows of matrix J 2 . Let's consider only the 1 st , 2 nd , 5 th and the 6 th rows of J 2 as.

J 2 =       1 0 0 0 0 1 0 0 ∂ L 2 f h 1 ∂ x 1 ∂ L 2 f h 1 ∂ x 2 ∂ L 2 f h 1 ∂ x 3 ∂ L 2 f h 1 ∂ x 4 ∂ L 2 f h 2 ∂ x 1 ∂ L 2 f h 2 ∂ x 2 ∂ L 2 f h 2 ∂ x 3 ∂ L 2 f h 2 ∂ x 4       (16)
The computation of the corresponding determinant gives:

∆ 2 = ∂ L 2 f h 1 ∂ x 3 . ∂ L 2 f h 2 ∂ x 4 - ∂ L 2 f h 2 ∂ x 3 . ∂ L 2 f h 1 ∂ x 4 = b 2 [-a 2 + am + 2k t (-cos(x 3 )x 1 (17) -sin(x 3 )x 2 )]x 4 -2b 2 x 3 4 -b 2 (a -m) ẋ4
From equation ( 17) the observability loss for the considered output and only its first and second derivatives is ∆ 2 = 0. The associated manifold of unobservability is given by Ω = {x : ∆ 2 (x) = 0 and ∆ 1 = 0}. Remark 4-3 In (17), at zero speed x 4 = 0, it is obvious that the SPMSM is locally weakly observable for ẋ4 = 0. This is less restrictive than condition ∆ 1 = 0 given by [START_REF] Ghanes | Observability Study and Observer-Based Interconnected Form for sensor-less Induction Motor[END_REF].

Case 2: SPMSM at zero speed and acceleration

The problem now it is to consider the particular case where ẋ4 = x 4 = 0 (zero speed and acceleration), and to look if possible to recover the observability of SPMSM by using the higher order derivatives (greater than 2) of the output. First: consider zero acceleration ( ẋ4 = 0) The model of the SPMSM used in this case is given by ( 3)-(4) in the form of [START_REF] Jang | Sensor-less drive of surfacemounted permanent magnet motor by high frequency signal injection based on magnetic saliency[END_REF] where the function f (x, u) is replaced by

f 0 (x, u) =     ax 1 + bx 4 sin(x 3 ) + cu α ax 2 -bx 4 cos(x 3 ) + cu β x 4 0     and h(x) = [x 1 , x 2 ] T ,
Consider now the vector of information generated by the output and its first, second and third derivatives:

O 3 =             h 1 h 2 L f 0 h 1 L f 0 h 2 L 2 f 0 h 1 L 2 f 0 h 2 L 3 f 0 h 1 L 3 f 0 h 2             (18)
The associated observability matrix is:

J 3 = ∂ ∂ x O 3 =             1 0 0 0 0 1 0 0 a 0 bx 4 cos(x 3 ) bsin(x 3 ) 0 a bx 4 sin(x 3 ) -bcos(x 3 ) a 2 0 c 5 d 5 0 a 2 c 6 d 6 a 3 0 c 7 d 7 0 a 3 c 8 d 8             (19)
with 2 4 cos(x 3 ) d 6 = -abcos(x 3 ) + 2bx 4 sin(x 3 ) c 7 = a 2 bcos(x 3 )x 4 + (-absin(x 3 )x 4bx 2 4 cos(x 3 ))x 4 d 7 = a 2 bsin(x 3 ) + abx 4 cos(x 3 )bx 2 4 sin(x 3 ) + (abcos(x 3 ) -2bx 4 sin(x 3 ))x 4 c 8 = a 2 bsin(x 3 )x 4 + (abcos(x 3 )x 4bx 2 4 sin(x 3 ))x 4 d 8 = -a 2 bcos(x 3 ) + abx 4 sin(x 3 ) + bx 2 4 cos(x 3 ) + (absin(x 3 ) + 2bx 4 cos(x 3 ))x 4 Remark 4-4 In ( 19), at zero acceleration ẋ4 = 0, it is obvious that the SPMSM is locally weakly observable for x 4 = 0.

c 5 = abx 4 cos(x 3 ) -bx 2 4 sin(x 3 ) d 5 = absin(x 3 ) + 2bx 4 cos(x 3 ) c 6 = abx 4 sin(x 3 ) + bx
Second: consider also x 4 = 0 (this corresponds to zero acceleration and speed) In this case the observability matrix J 3 [START_REF] Zaltni | Observability Analysis of PMSM[END_REF] becomes:

J 4 =             1 0 0 0 0 1 0 0 a 0 0 bsin(x 3 ) 0 a 0 -bcos(x 3 ) a 2 0 0 absin(x 3 ) 0 a 2 0 -abcos(x 3 ) a 3 0 0 a 2 bsin(x 3 ) 0 a 3 0 -a 2 bcos(x 3 )            
∂ ∂ x L k f 0 h i = a ∂ ∂ x L k-1 f 0 h i | x 4 =0 (21)
k = 2, 3, i = 1, 2. and can be generalized for higher derivatives. Thus, the higher derivatives of the output greater than three do not recover additional information. Remark 4-6: Equations ( 20) and [START_REF] Chen | An Extended Electromotive Force Model for Sensor-less Control of Interior Permanent-Magnet Synchronous Motors[END_REF] show that for the considered output (currents) and its derivatives at any order, the condition ẋ4 = x 4 = 0 (zero speed and acceleration) generates a structural subset of indistinguishability {x : x 4 = 0 and ẋ4 = 0}. A physical interpretation is related to have the Back Electromotive Forces (BEMFs) equal to zero at any time for x 4 = ẋ4 = 0, then any information with respect to the SPMSM rotor position is in the dynamics of stator currents.

V. THE OBSERVER DESIGN

In order, to overcome the observability problems of the SPMSM at zero speed, we propose here an Estimator/Observer swapping system based on the "Super Twisting Algorithm" (STA). Thus, in this section, the used STA is recalled and then applied to the SPMSM. The convergence of the developed observer is studied.

A. The Super Twisting Algorithm

The general form of the STA is defined as follows [START_REF] Fridman | Sliding modes of higher order as a natural phenomenon in control theory[END_REF]:

u(e 1 ) = u 1 + λ 1 |e 1 | 1 2 sgn(e 1 ) u1 = α 1 sgn(e 1 ) (22)
with e 1 = x 1 -x1 , λ 1 , α 1 > 0 are the observer parameters, u 1 is the output of the observer, x 1 is the estimated variable and:

sgn(e 1 ) =    1 i f e 1 > 0 -1 i f e 1 < 0 ∈ [-1 1] i f e 1 = 0

B. Application to Surface PMSM

Let e α and e β be the BEMFs. Consider only current dynamic equations of the SPMSM, we can write:

ẋ1 = ax 1 + x a + cu α ẋ2 = ax 2 + x b + cu β (23)
where e α = -ω e sin(θ e ) e β = ω e cos(θ e )

and

[x a x b ] = -b[e α e β ] (25)
[x a x b ] is the vector of unknown variables. Currents and voltages are assumed to be measurable. Applying the STA [START_REF] Zaltni | A HOSM Observer with an improved zero speed position estimation for Surface PMSM Control[END_REF] to system [START_REF] Hermann | Non linear Controllability and Observability[END_REF], we obtain systems ( 26) and ( 27):

ẋ1 = xa + ax 1 + cu α + λ 1 |e 1 | 1 2 sgn(e 1 ) ẋa = α 1 sgn(e 1 ) (26)
ẋ2 = xb + ax 2 + cu β + λ 2 |e 2 | 1 2 sgn(e 2 ) ẋb = α 2 sgn(e 2 ) (27)
Where e 1 = x 1 -x1 , e 2 = x 2 -x2 and λ 1 , λ 2 , α 1 , α 2 are positive constants that will be given later. xa and xb are the estimated values of the unknown variables x a and x b .

According to equations ( 23), ( 26) and ( 27), error dynamics of the observer are given by: Following the results proposed in [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF] and [START_REF] Perruquetti | Sliding mode Control in engineering[END_REF] with respect to the STA ( 22) dedicated to the observer design given by equations ( 26) and ( 27), we set:

ė1 = e a -λ 1 |e 1 | 1 2 sgn(e 1 ) ėa = f 1 (x b ) -α 1 sgn(e 1 ) (28)
Corollary: For any initial conditions x(0), x(0), there exists a choice of λ i and α i such that the error dynamics e 1 , e 2 , ė1 and ė2 converge to zero and by consequence xa -→ x a and xb -→ x b . (See proof in [START_REF] Zaltni | A HOSM Observer with an improved zero speed position estimation for Surface PMSM Control[END_REF]) Figure 1 shows the finite time convergence of the proposed observer. Parameters of the observer are given by :

α 1 > f + 1 and λ 1 > ( f + 1 + α 1 ) 2 α 1 -f + 1 (30) α 2 > f + 2 and λ 2 > ( f + 2 + α 2 ) 2 α 2 -f +

C. Position and speed estimation

Having the estimated value of x a and x b we can easily deduce the rotor position and speed using equations ( 24), [START_REF] Fridman | Sliding modes of higher order as a natural phenomenon in control theory[END_REF], [START_REF] Levant | Robust exact differentiation via sliding mode technique[END_REF] and [START_REF] Perruquetti | Sliding mode Control in engineering[END_REF]. When the motor operates out of the unobservable region, the rotor position can be calculated as:

θe = arctan 2 ( -xa xb ) (32)
However, as it is shown in section four, the rotor position is not observable at zero speed and acceleration because xa and xb are non existent in this condition and then we can not use the observer equation (32). For this reason, we propose here an Estimator/Observer swapping system which allows the use of the observer at high speed and swap automatically to the estimator when the speed becomes under a defined very low value. The estimated position is calculated as:

θe = t 0 | ωe |dt + cte (33)
Where

| ωe | = x2 a + x2 b b (34)
The initial value of the estimated position ( θe (0) = cte) is equal to the last value computed by the observer (32) before swapping to the estimator. The estimated speed is calculated as:

ωe = x2 a + x2 b b sgn( xa sin( θe ) -xb cos( θe )) (35)

VI. SIMULATION RESULTS

The used motor in the simulation testing is a three-phase SPMSM. The specifications and parameters are : Rated Power P n = 1.7kW ; Rated speed ω n = 157rad.s -1 ; Rated voltage U n = 380V; Rated current I n = 3.8A; Number of pole pairs P = 3; Stator inductance L 0 = 0.027H; Stator resistance R = 3.3Ω; Rotor flux φ m = 0.341; Rotor inertia J = 0.0026kg.m 2 ; Viscous friction f v = 0.0034kg.m 2 .s -1 . The proposed observer is tested in open loop to the benchmark trajectories [START_REF] Glumineau | www.irccyn.ec-nantes.fr/hebergement/bancEssai[END_REF] presented in Fig. 2. In this benchmark, two reference trajectories are defined: The reference rotor speed (Fig. 2(a)) and the load torque (Fig. 2(b)). In this work, we are interesting only to the observability of the SPMSM. Two tests are carried out. In the first test, we use only the observer ( Fig. 3). In this case, we remark that the estimated position and speed reach the real ones with good accuracy and robustness when the motor operates out of the unobservable condition. However, at zero speed and acceleration, the rotor position is not observable. In the second test, we use the proposed Estimator/Observer swapping system (Fig. 4). Thus, in this case, we show that the rotor position can be obtained at all range of speed.

VII. CONCLUSION

In this paper, the observability analysis of both the SPMSM and the IPMSM has been presented and discussed at different operating conditions. A necessary and sufficient observability condition has been presented. Furthermore, in order to improve the rotor position estimation at zero speed, an Estimator/ Observer swapping system has been designed for the SPMSM. The convergence of the observer is proved. Some simulation results has been presented to illustrate the performance of the proposed Estimator/Observer swapping system compared to the results obtained when using only the observer.

( 20 ) 4 - 5 :

 2045 Remark From the equation (20) a recurrence relation can be obtained

ė2 = e b -λ 2 |e 2 | 1 2

 1 sgn(e 2 ) ėb = f 2 (x a ) -α 2 sgn(e 2 ) (29) With e a = x axa , e b = x bxb , f 1 (x b ) = ω e x b and f 2 (x a ) = -ω e x a

2 =Fig. 1 .

 21 Fig. 1. The trajectory ė1 = f (e 1 ): Finite time convergence.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Benchmark trajectories: (a) Reference speed (rad/s), (b) Load torque(N.m)