Ramdane Tami
email: ramdane.tami@unicaen.fr

Driss Boutat

Gang Zheng

Frédéric Kratz

Rachid El Gouri

Rotor speed, load torque and parameters estimations of a permanent magnet synchronous motor using extended observer forms

This paper presents a new approach to estimate states and parameters of the permanent magnet synchronous motor (PMSM) in the presence of unknown load torque disturbance. Indeed, it highlights an auxiliary dynamics that is added to the PMSM model. Thereby, it supplies a change of coordinates that transforms the extended model (PMSM model together with the auxiliary dynamics) into the so-called extended nonlinear observer forms. It turns out that the obtained form provides two observer normal forms. One of them supports the well-know high gain observer which is used to estimate the speed and the unknown load torque. The second one which is affine in the parameters enables us to estimate simultaneously the states and motor parameters by means of an adaptive observer. Performance of the proposed method is illustrated by the simulations results.

Introduction

Permanent magnet synchronous motor (PMSM) is increasingly used in several industrial applications, like renewable energy and Maglev (derived from magnetic levitation) due to some important features as stability, high power density and large torque to current ratio (see [START_REF] Zhu | A nonlinear state observer for the sensorless control of a permanent-magnet ac machine[END_REF], [START_REF] Leonhard | Control of electrical drives[END_REF]). However, the control of the PMSM is a complex task due to the presence of nonlinearities in its model, load torque perturbations and wear occurring in mechanical sensors. This leads to, inter alia, to the development of sensorless control method which can be represented by three main approaches: the first one is based on back-emf technique (see [START_REF] Ogasawara | An approach to position sensorless drive for brushless dc motors[END_REF], [START_REF] Nahid-Mobarakeh | Mechanical sensorless control of pmsm with online estimation of stator resistance[END_REF], [START_REF] Gan | An adaptive nonlinear extended state observer for the sensorless speed control of a pmsm[END_REF]) whose advantage is the simplicity of implementation. However, it should be avoided in low speed region because on one hand, the velocity and the flux estimation are sensitive to the stator resistance and on the other hand, the noise amplitude is higher than back-emf. The second one is the model reference adaptive system (MRAS) method in which the performance depends on the accuracy of the used model (see [START_REF] Liang | Sensorless control of pm synchronous motors based on mras method and initial position estimation[END_REF], [START_REF] Rashed | A stable back-emf mras-based sensorless low-speed induction motor drive insensitive to stator resistance variation[END_REF]). Finally, the third one, that will be considered in 1 this paper, is based on the observer design as in ( [START_REF] Zheng | Sensorless control of pmsm based on extended kalman filter[END_REF], [START_REF] Boussak | Implementation and experimental investigation of sensorless speed control with initial rotor position estimation for interior permanent magnet synchronous motor drive[END_REF]).

A high performance of a control requires measurement and identification of unknown parameters. Therefore, an accurate estimation of unavailable state variables is needed. In order to reach this goal, numerous observer strategies have been developed to estimate states and parameters of PMSM. Among these strategies, for example, we find exact linearization method (see [START_REF] Chiasson | Nonlinear speed observer for the pm stepper motor[END_REF], [START_REF] Zhu | Speed tracking control of a permanent-magnet synchronous motor with state and load torque observer[END_REF]), adaptive observer [START_REF] Besanc ¸on | On adaptive observers for state affine systems[END_REF], sliding mode observer (see [START_REF] Ghanes | On sensorless induction motor drives: sliding-mode observer and output feedback controller[END_REF], [START_REF] Ali | Sliding mode state observation for non-linear systems[END_REF], [START_REF] Qiao | New sliding-mode observer for position sensorless control of permanent-magnet synchronous motor[END_REF]) and extended Kalman filter (see [START_REF] Bolognani | Sensorless full-digital pmsm drive with ekf estimation of speed and rotor position[END_REF], [START_REF] Janiszewski | Extended kalman filter based speed sensorless pmsm control with load reconstruction[END_REF]).

Another interesting approach can be found in [START_REF] Hamida | An adaptive interconnected observer for sensorless control of pm synchronous motors with online parameter identification[END_REF] where an adaptive interconnected observer has been proposed. Indeed, the dynamical system of PMSM was considered as an interaction between two sub dynamical systems: one contains the dynamics of current i q , the stator resistance R s and the second one is governed by the dynamics of current i d , angular speed Ω and load torque T l .

In the literature, several works dealt with observer design for PMSM on the one hand, to avoid the use of tachometer which is expensive, on the other hand to ensure redundancy of information. However, few solutions have been proposed to estimate simultaneously the states, disturbances and all parameters which are required for example in diagnosis and fault-tolerant control.

When a nonlinear dynamical system contain unavailable states for measurements, then it becomes a difficult task to directly apply the standard estimation methods, such as the Luenberger's observer or Kalman's filter. To overcome this problem, Krener and Isidori [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF] introduced the so-called linearization of a nonlinear dynamical system by means of a change of coordinates and output injection. This enables us to use the well-known Luenberger's observer for a class of nonlinear dynamical systems. Since, this concept has been used to the multiple outputs case in ( [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF] and [START_REF] Xia | Nonlinear observer with linearizable error dynamics[END_REF], [START_REF] Boutat | Observer error linearization multioutput depending[END_REF], [START_REF] Zheng | Multi-output dependent observability normal form[END_REF]). Afterwards, several variations have been developed in ( [START_REF] Keller | Nonlinear observer design by transformation into a generalized observer canonical form[END_REF], [START_REF] Phelps | On constructing nonlinear observers[END_REF], [START_REF] Rudolph | A block triangular nonlinear observer normal form[END_REF], [START_REF] Hou | Observer with linear error dynamics for nonlinear multi-output systems[END_REF], [START_REF] Marino | Nonlinear control design: geometric, adaptive and robust[END_REF], [START_REF] Boutat | New algorithm for observer error linearization with a diffeomorphism on the outputs[END_REF], [START_REF] Lynch | Nonlinear observers with approximately linear error dynamics: the multivariable case[END_REF] and [START_REF] Lopez | Linearization by completely generalized input ouput injection[END_REF], [START_REF] Zheng | Output dependent observability linear normal form[END_REF], [START_REF] Zheng | Synchronisation of chaotic systems via reduced observers[END_REF], [START_REF] Bououden | A triangular canonical form for a class of 0-flat nonlinear systems[END_REF]). Another interesting form is the so-called output depending observer normal form introduced by [START_REF] Respondek | Time scaling for observer design with linearizable error dynamics[END_REF], and then improved in [START_REF] Zheng | A single output dependent observability normal form[END_REF] and [START_REF] Wang | Observer design using a generalized time-scaled block triangular observer form[END_REF]. Recently, the so-called extended observer normal form was proposed in ( [START_REF] Jouan | Immersion of nonlinear systems into linear systems modulo output injection[END_REF], [START_REF] Noh | Nonlinear observer design by dynamic observer error linearization[END_REF], [START_REF] Back | Dynamic observer error linearization[END_REF], [START_REF] Boutat | Geometrical conditions for observer error linearization via 0[END_REF] and [START_REF] Boutat | On the transformation of nonlinear dynamical systems into the extended nonlinear observable canonical form[END_REF]). The idea behind this normal form is based on adding an auxiliary dynamics into the original system such that the augmented system fulfills conditions which guarantee the transformation of the nonlinear system into an extended observer normal form. Even now, the process of transforming nonlinear dynamical systems into nonlinear observer forms for a wide class of nonlinear systems keep going, as the extended output depending observer normal form is proposed in [START_REF] Tami | Extended output depending normal form[END_REF].

This paper presents a new approach to design nonlinear observer for PMSM by adding an auxiliary dynamics to the PMSM model such that the augmented dynamical system admits a change of coordinates that brings it into the so-called extended nonlinear observer form. This form enables us to effectively solve the problem of simultaneous estimation of PMSM states and parameters [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF]. The present work improves our previous work in [START_REF] Tami | Nonlinear observer for the pm synchronous motor[END_REF]. In fact, [START_REF] Tami | Nonlinear observer for the pm synchronous motor[END_REF] is focused only on reaching the observer normal forms for PMSM model. Hence simulations were rather did for the mathematical model without worrying on the physical model. Therefore it contains many gaps and defects. The present study is more complete because it addresses the problem of estimation of rotor position, rotor speed, load torque, currents and the parameters. The quality of the estimation, despite the presence of a varying load torque, shows the robustness of this method to the disturbances. The rest of this paper is organized as follows. Section 2 presents the dynamical model of PMSM and recalls the extended nonlinear observer form structure. Section 3 states an auxiliary dynamics and a change of coordinates that transforms the extended PMSM model into an extended nonlinear observer form. Furthermore, two cases will be considered. The first one assumes the knowledge of the parameters of PMSM and the second one does not. Section 4 provides simulation results that highlight the efficiency of the proposed approach.

Mathematical model of PMSM and Problem statement

In order to obtain a dynamical model that makes simpler the estimation and control procedures, the electrical and mechanical dynamics of PMSM are expressed in the direct and quadrature axis reference frame by transforming the original three-phase on two fictitious phase winding. The direct phase winding is parallel to the rotor magnet flux, and the quadrature phase winding is orthogonal to the direct phase winding. Based on states observer, the sensorless control system is shown in Figure 1. To avoid the flux weakening, a zero electric current is required in the direct phase, then the generated electromagnetic torque is given as product of constant permanent flux and electric current in direct axis. In this present study, we consider the cases of non-saliency rotor (L d = L q = L s ). The dynamical model of PMSM in (d -q) reference frame is governed by the following set of differential equations (see [START_REF] Chiasson | Modeling and High-performance Control of Electric Machines[END_REF], [START_REF] Petrovic | Interconnection and damping assignment approach to control of pm synchronous motors[END_REF]).

dθ dt = Ω (1)
dΩ dt = - f v J Ω + p J φ f i q - 1 J T l (2)
di d dt = - R s L s i d + pΩi q + 1 L s u d (3) di q dt = - R s L s i q -pΩi d -p 1 L s φ f Ω + 1 L s u q (4)
where [θ, Ω, i d , i q ] T represents the state vector of the rotor angular position, the rotor mechanical speed and the output currents respectively. u d and u q are the d-axis and q-axis stator voltages, respectively. T l is the load torque considered as an unknown exogenous input, R s is the winding stator resistance, L s is the stator winding inductance, φ f is the permanent magnet flux, p is the number of the pair poles, J is the rotor moment of inertia, and f v is the viscous friction coefficient. Within this work it is assumed that the available measurements are the currents in the (d -q) reference frame. Thus, the outputs of the above dynamical system are as follows

y 1 = h 1 = i d (5) y 2 = h 2 = i q (6)
Motivated by the design of robust controllers and faults diagnosis strategies for PMSM applications, as it has been widely mentioned above in the introduction, many works had been conducted to estimate states and parameters of PMSM. Hereafter, we will present a new approach which enables us to:

• estimate the rotor mechanical speed Ω and the load torque T l when all motor parameters are known;

• estimate the parameters R s , L s and φ f ;

• consequently, estimate the rotor angular position from the estimation of rotor mechanical speed.

In order to design an observer for PMSM model, it is necessary to study its observability property. This also enables us to effectively manage the loss of observability when it occurs. The rank observability conditions (see for example [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF]) for system (2)-( 4) together with the measurable outputs i d and i q is the rank of the following observability matrix

O =     dh 1 dh 2 dL f h 1 dL f h 2     =      0 1 0 0 0 1 pi q -Rs Ls pΩ -p Ls φ f + pi d -pΩ -Rs Ls     
A straightforward calculation shows that the rank of the above matrix is

rank(O) = 3 if and only if i d = - φ f
Ls or i q = 0. As stated in the introduction, we will use the well-known estimation approach based on the extended nonlinear observer normal form. In order to do so, let us recall the basic idea and some properties of this concept. Consider the following nonlinear system

ẋ = f (x) y = h(x) (7)
where x ∈ U ⊆ R n represents the state and y ∈ R denotes the outputs. We assume that the vector field f and the output function h are sufficiently smooth. It is also assumed that the pair (h, f ) satisfies the observability rank condition. Thus, the 1-forms θ i = dL i-1 f h, for 1 ≤ i ≤ n, are linearly independent where L i-1 f h is the (i -1) th Lie derivative of the output h along the vector field f and d the usual differential.

When system (7) cannot be transformed into an observer normal form by means of the socalled linearization by output injection [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], then we can overcome this limitation by using the immersion concept (dynamics extension). The idea is to seek for an auxiliary dynamics ẇ = η(y, w) with w ∈ R so that the following extended dynamical system

ẋ = f (x) (8) ẇ = η(y, w) (9) y = h(x) (10)
might be transformed by means of a change of coordinates (ξ T , w) T = φ(x, w) to the following extended observer normal form ξ = A(y, w)ξ + B(y, w)

(11) ẇ = B n+1 (y, w) (12) y = Cξ (13)
where w ∈ R is an auxiliary variable considered as an extra output, C = [0, ..., 0, 1], and

A (y, w) =        0 • • • 0 α 2 (y, w) . . . . . . 0 . . . . . . . . . 0 • • • 0 α n (y, w) 0        (14)
where α i (y, w) = 0 for 2 ≤ i ≤ n are functions depending only on the output y and the extra output w.

This extended nonlinear observer normal form supports the high gain observer to estimate the state of nonlinear dynamical system [START_REF] Busawon | A simple observer for a class of nonlinear systems[END_REF]. Moreover, in the case of PMSM model we will see that B(.) is affine on some parameters, therefore the adaptive observer can be used to estimate simultaneously the states and the parameters (see [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF], [START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear timevarying systems[END_REF]).

Change of coordinates and Extended dynamics

This section provides an auxiliary dynamics together with a change of coordinates that transforms the extended PMSM model ( 2)-( 4) into an extended nonlinear observer normal form.

Let us consider the equations ( 2)-( 4) together with an auxiliary dynamics presented as follows

ẋ1 = - f v J x 1 + p J φ f x 3 - 1 J T l (15) ẋ2 = - R s L s x 2 + px 1 x 3 + 1 L s u d ( 16
) ẋ3 = - R s L s x 3 -px 1 x 2 -p 1 L s φ f x 1 + 1 L s u q (17) ẇ = η (y, w) (18)
y 1 = x 2 y 2 = x 3 where [x 1 , x 2 , x 3 ] = [Ω, i d , i q ], y = [y 1 , y 2 ]
T and w is an auxiliary variable considered as an extra output.

Then the following result provides the sought auxiliary dynamics η (y, w) and the change of coordinates.

Theorem 1. For any function κ(w) = 0, consider the following auxiliary dynamics

ẇ = - f v J κ(w) (19)
Then, the following change of coordinates

z 1 = x 1 e -w 0 1 κ(s) ds (20) z 2 = x 2 (21) z 3 = x 3 (22)
transforms the nonlinear dynamical system ( 15)-( 17) into following form

ż1 = α 1 (w) T l + β 1 (y, w) (23)
ż2 = α 2 (y, w) z 1 + β 2 (y, w) + 1 L s u d ( 24
)
ż3 = α 3 (y, w) z 2 + β 3 (y, w) + 1 L s u q (25)
where • Function κ(w) = 0 is a free function. It has to be chosen in order to ensure the boundedness the auxiliary dynamics. For our goal, more details on this function are given in simulation section.

α 1 (y, w) = -1 J e -w 0
• As it can be seen, the PMSM outputs considered here are continuous-time measurements.

In practice, with high frequency sampling device, the measurement can be considered as continuous. Even with a very low sampling sensor (rarely), thanks to our observer normal form, it is easy to adapt the observer proposed in [START_REF] Farza | Adaptive observers for a class of uniformly observable systems with nonlinear parametrization and sampled outputs[END_REF] to work with sampled-outputs.

Proof. The time derivative of the state variable z 1 given in equation ( 20) leads to

ż1 = - 1 J e -w 0 1 κ(s) ds T l + p J φ f y 2 e -w 0 1 κ(s) ds . (26)
by setting

α 1 (y, w) = - 1 J e -w 0 1 κ(s) ds β 1 (y, w) = p J φ f y 2 e -w 0 1 κ(s) ds
hence equation ( 26) can be rewritten in the following form:

ż1 = α 1 (y, w)T l + β 1 (y, w)
Now by replacing the variables x 2 = z 2 and x 3 = z 3 in equations ( 16) and ( 17), we obtain

ż2 = py 2 e w 0 1 κ(s) ds z 1 - R s L s y 1 + 1 L s u d (27) = α 2 (y, w)z 1 + β 2 (y, w) + 1 L s u d and ż3 = -p y 1 + φ f L s e w 0 1 κ(s) ds z 2 - R s L s y 2 + 1 L s u q (28)
= α 3 (y, w)z 2 + β 3 (y, w).

Putting together equations ( 26), ( 28) and ( 29) to meet the stated nonlinear extended normal form of the above theorem. Now, to design an observer for PMSM relying on the extended nonlinear observer normal form obtained in Theorem 1, two case studies are to be considered:

• estimation of rotor mechanical speed z 1 = ω and the load torque T l by using equations ( 23) and ( 25) where all parameters of the system are assumed known,

• estimation of z 1 = ω, the load torque T l , and the parameters R s , L s and φ f by involving dynamics ( 23) and ( 24).

Remark 2. As proposed in [START_REF] Hamida | An adaptive interconnected observer for sensorless control of pm synchronous motors with online parameter identification[END_REF], we will use the following observer to estimate the rotor position

d θ dt = Ω + L θ i q -i q (29)
where L θ is a positive constant design parameter.

Estimation of motor speed and the load torque with the known parameters

In this subsection, it is assumed that all parameters of PMSM are known and the load torque T l is a piece-wise constant function ( Ṫl = 0). Now, in order to estimate the rotors speed and the load torque, we consider equation Ṫl = 0 together with dynamics ż1 and ż3 obtained from the extended nonlinear observer form given in Theorem 1. Thus, we consider the following nonlinear extended observer normal form:

  Ṫl ż1 ż3   =   0 0 0 α 1 (y, w) 0 0 0 α 3 (y, w) 0     T l z 1 z 3   +   0 β 1 (y, w) β 3 (y, w, u)   ẇ = -fv J κ(w) (30)
where

α 1 (y, w) = -e -w 0 1 κ(s) ds J , α 3 (y, w) = -p y 1 + φ f Ls e w 0 1 κ(s) ds , β 1 (y, w) = p J φ f y 2 e -w 0 1 κ(s) ds
and β 3 (y, w, u) = -Rs Ls y 2 + 1 Ls u q In order to design an observer for the above dynamical system, let us set:

A(y, w) =   0 0 0 α 1 (y, w) 0 0 0 α 3 (y, w) 0   and B(y, u) =   0 β 1 (y, w) β 3 (y, w, u)   .
Now, as in [START_REF] Busawon | A simple observer for a class of nonlinear systems[END_REF], let us consider this new dynamical system:

ξ = A(y, w) ξ + B(y, w, u) -Γ -1 (y) R -1 ρ C T (C ξ -y) 0 = ρR ρ + G T R ρ + R ρ G -C T C ( 31
)
where

G =   0 0 0 1 0 0 0 1 0   , Γ(y, w) =   α 1 α 3 0 0 0 α 3 0 0 0 1  
and from the last algebraic equation we can deduce that:

R ρ =   6 ρ 5 -3 ρ 4 1 ρ 3 -3 ρ 4 2 ρ 3 -1 ρ 2 1 ρ 3 -1 ρ 2 1 ρ   It is worth noting that the term -Γ -1 (y) R -1
ρ C T has the following simple expression:

-ρ 3 α 1 α 3 , -3 ρ 2 α 3 , -3ρ T .
It is well-known from [START_REF] Busawon | A simple observer for a class of nonlinear systems[END_REF] that if the output y and the extra-output w, functions α 1 (y, w) and α 3 (y, w) are bounded, then the dynamical system ( 31) is an observer for [START_REF] Krener | Nonlinear observers with linearizable error dynamics[END_REF]. Thus the error observation e = ξ -ξ which is governed by the following dynamics:

ė = ξ -ξ = A(y, w) -Γ -1 (y, w)R -1 ρ C T C e = A(y, w)e
where the matrix A(y, w) has the following expression A(y, w)

=    0 0 -ρ 3 α 1 α 3 α 1 0 -3 ρ 2 α 3 0 α 3 -3ρ    , is
exponentially stable by an appropriate choice of ρ.

As in many works that deal with high gain observers, in our normal form we don't need to do any further assumption (as Lipschitz condition) on B(y, w, u) because it depends only on measurable quantities.

Estimation of motor speed, the load torque and identification parameters

In this subsection, it is assumed that parameters φ f , L s and R s are unknown. Then, in order to deal with simultaneously estimation of states and parameters of dynamical system (2)-( 4), we consider equation Ṫl = 0 together with dynamics ż1 and ż2 obtained from the extended nonlinear observer forms given in Theorem 1. Thus, we consider the following extended nonlinear observer forms

         Ṫl = 0 ż1 = -1 J e -w 0 1 κ(s) ds T l + p J φ f y 2 e -w 0 1 κ(s) ds ż2 = py 2 e w 0 1 κ(s) ds z 1 -Rs Ls y 1 + 1 Ls u d ẇ = -fv J κ(w)
It is clear that this form is of the following adaptive observer form (affine on parameters)

ż = A(w, y, u)z + φ(w, u, y)δ y = Cz (32)
where

z = T l z 1 z 2 T , δ = φ f Rs Ls 1 Ls T A(w, y) =    0 0 0 -1 J e -w 0 1 κ(s) ds 0 0 0 py 2 e w 0 1 κ(s) ds 0    (33) and φ(w, u, y) = 
  0 0 0 p J y 2 e -w 0 1 κ(s) ds 0 0 0 -y 1 u d   
Now, as in [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF][START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear timevarying systems[END_REF], let us consider the following system:

˙ z = A(y, w) z + φ(w, u, y) δ + ΛS -1 δ Λ T C T + S -1 z C T Σ (y -C z) (34) ˙ δ = S -1 δ Λ T C T Σ(y -C z) (35)
where Λ, S z , S δ are respectively the solutions of the following equations:

Λ = A (y, w) -S -1 z C T C Λ + φ (w, u, y) Ṡz = -ρ z S z -A(y, w) T S z -S z A(y, w) + C T ΣC Ṡδ = -ρ δ S δ + Λ T C T ΣCΛ, S z (0), S δ (0) > 0
where ρ z and ρ θ are positive constants.

Under the following persistence of excitation condition: there exist λ > 0 and σ > 0 such that

λI ≤ t+T t Λ (τ ) T C T Σ (τ ) CΛ (τ ) dτ ≤ σI,
it has been proven in [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF][START_REF] Zhang | Adaptive observer for multiple-input-multiple-output (mimo) linear timevarying systems[END_REF] that dynamical system (34)-( 35) is an observer for dynamical system [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF]. Thus, it enables us to estimate exponentially the state z and the parameters δ for system [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF]. To validate the theoretical part of proposed approach, we will present the numerical results that reflect the good dynamics performance and fast estimation convergence to their real trajectories.

Parameter

Simulation results

The Table-I provides mechanical/electrical data for PMSM model used in simulation. The structure sensorless control system is depicted in Fig. 1. The structure sensorless control system is depicted in Fig. 1, where a zero electric current is setting in the direct phase i * d because the flux does not need to be controlled since it is constant in the case of non-saliency rotor and in order to optimize the torque-to-current ratio. In order to ensure the boundedness of extended dynamics [START_REF] Hamida | An adaptive interconnected observer for sensorless control of pm synchronous motors with online parameter identification[END_REF], we take κ(w) = sin 2 aw (aw) 2 = 0 for |w| < π a . The parameter a is chosen such that the simulation interval where w is bounded as wide as we want. In the present study, we take a = 0, 0005 to have an interval [-5, 5] (see Fig. 2 below). In order to verify the efficiency of the proposed approaches, we use the input proposed in [START_REF] Ezzat | Sensorless speed control of pmsm via adaptive interconnected observer[END_REF] to excite the motor under low and high transitions of rotor speed such that the observability and the PE conditions can be satisfied. Remark 3. Fig. 1 presents the general structure of observer-based control of motor. However, this paper deals only with the observer design problem and no controller has been investigated. Different kinds of controllers have already been proposed in the literature, such as observerbased sliding mode controller [START_REF] Traoré | Sensorless induction motor: highorder sliding-mode controller and adaptive interconnected observer[END_REF][START_REF] Hamida | An adaptive interconnected observer for sensorless control of pm synchronous motors with online parameter identification[END_REF], where the stability of the closed-loop system has been studied in the theoretical and practical situations. Interested reader can refer to the references therein. (aw) 2 .

Simulation results with high gain observer

When the parameters are assumed known, we consider the observer normal form [START_REF] Lopez | Linearization by completely generalized input ouput injection[END_REF] and the high gain observer [START_REF] Lynch | Nonlinear observers with approximately linear error dynamics: the multivariable case[END_REF] with ρ = 40. The efficiency of the proposed approach which estimates the rotor position, rotor speed and load torque is shown in Fig. 3 -Fig. 4.

To better highlight the role of the proposed observer normal form, we perform a comparison with sliding mode observer (SMO) [START_REF] Ali | Sliding mode state observation for non-linear systems[END_REF], [START_REF] Qiao | New sliding-mode observer for position sensorless control of permanent-magnet synchronous motor[END_REF]. Although, for some classes of nonlinear systems particularly in presence of faults and uncertainties, the SMO is combined to the high gain observer to bring robustness in the estimation process. Concerning the famous chattering phenomenon, it is well known that several methods can be used to deduce it, like filter [START_REF] Lee | Design of iterative sliding mode observer for sensorless pmsm control[END_REF], sigmoid function, discrete SMO [START_REF] Awan | Discrete-time observer design for sensorless synchronous motor drives[END_REF][START_REF] Mihoub | A chattering free second order discrete sliding mode observer: An experimentation on a chemical reactor[END_REF] and so on. However, those modifications provide as well some disadvantages. For example, the introduction of low-pass filter can definitely reduce the chattering, but it will slightly introduce delay. Replacing signum function by sigmoid function can reduce the chattering, but will the robustness of SMO. More details can be found in [START_REF] Edwards | Sliding Mode Control: Theory And Applications[END_REF]. However, in our work thanks to the obtained observer normal form (30), we reach a good estimation using only the high gain observer (HGO). The figures Fig. 3 -Fig. 4 show that for PMSM in d-q reference frame, the both methods (HGO, SMO) estimate successfully the states. Also, we note a less pronounced variation of estimation with the our method.

The load torque impact is noticeable on the figure Fig. 3 across the appearance of-overshoot when it changes the value. However the quality of the estimation is not affected. Therefore, we deduce that this observer is robust to external disturbance. Moreover, despite the difference of initial conditions between the simulation and estimation, it can be seen that the rotor shaft position value is correctly estimated. In other words, whatever the initial position of the rotor, the states of PMSM can be estimated.

Simulation results with adaptive observer

To simultaneously estimate the states and the parameters of PMSM, we consider the dynamical form [START_REF] Marino | Global adaptive observers for nonlinear systems via filtered transformations[END_REF] and the adaptive observer (34)- [START_REF] Nahid-Mobarakeh | Mechanical sensorless control of pmsm with online estimation of stator resistance[END_REF] with Σ = 0.05I d , ρ z = 77 an ρ δ = 60. The performances of this method are show in Fig. 5 -Fig. 8. Fig. 5 -Fig. 6 show a good agreement in starting up between the estimated and the real values despite the uncertainty of the parameters values and the difference of initial conditions. We also note in Fig. 5 that the estimated speed follows their references. Moreover, the results of parameter estimation are shown in Fig. 7 and Fig. 8. The obtained result proves that the extended adaptive observer form has a very satisfactory performance. The obtained result demonstrates that the proposed approach can be considered as a suitable alternative to other methods, particularly when the simultaneously stimulation of states and parameters is required.

However, as expected, the rapid transients of load torque (disturbance) affect momentarily the estimation process. This phenomena is depicted through the appearance of-overshoot on the figures. In other words, a good converging performance of the estimated states and parameters can be observed, despite the appearance of overshoots due to the variation of load torque. It is worth noting that those over-shoots in estimation need to be carefully treated when designing observer-based controller. More precisely, if the designed controller depends on those estimations with over-shoot, the saturation function should be applied to mitigate the harmful influence of closed-loop system. This saturation can be normally realized since the states of system are always bounded in practice.

Conclusions

This work proposes a sensorless PMSM control based on the concept of extended nonlinear observer normal forms. This enables us to apply a high gain observer or an adaptive observer to estimate the state and the parameters of PMSM. Compared to other methods (back-emf and MRAS methods) where the parameters should be well known, the extended adaptive observer approach, is easier to be implemented, and shows good dynamic performances in all speed ranges. Furthermore, the simultaneous estimation of states and parameters can have a big interest in the faults diagnosis and fault tolerant control based on auto-tuning and real time adaptation. Several simulation results have been presented to show the effectiveness of the proposed approach.

1 κ 1 κ 2 1 κ 2

 11212 (s) ds , α 2 (y, w) = py 2 e w 0 (s) ds , α 3 (y, w) = -p y 1 + (y, w) = -Rs Ls y 1 , β 1 (y, w) = p J φ f y 2 e -w 0 (s) ds , and β 3 (y, w) = -Rs Ls y Remark 1.

Fig. 1 :

 1 Fig. 1: Implementation of an Adaptive observer through a change of coordinates.

Fig. 2 :

 2 Fig. 2: Shape of sin 2 aw (aw) 2 .

Fig. 3 :Fig. 4 :

 34 Fig. 3: Estimation of the rotor mechanical speed Ω and the load torque T l .

Fig. 5 :Fig. 6 :

 56 Fig. 5: Estimation of the rotor mechanical speed the load torque T l .

Fig. 7 :

 7 Fig. 7: Estimation of the stator resistor.

Fig. 8 :

 8 Fig. 8: Estimation of the magnet flux linkage and the stator inductance.

Table 1

 1 Moment of inertia (J) 0.00003 kgm 2 Viscous friction coefficient (f v ) 0.0034 N m/s

		Value	Parameter	Value
	Stator resistance (R s )	2.875 Ω	Number of pole pairs (p)	3
	Stator inductance (L s )	8.5 mH	Rotor magnetic flux (φ f )	0.175 W b

TABLE MOTOR PARAMETERS

 MOTOR

Acknowledgments:

The authors are deeply grateful to the reviewers for their thorough review and their interesting comments and suggestions that significantly contributed to improving the scientific content and the writing of this work.


Gang Zheng Lei Yu

Driss Boutat

Adaptive observer for simultaneous state and parameter estimations for an output depending normal form

In this paper, we investigate the problem of simultaneous state and parameter estimation for a class of nonlinear systems which can be transformed into an output depending normal form. A new and simple adaptive observer for such class of systems is presented. Sufficient condition for the existence of the proposed observer is derived. A concrete application is given in order to highlight the effectiveness of the proposed result.

I. Introduction

Observer design problem has been widely studied in the literature since the last four decades. Many methods have been developed, and one famous approach is based on differential geometric method, which enables us to easily reuse the existing observers. The literature is vast about nonlinear observer normal form approach (see for example [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF][START_REF] Respondek | Time scaling for observer design with linearization error dynamics[END_REF][START_REF] Zheng | Multioutput dependent observability normal form[END_REF][START_REF]Signal output dependent observability normal form[END_REF][START_REF] Tami | Extended output depending normal form[END_REF][START_REF] Zheng | Output dependent observability linear normal form[END_REF][START_REF] Zheng | Synchronisation of chaotic systems via reduced observers[END_REF]).

For those proposed normal forms, many different types of observers are stated in the literature to estimate the states of the studied system [START_REF] Kreisselmeier | Adaptive observers with exponential rate of convergence[END_REF][START_REF] Cho | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF]. In [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF], a Kalman like adaptive observer was presented for state-affine systems with linear time-varying matrix, and a high gain observer was proposed in [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] for the nonlinear system with triangular form. After that, other adaptive observers are studied to estimate of the states for more general normal forms, including the output depending normal form [START_REF] Busawon | A simple observer for a class of nonlinear systems[END_REF][START_REF] Ghanes | On simultaneous parameter identification and state estimation for cascade state affine systems[END_REF].

On the other hand, there exist as well many works to simultaneously estimate the state and the parameters by using adaptive observers. In [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF], the authors proposed adaptive observers for a class of uniformly observable nonlinear systems with general linear/nonlinear parameterizations which is of Manuscript received L.Yu is with Signal Processing Laboratory, Electronic and Information School, Wuhan University, China. G. Zheng triangular form with linear constant part. [START_REF] Grip | Estimation of states and parameters for linear systems with nonlinearly parameterized perturbations[END_REF] treated the simultaneous of states and parameters for linear systems with nonlinear parameterized perturbations. Another adaptive observer was introduced in [START_REF] Tyukin | Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters[END_REF] in order to treat the nonlinear parameterization case. Except adaptive method, other approaches can be used as well to simultaneously estimate the state and the parameters, such as sliding mode technique [START_REF] Yan | State and parameter estimation for nonlinear delay systems using sliding mode techniques[END_REF].

However, the mentioned methods are almost for nonlinear systems which are either of the output injection normal form, or of the triangular normal form, and no results are reported for the output depending normal form. Since those proposed observers cannot be applied directly to such a form, thus this paper is motivated to propose a new and simple adaptive observer for the simultaneous estimations of the state and the parameters of this normal form. As an extension of [START_REF] Zheng | Adaptive observer for simultaneous estimation of state and parameter for a class of nonlinear systems[END_REF], the novelties of this paper are the following two aspects. Firstly necessary and sufficient conditions are deduced which guarantee the existence of a parameterindependent diffeomorphism transforming a class of nonlinear systems into a more general output depending nonlinear observer form with output injection. Secondly we show that if some sufficient conditions are satisfied, then a very simple adaptive observer can be designed to simultaneously estimate the state and parameter of the proposed normal form.

II. Problem statement

Consider the following nonlinear system:

ẋ = F(x) + G(x)θ + Q(x)u y = H(x) (1)
where x ∈ R n is the state vector, u ∈ R q is the input, θ ∈ R m is the constant parameter, y ∈ R is the output,

F : R n → R n , G = [G 1 , • • • , G m ] with G i : R n → R n , Q = [Q 1 , • • • , Q q ] with Q i : R n → R n
, and H : R n → R are smooth. It is supposed in this paper that F(0) = 0, G(0) = 0, Q(0) = 0 and H(0) = 0, and the pair (H, F) satisfies the observability rank condition, i.e. rank{dL i-1 F H, 1 ≤ i ≤ n} = n, where L k F H is the k th Lie derivative of H along F and dL i-1 F H stands for the associated differentiation for 1 ≤ i ≤ n with dL 0 F H = dH. Then, one can construct the well-known Krener & Isidori [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF] frame: τ = [τ 1 , • • • , τ n ] with the first vector field τ n given by the following algebraic equations:

dL i-1 F H (τ n ) = 0 for 1 ≤ i ≤ n -1 dL n-1 F H (τ n ) = 1
whereas the rest of vector fields are obtained by iterating on i:

τ i = [τ i+1 , F] for 1 ≤ i ≤ n -1 where [, ]
denotes the conventional Lie bracket.

In [START_REF] Krener | Nonlinear observer with linearizable error dynamics[END_REF], the commutativity of Lie bracket, i.e. [τ i , τ j ] = 0 for 1 ≤ i ≤ n and 1 ≤ j ≤ n, is the necessary and sufficient condition to transform system (1) with G(x) = Q(x) = 0 into the following well-known nonlinear observer form with output injection:

ż = Az + β (y) y = Cz
where β (y) : R → R n , and the matrices A and C are of the Brunovsky form.

This result was extended to study system (1) with input and parameter in [START_REF] Marino | Adaptive observers for single output nonlinear systems[END_REF], and the necessary and sufficient conditions are given which guarantee the existence of a diffeomorphism such that system (1) can be transformed as well into the above similar output injection form and an adaptive observer was proposed to estimate the state and parameter of the studied system. In that paper, the commutativity of Lie bracket condition was still imposed, i.e. [τ i , τ j ] = 0 for 1 ≤ i ≤ n and 1 ≤ j ≤ n.

The problem now is raised when this commutativity condition cannot be satisfied, then one cannot apply the method proposed in [START_REF] Marino | Adaptive observers for single output nonlinear systems[END_REF]. The main contributions of this paper are: 1) to propose a new output injection form; 2) to deduce the necessary and sufficient conditions to transform the studied system into the proposed form; 3) to design a very simple adaptive observer which simultaneously estimate the state and the parameters.

III. Assumptions, notations and preliminary results

Suppose that the the commutativity of Lie bracket condition is not satisfied for the family of vector fields

τ = [τ 1 , • • • , τ n ]
proposed by [START_REF] Krener | Nonlinear observer with linearizable error dynamics[END_REF], this implies that the studied system cannot be transformed into the simple output injection form with constant matrix A of the famous Brunovsky form, then according to [START_REF]Signal output dependent observability normal form[END_REF] we can construct another frame τ = [τ 1 , • • • , τ n ] from τ as follows:

τ n = π(y)τ n τ i = 1 a i (y) [τ i+1 , F] , for 1 ≤ i ≤ n -1 (2)
where

π(y) = n-1 ∏ i=1 a i (y), and a i (y) for 1 ≤ i ≤ n -1 being
non vanishing functions of the output to be determined.

Based on this new frame, one can state the following theorem, which is in fact an extension of our previous result in [START_REF]Signal output dependent observability normal form[END_REF].

Theorem 1 There exists a parameter-independent diffeomorphism z = φ (x) which transforms system (1) into the following output depending nonlinear observer form with output injection:

ż = A(y)z + β (y) + g(y)θ + q(y)u y = Cz (3)
where g(y) and q(y) are of the appropriate dimensions, A(y) is of the output depending Brunovsky form:

A(y) =       0 a 1 (y) • • • 0 0 . . . . . . . . . . . . . . . 0 0 • • • a n-2 (y) 0 0 0 • • • 0 a n-1 (y) 0 0 • • • 0 0       (4)
and C = [1, 0, • • • , 0], if and only if the following conditions are satisfied:

1.

[τ i , τ j ] = 0 for 1 ≤ i ≤ n and 1 ≤ j ≤ n; 2. [τ i , Q j ] = 0 for 1 ≤ i ≤ n -1 and 1 ≤ j ≤ q; 3. [τ i , G j ] = 0 for 1 ≤ i ≤ n -1 and 1 ≤ j ≤ m.
Proof 1 It has been proven in [START_REF]Signal output dependent observability normal form[END_REF] (see Theorem 3.1 in [START_REF]Signal output dependent observability normal form[END_REF]) that there exists a diffeomorphism z = φ (x) such that system (1) with G(x) = 0 and Q(x) = 0 can be transformed into the form (3) with q(y) = 0 and g(y) = 0 if and only if the first condition of Theorem 1 is satisfied. Following the same procedure, it is easy to prove that the second and the third conditions of Theorem 1 are necessary and sufficient to guarantee that G(x)θ and Q(x)u can be transformed into g(y)θ and q(y)u by the deduced diffeomorphism z = φ (x).

Remark 1 If a i = 1 in the matrix A(y) defined in (4), system (3) can be seen as an extension of the output injection normal form proposed in [START_REF] Krener | Nonlinear observer with linearizable error dynamics[END_REF], for which an adaptive observer was proposed in [START_REF] Marino | Adaptive observers for single output nonlinear systems[END_REF] to simultaneously estimate the state and the parameters. Besides, in [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF], the authors proposed the adaptive observers for a class of uniformly observable nonlinear systems with linear/nonlinear parameterizations. However, the proposed observer is complicated since it is based on the adaptive observer proposed by [START_REF] Zhang | Adaptive observer for multipleinput-multiple-output (mimo) linear time-varying systems[END_REF] whose gain depends on the solutions of two differential equations. Moreover, the proposed observer in [START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF] cannot be applied for (3) with the output depending matrix A(y) since it works only for a constant matrix A.

Due to the physical constraint, the control and the state values of the practical systems are always bounded. Therefore, in what follows, we make the following standard (see [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF][START_REF] Farza | Adaptive observers for nonlinearly parameterized class of nonlinear systems[END_REF], for example) assumption in the estimation theory on the boundedness of the state and the input for system [START_REF] Krener | Linearization by output injection and nonlinear observer[END_REF].

Assumption 1 For the studied system (1), it is assumed that the state x(t) is bounded for the given bounded parameter θ and the input u, i.e. x(t) ∈ X ⊂ R n under the given u(t) ∈ U ⊂ R q for any t ≥ 0 and θ ∈ Θ ⊂ R m where X , U and Θ are the compact sets.

Since it is assumed that system (1) can be transformed via a smooth diffeomorphism z = φ (x) : X → Z into the output depending normal form (3), therefore the state z and the output y of (3) are both bounded as well, i.e. z(t) ∈ Z ⊂ R n and y(t) ∈ Y ⊂ R for any t ≥ 0 where Z and Y are two corresponding compact sets. Therefore, for any a i (y) = 0 of system (3), there always exist positive constants a i > 0, a i > 0 and

σ i > 0 for 1 ≤ i ≤ n -1 such that 0 < a i <| a i (y) |< a i < ∞, ∀y ∈ Y (5)
and

| da i (y(t)) dt |< σ i < ∞, ∀y ∈ Y (6)
For the sake of simplicity and without loss of generality, let us define a = min 1≤i≤n-1 {a i } and a = max 1≤i≤n-1 {a i }.

Assumption 2 It is assumed that the vector-function g(y) in (3) is persistently exciting for all y ∈ Y , i.e. there exist positive constants b > 0 and T > 0 such that ˆt+T t g T (y(s))g(y(s))ds ≥ bI m×m

For the studied normal form, we define the following variables which will be used in the sequel. Denote

A 0 =      0 1 • • • 0 0 . . . . . . . . . . . . . . . 0 0 • • • 0 1 0 0 • • • 0 0      (7)
and

Γ(y, ρ) = diag{λ 1 , • • • , λ n } (8)
where λ 1 = 1, and

λ i = i-1 Π j=1 a j (y) ρ i-1
for 2 ≤ i ≤ n with positive constant ρ being freely chosen, which determines the gain of the proposed observer. Since 0 < a i <| a i (y) |< a i < ∞, then Γ(y, ρ) is always invertible and bounded. Define Λ(y(t), ẏ(t), ρ) = Γ(y(t), ẏ(t), ρ)Γ -1 (y(t), ρ) [START_REF] Cho | A systematic approach to adaptive observer synthesis for nonlinear systems[END_REF] where Γ(y(t), ẏ(t), ρ) = dΓ(y(t),ρ) dt = ẏ(t) ∂ Γ(y(t),ρ) ∂ y .

If Assumption 1 is satisfied, then Λ(y(t), ẏ(t), ρ) defined in ( 9) is bounded for all y(t) ∈ Y and t ≥ 0, and there always exists a positive constant λ such that each element of Λ(y(t), ẏ(t), ρ) satisfies |Λ i, j (y(t), ẏ(t), ρ)| < λ < ∞, ∀y ∈ Y for 1 ≤ i ≤ n and 1 ≤ j ≤ n. Let S ρ be the solution of the following equation:

0 = ρS ρ + A T 0 S ρ + S ρ A 0 -C T C (10)
with A 0 defined in [START_REF] Zheng | Synchronisation of chaotic systems via reduced observers[END_REF], where S ρ is symmetric positive definite [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF].

IV. Main result

Consider system (3) with Γ(y, ρ) and S ρ defined in the last section, let us denote

G = G T 1 , • • • , G T m T = g T (y) Γ(y, ρ)S ρ (11)
with the ith row in G noted as G i : R n → R n for 1 ≤ i ≤ m. Moreover, let us denote Y as the field of the output function, based on which we can define the module spanned by C over Y as follows:

Ω = span Y {dy} (12)
then the following assumption is imposed.

Assumption 3 With Γ(y, ρ) and S ρ respectively defined in ( 8) and [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF], it is assumed that G i ∈ Ω for 1 ≤ i ≤ m where G i and Ω are defined in [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] and (12), respectively.

Lemma 1 If Assumption 3 is satisfied, then there exists an output depending matrix

K(y) = g T (y) Γ(y, ρ)S ρ C T ( 13
)
such that g T (y) Γ(y, ρ)S ρ = K(y)C.

Proof 2 According to the definition of Ω in (12), if Assumption 3 is satisfied, then for each G i , there exists K i (y) such that

G i = K i (y)C for 1 ≤ i ≤ m. Hence we have g T (y) Γ(y, ρ)S ρ =    G 1 . . . G m    =    K 1 (y) . . . K m (y)   C = K(y)C where C = [1, 0, • • • , 0].
From the above equation we can prove that, if Assumption 3 is satisfied, then by choosing K(y) = g T (y) Γ(y, ρ)S ρ C T one has always the equality g T (y) Γ(y, ρ)S ρ = K(y)C.

Remark 2 For a class of nonlinear systems which can be transformed into the following output injection form:

ż = Az + β (y, u) + b ḡ(y, u)θ y = Cz ( 14
)
one can design a very simple adaptive observer, proposed in [START_REF] Marino | Adaptive observers for single output nonlinear systems[END_REF], if there exists a positive definite symmetric matrix P such that b T P = C. It is clear this sufficient condition is a special form of (13) since the form ( 14) is a special case of the form (3).

Theorem 2 If Assumption 1, 2 and 3 are satisfied, then there exist three positive constants γ > 0, ρ > 0 and η > 0 such that the following dynamics

   ż = A(y)ẑ + β (y) + g(y) θ + q(y)u +γΓ -1 (y, ρ)S -1 ρ C T (y -C ẑ) θ = ηK(y)(y -C ẑ) (15)
where K(y) is defined in [START_REF] Ghanes | On simultaneous parameter identification and state estimation for cascade state affine systems[END_REF], is an adaptive observer to simultaneously estimate the state z and the parameter θ of system (3).

Proof 3 Denote e z = zẑ and e θ = θ -θ , thus the observation error dynamics is governed by the following system:

ėz = A(y) -γΓ -1 (y, ρ)S -1 ρ C T C e z + g(y)e θ (16
) Since a i (y) = 0 for all y ∈ Y , then Γ(y, ρ) is always invertible. Therefore we can make the following change of coordinates ε z = Γ(y, ρ)e z and ε θ = e θ . According to the definition of Γ(y, ρ), we have

εz = Γ(y, ρ)A(y) -γS -1 ρ C T C Γ -1 (y, ρ)ε z +Γ(y, ρ)g(y)e θ + Γ(y, ρ)Γ -1 (y, ρ)ε z Since Γ(y, ρ)A(y)Γ -1 (y, ρ) = ρA 0 and CΓ -1 (y, ρ) = C, then one has εz = ρA 0 -γS -1 ρ C T C ε z + Λ(y, ẏ, ρ)ε z +Γ(y, ρ)g(y)ε θ (17)
which is due to the fact that Λ(y, ẏ, ρ) = Γ(y, ρ)Γ -1 (y, ρ).

Since ε θ = e θ and ε z = Γ(y(t), ρ)e z with Γ(y(t), ρ) being always bounded and invertible for all y ∈ Y , then the convergence of e z and e θ to zero are equivalent to the convergence of ε z and ε θ to zero. In what follows, we will prove the convergence of ε z and ε θ to zero.

For this, let us consider the following Lyapunov

function V (ε z , ε θ ) = ε T z S ρ ε z + ε T θ ε θ η , then one has V (ε z , ε θ ) = ε T z -2γC T C + ρC T C ε z + 2 ε T θ εθ η +ε T z -ρ 2 S ρ + 2ΛS ρ ε z + 2ε T z S ρ Γg (y) ε θ
Since Assumption 3 is supposed to be satisfied, then according to Lemma 1 we can find an output depending vector K(y) such that g T (y) ΓS ρ = K(y)C.

Due to the definition of θ in (15), we have

εθ = θ -θ = -ηK(y)(y -C ẑ) = -ηg T (y) ΓS ρ ε z
where θ is the constant parameters to be identified.

Then we have 2ε T z S ρ Γg (y) ε θ + 2 ε T θ εθ η = 0. Therefore we obtain the following equivalence: then we can conclude that V (ε z , ε θ ) ≤ -ρ 0 ε T z S ρ ε z . Integration of this inequality yields

V (ε z , ε θ ) = ε T z -2γC T C + ρC T C ε z +ε T z -ρ 2 S ρ + 2ΛS ρ ε z Since |Λ i, j (y(t), ẏ(t), ρ)| < λ < ∞,
V (ε z (t), ε θ (t)) -V (ε z (t 0 ), ε θ (t 0 )) ≤ -ρ 0 ´t t 0 ε T z (s)S ρ ε z (s)ds Since V (ε z (t), ε θ (t)) ≥ 0, then ρ 0 ˆt t 0 ε T z (s)S ρ ε z (s)ds ≤ V (ε z (t 0 ), ε θ (t 0 )) Due to the fact that V (ε z (t 0 ), ε θ (t 0 )) is bounded, thus ε z (t) ∈ L 2 .
ε z (t) = lim t→∞ εz (t) = 0, (17) implies lim t→∞ Γ(y, ρ)g(y)ε θ (t) = 0.
Due to the fact that Γ(y, ρ) is bounded and positive definite symmetric for all y ∈ Y , and g(y) is persistently exciting according to Assumption 2, then we can prove that lim t→∞ ε θ (t) = 0.

V. Application to batch reactor

Let us consider a real batch reactor system adopted from [START_REF] Respondek | Time scaling for observer design with linearization error dynamics[END_REF], with two chemicals: A k → B where k is the reaction rates. According to [START_REF] Respondek | Time scaling for observer design with linearization error dynamics[END_REF], the dynamics can be described as follows:

         dC A dt = -ke -E RTe C A dC B dt = ke -E RTe C A dT e dt = Jke -E RTe C A + ∆H V (T e ) y = T e (18)
where C A and C B are the concentrations, J, R, E are known constants, ∆H V (T e ) is a known function of the temperature T e . From the above equation, it can be seen that C A + C B = const which implies that the conservation law is satisfied, therefore, we need only to estimate C A . Suppose that system (18) is perturbed with an unknown constant disturbance θ in the following way:

     dC A dt = -ke -E RTe C A -e -E RTe -e -2E RTe θ dT e dt = Jke -E RTe C A + ∆H V (T e ) + e -2E RTe θ y = T e (19)
and the objective is to simultaneously estimate both the concentration C A and the unknown parameter θ . Following the procedure presented in [START_REF]Signal output dependent observability normal form[END_REF], one obtains the following diffeomorphism: z = φ (x) = [T e , JkC A + kT e ] T which transforms [START_REF] Krener | Nonlinear observer with linearizable error dynamics[END_REF] Since the studied system is a real batch reactor, thus the boundedness assumption of state is satisfied, and one can follow the proposed method to design an observer of the form [START_REF] Grip | Estimation of states and parameters for linear systems with nonlinearly parameterized perturbations[END_REF]. For the simulation settings, we set J = 1, k = 100,θ = -3, ρ = 15, η = 102500 and γ = 5. The estimation errors are presented in Fig. 1.

VI. Conclusion

This paper dealt with adaptive observer design for a special class of nonlinear systems, which can be transformed into an output depending normal form. Sufficient conditions were presented to guarantee the simultaneously asymptotic estimations of the state and the parameter for the studied system. The result was applied to estimate the simultaneous state and parameter estimations for a real batch reactor system.

VII. Acknowledgment

This work is partially supported by NSFC Grant 61401315, China Scholarship Council and the Projectsponsored by SRF for ROCS, SEM, under Grant

  into the following normal form:ż1 = e -E Ry z 2 + ∆H V (y) + e -2E Ry θ ż2 = k∆H V (y) + Jke

Fig. 1 .

 1 Fig. 1. Estimation errors for C A , T e and θ .

  is with Project Non-A, INRIA Lille-Nord Europe, 40 avenue Halley, 59650 Villeneuve d'Ascq, France. D. Boutat is with Loire Valley University, ENSI de Bourges, Laboratoire PRISME, 88, Boulevard de Lahitolle,18020 Bourges Cedex, France.

  ∀y ∈ Y , if we choose ρ > ρ 0 + λ /2 with any positive constant ρ 0 > 0 and γ > γ 0 + ρ 2 where γ 0 is any positive constant,

c 2015 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society Prepared using asjcauth.cls

  Moreover, since Λ(y, ẏ, ρ), Γ(y, ρ) and g(y) are all bounded for all y ∈ Y , then from[START_REF] Yan | State and parameter estimation for nonlinear delay systems using sliding mode techniques[END_REF] one can conclude that εz (t) ∈ L ∞ . Finally we have ε z (t) ∈ L 2 ∩ L ∞ and εz (t) ∈ L ∞ ,and according to Barbalat theorem, this implies that lim From (17), we can see that εz (t) is bounded and uniformly continuous, thus we have lim t→∞ εz (t) = 0 by applying Barbalat theorem. Since both lim

	t→∞

t→∞ ε z (t) = 0.

Asian Journal of Control, Vol. 00, No. 0, pp.1-6, Month 2015

c 2015 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society Prepared using asjcauth.cls