Tarek Ahmed-Ali
email: tarek.ahmed-ali@greyc.ensicaen.fr

Estelle Cherrier

Mohammed M'saad

Cascade high gain observers for nonlinear systems with delayed output measurement

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Cascade high gain observers for nonlinear systems with delayed output measurement

Tarek Ahmed-Ali, Estelle Cherrier and Mohammed M'Saad

Abstract-This work presents a set of cascade high gain observers for triangular nonlinear systems with delayed output measurement. A sufficient condition ensuring the exponential convergence of the observation error towards zero is given. This result is illustrated by some simulations.

Index Tenns-Time-delay systems, high gain observer, cas cade systems

I. IN TRODUCTION

This paper deals with the design of nonlinear observers in the presence of delayed output measurement. This problem is particulary challenging since time delays affecting output measurements arise in a variety of applications. One can cite for example systems which are controlled by a remote controller. In these systems, the output data are transmitted to the controller throughout a communication system which in troduces a time-delay between the process and the controller. The design of controllers for such systems can be viewed as an output feedback design based on state prediction system. In the linear case, this problem has been solved by the well known Smith predictor [l] and several predictive control algorithms [START_REF] Clarke | Generalized predictive control prut I and II[END_REF], [START_REF] Shaked | H-infinity static output-feedback control of linear continuous-time systems with delay[END_REF]. Recently, for the nonlinear case, a new kind of chained observers which reconstruct the state at different delayed time instants for drift observable systems has been presented in [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF]. The authors showed, by using Gronwall lemma, that under some conditions on the delay, exponential convergence of the chained observers is ensured. These conditions have been relaxed in [START_REF] Kazantzis | Nonlineru• observer design in the presence of delayed output measurements[END_REF] by using an approach based on a first-order singular partial differential equation. On the other hand, in [START_REF] Besan�on | Asymptotic state pre diction for contiunous-time systems with delayed input and application to control[END_REF] a novel predictor for linear and nonlinear systems with time delay measurement has been designed. This predictor is a set of cascade observers. Sufficient conditions based on linear matrix inequalities are derived to guarantee the asymptotic convergence of this predictor. In the present work, we design a set of cascade high gain observers for nonlinear triangular systems by considering a time delay in the output measurement. We will show that the general high gain observer design framework developed in [START_REF] Barnard | A high gain observer for a class of uniformly observable systems[END_REF], [START_REF] Gauthier | A simple obsever for nonlineru• systems: Application to bioreactors[END_REF], [9], to mention a few, for delay free output measurements can be extended to systems with delayed output. More precisely, we propose to use a suitable Lyapunov-Krasovskii functional and a sufficient number of high gain observers, in order to guarantee the exponential convergence of the estimated state at time t towards the true state at time t, even if the output is affected by any constant and known delay. We will also give an explicit relation between the number of observers and the delay. The present paper is organized as follows : In section II, we present the class of considered systems and the different assumptions. In the third one, we present the proposed observers and prove their convergence. In the last one, we illustrate our results throughout simulations on an academic example.

II. PRELIMINARIES AND NOTATIONS

First some mathematical notations which will be used throughout the paper are introduced.

• The euclidian norm on JR. n will be denoted by 11-11-• The matrix x r represents the transposed matrix of X. ith ,,..,,..

• es ( i) = (0, ... , o, 1 , o, ... , 0) E )R S , s ::::: 1 is the i th

) ( ¢1( � , u) ) ,p(x,u)= : <Pn(x, u) (1) (2)
(3)

The term T represents the measurement time delay, x(t) E JR. n is the vector state which is supposed unavailable. The output y ( t) E JR is a linear function of the state x at time t -T. The input u E U where U is a compact set in R The functions cp i , i = 1, ... , n are supposed smooth. This class represents the class of uniformly observable systems. It has been shown [START_REF] Barnard | A high gain observer for a class of uniformly observable systems[END_REF], [START_REF] Gauthier | A simple obsever for nonlineru• systems: Application to bioreactors[END_REF] that these models concern a wide variety of systems, such as bioreactors ... Throughout the paper, we assume that the following hypotheses are satisfied: H3. The time delay Ti s supposed constant and known.

H l .

Ill. MAIN RESULTS

In this work, we consider an arbitrary long time delay

T affecting the output measurement of system (1). The proposed nonlinear observer for system (1) is a set of m cascade high gain observers. Each one of them estimates a T delayed state vector with sufficiently small delay -.

m

In order to present the proposed observer, we use the following convenient notations adopted from [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF]:

. T Xj(t)= x(t -T+J-) m
, where j = 1, ... , m

Then the proposed observer can be written in the following form, for j = 1, ... , m:

ih Ax1 + <f,(x1) -et:,. -i s-1 c' C(x1 (t -!_) -x(t -T)) m Cx1 (t -!_) m Ax j + <t>(x j ) -0e:,.- 1 s- 1 c' c(x j (t -!_) -± j - 1 (t)) m Cx j (t -!_) = C± j -i (t) ( 7
)
m
where 0 is a positive constant satisfying 0 > 1. S is a symmetric positive definite matrix, solution of the following algebraic Lyapunov equation: [START_REF] Gauthier | A simple obsever for nonlineru• systems: Application to bioreactors[END_REF] and � is a diagonal matrix which has the following form : � = Diag (1 , ... , �1, ... , _____!__ 1 ) .

(9) g ig n -

We will show that the vector X j ( t) estimates the delayed state xj(t), j = 1, ... , m -1 and x m (t) estimates x(t).

Before proving the exponential convergence of the proposed chained observers, we consider the case when the delay T is sufficiently small. Then only one high gain observer is required to estimate the state of system (1).

x i)

Lemma 1: Consider the following observer:

Ax+ cp(x, u) -0� -1 s-1 c T C(x(t -T) -x(t -T)) Cx(t -T) ( 10
)
Then for sufficiently large positive 0, there exists a suffi ciently small positive constant T 1 such that VT ::; T 1 , observer (10) converges exponentially towards system (1).

Proof•

First let us denote the observation error as i = x -x. Then we will have:

i = Ax+cp(x, u)-cp(x, u)-0� -1 s-1 c T Ci(t -T) (11) If we apply the relation i(t) = i(t -T) + l � T i(s)ds (12)
and the change of coordinates x = �i, system (11) can be rewritten in the following manner:

x 0(A-s-1 c T C)x+�(cp(x,u) -cp(x,u)) + 0s-1 c' C l � T x(s)ds. ( 13
)
In order to derive an upper bound T 1 for the delay T, to ensure the exponential convergence to zero of the error x, we use the following Lyapunov-Krasovskii functional [10]:

W=x T Sx+ 1 � T ,1 t llx ( � ) ll 2 d � ds. ( 1 4 )
This functional can be written after some manipulations as follows: + -0x T c T cx + 20x T c T c l�T x(s)ds + T1llx(t)ll 2 -1�T1 llx(s)ll 2 ds.

W=x T sx+ 1 � T 1 (s -t+T1)11x(s)ll 2 ds ( 1
(

) 17
Note that by using the mean value theorem [11], we can write " ( f (X) -f ( .r )) �" (I en ( i)'en(j) ;;: ; ( ( ) ) " -'T (18) where� E Cov(x, x). Then we will have 2x T S�(rp(x)-rp(x)) = 2x T S�( f en(if en(j) !!i )� -l x i, j =l J

(19) Using the triangular structure and the Lipschitz properties of the functions ¢i , and the fact that 0 > 1, we deduce that where V = x T Sx and k 1 is a positive constant which does not depend on 0. Using the following property : where k 2 is also a positive constant which does not depend on 0.

Using this and equation ( 23), we will have: Then, the exponential convergence to zero of the observation error x is guaranteed if the following inequalities hold :

1 We can easily see, that by choosing T1 = 0 2 (since 0 > 1 we remark that condition (29) holds), the above inequalities will be equivalent to:

{ 0 � k1 + k 2 + {0 0 2 > 0 + k 2 + 17, - v0 (32)
It is obvious that the inequalities (32) are verified for sufficiently large values of 0. Note that it is sufficient to choose 0 � sup{2, (k1 + k 2 + v2)}.

(33)

Then, we can say that lemma 1 is verified for

■

To summarize Lemma 1, it gives the maximum delay supported by observer (10) which enables x(t) ----+ x(t), once 0 has been fixed according to conditions (34). To cope with a larger measurement delay, we propose in next paragraph a procedure to estimate x(t), based on a chain of high-gain observers: each observer will estimate the state at a given fraction of the output delay.

A. Cascade high gain observers

After proving that the convergence of the observer (10) requires a small delay, we will see that when the delay is arbitrary long, a set containing a sufficient number of cascade high gain observers [START_REF] Barnard | A high gain observer for a class of uniformly observable systems[END_REF] can reconstruct the states of system (1).

Theorem 1: Let us consider system (1), then for any constant and known delay T, there exist a sufficiently large positive constant 0 and an integer m such that the observer (7) converges exponentially towards the system (1).

Proof-The convergence of the cascade observer will be proved step by step :

Step 1: We consider the first observer in the chain:

Ax1 + ¢(x1) -et::. -i s -1 c ' C(x1(t -.!_) -x(t -T)) m f;i Cx1(t-.!_) (35) m
We remark that x(t -T) = x1(t --[;i) and consequently, if we choose 0 sufficiently large, and by choosing the integer m such that m ::::: 0 2 T, then x 1 ( t) converges towards

x1(t) = x(t -T + -[;i ) = x(t -(m -1) -[;i ).
Indeed, we are brought back to conditions of Lemma 1, T since the delay to handle with is now -, which is assumed 1 m smaller than 0 2 .

Step j: At each step (j = 2, ... , m) , we estimate the delayed state If we consider the following change of coordinates X j 6-.x J , we will have X j 0( A -s-1 c ' C)x j + 6-.(cp(x j ) -cp(x J ))

+ 0s -1 c ' C 1�-in X j (s)ds -0s -1 c ' Cx j _i-(38)
In order to prove by recurrence the convergence of the error x J , we suppose that the observation error X J -1(t) converges exponentially towards zero. Then we consider the following Lyapunov-Krasovskii func tional w j = xf Sx j + 1�-in (s -t + : ) I lxj(s)I l 2 ds (39) Then its time derivative satisfies the following inequality:

w j < -0xf Sx J + 2xf S6.(cp(x J ) -cp(x J )) + 0 -r c r c -20 -r c r c - -x j X j -x j X j -1 + 20xf c T c i�T X j (s)ds + T1llxJll 2 1�T1 llxj(s)ll 2 ds. (40)
As in the proof of the lemma (1), we will also have :

w j < -0½ + k� ½ + 0IJ c T ci j -T T - _._ 2 20x j C Cx j -1 + T1 I lx J 11 1�T1 llxj(s)ll 2 ds. ( 41
)
where ½ = xf Sx J , I J = JLT X j ( s )ds and k� is a positive constant which does not depend on 0 and k� ::::: k1. Now, by using Young's inequality, we derive the following inequalities (42) < where kf is a positive constant which does not depend on 0 and k; ::::: k 2 .

x( t -T + j-fn) by using the following observer:

Choosing Ti 1 0 2 , and using (41), ( 42) and (43), we 1

I T d . A x J + ¢(x J )-06. -1 S -CC(x J (t -m )-x J _1(t)) e n ve T . 1 Cx J (t -m ) = Cx J -i(t) (36) W J + v0 W J
It is not difficult to see that by considering the observation error vector i J = X J -X j , if we add and subtract the term 06.-1 s-1 c ' Cx J -i(t) in the previous equation, we obtain X j A i J + cp(x J ) -cp(x J ) -06.. -l s-1 c ' Ci J (t -: ) < + 06. -l S -1 c ' Ci j -l (t).

(37) Then, we can say that if

(44) (45)
we will have

<

Using the comparison lemma [12], we conclude that if X j -l converges exponentially towards zero, then X J converges also exponentially towards zero. Note that conditions (45), also ensure the convergence of the first observer (j = 1), then we deduce, recursively, that all observation errors converge exponentially towards zero.

■ IV. EXAMPLE

To illustrate the obtained results, consider the following nonlinear system, affected by delayed measurements:

{ i1(t) = x 2 (t) i 2 (t) = -2x 1(t) +O.5tanh(x1(t) +x 2 (t)) +x1(t)u(t) y(t) = X1(t -T) (47)
The input is u(t) = 0.1 sin(O. l t). System (47) belongs to the considered class of triangular systems with Lipschitz nonlinearities (1). The initial conditions for the system and for the (chained) observer(s) have been chosen as:

x(t) = ( � l ) , x(t) = ( � ), 'it E [-T, 0] (48) Simulations have been performed using Matlab, and the integration routine dde23, which specifically handles delay differential systems. The high gain parameter is set to 0 = 2. The following figures present the simulation results. Fig. 1 and Fig. 2 show the true states (solid) and the estimated states (dashed) when the output delay is set to O.25s, i.e. 1 T = 0 2 . For larger delays, one observer is not sufficient to estimate the states of system (47). For instance, if we set T = 0.5s, then two observers are necessary to reconstruct x 1 (t) and x 2 (t). This is reported by Fig. 3 and and Fig. 4. If the output delay T is still increasing, then four observers are required to achieve state estimation, as shown by Fig. 5 and Fig. 6.

V. CONCLUSION

In this paper, a novel predictor based on high gain observer has been presented. This observer can be applied to the class of nonlinear uniformly observable systems. The design of adaptive observers for nonlinear systems with delayed output is under investigation.

  basis of ]R s . • The convex hull of {x, y} is denoted as Co(x, y) = {Ax+ (1 -A)y, 0 ::; A ::; 1 }. • A m in ( S) and A max ( S) are the minimum and maximum eigenvalues of the square matrix S. In this paper, we consider the following class of nonlinear systems:

  The functions cp i (x,u) are triangular in x, i.e: 8cpi (x, u) ---= 0 k = i, ... , n -1 ( 5 ) 8xk+l H2. The functions cp i ( x, u) are globally Lipschitz, uniformly in u, i.e: :3,6 > 0 such that \f(x, y) E R n x R n , \fu EU llc/>i (x, u)c/>i ( Y , u) II ::; , Bll x -YII (6)

5 )

 5 If we compute its time derivative, we obtainw < ex r (A r s + SA-2c r c)x + 2x T S�(cp(x) -cp(x)) + 20x T c T c l � T x(s)ds + T1 llx (t) ll 2 -1 � T 1 llx (s) ll 2 ds (16)Using (8), we have W < -0x T Sx+2x T S�(¢(x) -¢(x))

  20x T C T C 1 t x(s)ds -0x T C T Cx t -T -0(Cx + C l�T x(s)ds)f (Cx + C l�T x(s)ds))+ 0(1�T x(s)dsf c T C(l �T x(s)ds).

+ 0 (

 0 1�T x(s)dsf c T C(l� T x(s)ds) + T1llx(t)ll 2 -1�T 1 llx(s)ll 2 ds (23) Now, let us remark that if we use equation (13), it comes: llx(t)ll 2 ::::; 0 2 k 2[ V + 11 l�T x(s)dsll 2 ] (24)

w

  ::; -0v +k1V +0Fc r c1 + T10 2 k 2[ V + IIIll 2 ] -1�T 1 llx(s)ll 2 ds.(25) where I= l t x(s)ds.t -TTo prove the above lemma (1), it is sufficient to find conditions which guarantee the following inequality . + k1V + Je + 0Fc r c1+ T10 2 k 2[ V + 11111 2 ] + T � l t llx(s)ll 2 ds V 0 t -T1 -1�T1 llx(s)ll 2 ds. (27)If we use the following Jensen's inequality :

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .

 1234 Fig. 1. Evolution of the first state for T = 0.25s with 1 observer

Fig. 5 .Fig. 6 .

 56 Fig. 5. Evolution of the first state for T = ls with a cascade of 4 observers


domain: spi phys chim

text:
Gerald Portemont

Thomas Fourest

Romain De Coninck

Experimental Method for Mode I Dynamic Fracture Toughness of Composite Laminates Using Double Cantilever Beam Specimens

Keywords: High loading rate, Fracture toughness, Double cantilever beam, Split Hopkinson Pressure Bar

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Impact loadings on laminated composite materials may damage the interlaminar phase and initiate the delamination process. Ideally, the material constituting the interlaminar phase must ensure overall cohesion and resist crack propagation. However, large decohesion may critically compromise the integrity of CFRPs lightweight structures. The capacity of the material to resist crack propagation is classically modelled by the energy release rate G which stands for the energy per unit area dissipated to form new crack surfaces [START_REF] Griffith | The phenomena of rupture and ow in solids[END_REF]. For a mode I crack opening, the critical energy release rate G IC refers to the initiation, and the dynamic energy release rate G ID is dedicated to the propagation stage. The interlaminar fracture energy is not a material property of the matrix because it depends on the thickness of the interlaminar phase and for on the substrate material [START_REF] Blackman | Measuring the mode I adhesive fracture energy GIC of structural adhesive joints: the results of an international round-robin[END_REF]. For mode I opening, the initiation value of the interlaminar fracture energy for CFRPs is classically determined by a Double Cantilever Beam (DCB) quasi-static test defined in ISO standard [START_REF]ISO: Fibre-reinforced plastic composites | Determination of mode I interlaminar fracture toughness GIC for unidirectionally reinforced materials[END_REF]. However, dynamic effects associated with fast crack propagations produce sharp oscillations in the recorded load, making it unrepresentative of the instant crack growth [START_REF] Blackman | The failure of fibre composites and adhesively bonded fibre composites under high rates of test,Part I : Mode I loading experimental studies[END_REF]. Consequently, no test standard has yet been defined for the mode I fracture toughness evaluation under high rates of loading. The general purpose of the present research is the definition of an experimental procedure to characterize the dynamic fracture toughness dependency on the crack velocity G ID (å) for carbon fiber-reinforced epoxy matrix composite. This work focuses on the description of the experimental protocol and the evaluation of the energy release rate without monitoring the crack tip.

Development of an Innovative Experimental Protocol

The present work is focused on the analysis of the loading rate effect on the mode I fracture of laminate carbon fiberreinforced epoxy matrix composite T700/M21. The mode I is classically characterized using a DCB configuration tests [START_REF] Astm D | Standard test method for mode I interlaminar fracture toughness of unidirectional fiber polymer matrix composites[END_REF]. Uniaxial dynamic loading have been performed using a Tensile Split Hopkinson Bar for opening rates between 4 m.s -1 G. Portemont ( ) • T. Fourest • R. De Coninck DMAS, ONERA, Lille, France e-mail: gerald.portemont@onera.fr and 17 m.s -1 . The opening displacement with respect to the loading axis was measured using an optical extensometer. The measurement is based on the tracking of a black-and-white target. The input and output bars are equipped with strain gauges to measure the tensile strain applied on the bars in order to evaluate the transmitted forces in the specimen. To measure the strain on the outer skin of the specimen, five strain gauges have been glued on each specimen spaced 10 mm apart. The kinematic field has been measured using 2D Digital Image Correlation (DIC) technique on the edge of the specimens. The images are captured by a high speed camera (Photron SA-X). Figure 22.1 illustrates the experimental setup.

To process the experimental results, the energy release rate G IC is expressed using the Linear Elastic Fracture Mechanics (LEFM) theory. However in this approach, the determination of G IC is very sensitive to the crack length measurement. Indeed, due to its singularity, the crack tip is intrinsically difficult to measure experimentally, especially for high loading rates for which the camera resolution is low with respect to those usually obtained for quasi-static tests. Therefore, in the present work, an analytic approach is proposed to locate the crack tip. This method consists in extending the Corrected Beam Theory with Effective crack length method to the crack propagation stage. The crack propagation stage is estimated using only the load and the opening displacement measurements. In this study, the energy release rate is expressed by the energy dissipation mechanism [START_REF] Hashemi | The analysis of interlaminar fracture in uniaxial fibre-polymer composites[END_REF]:

d (W ext -U el -U kin ) bda = G - dU kin bda .
where W ext is the external work, U el is the elastic energy and U kin is the kinetic energy. The crack is assumed to have a straight front of width b and an extension da of the crack length a is considered. This new approach allows the evaluation of the energy release rate G ID and the crash velocity (å) without monitoring the crack tip. Consequently, each energy terms was determined experimentally thanks to the measurement of the specimen displacement using Digital Image Correlation (Fig. 22.2), strain gauges measurements and the opening loading. An assessment of the proposed technique by comparison with ASTM standard approach applied to the low loading rates has been performed.

Conclusion

An experimental protocol has been developed to identify the dynamic energy release rate for delamination processes in composite materials in mode I. Double Cantilever Beam specimens were selected and tested on tensile Split Hopkinson Pressure Bars for several loading rates up to 17 m.s -1 . A new analysis method was developed using strain measurements and displacement fields evaluated with the Digital Image Correlation method on the edges of the sample.

Fig. 22. 1

 1 Fig. 22.1 Experimental setup and closed view of the DCB specimen

Fig. 22. 2

 2 Fig. 22.2 Measurement of the displacement field along the loading axis

Acknowledgements

The works presented in the communication have been funded by the Physafe project under the Directorate General of Civil Aviation (DGAC). The authors are also grateful to the French Ministry of Defense, the Elsat2020, the FEDER and the Région Haut de France for their financial support.


Tarek Ahmed-Ali
email: tarek.ahmed-ali@ensicaen.fr

Vincent Van Assche
email: vincent.vanassche@uni-caen.fr

Jean-François Massieu
email: jean-francois.massieu@unicaen.fr

Philippe Dorléans
email: philippe.dorleans@unicaen.fr

Continuous-Discrete Observer for State Affine Systems With Sampled and Delayed Measurements

Keywords: Continuous-discrete observer, delayed measurements, impulsive systems

The observation of a class of multi-input multi-output (MIMO) state affine systemswithbothsampledanddelayedoutput measurements is addressed. These two constraints disturb simultaneously the convergence of the observer. Assuming some persistent excitation conditions to hold, and by using Lyapunov tools adapted to impulsive systems, two classes of global exponential observers are proposed. Some explicit relations between maximum allowable delay and maximum allowable sampling period are given. An extension to some classes of nonlinear systems is also given.

I. INTRODUCTION

This work is devoted to observer design for a class of uniformly observable systems with sampled and delayed measurements. In the last decades, the design of nonlinear observers for continuous systems with sampled measurements has received a great attention. This interest is motivated by many engineering applications, such as network control systems (NCSs) in which the output is transmitted over a shared digital communication network, and is only available at discrete-time instants. For linear systems it is usually possible to design observers by using the discrete time model of the continuous time system. This is not always possible for nonlinear systems because the exact discrete time model is generally not available. In this case, there exist two main approaches dealing with this problem. The first one, is based on the design of a discrete observer by using a consistent approximation of the exact discretized model. This approach provides a semi-global practical stability of the observation error. More details on this method can be found in [START_REF] Arcak | A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation[END_REF] and its references. The second one is based on a mixed continuous and discrete design. This approach has been inspired by Jazwinski [START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF] who introduced the continuous-discrete Kalman filter to solve a filtering problem for stochastic continuous-discrete time systems. It consists on two steps. In the first one (which is called the prediction step), the observer is a copy of the model system, whereas in the second step, the value of the state estimate is updated using the newly available sampled measure. The exponential convergence of the observation error is then ensured under some sufficient conditions on the sampling period through the stability analysis of impulsive systems. In [START_REF] Deza | High gain estimation for nonlinear systems[END_REF] the authors use this approach to write a discrete-continuous version of the well known high gain observer [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF].

In [START_REF] Nadri | Design of a continuous-discrete observer for state affine systems[END_REF], observers for a MIMO class of state affine systems where the dynamical matrix depends on the inputs have been designed when the inputs are regularly persistent. This work was extended to adaptive observers in [START_REF] Ahmed-Ali | Continuous-discrete adaptive observers for state affine systems[END_REF]. In [START_REF] Astorga | Nonlinear continuous-discrete observers: Applications to emulsion polymerization reactors[END_REF], a similar method has been used for a larger class of systems and applied to the observation of an emulsion copolymerization process. The observation of a class of systems with output injection has been treated in [START_REF] Nadri | Observer design for continuous-discrete time state affine systems up to output injection[END_REF] and recently, in [START_REF] Hammouri | Constant gain observer for continuous-discrete time uniformly observable systems[END_REF], a high gain continuous-discrete observer has been developed by using constant observation gains. In [START_REF] Andrieu | Observer design fot Lipschitz systems with discrete-time measurements[END_REF], the authors extend the work of [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H performance analysis[END_REF] to the discrete-time measurements case. Recently, a hybrid sampled-data observer dedicated to a class of nonlinear systems has been presented in [START_REF] Karafyllis | From continuous-time design to sampled-data design of nonlinear observers[END_REF]. This scheme is based on an inter-sample time predictor which estimates the output between two sampling instants. The advantage of this algorithm is in the fact that the estimates remain continuous and only the predictor is re-initialized at each sampling instant. This algorithm has been extended to some networked control systems in [START_REF] Ahmed-Ali | High gain observer design for some networked control systems[END_REF] by using a Lyapunov Krasovskii approach. On the other hand, the design of observers for linear detectable systems with sampled and delayed measurements was treated in [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF] by using a descriptor system approach and a Lyapunov Krasovskii functional. The authors have proposed a hybrid observer for a class of linear systems and derive sufficient conditions based on linear matrix inequalities to guarantee exponential convergence of the observation error. This idea has also been used in [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF] and [START_REF] Van Assche | High gain observer design for nonlinear systems with time varying delayed measurements[END_REF] for some classes of nonlinear systems with nonuniformly sampled measurements.

In this technical note, we present two classes of observers. The first one is an extension to delayed measurements of the continuous-discrete observer developed in [START_REF] Nadri | Design of a continuous-discrete observer for state affine systems[END_REF]. The second one is based on the introduction of an estimator of the delayed output between two updating instants. It can be viewed as an extension of the work of [START_REF] Ahmed-Ali | Cascade high predictors for a class of nonlinear systems[END_REF] to sampled-data case by using a prediction of the output between two sampling instants following the ideas developed in [START_REF] Karafyllis | From continuous-time design to sampled-data design of nonlinear observers[END_REF]. The main advantage of the second class compared to the first one is in the fact that the states of the observer remain continuous and only the estimator is re-initialized. This property will simplify the implementation compared to the first class. For both classes, we give explicit sufficient conditions on the delay which is considered unknown but bounded and the sampling period, to ensure exponential stability of the observation error. This second observer structure is also extended to a wide class of nonlinear globally Lipschitz systems. We also emphasize the fact that the result presented in this technical note can be easily extended to the case of scheduling protocols [START_REF] Ahmed-Ali | High gain observer design for some networked control systems[END_REF]. The present note is organized as follows: in Sections II-VI, lies some notations are introduced, then the problem is stated; in the fourth section we present our main results and in Section V we extend the second observer structure to a wide class of nonlinear systems.

II. NOMENCLATURE

First, some mathematical notations are introduced. Let , , , and let be the Euclidean norm. For , , , , represents the set of real matrices of order and stands for the identity matrix of order . If and ,

X Y denotes the space of all continuous functions mapping Y. The notation , for , represents the -norm of and represent the transposed vector of . We say that where if and where and denote respectively the smallest and the biggest eigenvalues of the matrix S. For and , the notation denotes the left limit of at instant , if it exists. In all this study, the initial time is called .

III. PROBLEM STATEMENT

The following class of systems is considered:

(

where is the instantaneous state vector, the input vector ( is a compact set) and the output vector. The matrix , , , are known, with , , , and . In this work, we suppose that the measures of are sampled at instants and available for the observer only at instants . The notation represents a strictly increasing sequence such that , and represents the transmission delay. The sampling intervals are bounded with for all

. The transmission delays are unknown, only an upper bound is known for all . We also assume that . This assumption means that the measures sampled at are available for the observer before the next measures sampled at . We will design two different classes of observers for systems [START_REF] Arcak | A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation[END_REF], and provide for each of them upper bounds of the maximum allowable sampling period and the maximum allowable delay, so that the observation error converges globally exponentially towards zero.

IV. OBSERVERS DESIGN AND STABILITY STUDY

A. First Observation Structure

The following continuous-discrete observer is proposed. For ,

For , with

The notation denotes the estimate of the state , is the so-called observation gain, and the parameters and are some design parameters. Note that the existence of the inverse of matrix will be ensured in the following. The initial conditions are denoted and is symmetric positive definite. The observer (2) and ( 3) is composed of a predictor part (2) and a correction one (3). During , the state estimate has the same dynamics as system (1). The matrix is defined as the solu-tion of the third differential equation in [START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF] and is updated at instants to ensure that it remains positive definite (see for example in [START_REF] Nadri | Design of a continuous-discrete observer for state affine systems[END_REF]). At each instant , observer trajectory is corrected taking into account the measure obtained at instant . Thanks to (3), conditions on the inputs and restriction on , it will be shown that the matrix remains positive definite. The dynamical equation satisfied by the state observation error, , is, in view of (1), ( 2) and (3), for

Before stating the main results, some preliminary definitions and results are required. Definition 1: Consider the following system, for :

(

and the transition matrix associated to system [START_REF] Nadri | Design of a continuous-discrete observer for state affine systems[END_REF], such that . The bounded input is said to be regularly persistent if there exist , , , such that, for all

Regularly persistent inputs guarantee the system to be observable. For more details, see [START_REF] Bornard | Regularly persistent observers for bilinear systems[END_REF]- [START_REF] Hammouri | Chapter: Topological properties of observer's inputs[END_REF]. In the sequel, we will assume that the following hypothesis is fulfilled.

Hypothesis 1: The input is regularly persistent and the constants and are known. Remark 1: The constants and are preponderant for determining a bound on the admissible sampling period. Due to the difficulty to verify the condition on the observability Grammian, in practice, numerical tools can be used for this purpose.

1) Technical Results: The following proposition guarantees that the matrix remains positive definite for any , under conditions on the types of inputs applied to the system and on the sampling period . Proposition 1: Let be a regularly persistent input for system [START_REF] Arcak | A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation[END_REF], for all where , if , where is the unique positive term such that , then, for all symmetric positive definite, there exist constants , such that, for

The proof of Proposition 1 can be found in [START_REF] Nadri | Design of a continuous-discrete observer for state affine systems[END_REF].

• Since considered inputs belong to a compact set D, and because is continuous, is well defined. • There always exists such a ; indeed writing condition as , loosely speaking it is clear that, for positive values of , the polynomial term on the left-hand side will 'cross' the exponential on the right-hand side at a unique point.

• Considering regularly persistent inputs, the positive definition of the solution of equations ( 2) and (3) can only be ensured under condition on the sampling period, otherwise the solution of (2) may become non-positive definite [START_REF] Nadri | Design of a continuous-discrete observer for state affine systems[END_REF]. • Notice that , implicitly depend on the maximum allowable sampling period . 2) Stability Analysis: Based on the above hypotheses, we are now able to state our main result.

Theorem 1: Let us consider system [START_REF] Arcak | A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation[END_REF]. Assume that hypothesis 1 holds and that as in Proposition 1. Then for all [START_REF] Nadri | Observer design for continuous-discrete time state affine systems up to output injection[END_REF] and (9) with , system (2) and ( 3) is a global exponential observer for system [START_REF] Arcak | A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation[END_REF].

Proof: Note that condition (8) satisfies Proposition 1. This means that ( 7) is verified for all . Now, let us consider the following Lyapunov candidate function: [START_REF] Andrieu | Observer design fot Lipschitz systems with discrete-time measurements[END_REF] As in [START_REF] Nadri | Design of a continuous-discrete observer for state affine systems[END_REF], let us compute the time derivative of for , then we have [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H performance analysis[END_REF] For , first, by using the Leibniz integration formula then we can rewrite (4) as follows: [START_REF] Karafyllis | From continuous-time design to sampled-data design of nonlinear observers[END_REF] which leads to [START_REF] Ahmed-Ali | High gain observer design for some networked control systems[END_REF] Now, let us compute , then

where [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF] and In the sequel, , , , , and will be noted respectively , , , , and . Now let us consider the terms and . After some computations we have [START_REF] Van Assche | High gain observer design for nonlinear systems with time varying delayed measurements[END_REF] where [START_REF] Ahmed-Ali | Cascade high predictors for a class of nonlinear systems[END_REF] and [START_REF] Bornard | Regularly persistent observers for bilinear systems[END_REF] From equations ( 14)- [START_REF] Bornard | Regularly persistent observers for bilinear systems[END_REF], by gathering the terms in on the one hand and the terms in on the other, one can write with and

One can check that if the following inequality holds: [START_REF] Hammouri | Chapter: Topological properties of observer's inputs[END_REF] On the other hand, from [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H performance analysis[END_REF], we can write [START_REF] Besançon | Observer synthesis for a class of nonlinear control systems[END_REF] on . Using the property ( 7), then we have [START_REF] Rajamani | Observers for Lipschitz nonlinear systems[END_REF] Using the definition of and the fact that on , then we can write [START_REF] Karafyllis | Global exponential observers for two classes of nonlinear systems[END_REF] and from this [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF] from the fact that and ( 22), we deduce [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF] and ( 26)

Then, to ensure exponential stability, it is sufficient to have

(27)
which gives the following upper bound on the delay :

(28) with i.e., the second condition of the theorem.

B. Second Observation Structure

In this section, another class of observers structure is proposed. The main difference compared to the first class is in the introduction of an output delayed estimator which is re-initialized at each updating instant

(29a) (29b)
The observer (29) is composed of a classical observer (29a) and an output estimator (58b). Note that and are continuous on (the set of strictly positive real numbers) and only is re-initialized at each updating instant.

Theorem 2: Let us consider system (1). Assume that hypothesis 1 holds then for sufficiently large values of , there exist bounds on the delays and on the sampling period, such that system (29a)-(58b) is a global exponential observer for system [START_REF] Arcak | A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation[END_REF].

Proof: In view of (29a)

If we introduce the error due to the maximum allowable delay (31) then (30) can be rewritten

(32) (33)
Since and are both continuous on , the error is continuous on

. Before the study of the convergence of the observation error , let us recall the following technical result [21] Proposition 2: Let be a regularly persistent input for system [START_REF] Arcak | A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation[END_REF]. Then for all where , and for all symmetric positive definite, there exist constants , such that, for

where and does not depend on . Now, let us consider the following Lyapunov-Krasovskii candidate which is inspired from [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]:

(35)
where is a positive design parameter and is a piecewise differentiable function such that , , , , and , . In order to prove the exponential convergence of the observation error, it is sufficient to find conditions on the maximum allowable delay and the maximum sampling allowable period ensuring (

From the fact that the observation error is continuous and the error , then it is obvious that the inequality (37) is fulfilled. Now let us consider the derivative of on Then we can say that the exponential convergence of the observation error is ensured if following conditions hold:

(49a) (49b) (49c)
In order to derive the maximum allowable sampling period and the maximum allowable delay, we consider that . Assume that and , then (49a) becomes ( 50)

where . Furthermore one can choose as a sawtooth function with , and , . Thus, condition (49b) is verified when (51) But one must also have , . This implies that the sampling period must be smaller than . Together with (51), this leads to the following condition on the maximal allowable sampling period:

(52) From this, (49c) leads to a bound on (53) If conditions (50), (52), and (53) are fulfilled, then the three conditions (49) are fulfilled and this means that the observation error converges exponentially towards zero.

Remark 2: Notice that the free strictly positive value can be used to tune the lower bound of following (50) and the corresponding bounds of the sampling period and delay following (52) and (53).

V. EXTENSION TO GLOBALLY LIPSCHITZ NONLINEAR SYSTEMS

In this section we extend the result of Theorem This hypothesis means that the following system: (57) is a global exponential observer for systems (54). It has also to be noticed that hypotheses 2 and 3 are automatically satisfied for triangular systems (see [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]) and the class of nonlinear systems defined in [START_REF] Rajamani | Observers for Lipschitz nonlinear systems[END_REF].

Now we are able to give the result of this extension. Theorem 3: Let us consider systems (54) and suppose that hypotheses 2 and 3 hold. Then there exist bounds on the delay and sampling period such that the following system:

(58a) (58b)
is a global exponential observer for systems (54).

1) Sketch of Proof: The proof of this Theorem is similar to the proof of Theorem 2 by using the same Lyapunov-Krasovskii candidate function which is inspired from [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF] (59

)
where and is defined in hypothesis 3. Remark 3: Following the same methodology, it has to be noticed that the second structure used in the results of Theorems 2 and 3 can be easily extended to classes of observers contained in [START_REF] Karafyllis | Global exponential observers for two classes of nonlinear systems[END_REF].

Remark 4: The main contribution of this note is that we give several results concerning the observers design where both delay and sampling phenomena appear on the output measurements. This work must be compared to the result of [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF] where the authors consider this case for linear systems and without using predictor. The introduction of predictor will improve the bound of sampling period because the model of the output is used to predict it between two updating instants.

VI. CONCLUSION

In this technical note, the observation of a class of continuous MIMO systems with sampled and delayed measurements has been studied. Under some sufficient conditions, some classes of global exponential observers have been developed. This contribution can be obviously extended to the case of scheduling protocols in order to include a maximum number of communication constraints which often appear in the networked control systems. But for the class of finite-time observers developed in [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF] and [START_REF] Shen | Semi-global finite-time observers for multi-output[END_REF], this problem remains still open because the corrector term of the observer is not locally or globally Lipschitz.

  from (29a) and (33), we get Applying the Jensen's inequality, then we have handle the terms in , one can differentiate its expression (31) and get, through a direct application of the observer definition (29) to rewrite the upper bound on from (41), one gets(45)On the other hand from (32) we can easily deduce that (


Tarek Maatoug

Mondher Farza
email: mfarza@greyc.ensicaen.fr

Tarek Mohammed M'saad

T Ahmed-Ali

Adaptive output feedback high gain controller for a class of nonlinear systems

Keywords: Nonlinear system, Adaptive output feedback control, Admissible tracking capability, High gain control, Sliding mode control, Adaptive observer, Filtered integral action

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The past decade has witnessed many achievement in the design of adaptive controller for nonlinear systems with unknown parameters. Many significant results ( [START_REF] Kokotovic | Systematic design of adaptive controller for feedback linearizable systems[END_REF], [START_REF] Kanellakopouslos | Nonlinear and Adaptive Control Design[END_REF], [START_REF] Johansen | Robust adaptive control of minimum phase nonlinear systems[END_REF], [START_REF] Sastry | Adaptive control of linearizable systems[END_REF] and references therein) have been obtained. One of the important assumptions in these adaptive schemes is that the system nonlinearities are linear in the unknown parameters. In ( [START_REF] Hassan | Adaptive output feedback control of nonlinear systems represented by input-output models[END_REF]), an adaptive output feedback controller for a class of SISO nonlinear systems with unknown constant parameters is proposed. The controller uses a high gain observer for estimate the unmeasurable state and under an additional persistence of excitation, the unknown parameters are estimated. A globally stable adaptive output feedback controller was developed using high-gain adaptation for systems containing nonlinear parameterizations ( [START_REF] Marino | Global adaptive outputfeedback control of nonlinear systems-part ii: Nonlinear parameterization[END_REF]),which, however, is only applicable to set-point regulation control. In ( [START_REF] Santosuosso | Global adaptive output feedback controllers with application to nonlinear friction compensation[END_REF]), an output feedback controller for a class of nonlinear systems that consist of a set of unknown constant parameters and unmeasurable state variables was considered. Under the assumption that the dynamics of the unmeasurable state is asymptotically stable and us-ing parameter dependent filtered transformations, an output feedback controller was constructed. Recently ( [START_REF] Zhu | Adaptive controller and observer design of a class of nonlinear systems[END_REF]), a stable adaptive controller and observer for a class of nonlinear systems are designed using a parameters dependent lyapunov function.

In this paper, one proposes an adaptive output feedback controller for a class of nonlinear controllable and observable systems. More specifically, one will address an admissible tracking problem for systems without zero dynamics, allowing thereby a comprehensive presentation of the proposed control design framework. The adaptive output feedback controller is obtained by simply combining an appropriate high gain state feedback control with a high gain adaptive observer ( [START_REF] Farza | A set of adaptive observers for a class of mimo nonlinear systems[END_REF], [START_REF] Maatoug | Adaptive observer design for a class of mimo nonlinear systems[END_REF]). The state feedback control design was particularly suggested from the the high gain observer design bearing in mind the control and observation duality. Of particular interest, the controller gain involves a well defined design function which provides a unified framework for the high gain control design, namely several versions of sliding mode controllers are obtained by considering particular expressions of the design function. Furthermore, it is shown that a filtered integral action can be simply incorporated into the control design to carry out a robust compensation of step like disturbances while reducing appropriately the noise control system sensitivity. This paper is organized as follows. The problem formulation is presented in the next section. Section 3 is devoted to the state feedback control design with a full convergence analysis of the tracking error in a free disturbances case. The adaptive output feedback controller is presented in section 4 where the main result of this contribution is given. Section 5 emphasizes the high gain unifying feature of the proposed control design. The possibility to incorporate a filtered integral action into the control design is shown in section 6. Simulation results are given in section 7 to highlight the performances of the proposed controller.

PROBLEM FORMULATION

One seeks for an admissible adaptive tracking problem for MIMO systems which dynamical behavior can be de-scribed by the following state representation

ẋ = Ax + Bu + g(x) + Ψ(x)ρ y = Cx = x 1 (1) with x =      x 1 x 2 . . . x q      ; ρ =      ρ 1 ρ 2 . . . ρ m      ; g(x) =        g1 (x 1 ) g2 (x 1 , x 2 )
. . . gq-1 (x 1 , . . . , x q-1 ) gq (x)

       ; ΨT (x) =        ΨT 1 (x) ΨT 2 (x) . . . ΨT m (x)        , Ψj (x) =           Ψ1 j (x 1 ) Ψ2 j (x 1 , x 2 ) . . . Ψq-1 j (x 1 , . . . , x q-1 ) Ψq j (x)           A =          0 I p 0 0 . . . . . . I p 0 . . . . . . 0 0 . . . I p 0 . . . 0 0          (2)
C = [I p , 0 p , . . . ,

0 p ] (3)
B = [0 p , 0 p , . . . , I p ] (4)
where the output y ∈ IR p ; the state x ∈ ϑ an open subset R n with x k ∈ IR n1 , k = 1, . . . , q; the input u(t) ∈ U , a compact subset of IR s ; ρ ∈ IR m is a vector of unknown constant parameters, ρ i ∈ IR, i = 1, . . . , m; g(x) ∈ IR n with gk (x) ∈ IR p , k = 1, . . . , q; Ψ(x) is a n × m matrix and each Ψj (x) ∈ IR n , j = 1, . . . , m, denotes its j th column with Ψk j (x) ∈ IR p , k = 1, . . . , q. The notation I k where k is a positive number refers to the k × k identity matrix while 0 k and 0 k1×k2 , k, k 1 , k 2 being positive integers, denote the null matrix with dimension k × k and k 1 × k 2 , respectively. Please notice that system (1 is controllable and uniformly observable. The observer-based controller design requires some assumptions which will be stated in due courses. At this step, one assumes the following: (A1) For any bounded input u, i.e. ∀u ∈ U a compact subset of IR s , the state x(t) and the unknown parameters ρ are bounded, i.e. x(t) ∈ X, for t ≥ 0 and ρ ∈ Ω where X ⊂ IR n and Ω ∈ IR m are compacts sets. (A2) The matrix Ψ(x) is continuous on X. (A3) The functions g(x) and Ψ(x) are Lipschitz. Please notice that since the state is confined to the bounded set X, one can extend the nonlinearities g(x) and Ψ(x) into g(x) and Ψ(x) in such a way that the restrictions of g(x) and Ψ(x) respectively coincide with g(x) and Ψ(x) on X and that g(x) and Ψ(x) become global Lipschitz, i.e. Lispchitz on the whole space IR n . Indeed, let σ : IR n -→ X, x → σ(x) be any smooth bounded saturation function that coincides with x on X, i.e. σ(x) = x for all x ∈ X (see e.g. [START_REF] Shim | Semi-global observer for multi-output nonlinear systems[END_REF][START_REF] Conlon | Differentiable Manifolds[END_REF]). One defines the respective Lipschitz extensions, g(x) and Ψ(x), of g and Ψ as follows:

g(x) = g(σ(x)) Ψ(x) = Ψ(σ(x))
Now, consider the following dynamical system:

ẋ = Ax + Bu + g(x) + Ψ(x)ρ y = Cx = x 1 (5)
It is clear that system (5) coincides with system (1) for (x, u, ρ) ∈ X × U × Ω. Therefore, it does not make any difference that we consider system (5) instead of ( 1) for the observer-based controller synthesis. Indeed, system (5) shall be considered in the next section. Please notice that for any bounded input u ∈ U , g(x) and Ψ(x) are, by construction, globally Lipschitz with respect to x and are bounded for all x ∈ IR n .

The control problem to be addressed consists in an asymptotic tracking of an output reference trajectory that will be noted {y r (t)} ∈ IR p and assumed to be enough derived, i.e. lim t→∞ (y(t)y r (t)) = 0.

Taking into account the class of systems, it is possible to determine the system state trajectory {x r (t)} ∈ IR n and the system input sequence {u r (t)} corresponding to the output trajectory {y r (t)} ∈ IR p . This allows to define an admissible reference model as follows ẋr = Ax r + Bu r + g(x r ) + Ψ(x r )ρ

y r = Cx r (6)
The reference model state x r ∈ IR n and its input u r ∈ IR s can be determined as follows

           x 1 r = y r x k r = ẋr k-1 -g k-1 (x 1 r , . . . , x k-1 r ) -Ψ k-1 (x 1 r , . . . , x k-1 r )ρ f or k ∈ [2, q] u r = ( ẋr q -g q (x r ) -Ψ q (x r )ρ) (7)
By assuming that the reference trajectory is smooth enough, one can recursively determine the reference model state and input from the reference trajectory and its first derivatives, i.e. y

(i) r = d i y r dt i f or i ∈ [1, q -1], as follows.                                  ϕ 1 (y r ) = y r ϕ k y r , y (1)
r , . . . , y

(k-1) r = k-2 j=0 ∂ϕ k-1 ∂y (j) r y r , . . . , y (k-2) r y (j+1) r -g k-1 ϕ 1 (y r ) , . . . , ϕ k-1 y r , y (1)
r , . . . , y

(k-2) r -Ψ k-1 ϕ 1 (y r ) , . . . , ϕ k-1 y r , y (1)
r , . . . , y

(k-2) r ρ f or k ∈ [2, q]
The adaptive output tracking problem can be hence turned to a state trajectory tracking problem defined by

lim t→∞ e(t) = 0 (8)
where

e(t) = x(t) -x r (t) (9)
Such problem can be interpreted as a regulation problem for the tracking error system obtained from the system and model reference state representations ( 1) and ( 6), respectively.

   ė = Ae + B (u(x) -u r ) + g (x) -g (x r ) + (Ψ(x) -Ψ(x r )) ρ e m = y -y r (10)

STATE FEEDBACK CONTROL

As it was early mentioned, the proposed state feedback control design is particularly suggested by the duality from the high gain observer design proposed in [START_REF] Farza | A set of observers for a class of nonlinear systems[END_REF]. The underlying state feedback control law is then given by

             u (x) = u r + ν (e) u r = ẋq r -g q (x r ) -Ψ q (x r ) ρ ν (e) = -K c λ q B T S∆ λ e ∆ = -K c B T Sē (11)
where e is the tracking error defined by ( 9), ∆ λ is the block diagonal matrix defined by

∆ λ = diag I p , 1 λ I p , . . . , 1 λ q-1 I p ( 12
)
where λ > 0 is a positive scalar, S is the unique solution of the the following algebraic Lyapunov equation

S + A T S + SA = SBB T S (13)
and K c : IR p → IR p is a bounded design function satisfying the following property

∀ξ ∈ Ξ one has ξ T K c (ξ) ≥ 1 2 ξ T ξ ( 14
)
where Ξ is any compact subset of IR p .

I p 0 p . . . 0 p       (16)
Using some useful algebraic manipulations as in [START_REF] Farza | Observer design for a class of MIMO nonlinear systems[END_REF] yields

B T S = CS -1 T = [C q q I p C q-1 q I p . . . C 1 q I p ] (17)
The above state feedback control law satisfies the tracking objective [START_REF] Zhu | Adaptive controller and observer design of a class of nonlinear systems[END_REF] as pointed out by the following fundamental result Theorem 3.1 The state and output trajectories of the ( 1)-( 4) subject to the assumptions A1 et A2 generated from the input sequence given by ( 11)-( 14) converge globally exponentially to those of the reference model ( 6) for relatively high values of λ.

Proof.: due to lack of place, the proof is not given but its main outlines are similar to those given in [START_REF] Hajji | Observer-based output feedback controller for a class of nonlinear systems[END_REF].

Remark 3.2 Consider the case where the state matrix structure is as follows

A =          0 A 1 0 . . . 0 0 0 A 2 . . . 0 . . . . . . . . . . . . 0 . . . . . . . . . A q-1 0 . . . . . . 0 0         
where A i ∈ R p×p f or i ∈ [1, q -1] are invertible constant matrices. One can easily show that the corresponding control law ν(e) in the expression of the control law [START_REF] Shim | Semi-global observer for multi-output nonlinear systems[END_REF] is then given by ν(e) = -

q-1 i=1 A i -1 K c λ q B T S∆ λ Λe (18)
with

Λ = diag I p , A 1 , A 1 A 2 , . . . , q-1 i=1 A i (19)
To this end, let us consider the change of coordinates z = Λx, the system can be rewritten as follows

ż = ΛAΛ -1 z + ΛBu + Λϕ(x) y = CΛ -1 z = z 1 (20)
Taking into account the structure of the the system state realization as well as the transformation matrix, one gets

ΛAΛ -1 = 0 I n-p 0 0 ΛB = B q-1 i=1 A i and CΛ -1 = C (21)
One hence recovers the structure of the considered class of systems, i.e. equations ( 1) to ( 4), and naturally deduces the expression of the state feedback control law [START_REF] Narendra | Stable Adaptive Systems[END_REF].

ADAPTIVE OUTPUT FEEDBACK CONTROL

The adaptive output feedback control we are concerned by is obtained by invoking the certainty equivalence principle while using an adaptive high gain observer. Indeed consider the following dynamical system

u (x, ρ) = ẋq r -g q (x r ) -Ψ q (x r ) ρ + ν (ê) ( 22
)
where xr is the estimation of the reference trajectory, x r , and is computed as in ( 8) by replacing ρ by ρ and

ν (ê) = -K c λ q B T S∆ λ ê ∆ = -K c λ q B T S∆ λ (x -xr ) (23)
and

ẋ(t) = Ax + Bu (x, ρ) + g(x) + Ψ(x)ρ -θ∆ -1 θ S -1 + Υ P Υ T C T C (x -x) ρ(t) = -θP Υ T C T C (x -x) Υ(t) = θ (A -S -1 C T C) Υ(t) + ∆ θ Ψ(x(t)) with Υ(0) = 0 Ṗ (t) = -θP (t)Υ T (t)C T CΥ(t)P (t) + θP (t) with P (0) = P T (0) > 0 (24) where x =      x1 x2 . . . xq      ∈ IR n with xk ∈ IR p , k = 1, . . . , q; x
is the unknown trajectory of system (5); ρT = [ρ 1 ρ2 . . . ρm ] ∈ IR m ; S and C are respectively given by ( 15) and (3); ∆ θ is a diagonal matrix defined in a similar as the matrix ∆ λ (equation 12) for the positive scalar θ > 0; the notation P (0) = P T (0) > 0 means that the initial condition of the Ordinary Differential Equation (ODE) governing P is chosen SPD. The control law involves a gain depending on the bounded design function K c which is completely characterized by the fundamental property [START_REF] Gauthier | A simple observer for nonlinear systems -application to bioreactors[END_REF]. Some useful design functions have been given in [START_REF] Maatoug | A set of adaptive observers for a class of mimo nonlinear systems[END_REF] to emphasize the unifying feature of the proposed high gain concept.

Before stating our main result, one assumes the additional assumption : (A4) The inputs u are such that for any trajectory x of system (24) starting from (x(0), ρ(0)) ∈ X×Ω, the matrix CΥ(t) is persistently exciting i.e.

∃δ 1 , δ 2 > 0; ∃T > 0; ∀t ≥ 0 :

δ 1 I m ≤ t+T t Υ T (τ )C T CΥ(τ )dτ ≤ δ 2 I m
Please notice that assumption (A4) gives a certain excitation condition which is stated in a classical way [START_REF] Narendra | Stable Adaptive Systems[END_REF]. However, this assumption does not state how to generate the input u that ensures the realization of this condition.

In fact, up to our knowledge, except some particular cases (such as linear systems), the problem of characterizing the set of inputs ensuring the persistent excitation condition is still be open. Now, one states the main result. Indeed, the resulting control system achieves the required tracking performances as pointed out by the following Theorem.

η = θAη -θS -1 C T C ε + ∆ θ (g(x) -g(x)) +∆ θ Ψ(x) -Ψ(x) + θS -1 C T CΥρ ρ Set V 1 (η) = λ 2q η T Sη, V 2 (ρ) = λ 2q ρT P -1 ρ and let V o (η, ρ) = λ 2q (V 1 + V 2 )
be the Lyapunov candidate function. According to the algebraic Lyapunov equation ( 15), one has:

V1 = -θV 1 -λ 2q θη T C T Cη +2λ 2q η T S∆ θ (g(x) -g(x)) +2λ 2q η T S∆ θ (Ψ(x) -Ψ(x)) ρ ≤ -(θ -k 1 )V 1 + k 2 V 1 V 2 (25)
where k 1 and k 2 are positive constants which do not depend on θ ≥ 1.

Besides, one has:

V2 (t) = -θV 2 -λ 2q θ(Υρ) T C T CΥρ -2λ 2q θη T C T CΥρ (26)
Combining ( 25) and (26), one gets:

V o = V1 + V2 ≤ -(θ -k 1 )V 1 -θV 2 + k 2 V 1 V 2 (27)
The inequality (27) leads to:

V o (t) ≤ e -θ-k 1 2 t V o (0) (28)
The rest of the proof is not given, but is similar to that given in [START_REF] Hajji | Observer-based output feedback controller for a class of nonlinear systems[END_REF].

FILTERED INTEGRAL ACTION

One can easily incorporate a filtered integral action into the proposed state feedback control design, for performance enhancement considerations, by simply introducing suitable state variables as follows

σf = e f ėf = -Γe f + Γe 1 (29)
where Γ = Diag {γ i } is a design matrix that has to be specified according to the desired filtering action. The state feedback gain is then determined from the control design model

       ėa = A a e a + g a (x r , e a ) -g a (x r , 0) + B a (u a (e a ) -u r ) + (Ψ a (x r , e a ) -Ψ a (x r , 0)) ρ y a = σ f (30) e a =   σ f e f e   , A a =   0 I p 0 0 0 Γ 0 0 A   , B a =   0 p 0 p B   g a (x r , e a ) =   0 p -Γe f g(e + x r )   Ψ a (x r , e a ) =   0 p 0 p Ψ(e + x r )  
Indeed, the control design model structure (30) is similar to that of the error system [START_REF] Maatoug | Adaptive observer design for a class of mimo nonlinear systems[END_REF] and hence the underlying state feedback control design is the same. The adaptive output feedback control law incorporating a filtered integral action is then given by

                 ẋ = Ax + Bu (x) + g(x) + Ψ(x)ρ -θ∆ θ S -1 + ΥP Υ T C T C (x -x) ρ = -θP Υ T C T C (x -x) u(ê a , ρ) = ẋq r -g q (x r ) -Ψ q (x r )ρ + ν(ê a ) ν(ê a ) = -Γ -1 K c (λ q+2 B T a Sa ∆ aλ Λê a ) (31)
with êa =

  σ f e f ê   (32) ∆ aλ = diag I p , 1 λ I p , . . . , 1 λ q I p , 1 λ q+1 I p (33) Λ = diag (I p I p Γ . . . Γ) ( 34
)
where Sa is the unique symmetric positive definite matrix solution of the following Lyapunov algebraic equation

Sa + Sa Āa + ĀT a Sa = Sa Ba BT a Sa (35)
and K c is a bounded design function satisfying inequality [START_REF] Gauthier | A simple observer for nonlinear systems -application to bioreactors[END_REF]. It can be easily shown that the resulting adaptive output feedback control system is globally stable and performs an asymptotic rejection of state and/or output step like disturbances.

ILLUSTRATIVE EXAMPLE

Let consider an academic tracking problem for the nonlinear double integrator described by

       ẋ1 = x 2 + -x 3 1 + sinω 1 t ρ 1 ẋ2 = (2 + tanh(x 2 ))u -atan(x 2 ) ρ 1 + cosω 1 t 1 + x 2 2 ρ 2 y = x 1
where the state vector is x = [x 1 x 2 ] T ∈ IR 2 and the functions Ψ and g specify as follows:

Ψ(x) =   -x 3 1 + sinω 1 t 0 -atan(x 2 ) cosω 1 t 1 + x 2 2   and g(x) = 0 0 .
ρ 1 and ρ 2 are constant unknown parameters. The value of w 1 used in simulation is 20. The desired output reference trajectory is generated from a second order generator with unitary static gain and two equal poles p 1 = p 2 = -5 which input sequence is shown by the figure 1. In order to achieve the control objective using the observer-based controller proposed in the paper, we have defined the auxiliary input v = ((2 + tanh(x 2 ))u. Then, the real input, applied to system is obtained as u = intensive simulation study has been made using all the design functions that has been described above. As the performances were almost comparable, one will present only those obtained with the design function given by the expression K c (ξ) = k c tanh(k o ξ). The design parameters have been specified as follows :k c = 1, k o = 7, λ = 4.51, τ = 50 and θ = 15. The obtained results are given in figures 2 and 3. Two remarks are worth to be mentioned. Firstly, the proposed controller achieve the required tracking performances . Secondly, the obtained results clearly show the good performances of the adaptive observer in providing satisfactory estimates of the states as well as of the unknown parameters.

CONCLUSION

A unified high gain state feedback control design framework has been developed to address an admissible tracking problem for a class of controllable and uniformly observable nonlinear systems. Such a framework has been particularly suggested thanks to the duality from the high gain system observation. The unifying feature is provided through a suitable design function that allows to rediscover all those well known high gain control methods, namely the sliding modes control. Moreover, the proposed state

Remark 3 . 1

 31 Taking into account the structure of the matrices B et C and the fact that the following algebraic Lyapunov equationS + A T S + SA = C T C (15)has a unique symmetric positive definite solution S[START_REF] Gauthier | A simple observer for nonlinear systems -application to bioreactors[END_REF], one can deduce that equation (13) has a unique symmetric positive definite solution S which can be expressed as followsS = T S -1 T with T =      0p . . . 0 p I p . . . 0 p I p 0 p 0 p I p 0 p . . .

Theorem 4 . 1 Proof of Theorem 4 . 1 :

 4141 The control system corresponding to the adaptive output feedback controller (22)-(24) leads to an asymptotically exponentially vanishing tracking , i.e. lim t→∞ x(t)x r (t) = 0, provided that the assumptions A1 to A4 hold. one shall firstly show that the observation error converges exponentially to zero, i.e. lim t→∞ ε(t) = 0 and lim t→∞ ρ(t) = 0, and then conclude to the exponential convergence to zero of the tracking error estimate, i.e. lim t→∞ ê(t) = 0. The first part is established from a Lyapunov function using the error ε = ∆ θ ε and ρ. Define :η = ε -Υρ. One can show that:

Figure 1 .

 1 Figure 1. The reference generator input sequence

2 TIMEFigure 2 .Figure 3 .

 223 Figure 2. Performances of the adaptive output controller