V Van Assche

T Ahmed-Ali

C A B Hann

F Lamnabhi-Lagarrigue

High gain observer design for nonlinear systems with time varying delayed measurements

Keywords: Time-varying delay systems, sampled-data systems, high gain observer

This work presents a simple observer for triangular nonlinear systems with time varying delayed output measurement. A sufficient condition ensuring the asymptotic convergence of the observation error towards zero is given, as well as an explicit relation between the bound of the delay and the parameter of the observer. This result is illustrated by some simulations.

INTRODUCTION

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. For example, we can cite the systems which are controlled by a remote controller through a communication system as networked control systems, or when the measurements are intrinsically delayed as for example, in biochemical reactors. In the linear case, this problem has been solved by the well-known Smith predictor [START_REF] Smith | Closer control of loops with dead-time[END_REF] and several predictive control algorithms [START_REF] Clarke | Generalized predictive control-part I and II[END_REF], [START_REF] Shaked | H-infinity static output-feedback control of linear continuous-time systems with delay[END_REF]. Recently, a fundamental result has been derived for a class of nonlinear uniformly observable systems in [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF] with a constant delay. The authors presented a new kind of chained observers which reconstruct the states at different delayed time instants. The observer design is done step by step. At each step, the corresponding observer needs the states of all previous observers to converge. The authors showed, by using Gronwall lemma, that under some conditions on the delay, asymptotic convergence of these observers is ensured. These conditions have been relaxed in [START_REF] Kazantzis | Nonlinear observer design in the presence of delayed output measurements[END_REF] for a particular class of nonlinear systems. A change of coordinates based on a first-order singular partial differential equation is used, which puts the considered systems in output injection form with a stable linear part. On the other hand, in [START_REF] Besançon | Asymptotic state prediction for continuous-time systems with delayed input and application to control[END_REF] a predictor for linear and nonlinear systems with delayed input has been designed. Sufficient conditions based on linear matrix inequalities are derived to guarantee the asymptotic convergence of this predictor when the delay is constant. This predictor has been extended in [START_REF] Ahmed-Ali | Cascade high gain observers for nonlinear systems with delayed output[END_REF] to triangular systems by using the framework of high gain observer. On the other hand, the case of nonlinear systems with time varying delayed measurements has been recently treated in [START_REF] Cacace | An observer for a class of nonlinear systems with time varying observation delay[END_REF]. A nonlinear observer has been given for drift nonlinear systems. By using a Razumikhin approach, the authors derive some conditions ensuring the asymptotic convergence of the observation error. In the present work, we propose a nonlinear observer for nonlinear uniformly observable systems affected by a time-varying delay in the output measurements. We will see that the general high gain observer framework developed in [START_REF] Bornard | A high gain observer for a class of uniformly observable systems[END_REF], [START_REF] Gauthier | A simple obsever for nonlinear systems: Application to bioreactors[END_REF], to mention a few, for delay-free output measurements can be extended to systems with time varying observation delay. More precisely, if we suppose that the delay function is bounded and piecewise continuous, and without any information on its derivative, then by using a suitable Lyapunov-Krasovskii functional [START_REF] Fridman | On inputto-state stability of systems with time-delay: A matrix inequalities approach[END_REF], we provide simple conditions on the upper bound of the delay which ensure asymptotic convergence of our observer. The present paper is organized as follows: In section 2, we present the class of considered systems and the different assumptions. In the third one, we detail the proposed observer and prove its convergence. In the last one, we illustrate our result through simulations on an academic example.

PRELIMINARIES AND NOTATIONS

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

• The euclidian norm on R n will be denoted by . . • The matrix X T represents the transposed matrix of X. • λ min (S) and λ max (S) are the minimum and maximum eigenvalues of the square matrix S. • The set of natural numbers is represented by N.

In this paper, we consider the following class of nonlinear systems:

ẋ = Ax + φ(x, u) y = Cx(t -τ (t)) (1)
where

A =          0 1 0 . . . 0 0 0 1 0 . . . . . . 0 . . . 1 0 . . . . . . . . . . . . 1 0 . . . . . . . . . 0          (2) C = ( 1 0 . . . 0 ) (3) and φ(x, u) =    φ 1 (x, u)
. . .

φ n (x, u)    (4)
The term τ (t) represents the measurement time delay, x(t) = (x 1 , . . . , x k+1 , . . . , x n ) T ∈ R n is the vector state which is supposed unavailable. The output y(t) ∈ R is a linear function of the state x at time t-τ (t). The input u ∈ U where U is a compact set in R. The functions φ i , i = 1, . . . , n are supposed smooth. This class represents the class of uniformly observable systems. It has been shown that these models concern a wide variety of systems, such as bioreactors [START_REF] Bornard | A high gain observer for a class of uniformly observable systems[END_REF], [START_REF] Gauthier | A simple obsever for nonlinear systems: Application to bioreactors[END_REF]. We assume that the following hypotheses are satisfied:

• Hypothesis 1. The functions φ i (x, u) are triangular in

x, i.e ∂φ i (x, u) ∂x k+1 = 0 k = i, . . . , n -1 (5)
• Hypothesis 2. The functions φ i (x, u) are globally Lipschitz with respect to x, uniformly in u, i.e:

∃β > 0 such that ∀(x, x ′ ) ∈ R n × R n , ∀u ∈ U |φ i (x, u) -φ i (x ′ , u)| ≤ β x -x ′ (6)
• Hypothesis 3. τ (t) is a known piecewise-continuous time delay which satisfies 0 ≤ τ (t) ≤ τ 1 .

Note that the functions φ i (x, u) may contains linear parts and that ẋn depends on all x 1 , . . . , x n through φ n (x, u).

MAIN RESULT

In this section, we present the main result of this paper. Theorem 1. Consider the following observer:

ẋ = Ax + φ(x, u) -θ∆ -1 S -1 C T (C x(t -τ (t)) -y) ŷ = C x(t) (7)
where θ is a positive constant satisfying θ > 1, S is a symmetric positive definite matrix, which is a solution of the following equation:

SA + A T S -C T C = -S.
(8) and ∆ is a diagonal matrix which has the following form:

∆ = Diag(1, . . . , 1 θ i-1 , . . . , 1 θ n-1 ). ( 9
)
Then for sufficiently large positive θ, there exists a positive constant τ 1 such that ∀τ (t) ∈ [0, τ 1 ] and for any initial condition, lim t→+∞ x -x = 0.

Proof. First let us note the observation error x = x -x. Then we will have:

ẋ = Ax + φ(x, u) -φ(x, u) -θ∆ -1 S -1 C T C x(t -τ (t)) (10) If we apply the relation x(t) = x(t -τ (t)) + t t-τ (t) ẋ(s)ds (11)
and the change of coordinates x = ∆x, system (10) can be rewritten in the following manner:

ẋ = θ(A -S -1 C T C)x + ∆(φ(x, u) -φ(x, u)) + θS -1 C T C t t-τ (t)
ẋ(s)ds ( 12)

since C∆ = C∆ -1 = C.
In order to derive an upper bound τ 1 for the delay τ (t), to ensure the asymptotic convergence to zero of the error x, we use the following Lyapunov-Krasovskii functional from [START_REF] Fridman | On inputto-state stability of systems with time-delay: A matrix inequalities approach[END_REF]:

W = xT S x + t t-τ1 t s ẋ(ξ) 2 dξds. ( 13
)
This functional can be written after some manipulations as follows:

W = xT S x + t t-τ1 (s -t + τ 1 ) ẋ(s) 2 ds. (14)
If we compute its time derivative, we obtain

Ẇ ≤ θx T (A T S + SA -2C T C)x + 2x T S∆(φ(x, u) -φ(x, u)) + 2θx T C T C t t-τ (t) ẋ(s)ds + τ 1 ẋ(t) 2 - t t-τ1 ẋ(s) 2 ds. (15)
Using (8), we have

Ẇ ≤ -θx T S x + 2x T S∆(φ(x, u) -φ(x, u)) -θx T C T C x + 2θx T C T C t t-τ (t) ẋ(s)ds + τ 1 ẋ(t) 2 - t t-τ1 ẋ(s) 2 ds. ( 16
)
Using the triangular structure and the Lipschitz properties of the functions φ i , we can easily derive that if θ > 1, then

∆(φ(x, u) -φ(x, u)) ≤ √ nβ x . ( 17
)
From this, we deduce that there exists a constant

k 1 = 2 λ max (S) λ min (S) √ nβ, (18)
such that

2x T S∆(φ(x, u) -φ(x, u)) ≤ k 1 V (19) with V = xT S x.
Using the following property:

2θx T C T C t t-τ (t) ẋ(s)ds -θx T C T C x = -θ Cx -C t t-τ (t) ẋ(s)ds T • Cx -C t t-τ (t) ẋ(s)ds + θ t t-τ (t) ẋ(s)ds T C T C t t-τ (t) ẋ(s)ds , one gets 2θx T C T C t t-τ (t) ẋ(s)ds -θx T C T C x ≤ θ t t-τ (t) ẋ(s)ds T C T C t t-τ (t)
ẋ(s)ds .

From this, we will have

Ẇ ≤ -θV + k 1 V + θ t t-τ (t) ẋ(s)ds T C T C t t-τ (t) ẋ(s)ds + τ 1 ẋ(t) 2 - t t-τ1 ẋ(s) 2 ds. (20)
Using equation ( 12), one can write ẋ(t) ≤ θ α x + ∆(φ(x, u) -φ(x, u))

+ θλ max S -1 I , (21) where α = A -S -1 C T C , I = t t-τ (t)
ẋ(s)ds.

If θ > 1. Since, as proven by [START_REF] Gauthier | A simple obsever for nonlinear systems: Application to bioreactors[END_REF],

∆ φ(x, u) -φ(x, u) ≤ √ nβ x , one has ẋ(t) ≤ θ α + √ nβ x + θλ max S -1 I . ( 22
)
Using the Young inequality, one can write

ẋ(t) 2 ≤ θ 2 k 2 V + I 2 (23)
where

k 2 = 2 sup α + √ nβ λ min (S) 2 , λ 2 max S -1 , 1 . ( 24
)
Using this and equation ( 15), we will have:

Ẇ ≤ -θV + k 1 V + θI T C T CI + τ 1 θ 2 k 2 V + I 2 - t t-τ1 ẋ(s) 2 ds
If we use the following Jensen's inequality:

I 2 ≤ τ 1 t t-τ1 ẋ(s) 2 ds (25) Ẇ ≤ -(θ -k 1 -τ 1 θ 2 k 2 )V -1 -θτ 1 -τ 2 1 θ 2 k 2 t t-τ1 ẋ(s) 2 ds. (26) Let us choose τ 1 = 1 2k 2 θ . Then (26) becomes Ẇ ≤ - θ 2 -k 1 V -1 - 3 8k 2 t t-τ1 ẋ(s) 2 ds (27)
Since k 2 > 1 and θ > 1, then 1 -3 8k2 > 0. From this we can say that the inequality

Ẇ < - θ 2 -k 1 V ≤ 0 (28)
fulfilled under the following conditions;

θ > sup {1, 2k 1 } (29a) τ (t) ∈ 0, 1 2k 2 θ (29b)
Integrating ( 28) and using Barbalat's lemma, we conclude that lim t→+∞ x(t) = 0.

To summarize Theorem 1, it gives an estimation of the maximum delay supported by observer (7) which enables x(t) → x(t), once θ has been fixed according to the first inequality of conditions (29).

Application to sampled-data case

The result stated in Theorem 1, can be easily extended to the case of sampled measurements. Let the monotonically increasing sequence t k , k ∈ N with lim k→+∞ t k = +∞ which represents the sampling instants. The maximum sampling time interval is noted T e = max k∈N (t k+1 -t k ). Corollary 1. Let us consider the following observer:

ẋ = Ax + φ(x, u) -θ∆ -1 S -1 C T (C x(t k ) -y(t k )) ∀t ∈ [t k , t k+1 ) ŷ = C x(t) (30)
and suppose that the following conditions are fulfilled

θ > sup {1, 2k 1 } (31a) T e ≤ 1 2k 2 θ (31b)
then ( 30) is a global asymptotic observer for systems (1).

The proof of this corollary is obliviously derived from fact ∀t ∈ [t k , t k+1 ) we have t k = t -τ (t), with τ (t) = t -t k and τ (t) = 1. This properties allow us to use the result of Theorem 1.

EXAMPLES

Simulation with random varying delay

In order to illustrate the result presented above, consider the following system:

   ẋ1 (t) = c 1 x 2 (t) -l 1 x 1 (t) , ẋ2 (t) = c 2 sin(x 2 (t)) + c 3 cos(x 2 (t)) + c 4 u(t) , y(t) = x 1 (t -τ (t)) .
(32)

This system can be written in the form (1) with the following change of variable:

z 1 (t) = x 1 (t) , z 2 (t) = c 1 x 2 (t) ,
which leads to ż(t) = Az(t) + φ(z(t), u(t)) y(t) = z 1 (t -τ (t)) where

A = 0 1 0 0 φ(z(t), u(t)) = φ 1 (z(t)) φ 2 (z(t), u)) with φ 1 (z(t)) = -l 1 z 1 (t) φ 2 (z(t), u(t)) = c 1 c 2 sin(z 2 (t)/c 1 ) + c 1 c 3 cos (z 2 (t)/c 1 ) + c 1 c 4 u(t).
The function τ (t) is piecewise constant with values in the range [0, τ 1 ]. In the simulations below, the value of τ (t) is set randomly at regular time interval δ 1 = 0.1.

The values of the system parameters are set to c 1 = 1, c 2 = 0.02, c 3 = 0.02, c 4 = 8, l 1 = l 2 = 0.04, the initial conditions for the system are, ∀t ∈ [-τ 1 0], x 1 (t) = x 2 (t) = 50. The input function is u(t) = sin(0.35t) for t ≥ 0.

The solution of (8) for this system is

S = 1 -1 -1 2 ,
with λ min (S) = 0.382, λ max (S) = 2.618. One has β = 0.04, (18) leads to k 1 = 0.7755, and (24) leads to k 2 = 31.5377. The observer parameter is set to the smallest value allowed by (29a): θ = 1.55. The maximum value for the delay is then given by (29b): τ 1 = 10 -2 . The initial conditions for the observer are, ∀t ∈ [-τ 1 0], x1 (t) = x2 (t) = 0. As we can see, the simulations displayed in Figure 1 show clearly the asymptotic convergence of the observation error.

Simulation with sampled output

The simulations displayed in Figure 3 show also the exponential convergence of observer (30) in the case of a sampled output with sample period 0.01.

CONCLUSION

In this paper, a high gain observer with time varying delayed measurements has been presented. An explicit relation between the gain θ of the considered observer and the upper bound of the delay is given. This condition ensures asymptotic convergence of the observation error.

As we can see, this bound can be very small for high values of θ. This means that for relatively important delays, the considered observer can not guarantee an asymptotic convergence. To overcome this problem, we will present in a future work a cascade of high gain observers based on the delay bound computed here.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Evolution of the state variables and their estimate, random delay with τ 1 = 0.01.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Evolution of the state variables and their estimate in the case of a sampled output with sample period 0.01


Ahmed-Ali Tarek

Françoise Lamnabhi-Lagarrigue

High Gain Observer Design for Some Networked Control Systems

Keywords: High gain observer, networked control systems (NCSs), time-delay systems

New results on high gain observer design for networked control systems via an emulation-like approach are presented. By using a general framework and a Lyapunov approach, we derive some explicit conditions on the maximum allowable transmission interval that ensure an exponential convergence of the observation error for a large class of network protocols.

I. INTRODUCTION

The control and observation of systems called Networked control systems (NCSs) are currently attracting a lot of attention in the control community. In many applications, the interest for NCSs is motivated by many advantages they offer such as the ease of maintenance and installation, the greater flexibility and the low cost. For these reasons, many industrial control applications use a serial communication channel to connect sensors and controllers. In NCSs, the serial communication channel has many nodes (sensors and actuators) but the signals of these nodes cannot be transmitted at the same time. The rule that selects which node will use the network to transmit its data, is T. Ahmed-Ali is with GREYC UMR CNRS 6072, Caen Cedex 14050, France (e-mail: tarek.ahmed-ali@greyc.ensicaen.fr).

F. Lamnabhi-Lagarrigue is with CNRS, Laboratoire des Signaux et Systemes (INSIS-INS2I), European Embedded Control Institute (EECI) SUPELEC, Gifsur-Yvette 91192, France (e-mail: lamnabhi@lss.supelec.fr). called scheduling network protocol. In the present letter, only communication constraints on sensors are considered. It is well known that the stability of these systems is largely determined by the transmission protocol used and by the so-called maximum allowable transfer interval (MATI), i.e., the maximum allowable time between any two successive transmissions in the network. There exist several families of protocols in the literature, but in this study, only Uniformly Globally Exponentially Stable (UGES) protocols are considered. This class includes several protocols, such Round Robin (RR), Try-Once-Discard (TOD) and sampled data systems. For further details on NCSs, protocols and their properties, see the overview papers [START_REF] Walsh | Stability analysis of networked control systems[END_REF]- [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF].

The present note is focused on the implementation of a class of nonlinear observers on NCSs. Although the control of these systems has received considerable attention in the recent years, only few papers which deal with the observer design problem exist in the literature. For example, in deterministic case, the authors in [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] derived some conditions in order to maintain the observability of discrete-time linear systems under network communications. In [START_REF] Dačic | Observer design for wired linear networked control systems using matrix inequalities[END_REF], the authors proposed a method for the mutual design of protocols and observers for linear systems. The considered protocol gives the priority to the node where the error, between the measure and the estimate is the biggest, according to some weighted norms. Sufficient conditions for quadratic stability properties are then given using matrix inequalities. In [START_REF] Naghshtabrizi | Designing an observer-based controller for a network control system[END_REF], the authors consider the design of a linear observer for NCSs in the presence of time delay and sampling phenomenons without considering the constraints introduced by the protocols. They give sufficient conditions guaranteing asymptotic stability, by using a Lyapunov-Krasovskii approach. Recently, a small gain approach was used in [START_REF] Postoyan | Observers for classes of nonlinear networked control systems[END_REF] and [START_REF] Postoyan | A framework for the observer design for networked control[END_REF] for the observers design via an emulation-like approach for some classes of networked control systems. The results contained in [START_REF] Postoyan | Observers for classes of nonlinear networked control systems[END_REF] and [START_REF] Postoyan | A framework for the observer design for networked control[END_REF] state that if the observer has some robustness properties with respect to measurements errors then, for sufficiently small MATI, the stability in an appropriate sense of the observation error is guaranteed for several classes of network protocols. On the other hand, even without disturbances, the framework developed in [START_REF] Postoyan | A framework for the observer design for networked control[END_REF] does not allow asymptotic stability of the observation error when the network is in zero order hold (ZOH) fashion. In the present study, we focus on a class of nonlinear triangular systems without considering disturbances. More precisely, we provide a framework and use a Lyapunov approach to guarantee an exponential stability of the observation error and improve the bounds of MATI compared to small gain approaches. We specially show that by using the framework presented below and a suitable Lyapunov-Krasovskii functional, we can derive an exponential stability of the observation error when the network is in ZOH fashion. Note that the results presented in the present letter can be easily extended to several classes of observers such, for example, linear observers and the class of observers described in [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H performance analysis[END_REF]. The present note is organized as follows:

In Section II, we present the notations which will be used throughout the letter. The framework that we propose is formulated in Section III. In Section IV, we present our results on the emulation of high gain observers on a class of networks.

II. NOTATIONS AND PRELIMINARIES

First some mathematical notations are introduced. Let R = (01; 1), R+ =[0; 1) and N is the set of natural numbers. The notation N 3 denotes the set of strictly positive integers. The usual euclidian norm of any vector v will be noted by jvj and the identity matrix of size p is denoted by Ip . The matrix 0p is a square matrix of size p with all values equal to zero. The notations min (S) and max (S) denote respectively the minimum and maximum eigenvalues of the square matrix S . A continuous function 0:R+ ! R+ We consider the following class of nonlinear systems:

_ x = Ax + f(x) y = Cx = x 1 (1)
where x =( x 1 ;x 2 ; ...;x q ) T 2 R n with x k 2 R p and p:q

Throughout this letter, we assume that the following hypotheses are satisfied:

Hypothesis 1: The vector f(x) has the following form:

f(x)= f 1 (x 1
); ...;f s (x 1 ; ...;x s ); ...;f q (x)

T :

(4)

Hypothesis 2: The functions fs are globally Lipschitz, i.e: 90 > 0s u c ht h a t8(x1;x 2 ) 2 R p:s 2 R p:s jf s (x 1 ) 0 f s (x 2 )j 0 jx 1 0 x 2 j; (s =1 ; ...;q):

(5)

Using these hypotheses, the following observer has been proposed in [START_REF] Farza | Observer design for a class of mimo nonlinear systems[END_REF] for the class of systems (1) without considering network:

_ x = Ax + f(x) 0 1 01 S 01 C T (C x(t) 0 y) (6)
where is a positive constant satisfying 1. S is a symmetric positive definite matrix, solution of the algebraic Lyapunov equation

SA + A T S 0 C T C = 0S (7)
and 1 is a diagonal matrix which has the following form:

The exponential convergence of this observer has been derived by using the Lyapunov function V = x T S x, where x =1 x. The authors prove in [START_REF] Farza | Observer design for a class of mimo nonlinear systems[END_REF] that its time derivative satisfies _ V 0( 0 )V 0 x T C T C x where =2 p q max(S) min(S) 0 :

(9)

III. OBSERVERS FOR NETWORKED CONTROL SYSTEMS

In this section, we present a framework dedicated to implementation of a class of nonlinear observers on a class of networks. Let us consider the nonlinear systems [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to H performance analysis[END_REF] and suppose that there exists an observer described by the dynamics

_ x = f M (x) y = h M (x)
_ x = f o (x; (h M (x) 0 y)) (11)
observation error x =x 0 x which is represented by the system

_ x = f o (x; (h M (x) 0 y)) 0 f M (x) (12)
is globally asymptotically stable. Now, let us consider the case where all outputs of system (10) are connected to observer (11) via a serial communication channel. We suppose that the signals of these sensors cannot be transmitted to observer [START_REF] Farza | Observer design for a class of mimo nonlinear systems[END_REF] at the same time, then this problem can be modeled by the following framework:

_ x = f M (x) t 2 [t i01 ;t i ] x t + i = x(t i ) y = hM(x) _ x = fo(x; ŵ 0 w) t 2 [ti01;t i ] x t + i =x(t i ) _ w = g w (w; ŵ; x) t 2 [t i01 ;t i ] w t + i = ys(ti)+hy (i; ey(ti)) _ ŵ =ĝ ŵ (ŵ; w; x) t 2 [ti01 ;ti] ŵ t + i = h M (x(t i ))
_ y s = g y (y s ;x) t 2 [t i01 ;t i ) ys (ti )=y(ti ) i 2 N 3 ey = w 0 ys

(13)
The vector x is the continuous-time estimate of the system state

x. The monotonically increasing sequence tj;j 2 N represents the transmission instants. The vectors w and x (which are re-initialized at each instant tj) represent respectively most recently transmitted output values via network and the prediction of the observer output. The vector y s (which is also re-initialized at t j ) allows us the possibility to describe the sampling phenomena. More precisely, ys can be used to represent the output of the zero order hold (ZOH) device. This variable is specially useful when the network operates in ZOH fashion. In this case we replace conjointly the outputs of the system and the observer by their sampled signals before considering the constraints introduced by the network. As, we will see in the sequel, this framework allows us to derive an asymptotic stability of the observation error when the network operates in ZOH fashion. The functions g y , g w and ĝx represent prediction functions between two transmission instants. The error induced by the network is represented by the vector ey. The protocol hy is the algorithm by which the access to network of each node is determined. At each t j , the protocol h y selects which nodes k 2f1; ...;l g can transmit its data throughout the network. This algorithm is represented by the discrete time system: e y (t + j )=h y (j;e y (t j )) j 2 N.

Throughout this note, the variable =m a x ( t 0 t j ) represents the maximum allowable transfer interval (MATI) and MATI represents an upper bound of . To prevent zeno solution we suppose that the MATI satisfies >t j+1 0 tj >, where is an arbitrary positive constant.

Note that the sampled-data systems are a special case of NCSs since all sensor and control signals are transmitted at each transmission instant.

In this case, the protocol hy is equal to zero, and the maximum allowable time between any two successive transmissions is called maximum allowable sampling period MASP. Compared to [START_REF] Arcak | A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation[END_REF] where a framework based on discrete-time approximate models is developed for the observer design for sampled-data systems, we can say that the framework ( 13) is an alternative approach to [START_REF] Arcak | A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation[END_REF].

Hypothesis 3: We suppose that the protocol hy is UGES. This means that there exists a positive function, W : N 2 R n ! R + and some positive constants 2 [0; 1), a1, a2 such that, for all j 2 N and for all e y 2 R n a 1 je y jW (j; e y ) a 2 je y j

W (j +1 ;h y (j; e y )) W (j; e y ):

(15)

Hypothesis 4: We also assume that there exists a positive constant M 0 so that for all j 2 N and for all e y 2 R n @W(j; e y ) @e y M 0 :

(

) 16
This condition is satisfied by many protocols such Round Robin (RR) and Try-Once-Discard (TOD).

IV. HIGH GAIN OBSERVER FOR NCSS

In this section we focus on the implementation of high gain observer (6) over a network with UGES protocols. We present two examples: The first one is an implementation in ZOH fashion, whereas the second one, is based on the introduction of an output predictor between two transmission instants. In both cases we prove exponential convergence of the high gain observer for sufficiently small MATI.

A. High Gain Observer in ZOH Fashion

When the network operates in ZOH fashion, we will consider that all prediction functions in ( 13) are equal to zero. This means that the corrector term ŵ 0 w is held constant between two transmission instants.

Following framework [START_REF] Raff | Observer with sample-and-hold updating for lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF], we propose this observer:

_ x = Ax + f(x)
As we can see, system (18) is disturbed by the term C x(t + i01 ) 0Cx(t).

The Leibniz-Newton formula Provides

C x t + i01 0 C x(t)=0C t t0 (t) _ x(s1)ds1 (19)
with 1(t)=t 0 t + i01 for all t 2 (ti01 ;ti]. Note that we have 1(t) 2

[0;] and _ 1 =1 . This means that system (18) can be viewed as a time delay system with bounded time varying delay. This allows us to extend the approach developed in [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF], [START_REF] Raff | Observer with sample-and-hold updating for lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF] and [START_REF] Ahmed-Ali | Cascade high gain observers for nonlinear systems with delayed output[END_REF] to the problem considered in the present note. Theorem 1: Let us consider system (1) and suppose that hypotheses 1, 2, 3, 4 hold. Then, there exists a positive constant 0, such that for all > 0 system (17) is a global exponential observer for system

(1) if 2 [; MATI) whereMATI = min(min(S)=6supf[jA 0 S 01 C T Cj+ p q 0 ]; min (S) max (S 01 )g

2 ; 1 0 ).

Proof: Consider the change of coordinates

x =1 x, and from the fact that 1A1 01 = A and C T C1 01 = C T C, then (18) will be _ x = (A 0 S 01 C T C) x +1( f(x + x) 0 f(x))

+S 01 C T e y + C

In order to prove this theorem, let us consider the following candidate Lyapunov-Krasovskii functional:

U 1 = x T S x+ t t0 t s _ x(s 1 ) 2 ds 1 ds+ 1 1 (t)W 2
(j; e y ) (21)

where 1 > 0, j 2 N and 1 (t) is a bounded, positive function and decreasing between (t + i01 ;ti], with appropriate fixed values at instants t + i01 and t i . The functional U 1 has been used in [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF], [START_REF] Raff | Observer with sample-and-hold updating for lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF] and [START_REF] Ahmed-Ali | Cascade high gain observers for nonlinear systems with delayed output[END_REF] for observer design with sampled and delayed measurements without network constraints communications by considering W 2 (j; ey)=0.We will show below that this approach can be used here. More precisely, to prove exponential stability, it is sufficient to find MATI and 0 so that for > 0 the following inequalities hold:

_ U 1 + " 1 U 1 0 t 2 (t i01 ;t i ] U 1 t + i <U 1 (t i ) 8i 2 N 3 (22)
where "1 > 0. We can easily see that the second inequality of ( 22) can be satisfied by choosing 1 (t + i01 )=1and 1 (t + i01 + MATI )= 8i 2 N 3 where 2 (0; 1).

Let us compute the derivative of U1 for t 2 (ti01 ;ti], then we have (26)

_ U1 =2 x T S _ x + MATIj _
By choosing _ 1 = 0" 1 1 0 1 8t 2 (t i01 ;t i ]and 1 = a 2 1 2 +2MATI 2 k2 (27) we can say that the inequality _ U 1 + " 1 U 1 0 is ensured, if MATI satisfies both ( 27) and the following inequalities:

0 0 1 0 2MATI 2 k2 > 0 1 0 2 2 k 2 2
MATI 0 1 MATI 0 2 MATI >0 : (28) From this, and by integrating ( 27) on (t i01 ;t i01 + MATI ] and setting 1 ! 0, we derive the following conditions: MATI = min 1 4k ; 1 0 >s u p f2; 1g (29) using the fact that j xj(1= min(S)) U1(t) and from (22) then we deduce that under conditions (29) there exist c 1 > 0 and c 2 > 0 such that j xjc 1 exp 0c (t0t ) 8t t 0 . From this we can say that j xj converges exponentially towards zero.

B. High Gain Observer With an Output Predictor

The idea of using a predictor of the output was introduced in [START_REF] Karafyllis | From continuous-time design to sampled-data design of nonlinear observers[END_REF] for the design of sampled-data observers. More precisely, this predictor is used by the observer between two transmission instants and its model is a copy of the output system model. This gives this differential equation _ w =(@hM(x)=@x)fM(x) where the initial condition w(t0) is arbi- trary and at each transmission instant ti the value of w is reseated like this: w(t + i )=y(t i ). This idea has been to systems in and [START_REF] Postoyan | A framework for the observer design for networked control[END_REF] and some bounds of MATI have been derived using small gain approach. The aim of this section is to improve the bound of MATI for high gain observer when an output predictor is used. Following the framework (13), we can write the following observer:

_ x = Ax + f (x) 0 1 01 S 01 C T (ŵ 0 w) _ w =x 2 + f1(x 1 ) t 2 [ti01;ti] w t + i = ys(ti)+hy (i; ey(ti)) y s = y ŵ = C x(t) (30)
Let us consider the observation error x =x 0 x, then we have Proof: As in the above section, using the same change of coordinates then (31) will be _ x = (A 0 S 01 C T C) x +1(f(x + x) 0 f (x)) +S 01 C T ey t 2 [ti01;ti] _ ey = x 2 + f1(x 1 + x 1 ) 0 f1(x 1 ) t 2 [ti01;ti] e y t + i = h y (i; e y (t i ))

_ x =(A 0 1 01 S 01 C T C)x + f (x + x) 0 f (x) +1 01 S 01
(32)

Let us consider the following candidate Lyapunov function as in [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF]:

U (t)= x T (t)S x(t)+(t)W 2 (j; ey(t)) (33)
where j 2 N, and are two positive constants which will be computed below. The positive function (t) is bounded and decreasing on the transmission interval (t i01 ;t i ] i 2 N 3 . This function satisfies the following conditions: t + i01 = 01 and t + i01 + MATI = 8i 2 N 3 where 2 (0; 1), to guarantee that U (t + i ) U (ti). Now, let us com- pute the time derivative of U between (ti01;ti], then we have

_ U =2 x T S _
x + _ W 2 (i 0 1;ey)+2W(i 0 1;ey) _ W (i 0 1;ey)

(34) after some computations, we deduce that _ U 0 ( 0 )min(S)j xj 2 + jeyj 2 + _ W 2 (i 0 1;e y )+2W(i 0 1;e y ) @W(i 0 1;ey) @ey j _ e y j:

Let us remark that j _ e y jj x 2 j + f 1 (x 1 ) 0 f 1 (x 1 ) ( + 0 )j xj (35) using the property ( 16), we derive the following inequalities j@W (i 0 1;ey)=@eyjj_ eyjM0( + 0)j xj and _ U 0 ( 0 ) min (S)j xj 2 + je y j 2 + _ W 2 (i 0 1;e y )+2W(i 0 1;e y )M 0 ( + 0 )j xj:

Using the fact that the protocol h y is UGES, and by adding and subtracting the term 0 W 2 (i 0 1;e y ), where 0 > 0, thus we have _ U 0 ( 0 )min(S)j xj where 0 > 0. Note that the term 0W 2 (i 0

Using the fact that j xj(1= min(S)) U (t) and from (41), then we deduce that there exist 1 > 0 and 2 > 0 such that j xj 1exp 0 (t0t ) 8t t0. This means that

x converges exponentially towards zero, and the value of MATI is then derived by integrating (36) on (t i01 ;t i01 + MATI ], with 0 ! 0. This leads to MATI = lim !0( 42) and ( 47), then we have R =((=2) 0 2atan())=(1 0 ). Since 10 (=2)02atan() for all 2 [0; 1), then we deduce that R 1. This means that the bound derived from Lyapunov approach is larger than the one derived from small gain approach. This result means that we have also enlarged the interval of admissible values of the gain for any fixed MATI and therefore we have also improved the speediness of convergence. Note that in all formulas of MATI , it is clear that MATI depends inversely on . This obviously leads to small values of MATI for high values of the gain .

V. S IMULATIONS

In this part, we present some simulations on the flexible joint robotic arm used in [START_REF] Raff | Observer with sample-and-hold updating for lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF] _ Without unknown uncertainties (t), system (48) satisfies hypotheses 1, 2, 3, 4 with p = q =2and x =( x 1 ;x 2 ) where x 1 =( x 1 ;x 2 ) and x 2 =( x 3 ;x 4 ).

Constrained by the size of the present note, we focused on the simulations of observer (30) with Round-Robin (RR) protocols, that consists in granting access to each compenent of the vector ouputs after p transmission instants where p is the number of ouputs [START_REF] Nešic | Input-output stability properties of networked control systems[END_REF]. In the simulations presented below, we also suppose that unknown uncertainties (t) are represented by random function of Matlab with variance = 0:8. The following simulations are performed with u =0:1sin(t), =40and MATI =0 :01 (see Figs. 1 and2). As we can see, even with some uncertainties, the results remain good with =4 0and MATI =0 :01. This is not a general conclusion. Indeed the uncertainties can cause robustness problems especially for high values of .

VI. CONCLUSION

In this letter, some results on the design of high gain observer for networked systems with UGES protocols were presented. These results can be easily extended to other observers. In future work, we will present some results concerning other classes of protocols.

  I p 0 p ... 0 p ):

  1;e y):(23) Using (20), and Hölder inequality we can deduce that us set V = x T S x, after some computations we deduce that for t 2 (ti01;ti ],w eha v e _ U1 + "1U1 0V + V + "1V +2 jIj 2 + W

Theorem 2 :

 2 Let us consider system (1) and suppose that hypotheses 1, 2, 3, 4 hold. Then for all >maxf; 1g, system (30) is a global exponential observer for system (1) if 2 [; MATI ) where MATI =a1((=2)02atan()) min(S)( 0 )=M0( +0) p .

  e y )M 0 ( + 0 )j xj + 0 W 2 (i 0 1;e y ) If we choose _ = 0 ( 2 +1) 8t 2 (t i01 ;t i ] (36) thus, we will have _ U 0 ( 0 ) min (S)j xj 2 + M

1 )

 1 Comparison With Small Gain Approach: Let us consider the function V = x T S x, then after simple computations, we can easily derive the following inequality: where V (t 0 )= min (S)exp ((0(0))=2) (t0t 0 ) is a function of class KL.Now, let us define, ỹ = M0( + 0) x, then we derive the gain 0 from W to ỹ

Fig. 1 .

 1 Fig. 1. Evolution of the first and second states for

  x 1 = x 3 + (t) _ x2 = x4 + (t) _ x3 = 048:6x1 +48:6x2 0 1:25x3 +21:6u + (t) _ x 4 =19:5x 1 0 19:5x 2 0 3:3sin(x 3 )+(t)

Fig. 2 .

 2 Fig. 2. Evolution of the third and fourth states for

  is said to be of class K, if it is zero at zero, strictly increasing. It is of class K 1 , if it is of class K and unbounded. A continuous function ' : R + 2 R + ! R + is said to be of class 0KL,if '(:; t) is of class K for each t 0, and '(s; :) is non increasing and satisfies lim t!1 '(s; t)=0 . The notation e y (a + ) represents lim t!a;t>a e y (t) when it exists. The term W [t ;t) represents ess: sup s2[t ;t) jW(s)j. Throughout this letter the vectors x 2 R n and y 2 R p represent respectively the state vector and the outputs of the considered systems.

  0 1 01 S 01 C T

	_ ŵ =0

(ŵ 0 w) _ w =0 t 2 [t i01 ;t i ] w t + i = ys(ti)+hy (i; ey(ti)) t 2 [ti01 ;ti] ŵ t + i = C x(t i ) _ y s =0 t 2 [t i01 ;t i ) ys (ti )=y(ti ) ey = w 0 ys (17)

if we consider the observation error x =x 0 x, then we have

_ x =( A 0 1 01 S 01 C T C)x + f(x + x) 0 f(x) 01 01 S 01 C T C x t + i01 0 C x(t

) 0 ey t 2 [ti01 ;ti] _ ey =0 t 2 [ti01 ;ti] x t + i =x(t i ) e y + = h y (i; e y (t i ))

  C T ey t 2 [ti01;ti] _

ey =x 2 + f1(x 1 + x 1 ) 0 f1(x 1 ) t 2 [ti01;ti] x t + i =x(t i ) e y t + i = h y (i; e y (t i ))

  1;ey) is introduced to guarantee the exponential convergence. Choosing 2 (t i01 ;t i ] this means that for all t 2 (t i01 ;t i ]

	= =	M (+ ) (0) (S) + a + 0	(0)	(S)	(37)
	thus we can say that there exists a positive constant , so that _ U (t) 0U(t) U (t) U t + i01 exp 0 t0t (38)
	and
	U (t i ) U t + i01 exp 0 t 0t	(39)
	on the other hand, we have
	U t + i = V	x t + i	+ 01 W 2 ey t + i
	using the fact that the protocol is UGES, and	x(t + i )= x(ti), then we
	can write
	U t +
		i01 exp 0 t0t		8t 2 (ti01;ti]
	U t + i U t + i01 exp 0 t 0t

t i V ( x(ti)) + W 2 (ey(ti)) U (ti) i 2 N 3 (40)

thus, we derive

U (t) U t + i 2 N 3 :


Laurent Burlion

Elodie Duraffourg

Tarek Ahmed-Ali

Franc ¸oise Lamnabhi-Lagarrigue

Global asymptotic stabilization for some nonlinear models of flexible aerospace vehicles

New strategies are proposed to design GAS controllers for two significant classes of nonlinear flexible systems. Inspired by a non affine control technique, we have designed some novel nonlinear control laws which use the full state in order to enhance the damping of the flexible modes: this new design is all the more interesting than the classes of considered systems belong to non triangular nonlinear systems. Finally, the effectiveness of our approach is illustrated on simplified models of a flexible launcher and spacecraft.

I. INTRODUCTION

In the last years the stabilization problem of flexible nonlinear systems has received a great attention. This interest is motivated by the fact that flexible modes can generate a very bad transient even if the system is a minimumphase one. Several results concerning this topic exist in the literature : a typical design is to combine a nominal nonlinear controller with a vibration absorber controller. This 'vibration absorber' is often based on the well known method of Input Shaping [START_REF] Sorensen | A controller enabling precise positioning and sway reduction in bridge and gantry cranes[END_REF], [START_REF] Aribowo | Input shaping control to suppress sloshing on liquid container transfer using multi-joint robot arm[END_REF], [START_REF] Pridgen | Slosh suppression by robust input shaping[END_REF], [START_REF] Hu | Sliding mode and shaped input vibration control of flexible systems[END_REF] which has the major advantage to require merely relatively good approximation of modeling coefficients instead of direct measurements of the flexible modes : however, this useful method is by nature a linear method which can be used when the initial conditions are known and when the interaction between the flexible state and rigid ones is linear (although a few results exist in the nonlinear case e.g [START_REF] Gorinevsky | Nonlinear Input Shaping Control of Flexible Spacecraft Reorientation Maneuver[END_REF]). Another approach is to combine a nonlinear controller with a linear controller which uses the flexible modes and (only) locally damps them out [START_REF] Malekzadeh | A Robust Nonlinear Control Approach for Tip Position Tracking of Flexible Spacecraft[END_REF], [START_REF] Bartolini | Second-order sliding-mode control of container cranest[END_REF]. Open loop or robust feedforward methods can also be used to design a reference trajectory for the rigid state such that the oscillations of the flexible modes remain relatively low [START_REF] Graichen | Feedforward control design for finite time transition problems of non-linear MIMO systems under input constraints[END_REF], [START_REF] Gorinevsky | Nonlinear Input Shaping Control of Flexible Spacecraft Reorientation Maneuver[END_REF], [START_REF] Shkolnikov | Tracking in a class of nonminimum-phase systems with nonlinear internal dynamics via sliding mode control using method of system center[END_REF] but it seems tricky to apply these methods on a large nonlinear operating domain when the systems parameters are uncertain and when there is a lot of flexible modes. Recently, new nonlinear Lyapunov based approaches have been proposed for the stabilization problem of flexible aerospace systems when there is a large number of flexible modes and a strong nonlinear coupling between flexible and rigid modes [START_REF] Fiorentini | Nonlinear Robust Adaptive Control of Flexible Air-Breathing Hypersonic Vehicles[END_REF], [START_REF] Duraffourg | Nonlinear control of the longitudinal rotational dynamics of a flexible aircraft[END_REF], [START_REF] Reyhanoglu | Point-to-Point Liquid Container Transfer via a PPR Robot with Sloshing Suppression[END_REF], [START_REF] Reyhanoglu | Nonlinear dynamics and control of space vehicles with multiple fuel slosh modes[END_REF] : technically speaking, although these systems are minimum phase, they belong to the class of underactuated non triangular nonlinear systems [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF], which renders tedious the twofold objective of stabilizing the system and damping out the oscillations of the flexible modes. In this paper, we propose to enlarge these results by either adding more damping to the flexible dynamics [START_REF] Fiorentini | Nonlinear Robust Adaptive Control of Flexible Air-Breathing Hypersonic Vehicles[END_REF], [START_REF] Reyhanoglu | Point-to-Point Liquid Container Transfer via a PPR Robot with Sloshing Suppression[END_REF], [START_REF] Reyhanoglu | Nonlinear dynamics and control of space vehicles with multiple fuel slosh modes[END_REF] either studying larger classes of systems [START_REF] Duraffourg | Nonlinear control of the longitudinal rotational dynamics of a flexible aircraft[END_REF]. The control schemes developed in this paper are based on the result of [START_REF] Lin | Global asymptotic stabilization of general nonlinear systems with stable free dynamics via passivity and bounded feedback[END_REF] which is a systematic design approach to add Lyapunov decrease to non affine nonlinear systems whose free dynamics are stable. Moreover, the major difference of our approach compared to classical methods is the fact that the controllers are first designed to stabilize the flexible modes (i.e the zero dynamics) and then extended to stabilize the full state. The present paper is organized as follows : in section II, the class of nonlinear systems studied here are presented and motivated by some examples of flexible nonlinear systems. Then, we briefly recall the result of [START_REF] Lin | Global asymptotic stabilization of general nonlinear systems with stable free dynamics via passivity and bounded feedback[END_REF] used to determine the new control laws proposed in this paper : we show how one can both globally stabilize these classes of systems and add damping to the flexible dynamics. The two following sections illustrate our results on a flexible launch vehicle and a spacecraft model. We finally give our conclusions and some future research directions.

A. Notations

Let m ∈ N . Throughout the paper, the following notations will be used:

• . is the Euclidean norm in R m • Given a smooth function V : R 2m -→ R and a vector field g : R 2m -→ R 2m , L g V := ∂V ∂ x g denotes the Liederivative of V with respect to g • a continuous function σ : R >0 -→ R >0 belongs to class K it is monotonic increasing and σ (0) = 0 • a continuous function σ of class K belongs to class K ∞ if it is unbounded • given a symmetric matrix P, its minimal (resp. maximal) eigenvalue is denoted by λ min (P) (resp. λ max (P))

II. PROBLEM FORMULATION In this paper, we consider the stabilization problem of the following classes of minimum-phase nonlinear systems :

A. Class I        ẋ1 = x 2 ẋ2 = u η1 = η 2 η2 = A η η + B η u + P 1 η (x 1 ) + P 2 η (x 1 )x 2 (1)
This class of systems satisfies the following properties :

• x 1 , x 2 , u ∈ R and η := [η 1 ; η 2 ] ∈ R 2m , m ∈ N • 0 I m A η is a
       ẋ1 = x 2 ẋ2 = u η1 = η 2 η2 = A η η + B η u + f η (x 1 ) + g η (x 1 , η 1 )x 2 2 (2)
which verifies the following properties:

• x 1 , x 2 , u ∈ R and η := [η 1 ; η 2 ] ∈ R 2m , m ∈ N • 0 I m A η
is a real Hurwitz matrix and its eigenvalues are all complex conjugates • η 1 is a vector whose coordinates are called the flexible modes

• the dynamics of x = [x 1 ; x 2 ] (resp. [η 1 ; η 2 ]) is called the rigid (resp. flexible) dynamics • f η (resp.g η ) is a smooth and globally k f -(resp.
k g )Lipschitz function and all the e T i P η , i ∈ {1, m} are polynomials of degrees ≤ l , where l ∈ N It has to be noticed that there exist several flexible aerospace vehicles which are described by the above two classes of nonlinear systems. We can cite for instance :

• the rotational dynamics of a flexible hypersonic vehicle belongs to class I (see for more details the modeling in [START_REF] Bolender | Nonlinear Longitudinal Dynamical Model of an Air-Breathing Hypersonic Vehicle[END_REF] and some changes of coordinates leading to the class I form in [START_REF] Duraffourg | Nonlinear control of the longitudinal rotational dynamics of a flexible aircraft[END_REF]). • the longitudinal dynamics of a flexible launch vehicle belongs to class I (see [START_REF] Duraffourg | Longitudinal Modeling and Preliminary Control of a Non-linear Flexible Launch Vehicle[END_REF]) • the model of a spacecraft with a flexible attachment belongs to class II (see [START_REF] Reyhanoglu | Slewing maneuver of a flexible spacecraft using finite time control[END_REF]) • the rotational dynamics of space vehicles with propellant sloshing mode belongs to class II (see [START_REF] Reyhanoglu | Point-to-Point Liquid Container Transfer via a PPR Robot with Sloshing Suppression[END_REF], [START_REF] Reyhanoglu | Nonlinear dynamics and control of space vehicles with multiple fuel slosh modes[END_REF] when a x = a z = 0) It is well-known that if the zero dynamics of the above systems are supposed asymptotically stable and by considering the state x 1 as output, then there exist several classical methods(e.g Backstepping [START_REF] Sepulchre | Constructive nonlinear control[END_REF] and Input/Output Linearization [START_REF] Isidori | Nonlinear Control Systems[END_REF]), which solve the global asymptotic stabilization problem of these classes of systems. The most important drawback of these methods is that they do not integrate the zero dynamics in the design of the proposed controllers. This fact generates bad performances of the zero dynamics transients, which means that the flexible modes of several aerospace vehicles will exhibit some undesired oscillations. In the present work, we shall design some nonlinear controllers which simultaneously ensure a global asymptotic stability in closed loop and also improve the transient dynamics of the flexible modes. The major difference of the proposed schemes compared to the classical methods is in the fact that the controller is at first designed to stabilize the zero dynamics. This fact induces an improvement of the flexible modes behavior quality.

III. MAIN RESULTS

Before presenting our main results, we need to give the following important technical result which is largely derived from the work of [START_REF] Lin | Global asymptotic stabilization of general nonlinear systems with stable free dynamics via passivity and bounded feedback[END_REF] : Lemma 1[Wei Lin, [START_REF] Lin | Global asymptotic stabilization of general nonlinear systems with stable free dynamics via passivity and bounded feedback[END_REF]] : Consider a single-input non affine nonlinear system of the following form :

ż = f 0 (z) + g 0 (z)u + l ∑ i=2 g i (z)u i (3)
where u ∈ R, g i : R 2m -→ R 2m , 2 ≤ i ≤ l, are smooth functions. Suppose the unforced dynamics f 0 is such that there exists a function V : R 2m -→ R, α 0 of class K , and

α 1 , α 2 of class K ∞ such that ∀z α 1 ( z ) ≤ V (z) ≤ α 2 ( z ) (4)
V|u=0 = ∂V ∂ z f 0 (z) ≤ -α 0 (z) (5)
then the unforced system is GAS and the following control law

u(z) = ϕ(z) := -β 1 + ∂V /∂ z 2 ρ(z) 2 L g 0 V (z) 1 + (L g 0 V (z)) 2 (6)
where

ρ(z) ≥ l ∑ i=2 1 + g i (z) 2 (7)
and β ∈]0, 1[ still renders system (3) GAS such that :

V ≤ V|u=0 - (1 -β ) 1 + ∂V /∂ z 2 ρ(z) 2 (L g 0 V (z)) 2 1 + (L g 0 V (z)) 2 ≤ V|u=0 (8)
In this case, we say that u adds damping to system (3) since it achieves more Lyapunov decrease.

Now, we are able to give the following propositions:

A. Stabilization of Class I Proposition 1: let us consider the class of systems (1), then there exists a nonlinear feedback control law which simultaneously ensures the global asymptotic stability of the origin of the closed loop system and improves the transient of its corresponding zero dynamics.

Proof:

In order to prove the above proposition, let us consider the following change of coordinates:

η1 = η 1 -B η x 1 η2 = η 2 -B η x 2 (9)
then, we obtain :

       ẋ1 = x 2 ẋ2 = u η1 = η2 η2 = A η η + P 1 η (x 1 ) + P 2 η (x 1 )x 2 (10)
where :

P 1 η (x 1 ) := P 1 η (x 1 ) + A η B η 0 x 1 P 2 η (x 1 ) := P 2 η (x 1 ) + A η 0 B η
From this, we can easily see that system (10) is a nontriangular nonlinear system non affine in x 1 but affine in x 2 according to the classification of underactuated nonlinear systems proposed in [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF]. In order to remove x 2 in the flexible dynamics, let us use the following change of coordinates:

     z 1 = η1 z 2 = η2 -1 0 P 2 η (sx 1 )ds x 1 := η2 -P 3 η (x 1 )x 1 (11)
Indeed, one can check that :

d dt P 3 η (x 1 )x 1 = P 2 η (x 1 )x 2
and that P 3 η (x 1 )x 1 is a polynomial of degree ≤ d 2 , then we obtain

       ẋ1 = x 2 ẋ2 = u ż1 = z 2 + P 3 η (x 1 )x 1 ż2 = A η z + P 4 η (x 1 ) (12)
where

P 4 η (x 1 ) = A η 0 P 3 η (x 1 )x 1 + P 1 η (x 1 ) is of degree ≤ max{d 1 , d 2 }. Let us set z = [z 1 ; z 2 ].
As we can see from [START_REF] Reyhanoglu | Point-to-Point Liquid Container Transfer via a PPR Robot with Sloshing Suppression[END_REF], the zdynamics of the above system can now be viewed as a non affine triangular nonlinear system with respect to the input x 1 .

Since 0 I m A η is Hurwitz, then there exists

P z = P T z , Q z > 0 such that : 0 I m A η T P z + P z 0 I m A η ≤ -Q z . (13)
We thus propose to use V z = z T P z z and we compute ϕ according to [START_REF] Malekzadeh | A Robust Nonlinear Control Approach for Tip Position Tracking of Flexible Spacecraft[END_REF] where :

V := V z g 0 (z) = g 0 = P 3 η (0) d dx 1 P 4 η (0) g i (z) ∀i∈{2,l} = g i =   (i -1)! d i-1 dx i-1 1 P 3 η (0) i! d i dx i 1 P 4 η (0)   (14)
We choose

δ 1 = x 1 -ϕ(z) (15)
Using (8) and the fact that :

∀i ∈ N , x i 1 = [ϕ + δ 1 ] i = ϕ i + i-1 ∑ k=0 C k i ϕ k δ i-k 1 then we have V ≤ -z T Q z z - (1 -β ) 1 + ∂V /∂ z 2 ρ(z) 2 (L g 0 V (z)) 2 1 + (L g 0 V (z)) 2 +   (Lg 0 V ) + ∑ 2≤i≤l 0≤k≤i-1 C k i ϕ k (L g i V )δ i-k-1 1    δ 1
Now, in order to design our controller, we shall use the classical Backstepping design [START_REF] Sepulchre | Constructive nonlinear control[END_REF] and consider the following Lyapunov function:

W 1 = V (z) + 1 2 δ 2 1 + 1 2 δ 2 2 ( 16
)
where

δ 2 := x 2 -x2 and x2 = -k 1 δ 1 + φ -   (Lg 0 V ) + ∑ 2≤i≤l 0≤k≤i-1 C k i ϕ k (L g i V )δ i-k-1 1    (17)
where k 1 > 0. If we choose the following control law :

u = -δ 1 -k 2 δ 2 + d dt ( x2 ) (18)
where k 2 > 0, then we obtain

Ẇ1 = -z T Q z z - (1 -β ) 1 + ∂V /∂ z 2 ρ(z) 2 (L g 0 V (z)) 2 1 + (L g 0 V (z)) 2 -k 1 δ 2 1 -k 2 δ 2 2 (19)
which means the origin is GAS. This ends the proof.

B. Stabilization of Class II

Proposition 2: let us consider the class of systems (2) and define a bounded C 1 function χ and a positive constant

M χ > 0 such that ∀x 1 , |χ(x 1 )| ≤ M χ min(|x 1 |, 1) x 1 χ(x 1 ) ≤ -M χ x 1 ( 20
)
then there exists a nonlinear feedback control law which simultaneously ensures the global asymptotic stability of the origin of the closed loop system and improves the transient of the corresponding zero dynamics.

Proof:

Let us give step by step the control law design: 1) Firstly, as for the above proposition, let us introduce a change of coordinates to eliminate u from the flexible dynamics:

η1 = η 1 -B η x 1 η2 = η 2 -B η x 2 (21)
in order to obtain a non linear system of the following form :

       ẋ1 = x 2 ẋ2 = u η1 = η2 η2 = A η η + fη (x 1 ) + ḡ2 x 2 + ḡη (x 1 , η1 )x 2 2 ( 22
)
where ḡ2 :

= A η 0 B η , fη (x 1 ) = f η (x 1 ) + A η B η 0 and ḡη (x 1 , η1 ) = g(x 1 , η 1 ).
It is worth noting that such a system is a nontriangular nonlinear system non affine in x 1 and x 2 according to the classification of underactuated nonlinear systems proposed in [START_REF] Olfati-Saber | Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles[END_REF]. 2) Let us note z = [x 1 ; η1 ; η2 ] and let us use the fact that x 2 = χ(x 1 ) + (x 2 -χ(x 1 )). Then, let us write the dynamics of z : 23) where:

ż = f 0 (z) + g 0 (z)(x 2 -χ(x 1 )) + g 2 (z)(x 2 -χ(x 1 )) 2 (
f 0 (z) =    χ(x 1 ) η2 A η η + f (x 1 ) + ḡ2 χ + ḡη (x 1 , η1 )χ 2 g 0 (z) =    1 0 ḡ2 + 2 ḡη (x 1 , η1 )χ(x 1 ) g 2 (z) =    0 0 ḡη (x 1 , η1 )
3) then, given P z = P T z , Q z > 0 satisfying (13), we propose:

V = log 1 + ηT P z η + c 1 2 x 2 1 := log(1 +V η ) + c 1 2 x 2 1 ( 24
)
where c 1 , c 2 > 0 are given below. From this we obtain :

V = -ηT Q z η 1 +V η + c 1 x 1 χ(x 1 ) +2 ηT P z 1 +V η f (x 1 ) + ḡ2 χ(x 1 ) + ḡη (x 1 , η1 )χ 2 (x 1 ) +(L g 0 (z) V )(x 2 -χ(x 1 )) + (L g 2 (z) V )(x 2 -χ(x 1 )) 2 (25) 4) Using the fact that 2 ηT P z 1+V η = 2 ηT P 1/2 z 1+(P 1/2 z η) T (P 1/2 z η) P 1/2 z
and the assumptions on f , ḡ, and χ, then we have :

2 ηT P z 1 +V η f (x 1 ) = 2 ηT P 1/2 z 1 +V η P 1/2 z f (x 1 ) ≤ k f λ max P 1/2 z x 1 (26) 2 ηT P z 1 +V η ḡ2 χ(x 1 ) ≤ M χ g 2 λ max P 1/2 z x 1 (27) 2 ηT P z 1 +V η ḡη χ 2 (x 1 ) ≤ M 2 χ λ max P 1/2 z x 1 +2k g M χ x 1 (28)
So, let us choose c 1 > 0 such that :

c 1 = c1 + (k f + M χ g 2 + k g M 2 χ )λ max P 1/2 z + 2k g M χ M χ (29) with c1 > 0, then we obtain, V ≤ -ηT Q z η 1 +V η -c1 x 1 + (L g 0 (z) V )(x 2 -χ(x 1 )) +(L g 2 (z) V )(x 2 -χ(x 1 )) 2 (30) 5) then we use x 2 -χ(x 1 ) = (x 2 -χ(x 1 ) -ϕ(z)) + ϕ(z)
where ϕ is given by [START_REF] Malekzadeh | A Robust Nonlinear Control Approach for Tip Position Tracking of Flexible Spacecraft[END_REF]. After a few computations, we obtain :

V ≤ -ηT Q z η 1 +V η -c1 x 1 - (1 -β ) 1 + ∂V /∂ z 2 ρ(z) 2 (L g 0 V (z)) 2 1 + (L g 0 V (z)) 2 + (L g 0 V ) + 2ϕ(z)L g 2 V + (L g 2 V )δ 2 δ 2 (31)
where

δ 2 := x 2 -χ(x 1 ) -ϕ(z) (32) 6
) finally, if we use the following Lyapunov candidate function :

W 2 = V + 1 2 δ 2 2 ( 33
)
and by choosing the following control law :

u = -k 2 (x 2 -χ(x 1 )) -(L g 0 V ) -2ϕ(z)L g 2 V -(L g 2 V )δ 2 + d dt (χ(x 1 ) + ϕ(z)) (34)
where k 2 > 0, we derive that :

V ≤ -ηT Q z η 1 +V η -c1 x 1 -k 2 δ 2 2 - (1 -β ) 1 + ∂V /∂ z 2 ρ(z) 2 (L g 0 V (z)) 2 1 + (L g 0 V (z)) 2 (35)
from this we conclude that the origin of system (2) is GAS.

IV. APPLICATION TO A NONLINEAR MODEL OF A FLEXIBLE LAUNCH VEHICLE

A. Mathematical Model

From the full Mathematical model of a flexible launch vehicle obtained by using the Lagrange equations, we extract the rotational 'fast' dynamics (for more details, see ( [START_REF] Duraffourg | Longitudinal Modeling and Preliminary Control of a Non-linear Flexible Launch Vehicle[END_REF])):

     ψ = q q = - l aero I L L(ψ) + T I L (L T r -h)η 1 + T L T I L β η1 = -ω 2 η 1 -2ξ ω η1 + hrT η 1 + hT β (36)
where ψ, η 1 belong to R (We only consider the first flexible mode which is known to be the only one to be relevant to consider in the control design) and β ∈ R is the control. where the lift is a non-linear function of the attitude, given by:

L(ψ) = qS C 1 L ψ +C 2 L ψ 2 = L(ψ)ψ B.

Controller design

First we use the notation :

u := - l aero I L L(ψ) + T I L (L T r -h)η 1 + T L T I L β
By noting x 1 = ψ, x 2 = q, η1 = η 2 , we obtain a model which belongs to the Class I :

       ẋ1 = x 2 ẋ2 = u η1 = η 2 η2 = -ω2 η 1 -2ξ ωη 2 + hl aero L T L(x 1 ) + hI L L T u (37) with ω2 := ω 2 + hT L T (L T r -h) -hrT .
Applying the design method described in the proof of proposition 1, then we obtain : the following dynamics :

               ẋ1 = x 2 ẋ2 = u ż1 = z 2 -2ξ ω hI L L T x 1 ż2 = -ω2 z 1 -2ξ ωz 2 + h L T 4ξ 2 ω 2 I L -ω2 I L +l aero qSC 1 L x 1 + hl aero L T qSC 2 L x 2 1
We then numerically solve the Lyapunov equation by considering the equality (instead of the inequality) associated to [START_REF] Reyhanoglu | Nonlinear dynamics and control of space vehicles with multiple fuel slosh modes[END_REF]. We obtain :

V = z T P z z (38)
we then compute ϕ according to [START_REF] Malekzadeh | A Robust Nonlinear Control Approach for Tip Position Tracking of Flexible Spacecraft[END_REF] ; in this case, we use :

g 0 = h L T -2ξ ωI L 4ξ 2 ω 2 I L -ω2 I L + l aero qSC 1 L (39) g 2 = h L T 0 l aero qSC 2 L (40)
Then, given two gains k 1 , k 2 > 0 the proposed controller is :

u = -(x 1 -ϕ(z)) -k 2 (x 2 -x2 ) + d dt x2 (41)
where:

x2 = -k 1 (x 1 -ϕ(z)) + φ -L g 0 V -(L g 2 V )(x 1 -ϕ(z)) -2ϕ(z)(L g 2 V ) = -k 1 (x 1 -ϕ(z)) + φ -L g 0 V -(L g 2 V )(x 1 + ϕ(z)) (42)
This law is compared to the following 'nominal' Backstepping law :

u 0 = -x 1 -k 2 (x 2 -x20 ) + d dt x20 := -x 1 -k 2 (x 2 + k 1 x 1 ) - d dt (k 1 x 1 ) = -(1 + k 1 k 2 )x 1 -(k 1 + k 2 )x 2 (43)

C. Numerical results

The damping of the flexible mode is chosen very low (0.01) and the simulations are performed under the initial conditions ψ(0) = 20 deg, q(0) = 20 deg/s far from the origin. Figure [START_REF] Sorensen | A controller enabling precise positioning and sway reduction in bridge and gantry cranes[END_REF] shows in blue (resp. in orange) the evolution of both ψ and η 1 when the controller u (resp. u 0 ) is applied to the system. We observe that controller u enhances the transient of the flexible mode.

V. APPLICATION TO A NONLINEAR MODEL OF A FLEXIBLE SPACECRAFT

A. Mathematical Model

We consider the following model of a flexible attachment to a spacecraft which has been obtained after a partial feedback linearization [START_REF] Reyhanoglu | Slewing maneuver of a flexible spacecraft using finite time control[END_REF]

       ẋ1 = x 2 ẋ2 = u η1 = η 2 η2 = -ω 2 η 1 -2ξ ωη 2 + η 1 x 2 2 -αu (44)

B. Controller design

Applying the design method described in the proof of proposition 2, we first use the following change of coordinates :

z =    x 1 η1 η2 :=    x 1 η 1 + αx 1 η 2 + αx 2 (45)
We then numerically solve the Lyapunov equation by considering the equality (instead of the inequality) associated to [START_REF] Reyhanoglu | Nonlinear dynamics and control of space vehicles with multiple fuel slosh modes[END_REF]. We obtain :

V η = ηT P z η (46)

Then we choose (for instance)

χ(x 1 ) = -M χ tanh(x 1 ) (47)
where M χ > 0 and we rewrite the dynamics as follows :

ż = f 0 (z) + g 0 (z)(x 2 -χ(x 1 )) + g 2 (z)(x 2 -χ(x 1 )) 2 = f 0 (z) + g 0 (z)ϕ(z) + g 2 (z)ϕ 2 + g 0 (z)δ 2 v + 2g 2 (z)ϕ(z)δ 2 + g 2 (z)δ 2 2 ( 48
)
we then compute ϕ according to [START_REF] Malekzadeh | A Robust Nonlinear Control Approach for Tip Position Tracking of Flexible Spacecraft[END_REF] ; in this case, we use :

V = log(1 +V η ) + 1 2 c 1 x 2 1 f 0 =        χ(x 1 ) η2 -ω 2 η1 -2ξ ω η2 + ω 2

C. Numerical results

In order to give more importance to the nonlinear term in x 2 2 , we used a high coupling coefficient α = -1 (by comparison to the very small value of a real model [START_REF] Reyhanoglu | Slewing maneuver of a flexible spacecraft using finite time control[END_REF]). Moreover, the damping of the flexible mode is chosen very low (0.01) and the simulation model is run under the following initial conditions x 1 (0) = 45 deg, x 2 (0) = 0 deg/s far from the origin. Figure [START_REF] Aribowo | Input shaping control to suppress sloshing on liquid container transfer using multi-joint robot arm[END_REF] clearly shows the damping enhancement of the flexible mode when we apply our novel controller (in blue).

VI. CONCLUSIONS

New strategies have been proposed to design GAS controllers for two significant classes of nonlinear minimum phase systems. Inspired by a non affine control method [START_REF] Lin | Global asymptotic stabilization of general nonlinear systems with stable free dynamics via passivity and bounded feedback[END_REF], we have designed some novel nonlinear control laws which take into account the zero dynamics of the considered systems. In future work, we will extend our results to systems with uncertain parameters and unmeasured flexible states.

VII. FIGURES

αx 1 +2ξ ωα χ(x 1 ) + ( η1 -αx 1 )χ 2 (x 1 )-αx 1

 11211 ωα + 2( η1 -αx 1 )χ(x 1 )Our control law is finally given by (41) and compared to the following 'nominal' Backstepping control law :u 0 = -k 2 (x 2 -χ(x 1 )) + χ(x 1 ) = -k 2 (x 2 + M χ tanh(x 1 )) -M χ (1tanh 2 (x 1 ))x 2 (49)

Fig. 1 .

 1 Fig. 1. Application 1 : evolution of ψ(t) and η 1 (t) (below)

L. Burlion and E. Duraffourg are with Onera -The French Aerospace Lab, F-31055 Toulouse, France. Laurent.Burlion (at) onera.fr Elodie.Duraffourg (at) onera.fr

T. Ahmed-Ali is with Ecole Nationale Supérieure d'Ingénieurs de Caen (ENSICAEN), 6 boulevard du Marchal Juin 14050 Caen Cedex 4, France.tarek.ahmed-ali (at) greyc.ensicaen.fr

F. Lamnabhi-Lagarrigue is with L2S-CNRS, Supélec, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France. lamnabhi (at) lss.supelec.fr


Tarek Laurent Burlion

Franc ¸oise Lamnabhi-Lagarrigue

T Ahmed-Ali

Updating the gain of global finite-time high gain observers

Recently, new results about semi-global and global finite-time observation have been obtained by the use of continuous high gain observers. In this paper, we propose to extend these results by studying "time-varying high gain" observers and by providing new update laws : first, we adapt the law introduced by L. Praly in the case of asymptotic observation to the finite-time case and we prove that the updated high gain remains bounded. Secondly, we propose a new update law which guarantees the high gain's value tends asymptotically to 1.

I. INTRODUCTION

Over the past few decades, the problem of finite-time observation of non linear systems has received a lot of attention. Except a few other results, there are mainly two classes of finite-time nonlinear observers that have been widely studied. First, discontinuous finite-time sliding mode observers have deserved a lot of attention : a lot of papers using classical sliding observers can be found in the literature [START_REF] Drakunov | Sliding mode observers based on equivalent control method[END_REF] - [START_REF] Xiong | Sliding mode observer for nonlinear uncertain systems[END_REF]. Sliding mode observers is still an active field of research and more recently higher order sliding observers have also been introduced (see for instance [START_REF] Davila | Second-order slidingmode observer for mechanical systems[END_REF], [START_REF] Yu | HOSM Observer for a Class of Non-Minimum Phase Causal Nonlinear MIMO Systems[END_REF]). Secondly, spurred by the work of Bhat and Bernstein [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF] on the finite time stabilization of a double integrator, a continuous homogeneous observer has been proposed for a large class of nonlinear second order systems [START_REF] Hong | On an output feedback finite-time stabilization problem[END_REF]. This led to the development of the wide branch of continuous finitetime nonlinear observers. The homogeneous domination approach was soon introduced in order to deal with higher dimensional uncertain nonlinear systems : its principle is to dominate the nonlinearities by introducing a scaling gain into the homogeneous observer. This enabled to solve the problem of finite-time output feedback stabilization of more and more complex classes of nonlinear systems [START_REF] Li | Global finite-time stabilization of a class of uncertain nonlinear systems using output feedback[END_REF], [START_REF] Jia | Semi-global finite-time stabilization of a class of nonlinear systems by output feedback[END_REF]. Meanwhile, another type of finite-time observer was introduced for a class of linearizable nonlinear system [START_REF] Perruquetti | Finite-time observers : application to secure communication[END_REF] and soon extended to observe uniformly observable systems in a semi-global [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF] and global [START_REF] Menard | A global high-gain finitetime observer[END_REF] ways . In all these design methods, the gain of the observer is fixed and must be chosen sufficiently large (so in a conservative way). However, in the asymptotic stabilization case, there exist some results to adapt the gain of high gain observers when the unknown nonlinear functions have an unknown growth rate [START_REF] Lei | Universal adaptive control of nonlinear systems with unknown growth rate by output feedback[END_REF] or when this rate depends on the measured output [START_REF] Praly | Asymptotic stabilization via output feedback for lower triangular systems with output dependent incremental rate[END_REF].

Recently, in [START_REF] Shen | Uniformly observable and globally lipschitzian nonlinear systems admit global finite-time observers[END_REF], an update law whose gain is an exponential function with arbitrary growth rate has been combined with the semi-global finite time observer of [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF] to obtain a global result. Since, the resulted law is exponential and the uncertain nonlinear terms are bounded by a perfectly known Lipschitz condition, we think this observer deserves many extensions. Therefore, in this paper, we consider the problem of updating the gain of global finite time high-gain observers for a class of uncertain nonlinear system whose uncertain nonlinear terms have an unknown growth rate which depends on the output. We obtain two possible update laws : the first one is a natural extension of the law introduced in [START_REF] Praly | Asymptotic stabilization via output feedback for lower triangular systems with output dependent incremental rate[END_REF] at the difference that some terms of this law serve to dominate some homogeneous factors outside a compact set around the origin. It is proved that the high gain remains bounded when the state and input of the observed nonlinear system are bounded. Then, a second update law is proposed and provides a gain which not only remains bounded under the same conditions than above but also tends asymptotically to 1. This paper is organized as follows. Some basic notations and definitions followed by the problem formulation are given in Section 2. In Section 3, we present our observer and some hypotheses. Then, we provide the adaptation laws and prove our main results in Section 4. We then illustrate our results on a numerical example in Section 5 and compare our adaptation laws.

II. PRELIMINARIES

Let R (resp. N ) denote the set of real numbers (resp. natural integers). In this paper, we interest us to nonlinear SISO systems of dimension n ∈ N .

A. Notations

Given α ∈ R + \{0}, y ∈ R, we note :

⌈y⌋ α := |y| α sign(y)
For a given vector x := [x 1 , . . . , x n ] T ∈ R n , ∀j ≥ i, we note x i,j := [x i , x i+1 , . . . , x j ] T

We use the following notation for a given n × n diagonal matrix

diag(a 1 , . . . , a n ) :=    a 1 0 0 0 . . . 0 0 0 a n   

B. Finite-Time Stability

In this paper, we interest us to finite time stability and stabilization results which have the merit to provide continuous time observer and controller by using a mix between Lyapunov theory and geometric homogeneity. We invite the reader which is not familiar with the basic definitions of homogeneity to look at the work of [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] or [START_REF] Menard | A global high-gain finitetime observer[END_REF]. Nevertheless, let us briefly recall a basic result about Finite Time Stability (See [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] for more details) that will be useful in this paper. Theorem : Suppose there exist a possibly non Lipschitz vector field f on R n , a C 1 positive definite function V : R n → R and positive real numbers a > 0, 1 > β > 0 such that :

∀x ∈ R n , V (x) = L f V (x) ≤ -aV β (x)
then the origin is GFTS (Globally Finite Time Stable) under the vector field f .

C. Problem position

Throughout this paper, we consider the following class of nonlinear systems :

                       ẋ1 = x 2 + f 1 (x 1 ) ẋ2 = x 3 + f 2 (x 1 , x 2 ) . . . ẋi = x i+1 + f i (x 1 , x 2,i ) . . . ẋn = u + f n (x 1 , x 2,n ) y = x 1 (1)
where x = [x 1 , . . . , x n ] T ∈ R n , u ∈ R and y ∈ R are respectively the state, input and output of this nonlinear system ; where the unknown functions f i 's are characterized by the following assumption [H1] :

|f i (x 1 , x 2,i ) -f i (x 1 , x 2,i -z 2,i )| ≤ ∀i∈[2,n] γ(y) k=2:i |z k |
where γ ≥ 0 with γ(0) = 0.

Our aim is to design a global finite-time observer for this class of nonlinear systems.

III. GLOBAL FINITE-TIME OBSERVER

Let us note

ξ i := x i -xi
where x denotes the state of a nonlinear high gain observer defined by:

                     ẋ1 = x2 + f 1 (x 1 ) + ko,1 2 L(⌈ξ 1 ⌋ α1 + ξ 1 ) ẋ2 = x3 + f 2 (x 1 , x2 ) + ko,2 2 L 2 (⌈ξ 1 ⌋ α2 + ξ 1 ) . . . ẋi = xi+1 + f i (x 1 , x2,i ) + ko,i 2 L i (⌈ξ 1 ⌋ αi + ξ 1 ) . . . ẋn = u + f n (x 1 , x2,n ) + ko,n 2 L n (⌈ξ 1 ⌋ αn + ξ 1 )
• where [k o,1 , . . . , k o,n ] ∈ R n will be defined later

• where 1 > (α 1 , . . . , α n ) > 0 is decreasing and will be defined later.

• where L ≥ 1 is time varying ( L will be designed later) Let us also introduce the following change of coordinates :

∀i ∈ [1, n], ε i = ξ i L i-1+b
We note :

δf i := f i (x 1 , x 2,i ) -f i (x 1 , x2,i ).
We rewrite the hypothesis [H1] in the new coordinates (this will be useful in the proof of our main result)

∀i ∈ [2, n], |δf i | ≤ γ(y) k=2:i L k-1+b |ε k |
where b > 0 is defined later. In these coordinates, the dynamics of the observer error is written :

                                   ε1 = Lε 2 - ko,1 2 L( L (α1-1)b ⌈ε 1 α1 + ε 1 ) -b L L ε 1 ε2 = Lε 3 - ko,2 2 L( L (α2-1)b ⌈ε 1 α2 + ε 1 ) + δf2 L 1+b -(1 + b) L L ε 2 . . . εi = Lε i+1 - ko,i 2 L( L (αi-1)b ⌈ε 1 αi + ε 1 ) + δfi L i-1+b -(i -1 + b) L L ε i . . . εn = - ko,n 2 L( L (αn-1)b ⌈ε 1 αn + ε 1 ) + δfn L n-1+b -(n -1 + b) L L ε n (2)

A. Additional Hypotheses :

Let us note C = [1 0 . . . 0] and A the matrix defined by (A) i,j = δ i,j-1 (where δ i,j is the Kronecker delta). Let a > 0, we choose Q = Q T > 0 such that there exists 1 > q > 0 s.t

A T Q + QA -C T C ≤ -2aQ ; qI n ≤ Q ≤ I n (3)
We then define (k o,1 , . . . , k o,n ) > 0 by:

[k o,1 , . . . , k o,n ] T = Q -1 C T
Let us also note D = diag(0, 1, . . . , n -1). We suppose there exists b > 0 such that:

-bQ ≤ D T Q + QD ≤ bQ (4)
Remark : in practice, we solve the LMIs associated to hypothesis (3) and then we search b > 0 in order to satisfy hypothesis (4).

IV. MAIN RESULTS

Let us note :

|ε 1 | # := max{|ε 1 | α1 , |ε 1 | αn } c 1,n := min{3c 1 , c n } and c 1,n := max{3c 1 , c n } k B := 3n a max i∈[1,n] k o,i c B := (n -1) 2 √ q 1 + max i∈[2,n] 1 c i
Let us now present our first result :

Theorem 1: Let us suppose the state of system [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF] remains bounded by a given bounded controller u ; under the hypotheses [H1], ( 3), ( 4), there exists α < 1 such that system (2) is Globally Finite Time Stable when we use the following adaptation law :

L = - 1 b L a 3 (L -1 -L|ε 1 | # ) - c B c 1,n + 2 n -1 √ q γ(y) (5) with L(0) = 1.
Moreover, this law guarantees that L remains bounded.

proof : it is somewhat technical but relies on the fact that we prove these results in several steps, taking in mind the fact that if we prove global asymptotic stability and finite-time stability in a neighborhood of the origin, we obtain global finite time stability. (See for instance, Lemma 1 of [START_REF] Menard | A global high-gain finitetime observer[END_REF])

A. Global Asymptotic Stability

Under the hypotheses [H1], (3), ( 4), let us first study the asymptotic stability.

Let us consider the following candidate Lyapunov function

V o (ε) = ε T Qε
where Q satisfies (3). ∀i, α i ≤ 1, we interest us to the following quantity

B(ε) := ε T Q       ε 2 -k o,1 L (α1-1)b ⌈ε 1 ⌋ α1 . . . ε i+1 -k o,i L (αi-1)b ⌈ε 1 ⌋ αi . . . -k o,n L (αn-1)b ⌈ε 1 ⌋ αn       • When ε ∈ ε ∈ R n s.t V o (ε) > k B we have: B(ε) ≤ ε T QAε + n V o max i∈[1,n] k o,i |ε 1 | # ≤ ε T QAε + a 3 |ε 1 | # V o
• When ε is in the following complementary set

C 1 := ε ∈ R n s.t V o (ε) ≤ k B Since when α = 1, ∀ε ∈ R n , B(ε) ≤ 0 and C 1 is a compact set, we know from the tube Lemma [3] that there exists 1 > ᾱ1 > 0 such that ∀α ∈]1 -ᾱ1 , 1[, ∀ε ∈ C 1 , B(ε) ≤ 0
So we conclude, that there exists

1 > ᾱ1 > 0 such that ∀α ∈]1 -ᾱ1 , 1[, ∀ε ∈ R n , B(ε) ≤ max{0, ε T QAε} + a 3 |ε 1 | # V o
Therefore, when α ∈]1 -ᾱ1 , 1[, we can write :

Vo ≤ Lε T (QA -C T C)ε + LB(ε) +2ε T Q        0 . . . δfi L i-1+b -(i -1) L L ε i . . . δfn L n-1+b -(n -1) L L ε n        -2b L L ε T Qε ≤ Lε T (QA -C T C)ε + L max{0, ε T QAε} + a 3 L|ε 1 | # V o + 2 n -1 √ q γ(y)V o -2 L L ε T (QD + bQ)ε ≤ L max -aV o - 1 2 ε T C T Cε, -2aV o + a 3 L|ε 1 | # V o + 2 n -1 √ q γ(y)V o -2 L L ε T (QD + bQ)ε ≤ -a 1 - 1 3 |ε 1 | # L -2 n -1 √ q γ(y) V o - L L b 2 -sign L L V o
If we apply the adaptation Law (5), we get after a few computations (remark that it is easy to prove that this adaptation law guarantees that ∀t ≥ 0, L(t) ≥ 1 and so we can use -(2L + 1) ≤ -3):

Vo ≤ -aV o - c B c 1,n γ(y)V o
≤ -aV o so the system is GAS (and so, it will stay inside the ball B 1 defined below within a finite time)

B. Proof inside B(1) := {ε s.t |ε i | ≤ 1}
Secondly, we interest us to the ball B(1) 1) First subsystem: First we consider the following vector field f α :

                   ε1 = ε 2 -k o,1 L (α1-1)b ⌈ε 1 ⌋ α1 ε2 = ε 3 -k o,2 L (α2-1)b ⌈ε 1 ⌋ α2 . . . εi = ε i+1 -k o,i L (αi-1)b ⌈ε 1 ⌋ αi . . . εn = -k o,n L (αn-1)b ⌈ε 1 ⌋ αn
As in [START_REF] Perruquetti | Finite-time observers : application to secure communication[END_REF], we define, (r 1 , . . . , r n ) > 0 and (α 1 , . . . , α n ) > 0 such that :

   r i+1 = r i + d, 1 ≤ i ≤ n -1 α i = ri+1 r1 , 1 ≤ i ≤ n -1 α n = rn+d r1 (6)
These equations simply say that the vector field f α is homogeneous of degree d with respect to the weights (r 1 , . . . , r n ).

Let us take a real number 0 < α < 1.

We set: r 1 = 1, r 2 = α, d = r 2r 1 = α -1 and we recursively prove that:

r i = (i -1)α -(i -2) 1 < i ≤ n and, α i = iα -(i -1) 1 < i ≤ n.
Since r 1 > . . . > r n , in order to guarantee ∀i, r i > 0, we will also need to assume that :

α > n -2 n -1 = 1 - 1 n -1
Let us now define :V z (α, ε) := z T Qz where z i = ⌈ε i ⌋ Let us also note :

c i = 1 riπr It is obvious that V z is homogeneous of degree 2
πr with respect to the weights (r 1 , . . . , r n ). and that L fα V z is homogeneous of degree 2 πr + degree(f α ) = 2 πr + d = 2 πr + α -1 with respect to the weights (r 1 , . . . , r n ). By using Lemma 4.2 of [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF],

-β 1 (α, L)V β z (ε) ≤ L fα V z (ε) ≤ -β 2 (α, L)V β z (ε) where β := deg(L fα Vz) deg(Vz) = 1+ πr 2 (α-1) < 1 because α < 1 and where β 1 (α, L) := - min {z s.t Vz(α,ε)=1} L fα V z (α, ε) > 0 and β 2 (α, L) := - max {z s.t Vz(α,ε)=1} L fα V z (α, ε) > 0.
( we easily prove it when α = 1 (asymptotic case) and we then use the same proof than [START_REF] Shen | Uniformly observable and globally lipschitzian nonlinear systems admit global finite-time observers[END_REF]) 2) Full system: Since ∀ε, the following quantity is negative when α = 1 1) is compact, we know from the tube Lemma [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] that there exists 1 > ᾱ2 > 0 such that:

B z (ε) := z T Q       c 1 |ε 1 | c1-1 (ε 2 -k o,1 ε 1 ) . . . c i |ε i | ci-1 (ε i+1 -k o,i ε 1 ) . . . c n |ε n | cn-1 (-k o,n ε 1 )       moreover, since B(
∀ε ∈ B(1), ∀α ∈]1 -ᾱ2 , 1[, B z (ε) ≤ 0 (7) Therefore, if 1 > α > max 1 -1 n-1 , 1 -ᾱ2 , we can write : Vz ≤ -β 2 (α, L)V β z (ε) + LB z (ε) +2z T Q          c 1 |ε 1 | c1-1 -b L L ε 1 . . . c i |ε i | ci-1 δfi L i-1+b -(i -1 + b) L L ε i . . . c n |ε n | cn-1 δfn L n-1+b -(n -1 + b) L L ε n          ≤ -β 2 (α, L)V β z (ε) -2 L L z T (QD + bQ) ×    c 1 0 0 . . . . . . . . . 0 0 c n    z + 2z T Q         0 . . . c i |ε i | ci-1 δfi L i-1+b . . . c n |ε n | cn-1 δfn L n-1+b        
First, since the c i s are positive and increasing and by using (4), we have:

• if L L ≥ 0 -2 L L z T (QD + bQ)    c 1 0 0 . . . . . . . . . 0 0 c n    z ≤ -c n b L L V z • if L L ≤ 0 -2 L L z T (QD + bQ)    c 1 0 0 . . . . . . . . . 0 0 c n    z ≤ -3c 1 b L L V z
Due to space restriction, the right term of the last two inequalities is given by -bc

# L L V z where c # := c n + 1 2 1 -sign( L L ) (3c 1 -c n )
. By Young's inequality:

|ε i | ci-1 |ε k | ≤ c i -1 c i |ε i | ci + 1 c i |ε k | ci
Hence, ∀i ∈ [2, n]:

|ε i | ci-1 δf i L i-1+b ≤ γ(y) k=2:i L k-i-2 |ε i | ci-1 |ε k | ≤ γ(y) c i -1 c i (i -1)|ε i | ci + 1 c i i k=2 |ε k | ci ≤ γ(y) (n -1) √ z ′ z + 1 c i i k=2 |ε k | ci
Since, ∀i ≥ k, c i ≥ c k ≥ 1 and ε ∈ B(1), we have :

|ε k | ci ≤ |ε k | c k So, ∀i ∈ [2, n]: |ε i | ci-1 δf i L i-1+b ≤ γ(y) 1 + max i∈[2,n] 1 c i (n -1) √ z T z
Thus, we obtain:

Vz ≤ -β 2 (α, L)V β z (ε) -bc # L L V z +γ(y) (n -1) 2 √ q 1 + max i∈[2,n] 1 c i V z
We identify the coefficient c B and we thus obtain :

Vz ≤ -β 2 (α, L)V β z (ε) -bc # L L -c B γ(y) V z (8)

C. Conclusion of the proof of Theorem 1

To sum up, we applied the adaptation law (5):

L = - 1 b L a 3 (L -1 -L|ε 1 | # ) - c B c 1,n + 2 n -1 √ q γ(y)
with L(0) = 1. Moreover, there exists 1 > α > max 1 -1 n-1 , 1 -ᾱ1 , 1 -ᾱ2 > 0 such that:

• system (2) adapted by this law is GES.

• ∀ε ∈ B 1 , applying the adaption law to equation the [START_REF] Li | Global finite-time stabilization of a class of uncertain nonlinear systems using output feedback[END_REF], we obtain:

V z := z(ε) T Qz(ε) Vz ≤ -β 2 (α, L)V β z (ε) + a 3 c # (L -L|ε 1 | # -1)V z -2c 1,n n -1 √ q γ(y)V z ≤ -β 2 (α, L)V β z (ε) + ac 1,n 3 LV z
First, using the first Lyapunov function V o , we proved the system is GES so there exists a finite time such that the system enters the ball B 1 without leaving it anymore. Inside this ball :

• we need to consider the following set :

C 2 := ε s.t ac 1,n 3 V z (ε) ≤ min L≥1 β 2 (α, L) 2L V β z (ε)
If L remains bounded (as we will prove after), this set is not a zero measured set (i.e it is bigger than {0 n }).

Since the system is GES, there exists a finite time such that the system enters the set B 1 C 2 without leaving it anymore.

Since in this set we have:

Vz ≤ - β 2 (α, L) 2 V β z
we conclude that system (2) converges to the origin in a finite time. Thus, we proved the system is GFTS. Moreover, provided y remains bounded and since L(0) = 1, the adaptation law form guarantees that L remains bounded. Indeed, since ε 1 tends to 0 within a finite time, L can increase with a bounded rate within a finite time and after, L L < 0

if L > 1 + c B c 1,n + 2 n -1 √ q γ(y(t))
the right term is bounded because γ(y) remains bounded, so after a finite time L decreases if it is superior to a bounded value.

Let us now stem our second result which is an extension of the previous one : Theorem 2: Let us suppose the state of system (1) remains bounded by a given bounded controller u ; if the hypotheses [H1], (3), ( 4) are satisfied, there exists α < 1 such that system [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] is Globally Finite Time Stable when we use the adaptation law [START_REF] Lei | Universal adaptive control of nonlinear systems with unknown growth rate by output feedback[END_REF] when ε 1 = 0 and :

L = - L b a 3 (L -1) when ε 1 = 0 (9)
with L(0) = 1. Moreover, this law guarantees that L tends asymptotically to 1.

Sketch of the proof:

We use almost the same computations than the ones carried in the proof of theorem 1.

If we apply the adaptation Law (5) when ε 1 = 0 and (9) otherwise, after some computations, we get:

Vo ≤ -aV o if ε 1 = 0 Vo ≤ -a -2(n-1) √ q γ(y) V o if ε 1 = 0
When ε 1 = 0, the Lyapunov function can increase but the system can not remain on the set S c = {ε 1 = 0 ; ε 2,n = 0} during a non zero length time interval. We prove it by contradiction : suppose the system remains on this set during a non zero measured time interval, we have ε 1 = ε1 = . . . = ε (n-1) 1 = 0 but because of equation (2), this implies that ε 2 = . . . = ε n = 0 which contradicts the fact that ε ∈ S c . We say that the error system 'globally decreases at almost every time'.

As for Theorem 1, we will finally come to the following conclusions : there exists 1 > α > max 1 -1 n-1 , 1 -ᾱ1 , 1 -ᾱ2 > 0 such that: • system (2) adapted by the law of theorem 2 GA decreases at almost every time.

• ∀ε ∈ B 1 , applying the adaption law to equation ( 8), we obtain :

Vz ≤ -β 2 (α, L)V β z (ε) + ac 1,n 3 LV z if ε 1 = 0
If ε 1 = 0, we also have:

Vz ≤ -β 2 (α, L)V β z (ε) + ac 1,n 3 L + c B γ(y) V z ≤ -β 2 (α, L)V β z (ε) + L ac 1,n 3 + c B γ(y) V z
Let us define the following set:

C 3 := ε s.t ac 1,n 3 V z (ε) + c B max t≥0 γ(y(t)) ≤ min L≥1 β 2 (α, L) 2L V β z (ε)
Since the system GA decreases at almost every time, it will enter the set B 1 C 3 in a finite time without leaving it anymore. Since in this set, we have :

∀ε 1 , Vz ≤ - β 2 (α, L) 2 V β z (ε)
we conclude that system (2) converges to the origin in a finite time. Thus, we proved the system is GFTS. Moreover, provided y remains bounded and since L(0) = 1, we prove that L remains bounded as in the proof of theorem 1. Moreover, since ε 1 and its time derivatives are equal to 0 after a finite time, the adaptation law [START_REF] Shen | Semi-global finite-time observers for nonlinear systems[END_REF] guarantees that L tends to 1.

V. ILLUSTRATIVE EXAMPLE

We illustrate the effectiveness of our design on the following two dimensional nonlinear system:

   ẋ1 = x 2 + x 2 1 ẋ2 = u + x 2 1 cos x 2 y = x 1
Choosing γ(y) = y 2 , it is easy to see that assumption [H1] holds.

Then we choose α = 4 5 , a = 1 2 , we solve the LMIs given by (3) and (4) and we obtain : Q = 0.5865 -0.3787 -0.3787 0.6530 and q = 0.1054, k o1 = 2.7261, k o2 = 1.5811, b = 2.2646

We apply an input u such that the state and the input of the system we want to observe remain bounded.

• Figure 1 shows the observer error when we apply the adaptation 1 : the error goes to 0 in finite time and L remains bounded. More specifically, in this example, L oscillates like y (which oscillates between two bounds).

• Figure 2 shows the observer error when we apply the adaptation 2 : the error goes to 0 in finite time and the fact that γ(y) does not go to 0 does not prevent L from asymptotically tending to 1. (Remark : if the initial error of the observer is large, we will need to choose bigger value of α sufficiently close to 1 (but still inferior to 1) so that the observer still converges). So, this numerical result illustrates what we have theoretically proved in this paper.

VI. CONCLUSIONS AND FUTURE WORK

This paper has addressed the problem of updating the gain of a global finite-time high gain observer. Using the separation principle for this specific class of triangular systems and adding a few hypotheses on the f i s, it may be quite straightforward to build a finite time controller and to combine it with our observer in order to get an output feedback which renders the system GFTS and such that the time-varying high gains of both controller and observer are bounded (and can even tend to 1).

Fig. 1 .Fig. 2 .

 12 Fig. 1. With the adaptation Law 1