P N Kvieska

G Lebret
email: guy.lebret@irccyn.ec-nantes.fr

M Aït-Ahmed
email: mourad.ait-ahmed@univ-nantes.fr

Gain Scheduled LPV Systems -Global Vision and Stability Results

In this paper the different phases of Gain Scheduling are visited, with special attention to stability results for LPV systems. Given that the main reviews in the area date from 2000, more recent results were gathered. Stability is to be shown mainly with Parameter Dependent Lyapunov Functions, and a discussion on performance analysis is presented. At the end, a procedure for the Synthesis and Analysis of a Gain Scheduled LPV controller is presented based on the different results presented. This procedure is simpler than the purely theoretical ones, and still more reliable than pure simulation, as it is often the case in industrial applications.

I. INTRODUCTION

A. Preliminaries

Gain Scheduling is one of the most popular approaches to non-linear and time-varying systems. It is particularly appreciated in industrial applications, mainly because of possible intuitive approaches and also for its simplicity of implementation. However, Gain Scheduling applications usually lack of stability theoretical results, and most of the time, for simplicity, simulations are used to prove system's stability and performance. This is the main motivation for this work, and this is why stability results for Gain Scheduled LPV Systems were gathered together.

Given that after the main review in the area, by Rugh and Shamma [START_REF] Rugh | Research on gain scheduling[END_REF], new techniques were proposed we thought that a new review could be useful, and even though a thorough review is not our main goal in this present work, we will visit the different phases of Gain Scheduling for LPV Systems, giving special attention for what was done after 2000, mainly for stability results. Finally, a methodology for Gain Scheduling is proposed based on the concepts found in the literature, and it is, in our opinion, more reliable than simulations, as often used in industrial applications, and still simpler than the specific synthesis techniques.

A different approach of Gain Scheduling, with the velocity-based linearization, is given by [START_REF] Leith | Survey of gain-scheduling analysis and design[END_REF].

B. Motivation

Our current research is in the control of tension and frequency of embedded electrical networks. In this particular type of network, each load that is connected to the system may have and important influence over tension and frequency, which is not the case in large power systems, where these parameters are imposed by the network.

A major problem of these type of networks lies on the fact that the system dynamic model changes with load variation (the loads are part of the model). In an embedded electrical network, the power needed by the loads can vary within a very large range according to the environment and use of the system. For example, in a ship, stabilisators will demand power that varies according to the speed and sea state, lights will be more used during the night, restaurants work in welldefined periods, etc.

Given that the major electrical loads that demand power (electric machines in a ship for example) are non-linear, we have a tricky system to control: a dynamic evolving nonlinear model. Three approaches are possible:

• A Robust LTI controller, where the variations would be treated as uncertainties • A Non-Linear technique • Gain Scheduling The first (LTI Robust Controller), depending on the system, would have to be so robust that its performances would be degradated. The second (Non-Linear technique) would certainly be useful, but it can be complicated and given the dynamic model time variation, it is not always suitable. The third one will then be used.

In this review, Gain Scheduling methodologies will be explored, on theirs general form.

II. BASES

A. Gain Scheduling

"Machines that walk, swim, or fly are Gain Scheduled" [START_REF] Rugh | Research on gain scheduling[END_REF].

The general idea behind Gain Scheduling is very simple: a regulator that will evolve to match changes in the environment is gain scheduled. In a plane for example, speed and air pressure can be used as scheduling variables, and so the "optimum" regulator is chosen based in these parameters.

The main advantages of Gains Scheduling are [START_REF] Rugh | Research on gain scheduling[END_REF]:

• Use of linear design tools for non-linear problems • Few assumptions on the model (can even be used without an analytic model) • Linear intuition can be used for the synthesis (in contrast to non-linear techniques that may use variable changes) • Computation burden is usually inferior to non-linear design techniques. On the other hand, one of the drawbacks is mainly the fact that there are not always stability and performance proofs (apart from simulations). Also, there is not a well defined methodology for Gain Scheduling in its general form, the solution may depend on the specific problem.

A classical approach to Gain Scheduling is based on the "divide-and-conquer" strategy. First of all, it consists in having several local models (for operating points or linearisations of a non-linear system) and then proceed to the synthesis of local regulators. With these regulators, one must create a law that will link them according to the scheduling variables (Gain/Pole/Zero interpolation or simple commutation for example). This will create a Gain Scheduled non-linear regulator.

A more systematic approach can be given if at the beginning there is an LPV model. This will be the topic of our next paragraphs.

B. Linear Parameter Varying (LPV) Model

The LPV model is an useful representation of non-linear or time-evolving model systems. Once this model is obtained, the synthesis of a controller is straightforward (even though stability proofs are not). An LPV standard model has the following structure [START_REF] Scherer | Linear Matrix Inequalities in Control[END_REF]:

  ẋ z p y   =   A(δ(t)) B p (δ(t)) B(δ(t)) C p (δ(t)) D p (δ(t)) E(δ(t)) C(δ(t)) F (δ(t)) 0     x w p u   (1)
The signals have the usual representation: u is the command, y the measured output and w p → z p a performance channel. This represents a family of systems with δ : [0, ∞) → R l with ∀t ≥ 0. Each one of this systems (for a fixed value of δ(t)) is linear, hence the name Linear Parameter Varying. The hypothesis is that δ(t) is not know à priori but can be measured (or at least evaluated) in real time. If the parameter δ(t) depends on at least one of the states variables, it is called a quasi-LPV model. This can be used to "disguise" non-linearities as a parameter to create an LPV model.

As this is a crucial phase of the procedure, it should not be neglected (as it sometimes is). This problem is addressed in [START_REF] Bruzelius | Linear parameter-varying descriptions of nonlinear systems[END_REF], where two approaches are proposed to obtain a closer relation between the non-linear system and the LPV Model.

The first one is the Deviation Approach: it's logical to think that all the trajectories of the real system will also be a trajectory of the LPV model. However, the inverse is not always true: the LPV model is larger than the original. The Deviation Approach will try to minimize this difference.

The second one is the Sensibility Approach, which has as main idea the reduction of the influence of the parameter δ(t) in the LPV model. The reader is encouraged to refer to [START_REF] Bruzelius | Linear parameter-varying descriptions of nonlinear systems[END_REF] for further details.

III. STABILITY ANALYSIS

In this section the stability problem of closed-loop systems will be addressed, and all the results will be presented as LMI problems. This phase can be the most important of the synthesis procedure, given that it is possible to use linear techniques over a linearized system, which does not assure, à priori, any stability. The main results for Gain Scheduled LPV systems are given by the Lyapunov theory. There are, however, other results that will also be explored in this section.

As a motivation for the stability study, the following example is given [START_REF] Rugh | Research on gain scheduling[END_REF]:

Example 3.1: Local Stability x Global Stability When parameters (or uncertainties) vary with time, the fact that all the locals systems are stable does not imply global stability. This is not intuitive, but it is sadly true. This example illustrates this situation, as seen in Fig. [START_REF] Biannic | Robust Control of Linear Parameter Varying Systems[END_REF]. The dynamics are given by the matrix:

ẋ = 0 1 -(1 + σ(t)/2) -0.2 x (2)
For all σ(t) ∈ [-1, 1] all the eigenvalues of the system are stable, which implies that all the local systems are stable. However, for σ(t) = cos 2t through simulation it is obvious that there is an unstable dynamic for non-zero initial conditions. A physical explanation is that the stiffness increases as it is contracting and decreases as it is expanding. This way, oscillations tend to grow (Fig. ( 2)).

A. Stability in the Sense of Lyapunov

The principal stability results for LPV systems are in the sense of Lyapunov. In this paper results will be presented gradually, having as the starting point the classical Lyapunov theorems and definitions. First of all, we state Lyapunov's Theorem [START_REF] Scherer | Linear Matrix Inequalities in Control[END_REF]: Theorem 3.1: (Lyapunov Theorem) For the differential equation ẋ(t) = f (x(t), t), considering x * ∈ X an equilibrium point in the interior of the set S:

(a)
If there exists a positive-definite, continuously differentiable function

V : S × T → R with V (x * , t) = 0 and V ′ negative semi-definite, then x * is stable. If V is decrescent, then x * is Uniformly Stable. (b)
If there exists a positive-definite decrescent and continuously differentiable function V : S×T → R with V (x * , t) = 0 and V ′ negative-definite, then x * is Uniformly Asymptotically Stable. This result is very well known, and it is presented here as the basis of the following sub-sections. Fig. 1. Spring-Mass system with a time varying stiffness spring 1) Lyapunov Stability for LTI Systems: Theorem 3.2: Lyapunov Stability for LTI Systems An LTI System is stable if and only if, there exists a symmetric positive definite matrix P (P ∈ S + ) such that, for the system ẋ = Ax:

A ′ P + P A < 0 (3)
where the function

V (x) = x ′ P x is the Lyapunov Func- tion
This condition is necessary and sufficient.

2) Lyapunov Stability for LPV Systems -Fixed Lyapunov Function: 1 A similar theorem for LPV Systems can be stated, which is a natural extension of the latter.

Theorem 3.3: Lyapunov Stability for LPV Systems An LPV System is stable if there exists a symmetric positive definite matrix P (P ∈ S + ) such that, for the system ẋ = A(δ(t))x:

A(δ(t)) ′ P + P A(δ(t)) < 0 (4)
where the function

V (x) = x ′ P x is the Lyapunov Func- tion
This condition is now just sufficient [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF], given that this is no longer the LTI case. No assumptions were made about the variation rate of the function, that is therefore unbounded. This is actually the case of Switched Systems, and a discussion on stability results for these systems is given in [START_REF] Montagner | Lmi approach for hinfinity linear parameter-varying state feedback control[END_REF].

3) Lyapunov Stability for LPV Systems -Parameter-Dependent Lyapunov Function: If the derivative of the parameter is not considered for the search of a Lyapunov function, this search will concern a very large class of system, given that, in this case, this derivative can be unbounded. This conservatism may be responsible for the impossibility of finding a matrix P that verifies [START_REF] Chadli | Stability and control of multiple model: LMI approach[END_REF].

A less conservative approach is what is called a Parameter Dependent Lyapunov Function 2 . The idea is to have, rather 1 For this work LPV systems are considered, but this proofs hold for uncertain systems as well. 2 One of the first works in this sense is the PhD dissertation [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF]. Fig. 2. Mass position as a function of time for σ(t) = cos(2t) and non-zero initial conditions than a matrix P , a matrix P (δ(t)). In this case, Lyapunov's condition comes to [START_REF] Wu | Control of Linear Parameter Varying Systems[END_REF]:

A ′ (δ(t))P (δ(t)) + P (δ(t))A(δ(t)) + dP dt < 0 (5)
which is equivalent to:

A ′ (δ(t))P (δ(t)) + P (δ(t))A(δ(t)) + δ dP dδ < 0 (6)
This condition is once again just sufficient, but it is less conservative than the last. This is the main stability result for LPV systems using Lyapunov's Theory, and can be stated as follows [START_REF] Scherer | Linear Matrix Inequalities in Control[END_REF]: Theorem 3.4: Parameter Dependent Lyapunov Theorem: Assuming that the function P : ∆ → S p is continuously differentiable within a compact set ∆ and that it verifies:

P (δ) ≻ 0 ∀δ ∈ ∆ (7) ∂ δ P (δ)λ + A(δ) T P (δ) + (8) +P (δ)A(δ) ≺ 0 ∀δ ∈ ∆ and λ ∈ Λ (9)
Then the origin of the system ẋ = A(δ(t))x(t) is exponentially stable for a variation of δ R → R l that verifies for the subsets ∆ and Λ of R l :

δ(t) ∈ ∆ et λ ∈ Λ ∀t ∈ R (10)
In this case, the function V (x, δ) := x T P (δ)x is the Quadratic Parameter Dependent Lyapunov Function for the system.

In this theorem, λ is the maximum value of the derivative of the parameter. A problem of this theorem in particular, is that as the domains are continuous, there is an infinity of LMIs to evaluate. This is obviously impossible, so there are two possible solutions: make some structural assumption (such as polytopic domains, LFT structure, etc) or make a grid and test the conditions in all the chosen points. For the latter approach, the more points are used, the more reliable the result will be, but it is not an analytical result.

An interesting approach for analysis and synthesis of LPV systems with polynomial dependence is given in [START_REF] Wu | A new solution approach to polynomial lpv system analysis and synthesis[END_REF]. It is based in the Sum of Squares (SOS) decomposition. A polynomial f (x) is SOS if there exists polynomial functions

f 1 (x), ..., f n (x) such that f (x) = n i=1 f 2 i (x)
. This directly implies that f (x) ≥ 0 for all x ∈ R n . The SOS decomposition assures a sufficient condition for non negativity of the multivariate polynomial and is equivalent to the existence of a positive semi-definite matrix Q and a properly chosen monomial vector Z(x) such that f (x) = Z ′ (x)QZ(x). This could help to provide a coherent methodology for the synthesis of a Lyapunov function for non-linear systems.

B. Input/Output Stability: Small Gain Theory

Given that this work is focused on the Lypunov approach, this theory will not be explored. Nevertheless this is an interesting approach and specific LPV results are available in the literature. As it is an Input/Output approach, no assumptions are made on the model, that can be linear/nonlinear, stationary/non-stationary, etc. The work [START_REF] Biannic | Robust Control of Linear Parameter Varying Systems[END_REF] is a good starting point for further informations.

C. Multiple Models -Interpolated Systems

The works of Mohammed Chadli ([3], [START_REF] Chadli | Stability and control of multiple model: LMI approach[END_REF]) state stability results for Multiple Models, which are in fact a general form of interpolated systems. This is, from our point of view, very useful, because it can be an analytical proof for systems without a well-defined model (mainly for systems without an LPV model). Here the different steps of the procedure will be visited, and the main result will be stated.

First of all, considering the non-linear system:

ẋ(t) = f (x(t), u(t)) (11)
For n arbitrary points (x i , u i ) ∈ R p × R m , the system can be linearized, and with a law that will be described in the following, a multiple model is defined as:

ẋ(t) = n i=1 µ i (z(t))(A i x(t) + B i u(t)) (12)
where µ i (z(t)) are the activation functions and z(t) is the decision variables vector, depending on the measurable state variables and eventually the command u(t). The number of local models n depends on the desired accuracy, complexity of the non-linear system and on the structure of the activation functions. This is a general form of an interpolated system, because for each point there is a dependency on all the other local models, and not only on the ones close to the point.

Two hypothesis are stated: Hypothesis 3.1: The activation functions µ i (z(t)) are continuously differentiable.

Hypothesis 3.2: The activation functions have the following properties:

µ i (z(t)) ≥ 0 ( 13
) n i=1 µ i (z(t)) = 1 (14)
Stability is shown using polyquadratic Lyapunov functions depending only on the system's states. Consider the function:

V (x(t)) = x(t) T P (x(t))x(t) (15)
with

P (x(t)) = n i=1 µ i (x(t))P i , P i > 0 (16)
The time derivative of the function (15) leads to

V (x(t)) = ẋ(t) T P (x(t))x(t)+ +x(t) T P (x(t)) ẋ(t) + x(t) T Ṗ (x(t))x(t) (17)
There are the two "classical" Lyapunov terms and another one that depends on the derivative of P. This last one can be bounded as follows:

x(t) T Ṗ (x(t))x(t) = x(t) T n i=1 ∂µi(x(t)) ∂x(t) , ∂x(t) ∂t P i x(t) ≤ x(t) T n i=1 ∂µi(x(t)) ∂x(t) ẋ(t) P i x(t) (
ẋ(t) ≤ υ, ∀x(t) ∈ R p , i ∈ I n .
With these results it is finaly possible to state the theorem concerning stability of multiple models. From our point of view, even though there are several inequalities to verify, it is impressive to have an analytical proof for an interpolated system, and this is why we mention it in this present review.

Theorem 3.5: If there exists positive definite symmetric matrix Q and P i , i ∈ I n and symmetric matrix U and V such the following LMIs are verified:

P i > P j+r , i ∈ I n , j ∈ I n-r (19) A T i P i + P i A i ≤ U, ∀i ∈ I n (20) A T i P j + P j A i + A T j P i + P i A j ≤ 2V, ∀(i, j) ∈ I 2 n , i < j (21) U -V ≤ 0 (22) V + r -1 (U -V ) + υ r i=1 P i < -Q (23)
with µ i (z(t))µ j (z(t)) = 0, r the maximal number of local models activated and υ an upward bound for the state variation (hypothesis (3.3)). Then, the equilibrium point of the multiple model ( 12) is globally exponentially stable.

Further details on stability results for discrete multiple models and applications are given in [START_REF] Chadli | Stability and control of multiple model: LMI approach[END_REF].

D. Contraction Analysis and δ stabilities

A last approach worth of mention is the Contraction Analysis. A review is given by [START_REF] Lohmiller | On contraction analysis for nonlinear systems[END_REF], and several works were done by Jérôme Jouffroy, the base of this section being [START_REF] Jouffroy | A simple extension of contraction theory to study incremental stability properties[END_REF]. Here just the main idea of the theory will be stated. According to the author, it is a more general approach than Lyapunov's theory. The stability is shown not with equilibrium points, but equilibrium trajectories.

The main idea is to consider two neighbors trajectories initially separated by an infinitesimal distance. If they converge one toward the other, it is possible to imply the convergence of two trajectories separated by a finite distance. This is the base of the contraction analysis, to show the convergence toward an equilibrium trajectory. Later, with a δ stability (also called incremental stability) technique, it is possible to show the stability of the system.

IV. PERFORMANCE ANALYSIS Once stability is verified, it is natural to search for the performances properties of the LPV system. Here the classical results based on the H 2 and H ∞ norms will be explored, as well as an unified approach for stability and performance.

It is important to keep in mind that local results for performance are simpler, but not always accurate for the global system (as seen in example (3.1) for stability). However, local results are indeed a good indication of the global performance, and it is intuitive to say that the better are the local performances, the better will the global be, even though this is not necessarily true. In the next section the extensions of these norms for LPV Systems will be explored.

A. H 2 and H ∞ Norms

An interesting method for performance evaluation is the norm between the input and output signals. It is possible to measure performance aspects such as trajectory tracking, noise rejection, control saturation, etc.

1) H ∞ norm extension for LPV systems: Let [START_REF] Rugh | Research on gain scheduling[END_REF]:

z = T δ(.) zυ υ (24)
be the input/output relation of a closed-loop for an LPV system with zero initial conditions. The signal υ is an exogenous signal and z is the error measure. The induced norm is then defined as [START_REF] Rugh | Research on gain scheduling[END_REF]:

T zυ i,2 = sup δ(.)admissible sup υ(.) ∞ 0 z T (t)z(t)dt 1 2 ∞ 0 υ T (t)υ(t)dt 1 2 (25)
This is related to the "worst case scenario". It is not the usual system performance, but an inferior bound for it, as it is shown by the sup of all admissible trajectories of the sup of all admissible values (all the trajectories that verify (10)).

2) H 2 norm extension for LPV systems: The definition of the H 2 norm here developed [START_REF] Biannic | Robust Control of Linear Parameter Varying Systems[END_REF] is the largest amplification caused by the impulse response for δ that verifies [START_REF] Montagner | Lmi approach for hinfinity linear parameter-varying state feedback control[END_REF]. Consider the m input LPV System:

S : ẋ = A(δ)x + [b 1 (δ), . . . , b m (δ)]w z = C(δ)x (26)
It is possible to decompose this system in m subsystems, S i where each input will be replaced for a non-zero initial condition that is equivalent to an impulse input.

S i : ẋi = A(δ)x i , x i (0) = b i (θ 0 ) z i = C(δ)x i ( 27
)
The extension of the H 2 norm is then:

γH 2 (S) = sup δ(t)∈∆ m i=1 ∞ 0 z T i (t)z i (t)dt (28)
As this concerns the output energy, it measures an "usual" performance of the system, the one that will be presented most of the time. These two last definitions generalize to LPV systems the two principal results for performance evaluation.

B. Stability and Performance Tests

Usually stability and performance are treated in different analysis. This is indeed an interesting approach, given that it is possible to make a rigorous stability analysis and a quicker local analysis for performance, for example. However, it is of course possible to evaluate both in a single inequality. This is done in several of our references, and an interesting approach is given to the problem of regulators interpolation that conserve the H ∞ performance in [START_REF] Yu | Gain scheduled output feedback control based on lti controller interpolation that preserves lpv hinf performance[END_REF].

Consider a closed-loop LPV system:

ẋ(t) = A(δ(t))x(t) + B(δ(t))w(t) z(t) = C(δ(t))x(t) (29)
A sufficient condition for this closed-loop to have a H ∞ performance of γ and to be stable in the sense of Lyapunov is given by [START_REF] Yu | Gain scheduled output feedback control based on lti controller interpolation that preserves lpv hinf performance[END_REF]: Definition 4.1: For the closed-loop system (29), with δ ∈ ∆ ⊂ R l , if there exists a symmetric positive semi-definite continuous matrix function P (δ) that verifies:

A T (δ)P (δ) + P (δ)A(δ) + d dt P (δ)+ +γ -2 C T (δ)C(δ) + P (δ)B(δ)B T (δ)P (δ) < 0 (30)
then the closed looped system (29) has the H ∞ performance γ.

In our opinion, this is a very "reassuring" condition: it assures stability and a minimum performance level. If this condition is verified, it is possible to simulate the system to justify its performance in particular situations, because there is the certainty that even in unexpected situations, the worst performance possible is proved to be γ.

V. SYNTHESIS

There are several possible strategies for the synthesis of a Gain Scheduled LPV controller, and here three different approaches are presented, one in detail. There is no "better" or "worse" between the two, it all depends on the type of problem being treated, the system regulation requirements and computational tools available.

A. LFT Representation

This is a very useful model for LPV Systems, with structured and efficient algorithms for synthesis, where the complexity is comparable to the synthesis of a LTI controller [START_REF] Rugh | Research on gain scheduling[END_REF]. However, one should be careful with this method, seen that it can be restrictive in several practical applications. An interesting recent approach for LFT systems is given by [START_REF] Wu | Gain-scheduling control of lft systems using parameter-dependent lyapunov functions[END_REF] where Parameter-Dependent Lyapunov functions are used for the synthesis.

B. LPV specific Synthesis Technique

LPV specific techniques can be found in several works, even though the approach is often different from one to another. The most appropriated from our point of view is the one in [START_REF] Scherer | Linear Matrix Inequalities in Control[END_REF]. There are drawbacks as a possible dependence on the derivative of the parameter, which can be a problem for several applications.

The development of this technique is long and as this is not the goal of this present work, it will not be done here. But to make the synthesis of a LPV regulator, there are several functionnal inequalities to verify (in the general case, polytopic domains reduce these to a finite set), and assumptions must be made to avoid the dependence on the derivative.

The advantage of this method is obviously the fact that there are no iterations, and performance and stability specifications are considered from the start. The inconvenient is that is is more complex and computationally heavier than the Ad Hoc techniques.

C. Ad Hoc Techniques

The Ad Hoc techniques are those for which the performance and stability specifications are not considered à priori, but should be verified after the synthesis procedure. As it is intuitive to see, there is not an exact procedure for doing this, and it may vary with the experience of the engineer. From our point of view, this type of method shows the interest of Gain Scheduling: with very simple methods it is possible to make the synthesis of very good non-linear regulators.

Even if there are no models, it is possible to make some regulators with data tables issued from the plant behavior, but in this way there will not be analytical proofs of stability or performance. For this reason, in this section it is assumed in this section that the LPV Model was already obtained.

The difference between the procedure here described and the specific one resides mainly in a simpler synthesis phase. In addition, there is a greater choice of techniques for local regulator synthesis, where experience for particular applications can be used. Surprisingly, the procedure here described is not the "natural" procedure found in the literature, where usually one of the two cases arises: applications are justified with simulations or a specific LPV technique is used.

Once the LPV model is defined (1), a controller with the following structure is to be searched:

K(δ(t)) : ẋK (t) = A K (δ(t))x K (t) + B K (δ(t))y(t) u(t) = C K (δ(t))x K (t) (31
) So any non structured or non specific method that results in this representation, is an Ad Hoc technique. This choice is obviously vital for the global system, so, what is propose in this paper is the following technique. First, the synthesis phase:

• Depart from an open-loop LPV model • Grid the scheduling variable domain for n local models (this should not be confused with griding a non-linear plant for the LPV modeling). The more points are used, the more the global performance will be close to the local evaluations, seen that the system will be closer to the synthesis points • Perform the synthesis of n local regulators using a linear technique. The same topology shall be imposed to all the local regulators for the next step. For example, use a pole-placement technique for each chosen local model with the same number of poles/zeros. • Proceed to the actual Gain Scheduling, the Scheduling law. As the same topology for each local model is imposed, a simple Gain/Pole/Zero interpolation results in an appropriated non-linear controller of the form (31). With this assumption there is no commutation, the function is continuous. • Calculate the closed-loop. This is a very light computational burden. But so far there is no proof of any kind. Simulations could be used for noncritical applications, but this is not the goal here. So, there is the analysis phase:

• Stability can be verified with a Parameter Dependent Lyapunov Function (Theorem (3.4)), seen that there is a closed-loop LPV model. In polytopic domains, the number of LMI is finite, and this is often the case. For example, as seen in [START_REF] Tanguy | Interpolated versus polytopic gain scheduling control laws for fin/rudder roll stabilisation of ships[END_REF], in a ship the domains of a two-dimensional Scheduling variable based on speed and angle between the ship and the current form a rectangle, which is a polytopic domain. This particular paper is an example of application without theoretical proof, where this particular procedure could had been used. Several other examples can be found for real applications, showing that this method of verification is suited. • Performance is second to stability, given that stability is primordial. Three strategies are possible, and from our point of view the third one is the most appropriated: Verify only local stability and simulate the global system. Global stability depends on local models, but this can be misleading, so for critical applications this is not the best choice, there can be an unexpected input that throws system's performance away from expected values. -Use the norms extensions given by ( 25) and (28). This is the ideal test, but can anyway be complicated to evaluate. -Use equation (30), in the first step, given that this test is based on a Parameter Dependent Lyapunov Function. With this test there is a minimum performance assurance. If the conditions are met for this test, it is possible to proceed to the simulation of the global system for the expected inputs, verifying if the outputs are reasonable. If the analysis fails an iteration with another synthesis is necessary. Most of the time, global results will depend on the local models, so it is possible to know the direction to go during the local synthesis.

The obvious advantage of this method is the fact that with simple techniques, it is possible to build an analytically proved Gain Scheduled controller. An also obvious inconvenient is that it may take several iterations, and in the worst case scenario, one may never achieve the synthesis of the expected regulator. In this case, if possible, relaxing the system requirements may be a solution.

Anyway, as described in the preceding subsections, this technique is simpler than the specific one for LPV Systems. And in addition, it is better than what is currently used in industrial applications, where often simulations are the only proofs.

VI. GAIN SCHEDULING INDUSTRIAL APPLICATIONS

Several applications of Gain Scheduling can be found through literature, in different domains. A robotic directdrive manipulator is explored in [START_REF] Yu | Gain scheduled output feedback control based on lti controller interpolation that preserves lpv hinf performance[END_REF], a ship control problem in [START_REF] Wu | Gain-scheduling control of lft systems using parameter-dependent lyapunov functions[END_REF], the rotating stall and surge control problem in [START_REF] Giarre | Lpv model identification for gain scheduling control: An application to rotating stall and surge control problem[END_REF] and an aerospace launch vehicle control problem in [START_REF] Clement | Aerospace launch vehicle control: A gain scheduling approach[END_REF], to cite a few. From these few examples several domains are cited: robotics, naval and aerospatial industries and fluids mechanics.

It shows why it is important to find theoretical results for this approach of non-linear control.

VII. CONCLUSION AND FUTURE WORKS

In this work we were able to explore the different aspects of the Gain Scheduling methodology, from modeling to synthesis and analysis. Modeling is a crucial phase that we did not detail, but should not be neglected.

The procedure we proposed is something in between of the theoretical tools and simulations, and from our point of view it is useful. For our research in the control of Electrical Networks, which is a very particular type of system, we find this technique suitable.

Finally, we see that our efforts are justified throughout the large number of industrial applications of this concept.


Jean-Yves Georges
email: jean-yves.georges@c-strasbourg.fr

Sabrina Fossette

Estimating body mass in leatherback turtles Dermochelys coriacea accepted 12 Jan 06 in Marine Ecology Progress Series

Body mass is a major life history trait and provides a scale for all living processes of organisms. Unfortunately body mass cannot be easily measured for many species, because of the logistical difficulties involved in actually catching and weighing them. This is particularly true for sea turtles which are large vertebrates that spend most of their life at sea. Here we developed a general linear model to predict body mass from 17 morphometric measurements obtained from 49 leatherback turtles nesting in Awala Yalimapo beach, French Guiana, South America. A stepwise backward analysis removing independent parameters with P > 0.001 indicated that body mass can be estimated with 93% accuracy by Body Mass (kg) = -709.146 + 3.391MedianBodyCirc (cm) + 2.664SCCL (cm), where SCCL is the standard curvilinear carapace length and MedianBodyCirc is body circumference at half of SCCL. A sensibility test showed that this parsimonious model is robust, as estimated body mass may change by 0.7-1.3% for 1-2 cm changes in SCCL, by 0.9-1.7% for 1-2 cm changes in circumference.

Leatherback turtles from French Guiana were larger and heavier than in all other nesting sites studied so far, suggesting either that gravid leatherbacks feed during the nesting season in French Guiana, and/or that this species may exhibits site-specific growth strategies. Further studies are required to test these hypotheses, including implementation of similar simple models for other nesting populations, in order to better understand the life history of this endangered species.

Introduction

Body mass is a major life history trait (Stearns 1992) and provides a scale for all living processes of organisms (Schmidt-Nielsen 1984) including their interactions with their habitat.

For example, body mass contributes to the capacity with which organisms can buffer changes in food availability. A larger body mass enables species and/or individuals to better buffer a substantial decrease in food abundance compared with smaller species and/or individuals.

This has important consequences for the way organisms reproduce. In animals, larger species have been reported to store body reserves prior to reproduction and rely on them during reproduction (capital breeders, Jonnson 1997). In contrast, smaller species adjust their food intake during the breeding season according to requirements associated with reproduction (income breeders, Jonnson 1997). Body mass thus provides valuable information concerning the potential survival and the reproductive success of a population (Clutton-Brock 1988[START_REF] Fossette | Dispersion and diving pattern in gravid leatherback turtles during the nesting season in French Guiana[END_REF].

Unfortunately, body mass may be difficult to be measured for many species, particularly for large marine vertebrates such as sea turtles and marine mammals, which spend most of, if not their entire, life at sea and whose adult body mass frequently exceeds several hundred kilograms. Although mean body mass is often estimated from life tables and growth curves (e.g. Trites & Pauly 1998), such life history data do not consistently exist for sea turtles. A few, rare studies investigated the relationship between body mass and morphometrics in sea turtles (e.g. [START_REF] Boulon | Leatherback turtles (Dermochelys coriacea) on St Croix, U.S. Virgin Islands: fifteen years of conservation[END_REF], Leslie et al. 1996, Hays et al. 2002). These studies focused on body length, probably because it is the most common and one of the easiest measurement to be taken in the field. However, the weak relationship so far reported between body mass and body length indicates that body length is not a good predictor of body mass in sea turtles, particularly in leatherback turtles [START_REF] Boulon | Leatherback turtles (Dermochelys coriacea) on St Croix, U.S. Virgin Islands: fifteen years of conservation[END_REF], Leslie et al. 1996). Hence, accurate estimates of key life history traits, such as body mass are crucial for improving our knowledge of the basic biology of this critically endangered species.

Here we propose a method for accurately estimating body mass in gravid leatherback turtles taking simple measurements that can be easily achieved in the field at low cost. We developed a sensibility test to take into account that field work in sea turtles is performed at night, potentially resulting in errors of these field measurements. Finally we compare our results with previous studies and suggest new areas of investigation in sea turtle research.

Material and Methods

This study was conducted during the entire 2005 nesting season at Awala Yalimapo beach (5.7°N -53.9°W), French Guiana, South America. Awala Yalimapo beach is situated on the French side of the Maroni River, separating French Guiana and Surinam, where more than 50% of the world population of leatherback turtles nest (Girondot & Fretey 1996, Spotila et al. 1996). Monitoring programs at this beach have been in place since the late 1970s (Girondot & Fretey 1996, Rivalan et al. 2005a). Nesting individuals have been extensively tagged using external metal tags (Monel tags, National Band and Tags, Newport, Kentucky, USA) and, more recently, with internal Passive Integrated Transponders tags (Trovan Euroid, Courbevoie, France), allowing individual identification. All individuals we considered in the present study were already tagged, so that their identity and their history were known.

Between April 4 and July 21, 2005, a 4 km stretch of the beach where most of nesting events occur (Girondot & Fretey 1996), were continuously patrolled every night from 18:00 to 07:00 hours. All tagged turtles encountered during these patrols were individually identified to ensure that individuals used in the present analyses were considered only once. Additionally, for individuals captured more than once during the study season, we only considered measurements collected during the first capture except for body size (see below). This was necessary to avoid pseudo-replication for statistical analysis. Turtles were measured during ovi-position and weighed afterwards, when returning to the sea. This ensured that disturbance was minimal to the turtles, as suggested by the observation that no turtle ended ovi-position prematurely and all of them came back at least once for a subsequent laying.

In total, 182 individual turtles were opportunistically sampled during our patrolling. Among these 182 individuals, 72 turtles were captured more than once throughout the study period.

This permitted to estimate the reliability of our measurements of conservative biometric parameters, such as body length, in order to develop a test of sensibility of our models. During the first month, 49 individuals were captured and measurements of 17 biometrical parameters (see Figure 1 for details) were obtained using a flexible measuring tape (± 0.5 cm), except head and body heights were obtained using a custom-made calliper rule (± 1cm) when the turtle was lifted for weighing. This permitted to develop in the field a first model identifying the most relevant biometric measurements to be performed during the rest of the study while reducing substantially the time spent per animal. All turtles considered in this study had an intact caudal peduncle to avoid potential biases in body size measurements [START_REF] Godfrey | Dermochelys coriacea (leatherback sea turtle) size[END_REF]. Standard curvilinear carapace length (hereafter referred to SCCL) was measured on the midline of the shell, from the nuchal notch of the carapace to the end of the caudal peduncle (Figure 1). For individual turtles captured more than once during the season, we averaged multiple measurements of SCCL to obtain the most accurate value for body length for the analyses, assuming that no detectable linear growth occurred during the nesting season [START_REF] Broderick | Variation in reproductive output of marine turtles[END_REF], Price et al. 2004, Georges et al. unpublished data). Body circumference measurements were obtained during ovi-position by passing the flexible measuring tape through narrow tunnels below the turtle that were burrowed by hand. Thanks to consistency in the shape of the carapace among individuals, circumferences were measured at the maximum body width (Anterior circumference), at half of the SCCL by marking the midway point when measuring SCCL as a guideline (Median Body Circumference) and at the frontal insertion point of the rear flippers (Posterior Body Circumference) for consistency among individuals. As each turtle was returning to the sea, a custom made harness with four 10-cm wide nylon straps was placed on the sand in front of the turtle, which was secured when the turtle was lying on it. A 4.5 m tall carbon-fibre tripod was immediately placed above the securely harnessed turtle. The turtle was then lifted using a hoist on which a electronic spring scale (600 ± 0.1 Kg, Kern Ltd, Germany) was fixed for weighing. The spring scale was calibrated against a known weight before at the start and at the end of the experiment, and did not show any deviance through time. Additionally, a second measurement of body circumferences was made when the turtle was lifted and did not differ significantly from the measurement made through the tunnels during ovi-position.

During the rest of the nesting season additional turtles were weighed and measured to increase sample size for further analyses. Morphometric measurements (including tunnel digging) and weighing (including harness attachment, turtle lifting and release) did not exceed 10 minutes in total.

Models developed in this study were statistically compared using information statistics rather than simply comparing models for lack of fit according to the Akaike's information criterion (AIC, [START_REF] Burnham | Model selection and inference: a practical informationtheoretic approach[END_REF].

Results

Estimating body mass

Between April 4 and 29, 2005, a total of 49 females were extensively measured when laying eggs and weighed after nesting. Body mass after ovi-position (hereafter referred to body mass) ranged from 275.6 to 562.7 kg (mean ± SD: 408.1 ± 63.4 kg, n = 49 turtles). A stepwise backward analysis was performed on a general linear model (GLM) incorporating 17 morphometric variables (Figure 1) to identify the most parsimonious model explaining body mass. A first GLM was performed followed by a stepwise backward analysis indicated that the best and most simple model estimating body mass was Body Mass (kg) = -699.352 + 3.469MedianBodyCirc (cm) + 2.481SCCL (cm), F 2,46 = 319.16, r² = 0.933, P < 0.001, AIC = 421, [model 1]

where SCCL is the standard curvilinear carapace length (159.1 ± 7.0 cm, range = [146 -179 cm], n = 49) and MedianBodyCirc is the body circumference at half of SCCL (205.5 ± 14.6 cm, range = [173 -237 cm], n = 49 Figure 1 and Table 1).

MedianBodyCirc and SCCL accounted for 75%, and 25% of the variance of model 1, respectively. Comparisons of Akaike weights indicate that model 1 was significantly better than the simple linear regression between body mass (y, in kg) and SCCL (x, in cm: y = 6.217x -580.665, r² = 0.467, P < 0.001, AIC = 521, P > 0.99), or between body mass and SCCL 3 (y = 8.054E -5 x + 82.155, r² = 0.458, P < 0.001, AIC 522, P > 0.99) commonly reported in previous studies (e.g. [START_REF] Boulon | Leatherback turtles (Dermochelys coriacea) on St Croix, U.S. Virgin Islands: fifteen years of conservation[END_REF], Leslie et al. 1996, James et al. 2005), but also between body mass and MedianBodyCirc (y = 4.08x -430.402, r² = 0.878, P < 0.001, AIC = 449, P > 0.99).

In order to increase sample size and analysis power, we then assumed that body mass As for model 1, model 2 calculated throughout the nesting season was significantly better than the simple linear regression between body mass and SCCL (y = 6.517x -648.268, r² = 0.514, n = 182, P < 0.001, AIC = 1998, P > 0.99), or between body mass and SCCL 3 (y = 9E - 5 x + 39.458, r² = 0.52, n = 182, P < 0.001, AIC = 1891, P > 0.99), or between body mass and MedianBodyCirc (y = 4.107x -427.255, r² = 0.862, P < 0.001, AIC = 1664, P > 0.99).

Body mass estimated from model 1 (BMest 1 ) was strongly correlated with body mass estimated from model 2 (BMest 2 ): BMest 2 = 0.997BMest 1 + 4.762 (Figure 3, F 1,180 = 521390, r² = 0.9997, P < 0.001). The slope of the regression did not differ from 1 (t-test = 0.004, P > 0.99) but the intercept with the origin differed significantly from 0 (t = 345, P < 0.001), indicating a slight (1%) difference when considering individuals early in the season compared to individuals sampled throughout the nesting season.

Sensibility test

Among the 182 individual turtles considered in this study, 72 of them have been captured more than once. This enabled to estimate the reliability of our measurements, considering that misreading can occur particularly when field work is performed at night, as is the case when working with nesting sea turtles. Measurements of the SCCL made on individual turtles during at least two consecutive nesting events differed by 1.5 ± 1.2 cm (n = 209 measurements performed on 72 turtles), ranging from zero up to 5 cm in some cases. We thus implemented a sensibility test to assess how changes in MedianBodyCirc and/or in SCCL would affect body mass estimates based on model 2, considering an hypothetical turtle of 160 cm SCCL and 200 cm MedianBodyCirc, and 395.3 kg (according to model 2). Not surprisingly, body mass estimates from model 2 were more affected by changes in MedianBodyCirc than by changes in SCCL (Figure 4). Changes of 1, 2, and 5 cm in MedianBodyCirc around 200 cm resulted in relative variations in body mass of 0.9, 1.7, and 4.3%, respectively (corresponding to 3.4, 6.8, and 16.9 kg, respectively). Similar changes in SCCL around 160 cm would result in relative variations in estimated body mass of 0.7, 1.3, and 3.3%, respectively (i.e. 2.7, 5.7, 13.3 kg, respectively). Relative variations in estimated body mass associated with concurrent changes in MedianBodyCirc and SCCL were additive, i.e. concurrent changes of 1, 2, and 5 cm around 200 cm of MedianBodyCirc and 160 cm of SCCL would result in relative variations in body mass of 1.5, 3.1, and 7.7%, respectively.

Discussion

Leatherback turtles rank among the largest living reptiles [START_REF] Eckert | Death of a giant[END_REF], making them logistically difficult to weigh. This problem is compounded by the fact that only females are accessible on land and then only for a period of about 24 hours in total every 2 to 3 years while nesting (Girondot & Fretey 1996, Rivalan et al. 2005b). Body mass is however a key life history trait and has been reported as the most appropriate parameter for assessing somatic growth, superior to the commonly used measurement of carapace length [START_REF] Bjorndal | Growth rates of immature green turtles, Chelonia mydas, on feeding grounds in the southern Bahamas[END_REF][START_REF] Chaloupka | Age, growth, and population dynamics[END_REF], Price et al. 2004). Body mass is also required for estimating food consumption of a population (e.g. Trites et al. 1997) and to assess the trophic rule of a species on its food web. Body mass is thus required for assessing energy fluxes through trophic levels, and for ultimately better understand ecosystem functioning.

The aim of this study was to provide a method, easily applicable in the field by both specialists and non specialists, to accurately estimate body mass in leatherback turtles. We found that in French Guiana, body mass of gravid leatherback turtles, individually captured throughout the nesting season, can robustly be estimated with 93% accuracy from standard curvilinear carapace length (SCCL) and body circumference at half of SCCL (MedianBodyCirc). This model may apply for male individuals as well, as James et al. (2005) found no sex-specific relationship between body mass and SCCL between 7 females and 6 males feeding off Canada. Therefore additional data are needed to establish whether this relationship also holds true for males in French Guiana. Both SCCL and MedianBodyCirc can be taken with relative ease using flexible measuring tape. Body mass was mainly predicted by body circumference, a parameter simple to measure by digging a narrow tunnel below the turtle's body. Although, body circumference can only be measured when the turtle is almost immobile during nest digging and ovi-position, these activities occur over a span of approximately 50 minutes [START_REF] Billes | Etude de la nidification de la tortue luth Dermochelys coriacea en Guyane Française[END_REF], allowing more than sufficient time to dig a tunnel. SCCL is the most commonly measured parameter in all sea turtle monitoring programs throughout the world because of its practicability, yet it contributed to less than 30% of our model, and explained only about 50% of the variance of body mass when considered alone.

Similar low r² values have previously been reported for body mass -SCCL regressions of gravid leatherback turtles nesting in St Croix Island, Caribbean Sea, and along the Atlantic coast of Costa Rica (r² = 0.551, and r² = 0.704, respectively; [START_REF] Boulon | Leatherback turtles (Dermochelys coriacea) on St Croix, U.S. Virgin Islands: fifteen years of conservation[END_REF], Leslie et al. 1996), indicating that accurate estimates of body mass in leatherback turtles require more than a single parameter. Our study proposes a simple, yet robust and powerful way (Figure 3) to improve such estimates of body mass. The fact that slopes and intercepts of body mass -SCCL regressions are slightly different between nesting populations (Figure 5, [START_REF] Boulon | Leatherback turtles (Dermochelys coriacea) on St Croix, U.S. Virgin Islands: fifteen years of conservation[END_REF], Leslie et al. 1996, Southwood et al. 1999, this study) emphasises the importance of similar studies for other nesting sites to test if body mass in other populations can also be estimated from relative simple models.

Differences in the relationship of body mass and SCCL between studies may result from different environmental trophic conditions and/or from different growth strategies. Atlantic leatherback turtles were 18-25% heavier (for a given length) when feeding off Canada than when nesting (Figure 5). This is consistent with the fact that gravid sea turtles are commonly considered as not feeding substantially during the nesting season (Miller 1997). Heavier individuals in cold waters may also result from water retention related to osmotic regulation and to the optimisation of body shape in this extended migrant. Interestingly, leatherback turtles nesting in French Guiana in 2005 were about 70 to 120 kg heavier than those from other nesting sites (Figure 5, [START_REF] Boulon | Leatherback turtles (Dermochelys coriacea) on St Croix, U.S. Virgin Islands: fifteen years of conservation[END_REF], Leslie et al. 1996, Southwood et al. 1999) and than individuals weighed in French Guiana in 1987(Girondot & Fretey 1996). Our study indicates that for an average leatherback turtle of 160 cm SCCL and 200 cm body circumference, changes of 5 cm in SCCL (all remaining similar otherwise) results in a change in body mass of about 13 kg (Figure 4). If this tendency holds for other populations too, the large differences in body mass observed between populations may not only be attributed to the slight inter-population differences in SCCL, but are more likely to result from different growth patterns among populations. However, growth patterns are poorly documented in leatherback turtles (but see Zug & Parham 1996, Price et al. 2004) and should be further investigated so that we can test the hypothesis of population-specific growth patterns, but also to test potential sex-specific growth patterns.

Interestingly, the slope of the body mass -SCCL regression we found for leatherbacks nesting in French Guiana was more similar to the slope for individuals feeding off Canada Leslie et al. 1996). This suggests that leatherback turtles nesting in French Guiana feed during the nesting season but to a lesser extend than turtles observed in their feeding grounds off Canada by James et al. (2005) and in other sites, turtles may not feed during nesting or to a lesser extent than in French Guiana.

Although this analysis may provide indirect evidence that leatherback turtles may feed during the nesting season in French Guiana (Georges et al. submitted, Fossette et al. submitted) further investigations using direct measurements of body mass changes during the nesting season are required to provide confirmation. In addition, leatherbacks feeding off Canada were on average about 7-8 cm smaller than individuals nesting in French Guiana (estimated from James et al. 2005, this study), yet within the range of body lengths observed in the nesting population of French Guiana. This could also suggest that relatively small leatherbacks observed feeding off Canada might be individuals that originated from, and/or nested in French Guiana. This is supported by recent satellite tracking data (Ferraroli et al. 2004, Georges et al. unpublished data) but further investigations, including genetic analyses, are required to confirm the actual origin of sea turtles opportunistically or accidentally captured on their feeding grounds.

Finally, the relatively large size of gravid leatherback turtles in French Guiana compared to other sites may permit them to better buffer potential changes in food availability during their trans-oceanic migration [START_REF] Ferraroli | Where leatherback turtles meet fisheries[END_REF]). This may contribute to the relatively healthy population in French Guiana and possibly on the Guiana shield (Guiana, Suriname, French Guiana), compared to the declining population in the Pacific Ocean (Spotila et al. 2000), where the smallest specimens have been reported (Figure 5). In addition, leatherbacks measured in our study were about 5 cm longer than individuals measured on the same site 18 years earlier (154.6 ± 9.0 cm, n = 15, Girondot & Fretey 1996). Such size difference is consistent with the growth rate estimated to about 0.2 cm/year in adult gravid leatherbacks (Price et al. 2004). Although size is not a good predictor of age in sea turtles, the relatively large size of individuals observed in French Guiana during this study may thus be explained by a relatively high proportion of old individuals within this population. This would imply a different population structure which may also cause different dynamics within this population, compared with other sites.

Managing sea turtle populations requires estimates of vital parameters (growth, mortality, fertility, etc.) that are either hard or impossible to obtain by direct sampling. Greater effort should therefore be directed toward deriving empirical models from easy-to-measure parameters to obtain estimates of parameters which are hard-to measure (such as mean body mass of an individual in an age-structured population). Improving estimation of other parameters (related to physiology, morphology or population dynamics) in leatherback turtles should lead to a better understanding of their life history, trophic ecology and population dynamics, which is urgently needed for developing more appropriate conservation strategies to preserve this critically endangered species.

  obtained in 182 turtles (which included the 49 individuals used for model 1) measured throughout the nesting season was related to MedianBodyCirc and SCCL. Body mass measured throughout the nesting season ranged from 275.6 to 567.3 kg (mean ± SD: 389.7 ± 61.9 kg, n = 182) and was significantly related to MedianBodyCirc and SCCL according to the following equation: Body Mass (kg) = -709.146 + 3.391MedianBodyCirc (cm) + 2.664SCCL (cm), F 2,179 = 1054.476, r² = 0.922, P < 0.001, AIC = 1563, [model 2], with SCCL (159.3 ± 6.8 cm, range = [142 -179 cm], n = 182) and MedianBodyCirc (198.9 ± 14.0 cm, range = [170 -237 cm], n = 182) accounting for 28% and 72% of the variance of the model, respectively.

(

  Body mass [kg] = 6.8CCL [cm] -600, estimated from James et al. 2005) than to the slopes previously reported for other nesting Atlantic populations (St Croix Island, Caribbean Sea: Body mass [kg] = 5.208SCCL [cm] + 468.84, Boulon et al. 1996; Atlantic coast of Costa Rica: Body mass [kg] = 5.032SCCL [cm] -417.7,

Figure 1 .

 1 Figure 1. Schematic representation of morphometric measurements collected on gravid leatherback turtles (left side) nesting in French Guiana. The carapace is presented in solid lines, in contrast to the soft tissues that are shown in dashed lines, except for the tail (filled in black) which was considered in the analysis. Numbers in classic and in italic types correspond to straight and curvilinear measurements, respectively. 1 head length, from the distal point of the nose to the tip of occipital bone; 2-3 head height and head circumference at the level of the occipital bone; 4 neck length from the occipital bone to the nape notch of the carapace; 5 neck circumference, measured at the maximum circumference during apnea, 6 anterior carapace width at the level of maximum of maximum width back of the fore-flippers, 7 median carapace width at half of 9 (SCCL), 8 posterior carapace width at the frontal insertion point of the rear-flippers, 9 standard curvilinear carapace length (SCCL) measured on the midline of the shell, from the nape notch of the carapace to the distal point of the peduncle, 10 tail length measured from below of the tail when stretched during ovi-position, 11 peduncle length measured from below of the peduncle from the posterior end of the soft body to the distal point of the peduncle, 12-13-14 anterior-median-posterior body height, respectively, measured at the same levels as 6-7-8 when the turtle was lifted for weighing, 15-16-17 anterior-median-posterior body circumference, respectively, measured at the same levels as 6-7-8.

Figure 2 .

 2 Figure 2. Final general linear model (after stepwise backward analysis) of body mass in gravid leatherback turtles performed on 182 individuals nesting in French Guiana between April 4 and July 27, 2005. Body Mass (kg) = -709.146 + 3.391MedianBodyCirc (cm) + 2.664SCCL (cm), F 2,179 = 1054.476, r² = 0.922, P < 0.001 (model 2 in the text), where SCCL is the standard curvilinear carapace length and MedianBodyCirc is the body circumference at half of SCCL (see Figure 1).

Figure 3 .

 3 Figure 3. Comparison of the estimates of body mass from standard curvilinear carapace

Figure 4 .

 4 Figure 4. Sensibility test of the model estimating body mass from standard curvilinear

Figure 5 .

 5 Figure 5. Relationship between mean body mass and mean standard curvilinear carapace

Table 1 .

 1 Analysis (general linear model) of body mass in gravid leatherback turtles in relation to 17 morphometric variables (see Figure 1, complete model : F 17,32 = 87.869, r² = 0.98, P < 0.001). Analysis was run for 49 individuals sampled once when nesting in French Guiana between April 4 and 29, 2005.

	Independent variable	t-value	P-value	b*

Acknowledgements

We thank G Alberti, M-H Baur, C Blanc, M Bouteille, N Lapompe-Paironne and I Van der Auwera for their assistance in the field, M Girondot for statistical advices and M Enstipp for correcting the English. We are grateful to all Awala-Yalimapo inhabitants for their hospitality and all people involved in the sea turtle monitoring programmes through Kulalasi NGO and the Réserve Naturelle de l'Amana for their logistical contribution in the field. Funding was provided by grants to JYG from Programme Amazonie du CNRS. SF was supported by a studentship from the French Ministry of Research. This study was carried out under CNRS-CEPE institutional license (B67-482 18) and JYG individual license (67-220 and 04-199) and adhered to the legal requirements of the country in which the work was carried out, and all institutional guidelines.

Hays GC, Broderick AC, Glen F, Godley BJ (2002)


Emmanuelle Anceaume

Yann Busnel

Estimer la corrélation à la volée entre flux massifs est possible avec très peu de mémoire

Keywords: Flux de données, métrique de corrélation, algorithme distribué d'approximation probabiliste

L'analyse à la volée de flux massifs potentiellement infinis est fondamental dans un grand nombre d'applications de monitoring pour calculer des statistiques, détecter des tendances ou prédire des comportements déviants. En particulier, détecter la corrélation entre flux distribués semble être un bon indicateur de la présence d'attaques dans un réseau. Dans cet article, nous présentons une nouvelle métrique permettant d'évaluer la corrélation entre des flux distribués. Nous présentons un algorithme étonnamment simple et très peu couteux en espace permettant d'estimer à la volée la corrélation entre des flux. Nous dérivons des bornes sur la qualité de l'évaluation, et validons notre approche sur des traces réelles.

Introduction

Performance of many complex monitoring applications, including Internet monitoring applications, data mining, sensors networks, network intrusion/anomalies detection applications, depend on the detection of correlated events [START_REF] Shuyuan | A covariance analysis model for ddos attack detection[END_REF] . To cope with the complexity and the amount of raw data, current network management tools analyze their input streams in isolation. The point is that, in all these monitoring applications, data streams arrive at nodes in a very high rate and may contain up to several billions of data items per day. Thus computing statistics with traditional methods is unpractical due to constraints on both available processing capacity and memory. Two main approaches exist to monitor in real time massive data streams. The first one consists in regularly sampling the input streams so that only a limited amount of data items is locally kept. Accuracy of this computation fully depends on the volume of data items that has been sampled and their order in the stream. Furthermore, an adversary may easily take advantage of the sampling policy to hide its attacks among data items that are not sampled, or in a way that prevents its "malicious" data items from being correlated. In contrast, the streaming approach consists in scanning each piece of data of the input stream on the fly, and in locally keeping only compact synopses or sketches that contain the most important information about these data, deriving some data streams statistics with guaranteed error bounds without making any assumptions on the order in which data items are received at nodes. On the other hand, very few works have tackled the distributed streaming model, also called the functional monitoring problem [START_REF] Cormode | Algorithms for distributed functional monitoring[END_REF], which combines features of both the streaming model and communication complexity models, where each node receives an input data stream, performs some local computation, and communicates only with a coordinator who wishes to continuously compute or estimate a given function of the union of all the input streams. The challenging issue in this model is for the coordinator to compute the given function by minimizing the number of communicated bits [START_REF] Cormode | Algorithms for distributed functional monitoring[END_REF] In this paper, we go a step further by studying the dispersion matrix of distributed streams. Specifically, we propose a novel correlation metric, called the sketch codeviation, that allows to quantify in real time how observed data items, between distributed and massive streams, change together, and in which proportion. We give upper and lower bounds on the quality of this approximated metric with respect to the codeviation.

We then provide a distributed algorithm that additively approximates the codeviation among n data streams σ 1 , . . . , σ n by using O ((1/ε) log(1/δ) (log N + log m)) bits of space for each of the n nodes, where N is the domain size from which items values are drawn, and m is the largest size of these data streams. We guarantee that for any 0 < δ < 1, the maximal error of our estimation is bounded by εm/N, by sending

O (n(1 + (1/ε) log(m/2) log(1/δ))
) bits among the whole network.

Model and Metrics

Model We consider a set of n nodes S 1 , . . . , S n such that each node S i receives a large sequence σ i of data items or symbols. We assume that streams σ 1 , . . . , σ n do not necessarily have the same size and support (which are unknown). Items arrive regularly and quickly, and due to memory constraints, need to be processed sequentially and in an online manner. Each data item j is drawn from the universe Ω = {1, 2, . . . , N}, where N is very large. Nodes cannot communicate among each other. On the other hand, there exists a specific node, called the coordinator in the following, with which each node may communicate. We assume that communication is instantaneous. A natural approach to study a data stream σ i of length m i is to model it as a fingerprint vector (or item frequency vector) over the universe Ω, given by X i = (x 1 , x 2 , . . . , x N ) where x j represents the number of occurrences of data item j in σ i . Note that 0 ≤ x j ≤ m i .

Codeviation We propose a metric over fingerprint vectors of items, which is inspired from the classical covariance metric in statistics. Such a metric allows us to qualify the dependance or correlation between two quantities by comparing their variations. As will be shown in Section 4, this metric captures shifts in the network-wide traffic behavior when a DDoS attack is active. The codeviation between any two fingerprint vectors X = (x 1 , x 2 , . . . , x N ), and Y = (y 1 , y 2 , . . . , y N ) is the real number denoted cod(X,Y ) defined by cod(X,Y ) = 1

N ∑ i∈Ω (x i -x)(y i -y) = 1 N ∑ i∈Ω x i y i -x y where x = 1 N ∑ i∈Ω x i and y = 1 N ∑ i∈Ω y i .
(1)

Distributed Codeviation Approximation Algorithm

We propose a statistic tool, named the sketch codeviation, which allows to approximate the codeviation between any two data streams using compact synopses or sketches. This paper presents the main results of our study. For space limitation reasons, proofs are presented in the companion paper [START_REF] Anceaume | Deviation estimation between distributed data streams[END_REF].

Définition 3.1 (Sketch codeviation) Let X and Y be any two fingerprint vectors of items, such that X = (x 1 , . . . , x N ) and Y = (y 1 , . . . , y N ). Given a precision parameter k, we define the sketch codeviation between X and Y as

cod k (X,Y ) = min ρ∈P k (Ω) cod X ρ , Y ρ = min ρ∈P k (Ω) 1 N ∑ a∈ρ X ρ (a) Y ρ (a) - 1 N ∑ a∈ρ X ρ (a) 1 N ∑ a∈ρ Y ρ (a)
where ∀a ∈ ρ, X ρ (a) = ∑ i∈a x i , and P k (Ω) is a k-cell partition of Ω, i.e., the set of all the partitions of the set Ω into exactly k nonempty and mutually disjoint sets (or cells). Proposition 3.2 Let X = (x 1 , . . . , x N ), and Y = (y 1 , . . . , y N ) be any two fingerprint vectors. The sketch codeviation is a function of the codeviation. We have cod N (X,Y ) = cod(X,Y ) and

cod k (X,Y ) = cod(X,Y ) + E k (X,Y ) where E k (X,Y ) = min ρ∈P k (Ω) 1 N ∑ a∈ρ ∑ i∈a ∑ j∈a {i}
x i y j .

We have shown in Theorem 4 of the companion paper [START_REF] Anceaume | Deviation estimation between distributed data streams[END_REF] that the sketch codeviation matches exactly the codeviation if k ≥| supp(X) ∩ supp(Y ) | +1 supp(X) supp(Y ) + 1 supp(Y ) supp(X) , where supp(X), respectively supp(Y ), represents the support of distribution X, respectively Y (i.e., the set of items in Ω that have a non null frequency x i = 0, respectively y i = 0, for 1 ≤ i ≤ N), and notation 1 A denotes the indicator function, which is equal to 1 if the set A is not empty and 0 otherwise. We have also characterized the upper bound of the overestimation factor, i.e., the error made with respect to the codeviation, when k is strictly less than this bound.

Estimer la corrélation à la volée entre flux massifs est possible avec très peu de mémoire Theorème 3.3 (Upper bound of E k (X,Y )) Let k ≥ 1 be the precision parameter of the sketch codeviation. For any two fingerprint vectors X ∈ X and Y ∈ Y , let E k be the maximum value of the overestimation factor E k (X,Y ). Then, the following relation holds.

E k = max X∈X ,Y ∈Y E k (X,Y ) =    m X m Y N if k = 1, m X m Y N 1 k -1 N if k > 1.
Distributed approximation algorithm We compute the codeviation between a set of n distributed data streams, so that the number of bits communicated between the n sites and the coordinator is minimized. This amounts for the coordinator to compute an approximation of the codeviation matrix Σ, which is the dispersion matrix of the n data streams. Specifically, let X = {X 1 , X 2 , . . . , X n } be the set of fingerprint vectors X 1 , . . . , X n describing respectively the streams σ 1 , . . . , σ n . We have

Σ = cod(X i , X j ) 1≤i≤n,1≤ j≤n
.

We propose a one-pass algorithm that computes the sketch codeviation between any two large input streams. By definition of the Sketch codeviation metric, we need to generate all the possible k-cell partitions. The number of these partitions follows Stirling numbers of the second kind, which is equal to

S(N, k) = 1 k! ∑ k j=0 (-1) k-j k j j N .
Therefore, S(N, k) grows exponentially with N, which is unreasonable in term of space complexity as N is supposed to be very large. Actually generating t = log(1/δ) random k-cell partitions, where δ is the probability of error of our randomized algorithm, is sufficient to guarantee good overall performance of the sketch codeviation metric [START_REF] Anceaume | Deviation estimation between distributed data streams[END_REF]. Our approximation algorithm uses an elegant data structured inspired by [START_REF] Cormode | An improved data stream summary : the count-min sketch and its applications[END_REF] to generate those t partitions on the fly. The algorithm proceeds in rounds until all the data streams have been read in their entirety. In the following, we denote by σ (r) i the substream of σ i received by S i during the round r, and by d r the number of data items in this substream. During each round, each site S i computes a single sketch C σ i of the received data stream σ i , until receiving d r items (See [START_REF] Anceaume | Deviation estimation between distributed data streams[END_REF] for details on the data structure C σ i ). Then node S i sends C

σ i ← C σ i + C σ (r) i
and updates the sketch codeviation matrix Σ = cod(X i , X j ) 1≤i≤n,1≤ j≤n on every couple of sketches, such that the element in position i, j represents the sketch codeviation between streams σ i and σ j . As the codeviation is symmetric, the codeviation matrix is a symmetric matrix, and thus only the upper-triangle and the diagonal need to be computed. In order to make this algorithm self-adaptive to the unknown lengths of the streams, d r is more or less doubled at each round. Note that during round r, S i regularly computes cod σ (r-1) i , σ (r) i to detect whether significant variations in the stream have occurred before having received d r items. This allows to inform the coordinator as quickly as possible that some attack might be undergoing, by sending its current sketch C ( X i 1 X j 1 -X i X j 1 ) ≤ δ.

using O ((1/ε) log(1/δ) (log N + log m)) bits of space for each n nodes, and O (n log m (1/ε log(1/δ) + n)) bits of space for the coordinator, where m is the maximum size among all the streams, i.e., m = max i∈[n] X i 1 . Moreover, the distributed sketch codeviation algorithm gives an approximation of matrix Σ by sending O (rn(1 + (1/ε) log(m/2) log(1/δ))) bits, where r is the number of the last round.

  i , and starts a new round r + 1. Upon receipt of C σ (r) i from any S i , the coordinator asks all the n -1 other nodes S j to send their own sketch C σ (r) j . Once the coordinator has received all C σ (r) i , the coordinator locally updates the n sketches such as C

  The approximated codeviation matrix Σ returned by the distributed sketch codeviation algorithm satisfies Σ ≥ Σ andP Σ -Σ ≥ ε N max i, j∈[n]

This work was partially funded by the French ANR project SocioPlug (ANR-13-INFR-0003), and by the DeSceNt project granted by the Labex CominLabs excellence laboratory (ANR-10-LABX-07-01).

Performance Evaluation

We have implemented the distributed sketch codeviation algorithm and have conducted a series of experiments on different types of streams and for different parameters settings. We have fed our algorithm with both real-world data sets and synthetic traces. Due to space constraints, we only describe one of these experiments, which is mainly representative of our algorithm's accuracy. The interested reader is invited to consult [START_REF] Anceaume | Deviation estimation between distributed data streams[END_REF] for a complete overview of our evaluation, which clearly shows that our distributed algorithm is capable of efficiently and accurately quantifying how observed data streams change together and in which proportion whatever the shape of the input streams. Figure 1 shows how efficiently our approximation distributed algorithm detects different scenarii of attacks in real time. Specifically, we compute at each round of the distributed protocol, the distance between the codeviance matrix Σ constructed from the streams under investigation and the mean of covariance matrices E(Σ N ). This distance has been proposed in [START_REF] Shuyuan | A covariance analysis model for ddos attack detection[END_REF]. Specifically, given two square matrices M and M of size n, consider the distance

We evaluate at each round r, the variable d r defined by

Based on this distance, we have fed our distributed algorithm with different patterns of traffic. In Figure 1, distance is depicted, as a function of time, when the codeviance is exactly computed and when it is estimated with our distributed algorithm with different values of k. What can be seen is that, albeit there are up to two orders of magnitude between the exact codeviance matrix and the estimated one, the shape of the codeviance variations are for most of them similar. Different attack scenarii are simulated. From round 0 to 10, all the 10 synthetic traces follow the same nominal distribution (e.g., a Poisson distribution). Then from round 10 to 20 a targeted attack is launched by flooding a single node (i.e., one among the ten traces follows a Zipfian distribution with α = 4). This gives rise to a drastic and abrupt increase of the distance. As can be shown, the estimated covariance exactly follows the exact one, which is a very good result. Then after coming back to a "normal" traffic, half of the traces are replaced by Zipfian ones (from round 30 to 40), representing a flooding attack toward a group of nodes. As for the previous attack, the covariance matrices are highly impacted by this attack. From round 50 to 60, traces follow a Zipfian distribution with α = 1 which represents unbalanced network traffic but should not be completely representative of attacks. On the other hand, in the fourth and fifth attack periods, all the traces follow a Zipfian distribution with different values of α ≥ 2, which clearly shows a flooding attack toward a group of targeted nodes. The main lesson drawn from these results is the good performance of our distributed algorithm whatever the pattern of the attack.