Hugues Sert
email: hugues.sert@isen.fr

Annemarie Kökösy
email: nemarie.kokosy@isen.fr

Wilfrid Perruquetti
email: wil-frid.perruquetti@inria.fr

A single landmark based localization algorithm for non-holonomic mobile robots

This paper proposes a single landmark based localization algorithm for non-holonomic mobile robots. In the case of a unicycle robot model, the localization problem is equivalent to the system observability. Based on this observation, the proposed localization method consists in finding a vector function which depends on the measurement vector and its derivatives. In order to compute estimates of the successive derivatives of the measurement vector, we will use a numerical differentiation method. When the robot is able to only measure the relative angle between itself and the landmark in 2D case, the algorithm estimates the posture of the robot, under the hypothesis that control inputs are known. But, sometimes it is also useful to be able to estimate the control input (for example when the robot slips). This is possible with the proposed algorithm by using a landmark in dimension three. The simulation results will be given in order to show the effectiveness of the proposed algorithm. Moreover, these results are compared with those obtained by an Extended Kalman Filter in order to underline the advantages of the new algorithm.

I. INTRODUCTION

Localization is one of the most important issues for mobile entity autonomous navigation [START_REF] Borenstein | Where am I? Sensors and methods for mobile robot positioning[END_REF]. Indeed it is impossible to look for an intelligent mobile robot navigation strategy if the robot has no self-localization capabilities. This localization problem has focused researchers' efforts for a long time. It admits two sub-classes:

1) absolute localization: the collected data allow to localize the robot in the global environment (using for example GPS), 2) relative localization: the collected data allow to localize the robot with respect to the current situation (using for example odometers or other proprioceptive sensors). In an indoor environment it is almost very difficult to get absolute localization because it is impossible to use GPS (satellite signals are no more efficient indoors). Odometers or similar proprioceptive sensors have an important drawback because of drift issues [START_REF] Borenstein | Control and kinematic design of multi-degree-of freedom mobile robots with compliant linkage[END_REF]. Thus, in order to solve the localization problem, the researchers have developed landmarks based localization methods: landmarks are points of known position in the environment which can be "seen" by the robot. Then the localization problem can be formulated as an observability problem (in the sense of the automatic control community): see [START_REF] Conticelli | Observability and nonlinear observers for mobile robot localization[END_REF], [START_REF] Bicchi | On the observability of mobile vehicles localization[END_REF] and [START_REF] Martinelli | Observability analysis for mobile robot localization[END_REF] for one or more robots. One main drawback of such obtained results is that the system is observable only for more than three landmarks. In some environment it could be difficult to find so much landmarks (see for example [START_REF] Bais | Single landmark based selflocalization of mobile robots[END_REF]), so other authors have developed single landmark based methods which require to compute a scale factor. [START_REF] Jang | Metric localization using a single artificial landmark for indoor mobile robots[END_REF] and [START_REF] Lemaire | A practical 3D bearing-only SLAM algorithm[END_REF] use the shapes of the landmarks which require to use artificial landmarks of known position and shape. If the shapes of the landmarks are not known, it is possible to use information about the robot movement in order to estimate the scale factor (see [START_REF] Bais | Single landmark based selflocalization of mobile robots[END_REF]). [START_REF] Dao | Visual navigation for indoor mobile robots using a single camera[END_REF] uses the odometers to estimate the scale factor. The greatest drawback of this method is the robustness. In this paper a new localization method is proposed which uses only the position of the landmarks and the relative angle between the landmarks and the robot to estimate, the linear speed, the angular speed and the position of the robot with only one landmark.

II. PROBLEM FORMULATION OF A NEW SINGLE LANDMARK BASED LOCALIZATION

A. Notations

In this paper the following notations are used (see Fig. 1):

• P = [x, y, θ] is the robot posture which contains the coordinates [x, y] of the robot and its orientation θ,

• [x Ai , y Ai , z Ai ] is the coordinates of the landmark,

• the relative coordinates robot-landmarks:

x r = x Ai -x, y r = y Ai -y, (1)
• α and β the relative angles robot-landmarks A i ,

• the distance between the robot and the landmark:

-d RAi = x 2 r + y 2 r + z 2 Ai in space, -d RAisol = x 2
r + y 2 r in plane, • for a physical variable v, v m will denote its measured quantity (usually v m = v + η v where η v is an additive noise) and v f will denote a filtered value of the measured quantity v m which should be quite similar to v in the ideal case.

• x will denote the estimate value of variable x.

B. Problem

Let

Ẋ = F (X, V ) Z = H(X) (2)
t β j j j t 6 - 1 θ α arctan y r x r t A i : Landmark A i Top View Three Dimensional View x y A i   x Ai y Ai z Ai   A i   x Ai y Ai z Ai   d RAisol d RAi
Fig. 1. Robot and landmark notation for the localization be a state model system of a mobile robot, where X ∈ R n is the state vector containing the mobile robot posture P = [x, y, θ], V ∈ R m is the control input vector and Z ∈ R p is the measurement vector. Since localization consists in finding the posture of the robot (which is a part of the state vector) from the measured output, it is clear that such a problem is closely linked to the observability problem (in fact it is a partial observability problem). Such a problem has two main points of view: a differential geometric one and an algebraic one. Let us recall here, one of the most important results obtained by Michel Fliess about observability in an algebraic framework:

Theorem 1 (see [START_REF] Diop | Nonlinear observability, identifiability, and persistent trajectories[END_REF] and [START_REF]On nonlinear observability[END_REF]): A system variable x ∈ F is said to be observable if, and only if, it is algebraic over f u, y . An input-output system F/f is said to be observable if, and only if, the extension F/f u, y is algebraic. Where the following definitions are used: for F which is a differential field, the differential field extension L/F is given by two differential fields F, L, such that the derivation of F ⊂ L is the restriction to F of the derivation of L. An element of L is said to be differentially algebraic over F if, and only if, it satisfies an algebraic differential equation with coefficients in F . F S , where S is a subset of L, the differential subfield of L generated by F and S.

Thus the above mentioned result shows that a system variable x ∈ F is said to be observable, if and only if there is an algebraic equation linking x, the outputs, the inputs and a finite number of the time derivatives of the inputs and outputs. Let's use this algebraic criteria for our localization problem.

Theorem 2: A mobile robot with state dynamics (2) is said to be localizable if, and only if, there is an algebraic equation linking posture P , the measured output Z, input V and a finite number of the time derivatives of Z and V .

Thus, based on this result, our proposed method of localization consists in finding the vector function K(Z, Ż, Z, ..., Z (q) ) such that: P = K(Z, Ż, Z, ..., Z (q) ).

(

Where Z (q) is the q-derivative of Z.

For our case study, a unicyle robot, whose posture is exactly state X, thus its localizability is equivalent to the system observability (see [START_REF] Diop | Nonlinear observability, identifiability, and persistent trajectories[END_REF] and [START_REF] Leiva | A necessary algebraic condition for controllability and observability of linear time-varying systems[END_REF] for additional results on the observability of non linear systems). The knowledge of the formal expression of K and an estimation of the successive derivatives of Z, will lead to the reconstruction of posture P for any time t: this is the point of view used in this paper which is closely linked to the new algorithms developed within the ALIEN -INRIA project for numerical differentiation estimation (see [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF]). Let us emphasize that these methods, which are algebraic and non-asymptotic, exhibit good robustness properties with respect to corrupting noise, without the need of knowing their statistical properties.

To sum up, the localization (reconstruction of P = X (for our case study)) is then obtained through three steps: 1) get a formal expression of K (see theorems 3 and 4), 2) compute the estimates of the successive derivatives of Z (see sub-section II-E) (for this case study, there is no need to estimate the derivatives of V ), 3) in K, substitute the derivatives by their estimates.

C. Robot State Model

The mobile robot considered is of unicycle type, with two driving wheels mounted on the same axis and independently controlled by two actuators (DC motors). The robot is fully described by a three dimensional vector of generalized coordinates X constituted by the coordinates [x, y] of the midpoint between the two driving wheels and by the orientation angle θ with respect to a fixed frame. This non-holonomic robot is assumed to observe landmarks in two or three dimensions (both cases will be dealt with). The kinematic model of this type of robot is given by:

Ẋ = F (X, V ) =   u cos (θ) u sin (θ) ω   , (4)
where X = x y θ T is the posture of the robot and V = u ω T is the control input (linear velocity, angular velocity).

D. Measurement Model

We are going to distinguish the 2D and 3D cases. The measurement model

• in the 2D case, is given by equations ( 5) and ( 6):

Z = H(X) = α m = α + η α (5) α = arctan y r x r -θ (6)
where x r and y r are the relatives positions between the landmarks and the robot defined by ( 1) and η α is a measurement noise.

• in the 3D case, is given by equations ( 7)-( 9):

Z = H(X) = α m = α + η α β m = β + η β (7) α = arctan y r x r -θ (8)
β = arctan z Ai x 2 Ai + y 2 Ai ( 9
)
where η α and η β are additive noise measurements.

E. Estimation of the successive time derivatives of Z

This algebraic setting for numerical differentiation of noisy signals was introduced in [START_REF] Fliess | An algebraic framework for linear identification[END_REF] and analyzed in [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF]. Consider a signal v m = v + η v . We want to estimate the derivative of v.

1) Continuous version of the derivative:

The continuous version of the n th time derivative estimate of the variable v is given by: [START_REF] Diop | Nonlinear observability, identifiability, and persistent trajectories[END_REF] where v m is the measured quantity of v (see notations) and

v (n) f (T t; κ; µ; N ) = 1 0 g(τ, κ, µ, N )v m (t -τ )dτ
g(τ ) = q l=0 λ l h κ+q-l,µ+l (τ ), (κ; µ) ∈ N, q = N -n (11)
with λ l = (-1) q-l p + q -l p p + q + 1 l , l = 0, ..., q

) where q = N -n and p = n + κ and

h κ,µ (τ ) = (-1) n γ κ,µ,n T n rect(τ ) d n dτ n ω κ,µ (τ ) (13) rect(τ ) = 1 if τ ∈ [0; 1] 0 ortherwise (14) ω κ,µ (t) = t κ+n (1 -t) µ+n (15) γ κ,µ,n = (µ + κ + 2n + 1)! (µ + n)!(κ + n)! (16)
See [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] for details about the choice of the different parameters involved in this differentiation estimation method. Remark 1: Let us note that this formula is still valid for n = 0 and thus gives a filtered estimate of the measured variable. This will be used to obtain the following filtered quantities:

α f , β f , u f , ω f , θ f , . . ..

2) Discrete version:

The n th order time derivative estimate in the discrete case is obtained as the output of the finite impulse response filter (FIR):

v (n) (lT s ; κ; µ; N ) ≈ M j=0 W j g j v m,l-j (17)
where v m,i = v m (iT s ), T s is the sampling period, M is the number of coefficients of the filter and c j = W j g j , j = 0, ..., M -1 is its impulse response. Coefficients W j correspond to the chosen method of integration of the signal between 0 and 1. Here, for the trapezoidal method, the coefficients are given by:

W 0 = W M = 1 2M W j = 1 M , j = 1, ..., M -1 (18)
Coefficients g j are such as g j = g(jT s /T ), j = 0, ..., M , with T = M T s the first moment of estimation.

III. MAIN RESULTS

A. New 2D and 3D robot localization algorithms 1) 2D case: A new single landmark based localization algorithm:

Hypothesis 1: The following data are supposed to be known:

• [x Ai , y Ai , z Ai ], the landmark position, • u, the linear speed of the robot,

• ω, the angular speed of the robot. The following data are supposed to be measured:

• θ m , measured with a compass (a noisy measurement),

• α m the robot-landmark relative angle (a noisy measurement). Theorem 3: Let us consider the robot state model, ( 4), the measurement model, ( 5) and ( 6) and the relative coordinates between the robot and the landmark x r and y r given by (1). Under hypothesis 1, the relative coordinates of the robot, X r = [x r , y r ] T are estimated by:

Xr = xr ŷr =     u f sin (α f ) cos (α f + θ f ) ( αf + ω f ) u f sin (α f ) sin (α f + θ f ) ( αf + ω f )     (19)
where u f , ω f , α f , αf and θ f are the filtered quantities (see remark 1).

Remark 2: This algorithm is suitable only when the nonholonomic constraint is satisfied (no slipping, no skidding). More over the following condition should be in force α+ω = 0 : for example when the landmarks is far away from the robot α ≃ 0 and the robot is not spinning (in line) ω ≃ 0 or when the robot is at rest. Proof: By taking the time derivative of (1) combined with (4), the following equation is obtained: ẋr = -ẋ = -u cos (θ) and ẏr = -ẏ = -u sin (θ). (20) By using equation [START_REF] Bais | Single landmark based selflocalization of mobile robots[END_REF], after the filtering process (i.e. removing the noise effect η):

sin (α f + θ)x r -cos (α f + θ)ŷ r = 0 (21)
The time derivative of this equation combined with (20), leads to:

( αf + ω) cos (α f + θ)x r -sin (α f + θ)u cos (θ) = -( αf + ω) sin (α f + θ)ŷ r -cos (α f + θ)u sin (θ) (22)
( αf + ω) [cos (α f + θ)x r + sin (α f + θ)ŷ r ] = sin (α f + θ)u cos (θ) -cos (α f + θ)u sin (θ) (23)
So: 21) and ( 24) can be rewritten into a matrix form with xr and ŷr as unknown data:

( αf + ω) [cos (α f + θ)x r + sin (α f + θ)ŷ r ] = u sin (α f ) (24) Now (
sin (α f + θ) -cos (α f + θ) ( αf + ω) cos (α f + θ) ( αf + ω) sin (α f + θ) xr ŷr = 0 u sin (α f )
Solving this equation it follows that:

Xr = xr ŷr =     u sin (α f ) cos (α f + θ) ( αf + ω) u sin (α f ) sin (α f + θ) ( αf + ω)     .
Then u, ω and θ are also filtered in order to increase the robustness of the estimation process with respect to the noise which could affect the control inputs and measurements. Remark 3: We can remark in hypothesis 1 that, in order to estimate Xr by using a 2D landmark, it is necessary to know the linear and angular speed of the robot. If this hypothesis is too restrictive, it can be relaxed by using a three-dimensional landmark. In this case, it is possible to estimate the linear and the angular speed of the robot.

2) 3D case: A new single landmark based localization algorithm without the knowledge of the linear and angular speed: Contrary to hypothesis 1 in the 2D case, when a 3D landmark is used, it is not necessary to know the linear and angular speeds of the robot.

Hypothesis 2: The following features are assumed to hold:

• landmark position [x Ai , y Ai , z Ai ] is known, • relative angles α m and β m are measured (noisy measurement),

• θ m , the robot orientation is measured with a compass (a noisy measurement). Theorem 4: Let us consider the robot state model, ( 4), the measurement model, ( 7)-( 9) and the relative coordinates between the robot and the landmark x r and y r given by equation [START_REF] Borenstein | Where am I? Sensors and methods for mobile robot positioning[END_REF]. Under hypothesis 2 the relative coordinates of the robot, X r = [x r , y r ] T and the velocities (linear u and angular ω) are estimated by:

û = z Ai βf sin 2 β f cos α f (25) ω = 2 tan (α f ) sin (2β f ) βf -αf (26)
Xr = xr ŷr =     û sin (α f ) cos (α f + θ f ) ( αf + ω) û sin (α f ) sin (α f + θ f ) ( αf + ω)     (27)
where α f , αf , β f , βf and θ f are the filtered quantities (see remark 1).

Proof: The only part to be proved is the estimation of the velocities (for the proof of equation ( 27) see the 2D case). Estimation of the linear speed u Using Fig. 1:

x r y r = d RAisol cos(α + θ) sin(α + θ)
Using the filtering process to eliminate the noise measurement, this equation becomes:

xr ŷr = d RAisol cos(α f + θ f ) sin(α f + θ f )
so:

d RAisol = xr cos(α f + θ f ) + ŷr sin(α f + θ f ) (28)
By differentiating this equation and by using equation ( 20), the following equation is obtained:

ḋRAisol = -x r ( αf + ω) sin(α f + θ f ) -û cos θ f cos(α f + θ f ) +ŷ r ( αf + ω f ) cos(α f + θ f ) -û sin θ f sin(α f + θ f ) , which combined with equation (21) leads to: ḋRAisol = -û cos(α f ). (29)
Using equation ( 9):

tan β f = z Ai d RAisol ,
which can be rewritten in the following expression:

d RAisol sin β f -z Ai cos β f = 0. ( 30
)
The following equation can be obtained by differentiating this last one:

ḋRAisol sin β f + βf d RAisol cos β f + z Ai βf sin β f = 0.
In this relation, by replacing ḋRAisol by equation ( 29):

-û cos α f sin β f + βf (d RAisol cos β f + z Ai sin β f ) = 0.
(31) And by using Fig. 1:

d RAisol z Ai = d RAi cos β f sin β f so: d RAi = d RAisol cos β f + d RAisol sin β f ( 32
)
By replacing this equation in (31) and by knowing that z Ai = d RAi sin β f , (Fig. 1):

û = βf d RAi cos α f sin β f = βf z Ai cos α f sin 2 β f (33)
Estimation of the angular speed ω By using the expression of d RAisol found in (28) in ( 24):

( αf + ω)d RAisol = û sin α f
So by using equation ( 9):

ω = û sin α f tan β f z Ai -αf
By replacing û found in equation ( 33), the result for ω is demonstrated:

ω = βf tan α f sin β f cos β f -αf = 2 βf tan α f sin(2β f ) -αf

B. Experimental results and comparisons with an extended Kalman filter

In order to show the effectiveness of the here proposed method, the obtained results from theorem 4 were implemented on Matlab (hereafter called ALIEN algorithm). Results from theorem 3 are not implement as they are subparts of theorem 4. Results obtained using ALIEN algorithm are compared with those obtained by an EKF (Extended Kalman Filter) for two parameters setting. In the first one, the statistical noise characteristics (R matrix) are known by the EKF and in the second one they are unknown (the covariance noise matrix (R matrix) is so set to high values). In both cases the new algorithm has no information about the noise characteristics, and EKF is initialized at 10cm of the true initial position. Due to paper limitation the EKF is not developed here (for more details see [START_REF] Sorenson | Kalman filtering: theory and application[END_REF]).

Tab. I summaries the required inputs (informations) for each algorithm (ALIEN and EKF). measurements of α and β, the landmarks position and the orientation of the robot to get an estimation of the velocities and of the position of the robot contrary to the EKF which needs further noise characteristics (matrix R), and an initialization (P 0 ). Furthermore, EKF needs the velocities input contrary to the ALIEN algorithm which does not need this information, as it is able to estimate it. Moreover, the ALIEN algorithm only uses information about a single landmark to get an estimation of the robot velocities and position contrary to the EKF which needs at least three landmarks to get an estimation of the position of the robot. In the case of more than one landmarks, the ALIEN algorithm uses a mathematical means to fuse the different estimations obtained by using each landmark separately. 1) Comparative results: The algorithms are tested for 1, 5, 10, 50 and 100 landmarks. For each case, the experiment is repeated 50 times and an uniformly distributed noise on the interval [-0.5; 0.5] is added to the angular measurements. The initial covariance matrix (P 0 ) for the EKF is set at 0.1m on the diagonal for the position and 0rad on the diagonal for the orientation and 0 elsewhere. a) First experimentation: EKF knows the true noise characteristics: Tab. II shows the results obtained by both algorithms. For each algorithm the mean error and the variance of error are given. We can notice that there are no significant differences between the two algorithms in terms of mean estimation error and variance of estimation error, but the ALIEN algorithm computing time (t ALIEN ) is really competitive with respect to the EKF one (t EKF ) when the number of landmarks is below 4 or increases a lot. Fig. 3 shows the results obtained for one run and for one landmark.

b) Second experimentation: EKF does not know the noise characteristics: Tab. III shows the results obtained by both algorithms. For each algorithm the mean estimation error and the variance estimation error are given. EKF gives bad estimation results due to a bad initialisation which could not be compensated by the measurements because the algorithm has no knowledge about the noise characteristics. The ALIEN algorithm does not have this drawback because it does not need the noise characteristics in the estimation process.

IV. CONCLUSION

This paper has proposed a new landmark based localization algorithm for a unicycle mobile robot. One of the most important advantage of this algorithm is that it is able to localize the robot with only one landmark. If the robot can only measure the relative angle between itself and the landmark in a 2D case, the proposed solution allows to localize the robot with respect to the landmark. In this case the control input (velocities: linear and angular ones) have to be known. Due to the drift problem in robotic navigation it can be useful to also estimate the linear and angular speed of the robot. This paper has proposed a solution to this problem by using only the relative angles between the robot and a single landmark in the 3D case. The theoretical development is implemented in Matlab in order to show the effectiveness of the proposed solution. Moreover, these results have been compared with those given by an Extended Kalman Filter, which is a reference in the mobile robotics community. A number of conclusions can be drawn from these experiments. First of all the more landmarks there are the more the ratio between the computing time necessary for the localization using the new algorithm and that using the EKF becomes favorable to the proposed algorithms. The new proposed algorithm requires no knowledge of velocities (linear velocity and angular velocity), unlike the EKF which needs to know these inputs. Nevertheless, it is possible to extend the EKF to this case, but it requires a number of new theoretical developments, because there is no model of evolution for the velocities. Moreover, for similar results in terms of error and variance, the paper proposes a new algorithm which is simpler to implement and more efficient in terms of computing time than the EKF one. However, the advantage of the EKF is its ability to give a confidence interval estimation. It is not yet possible for the proposed algorithm to give such a confidence interval, but it will the main subject of our future work. Concerning the initialization of the algorithm, it is not possible to conclude, because the EKF can be initialized with a least squares estimator using fifty measurements, in the same way as the ALIEN algorithm needs fifty measurements to give a first estimate of the position. Another great advantage of the new algorithm is the lack of statistical assumptions on noise measurements, which makes it more robust with respect to any type of sensors. The future works concerns the study of the placement of the landmark and the choice of the landmarks in function of the current situation.

  the control input NO YES Needs to know the noise characteristics NO YES Needs to know the orientation of the robot YES NO Needs to be initialized NO YES

Fig. 2 Fig. 2 .

 22 Fig.2shows the inputs and outputs for each algorithms. One can see that the ALIEN algorithm only needs the

Fig. 3 .

 3 Fig. 3. Results for one run and for one landmark

TABLE I COMPARISON

 I OF TWO LOCALIZATION ALGORITHMS HYPOTHESIS

This work was partially supported by ALIEN INRIA -LNE, Mission Insertion Thales and the INTERREG IV A 2 Mers Seas Zeen Cross-border Cooperation Program 2007-2013 under SYSIASS 6-20 project.


C Fiter

L Hetel

W Perruquetti
email: wilfrid.perruquetti@ec-lille.fr

J.-P Richard

State Dependent Sampling: an LMI Based Mapping Approach ⋆,⋆⋆

Keywords: networked/embedded control systems, state dependent sampling, self-triggered control, linear matrix inequality, convex polytope

The uprising use of embedded systems and Networked Control Systems (NCS) requires reductions of the use of processor and network loads. In this work, we present a state dependent sampling control that maximizes the sampling intervals of state feedback control. We consider linear time invariant systems and guarantee the exponential stability of the system origin for a chosen decay rate α. The proof of the α-stability is based on a quadratic Lyapunov function which is computed, thanks to LMIs, so as to optimize some performance criterion on the sampling intervals. A mapping of the state space is then designed offline: it computes for each state of the state space the maximum allowable sampling interval, which makes it possible to reduce the number of actuations during the real-time control of the system.

INTRODUCTION

In the last decades, a large attention has been given to embedded and Networked Control Systems [START_REF] Zhang | Stability of networked control systems[END_REF]). Such systems present several advantages such as: reduced system wiring, plug and play devices, increased system agility, and ease of system maintenance. However, from the control theory point of view, they bring up new challenges: these systems are often required to share a limited number of computational and transmission resources. In practice, this often leads to fluctuations of the sampling interval, which may have a destabilizing effect if it is not properly taken into account.

Several studies about robust stability with respect to sampling period variations have been made (see [START_REF] Fridman | A refined input delay approach to sampleddata control[END_REF], [START_REF] Seuret | Stability analysis for sampled-data systems with a time-varying period[END_REF], [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF], [START_REF] Fridman | Robust sampleddata stabilization of linear systems: An input delay approach[END_REF], and [START_REF] Cloosterman | Controller synthesis for networked control systems[END_REF]). Also, intensive research has been conducted to adapt dynamically the sampling in order to ensure the desired control performances. Two main approaches exist in the literature: Event-triggered control [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], [START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF], [START_REF] Lunze | A state-feedback approach to event-based control[END_REF]), in which sensors are equiped with special intelligence so that information is sent to the controller only when special events occur (i.e. crossing a frontier of the state space, or a level of a Lyapunov function). However, the main drawback of this approach is that it generally requires dedicated hardware to continuously monitor the plant state and check the defined stability conditions. Self-triggered control [START_REF] Velasco | The self triggered task model for real-time control systems[END_REF], [START_REF] Mazo-Jr | An iss selftriggered implementation of linear controllers[END_REF], [START_REF] Anta | To sample or not to sample: selftriggered control for nonlinear systems[END_REF], [START_REF] Wang | Self-triggering under stateindependent disturbances[END_REF]), in which at each sampling instant one computes a lower bound estimation of the next largest admissible sampling interval, so as to emulate event-triggered control without resorting to extra hardware. However, in these works, the computations for the next sampling times are made online. Moreover, most of the works found in the literature require the use of a Lyapunov function, but there is no method to compute such a Lyapunov function guaranteeing stability while optimizing a performance criterion on the sampling intervals.

In this paper, we design offline a state dependent sampling function maximizing the sampling intervals under some Lyapunov exponential stability conditions. The approach is based on a mapping of the state space defining the sampling intervals. Moreover, we provide a formal method based on LMIs to compute the adequate Lyapunov function, in order to maximize the lower bound of the sampling function.

The paper is organized as follows. First, we introduce the studied problem in Section 2. Then, Sections 3 and 4 describe the proposed method and the guaranteed performances. Finally, some simulation results are shown in Section 5 before concluding in Section 6. The proof of the first theorem is given in Appendix 7 along with two lemmas.

Notations: Throughout the paper, the superscript 'T ' stands for matrix transposition. M n (R) is the set for all n × n matrices, and the notation P ≻ 0 (resp. P 0) for a symmetric matrix P ∈ M n (R) means that P is definite positive (resp. semi-definite positive). The set of eigenvalues of a matrix M ∈ M n (R) will be written eig(M ). We also denote by ⌊x⌋ the floor of x (i.e. the largest integer n not greater than x: x -1 < n ≤ x). Eventually, the notations . and . ∞ will stand for the Euclidean and the infinity norm respectively. We recall that for a bounded function f : R p → R q , f ∞ = Sup x∈R p f (x) .

PROBLEM STATEMENT

Consider the linear time invariant (LTI) system

ẋ(t) = Ax(t) + Bu(t), ∀t ∈ R + x(t) = x 0 , ∀t ≤ 0, (1)
where x : R → R n and u : R → R m represent the system state and the control function, and the matrices A and B are constant and of appropriate dimensions. The control function is defined as a piecewise constant state feedback

u(t) = -Kx(t k ), ∀t ∈ [t k ; t k+1 ), (2) where 0 = t 0 < t 1 < • • • < t k < • • • are the sampling instants satisfying lim k→∞ t k = ∞ and defined by t k+1 = t k + τ (x(t k )), ∀k ∈ N,
(3) with a state dependent sampling function τ : R n → R + .

The feedback control matrix K is supposed to be fixed such that the continuous state feedback u(t) = -Kx(t) stabilizes the ideal control loop system (A -BK is Hurwitz).

The objective of this work is to study how to update the given control law as few times as possible, while insuring the exponential stability of the system origin for a chosen decay rate α. At each sampling instant, we want to determine for how long it is possible to let the control input unchanged, without additional sampling. A first, literal problem formulation can be expressed as:

General Problem: Given the LTI system (1) and the linear state feedback control (2), find the state dependent sampling function τ that maximizes the sampling intervals while insuring the exponential stability of the system origin for a chosen decay rate α.

Before computing the state dependent sampling function, we need to decide how to check the system exponential stability. In this respect, the Lyapunov stability theory has proved to be a very useful tool. In this paper, in order to keep things simple and easy to read, we will work exclusively with quadratic Lyapunov functions. We will use the following well known property: Proposition 1. Let V : R n → R + be a quadratic Lyapunov candidate function satisfying V (x) = x T P x, ∀x ∈ R n , with P = P T ≻ 0. If the condition V (x) + 2αV (x) ≤ 0 (4) is satisfied for all trajectories of (1), for a given scalar α > 0, then the system origin is globally α-stable (i.e. there exists a scalar β such that the trajectories satisfy x(t) ≤ βe -αt x 0 for any initial condition x 0 ).

Throughout this work, we will focus on solving two main problems. The first one concerns the design of the sampling function and is formulated as:

Problem 1: Given system (1), the state feedback control (2), and a Lyapunov function V (x(t)), find the state dependent sampling function τ satisfying the stability condition (4) from Proposition 1 that maximizes the sampling time τ (x) for all x ∈ R n .

One can see in that formulation that the Lyapunov function is supposed to be given, which makes us wonder if there is a clever way to choose it. Since the objective is to sample as few times as possible, one will also want to make sure the minimal sampling interval is as big as possible by solving the following problem:

Problem 2: Given system (1) and the state feedback control (2), find a Lyapunov function V (x(t)) such that there exists a sampling function τ satisfying the stability condition (4) from Proposition 1 and maximizing the minimal sampling time τ = inf x∈R n τ (x).

A GENERIC STABILITY PROPERTY

The goal of this section is to provide checkable stability conditions from Proposition 1. For that purpose, we prove the following Lemma that will be used as a stability condition basis throughout the whole here reported work: Lemma 2. Given a scalar α > 0, if there exist a matrix

P = P T ≻ 0 ∈ M n (R) and a bounded function τ : R n → R + such that for all x ∈ R n and σ ∈ [0; τ (x)]: x T Φ P,α (σ)x ≤ 0, (5) with
Φ P,α (σ) = Λ(σ) I T A T P + P A + 2αP -P BK -K T B T P 0 Λ(σ) I (6) and Λ(σ) = I + σ 0 e sA ds(A -BK), (7)
then the origin of system (1) ( 2) is globally α-stable for any time varying sampling function τ : R + × R n → R + that defines sampling instant sequences by the sampling law

t k+1 = t k + τ (t k , x(t k )), k ∈ N and which satisfies 0 < τ (t, x) ≤ τ (x) for all t ∈ R + and x ∈ R n .
Proof: Let α > 0 be given. Let P = P T ≻ 0 ∈ M n (R) and V (x) = x T P x the associated quadratic function. For the studied system, the stability condition (4) from Proposition 1 can be written as: for all k ∈ N, for all t ∈ [t k ; t k+1 ),

x(t) x(t k ) T A T P + P A + 2αP -P BK -K T B T P 0 x(t) x(t k ) ≤ 0.
Let us then take a bounded function τ : R n → R + and a sampling function τ : R + ×R n → R + that defines sampling instant sequences from the sampling law

t k+1 = t k + τ (t k , x(t k )), k ∈ N and which satisfies 0 < τ (t, x) ≤ τ (x)
for all t ∈ R + and x ∈ R n . For a given trajectory of (1), k ∈ N, and for t ∈ [t k ; t k+1 ), when using the notations

x = x(t k ) and σ = t -t k ≤ τ (t k , x(t k )) ≤ τ (x(t k )), one can write x(t) = Λ(σ)
x, with Λ(σ) defined in (7), so that the stability conditions can be rewritten as: for all x ∈ R n , for all σ ∈ [0; τ (x)], x T Φ P,α (σ)x ≤ 0, with Φ P,α (σ) defined in ( 6). This ends the proof. 2

Remark 1: Since the LTI system (1) is supposed to be asymptotically stable with u(t) = -Kx(t), one can show that there always exist a couple of parameters α, P satisfying Φ P,α (0) = (A -BK) T P + P (A -BK) + 2αP ≺ 0 and that, for such parameters, we can find sampling functions τ satisfying the stability conditions of Lemma 2 that are lower-bounded by a strictly positive scalar, hence avoiding any Zeno phenomenon issue.

Remark 2: This lemma also says that if a state dependent function τ : R n → R + satisfies the conditions from Lemma 2, with given P and α, then any time varying sampling law τ k = t k+1t k satisfying at each sampling instant 0 < τ k ≤ τ (x(t k )) will make the system origin globally α-stable. In particular, if τ is a constant function, τ (x) = τ * , ∀x ∈ R n , the system origin will be globally α-stable for any varying sampling interval bounded by τ * .

Remark 3: For any given state x = 0, the stability conditions from Lemma 2 are the same for any state y = λx, λ ∈ R * . Therefore it is sufficient to work with homogeneous state dependent sampling functions of degree 0 (i.e. satisfying τ (λx) = τ (x) for all x ∈ R n , λ ∈ R * ) and to check Lemma 2 stability conditions on the n-dimensional sphere when attempting to solve Problem 1.

Lemma 2 gives some preliminary stability conditions for a state feedback control system with a state dependent sampling. However, one can see that there is an infinite number of inequalities to check because of both temporal and spatial dependencies in the stability conditions.

A NUMERICAL METHOD TO DERIVE A FINITE NUMBER OF STABILITY CONDITIONS

Here a two step tractable methodology is proposed to derive a finite number of stability conditions from Lemma 2:

Convex embedding according to time: The matrix function Φ P,α is replaced by a finite number of constant matrices whose convex hull embed this matrix: for this, a Taylor expansion of Φ P,α is used. Space discretization: Finally, the state space is divided into conic regions in order to design a state dependent sampling period such that Lemma 2 holds.

A finite number of LMI stability conditions will then be derived so to compute the Lyapunov function V (x) =

x T P x solving Problem 2 and build offline (once for all) the associated state dependent sampling function τ solving Problem 1.

Convex embedding -Technical result

In this part, the objective is to obtain a finite number of sufficient conditions to satisfy x T Φ P,α (σ)x ≤ 0, ∀σ ∈ [0; σ] for given state x, scalar σ and parameters P , α and τ in order to get rid of the time dependency of the stability conditions in Lemma 2.

The idea behind the convex embedding is to use the knowledge we have of the system to predict the evolution of the state in order to design a convex polytope around the function Φ P,α (.) and derive a finite number of stability conditions on the vertices. The method is proposed as follows:

Theorem 3. Let scalars α > 0, σ > 0 and integers N ≥ 0, l ≥ 1 be given.

If there exist a matrix P = P T ≻ 0 ∈ M n (R) and a bounded function τ : R n → R + satisfying τ ∞ ≤ σ and such that for all x in R n , for all i ∈ {0; • • • ; N } and for all j ∈ {0; • • • ; ⌊ τ (x)l σ ⌋}, the conditions x T Φ i,j x ≤ 0 are satisfied, with Φ i,j = Φi,j + νI, (8)

           Φi,j = i k=0 L k,j σ l k if j < ⌊ τ (x)l σ ⌋, Φi,j = i k=0 L k,j τ (x) - j σ l k otherwise, (9)
                       L 0,j = Π T 3,j Π 1 Π 3,j -Π T 3,j Π 2 -Π T 2 Π 3,j , L 1,j = Π T 4,j (Π 1 Π 3,j -Π 2 ) + (Π T 3,j Π T 1 -Π T 2 )Π 4,j , L k≥2,j = Π T 4,j (A k-1 ) T k! (Π 1 Π 3,j -Π 2 ) + (Π T 3,j Π T 1 -Π T 2 ) A k-1 k! Π 4,j + Π T 4,j k-1 i=1 (A i-1 ) T i! Π 1 A k-i-1 (k -i)! Π 4,j , (10)
Π 1 = A T P + P A + 2αP , Π 2 = P BK, Π 3,j = I + M j (A -BK), Π 4,j = N j (A -BK), (11)
M j = j σ l 0 e As ds, N j = AM j + I, (12)
and

ν ≥ max σ ′ ∈ [0; σ l ] r ∈ {0; • • • ; l -1} max λ∈eig(ΦP,α (σ ′ +r σ l )-ΦP,α,N,r (σ ′ ))
λ , (13) with the function ΦP,α,N,r defined on [0; σ l ] as

ΦP,α,N,r (σ ′ ) = N k=0 L k,r σ ′k , (14)
then the origin of system (1) is globally exponentially stable with a decay rate α regarding the control (2) for any time varying sampling function τ : R + × R n → R + that defines sampling instant sequences by the sampling law t k+1 = t k + τ (t k , x(t k )), k ∈ N and which satisfies for all t in R + and for all x in R n , 0 < τ (t, x) ≤ τ (x).

The proof, which describes the principle in details, can be found in the Appendix. Compared to Lemma 2, Theorem 3 reduces the number of α-stability conditions. They depend on a finite number of matrices Φi,j representing the vertices of a convex polytope built around a polynomial approximation of the function Φ P,α (.), and on a scalar ν which bounds the approximation error. It is possible to compute an approximation of such a bound ν by using a gridding. In this theorem, N represents the order of the polynomial approximation, while l is the number of polytope subdivisions, as described in the first step of the proof as well as in Figure 4, in the Appendix.

The number of α-stability conditions to check has been reduced, but there is still an infinite number of conditions regarding the state x.

Main result

Remember that when trying to solve Problem 1, it is sufficient to work with homogeneous state dependent sampling functions of degree 0 (see Remark 3). Therefore, in order to derive a finite number of conditions, one will want to divide the state space into a finite number of subspaces R s defined by conics centered on the origin and try to find for each subspace its maximum allowable sampling time τ s , as shown for a 2 dimensional system in Figure 1. For higher dimensions, one can use the generalized spherical coordinates in R n , each region R s being associated to some range of the (n -1) angular coordinates θ i :

θ i ∈ [θ - i,s , θ + i,s ].

Fig. 1. Dividing the space into conic subspaces

The following theorem gives a finite number of conditions to solve this problem. Theorem 4. Let a matrix P = P T ≻ 0 ∈ M n (R), scalars α > 0 and σ > 0 and integers N ≥ 0 and l ≥ 1 be given. Let us divide the state space into a partition of q conic subspaces R s , s ∈ {1; • • • ; q}, defined for all s ∈ {1;

• • • ; q} as R s = {x ∈ R n , x T Q s x ≥ 0}, with Q s = Q T s ∈ M n (R)
, and let us define sampling times for these subspaces, τ 1 , • • • , τ q , satisfying τ s ≤ σ for all s ∈ {1; • • • ; q}, and a bounded function τ :

R n → R + satisfying τ (x) = τ s for all x ∈ R s , s ∈ {1; • • • ; q}. If there exist scalars ε s,i,j ≥ 0, for s ∈ {1; • • • ; q}, i ∈ {0; • • • ; N } and j ∈ {0; • • • ; ⌊ τsl σ ⌋}, such that the LMI conditions Φ i,j,s + ε s,i,j Q s 0 are satisfied for all i ∈ {0; • • • ; N }, s ∈ {1; • • • ; q} and j ∈ {0; • • • ; ⌊ τsl σ ⌋}, with Φ i,j,s = Φi,j,s + νI, (15)            Φi,j,s = i k=0 L k,j σ l k if j < ⌊ τ s l σ ⌋, Φi,j,s = i k=0 L k,j τ s - j σ l k otherwise, ( 16
)
with the L k,j and ν defined by the equations ( 10) to ( 14), then the origin of system ( 1) is globally α-stable regarding control (2) for any time varying sampling function τ : R + × R n → R + that defines sampling instant sequences by the sampling law t k+1 = t k + τ (t k , x(t k )), k ∈ N and which satisfies 0 < τ (t, x) ≤ τ (x) for all t ∈ R + and x ∈ R n .

Proof: Let x be in R n . There exists a subspace

R s = {x ∈ R n , x T Q s x ≥ 0}, s ∈ {1; • • • ; q}, Q s = Q T s , such that x ∈ R s and τ (x) = τ s .
Using the lossless version of the S-procedure, one can see that for any i ∈ {0; • • • ; N } and any j ∈ {0; • • • ; ⌊ τsl σ ⌋} the condition x T Φ i,j,s x ≤ 0, x ∈ R s is satisfied if and only if there exists a scalar ε s,i,j ≥ 0 such that Φ i,j,s +ε s,i,j Q s 0. As a consequence, if the condition Φ i,j,s + ε s,i,j Q s 0 is satisfied for all i ∈ {0; • • • ; N }, for all s ∈ {1; • • • ; q} and for all j ∈ {0; • • • ; ⌊ τsl σ ⌋}, then the stability conditions from Theorem 3 are satisfied for any x ∈ R n , which ends the proof. 2 Corollary 5. Let scalars α > 0 and σ > 0 and integers N ≥ 0 and l ≥ 1 be given. Let us define a sampling time τ * ≤ σ for the whole state space: the sampling function τ : R n → R + satisfies τ (x) = τ * for all x ∈ R n . If there exist a matrix P = P T ≻ 0 ∈ M n (R) such that the LMI conditions Φ i,j 0 are satisfied for all i ∈ {0; • • • ; N } and j ∈ {0; • • • ; ⌊ τ * l σ ⌋}, with Φ i,j defined by the equations ( 8) to ( 14), then the origin of system (1) with control ( 2) is globally α-stable for any time varying sampling bounded by τ * .

Proof: This comes naturally from Theorem 4 when working with a single subspace: R n itself. 2

General algorithm

Theorem 4 and Corollary 5 are solutions to Problem 1 and 2 respectively. While Corollary 5 gives a way to compute the Lyapunov function parameter P maximizing the lower bound τ * of the sampling function τ under the stability conditions of Proposition 1, Theorem 4 gives a way to maximize the sampling function on state subspaces for a given P . A method to apply the proposed technique is the following: First, use Corollary 5 with ν = 0 at first to compute (using a line search algorithm) an approximation τ * of the maximal admissible τ * as well as the Lyapunov function parameter P that enables such a bound.

Then, compute the variable ν corresponding to that Lyapunov function. It is possible, though not needed, to compute the real lower bound τ * verifying the stability conditions of Corollary 5 using the calculated P and ν.

Finally, divide the state space into a partition of q conic subspaces R s , s ∈ {1; • • • ; q} and use the LMI conditions from Theorem 4 with the computed values of P and ν to the maximal admissible sampling intervals τ s for each subspace R s (again using a line search algorithm). The state dependent sampling function τ can then be defined as τ (x) = τ s , for all x ∈ R s , s ∈ {1, • • • , q}.

NUMERICAL EXAMPLE

Consider the standard double integrator:

ẋ(t) = 0 1 0 0 x(t) - 0 1 Kx(t k ), K = (2 3) .
After setting the polynomial approximation degree term N = 5, and the number of polytopic subdivisions l = 100, we can obtain a mapping of the state space that gives the maximal allowable sampling time for each state for a given decay rate α > 0 thanks to Corollary 5 and Theorem 4. State dependent sampling functions obtained offline and insuring the exponential stability of the system origin for different decay rates α are presented Figure 2, using the spherical coordinate angle of the state, θ. These state dependent sampling functions were obtained using the partition of the state space shown in Figure 1 with a number of q = 1000 conic subspaces (which explains this impression of having smooth curves). Note that for constant samplings greater than T max = 0.67s the discrete time model is not Schur anymore, so the system becomes unstable. However we can see that with the proposed technique, for some subspaces of the state space we can go beyond that limit T max .

Figure 3 shows simulation results with α = 0 and a random initial state. It first shows the sampling intervals (in blue), with the lower bound (in red) of the offline computed state dependent sampling function, and the "Schur limit" T max (in magenta), before showing the decreasing Lyapunov function, and the control input.

CONCLUSION

We have introduced a method based on Lyapunov stability conditions to design a state-dependent sampling function τ that insures global exponential stability with a chosen decay rate α for linear state feedback control systems. It presents two main advantages.

The first advantage of the method is that it makes it possible to maximize the minimal sampling time τ * of the state dependent sampling function τ and to compute the associated quadratic Lyapunov function.

The second advantage is that the method makes it possible to compute offline a mapping of the state space with a maximum allowable sampling time for each subspace. The online number of computations are then reduced to the minimum since at each sampling instant t k one only needs to compute the spherical coordinates of the state x(t k ), check and memorize the offline computed associated sampling time τ (x(t k )) and compute the control input u(t) = -Kx(t k ).

Extensions to perturbed, delayed, and nonlinear systems are currently being studied.

APPENDIX

Proof of Theorem 3: In order to prove Theorem 3, we first need to introduce two important lemmas. Lemma 6. (From Hetel et al. [2007]) Consider the matrix polynomial function

L(σ) = L 0 + L 1 σ + • • • + L N σ N such that the variable σ is positive bounded: 0 < σ < σ.
Then we can find a convex polytope formed by N + 1 vertices which envelopes the matrix polynomial function L(σ), i.e. there exists an indexed family µ

i (σ) > 0, i = 0..N , verifying N i=1 µ i (σ) = 1, and such that L(σ) = N i=1 µ i (σ)U i
where the matrices U i represent the vertices of the polytope and are given for all i = 0..N by

U i = i k=0 σ k L k .
Lemma 7. Consider a state x ∈ R n , scalars σ > 0, 0 < σ ≤ σ, integers N ≥ 0, l ≥ 1, and parameters α > 0, P = P T ≻ 0 ∈ M n (R). If the conditions x T Φ i,j x ≤ 0 are satisfied for all i ∈ {0; • • • ; N } and j ∈ {0; • • • ; ⌊ σl σ ⌋}, with Φ i,j defined by the equations ( 8) to ( 14) (with σ replacing τ (x) in the equations), then for all σ ∈ [0; σ], x T Φ P,α (σ)x ≤ 0, with Φ P,α (σ) defined in (6).

Proof of Lemma

7: Let x ∈ R n , σ > 0, 0 < σ ≤ σ, N ≥ 0, l ≥ 1, P = P T ≻ 0 ∈ M n (R) and α > 0.
The proof of the Lemma is divided into 4 steps.

(1) First, we divide the time interval [0; σ] into l subdivisions and take a time σ ≤ σ into one of these subdivisions. The aim of this step is to prepare the field to compute a precise estimation of Φ P,α (.) by building up to l small convex embeddings around it instead of building one big one (see Figure 4). (2) Then, we compute a polynomial approximation of Φ P,α (.) for the chosen time interval subdivision. (3) Afterwards, we bound the error term from this polynomial approximation with a constant term. (4) Finally, we build a convex polytope around the polynomial approximation, using the method proposed in [START_REF] Hetel | Lmi control design for a class of exponential uncertain systems with application to network controlled switched systems[END_REF] (see Lemma 6), to obtain the desired sufficient finite number of conditions. Step 2: In order to have lighter equations, let us define Π 1 = A T P + P A + 2αP and Π 2 = P BK. From equations ( 6) and ( 7), we deduce that Φ P,α (σ) = Λ(σ) T Π 1 Λ(σ) -Λ T (σ)Π 2 -Π T 2 Λ(σ). ( 17) In order to derive a useful expression of Λ(σ) as a function of σ ′ , we use the following equation obtained with some computations: (18) with M j = j σ l 0 e As ds, N j = AM j + I, Π 3,j = I + M j (A -BK), and Π 4,j = N j (A -BK). Then, note that

σ ′ 0 e As ds = ∞ i=1 A i-1 i! σ ′i . (19)
Combining equations ( 17), ( 18) and ( 19), one can compute

Φ P,α (σ) = ∞ k=0 L k,j σ ′k , (20)
with the L k,j defined in (10). It is then possible to express a polynomial approximation of order N of Φ P,α on the temporal interval subdivision [j σ l ; (j + 1) σ l ] as ΦP,α,N,j (σ ′ ) = N k=0 L k,j σ ′k , σ ′ ∈ [0; σ l ].

Step 3: Let us denote the approximation error term R P,α,N,j (σ ′ ) = Φ P,α (σ) -ΦP,α,N,j (σ ′ ). If we can compute a bound with a scalar ν independent of σ ′ such that R P,α,N,j (σ ′ ) νI then the condition x T ( ΦP,α,N,j (σ ′ ) + νI)x ≤ 0 will imply that x T Φ P,α (σ)x ≤ 0. For a given σ ′ , since R P,α,N,j (σ ′ ) = Φ P,α (σ) -ΦP,α,N,j (σ ′ ) is symmetric, then if we denote λ σ ′ the maximal eigenvalue of R P,α,N,j (σ ′ ), we have R P,α,N,j (σ ′ ) λ σ ′ I. As a consequence, we can write R P,α,N,j (σ ′ ) νI with ν a constant defined in (13).

Step 4: Since the function ΦP,α,N,j (.) + νI : [0; χ] → M n (R) is polynomial, we can use the convex polytope given in Lemma 6 to prove that if x T Φ i,j x ≤ 0 for all i ∈ {1; • • • ; n}, with Φ i,j = i k=0 L k,j χ k + νI, then x T ( ΦP,α,N,j (σ ′ ) + νI)x ≤ 0, and therefore x T Φ P,α (σ)x ≤ 0, which ends the proof of Lemma 7. 2

The proof of Theorem 3 is obtained by implementing Lemma 7 conditions to satisfy x T Φ P,α (σ)x ≤ 0, ∀σ ∈ [0; σ] in Lemma 2 with σ ≥ τ ∞ and σ = τ (x), for all x ∈ R n . 2

⋆

  This work was supported by Ministry of Higher Education and Research, Nord-Pas de Calais Regional Council and FEDER through the 'Contrat de Projets Etat Region (CPER) 2007-2013'. ⋆⋆ The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement n o 257462: HYCON2 Network of Excellence "Highly-Complex and Networked Control Systems".

Fig. 2 .

 2 Fig. 2. Example 1: State-angle dependent sampling function τ for different decay rates α.

Fig. 3 .

 3 Fig. 3. Example 1: Inter-execution times τ (x(t k )), Lyapunov function V (x) = x T P x evolution, and control input u(t) for a decay rate α = 0.

Fig. 4 .

 4 Fig. 4. Building polytope subdivisions (here with l = 6)Step 1:Let us divide the time interval [0; σ] into l subdivisions [j σ l ; (j + 1) σ l ], with j ∈ {0; • • • ; l -1}. Let σ ∈ [0; σ]. There exists j ∈ {0; • • • ; ⌊ σl σ ⌋} such that j σ l ≤ σ ≤ (j + 1) σ l . Let us then define σ ′ = σj σ l (σ ′ ∈ [0; χ], with χ = σ l if j < ⌊ σl σ ⌋, and χ = σ -j σ l

e 0 e 0 e

 00 As ds + I , which is satisfied for any scalars a and b, in order to getΛ(σ) = I + M j + σ ′ As dsN j (A -BK) = Π 3,j + σ ′ As dsΠ 4,j ,


Sylvain Kubler

William Derigent

André Thomas

Eric Rondeau

Éric Rondeau

Key Factors for Information Dissemination on Communicating Products and Fixed Databases

come

Introduction

Intelligent products or products carrying their own information are more and more present nowadays. [START_REF] Ley | Becta[END_REF] quotes the example of clothes able to carry their own information and thus enabling the washing machine to automatically adapt its washing program. In one of our previous works, [START_REF] Kubler | Problem definition methodology for the "Communicating Material" paradigm[END_REF] highlight several possible scenarios in different sectors: Supply Chain Management, healthcare [START_REF] Ausen | Fobis: Foresight biomedical sensors[END_REF], home automation. Such applications rely on ever more complex information systems using a multitude of information vectors, in order to allow product information to be available anywhere and at anytime. These vectors may be fixed (desktop computers) or mobile devices (PDA, laptops, sensors, RFID technologies. . . ), short-lived or even invisibles (concept of disappearing computer, ubiquitous computing). More generally, the concept of the Internet of Things [START_REF] Gershenfeld | The internet of things[END_REF] based on the RFID usage enables to access to information disseminated on any kind of physical object and to develop new smart services and applications.

According to Meyer [START_REF] Meyer | Intelligent products: A survey[END_REF], in the context of supply chain management, few researches has been conducted on "intelligence at object", i.e products carrying their own information and intelligence. In fact, most of the time, products are only given an identifier (stored in a RFID tag) referring to a software agent or a database (approach used by [START_REF] Morel | Special issue on intelligent manufacturing[END_REF]). This mode of information management is diametrically opposed to works initiated since 2003 by the PDMS (Product-Driven Manufacturing Systems) community, which advocates a physical information distribution on the product. In that case, a product carries physically a part, or even the totality of the information needed for its manufacturing or to manage its evolution all along its life cycle. Our previous work [START_REF] Kubler | Prototyping of a communicating textile[END_REF] aimed at prototyping a new type of materials, in which it is possible to write a significant quantity of information by inserting thousands of micro RFID tags. This new type of material is then referred to "communicating material". We developed an industrial process to produce a communicating textile with up to 1500tags/m 2 . Meyer concurs with the PDMS community by stressing the fact, in an increasingly interconnected and interdependent world involving many actors issued from different domains, supply chain information should not be stored in a single database but should be distributed all over the supply chain network. In fact, substantial information distribution improves data accessibility and availability, compared to centralized architectures. However, update mechanisms of the distributed information are needed in order to avoid problems related to data consistency and integrity. This type of architecture is thus more complex to design than centralized architectures. As a result, product information can be spread out on mobile or fixed devices or even directly on the product, via simple RFID tags or communicating materials. Centralized architectures or highly distributed architectures can be employed. One might then wonder what the optimal information distribution is. The present paper aims to study the different ways to distribute information over a network composed of centralized, distributed databases and "communicating products", which may store information fragments as well. This study will determine the key factors which lead to improve information distribution performance. The performance is analyzed regarding the time required for accessing to the information system during the product life cycle. Based on this influent factors determination, in a further work, we will be able to implement an experimental design leading us to control the best way to disseminate information on the informational vectors. This question is addressed in several steps. First, the data distribution is introduced, and then an overview on conducted researches on distributed databases over fixed and mobile devices is presented in section 2. Then, a case study extracted from this overview and adapted to our context is detailed in section 3. It only considers two types of informational vectors: fixed computers and communicating products. This case study is then used as a basis of comparison and evaluation between two different architectures of information distribution (one forbids data allocation on products while the other allows it). The evaluation process relies on several specific tools and a methodology using jointly two discrete-even simulators, one using Petri Nets (CPN tools) and the other dedicated to network protocol development and measurement (OPNET). This simulator looks for evaluating the manufacturing lead-time of a given number of communicating products all along the supply chain, by taking into account manufacturing run times, network delays, times to read/write information for both distributed databases and communicating products. This piece of software is presented in the section 4. Finally, the section 5 presents the results obtained with the case study and an analysis of the main factors impacting on the performance of the information distribution.

Distributed Database Systems

General distribution framework

During the product lifecycle, users could have to access to product information for diverse reasons, either in the design phase, the usage phase or still the recycle phase. As exposed before, information can be stored both on the product andor on fixed databases. Information are therefore bind to one or more relational data models, which have to be fragmented and distributed by the best way on these informational vectors. One example retracing briefly a bobbin lifecycle is presented in the figure 1. We can see 5 data fragments [F 1..F 5] distributed between the product and the database (F 1, F 4, F 5 allocated to the database system and F 2, F 3 to the product). By reconsidering the example given by [START_REF] Ley | Becta[END_REF], the washing machine could access to data fragments located both on the product and on the database according to its queries. In our researches, we are looking for assessing different distribution patterns between both informational vectors (manufactured products and fixed databases) by taking into account the access times for reaching information. Work on distributed databases considering fixed and mobile environments are introduced in the next section.

Fixed database

F 1 F 4 F 5 F 2 F 3

Manufacturing parties Users Recyclers

Beginning of Life Middle of Life End of Life

F 3 F 2 F 2 F 3 F 2 F 3 F 3
Fig. 1. Information distribution on products and fixed databases

Distributed databases through literature

The main objective of the data dissemination in an information system is to make the dissemination process transparent for users: location, partitioning and replication transparency. Indeed, no matter why, where and how the data repartition is achieved from a user's point of view. The structured data distribution regarding relational data models (within distributed database systems framework) is carried out in two steps: The partitioning of the data model follows up by the allocation phase of the resulting fragments. Many approaches and mechanisms exist for ensuring the best partitioning and allocation of the relational model regarding to the environment and some applicative constraints. Basically, the partitioning aim is to subdivide the concerned relational data model. Thus, the resulting fragments will be allocated at specific informational vectors in order to improve system performance. Three types of fragmentation exist: vertical [START_REF] Navathe | Vertical partitioning algorithms for database design[END_REF], horizontal [START_REF] Apers | Data allocation in distributed database systems[END_REF], mixed/hybrid [START_REF] Navathe | A mixed fragmentation methodology for initial distributed database design[END_REF]. The vertical fragmentation aims to break up a relation into a set of relations. It consists in dividing the attributes of a relation (i.e the columns of a relational table). The horizontal fragmentation aims to break the large number of object instances into disjoint subsets. It consists in partitioning the tuples of a relation (i.e the rows of a relational table). The hybrid fragmentation first divides the relation horizontally, and then splits each of the obtained fragments vertically or vice versa.

As stated previously, the allocation phase takes place subsequently to the fragmentation phase and the aim is to establish the optimal fragment assignation on the databases. Usually, methods tend to assign fragments to the clients requesting them mostly via objective functions that we attempt to minimize or maximize [START_REF] Huang | Fragment allocation in distributed database design[END_REF][START_REF] Hababeh | A method for fragment allocation design in the distributed database systems[END_REF]. Let us note that it is possible to perform data replications, or in other words, to replicate a same fragment on several databases. This has the dual benefit of maintaining the system reliability and of increasing performance (e.g reduction of the overload traffic) [START_REF] Padmanabhan | A survey of data replication techniques for mobile ad hoc network databases[END_REF]. However, replication mechanisms are necessary for handling both the modification broadcast (updates) on replica and also the information access rights (to authorize one site or one group of sites to modify replica). The applicative expectations have an influence on the mechanism to implement and actually two parameters have to be characterized: When and Where? When do the updates have to be propagated? Two modes are available: Synchronous (S) and Asynchronous (As). The As mode makes it possible to carry out local modification without needing to inform its peers (contrarily to the S mode). Where do the updates have to be performed ? Two principles exist: Update everywhere (Ue) and Primary copy (Pc). The Pc principle allows one site to perform modifications on a data fragment contrarily to the Ue mode which allows one group of sites. Finally, four types of replication may be considered: Ue-S, Ue-As, Pc-S and PC-As. Also note that the memory storage limitation of mobile devices is a problem frequently encountered in the literature. Accordingly, some authors focus on the data summarization [START_REF] Chan | Context-sensitive mobile database summarisation[END_REF][START_REF] Lubinski | Small database answers for small mobile resources[END_REF] (subclass of the data mining) whose primary aim is to reduce the information somehow. [START_REF] Chan | Summarisation for Mobile Databases[END_REF] list the summarization methods used for distributed database systems and mention the fragmentation/allocation method (used in our study).

A multitude of interesting approaches are proposed in the literature, we therefore feel it is necessary to confront our proposition with them in order to compare and assess our distribution models. In this sense, works reported by Hababeh [START_REF] Hababeh | A method for fragment allocation design in the distributed database systems[END_REF] seem interesting as basis of comparison. Indeed, a fragment distribution method is developed and then applied on a case study, which can be easily extended to our application. In what follows, two distribution architectures will be defined, the first one does not consider the presence of communicating products able to store data fragments, i.e all information is located on databases. In fact, we rely on the distribution defined by Hababeh. The second one considers communicating products able to store data fragments, thus, diverse distribution patterns of fragments between the product and databases will be possible. The next section introduces this case study and then the adaptation realized in this paper.

3 Case study presentation

Reference distribution pattern

Hababeh proposes a fragment distribution approach based on a two step process: first, the sites (clients and databases) are clustered according to communication costs, and then data fragments are allocated to the different clusters via an optimization function. This approach is applied on a specific case study, including 3 databases, 3 clients which perform read and write accesses on a set of fragments (8 in total: [F 1..F 8]). The resulting optimal allocation [START_REF] Hababeh | A method for fragment allocation design in the distributed database systems[END_REF] is depicted on the figure 2, the access pattern to the 8 fragments performed by each client is specified, too (number in brackets indicates the number of bytes). The next section formulates the adaptation of this case study to our logistic scenario. In fact, we match parameters and data defined by Hababeh with the supply chain tasks, actors: number of databases and clients, query patterns, data fragments. . .

Allocated fragments: F 1, F 5, F 8 Cluster-Location 1 client 1 DB 1 machine 1 Read F 1(540) F 4(90) F 5(54) F 8(720) Upd F 1(286) F 3(110) F 4(220) F 5(110) F 8(220) Allocated fragments: F 4, F 7, F 8 Cluster-Location 2 client 2 DB 2 machine 2 Read F 2(180) F 4(45) F 5(180) F 7(315) F 8(540) Upd F 2(66) F 4(132) F 5(110) F 7(110) F 8(110) Allocated fragments: F 2, F 3, F 5, F 6, F 7 Cluster-Location 3 client 3 DB 3 machine 3 Read F 1(315) F 2(45) F 3(360) F 5(315) F 6(225) F 7(90) Upd F 1(55) F 2(330) F 3(330) F 5(110) F 6(55) F 8(220)
Network Fig. 2. Optimal distribution architecture established by [START_REF] Hababeh | A method for fragment allocation design in the distributed database systems[END_REF]

Adaptation of the logistic process

A supply chain process consists of a set of tasks in a planned pattern or sequence (rout sheet). These tasks may correspond to manufacture operations, transport phases. . . and can be performed on different physical locations (e.g supplier). These locations may dispose of local databases and can implement their own information systems (related to their tasks), but they can also access to remote databases if a collaboration between actors exists. As a matter of fact, databases are distributed (or federated) through one or more relational data models. Inspired by our current researches, the applicative framework considered is related to a supply chain management process dedicated to the textile industry.

In order to adapt the case study described previously to our logistic scenario, we consider three remote locations corresponding to the clusters 1, 2 and 3 introduced on the figure 2. One physical transformation is carried out on each location: the operation 1 and the operation 2 (performed on the location 1 and 2 respectively) consist of cutting a set of bobbins 1 and bobbins 2 respectively, the operation 3 (performed on the location 3) consists of sewing the textile pieces resulting from the two previous operations. Each location has one machine to achieve its own operation, this machine requires information after the arrival of products (range of product, production order. . . ) and updates some of this information (notifications. . . ). Therefore, the applicative characteristics defined for each client in Hababeh will be matched to each machine (located in each cluster). In other words, the machine 2 will have the same fragment access pattern (read and write accesses) than the client 2 defined in Hababeh and so on. Likewise, each location disposes of a local database and shares a relational data model, this data model has to be distributed on the three databases. Taking into account of the input parameters defined in Hababeh (query pattern, architecture. . . ), the optimal distribution considered in our paper may be defined as shown in the figure 2: F 1,F 5,F 8 allocated to DB1, F 4,F 7,F 8 to DB2. . . The architecture described in the previous paragraph corresponds to the Optimal Distributed Architecture (ODA), without any possibility to allocate data fragments 1 on communicating products. In the second stage, we allow data fragment allocation on the products (according to their memory capacity and characteristics). We denote this architecture: ODAP (Optimal Distributed Architecture considering communicating Products). The idea is to highlight the potential benefits that could be achieved regarding a supply chain via such an ODAP architecture, where manufactured products act as mobile databases as opposed to a classic architecture (ODA). The figure 3 illustrates in form of Petri Nets the synoptic of the part of the supply chain peculiar to our application. It consists of the ODA architecture implementing classic product and of the ODAP architecture implementing communicating product. Let us note that we design the Petri Net model by working on hierarchical views. Consequently, the distribution aspect will be detailed in the lower views (in the section 4.3). 4 ODA and ODAP architecture modeling

Architecture

A description of how the assessment and the comparison are undertaken of both architectures (ODA and ODAP) is proposed in this section. The evaluation architecture relies on two discrete events simulators and its usage process is depicted on the figure 4. This architecture is composed by two sub-systems. The first one is a tool for editing, simulating, and analyzing Colored Petri Nets (CPN tools). As shown on the figure 4, the supply chain behavior is simulated via this tool, or at least the considered part which has been exposed in the section 3.2 (operation 1, 2 and 3). It allows to deal with sharing of physical resources taking place into the system (databases, machines, manufactured products. . . ), operation times, queuing tasks, times for reading/writing information on databases or still on manufactured products (considering the ODAP architecture) and so on. The ODA and ODAP distribution patterns are specified in this tool. Let us note that for the ODAP architecture, all the possible combination of distribution between the product and the distributed fixed databases are realized (i.e 2 k possibilities with k the total number of fragments). The second tool is the OPNET network simulator which is primarily aimed at developing and validating network protocols. However, it allows estimating various parameters on specific case studies, such as network times, overload traffic, equipment processing times, battery life, etc. In our study, the OPNET tool is used for assessing the round trip time 2 to achieve read/write queries on databases, considering the ODA architecture (we do not take into account the possibility to read and write the product). To do this, the physical architecture and the distribution adopted in the section 3 have to be specified in OPNET. The resulting times are then injected into CPN Tools. The next sections introduce the methodology employed for assessing the ODA and ODAP architectures. First, the network interconnecting the client machines and the fixed databases is defined in OPNET (see figure 4). Thereafter, it is necessary to create system partitions on each server in order to allocate the data fragments (i.e the ODA distribution specified in the figure 2). Therefore, a replicating protocol has to be implemented owing to the replication of F 5, F 7 and F 8. In our application, we implement Synchronous and Primary copy mechanisms described in the section 2. Subsequently, it is necessary to specify the applicative exchanges between equipments, i.e the query pattern (read/write) performed by each client machine on the databases. To do this, three models are implemented in OPNET: Task, Application and Profile models. Finally, it is possible to estimate the round trip time for a specific query sent from a client to a database. Statistical tools are available in OPNET for computing averages, variances or still confidence intervals based on a set of simulations. In our study, 50 simulations have been running for a same scenario and then, both the round trip time average and the statistical variance have been extracted for each query. For instant, the table 1 gives the round trip time induced by a read (R) or write (W) query on F 1 (fragment allocated to DB1). The machine 1 requires 3.6ms on average with a variance of 9µs to access to this fragment and spends 7.6ms and 7µs respectively to write it.

Petri Nets: ODA and ODAP architectures

As illustrated on the evaluation architecture (figure 4), the estimated round trip times are injected into CPN Tools and more exactly, they shall be set on timed transitions which reflect the read/write actions on databases. Second views describe the considered operations. For instance, the second view of the cutting 1 operation (introduced section 3.2) is presented in the figure 5. Both views of this operation, related to the ODA and ODAP architectures, are shown in the same figure because they are almost similar except for the read and write phases which will be discussed in more detail below. Three places from these two Petri Nets are bound to the first view (see figure 3), namely the place Bobbin1(c) (port:

In), the place DB in which the three databases are defined (port: I/O since they are shared resources between the three operations) and the place pt1(c) in which cut pieces are stored (port: Out).

Let us focus now on the Petri Net structure of the cutting 1 operation (see figure 5): when a bobbin1/c (i.e bob1/c) arrives into the queue for being cut (i.e places Bobbin 1/Bobbin 1c), we generate straight away the data fragments needed by the machine 1 for starting the operation (fabrication orders. . . ). Let us remind the machine 1 needs to read the following fragments: F 1, F 4, F 5 and F 8. Considering the ODA architecture, these fragments are all allocated on the fixed databases, i.e into the place Info DB bef. read. Considering now the ODAP architecture, one part of these fragments can be allocated on the product and another part on databases. To do this, we add to the ODAP view the place Info AP bef. read to indicate that fragments should be read on the product rather than on databases. Thus, several distribution patterns between product and fixed databases can be defined. One possible combination might be to allocate F 1, F 5 to the product and F 4, F 8 to the databases. After having read the fragments, the cutting task (denoted "cut" transition) can start. The write phase is similar to the read phase. This Petri Net structure is used for the other operations: cutting 2 and sewing.

Views of the third level deal with the read and write phases of fragments, either on databases and on products, i.e the transitions denoted read/write DB and AP: figure 5). With regard to the read/write DB transitions, we added the statistical round trip times extracted from OPNET taking into account the client machines, the databases and the fragments (see table 1). With regard to the read/write PA transitions, we define several throughputs in the next section.

Results and Analysis

Considering the ODA architecture as our reference model, the purpose of our experimentation is then to determine the factors impacting positively or negatively on the ODAP architecture performance, and to identify configurations where ODAP should be benefit. In this article, we aim to study the influences of two parameters which are the communicating product throughput and the fragment distribution pattern. In fact, communicating products can exchange data with their environment at a given throughput. For our experiments, we consider 4 levels of throughput: 100M bps,54M bps, 11M bps and 1M bps. A fragment distribution pattern indicates the simulator how to place the different fragments, either on the distributed database or on the product as explained in the previous section. It is composed of eight boolean values [F 1, F 2. . . F 8]; F i meaning that the fragment i is located on the database and F i meaning the fragment i is located on the product. For example [F 1 F 2 F 3 F 4 F 5 F 6 F 7 F 8] informs the simulator that only fragments F 8 and F 6 should be placed on products and the others let on the database. In practice, all the different possible dissemination patterns are tested for a given throughput, which leads to 256 (2 8 ) experiments per throughput. Each experiment is simulated 10 times and the mean times needed to produce 85 headdresses using ODA and ODAP architectures are recorded. This number has been defined arbitrarily.

Table 2 summarizes the results obtained during the experimentation. Each line of the table corresponds to a given throughput value and each column to a specific configuration of the data distribution pattern: ODA (all fragments are allocated to the databases), full ODAP (all fragments are located on the product), and best hybrid ODAP configuration (some fragments are on the database, others on the product). Time values obtained for a given throughput and configuration are then reported in the table. As can be seen, for our scenario, full ODAP and ODA are quite similar in terms of performance, when considering 100M bps, 54M ps and 11M bps throughputs. Therefore, it might appear that disseminating information all over the different informational vectors has no influence on the manufacturing time if the product throughput is high enough. With a correct throughput, it is then possible to imagine an information system completely distributed on a product network. When decreasing, the throughput yet acts as a very important constraint and full ODAP is clearly a bad solution. However, the best hybrid configuration always gives good results, which means some data can be stored on the product no matter what the throughput is.

In the figure 6 are plotted two curves representing the ODA and ODAP times (y-axis) for producing 85 headdresses, with a throughput of 1M b/s. On the x-axis are represented the 2 8 possible combinations of distribution. Clearly, the distribution pattern has a very important effect in that case. In fact, the time needed to complete the production varies from 2 ′ 25 ′′ to 17 ′ 26 ′′ . Based on these observations, it might be interesting to determine if each fragment has the same impact on the manufacturing time and if not, to identify the important fragments and evaluate their influence. To do so, a statistical analysis of the experiments done with a 1M bps throughput has been initiated and showed that all fragments are important, but some of their interactions as well. An interaction between 2 fragments means that the impact of these fragments all together on the product is different from the sum of the impacts of each fragment. In our case, interactions up to level 3 (interactions of three fragments) have to be taken into account. In the following, we use the term "factors" to denote significant fragments and interactions. For our scenario, there are up to 51 factors, as reported on the histogram x-axis in the figure 7. The impact of each factor is then estimated thanks to a multiple regression analysis that determines for each factor a coefficient, related to the linear regression equation. The value of a factor coefficient could be roughly considered as the effect of this factor on the manufacturing time. The higher the coefficient value, the more the manufacturing time increases. The figure 7 clearly shows that some fragments have a very important effect (F 1, F 3, F 5), and others have a very moderate one, sometimes equal to interactions of level 2 (e.g the effect of F 4 is almost the same as F 7 * F 8). As a result, our study demonstrates there are lots of factors with different effects to take into account when setting up an ODAP.

conclusion

A multitude of informational vectors take place in Supply Chain environments as fixed databases or manufactured products on which we are able to embed significant proportion of data. By considering distributed database systems, specific data fragments can be embedded/allocated on these products (for example, data useful for their life cycle). The paper analyzes three specific distribution patterns to determine the different parameters impacting on the manufacturing system performance and shows that choosing a good pattern is complex. In a further work, we are willing to implement an experimental design leading us to control the best way to disseminate information on the informational vectors.

Fig. 3 .

 3 Fig. 3. Global view of the supply chain process 1 Products are only given an identifier (stored in a RFID tag) referring to a database.

Fig. 4 .

 4 Fig. 4. Usage process of evaluation architecture

Fig. 5 .

 5 Fig. 5. Petri Net structure of the operation: cutting 1

Fig. 6 .

 6 Fig. 6. Comparison between ODA and ODAP (product throughput= 1M bps)

Fig. 7 .

 7 Fig. 7. Significant factors and their respective coefficient values

Table 1 .

 1 Evaluated times regarding access query patterns: S-Ue

		Machine 1		Machine 2	Machine 3
		DB1	DB2 DB3 DB1 DB2 DB3	DB1	DB2 DB3
	F1	R 3.6ms, 0.009µs × W 7.6ms, 0.007µs ×	× ×	× ×	× ×	× 7.8ms, 0.012µs × × 11.3ms, 0.015µs ×	× ×

Table 2 .

 2 Times obtained for 3 fragment distributions according to 4 throughputs ′ 27 ′′ 2 ′ 28 ′′ 2 ′ 27 ′′ (F 5,F 7) 54Mbps 2 ′ 27 ′′ 2 ′ 28 ′′ 2 ′ 27 ′′ (F 5,F 6,F 7,F 8) 11Mbps 2 ′ 27 ′′ 2 ′ 52 ′′ 2 ′ 27 ′′ (F 7) 1Mbps 2 ′ 27 ′′ 17 ′ 26 ′′ 2 ′ 28 ′′ (F 7)

	Product	ODA	Full ODAP	Best hybrid distribution
	throughput	distribution	distribution
	100Mbps	2