domain: spi phys math

text:
Guillaume Hardouin

Sylvain Bertrand

Hélène Piet-Lahanier

Distributed event-triggered

Keywords: multi-agent systems, consensus, event-triggered control, distributed, measurement noise, bounded-error measurement, guaranteed enclosure

INTRODUCTION

Consensus determination constitutes a major topic of multi-agent coordination Olfati-Saber and [START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF]; [START_REF] Ren | Distributed multi-vehicle coordinated control via local information exchange[END_REF]. Moreover, due to exchange limitation, it is required that reaching the consensus should be performed using limited exchange of information. Eventtriggered approaches have been designed to tackle this issue for simple [START_REF] Dimarogonas | Distributed event-triggered control for multiagent systems[END_REF]; [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF]) or more general dynamics [START_REF] Garcia | Decentralized event-triggered consensus with general linear dynamics[END_REF]; [START_REF] Viel | New state estimator for decentralized eventtriggered consensus for multi-agent systems[END_REF]). Most of the developed techniques do not address the problem of noise in exchanged information. However, existing methods consist in designing proper stochastic approximation type consensus protocols to reduce the noise effect [START_REF] Huang | Coordination and consensus of networked agents with noisy measurements: stochastic algorithms and asymptotic behavior[END_REF]). This early work allows [START_REF] Hu | Eventtriggered consensus of multi-agent systems with noises[END_REF] to introduce eventtriggered mechanism for mean-consensus. In [START_REF] Ge | Event-based set-membership leader-following consensus of networked multi-agent systems subject to limited communication resources and unknown-but-bounded noise[END_REF], a set-membership leader-follower event-triggered consensus is described that provide bounding ellipsoidal set containing the states of all followers of a leader using recursive convex optimization.

In this paper, we consider bounded uncertainties on the state estimates of agents. An event-triggered strategy is described to enable the MAS to converge to a guaranteed consensus region that should be updated during the trajectory. This work is inspired from [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF]. Section 3 presents the problem definition. Because of the noise, the self-error from true state is no more accessible for agent, then new communication triggering conditions (CTCs) based on error's bound are introduced for singleintegrator (Section 4) and double-integrator (Section 5) dynamics. The presence of bounded noise results in defining a region of consensus contrary to classical noise free case. Thus, Sections 4.3 and 5.3 expose strategies to obtain guaranteed interval bounds for consensus values from the communication mechanism and using lower and upper bounding dynamical systems for the uncertain systems [START_REF] Kieffer | Guaranteed nonlinear state estimation for continuous-time dynamical models from discrete-time measurements[END_REF]). Simulations of theoretical results are exposed for each type of dynamics. Finally, conclusions are drawn in Section 6.

PRELIMINARIES 2.1 Graph Theory

The interaction topology of a network of N agents is represented using a graph G = (V, E) with the set of nodes V = {1, 2, ..., N } and edges E ⊆ V × V. E ij is the edge between the nodes i and j, with i, j = 1, ..., N . The adjacency matrix A is defined by a ij = 1 if i and j are adjacent and a ij = 0 otherwise. The set of neighbors of a node i is N i = {j ∈ V|(i, j) ∈ E, i = j} and N i its cardinal number. If there is a path from i to j, then i and j are called connected. If all pairs of nodes in G are connected, then G is called connected. The degree matrix D of G is the diagonal matrix with elements d i equal to the cardinality of node i's neighbor set N i . The Laplacian matrix L of G is defined as L = D -A. L is symmetric and positive semi-definite iff G is undirected. Moreover, in that case, every row and every column of L sum to zero, which means L satisfies L1 N = 0. L has only one null eigenvalue λ 1 (L), and all its other non-zero eigenvalues λ 2 (L) ≤ λ 3 (L) ≤ . . . ≤ λ N (L) are strictly positive. Lemma 1. from [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF]. Suppose L is the Laplacian of an undirected, connected graph G. Then for all t ≥ 0 and all vectors v ∈ R N with 1 T v = 0, it holds that e -Lt v ≤ e -λ2(L)t v . Lemma 2. from [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF]. Suppose L is the Laplacian of an undirected, connected graph G. Define µ > 0 and Γ = 0 I N -L -µL Then, for all t ≥ 0 and all

vectors v ∈ R 2N with [1 T 0 T ]v = [0 T 1 T ]v = 0, it holds that e Γt v ≤ e Re(λ3(Γ))t c v v , with c v = V -1 V and
where V is a non singular matrix defined as in [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF] from the eigen vectors of Γ.

Interval Analysis

A real interval [x] is a connected subset of R. The lower bound lb ([x]) of an interval [x], also denoted by x, is defined

as x = lb([x]) ∆ = sup {a ∈ R ∪ {-∞, ∞} | ∀x ∈ [x], a ≤ x}.
Its upper bound ub ([x)], also denoted x, is defined as

x = ub([x]) ∆ = inf {b ∈ R ∪ {-∞, ∞} | ∀x ∈ [x], x ≤ b}. The width of any non-empty interval [x] is w([x]) = x -x.

PROBLEM DEFINITION

Consider a multi-agent system (MAS) of N agents whose communication topology can be described by an undirected fully-connected and time-invariant graph G. It is also assumed that there is no communication delay. Let x i the state of each Agent i ∈ V and u i its control input. It is assumed that each Agent i is able to compute or has access to an estimate xi of its own state x i . Let t i k be the time instant at which the k-th message is broadcast by Agent i. This message is composed of its state's estimate:

xi (t) = x i t i k + w i (t i k ), ∀t ∈ t i k , t i k+1 (1)
where w i is some additive bounded noise. xi kept constant between two communication instants and equal to the last broadcast value. It is also assumed that Agent i is able to broadcast xi to all other neighbor agents j, j ∈ N i . Due to the presence of noise in the state estimates, only a bounded consensus of the MAS can be obtained, that is

∃ > 0 s.t. lim t→+∞ x i (t) -x j (t) ≤ , ∀(i, j) ∈ E.
Define w as:

w = sup i∈V sup k∈N * |w i t i k | (2)
The trigger function f i (.) for Agent i takes values in R and relies on local information only. A communication is triggered when the CTC f i (.) > 0 is satisfied. As soon as an agent sends its state value or receives other agent state value, it recomputes its control u i immediately.

Problem:

The problem addressed here is to design distributed control laws and CTCs requiring only local information available to allow the MAS to achieve a bounded consensus. It also provides boundary values on the consensus reflecting the bound on the noise and the broadcast strategy.

SINGLE-INTEGRATOR AGENTS

Consensus & communication triggering condition

Consider first the noise-free case i.e w i (t) = 0, for agents with single-integrator dynamics described as

ẋi (t) = u i (t) (3)
with i ∈ V, where x i (t) ∈ R is the state of Agent i and u i (t) ∈ R its control input. The dynamics of the MAS can be described in a matrix form as ẋ (t) = u (t), with the state vector [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF] proposed an event-based implementation,

x (t) = [x 1 (t) , ..., x N (t)] T , x(0) = x 0 ∈ R N and the control vector u (t) = [u 1 (t) , ..., u N (t)] T .
u i (t) = j∈Ni (x j (t) -xi (t)) , ( 4
)
or in matrix form u (t) = -Lx (t), where Agent i does not use its true state x i (t) but the last broadcast estimate xi (t) to guarantee the zero average of the control u (t) so that a consensus can be achieved. As the MAS evolve, the state x i will drift from the last broadcast value xi . Therefore, an estimation error e i can be defined for Agent i as

e i (t) = xi (t) -x i (t) . ( 5
)
Let e (t) = [e 1 (t) , ..., e N (t)] T be the collection of MAS errors. Let us recall from Olfati-Saber and [START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF] the disagreement vector of the MAS δ (t):

δ (t) = x (t) -a(t)1 N , (6)
with a (t) = (1/N ) 1 T N x (t), so that 1 T N δ (t) = 0. In case there are no measurement noises, a CTC is defined in Theorem 3.2 of [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF] as and0 < α < λ 2 ( L). Then, for all initial condition x 0 ∈ R N , the disagreement vector δ of the closed-loop system converges to a ball centered at the origin with radius r = L √ N c 0 /λ 2 (L). Moreover, the closed-loop system does not exhibit Zeno behavior.

f i (t, e i (t)) = |e i (t) | -c 0 + c 1 e -αt > 0, (7) with constants c 0 0, c 1 0, c 0 + c 1 > 0,

Presence of unknown but bounded measurement noise

Consider now that Agent i's state estimate is defined as (1). Let us define a bound on the error, that can be evaluated by Agent i, as

ẽi (t) = w + t t i k |u i (s) | ds, ∀t ∈ t i k , t i k+1 . ( 8
)
Based on this definition, let us also introduce the following new communication triggering condition

f i (t, ẽi (t)) = ẽi (t) -c 0 + c 1 e -αt > 0 (9) with constants c 0 > w 0, c 1 0, c 0 + c 1 > 0, and 0 < α < λ 2 (L).
Theorem 3. Consider the multi-agent system (3), with control law (4) and state estimate (1) for each Agent i of the MAS. If communications are triggered when the CTC ( 9) is verified then, for all initial conditions x 0 ∈ R N , a bounded consensus is obtained for the MAS and the disagreement vector δ converges to a ball centered at the origin with radius r = L √ N c 0 /λ 2 (L) . Moreover the MAS does not exhibit Zeno behavior.

Proof. From ( 6), one has δ (t) = ẋ (t)-ȧ(t)1 N . Assuming that the same information is received by all its neighbor agents when each Agent i broadcasts a message (no time delay, no packet loss) and that the communication graph G is balanced, then it can be shown that

N i=0 u i = 0. In this case, ȧ(t) = 0 and δ(t) = ẋ(t). Since ẋ (t) = -Lx (t) = -L (x (t) + e (t)), one obtains δ (t) = -Lδ (t) -Le (t) (10)
and thus δ (t) = e -Lt δ (0) -t 0 e -L(t-s) Le (s) ds. Knowing the estimation error is bounded by the trigger function due to the CTC, |e i (t) | ≤ ẽi (t) ≤ (c 0 + c 1 e -αt ) and using Lemma 1, an upper bound for the disagreement vector δ can be obtained similarly to [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF]:

δ (t) ≤ e -λ 2 t δ (0) -L √ N c 0 λ 2 + c 1 λ 2 -α + e -αt L √ N c 1 λ 2 -α + L √ N c 0 λ 2 ≤ δ (0) + L √ N c 0 λ 2 + c 1 λ 2 -α = δ (11)
From (5), we have

e i (t) = xi (t) -x i (t) = x i t i k + w i t i k -x i (t)
. The aim here is to show that the inter-event times can be lower bounded by a strictly positive constant τ . If i triggers at time t * > 0, then e i (t * ) = w i (t * ), ẽi (t * ) = w and f i (t * , ẽi (t * )) ≤ 0. Knowing this and the time-derivative of e i (t) given by ėi (t) = -ẋi (t) = -u i (t), for t between two event times,

|e i (t) | ≤ |w i (t * ) | + t t * |u i (s) | ds ≤ w + t t * |u i (s) | ds = ẽi (t) (12)
the control is bounded by [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF])

|u i (t) | ≤ u (t) ≤ L δ + √ N (c 0 + c 1 ) = ũ. ( 13
)
We assume here c 0 = 0. For t ≥ t * and before the next event time, ( 12) and ( 13)

gives |e i (t) | ≤ ẽi (t) ≤ w + (t -t * ) ũ.
The event is triggered when CTC is satisfied i.e. not before ẽi (t) > c 0 , and therefore not before w + (t -t * ) ũ > c 0 . For c 0 > w, the inter-event times are lower bounded by τ (t * ) = c0-w ũ > 0. This bound holds for all event times t * and all agents i. Thus we get the condition c 0 > w ≥ 0 (14) for exclusion of the Zeno behavior. Moreover, since e (t) is piecewise continuous and the right hand side of ( 10) is globally Lipschitz in δ, existence and uniqueness of the solution is guaranteed as said in [START_REF] Khalil | Nonlinear systems[END_REF]. Thus, δ (t) converges exponentially to a ball of radius r = L √ N c0 λ2 as t → ∞.

Guaranteed bounds on the state of the MAS and on the consensus

Assuming that the bound w is known by all the agents, this section aims at exploiting this information along with the constraint imposed by the CTC to derive bounds on the state of the agents and on the consensus value. These bounds are computed in a distributed way by all the agents, and updated dynamically based on the application of Muller's theorem [START_REF] Müller | Über das fundamentaltheorem in der theorie der gewöhnlichen differentialgleichungen[END_REF]), by [START_REF] Kieffer | Guaranteed nonlinear state estimation for continuous-time dynamical models from discrete-time measurements[END_REF] between two triggering instants inside the MAS.

Let us also define by K the index of the K-th message broadcast in the MAS, whatever the broadcasting agent. Consider t ∈ [t K , t K+1 [. Each Agent i of the MAS will compute a lower bound x i (t) and an upper bound x i (t) of x(t) using ẋi (t) = -Lx i (t), ẋi (t) = -Lx i (t) (15) At the trigger instant t K , Agent i updates its control input u i and the j-th component of x i and x i by using the information xj (t K ) sent by the Agent j that triggered the communication at t K :

x i j (t K ) = xj (t K )-h(t K ), x i j (t K ) = xj (t K )+h(t K ) (16) with h(t K ) = c 0 + c 1 e -αt K .
Since the graph G is fully connected and there is no communication delay, using this proposed protocol, all the agents update simultaneously the same component of the lower and upper bound of x. Therefore the x i will be identical at t K and ∀t ∈ [t K , t K+1 [ for all the Agents i of the MAS. The same consideration stands for the upper bound x i . Let us denote by x(t) and x(t) these vectors over [t K , t K+1 [. It is assumed that all agents trigger a communication at the initial instant t = 0. Consider the CTC (9) of triggering Agent j.

At time t K one has ẽj (t K ) -h(t K ) ≤ 0. Since |e j (t)| ≤ ẽj (t), one can deduce -h(t K ) ≤ e j (t K ) ≤ h(t K ).
Recalling that e j (t) = xj (t) -x j (t), one obtains

x j (t K ) = xj (t K ) -h(t K ) ≤ x j (t K ) ≤ xj (t K ) + h(t K ) = x j (t K ) (17) Therefore x(t K ) ≤ x(t K ) ≤ x(t K ).
(18) In addition, it can be shown that (15) define a lower and upper dynamical system of ẋ = -Lx, in the sense that ẋ = -Lx ≤ ẋ = -Lx ≤ ẋ = -Lx (19) where the inequalities are interpreted component by component. Using Muller's Theorem between two triggering instants, one can conclude that x(t) is bounded by x(t) and x(t) as

x(t) ≤ x(t) ≤ x(t), ∀t ∈ [t K , t K+1 [ . ( 20
) These bounds allow to find an estimation of the state of the MAS, which can be computed in a distributed way by each agent. Based on this knowledge, we derive bounds on the value of the consensus. As shown in the proof of Theorem 3, for t ∈ [t K , t K+1 [, one has ȧ(t) = 0. Then the agreement value a of the consensus remains invariant between two triggering instants and can be defined as

a(t) = a(t K ) = (1/N )1 T N x(t K ).
Extension of the proof results to a(t) and a(t) is straight forward and we can define a(t

K ) = (1/N ) 1 T N x (t K ) and a(t K ) = (1/N ) 1 T N x (t K ). From (18), one has a(t K ) ≤ a(t K ) ≤ a(t K ).
(21) Let us assume that for each agent of the MAS, the realization of its corresponding noise at each triggering instant is such that there exists at least one agent of the MAS for which the control input, as computed by (4), is non zero1 . For such an Agent i, one has therefore |u i (t K ) | > 0. Then, as t increases, ẽi (t) evaluated from (12) also increases and reaches the broadcast threshold h(t) which is decreasing with time. The CTC (9) is hence verified and another communication is triggered. Therefore the number K of triggered communications inside the MAS increases with time t. As t → ∞, h(t) → c 0 , and one can find some integer m > 0 such that ∀K ≥ m, h(t K ) ∼ c 0 . In this case, using (17), one has

a(t K ) = 1 N 1 T N x(t K ) ∼ 1 N 1 T N x(t K ) -c 0 (22) a(t K ) = 1 N 1 T N x(t K ) ∼ 1 N 1 T N x(t K ) + c 0 (23)
It can be concluded that, as t → ∞, the width of the interval [a(t K ), a(t K )] tends to 2c 0 .

Simulation example

A simulation example is proposed in this section to illustrate the proposed approach. A network of five agents with a fully-connected communication graph G is considered. Initial conditions are chosen randomly. An uniformly distributed random noise within [-w, w] is used for the perturbation w i affecting the state estimate of each Agent i, with w = 9.10 -5 . Triggering function parameters are set as c 0 = 0.0001 = 1.1 w, c 1 = 0.2499, α = 0.9λ 2 (G) = 0.45.

Figure 1 shows evolution of agents' states to the consensus, and the triggering instants defined from the evaluation of the distributed CTC (9). A set of 200 simulations gives an average number of communications of 8.64% using the proposed new CTC and in presence of noise -with 100% being the percentage in case periodic communications at each sampling period. Figure 2 presents the evolution of the bounds x 1 and x 1 on the state of Agent 1. Similar behaviour is obtained for all the agents. Figure 3 shows the evolution of the consensus value and its bounds. As can be observed, the bounds on the consensus value tend to an interval of width close to 2c 0 .

DOUBLE-INTEGRATOR AGENTS

Consensus & communication triggering condition

Consider first the noise-free case for agents with doubleintegrator dynamics described as ẋi (t) = 0 1 0 0

x i (t) + 0 1 u i (t) , ( 24
)
where

x i (t) = [ξ i (t) ζ i (t)]
T ∈ R 2 is the state of Agent i and u i ∈ R its control input. [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF] proposed the event-based implementation

u (t) = -L ξ (t) + diag(t -t 1 k , ..., t -t N k ) ζ (t) + µ ζ (t) (25)
with µ > 0 and with the stack vectors u = [u 1 , . . . , u N ] T , ξ = [ ξ1 , . . . , ξN ] T and ζ = [ ζ1 , . . . , ζN ] T and where estimates ξi (t) and ζi (t) are defined between two triggering instants by ξi (t) = ξ i (t i k ) and ζi (t

) = ζ i (t i k ), ∀t ∈ [t i k , t i k+1 [. Introducing the measurement errors e ξ (t) = ξ (t) + diag(t -t 1 k , ..., t -t N k ) ζ (t) -ξ (t) e ζ (t) = ζ (t) -ζ (t) ( 26
)
allows to re-write (25) as u (t) = -L (ξ (t) + µζ (t) + e ξ (t) + µe ζ (t)). Using this control law, the closed-loop dynamics of the second-order multi-agent system are

ξ ζ = Γ ξ ζ - 0 0 L L e (t) ( 27
)
with e = [e T ξ µe T ζ ] T and Γ defined as in Lemma 2. A CTC is defined in Th. 5.2 of [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF] as |, for all initial conditions ξ 0 , ζ 0 ∈ R N . Moreover, the MAS does not exhibit Zeno behavior. This result is obtained in the noise-free case.

f i (t, e ξ,i (t) , e ζ,i (t)) = e ξ,i (t) µe ζ,i (t) -c 0 + c 1 e -αt (

Presence of unknown but bounded measurement noise

Considering now that state estimates are defined as ξi

(t) = ξ i (t i k ) + w ξ,i (t i k ), ζi (t) = ζ i (t i k ) + w ζ,i (t i k ) (29) ∀t ∈ t i k , t i k+1
for each Agent i of the MAS, where w ξ,i and w ζ,i are some additive unknown but bounded noises. Let us denote wξ = sup i∈V ( sup

k∈N * |w ξ,i t i k |) and wζ = sup i∈V ( sup k∈N * |w ζ,i t i k |
) the respective bounds on all the possible realizations of w ξ,i and w ζ,i for all agents of the MAS and all communication times. To introduce a new CTC for this problem, let us first define

|U i (t)| = µ t t i k -u i (s)ds , |V i (t)| = wζ (t -t i k ) + t t i k |U i (s)|ds, ( 30
) e ζ,i (t) = wζ + |U i (t) |, e ξ,i (t) = wξ + |V i (t) | and e i = [e ξ,i , µe ζ,i ].
Based on these notations, a new CTC to be evaluated by Agent i is introduced:

f i (t, e i (t)) = e i (t) ∞ -c 0 + c 1 e -αt > 0 (31) with constants c 0 > max( wξ , wζ ) ≥ 0, c 1 ≥ 0, c 0 + c 1 > 0 and 0 < α < |Re(λ 3 (Γ))|.
Theorem 4. Consider the multi-agent system (24) with the control law (25) and state estimates (29). If communications are triggered when the CTC (31) is verified then, for all initial conditions ξ 0 , ζ 0 ∈ R N , a bounded consensus is obtained for the MAS and the disagreement vector δ converges to a ball centered at the origin with radius

r = c 0 c v √ 2N L /|Re (λ 3 (Γ)) |
, where c v is defined as in Lemma 2. Moreover the MAS does not exhibit Zeno behavior.

Proof. Similarly to the case with single-integrator agents, the disagreement dynamics is defined as

δ (t) = Γδ (t) - 0 0 L L e (t) (32)
and its analytical solution is

δ (t) = e Γt δ (0) - t 0 e Γ(t-s) 0 0
L L e (s) ds. Similarly to Seyboth et al. ( 2013), Lemma 2 yields δ (t) ≤ k 1 + k 2 e -αt + k 3 e Re(λ3(Γ))t , with Re (λ 3 (Γ)) < -α < 0 and

k 1 = c0cv √ 2N L |Re(λ3(Γ))| , k 2 = c1cv √ 2N L |Re(λ3(Γ))+α| , k 3 = c v δ (0) positive constants.
An upper bound u (t) of the control u (t) is obtained as (see [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF] > 0. This bound holds for all event times t * and all agents i. Thus we get the condition c 0 > max ( wξ , µ wζ ) for exclusion of the Zeno behavior. Moreover, e (t) is piecewise continuous and the right hand side of ( 32) is globally Lipschitz in δ, existence and uniqueness of the solution is guaranteed as said in [START_REF] Khalil | Nonlinear systems[END_REF]. Thus, δ (t) converges exponentially to the ball δ < k 1 as t → ∞.

): u (t) ≤ 1 + µ 2 L (k 1 + k 2 + k 3 ) + √ 2 L (c 0 + c 1 ) = ũ.

Guaranteed bounds on the state of the MAS and on the consensus

As for the single-integrator case, one assumes the graph G is fully connected and there is no communication delay. Denote by K the index of the K-th message broadcast in the MAS, whatever the broadcasting Agent.

ξ ζ = Γ ξ ζ - 0 0 L L H (t) , ξ ζ = Γ ξ ζ + 0 0 L L H (t) (36)
with H (t) a vector of size 2N which all components are h (t). And because G is fully-connected, (36) becomes

ξ ζ = Γ ξ ζ , ξ ζ = Γ ξ ζ (37)
At the trigger instant t K , Agent i updates its control input u i and the j-th component of ξ, ζ and ξ, ζ by using the information ξj (t K ), ζj (t K ) sent by the Agent j that triggered the communication at t K :

ξ j (t K ) = ξj (t K ) -h(t K ), ξ j (t K ) = ξj (t K ) + h(t K ) ζ j (t K ) = ζj (t K ) -h(t K )/µ, ζ j (t K ) = ζj (t K ) + h(t K )/µ (38) with h(t K ) = c 0 + c 1 e -αt
(t K )|, µ|e ζ,j (t K )|) -h(t K ) ≤ 0. Since max(|e ξ,j (t K )|, µ|e ζ,j (t K )|) ≤ max(|e ξ,j (t K )|, µ|e ζ,j (t K )|), one can deduce -h(t K ) ≤ e ξ,j (t K ) ≤ h(t K ) and -h(t K ) ≤ µe ζ,j (t K ) ≤ h(t K ). Therefore ξ j (t K ) = ξj (t K ) -h(t K ) ≤ ξ j (t K ) ≤ ξ j (t K ) = ξj (t K ) + h(t K ) ζ j (t K ) = ζj (t K ) -h(t K )/µ ≤ ζ j (t K ) ≤ ζ j (t K ) = ζj (t K ) + h(t K )/µ (39) so ξ(t K ) ≤ ξ(t K ) ≤ ξ(t K ) and ζ(t K ) ≤ ζ(t K ) ≤ ζ(t K ).
In addition, it can be shown that (37) define a lower and upper dynamical system for ( 27), in the sense that

ξ ζ = Γ ξ ζ ≤ ξ ζ - 0 0 L L e (t) ≤ ξ ζ = Γ ξ ζ ( 40
)
where the inequalities are interpreted component by component. Using Muller's Theorem between two triggering instants, one can conclude that

ξ ζ ≤ ξ ζ ≤ ξ ζ , ∀t ∈ [t K , t K+1 [ (41)
These bounds allow to find an estimation of the state of the MAS, which can be computed in a distributed way by each agent. Based on this knowledge, one would like now to derive some bounds on the value of the consensus. From [START_REF] Seyboth | Event-based broadcasting for multi-agent average consensus[END_REF]), for t between two events t K and t K+1 , the state vector can be expressed ξ

(t) = a(t)1 N + b(t)t1 N + δ ξ (t) and ζ (t) = b(t)1 N + δ ζ (t)
where δ ξ and δ ζ are block vectors composing δ such that 1 T δ ξ (t) = 0 and 1 T δ ζ (t) = 0, and where b

(t) = b(t K ) = (1/N )1 T N ζ(t K ) and a(t) = a(t K ) = (1/N )1 T N ξ(t K ) are invariant quantities i.e for t ∈ [t K , t K+1 [, ζ i (t) ∼ b(t K ) and ξ i (t) ∼ a(t K ) + b(t K )t as t → ∞.
Using the same scheme as in Section 4.3, it can be shown from (39) that:

a(t K ) ≤ a(t K ) ≤ a(t K ) b(t K ) ≤ b(t K ) ≤ b(t K ). ( 42
)
and therefore

b(t K ) = 1 N 1 T N ζ(t K ) ∼ 1 N 1 T N ζ(t K ) -c 0 /µ b(t K ) = 1 N 1 T N ζ(t K ) ∼ 1 N 1 T N ζ(t K ) + c 0 /µ (43) a(t K ) = 1 N 1 T N ξ(t K ) ∼ 1 N 1 T N ξ(t K ) -c 0 a(t K ) = 1 N 1 T N ξ(t K ) ∼ 1 N 1 T N ξ(t K ) + c 0 (44)
So as t → ∞, the width of intervals [b(t K ), b(t K )] tends to 2c 0 /µ and [a(t K ), a(t K )] tends to 2c 0 . triggering function (31) allows to reach the consensus and that communications are triggered regularly. The average rate of communication is of 9.99% for 3 agents. Figure 5 presents the evolution of the bounds ξ 1 , ξ 1 and ζ 1 , ζ 1 on the state of Agent 1. Similar behaviour is obtained for all the agents. Figure 6 shows evolution of the consensus values and their bounds.

Simulation example

CONCLUSIONS

A new distributed event-triggered communication method based on an existing work allowing multi-agent systems to reach the consensus in case where agent's state measurements are subject to unknown but bounded noise has been introduced. Based on bound on measurement error, conditions for design of a new CTC has been determined. From this mechanism, guaranteed bounding for consensus has been obtained using upper and lower dynamical systems.

Simulations have shown the efficiency of the new strategy.

Further works include considering more complex CTCs or other state estimators as in [START_REF] Viel | New state estimator for decentralized eventtriggered consensus for multi-agent systems[END_REF].

Figure 1 .

 1 Figure 1. Evolution of the agents' states to the consensus and their triggering instants.

Figure 2 .

 2 Figure 2. Evolution of x 1 (t), x 1 (t) and x 1 (t).

Figure 3 .

 3 Figure 3. Evolution of a(t), a(t) and a(t).

  28) where e ξ,i and e ζ,i respectively stand for the i-th component of e ξ and e ζ and with constants c 0 ≥ 0, c 1 ≥ 0, c 0 + c 1 > 0 and 0 < α < |Re (λ 3 (Γ)) |. Using this CTC, Th. 5.2 of Seyboth et al. (2013) ensures that the disagreement vector δ of the closed loop system converges to a ball centered at the origin with radius r = c 0 c v √ 2N L /|Re (λ 3 (Γ))

Figure 4 .

 4 Figure 4. Simulation result for single-integrator agents.

Figure 5 .

 5 Figure 5. Evolution of ξ 1 (t), ξ 1 (t), ξ 1 (t) and ζ 1 (t), ζ 1 (t), ζ 1 (t).

Figure 6 .

 6 Figure 6. Evolution of b(t), b(t), b(t) and a(t), a(t), a(t).

  Againwe have to prove that the Zeno behavior is excluded. For that purpose, let us compute a bound on e i (t) ∞ for each Agent i and for t between two triggering instants. Denote by t * the last triggering instant.Let us define e ζ,i (t) = wζ + |U i (t) | and ẽζ,i (t) = wζ + ũ (t -t * ). Using the fact that ėξ,i (t) = e ζ,i (t), + c 1 ) (t -t * ). We get from (33):e i (t) ∞ ≤ max e ξ,i (t), µe ζ,i (t) ≤ max (ẽ ξ,i (t), µẽ ζ,i (t))Here, we assume c 0 = 0. The event is triggered when CTC is satisfied i.e. not before e i (t) ∞ > c 0 , and therefore not before max (ẽ ξ,i , µẽ ζ,i ) > c 0 . The inter-event times are lower bounded by τ (t * ) = max

			t		t
	e i (t) ∞ = max	ėξ,i (s) ds , µ	ėζ,i (s) ds	(33)
			t *	t *
	Using the fact that µ ėζ (t) = -µu(t), one has
	t		t
	µ	ėζ,i (s) ds = µ	-u i (s) ds
	t *		t *
	µ|e ζ,i (t) -w ζ,i (t t	t	t	t
	ėξ,i (s) ds =	e ζ,i (s) ds ≤	wζ ds +	|U i (s) | ds
	t *		t *	t *	t *
	|e ξ,i (t) -w ξ,i (t * )| ≤ |V i (t) |
	|e ξ,i (t) | ≤ wξ + |V i (t) | ≤ wξ + (c 0 + c 1 ) (t -t * )
					(35)
	Let us introduce e ξ,i (t) = wξ + |V i (t) | and ẽξ,i (t) = wξ +
	(c 0 c0-wξ c0+c1 ,	c0-µ wζ µũ

* )| = µ |U i (t) | µ|e ζ,i (t) | ≤ µ wζ + |U i (t) | ≤ µ wζ + ũ (t -t * )

(34)

  Consider t ∈ [t K , t K+1 [. Each Agent i of the MAS will compute lower bounds ξ(t), ζ(t) and upper bounds ξ(t), ζ(t) of ξ(t), ζ(t), which are identical for all agents, using

  K . It is assumed that all the agents trigger a communication at the initial instant t = 0. Consider the CTC (31) of triggering Agent j. At time t K , max(|e ξ,j

In practice, this assumption is likely to be verified, e.g. if the w i are independent randomly distributed variables.


Daniel Maghanjo Mwamidi

Pablo Domínguez

Can

Can Pastoral Communities Offer Sustainable Ecological Management Solutions? The case of Mwanda-Marungu Pastoral Commons in Taita Hills, Kenya

Introduction

Indigenous peoples' and Community Conserved Areas and territories (ICCAs) provide myriad of ecosystem services to millions of people worldwide and are now emerging as one of the key paradigms of 'Other Effective area-based Conservation Measures' (OECMs) promoted by policy makers like the IUCN among others. Despite emerging interests in ICCAs, few studies in the past dedicated to study pastoral commons, yet pastoralism is practiced in 70% of Kenya's landmass whilst contributing to over 13% of Kenya's economy [START_REF] Fratkin | East African Pastoralism in Transition: Maasai, Boran, and Rendille Cases[END_REF], and pastoralism being a typical system for the commons. Lack of recognition to pastoral commons is linked to the assumption that pastoralism harms the environment through overgrazing because local communities can't manage sustainably (eg. Hardin 1969;Stiles, 1984). Nevertheless, we have seen that pastoralism can contribute environmentally wise (e.g. forest regeneration through seed dispersal by livestock, carbon sequestration, promoting nutrient cycling across tropical savanna, etc. [START_REF] Mcgahey | Pastoralism and the Green Economya natural nexus?[END_REF]. Consequently, we set to test two hypotheses: 1.-pastoral commons exist in an abundant manner in Kenya despite the surprisingly little scientific attention they have received up to now. 2.-inversely as the State thinks, they can support socio-ecological benefits (e.g. environmental conservation, banks of economic resources, cultural identity..).

We focused a first fieldwork among the Daasanach tribe in North-West Kenya in 2016 and proofed intense existence of pastoral commons with a myriad ecosystem services, at the same time as how State conservation policies had damaged severely these systems, for example through the creation of the Sibiloi National Park, results which we published in the Journal PARKS (2017). In 2018, we moved to the opposite extreme of the country in South-East Kenya to explore the possible existence of new pastoral commons in the Taita tribe, demonstrating again their important presence, and confirming our first and second hypotheses. In the subsequent section, we present our methods and results in our latest field of Taita that culminated to the conclusion that "Kenya is possibly home to numerous pastoral commons even if underrepresented scientifically, and could be potentially recognized as 'Other Effective area-based Conservation Measures' OECMs).

th Egerton University International Conference on 24 th to 26 th ,

November, 2020, Nakuru, Kenya.

Materials and methods

. Ethnographic methods: participant observation and semi-structured interviews with validation of data conducted through fifteensessions of focus group discussions of between 6 to 10 respondents per session. Sample size n = 234, data organized in themes/domains as described by CBD's Aichi Target 11 guidelines and explored whether the identified pastoral commons fit the criteria of ICCAs and OECMs. This included field visits to the common territories themselves and de visu landscape analysis

Conclusion

We conclude that more research is necessary to find ways of strengthening customary management regimes such as these pastoral commons, which seem important to conserve the present pastoral landscapes we studied. The goal would be to promote the present sustainable provision of ecosystem services and socio-cultural identity attached to them, and which seem to guarantee a certain environmental conservation whilst assuring livelihoods and even possible future human development (social, economic, cultural, etc.). They are in fact facing real urgent challenges that I have been able to verify on the ground, such as generational shift, acculturation, climate change and even explicit landgrabbing. International organizations such as the ICCA Consortium and the WCMC through the International ICCA Registry that they promote or the IUCN through the OECM approach that it is also putting forward, could maybe contribute to reinforce these so valuable (bio-physically and socio-culturally) African systems.

Results

We identified three pastoral Commons (Map 3), but in this first phase focused principally on Mwanda-Marungu commons which we found the one having most elements described by ICCAc, as well as meeting the Aichi's Target 11 guidelines on OECMs. Mwanda-Marungu pastoral resources are managed by a Council of Elders (Waghosi Wa Kireti meaning "Elders of Wilderness"). These Elders have overall rights over three villages, and each village abides by the rules and customary norms set by them. But within villages, there are other elders with lower degree who report to Waghosi wa Kire. These three villages include Itinyi and Maranu gha Mka Mwasi, located on higher ecological zone, and Marungu located on the plains which borders Tsavo west National Park.

Elements that we found to go in tandem with ICCA/OECM criteria in Mwanda-Marungu Pastoral commons: i) Environmental conservation; 2) Good community governance structure of commons and 3) The commons offer a myriad of socio-ecological benefits to local community.

At the same time, rules consist on: 1) No fire burning/charcoal/smoking cigarettes within community protected areas. 2) Grazing on the upper/hilly ecological zone (in Itinyi and Maranu gha Mka Mwasi villages) is prohibited during wet seasons and all herders are to graze in Marungu village at that time. 3) Herders have their specific days to take livestock to the determined water points and saltlicks. 4) Elders report that they have set rules that demands all herders and livestock to have vacated water/saltlick points by 3pm afternoon so as to pave way for wildlife, thus minimizing human-wildlife conflicts. 5) No non-village member is allowed to graze inside the community land. 6) The use of religious shrines called fighis that all together help to conserve forests and pastoral habitats. 7) No grazing in one area for over 2 days so as to minimize pasture degradation and 8) Resources (wild vegetables, fruits, insects, herbal medicine, water, trees, wildlife) found in their territory belongs to all members and none has more utility rights over the other, hence being obliged to protect them from husbandry depredation.

Discussion

Our study, contrary to what the literature seems to suggest up to present, confirms the important presence of pastoral commons in Kenya through a fine grain ethnographic verification of their existence in two very separated areas of the country, at the same time as our knowledge of their presence in many other sites of Kenya. With elaborate form of customary management strategies that are aimed at environmental protection, wildlife co-existence, governance structure that guarantee sustainable functioning of these commons and consensus on utilization of resources in their community landscape, it gives at the same time sense of identity and existence. This confirms Kenyan pastoral commons as deeply enrooted in local religion/culture being key for communities' material and immaterial reproduction. All of it reinforces the hypothesis that Taita commons could very probably be considered OECMs (currently under study).

Fig: 1

 1 Fig: 1 and 2: Pastoral commons of Mwanda-Marungu in Taita hills, Southern Kenya © Daniel Mwamidi

Fig 3 and 4 :

 4 Fig 3 and 4: Livestock on a 'common' salt lick site (Left: Taita, Southern Kenya; Right: Daasanach, Northern Kenya) © Daniel Mwamidi

Map1:

  Map indicating our first area of study in 2016 among the Daasanach, Northern Kenya © Daniel Burgas; Fig. 5: Aerial view of Daasanach riverine ecosystem which is protected by the community © Daniel Mwamidi

  Map 2 and 3: The map of Kenya indicating the study area (on yellow spot); and GoogleEarth map image of Taita hills indicating three ICCAs © Google

Figure 7 and 8 :

 8 Figure 7 and 8: From left to right: Semi structured interviews; and Focus group discussion with the local community members © Daniel Mwamidi

Table 1 : Respondents structure in our study in Taita hills © Field work.

 1

Can Pastoral Communities Offer Sustainable Ecological Management Solutions? The case of Mwanda-Marungu Pastoral Commons in Taita Hills, Kenya Daniel

  Maghanjo Mwamidi : PhD candidate at Institute of Environmental Science and Technology (ICTA), Autonomous University of Barcelona, Spain. Pablo Dominguez : Senior Researcher at Laboratory of Geography of the Environment (GEODE), Centre National de la Recherche Scientifique (CNRS), France / Linked researcher of the Institute of Environmental Science and Technology (ICTA) and Social and Cultural Anthropology Department, Autonomous University of Barcelona, Spain.

	Men	Women			Age	Total
			< 31	31-50	> 50
	Semi-structured interviews	97	40	30	85	22	137
	Focus group discussions (15)	82	27	23	58	28	109


domain: spi phys

text:
Thierry Nuns

Christophe Inguimbert
email: christophe.inguimbert@onera.fr

Juan Barbero

Juan Moreno

Samuel Ducret

Alexandru Nedelcu

Bjorn Galnander

Elke Passoth

Displacement Damage Effects in InGaAs Photodiodes due to Electron, Proton, and Neutron Irradiations

Keywords: Space Environment, Displacement Damage, gamma rays, Non Ionizing Energy Loss. I

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

the number of produced defects is proportional to the nonionising dissipated energy. The incident particles interact with nuclei of the target material thanks to nuclear interactions (Coulombian, nuclear elastic and inelastic). According to binary collision approximation, the number of displaced atoms is equal to the dissipated energy divided by twice the minimum energy required to displace an atom (T d ). To the first order, the final number of defects is shown to be proportional to the number of atomic displacements and thus to energy dissipated in terms of nuclear interactions, i.e. Displacement Damage Dose (DDD). The displacement damage dose deposited per incident particle is given by Non Ionizing Energy Loss (NIEL) [START_REF] Messenger | Non ionizing energy loss (NIEL) for heavy ions[END_REF]- [START_REF] Boschini | Nuclear and non-ionizing energy-loss for Coulomb scattered particle from low energy up to relativistic regime in space radiation environment[END_REF], which is shown to be the best parameter to scale degradations induced in microelectronic devices by atomic displacements. The wellknown "NIEL scaling approach", that consists in considering that damage factor is proportional to NIEL, is shown to work quite well in most of cases [START_REF] Srour | Review of displacement damage effects in silicon devices[END_REF]- [START_REF] Messenger | Modeling solar cell degradation in space: A comparison of the NRL displacement damage dose and JPL equivalent fluence approaches[END_REF]. Apart from some deviations observed for electrons [START_REF] Thommen | Recovery of low temperature electron irradiationinduced damage in n-type GaAS[END_REF]- [START_REF] Allam | Gamma and Electron NIEL Dependence of Irradiated GaAs[END_REF] in both Si and GaAs, high energy protons in GaAs [START_REF] Summers | Displacement damage in GaAs structures[END_REF]- [START_REF] Barry | Energy dependence of lifetime damage constants in GaAs LEDs for 1-500 MeV protons[END_REF] and neutrons in GaAs [START_REF] Ougouag | Differential displacement kerma cross sections for neutron interactions in Si and GaAs[END_REF], the NIEL scaling approach is proven to be effective for silicon and gallium arsenide material, for which it is widely used [START_REF] Srour | Review of displacement damage effects in silicon devices[END_REF]- [START_REF] Messenger | Modeling solar cell degradation in space: A comparison of the NRL displacement damage dose and JPL equivalent fluence approaches[END_REF].

It is with electrons that deviation from NIEL scaling is most frequently observed, whatever the semiconductor Si [START_REF] Srour | A framework for understanding displacement damage mechanisms in irradiated silicon devices[END_REF] [START_REF] Summers | Damage correlations in semiconductors exposed to gamma, electron and proton radiations[END_REF] [START_REF] Summers | displacement damage analogs to ionizing radiation effects[END_REF] [START_REF] Arnolda | NIEL scaling: comparison with measured defect introduction rate in silicon[END_REF] and GaAs [START_REF] Thommen | Recovery of low temperature electron irradiationinduced damage in n-type GaAS[END_REF]- [START_REF] Xapsos | Co60 Gamma Ray and Electron Displacement Damage Studies of Semiconductors[END_REF][18] [START_REF] Warner | A comparison between p+n and n+p GaAs displacement damage coefficients following proton irradiation[END_REF] [START_REF] Allam | Gamma and Electron NIEL Dependence of Irradiated GaAs[END_REF].

Furthermore, space applications in infrared domain have been growing rapidly, requiring good knowledge of resistance to radiation of semiconductor materials such as II-VI (HgCdTe), or III-V (InGaAs). Compared to Si and GaAs, InGaAs material has not been extensively studied. Extracting general rules requires large set of data obtained with various particle species and energies [START_REF] Gilard | Damage Factor for Radiation-Induced Dark Current in InGaAs Photodiodes[END_REF].

Useful data in the literature comes from devices having different technological characteristics that can affect measured damage factor. Moreover, damage factors associated to different electrical parameters (dark current, diffusion length) can be mixed and the response to radiation varies from one parameter to another. For example, in solar cells degradation, short circuit current evolution differs from power loss. In addition, test conditions are often not identical. For instance, temperature and annealing time impact directly measurements.

The aim of this work is to measure the dark current degradation of photodiodes from a single set of devices Displacement damage effects in InGaAs photodiodes produced by electrons, protons and neutrons irradiations T. Nuns 1 , C. Inguimbert 1 , J.Barbero 2 , J. Moreno 2 , S. Ducret 3 , A. Nedelcu 3 , B. Galnander 4 , E. Passoth 4 E (batch), irradiated and tested with identical conditions. A large set of irradiation conditions is applied on this set of devices. It is then possible to analyse validity of NIEL scaling approach for InGaAs material, based on a large set of irradiation data. The detailed structure of each photodiode is not known (confidential data from manufacturer). Therefore only a relative comparison between these different devices is proposed. We should also remark that dark current increase is not the only degradation that may occur on such devices. Indeed, quantum efficiency may be affected. This parameter is related to the carriers' lifetime and recombination processes in regions where their concentration is in excess, whereas our study focused on dark current in depleted regions.

Three device types coming from different manufacturers (Excelitas, OSI, and LYNRED) have been tested with electrons of energies 0.5, 1.5, 6, 12, and 20 MeV. They have also been irradiated with protons of 60, 100 and 170 MeV, and with atmospheric-like neutron spectrum (equivalent damage energy of ~2 MeV). Damage factor has been extracted and compared to NIEL calculated with NEMO [START_REF] Inguimbert | NEMO: A code to compute NIEL of protons, neutrons, electrons, and heavy ions[END_REF] package of ONERA implemented in OMERE [START_REF] Omere | Outil de Modélisation de l'Environnement Radiatif Externe[END_REF] toolkit.

Section II briefly describes the method for calculating NIEL with a focus on neutrons case. Section III presents the experimental setup, describing devices, irradiation facilities, test and irradiation conditions. Section IV presents dark current increase and section V compares dark current damage factor with NIEL.

II. NIEL CALCULATION

NIEL, which is average energy loss imparted in atomic displacement, is classically given by [START_REF] Inguimbert | NEMO: A code to compute NIEL of protons, neutrons, electrons, and heavy ions[END_REF]:

𝑁𝐼𝐸𝐿 =  • ∫ 𝑑𝜎 𝑑𝑄 • 𝐺(𝑄) • 𝑄 • 𝑑𝑄 𝑄 𝑚𝑎𝑥 𝑇 𝑑 (1)
Where  is the atomic density of target material, T d threshold displacement energy and Q max maximum energy level that the incident particles can yield to target's nuclei. d/dQ is the differential interaction cross section per unit of transferred energy, thus Q is the recoil energy imparted to target atoms. G(Q) is the energy partition function that provides part of recoil energy that goes into atomic displacements. The energy partition function of Lindhard [START_REF] Robinson | The dependence of radiation effects on the primary recoil energy[END_REF] is used in our calculation [START_REF] Inguimbert | NEMO: A code to compute NIEL of protons, neutrons, electrons, and heavy ions[END_REF], with threshold displacement energies of 10 eV, 10 eV and 15 eV for respectively Ga, As and In. Other key parameters of the calculation are interaction cross sections d/dQ. For protons, elastic interaction cross sections (Coulombian + nuclear elastic) have been estimated following the partial wave method [START_REF] Inguimbert | NEMO: A code to compute NIEL of protons, neutrons, electrons, and heavy ions[END_REF]. For electrons the Mc Kinley & Fesbasch-screened coulomb cross section expression is employed [START_REF] Mckinley | The Coulomb Scattering of Relativistic Electrons by Nuclei[END_REF]. For nuclear reactions induced by protons and neutrons, GEANT4 10.patch02 library has been used [START_REF]GEANT4 a simulation toolkit[END_REF].

In case of protons and neutrons, but more generally for hadrons above some tens of MeV, nuclear reactions take place quite abundantly and must be taken into account in NIEL calculations. For protons and ions below ~10 MeV, these nuclear reactions can be ignored. Indeed, the Coulombic potential barrier prevents protons from inducing such reactions. This is not the case for neutrons that are able to produce nuclear reactions even at very lower energy levels. The neutron NIEL has been calculated with GEANT4 using FTFP_BERT_HP physic option [START_REF]GEANT4 a simulation toolkit[END_REF]. This option is based on the Bertini Cascade model valid in range [~MeV, [START_REF] Xapsos | Co60 Gamma Ray and Electron Displacement Damage Studies of Semiconductors[END_REF] GeV] thanks to the use of a pre-compound model below ~100 MeV [START_REF] Wright | The Geant4 Bertini Cascade[END_REF]. The High Precision (HP) option allows an accurate simulation of nuclear reactions (elastic + inelastic) thanks to the use of G4NDL nuclear reaction database. To summarize, in the energy range of interest [~100 keV, ~1 GeV] FTFP_BERT_HP physical option of GEANT4 uses the reference Bertini cascade model above 75 MeV and switch to the pre-equilibrium and nuclear evaporation models native to the Bertini code down to 20 MeV. Below 75 MeV, the former offers better precision than intra-nuclear cascade model better suited for higher energies. G4NDL database is used thanks to the HP option below 20 MeV. G4NDL data comes from different releases of evaluated data libraries (e.g. ENDF/B-VII.1, JEFF-3.3, JENDL-4.0, etc.), which have been converted into Geant4 format [START_REF]GEANT4 a simulation toolkit[END_REF]. HP database (ENDF files) reconstructs the resonance cross section observed at low energy according to procedure similar to NJOY [34] for both elastic and inelastic reactions [START_REF]GEANT4 a simulation toolkit[END_REF]. Validations of those models can be found in [START_REF] Catanesi | Large-angle production of charged pions with 3-12.9 GeV/c incident protons on nuclear targets[END_REF].

NIEL calculations have been validated by comparison with neutron NIEL from SR-NIEL code [START_REF]Screened Relativistic (SR) Nuclear Stopping Power Calculator[END_REF] and ASTM standard [START_REF]Standard Practice for Characterizing Neutron Fluence Spectra in Terms of an Equivalent Monoenergetic Neutron Fluence for Radiation-Hardness Testing of Electronics[END_REF]. As can be seen in Fig. 1, agreement is very good for silicon material. For compound materials A x B y comprising two elements A and B with respective stoichiometry x and y, NIEL is simply the weighted sum of NIELs of single elements constituting the material:

𝑁𝐼𝐸𝐿 𝐴 𝑥 𝐵 𝑦 = 𝑥 𝑀 𝐴 𝑀 𝐴 𝑥 𝐵 𝑦 𝑁𝐼𝐸𝐿 𝐴 + 𝑦 𝑀 𝐵 𝑀 𝐴 𝑥 𝐵 𝑦 𝑁𝐼𝐸𝐿 𝐵 (2)
Where M A , M B and M AxBy are respectively the molar mass of elements A, B. Calculations have been performed for GaAs and InGaAs materials. For GaAs, similarly to silicon material, comparison has been performed with SR-NIEL code [START_REF]Screened Relativistic (SR) Nuclear Stopping Power Calculator[END_REF] and ASTM standard [START_REF]Standard Practice for Characterizing Neutron Fluence Spectra in Terms of an Equivalent Monoenergetic Neutron Fluence for Radiation-Hardness Testing of Electronics[END_REF] data. The agreement with KERMA of both references is quite good (cf. Fig. 2). A maximum difference of ~30 % is observed with our calculations. This is [START_REF] Ougouag | Differential displacement kerma cross sections for neutron interactions in Si and GaAs[END_REF], [START_REF]Standard Practice for Characterizing Neutron Fluence Spectra in Terms of an Equivalent Monoenergetic Neutron Fluence for Radiation-Hardness Testing of Electronics[END_REF] and with damage function of ASTM E722-14 calculated with SR-NIEL code [START_REF]Screened Relativistic (SR) Nuclear Stopping Power Calculator[END_REF]. In order to compare different functions in absolute terms, the damage function is presented in the figure divided by normalization factor 2.2 used in ASTM. Calculations have been next extended to InGaAs material (Fig. 3) for a chemical composition In x Ga 1-x As with x = 0.53. As can be seen in Fig. 3, NIEL of GaAs and InGaAs are very close to each other.

InGaAs NIEL was compared with experimental data (cf. section V). Some devices have been irradiated with atmospheric like ANITA neutron spectrum of the TSL facility. This spectrum is presented in section III. For comparison with experimental measured damage factors, the equivalent damage energy E q with ANITA spectrum has been calculated according to the following formula:

∫ 𝑑𝛷 𝑑𝐸 𝐸 𝑚𝑎𝑥 𝐸 𝑚𝑖𝑛 (𝐸)) 𝐴𝑛𝑖𝑡𝑎 •𝑁𝐼𝐸𝐿 𝑛,𝐼𝑛𝐺𝑎𝐴𝑠 (𝐸)•𝑑𝐸 ∫ 𝑑𝛷 𝑑𝐸 𝐸 𝑚𝑎𝑥 𝐸 𝑚𝑖𝑛 (𝐸)) 𝐴𝑛𝑖𝑡𝑎 = 𝑁𝐼𝐸𝐿 𝑛,𝐼𝑛𝐺𝑎𝐴𝑠 (𝐸 𝑞 ) (3)
The equivalent energy of ANITA spectrum has been estimated to E q ≈ 2 MeV and used to scale measured induced damage factors in InGaAs.

III. EXPERIMENTAL SETUP

A. Tested photodiodes

Photodiodes tested in this study come from three different manufacturers: Lynred, Excelitas and OSI.

Lynred [START_REF]Lynred[END_REF] provided test vehicles manufactured in the same technology as SNAKE device. This device is a new generation of VGA (Video Graphics Array) InGaAs FPA (Focal Plane Array) sensor with high level of sensitivity and resolution dedicated to low flux applications. It operates in the 0.9 µm to 1.7 µm wavelength domain. Each test vehicle includes a set of InGaAs PIN photodiodes having different topologies. Some of them are circular single photodiodes with different dimensions (diameter in the range [4 µm, 300 µm]). Some of them are 10x10 sub-arrays of pixels with various pitches (in the range [10 µm, 30 µm]). Whereas circular photodiodes are addressed separately, all pixels of sub-array are addressed in parallel (all together) so that the measured current is the sum of the contribution of all photodiodes in the sub-array. We limited our measurements to six photodiodes or sub-arrays per test vehicles, selected within the available topologies. We did not have enough identical test vehicles in order to cover all irradiation conditions. Some slight differences occur from one sample to another, in terms of photodiode topology. As a consequence, measured photodiodes may not be exactly the same for each irradiation. Anyway, these differences allow us to focus on damage factor and influence of the photodiode sizes on degradation. All measurements presented here are performed at -5 V.

The second tested device is PIN photodiode C30618 from Excelitas [START_REF]EXCELITAS: Excelitas technologies[END_REF]. This high speed single InGaAs circular photodiode is designed for use in OEM fiber-optics communications systems and high-speed receiver applications. It operates between 1000 nm and 1600 nm and has a diameter of 350 µm and maximum intrinsic dark current of typical ~1 nA with an applied voltage of -10V (bias at which current measurements were made).

The last tested device is FCIQ1000 from OSI Optoelectronics [41]. FCI-InGaAs-QXXX series are large active circular area InGaAs photodiodes segmented into four quadrants. As a consequence, each device has four photodiodes, all measured independently in our study. The Q1000 device active diameter is 1000 µm and presents a 0.5 nA dark current at -5V (bias for the current measurements). The Q1000 photodiode is optimized for good responsivity from 1100 nm to 1620 nm.

The two latter devices have been selected because of their relatively large active areas with several hundreds of micrometers of diameter, and present relatively low intrinsic dark currents. The goal was to have optimal conditions in order to get the greatest sensitivity to radiation. Because all OSI or Excelitas irradiated samples are identical whatever irradiation conditions (in the contrary of SNAKE test vehicles), obtained data are used for direct comparison of degradation with bias, annealing, particle type and energy, bias conditions effects.

B. Irradiation facilities

The three devices presented in previous section have been irradiated with protons, electrons, neutrons and gammas thanks to five different facilities. Two electron energies (0.5 and 1.5 MeV) were performed at GEODUR facility of ONERA (Toulouse, France) [START_REF] Geodur | Chambre d'irradiation électrons multi applications (matériaux, composants[END_REF]. Van de Graaff accelerator provides mono-energetic beams and fluxes can be selected from 1 up to 100 nA/cm².s (10 9 -10 12 e -/cm².s) with nonuniformity lower than 10% within a beam diameter of 16 cm. Flux is measured thanks to Faraday cups. Global uncertainty of deposited fluence over samples is better than ±20%. Three higher electron energies (6, 12 and 20 MeV) were performed at RADEF facility (LINAC of University of Jyväskylä, Finland) [START_REF]Jyväskylän Yliopisto[END_REF]. This pulsed electron beam provides average dose rates ranging from 1 up to 10 Gy(water)/min. The beam non-uniformity and fluence measurement discrepancy is lower than ±5% over 20 cm x 20 cm. Some devices have also been irradiated with protons of 60, 100, and 170 MeV, as well as with atmospheric-like neutron spectrum of The Svedberg Laboratory (TSL, Uppsala, Sweden) facility [START_REF]The ANITA neutron beam facility[END_REF]. Protons are produced with Gustaf Werner cyclotron providing beam intensities from 10 µA down to few protons per second to PAULA user facility (Proton fAcility in UppsaLA). Uncertainty on the deposited fluence due to dosimetry and beam non-uniformity was ±15% for the largest beam diameter (~20 cm). Neutrons are delivered to Quasi-Monoenergetic Neutron (QMN) facility and ANITA facility (Atmospheric-like Neutrons from thIck Target). Spallation neutrons from ANITA spectrum are created with tungsten target. Neutron beams have continuous spectrum (Fig. 4) going from thermal energies up to ~180 MeV. The neutron flux was in the order of few 10 6 cm - 2 s -1 with a beam spot non-uniformity of ±5% within ~20 cm diameter. The part of thermal neutron flux represents less than 1% of the total integrated flux. Global uncertainty of deposited fluence was ±10%. Based on NIEL of InGaAs material, the equivalent damage energy of the ANITA spectrum has been estimated to be around 2 MeV. This value is subjected to uncertainties from dosimetry (energy distribution and fluence level) and InGaAs NIEL estimation related to the choice of the nuclear database.

Fig. 4: Atmospheric-like neutrons spectrum provided by ANITA facility. Depending on tested material, the equivalent damage energy is found to be in the order of 2 MeV.

Some Total Ionizing Dose (TID) tests have also been performed with the CNA 60 Co facility (Centro Nacional de Aceleradores, Sevilla, Spain [42]), in order to check the sensitivity of devices to TID effect. Irradiation took more than four months to reach the final 5240 Gy level so that we did not perform the 24 h-annealing step recommended in ESA ESCC22900 specification, but devices were measured after two months annealing at room temperature. Irradiated devices have been placed inside lead/aluminum box. They have been covered by a sheet of PMMA (poly(methyl methacrylate)) in order to achieve conditions of equilibrium. Dosimetry was performed before irradiation using ionization chamber placed in the same conditions (i.e. inside a lead box). The accuracy of the dosimetry is within 5%.

C. Test conditions

Photodiodes have been irradiated with two different bias conditions. Some devices have been shorted and others irradiated with a reverse bias. This is same bias conditions as those applied during dark current measurements: -10 V for the C30618 (Excelitas) and -5 V for both SNAKE (SOFRADIR) and FCIQ1000 (OSI). These biases correspond to large depletion regime dominated by generation current. Irradiations have been performed at ambient temperature (~20°C). One device is irradiated per bias and energy. All devices are irradiated after removing the lid (no shielding between the beam and the photodiode). For each step of measurement, one non-irradiated sample was measured as a reference. Electrical measurements were performed with devices in complete dark condition inside a box with ambient temperature close to 24°C. Excelitas and OSI devices were inserted in aluminium block machined to ensure good thermal contact with devices. This block was equipped with a thermistor and TEC, stabilizing temperature to exactly 24°C and ensuring variation is lower than 0.1°C between measurements. SNAKE test vehicles were just put into the box, leading to larger uncertainty of the device temperature during measurements and much worse reproducibility. For the InGaAs gap, considering SRH carrier generation with activation energy close to mid-gap, an error of 1°C around 24°C corresponds to 5% error on dark current (close to 10% for 2°C).

In this paper, dose values are given for InGaAs unless otherwise stated. As can be seen in TABLE I, applied fluences correspond to quite high levels of irradiation compared to most space radiation environments. The goal is to have a significant dark current increase that ensures some relevant measurements. But we avoid reaching and exceeding fluence of 10 12 particle/cm² in order to prevent any significant effect of carriers' removal. This phenomenon, which occurs for high fluence and subsequent high displacement damage doses, should not impact directly on dark current increase of photodiodes. For protons and high energy electrons, the fluences have been chosen in order to deposit same TID levels (~320 Gy). Hence, the final fluence and displacement damage dose are not identical from one irradiation to another. For low

IV. EXPERIMENTAL RESULTS

A. Dark current increase with fluence or ionizing dose

Shown in Fig. 5 are plots of OSI photodiodes responses for three different irradiation conditions: 170 MeV protons; neutrons from ANITA; Co-60 gammas. These three irradiations are selected because several fluences or dose levels are available. Initial dark current is between 0.1 and 0.5 nA over different samples. We can see that dark current increase is linear with fluence or TID. This trend is true for the three types of photodiodes. This linear dependence will allow extraction of damage factor as described in section V.

Annealing

Final TID level for gamma irradiations (5240 Gy) is much higher than TID reached after proton and electron irradiations (close to 320 Gy for protons and up to 500 Gy for electrons). Dark current increase is much lower for 500 Gy gamma than for other irradiations, except for 0.5 MeV electron where degradation is similar. The conclusion is that, for 1.5 MeV or higher electrons energies, protons and neutron irradiations, dark current degradation is dominated by displacement damage effects rather than TID effects.

The comparison between 0.5 MeV electrons and gamma irradiations will now be studied in more details for OSI and Excelitas devices (identical irradiated samples). TABLE III gives dark current increase per device after 0.5 MeV 500 Gy (final step) and gamma dose 430 Gy (closest step). TABLE III compares absolute values and variations for biased and unbiased devices. We can see that electrons and gamma rays induce similar dark current increases (difference of 30% before annealing and less than 10% after two month annealing). We must note that gamma irradiation can deposit displacement damage dose thanks to the interaction of secondary electrons with matter. This DDD has been evaluated using GaAs NIEL (9×10 -8 MeV.cm²/g [START_REF] Allam | Gamma and Electron NIEL Dependence of Irradiated GaAs[END_REF] and TABLE I). Indeed, because NIEL of electrons in InGaAs and GaAs are close to each other, NIEL of gamma rays in these two materials should be close also. Both 500 Gy electron and 430 Gy gamma doses depict almost the same dark current increase while deposited DDD is five time larger for incident electrons (1.7×10 +07 Gy vs. 3.3×10 +06 Gy). We conclude here that ionizing effects are dominant for gamma irradiation.

We should note that dose rates between gamma and electrons are different (500 to 1,000 times higher for electron) and that irradiation yields are also slightly different. Nevertheless, we conclude that TID effects are not negligible for 0.5 MeV electron irradiation, and may be the dominant contribution to dark current increase. This will be developed when comparing damage factors with NIEL in section V.

B. Bias effect

Bias effect is only observable when both ON and OFF irradiations are made for a given particle type and energy. Thus, this analysis was performed for the three device types exposed to neutrons, protons, and gamma while only for OSI and Excelitas photodiodes with low energy electron irradiations (0.5 and 1.5 MeV, see Recovery processes at room temperature have been shown to be negligible even two months after irradiation. This is also shown on Fig. 5. The detail of annealing percentage is given in TABLE V. Most of the time, annealing is close to 5% of the final dark current level just after irradiation for OSI and Excelitas devices. Two inconsistent annealing values are observed with low energy electrons. We suspect an error in thermal regulation during annealing measurements that were made at the same time on the same facility. Annealing of SNAKE samples is close to 10%, which is within the uncertainty of the device temperature. Results presented in the rest of the paper correspond to measurements two months after irradiation.

C. Annealing effect

D. Effect of the geometrical dimensions of photodiodes

In this section, we specifically investigate the relative degradation of SNAKE photodiodes with respect to active surface. Indeed, all photodiodes on test vehicles are made in the same technology, i.e. vertical dimensions of different photodiodes are the same. Main difference is linked to topology (circular or square shapes), surface and structure in the vicinity (for sub-arrays of pixels). So, if dark current increase is due to defects in semiconductor volume, it should be proportional to the surface of the different elements.

The comparison with OSI and Excelitas devices is not possible here because we do not have any information on device structure, except the active surface. As a consequence, any differences in doping concentrations, vertical dimensions… will induce difference within active volume. I(V) curves of all patterns from two test vehicles irradiated with 60 MeV protons (biased and unbiased) were measured several months after irradiation. The two test vehicles, labelled "Sample 1" and "Sample 2", come from two different InGaAs wafers. Fig. 6 shows the normalized dark current degradation: dark current after irradiation, much larger than dark current before irradiation (see Fig. 5), is divided by diode surface (area of the P-type region). The figure presents responses curves of both biased and unbiased test vehicles (during irradiation). All normalized curves follow the same trend within a factor of two. This confirms that increase of dark current is due to defects produced in the volume of photodiodes. Note that measurements of dark current presented in previous sections on SNAKE test vehicles were made at -5 V. We should note that dark current increase is due to both ionizing (Δ𝐼 𝑑𝑎𝑟𝑘,𝑖 ) and non-ionizing (Δ𝐼 𝑑𝑎𝑟𝑘,𝑛𝑖 ) effects:

Δ𝐼 𝑑𝑎𝑟𝑘 (Φ) = Δ𝐼 𝑑𝑎𝑟𝑘,𝑖 (Φ) + Δ𝐼 𝑑𝑎𝑟𝑘,𝑛𝑖 (Φ) (5)
As a consequence, the measured damage factor is the sum of ionizing and non-ionizing contributions:

𝐾 𝐼 𝑑𝑎𝑟𝑘 = 𝐾 𝐼 𝑑𝑎𝑟𝑘,𝑖 + 𝐾 𝐼 𝑑𝑎𝑟𝑘,𝑛𝑖 [START_REF] Srour | Review of displacement damage effects in silicon devices[END_REF] with i and ni referring respectively to ionizing and nonionizing effects. 𝐾 𝐼 𝑑𝑎𝑟𝑘,𝑖 can be defined because dark current increase is proportional to TID for gamma irradiations, where ionizing effects are dominant.

We are interested here in the estimation of 𝐾 𝐼 𝑑𝑎𝑟𝑘,𝑛𝑖 , hence any ionizing contribution in the measurement of Δ𝐼 𝑑𝑎𝑟𝑘 (Φ) introduces an error in this estimation. Nevertheless, for all irradiations but 0.5 MeV electrons, dark current increase was dominated by displacement damage. In these cases, 𝐾 𝐼 𝑑𝑎𝑟𝑘,𝑖 ≪ 𝐾 𝐼 𝑑𝑎𝑟𝑘,𝑛𝑖 and 𝐾 𝐼 𝑑𝑎𝑟𝑘 ≃ 𝐾 𝐼 𝑑𝑎𝑟𝑘,𝑛𝑖 . For 0.5 MeV electrons, 𝐾 𝐼 𝑑𝑎𝑟𝑘,𝑖 is not negligible anymore. Because it is difficult to determine ionizing contribution in dark current increase, we can only remark that 𝐾 𝐼 𝑑𝑎𝑟𝑘,𝑛𝑖 < 𝐾 𝐼 𝑑𝑎𝑟𝑘 . As a conclusion, the evaluation of 𝐾 𝐼 𝑑𝑎𝑟𝑘,𝑛𝑖 is not distorted by ionizing effects for all irradiations except the 0.5 MeV electron irradiation where it is overestimated.

Annealing after irradiation at room temperature in silicon device has been previously reported and synthesis can be found in [START_REF] Srour | Review of displacement damage effects in silicon devices[END_REF]. It shows that annealing process decreases with time and is very slow after one month following irradiation. Even if few studies report significant annealing at room temperature in InGaAs materials, Yue et al. [START_REF] Yue | Study on annealing effects of irradiated Ga0.5In0.5P/GaAs/Ge solar cell by 170 keV proton[END_REF] shows that annealing occurs in III-V multi-junction solar cells at higher temperatures. In order to take into account possible annealing processes occurring shortly after irradiation and considering test conditions in [START_REF] Srour | Review of displacement damage effects in silicon devices[END_REF] and [START_REF] Srour | A framework for understanding displacement damage mechanisms in irradiated silicon devices[END_REF], damage factors have been evaluated months after irradiation. Anyway, the previous section showed that annealing is small for our devices.

To evaluate damage factors, dark current increase has been divided by the surface of photodiodes. For OSI and Excelitas devices, optimal surface provided in datasheet have been used (350 µm diameter, 9.62 10 -4 cm² for Excelitas, 4 quadrants of one 1000 µm diameter, 7.85 10 -3 cm² each). For SNAKE test vehicles, surfaces of photodiodes have been provided by LYNRED. These values are smaller than the pitch of subarrays (this point will be discussed in paragraph V.B). Bulk damage are relative to the volume of the active region of devices. This volume was known for SNAKE devices, but not for commercial OSI and Excelitas photodiodes. Indeed, type and concentrations of dopants and layer thicknesses are probably different from one manufacturer to another. This prevented any absolute comparison of damage factors between the SNAKE and two other devices. Nevertheless, assuming a common depth of the active region for all photodiodes inside a SNAKE test vehicle, normalizing dark current increase with surface is relevant in order to compare the different topologies of SNAKE photodiodes. This last section is devoted more specifically to the analysis of NIEL scaling approach validity. The Experimental damage factors for different particle types and energies are compared to their corresponding InGaAs NIEL. This comparison is performed relatively to one given particle type and energy (Part 0 , E 0 ). The damage factors of a given particle (Part, E). are then normalized relatively to reference particle NIEL (Part 0 , E 0 ) according to following formula: 𝑟𝑒𝑙 𝐾 𝐼 𝑑𝑎𝑟𝑘 (𝑃𝑎𝑟𝑡. , 𝐸) = 𝐾 𝐼 𝑑𝑎𝑟𝑘 (𝑃𝑎𝑟𝑡.,𝐸) 𝐾 𝐼 𝑑𝑎𝑟𝑘 (𝑃𝑎𝑟𝑡 0 .,𝐸 0 ) NIEL(𝑃𝑎𝑟𝑡 0 . , 𝐸 0 ) [START_REF] Srour | A framework for understanding displacement damage mechanisms in irradiated silicon devices[END_REF] Where 𝑃𝑎𝑟𝑡 0 and 𝐸 0 are the particle type and energy used as reference. The chosen reference particle is 2.07 MeV neutron because it is the equivalent energy of ANITA spectrum (NIEL 𝑛 0 ,2.07 𝑀𝑒𝑉 = 5.71×10 -4 MeVcm²/g). This choice has been made because no ionizing effects are expected for this kind of particle. More precisely, 𝐾 𝐼 𝑑𝑎𝑟𝑘 (𝑛°, 2.07 𝑀𝑒𝑉) is the mean damage factor averaged for each device type (over different bias and topologies). Nevertheless, because equivalent fluence for neutron irradiations is subject to uncertainties (as presented in III.B), and in order to better correlate damage factors with both protons and electrons NIELs, the chosen normalisation factor was half the 2.07 MeV neutron NIEL (2.65×10 - 4 MeVcm²/g). This will be detailed in paragraph V.E.

Assuming a maximum error of dosimetry of ±20% and error in dark current measurement of ±10%, we evaluate the error on damage factor as the quadratic sum of these two errors, close to ±22% (error bars in Fig. 7 to Fig. 10).

B. SNAKE relative damage factor

SNAKE devices present the advantage to propose different topologies. That leads to a scatter of dark current measurements. As already explained, dark current increase has been normalized to the surface of depleted region in order to achieve a proper comparison of data, regardless of photodiodes size. This point is highlighted in first part of this section. Fig. 7 (a) presents the damage factor of all available data. As explained at the end of section V.A., experimental relative damage factors have been normalized to half of the 2.07 MeV neutron NIEL in order to better scale data with proton NIEL. Dispersion within data can be observed in Fig. 7 (a). Dispersion is quite significant for low energy electrons (factor of five). Unlike to OSI and Excelitas samples that have both large dimensions, some SNAKE samples have small diffusion diameters down to 4 µm. Any error in the evaluation of diffusion diameter could lead to error in damage factor. Such a difference could occur if theoretical and effective diameters differ due for example to manufacturing processes.

In Fig. 7 (b), an offset of 2.5 µm has been added to the diffusion diameter. This correction was suggested by the manufacturer. Indeed, this small value has great impact on small dimensions and the effect is clearly homogenization of the results. For the smallest photodiodes (4 µm diameter), an error of 2.5 µm in diameter induces an error of 164% in the photodiode surface. This of course directly impacts on the comparison of damage factors from one topology to another. But for large photodiode dimensions (SNAKE, OSI and Excelitas), the diffusion area is close to optimal diameter of the topology and these two surfaces are almost the same. An error of 3 µm in diameter induces an error smaller than 2% in photodiode surface. The offset of 2.5 µm will be applied in the rest of the analysis for the SNAKE photodiodes.

C. Proton relative damage factor for all device types

In Fig. 8, damage factors are compared with NIEL for protons. The same normalization factor as the one used in Fig. 7 is applied (i.e. half of the neutron NIEL for all devices). One can see that, for the three device types (OSI, Excelitas, SNAKE), damage factors follow total NIEL scaling. This is a difference compared to other publications that reported damage factor decreasing with proton energy above some tens of MeV, sometimes following the curve of NIEL Coulombian part.

D. Electron relative damage factor for all device types

In Fig. 9, the case of electrons is shown. We should note that, for such a comparison, we used different scale factors for OSI and Excelitas devices than for SNAKE. This can clearly be noticed in the following section where all damage factors are presented together. The consequence here is that one proposes a comparison of the damage factors variation with NIEL. On the studied domain of energy, variation of the experimentally measured damage factors is greater than the one of electron NIEL. Indeed, the amplitude of damage factors variation between 0.5 MeV and 20 MeV is close to two decades, whereas variation of NIEL is close to ten. The measured damage factors behaviour is close to NIEL², which is also represented on the figure (dashed line). Such behaviour has been observed for silicon devices [START_REF] Srour | Review of displacement damage effects in silicon devices[END_REF]. An interpretation was given in [START_REF] Inguimbert | Effective NIEL in Silicon: Calculation using molecular dynamic results[END_REF], related to worse calculation of low NIEL values. Finally, Fig. 10 compares damage factors all together. The normalization factor is the same for the three device types, i.e. half of the 2.07 MeV InGaAs neutron NIEL. In this case, we see that 1) proton data fit with total NIEL, 2) neutron data are lower than neutron NIEL and 3) electron data are spread around electron NIEL. For the best fit, we could have chosen to normalize data with electron or neutron results. This would have shifted data up or down. But at this stage, it is difficult to conclude on the best way to normalize data. Anyway, because of the complex spectrum of neutron irradiations and the uncertainties in evaluation of equivalent energy and fluence, the difference between neutron NIEL and damage factors should not be considered as abnormal (factor two). But one must retain that the gap between proton and electron damage factors is not the same for OSI and Excelitas devices on one side, and SNAKE test vehicles on the other side. Fig. 10 shows that a factor close to four between all these data remains after normalization. One possible explanation could be due to different structures between three device types. But, as presented in first section, these structures are not available easily so that it is difficult at this stage to investigate deeper these discrepancies.

E. Relative damage factors for all devices and particle types

VI. CONCLUSION

This work shows a large set of irradiation data i.e. dark current increase in different InGaAs photodiodes. The current increases linearly with fluence or TID. The degradation of different photodiode topologies is consistent with degradation in the volume of devices. Experimental damage factors are compared with NIEL for InGaAs material. Results show some differences in the ratio between proton and electron results from device type to the next. For high energy protons, the damage factors follow the total NIEL scaling. For electrons, NIEL overestimates low energy damage factors which seem to fit with a square function of NIEL.

Fig. 1 :

 1 Fig.1: NIEL of neutrons in silicon. Comparisons of our GEANT4 calculation with ASTM KERMA[START_REF]Standard Practice for Characterizing Neutron Fluence Spectra in Terms of an Equivalent Monoenergetic Neutron Fluence for Radiation-Hardness Testing of Electronics[END_REF].

Fig. 3 :

 3 Fig. 3: Comparison of NIEL of neutrons in GaAs and InGaAs. Calculations are performed with our GEANT method.

Fig. 5 :

 5 Fig. 5: Dark current of photodiodes FCIQ1000 (OSI) irradiated with (a) 170 MeV protons, (b) ANITA neutron spectrum and (c) gamma rays. Each figure details mean response over the four photodiodes of each device, irradiated biased or unbiased. The caption indicates bias condition during irradiation. Responses are linear as a function of DDD or TID.

Fig. 6 :

 6 Fig. 6: I(V) curves of the different photodiode patterns. The current is normalized by the surface of diodes (i.e. the diffused P-type area). Measurements at 22°C. V. DARK CURRENT DAMAGE FACTOR A. Definition and evaluation Damage factor is commonly used to compare degradation of devices for different particle types and energies due to displacement damage effects [6]-[8][16][18][19][23]. In our case, damage factor is calculated by dividing dark current increase by the applied fluence Φ. 𝐾 𝐼 𝑑𝑎𝑟𝑘 = 𝐼 𝑑𝑎𝑟𝑘 (Φ)-𝐼 𝑑𝑎𝑟𝑘 (0) Φ

Fig. 7 :

 7 Fig. 7: Comparison between dark current increase experimental damage factors for SNAKE test vehicles and NIEL of InGaAs. (a) using the diffusion surface evaluated by LYNRED and (b) correcting the diameter by 2.5 µm.

Fig. 8 :

 8 Fig. 8: Comparison between dark current increase experimental damage factors and NIEL of InGaAs for protons. SNAKE data correspond to mean value over different sizes of photodiodes.

Fig. 9 :

 9 Fig. 9: Comparison between dark current increase experimental damage factors and NIEL of InGaAs for electrons. SNAKE data correspond to mean value over different sizes of photodiodes.

Fig. 10 :

 10 Fig. 10: Comparison between dark current increase experimental damage factors and NIEL of InGaAs for neutrons, protons and electrons. SNAKE data correspond to mean value over different sizes of photodiodes.

Non Ionizing Energy Loss (MeV/g.cm²) Energy (MeV)

	not surprising since the different calculations employed
	different nuclear reaction databases.
	Fig. 2: NIEL of neutrons in GaAs. Comparison of our GEANT4 calculations
	with ASTM KERMA
	1,E-02 10 -2
	1,E-03 10 -3				Silicon
	1,E-04 10 -4
	1,E-05 10 -5
					Si, ASTM E722
	1,E-06 10 -6				Si, this work with GEANT4
	0,001 10 -3	0,01 10 -2	0,1 10 -1	1 10 0	10 10 1	100 10 2	1000 10 3

  TABLE I and TABLE II summarize different irradiation steps.

  energy electrons and gamma irradiations, the final dose is higher. But we provide intermediate steps at 300 Gy for electron and 450 Gy for gamma irradiations so that a comparison of degradation at same ionizing dose remains possible. In a global point of view, intermediate fluence steps with dark current measurements in between have been performed for neutron spectrum, 170 MeV protons, gamma rays, 0.5 MeV and 1.5 MeV electrons irradiations. These measurements were performed in order to check linearity of dark current increase as a function of fluence. The maximum applied displacement damage dose has been deposited with ANITA neutron spectrum (~2.4 10 +10 MeV/g). Annealing has been investigated thanks to measurements performed within one hour after the end of irradiation and two months later. The first step depends on access time to irradiated samples (venting and radio activation relaxation).

		1,E+10 10 10
	(neutrons/cm²/MeV)	1,E+08 1,E+09 10 9 10 8		Anita neutron differential spectrum
	distribution	1,E+07 10 7
	Differential	1,E+06 10 6		energy in term of displacement damage for GaAs, InGaAs, GaP, Area representing the equivalent
		1,E+05 10 5		InGaAsP, InGaAlP, InGaN, AlGaN
		0,1 10 -1	1 10 0	10 10 1	100	1000 10 3
				Energy (MeV)

TABLE I :

 I DDD LEVELS APPLIED ON PHOTODIODES FOR ALL IRRADIATIONS.

	Particle	Energy	Fluence	Fluence	TID	DDD
	type	(MeV)	(p./cm²)	accuracy	level	level
					(Gy)	(MeV/g)
	Neutrons	Spectrum	3 10 +11	10%	NA	6.9 10 +8
	(ANITA)		3 10 +12			6.8 10 +9
			10 +13			2.4 10 +10
	Protons	60	3 10 +11	15%	320	1.1 10 +9
	(PAULA)	100	4.3 10 +11		320	1.5 10 +9
		170	3 10 +11		152	8.7 10 +8
			5 10 +11		254	1.4 10 +9
			6.3 10 +11		320	1.8 10 +9
	Electrons	0.5	10 +12	20%	200	6.7 10 +6
	(ONERA)		1.5 10 +12		300	1.0 10 +7
			2.5 10 +12		500	1.7 10 +7
		1.5	5 10 +11		100	1.5 10 +7
			10 +12		200	2.9 10 +7
			1.5 10 +12		300	4.3 10 +7
			2 10 +12		400	5.8 10 +7
	Electrons	6	1.18 10 +12	5%	325	7.1 10 +7
	(RADEF)	12	9.22 10 +11		333	7.1 10 +7
		20	7.16 10 +11		339	6.4 10 +7
	60 Co (CNA)	1.25	up to ~4.5 10 +14 (*)	5%	100 430	up to ~4 10 +7 (*)
					1050
					2720
					5240

(*) value estimated using GaAs NIEL instead of InGaAs one.

TABLE II :

 II LIST OF IRRADIATIONS PER DEVICE TYPE.

	Particle	E (MeV)	SNAKE	OSI		Excelitas
			OFF	ON	OFF	ON	OFF	ON
	n°	Spectrum	X	X	X	X	X	X
	P+

Gamma 1.25 MeV

				Protons 170 MeV
		6,0E-07 6×10 -7						Mean OFF
		5,0E-07 5×10 -7						Mean ON
		4,0E-07 4×10 -7
	(A)
	Idark	3,0E-07 3×10 -7
		2,0E-07 2×10 -7
		1,0E-07 1×10 -7
		0
		0	1E+11 2E+11 3E+11 4E+11 5E+11 6E+11 7E+11 8E+11 9E+11 1×10 11 2×10 11 3×10 11 4×10 11 6×10 11 7×10 11 Annealing 5×10 11
				Fluence (p./cm²)
				Neutrons spectrum
		3,0E-06 3.0×10 -6						Mean OFF
		2,5E-06 2.5×10 -6						Mean ON
		2,0E-06 2.0×10 -6
	(A)
	Idark	1,5E-06 1.5×10 -6
		1,0E-06 1.0×10 -6
		5,0E-07 0.5×10 -6
		0,0E+00 0
		0		5E+12 5×10 12		1E+13 1×10 13	Annealing	1,5E+13
				Fluence (p./cm²)
		1,4E-08 1.4×10 -8						Mean OFF
		1,2E-08 1.2×10 -8						Mean ON
		1,0E-08 1.0×10 -8
	(A)	8,0E-09 0.8×10 -8
	Idark	6,0E-09 0.6×10 -8
		4,0E-09 0.4×10 -8
		2,0E-09 0.2×10 -8
		0,0E+00 0
		0	100	200	300	400	500	600	700
					Dose (krad)

TABLE III :

 III COMPARISON OF 430 GY GAMMA AND 500 GY 0.5 MEV ELECTRON IRRADIATIONS FOR OSI AND EXCELITAS PHOTODIODES DARK CURRENT.

	Device	Gamma	0.5 MeV e-500 Gy	0.5 MeV e-500 Gy
		430 Gy	before annealing	after annealing
		I (nA)	I (nA)	Comp. 	I (nA)	Comp. 
	Excelitas	1.58	2.28	30.7%	1.52	-4.1%
	OFF
	Excelitas	1.17	1.73	32.4%	1.27	7.8%
	ON
	OSI OFF	1.13	1.60	29.7%	1.06	-6.8%
	(*)
	OSI ON	0.86	1.28	32.7%	0.96	9.8%
	(*)
	(*) Mean value over the 4 samples of each device.

  TABLE II). Comparison is given in TABLE IV. For proton and neutron irradiations, difference is small and probably within the uncertainty of measurements and temperature control. Difference is more obvious after gamma and low energy electron irradiations where it can reach up to 25%. Overall a lower degradation level is observed for biased devices during gamma irradiations.

TABLE IV :

 IV COMPARISON OF THE EFFECT OF BIAS CONDITIONS BEFORE

				ANNEALING.
	Particle	E (MeV)		OSI(*)			Excelitas
			I (nA)	%	I (nA)	%
			OFF	ON	OFF/	OFF	ON	OFF/
					ON			ON
	n°	Spectrum	2070	2030	2%	2460	2340	5%
	P+	60	202	213	-6%	233	218	6%
		100	289	307	-6%	301	277	8%
		170	454	406	11%	476	419	12%
	e-	0.5	1.6	1.3	20%	2.3	1.7	26%
		1.5	10.6	10	5%	15.5	12.5	18%
	60 Co	1.25	10.9	8.2	25%	14.6	10.7	27%
	(*) Mean value over the 4 samples of each device.

TABLE V :

 V EFFECT OF TWO MONTHS ANNEALING AT ROOM TEMPERATURE.

	Particle	E (MeV)	SNAKE		OSI	Excelitas
			OFF	ON	OFF	ON	OFF	ON
	n°	Spectrum	1.6%	0.6%	-6.0% -9.3% -3.1%
	P+	60	11.5	8.6%	-2.5% -6.5%	0.9%	-4.1%
			%
		100			1.9%	-1.5%	0%	-4.0%
		170	12.4	9.6%	0.5%	-2.6% -1.5% -4.6%
			%
	e-	0.5	18.3		27.4	17.4	33.4	26.7
			%		%	%	%	%
		1.5	10.8		19.5	15.5	18.7	15.0
			%		%	%	%	%
		6	6.2%		6.0%		5.5%
		12	-1.1%
		20	-3.9%		3.8%		2.9%
	60 Co	1.25	-2.4%	0.3%	4.5%	1.9%	2.2%	0.6%

X X X X X X 100 X X X X 170 X X X X X X e-0.5 X X X X X 1.5 X X X X X 6 X X X 12 X 20 X X X 60 Co 1.25 X X X X X X

ACKNOWLEDGEMENTS

Results presented here were obtained in the framework of the EDA contract ROVER A-1341-RT-GP. Authors would like to thanks the European Defence Agency for supporting this study [46].