Aneel Tanwani

Error Bounds for Locally Optimal Distributed Filters with Random Communication Graphs

Keywords: Stochastic hybrid system, distributed filtering, graph theory, random communication

We consider the problem of analyzing the performance of distributed filters for continuous-time linear stochastic systems under certain information constraints. We associate an undirected and connected graph with the measurements of the system, where the nodes have access to partial measurements in continuous time. Each node executes a locally optimally filter based on the available measurements. In addition, a node communicates its estimate to a neighbor at some randomly drawn discrete time instants, and these activation times of the graph edges are governed by independent Poisson counters. When a node gets some information from its neighbor, it resets its state using a convex combination of the available information. Consequently, each node implements a filtering algorithm in the form of a stochastic hybrid system. We derive bounds on expected value of error covariance for each node, and show that they converge to a common value for each node if the mean sampling rates for communication between nodes are large enough.

I. INTRODUCTION

When dealing with systems over networks, the information available for decision making is constrained by the communication channels and the underlying interconnections. Recent advancements in networked systems have led to the motivation for distributed estimation and filtering problems. In this paper, we consider the analysis and design of filters under certain information constraints induced by (a) distributed measurements described by a graph; and (b) frequency of communication between the graph nodes. In the literature on stochastic systems, the filter design, or analysis, has received continued attention. An important question addressed in these works is to quantify the performance of the filters by analyzing the bounds on the evolution of error covariance matrix [START_REF] Taghvaei | Error analysis of the stochastic linear feedback particle filter[END_REF]. Several design techniques have now emerged in the literature with varying degree of analysis. For discrete-time deterministic systems, the estimation problem in distributed setting is studied in [START_REF] Mitra | Distributed observers for LTI systems[END_REF], [START_REF] Park | Design of distributed LTI observers for state omniscience[END_REF]. For continuoustime deterministic systems, one may refer to [START_REF] Han | A simple approach to distributed observer design for linear systems[END_REF], [START_REF] Wang | A distributed observer for a time-invariant linear system[END_REF]. The focus in these works is on proposing estimation algorithms and provide design criteria for each unit which uses minimal information about the centralized system dynamics. In certain papers, distributed state estimation is studied as an application of synchronization of multi-agent systems [START_REF] Kim | Robustness of synchronization of heterogeneous agents by strong coupling and a large number of agents[END_REF], [START_REF] Lee | A tool for analysis and synthesis of heterogeneous multi-agent systems under rank-deficient coupling[END_REF],

A. Tanwani is affiliated with CNRS -LAAS, University of Toulouse, CNRS, 31400 Toulouse, France (e-mail: aneel.tanwani@laas.fr).

This work is sponsored by the project CYPHAI, financed by ANR-JST CREST program with grant number ANR-20-JSTM-0001. [START_REF] Panteley | Synchronization and dynamic consensus of heterogeneous networked systems[END_REF]. Distributed state estimation for deterministic continuoustime systems with communication at discrete times has been studied in [START_REF] Dorfler | Continuous-time distributed observers with discrete communication[END_REF], [START_REF] Li | Robust distributed estimation for linear systems under intermittent information[END_REF]. For stochastic systems, we see consensus based approaches for scalar systems in [START_REF] Carli | Distributed Kalman filtering based on consensus strategies[END_REF]. Distributed filtering with more general probability distributions and application to Gaussian distributions with linear discrete-time systems is studied in [START_REF] Battistelli | Kullback-Leibler average, consensus on probability densities, and distributed state estimation with guaranteed stability[END_REF]. Prior to that, several algorithms for distributed filtering have been proposed in discrete-setting [START_REF] Olfati-Saber | Distributed Kalman filtering for sensor networks[END_REF], [START_REF]Kalman-Consensus Filter : Optimality, stability, and performance[END_REF]. On the other hand, some recent dissertations have addressed the problem of distributed optimization and control over random graphs in discrete-time setting [START_REF] Touri | Product of random stochastic matrices and distributed optimization[END_REF], [START_REF] Etesami | Potential-based analysis of social, communication, and distributed networks[END_REF].

An important issue from the point of implementation of filters over networks is to make the algorithms compatible with the underlying communication protocol. In communicating messages over the networks, the transmission may break down at some time instants, or the messages may not be transferred at exact scheduled times. An abstract way to model such scenarios is to assume that the underlying protocols transmit messages only at randomly drawn discrete time instants. With this motivation, certain works in the literature have studied the problem of stabilization and control with randomly sampled measurements: The reader may refer to [START_REF] Adès | Stochastic optimal control under Poisson-distributed observations[END_REF] for optimal control and the papers [START_REF] Hespanha | Stochastic impulsive systems driven by renewal processes[END_REF], [START_REF] Antunes | Volterra integral approach to impulsive renewal systems: Application to networked control[END_REF] for stability analysis of such systems. A recently published book chapter [START_REF] Tanwani | Stabilization of continuous-time deterministic systems under random sampling: Overview and recent developments[END_REF] provides an overview of such results and some recent developments. There have been relatively fewer works which have addressed filtering problem in the presence of measurement errors or communication uncertainties. The papers [START_REF] Matveev | The problem of state estimation via asynchronous communication channels with irregular transmission times[END_REF], [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF], [START_REF] Huang | Stability of Kalman filtering with Markovian packet losses[END_REF] consider a discrete-time linear dynamical system and associate randomness with the transmission times of the output measurements. A different toolset, based on relative entropy, is adopted in [START_REF] Mcdonald | Stability of non-linear filters, observability and relative entropy[END_REF] to study the stability and convergence of filters under relaxed assumptions on observation channels. For continuous-time dynamical system driven by white noise, centralized continuous-discrete observer is proposed in [START_REF] Jazwinski | Stochastic processes and filtering theory[END_REF]. Some analytical results on the performance of centralized continuous-time systems under random sampling appear in [START_REF] Tanwani | Error covariance bounds for suboptimal filters with Lipschitzean drift and Poisson-sampled measurements[END_REF].

In this article, we consider filtering problem in distributed setting for continuous-time linear stochastic systems. Our objective is to propose filtering algorithms, and analyze their performance, when the centralized measurements are not available, and the agents can only communicate their own state estimates to their neighbors at time instants determined by a Poisson counter. Compared to above mentioned references, the derivation and analysis of the expectation of error covariance bounds with respect to random (Poisson) sampling in distributed setup is a unique feature of our work. The results presented in this article build on our recent work in [START_REF] Tanwani | Suboptimal filtering over sensor networks with random communication[END_REF]. For each edge describing the communication link between two sensor nodes, we model communication times by an independent Poisson counter, and derive the error covariance bounds quantifying the performance of these individual filters. In comparison to [START_REF] Tanwani | Suboptimal filtering over sensor networks with random communication[END_REF], instead of using the constant linear injection gains in the filters, we use timevarying gains that depend on error covariance bounds which makes the filters nonlinear. This motivates the use of term "locally optimal" in the title since we choose injection gains to minimize error covariance between two communication times. Due to this change in filter dynamics, even though the results remain qualitatively similar, the proofs of the main results in this paper require some additional technicalities.

II. LOCALLY OPTIMAL DISTRIBUTED FILTERING

Let us begin with the description of the system class and the formulation of the distributed filtering problem studied in this paper. In the process, we describe the graph structure representing the interconnection between sensor nodes, and the sampling process at which the connected nodes (or the neighbors) exchange information.

A. System Class

We consider dynamical systems modeled by linear stochastic differential equations of the form

dx = Ax dt + B dω (1)
where (x(t)) t≥0 is an R n -valued diffusion process describing the state. Let (Ω, F, P) denote the underlying probability space. It is assumed that, for each t ≥ 0, (ω(t)) t≥0 is a zero mean R m -valued standard Wiener process adapted to the filtration F t ⊂ F, with the property that E[dω(t) dω(t) ] = I m dt, for each t ≥ 0. The matrices A ∈ R n×n and B ∈ R n×m are taken as constant with (A, B) controllable, and the process (ω(t)) t≥0 does not depend on the state. The solutions of the stochastic differential equation ( 1) are interpreted in the sense of Itô stochastic integral.

The centralized output measurement associated to the process (1) is of the form

dy = Hx dt + dv (2)
where H ∈ R p×n is a constant matrix, with (A, H) being observable, and (v(t)) t≥0 is a zero mean R p -valued standard Wiener process. The conventional filtering problem, with initial state having Gaussian distribution, deals with constructing a mean-square estimate of the state x t , denoted by

x t so that E[|x t -x t | 2 | (dy(s)) s≤t ] is minimized.
The optimal estimate which achieves this minimum value is E[x t | (dy(s)) s≤t ] and is computed recursively using a Kalman-Bucy filter. For the problem studied in this paper, it is assumed that the centralized measurements are not available and we address the filtering problem with similar assumptions on system data, but under different information constraints which are described next.

B. Information Structure

The measurements associated with system (1) are obtained from a set of N sensors which are distributed in their localization. Each of these sensors provides a partial measurement about the state described as,

dy i = H i x dt + dv i , i = 1, . . . , N, (3)
where H i ∈ R pi×n , and

N i=1 p i = p.
That is, for each node, (y i (t)) t≥0 describes an R pi -valued continuous-time observation process. In the observation equation ( 3), v i (t) is a zero mean F t -adapted standard Wiener process, taking values in R pi , and E[dv i (t) dv i (t) ] = V i dt, with V i ∈ R pi×pi assumed to be positive definite. The optimal filter which minimizes the mean square estimation error conditioned upon the information available through the measurements {dy i (s)|s ≤ t} is,

d x i (t) = A x i (t)dt + P i (t)H i V -1 i (dy i (t) -H i x i (t)dt) (4a) Ṗi = AP i + P i A -P i H i V -1 i H i P i + BB , (4b) with x i (0) = E[x(0)], and P i (t) is exactly the error cov- ariance E[(x i (t) -x i (t))(x i (t) -x i (t)) | dy i (s), s ≤ t] if P i (0) = E[(x i (0) -x i (0))(x i (0) -x i (0)) ].
a) Communication Graph: The sensor nodes are connected via a graph G = (V , E ), where V = {1, . . . , N } is the set of graph nodes, and E contains all the edges defined by a subset of the pairs (i, j), i = j, i, j ∈ V . We assume that the graph is undirected and connected. The neighbors of a node i ∈ V are denoted by N i and we adopt the convention that i ∈ N i . The adjacency matrix A := [α ij ] ∈ {0, 1} N ×N of the graph, which is symmetric, provides the information about which sensor nodes can communicate with each other, that is, if α ij = 1 then sensor i and j can communicate, whereas α ij = 0 means there is no communication possible between those sensors. The degree of a node i ∈ V is defined as |N i |, that is, the cardinality of the set N i . The diagonal matrix D = [d ii ], with d ii = |N i | is therefore the degree matrix. We associate a Laplacian L with this graph, defined as, L = D -A. For our purposes, the nonnegative matrix

Π = [π ij ] ∈ R N ×N , defined as Π := I N -L (5)
where 0 < ≤ min i∈V 1 |Ni| plays an important role. Note that, by construction, Π is a doubly stochastic matrix, that is, for each row and each column, the sum of their entries equals one.

b) Random Sampling: The next main ingredient of our problem formulation is the description of the time instants at which the communication takes place between two sensor nodes connected by an edge. Corresponding to each edge (i, j) ∈ E , it is stipulated that there is an increasing and divergent sequence (τ ij k ) k∈N ⊂ [0, +∞[ with τ ij 0 := 0, and

• for each (i, j) ∈ E , the sensor nodes i, j ∈ V transmit the value of their state estimate to each other at τ ij k , k ∈ N. In this article, we are interested in the case where the sampling times (τ ij k ) k∈N are generated randomly. Formally, we define

N ij t := sup k ∈ N τ ij k ≤ t for t ≥ 0 (6)
and assume in addition that, tor each

(i, j) ∈ E , (N ij t ) t≥0 is a continuous-time stochastic process such that τ ij N ij t - → + ∞
almost surely as t → +∞. The map t → N ij t increments by 1 at random times, and it provides a description of the number of times the nodes i, j communicate with each other up to and including time t. For the sake of computational tractability, it is stipulated that

• For each (i, j) ∈ E , (N ij t ) t≥0 is an independent Poisson process of intensity λ ij > 0. That is, (N ij t )
t≥0 is a Markov process taking values in N, has independent increments, and satisfies N ij 0 = 0, and for h 0 and t ≥ 0,

P N ij t+h -N ij t = k N ij t =      1 -λ ij h + o(h) if k = 0, λ ij h + o(h) if k = 1, o(h) if k ≥ 2,
where the terms o(h) do not depend on t.

Because of the arrival of new information at random times, the estimate x i , i ∈ V , gets updated. To describe this update rule, we associate with each node i ∈ V , the process

N i t , N i t := j∈Ni N ij t
so that N i t increments by one whenever node i ∈ V exchanges information with any of its neighbor. We recall that N i t is also a Poisson process of intensity λ i := j∈Ni λ ij . The times at which N i t gets incremented are denoted by τ N i t . We can now introduce the activation set A i t ,

A i t := j ∈ N i N ij t -N ij t = 0, t = τ N i t -1
, so that, at communication times t c = τ N i t , the set A i tc describes the neighbors of node i ∈ V that communicate their estimate to node i ∈ V . Consequently, at t c = τ N i t , we update the state estimate as follows:

x i (t + c ) = j∈A i tc π ij x j (t - c ) + 1 - j∈A i tc π ij x i (t - c ), (7)
where π ij are the elements of the matrix Π introduced in [START_REF] Wang | A distributed observer for a time-invariant linear system[END_REF].

If e i := x -x i denotes the estimation error, then because of this update rule, it is observed that,

e i (t + c )e i (t + c ) ≤ 1 - j∈A i tc π ij e i (t - c )e i (t - c ) + j∈A i tc π ij e i (t - c )e i (t - c )
which is a direct consequence of the following lemma, whose proof appears in [27, Lemma III.3]:

Lemma 1. Let m be a positive integer, and let x 1 , . . . ,

x m ∈ R n . If z := m j=1 γ j x j for some γ j ∈ [0, 1], then zz ≤ m j=1 γ j x j x j . (8)

C. Summary of Filtering Algorithm

So far, we have specified the information available to each sensor node and a filtering algorithm, (4), [START_REF] Lee | A tool for analysis and synthesis of heterogeneous multi-agent systems under rank-deficient coupling[END_REF], which uses this available information. If Y i t denotes the information available to sensor node i ∈ V up till time t ∈ [0, +∞[, then we can write

Y i t = {(dy i (s), x j (τ N i s )) | s ≤ t, j ∈ N i }.
Our goal is to quantify the performance of these distributed filters by computing a bound on expected value of the error covariance matrices, that is,

E[E[(x(t) -x i (t))(x(t) -x i (t)) | Y i t ]
], for t ≥ 0. Here, the outer expectation is with respect to the random update times, and the inner expectation is with respect to the noise process in the state and output equation. The estimate computed by each node i ∈ V is obtained by executing the following steps:

• Integrate (4a) and (4b) over the interval [τ

N i t , τ N i t +1 [, • At t c = τ N i
t , reset the state x i via (7), and set

P i (t + c ) = 1 - j∈A i tc π ij P i (t - c ) + j∈A i tc π ij P j (t - c ) (9)
with

P i (0) ≥ E[(x(0) -x i (0))(x(0) -x i (0))].
As a first step in obtaining the desired bounds, one immediately observes that, for each i ∈ V , if we fix the times at which node i ∈ V communicates with its neighbors, then

E[(x(t) -x i (t))(x(t) -x i (t)) | Y i t ] ≤ P i (t)
, t ≥ 0 (10) where P i is described by (4b), [START_REF] Dorfler | Continuous-time distributed observers with discrete communication[END_REF]. The remaining task therefore is to compute expectation with respect to the distributions assigned to the times at which information between sensor nodes takes place.

III. MAIN RESULTS

The basic problem studied in this paper is the performance of the distributed filters proposed in the previous section. In particular, we want to relate the mean sampling rates λ ij , corresponding to the edges (i, j) ∈ E , with the bounds on the error covariance. As our first main result, we compute an upper bound on the expectation (with respect to sampling process N i t ) of the error covariance matrices

E[E[(x(t) - x i (t))(x(t) -x i (t)) | Y i t ]
], for t ≥ 0. Theorem 2. Consider system (1) with distributed measurements (3) and the corresponding hybrid filters (4), ( 7), ( 9) linked together by an undirected and connected graph G = (V , E ). For an edge (i, j) ∈ E , if the communication between nodes i, j ∈ V takes place at random times generated by a Poisson process of intensity λ ij > 0, then

for each i = 1, • • • , N , it holds that E[E[(x(t) -x i (t))(x(t) -x i (t)) | Y i t ]] ≤ P i (t), ( 11
)
where the matrix-valued function

P i : [0, ∞[→ R n×n satisfies the differential equation Ṗi = AP i + P i A -P i H i V -1 i H i P i + BB + j∈Ni λ ij π ij (P j -P i ). (12)
The proof of Theorem 2 is very similar to the proof of [27, Theorem III.1, item 1)] and is omitted due to space constraints. The result of Theorem 2 provides a bound on the evolution of error covariance for each node in terms of a differential equation. These equations are quadratic (and hence nonlinear), driven by a constant term which corresponds to the noise level in the system, and are interconnected by some coupling term. Such systems in the literature are studied under the framework of heterogenous multi-agent systems since the dynamics of P i are different for each i ∈ V . In contrast to homogenous agents, consensus in heterogenous agents is not possible in general. However, one can get the states of all the agents close to desired accuracy by increasing the coupling strength. The next result relates to the asymptotic behavior of the coupled differential equations ( 12). Theorem 3. For i = 1, . . . , N , consider the matrix-valued equations [START_REF] Battistelli | Kullback-Leibler average, consensus on probability densities, and distributed state estimation with guaranteed stability[END_REF] and assume that λ ij = λ is the same for each (i, j) ∈ E . Let S ∈ R n×n be symmetric positive semidefinite matrix satisfying

0 = AS + SA - 1 N S N i=1 H i V -1 i H i S + BB . ( 13
)
Then for every δ > 0, there exists λ > 0 sufficiently large, such that the corresponding solution of (12) satisfies 1 lim sup

t→∞ P i (t) -S ≤ δ. ( 14
)
The proof of Theorem 3 is carried out in Section IV. To conclude this section, we provide some remarks about our main results. Remark 1. The injection used in (4) over an interval

[τ N i t , τ N i t +1
[ do not need any information about how the other filters in the network choose their gains. Moreover, they minimize the value of P i in (4b) over the class of linear time-varying gains. The latter statement follows from the fact that

(A-L i H i )P i +P i (A i -L i H i ) +L i V i L i = (A-L i H i )P i + P i (A i -L i H i ) + L i V i L i -(L i -L i )V i (L i -L i ) (15)
for any constant matrix L i ∈ R n×pi , and

L i = P i H i V -1
i . This shows that the filters (4) perform better than the constant linear gains proposed by the author in [START_REF] Tanwani | Suboptimal filtering over sensor networks with random communication[END_REF]. Remark 2. In Theorem 3, we basically study convergence of the differential equations ( 12) which contain quadratic 1 When taking the norm of a matrix, we refer to Frobenius norm.

nonlinearities. In general, such nonlinearities result in semiglobal convergence, that is, the solutions converge starting from initial conditions in a compact set. However, because of the minimum property described in Remark 1, we get global convergence with no restrictions on the initial condition. However, the convergence is not necessarily asymptotic, but only up to a neighborhood of a fixed point, which we often call practical convergence. The practical aspect of the convergence is unavoidable since we allow different noise covariance levels for each filter. Remark 3. In the formulation of Theorem 2, since we associate a different Poisson process to each link, the nodes communicate with each other at different times. However, we associate the same sampling rate λ with each edge (i, j). The motivation for doing so is that, when we write the collective dynamics for each node, the last term in ( 12) is written as a scalar multiple of the Laplacian. If we assume that the process N ij t associated with edge (i, j) has intensity λ ij , then the arguments required for establishing practical convergence are more involved and are not carried out in this paper. One would expect that if each λ ij > 0 is large enough, then we do get practical convergence.

IV. PROOF OF THEOREM 3

For the proof of Theorem 3, we introduce the following notation:

L i (t) := P i (t)H i V -1 i , A i (t) := A -L i (t)
H i where we will drop the time argument in the sequel. Furthermore, let

F i (P i ) := A i P i +P i A i ,
and G i (P i ) := BB +L i V i L i so that, Ṗi = F i (P i ) + G i . Also, we let

P :=    P 1 . . . P N    , F (P) :=    F 1 (P 1 )
. . .

F N (P N )    , G(P) :=    G 1 (P 1 ) . . . G N (P N )    .
Then, we can write

Ṗ = F (P) + G(P) + λ(Π ⊗ I n )P -λ(I N ⊗ I n )P = F (P) + G(P) -λ (L ⊗ I n )P, (16)
where ⊗ denotes the Kronecker product, and we recalled the definition of Π to write Π -I N = -L.

Proposition 4. The solutions of (16) stay bounded over the interval [0, +∞[.

The proof of Proposition 4 is based on the observation made in Remark 1 combined with the fact that the convergence of solutions of ( 16), with time-invariant L i , has been established in author's work [START_REF] Tanwani | Suboptimal filtering over sensor networks with random communication[END_REF]Theorem III.1]. In the sequel, we establish the practical convergence of the solutions of [START_REF] Etesami | Potential-based analysis of social, communication, and distributed networks[END_REF]. The analysis is based on rewriting the dynamics in new coordinates and providing a Lyapunov function in those coordinates.

To describe the coordinate transformation, we recall that, for an undirected connected graph with Laplacian L, we can find a matrix U such that U LU = diag (0, Λ). Let v 1 denote the normalized eigenvector corresponding to the eigenvalue σ 1 (L) = 0, so that

v 1 = 1 √ N 1 1 • • • 1 ∈ R 1×N , and v 1 L = 0.
There exists a matrix U ∈ R N ×(N -1) , with

U U = I N -1 , U • 1 N = 0, such that U = v 1 U , U -1 = U = [v 1 , U † ]
where 1) is the matrix satisfying

U † ∈ R N ×(N -
U U † = I N -1 , and
1 N U † = 0.
Introduce the coordinate transformation

Q = 1 √ N (U ⊗ I n )P = 1 N (1 N ⊗ I n ) 1 √ N ( U ⊗ I n ) P =: Q 1 Q
and the inverse of this transformation gives,

P = √ N (U ⊗ I n ) -1 Q = √ N (U -1 ⊗ I n )Q = √ N (v 1 ⊗ I n ) ( U † ⊗ I n ) Q = (1 N ⊗ I n )Q 1 + √ N ( U † ⊗ I n ) Q.
Let S denote the solution of the differential equation

Ṡ = AS + SA - 1 N S N i=1 H i V -1 i H i S + BB = AS + SA -SH V -1 HS + BB
where we observe that 1

N N i=1 H i V -1 i H i = H V -1 H.
Under the assumption that (A, H) is observable and (A, B) is controllable, it holds that lim sup t→∞ S(t) = S, where S is positive semidefinite matrix satisfying [START_REF] Olfati-Saber | Distributed Kalman filtering for sensor networks[END_REF]. In particular, S(t) is uniformly bounded for t ≥ 0. We introduce the variable E to denote the difference between the mean value of P i , i = 1, . . . , N , and S, that is,

E := Q 1 -S = 1 N N i=1 P i -S,
and it is observed that

Ė = Q1 -Ṡ = 1 N N i=1 Ṗi - Ṡ = AE + EA -EH V -1 HE + g 1 (E, S, Q) (17)
for some matrix-valued function g 1 (•). Due to the boundedness of P i and S, it can be shown that g 1 (E, S, Q) ≤ M 1 Q for some M 1 ≥ 0. Also, we can write

˙ Q = -λ (Λ ⊗ I n ) Q + 1 √ N ( U ⊗ I n ) (F (P) + G(P)) = -λ (Λ ⊗ I n ) Q + 1 √ N ( U ⊗ I n ) (F + G)• ((1 N ⊗ I n )Q 1 + √ N ( U † ⊗ I n ) Q) = -λ (Λ ⊗ I n ) Q + g 2 (E, S, Q), (18)
for some matrix-valued function g 2 (•). Once again, due to the boundedness of the solutions, we can find a constant

M 2 ≥ 0 such that g 2 (E, S, Q) ≤ M 2 E + M 3 .
For stability analysis of the process P, we equivalently analyze the stability of ( 17) and [START_REF] Hespanha | Stochastic impulsive systems driven by renewal processes[END_REF]. Towards this end, we recall that, if the matrix S that satisfies (13) then (A -SH V -1 H) is a Hurwitz matrix, and there exists a positive definite matrix R such that

(A -SH V -1 H)R + R(A -SH V -1 H) ≤ -I. (19)
Let us introduce the functions

V 1 (E) := 1 2 tr(R 1/2 ERER 1/2 ), V 2 ( Q) := 1 2 tr ( Q Q ).
To analyze the evolution of V 1 along the solutions of ( 17), we compute the bound on the derivative of V 1 . Using the cyclic property of the trace operator, we get

V1 = 1 2 tr Sym((RA + A R)ERE) + tr Sym RERg 1 (E, S, Q) ≤ -tr(ERE) + C 0 E g 1 (E, S, Q) ≤ -σ 1 (R) E 2 + C 0 M 1 E Q
where we used the bounds on the trace of products from [START_REF] Fang | Inequalities for the trace of matrix product[END_REF], tr(ERE) = tr(RE 2 ) ≥ σ 1 (R) tr(E 2 ), with σ 1 (R) being the smallest eigenvalue of R and C 0 ≥ 0 is some constant.

On the other hand, using similar techniques, we can bound the derivative of V 2 along the solutions of (18) as follows:

V2 ≤ -λ σ 2 (L) tr( Q Q ) + C 1 tr( Q Q ) + C 2 M 2 E Q + C 3 M 3 Q
for some appropriate positive scalars C 1 , C 2 , C 3 ≥ 0. For the combined dynamical system (17), [START_REF] Hespanha | Stochastic impulsive systems driven by renewal processes[END_REF], we now consider the proper, positive definite Lyapunov function

V(E, Q) = V 1 (E) + V 2 ( Q)
and observe that

V ≤ -σ n (R) E 2 -(λ σ 2 (L) -C 1 ) Q 2 + (C 0 M 1 + C 2 M 2 ) E Q + C 3 M 3 Q .
To get a bound on V (E, Q), bet us recall the following lemma:

Lemma 5 ([6]). Consider a function W : R ≥0 × R ≥0 × R ≥0 → R, and a function g : R ≥0 → R ≥0 , such that W (t, r, s) ≤ -a 1 r 2 + a 2 rs -a 3 s 2 + g(t)s for some positive scalars a 1 , a 2 , a 3 > 0. Then, there exists c > 0 such that W (t, r, s) ≤ -c(r 2 + s 2 ), if r 2 + s 2 ≥ g 2 (t)χ 1 a 3

where χ is a continuous nondecreasing function and χ(s) = O(s) as s → 0.

Applying Lemma 5, it follows that for every δ > 0, there exists λ > 0 such that

lim sup t→∞ E(t) 2 + Q(t) 2 ≤ δ 2 √ N
For S satisfying (13), we have lim t→∞ S(t) = S, which leads to the following bound for each i ∈ V , Hence, we obtain [START_REF]Kalman-Consensus Filter : Optimality, stability, and performance[END_REF] and this completes the proof of Theorem 3.

lim

V. CONCLUSIONS AND PERSPECTIVES

We considered the problem of distributed filtering over undirected and connected graphs. The sensor nodes exchange information about their estimate, and the the communication between the sensor nodes, along each edge, is driven by an independent Poisson process. The algorithm implemented by each node is in the form of a stochastic hybrid system. We obtain pointwise and asymptotic bounds on the covariance of the estimation error for each sensor node, and show that these covariances converge practically to a constant matrix if the mean sampling rate is large enough.

A possible extension is to generalize the main results for a broader class of random processes for the communication between nodes. While the memoryless Poisson counter treated in this paper results in diffusive coupling between the dynamics of error covariances, it remains to be seen what kind of interconnection is obtained from other communication processes.


V Cazajus

B Lorrain

H Welemane

Y Paranthoen

M Karama

Thermo-mechanical behaviour of ceramic metal brazed assemblies

Keywords: ceramic-metal assembly, residual stresses, thermo-mechanical analysis, experimental tests, numerical simulations

This study investigates the effects of residual stresses developed during brazing on the performance of brazed ceramic metal joints. The thermal expansion gradient between ceramic and metallic materials leads to the development of such stresses during the cooling phase of the brazing process, which consequently reduce the strength of these composite structures. Here, the objective is to compare the failure behaviour of various assemblies observed during experimental tests and obtained through numerical simulations. In order to get a representation consistent with the physical mechanisms involved, these simulations must account for the brazing phase giving rise to residual stresses before applying in use solicitations to brazed joints. This paper focuses on the tensile strength of ceramic metal joints, for which two brittle failure modes (within the ceramic or at the interface) are observed during the experimental tests.

INTRODUCTION

The association of materials with dissimilar thermal, mechanical, electronic, and chemical behaviour is especially interesting for high-technology applications (medical, aircraft, spatial, electronic or nuclear). Among joining processes, brazing is one of the most efficient and widespread to produce ceramic to metal assemblies. However, the difference between the thermal expansion coefficients (CTE) of the base materials leads to the development of residual stresses within the assemblies during the cooling phase of the brazing process. Such residual stresses clearly reduce the strength of the brazed joints and can cause cracking inside the ceramic and/or debonding at the interface, even during the brazing process itself [START_REF] Evans | On the mechanics of failure in ceramic/metal bounded systems[END_REF][START_REF] Nicholas | Active metal brazing[END_REF][START_REF] Loehman | Joining of ceramics[END_REF][START_REF] Tanaka | Joining of ceramics to metals[END_REF].

The prediction of these failure mechanisms under any kind of solicitation is an important task for both practical and fundamental applications. Especially, associated problems often limit the use of ceramics in spite of their attractive high-temperature properties. In this way, many analytical and numerical approaches have been proposed to estimate the residual stresses field induced by the brazing process [START_REF] Hsuech | Residual stresses in metalceramic bonded strips[END_REF][START_REF] Zhang | Effect of residual stress on the strength of an alumina-steel joint by partial transient liquid phase (PTLP) brazing[END_REF][START_REF] Yu | Residual stresses in ceramic-interlayer-metal joints[END_REF][START_REF] Iancu | Residual stress state of brazed ceramic/metal compounds[END_REF][START_REF] Santacreu | Sur la mécanique des assemblages: calcul des contraintes résiduelles dans les assemblages céramique-métal obtenus par brasage réactif et caractérisation mécanique des interfaces[END_REF][START_REF] Munz | Thermal stresses in ceramic-metal joints with an interlayer[END_REF][START_REF] Levt | Thermal residual stresses in ceramic to metal brazed joints[END_REF][START_REF] Lovato | Rhéologie des joints brasés: étude expérimentale et détermination par méthode inverse[END_REF].

This paper aims at determining through numerical simulations these residual stresses within standard tensile specimens of two different ceramic metal assemblies. Then, these features are taken into account to determine the damage development and the consecutive bond strength of such composite structures under standard tensile tests. In particular, this approach stands out by the introduction of the brittle behaviour of the ceramic metallization. Numerical and experimental results are finally compared and discussed.

EXPERIMENTAL AND NUMERICAL TECHNIQUES

The tensile test is generally used by the SCT Company for tension and vacuum testing of metallized ceramic seals. Indeed, this quite severe solicitation leads to a representative idea of their in-use performance.

Accordingly, the objective is to compare in what follows the mechanical response of ceramic-metal assemblies (i.e. after the brazing process) during such a test obtained by experiments and simulations.

Materials and brazing conditions

In this case, the ceramic part of assemblies is an alumina with a purity of 97.7 per cent which exhibits linear elastic brittle behaviour. Its elastic properties, which do not depend on temperature for the temperature range of the brazing process (20-850 • C), have been identified through ultrasonic tests at room temperature (Table 1). If the failure of alumina material depends on the solicitation applied, the behaviour under tensile solicitations is purely brittle with a tensile strength around 250 MPa [START_REF] Grujicic | Optimization of 316 stainless steel: alumina functionally graded material for reduction of damage induced by thermal residual stresses[END_REF].

Two metals have been joined to the ceramic:

(a) stainless steel AISI 304L; (b) Kovar (nickel-cobalt-ferrous alloy).

The filler metal (or braze alloy), which is introduced between base materials to produce the joint, is in both cases a silver-copper eutectic alloy. All these metallic materials exhibit elastoplastic behaviour depending on temperature. The evolution of their yield stress with temperature given in Table 1 is obtained from reference [START_REF] Lovato | Rhéologie des joints brasés: étude expérimentale et détermination par méthode inverse[END_REF]. Plasticity damage within these metallic parts has been neglected because tensile tests rather show a brittle mode of failure of specimens. Finally, note that CTE of base materials and filler metal according to temperature have been identified through dilatation test (Table 1).

If the attraction of the filler metal by the metallic part occurs naturally, it is necessary to apply formerly on the ceramic surface a low thickness metallization to increase wettability by the filler metal at the liquid state and allow capillary attraction (molymanganese process [START_REF] Klomp | Interfacial reaction between metals and oxides during sealing[END_REF]). Assuming a brittle linear elastic behaviour, nano-indentation tests on the ceramic metallization part of brazed assemblies have been performed to determine the elastic properties of such material. The Young's modulus mean value is about 230 GPa [START_REF] Cazajus | Approches numérique et expérimentale de la liaison céramique metal réalisée par brasage en vue de la création d'un outil d'aide à la conception[END_REF] and the Poisson ratio is equal to 0.22. Besides, the tensile strength of this brittle material has been deduced from standard tensile specimens on ceramic-to-ceramic assemblies, leading about to 100 MPa.

The brazing process comprises heating the assembly of materials to a suitable temperature above the liquidus temperature of the filler metal. Then, the filler metal solidification that occurs during the cooling phase to room temperature leads to the final seal of the composite structure. Such procedure is realized in a vacuum atmosphere to protect the metallic part from oxidation and the heating (around 4 • C/min) and cooling (around 2 • C/min) rates are defined to protect the brittle parts from thermal shock. Base materials and filler metal are not submitted to any metallurgical transformation during the cooling phase; moreover, there is only thermal loading on the ceramic-metal assembly during the brazing process.

The microstructure of a ceramic-metal assembly after brazing is depicted in Fig. 1. Three domains: Fig. 1 Microstructure of the ceramic to metal brazed assembly metal, joint, and ceramic are observed. The joint is composed of the filler metal, the metallized surface of the ceramic, and the interfaces generated at high temperature by diffusion. Finally, note that the figure shows the very tiny thickness of the metallization layer after brazing, with a value of about 15 µm. This value is close to the mean metallization thickness measured by the SCT Company after the molymanganese process and before the brazing process.

Tensile test

The experimental procedure follows the standard test method described by the American Society for Testing and Materials (ASTM F19-64) [START_REF] Astm) | Standard test method for tension and vacuum testing metallized ceramic seals[END_REF]. The specimen is composed of two identical washers of filler metal, which join a metal washer situated at the centre of the specimen with two identical ceramic components. The geometrical dimensions of the ceramic component are shown in Fig. 2, the metal washer thickness is equal to 0.3 mm and the filler metal washer thickness 0.06 mm. The gripping device used to apply the load (rate of 2 mm/min) allows a self-aligning of the test specimen with the axis of the test machine heads. The results dispersion for the identification of the tensile strength can be significantly reduced with this care.

Numerical model

Numerical simulations have been performed with the finite-element ABAQUS code. The model is defined in accordance with the standard specimen geometry defined in Fig. 2 and with metallization thickness observed after brazing in Fig. 1. The simulation is divided into two parts: the first step is the numerical simulation of the brazing process and the second one is the simulation of the tensile test.

For both models:

1. The structure is two-dimensional axisymmetric and symmetric with the median plane. 2. The mesh of the assembly has been done with an eight-node quadratic axisymmetric quadrilateral element. According to the failure modes observed, the main residual stresses are essentially located near the interfaces and the free side surface, the element size is then taken lowest on these zones (Fig. 3). A converging study on maximum principal stresses has shown that 5210 nodes leads to an accurate estimation of the mechanical response after brazing process and during the tensile test. 3. The interfaces between the base materials, filler metal, and metallization are supposed to be perfect.

For the brazing process model:

1. The starting point of the simulation is the solidification of the filler metal during the cooling phase of the brazing process and the simulation ends when the ceramic-metal assembly temperature is equal to the room temperature. 2. There is no mechanical loading on the ceramicmetal assembly. 3. The thermal loading is the cooling cycle from the filler metal solidus temperature to room temperature with the uniform cooling hypothesis: the temperature of all the nodes is assimilated to the furnace temperature according to time.

For the tensile test model:

1. The stress state within the specimen is equal to the residual stress state given by the brazing process mode. 2. A negative vertical displacement is applied on the lower surface of the specimen (Fig. 3(b)). 3. As the ceramic part is only concerned by tensile solicitations, a simple Rankine criterion is used to detect crack initiation. This criterion states that a Fig. 3 Mesh of the specimen: (a) brazing process and (b) tensile test crack forms when the maximum principal tensile stress exceeds the tensile strength of the alumina and leads to a loss of the material stiffness in the related principal direction. 4. The metallization layer exhibits a negligible thickness in comparison with other materials; accordingly, its damage evolution is analysed with the cohesive zone formulation of ABAQUS introduced by Needleman [START_REF] Needleman | A continuum model for void nucleation by inclusion debonding[END_REF]; especially, the brittle failure of such an interface occurs when the normal stress reaches the tensile strength of the considered material.

RESULTS

Experimental results

Two different types of assemblies depending on the choice of the metal have been tested, one with the stainless steel and the other one with the Kovar. Five specimens for each type of assembly were prepared and tested for this study. Figure 4 shows the evolution of the load applied on the specimen according to the displacement of the grip for each sample.

It is observed that the load increases slowly during the first part of the curve that represents the self-aligning of the grip and also the compression of the elastomer (split TFE-fluorocarbon) washer. Then a linear increase takes place until the brittle failure of the sample that occurs within the ceramic part near the bond interface for the alumina-stainless steel specimens and within the ceramic metallization for the alumina-Kovar specimens (Fig. 5). Although the linear coefficient of the elastic response is quite the same for each specimen, the applied load that leads to the failure is quite different between the five samples. The average of these five values is equal to 10.7 kN for the alumina-Kovar assembly and 12.2 kN for the alumina-stainless steel assembly with a standard deviation of 1 kN in both cases. The tensile strength deviation of the brittle materials (alumina and metallization) could explain such deviation of the tensile strength measurements on the assemblies.

Numerical results

The stress evolution during the brazing process and during the tensile test within two types of specimen has been studied.

Brazing process

In this case, the base materials and the metallization remain in the elastic regime at the end of the brazing process. For the mechanical reliability of assemblies, it is essential to examine residual stresses developed within brittle materials. As an illustration, it is chosen to focus here on the ceramic part for which Fig. 6 shows the spatial distribution of the maximum principal stress (failure governed by the Rankine criterion). Especially, the maximum value is located for Fig. 4 Tensile tests on ceramic metal assemblies Fig. 5 Tensile specimens after failure Fig. 6 Maximum principal stress distribution within the specimen after brazing both types of assemblies on the radial free edge of the specimen near the ceramic filler metal interface. Nevertheless, its intensity during the brazing process clearly differs according to the metal of the assembly (Fig. 7): residual stresses are negligible for the alumina-Kovar assembly since CTE of base materials are very close; on the contrary, the thermal expansion of the stainless steel is much bigger than alumina and leads to greater values of residual stresses. Finally, note that the above conclusions can be done also

Tensile test

Figure 8 shows the failed elements of the specimen at the end of the tensile test (after global failure) for each kind of assemblies. A failed (respectively active) element is such that the failure criterion (Rankine for ceramic or normal stress for metallization) has been reached (respectively not reached). In accordance with experimental observations, numerical results clearly establish a marked difference in the failure mode depending on whether the coefficient of thermal expansion of the ceramic and metal are matched or not: for similar coefficients (alumina-Kovar), failure occurs within the metallization layer, whereas if coefficients are very much different (alumina-stainless steel), failure concerns the ceramic part close to the interface. The influence of residual stress near the ceramic metal interface explains this failure type variation. In this sense, it demonstrates the importance of considering residual stresses within numerical simulations and good predictive ability of the model to account for the assembly failure mode: without residual stresses, stainless steel assemblies would have behaved as Kovar assemblies, namely with a failure at the interface (within the metallization layer).

Concerning the quantitative analysis, Fig. 8 shows for the two types of assemblies the evolution of the Fig. 8 Status elements for each type of assembly at the end of the tensile test load measured at the central symmetry on the specimen according to the displacement applied at its bottom. The tensile strength obtained through simulation is then equal to 11.1 kN for the alumina-Kovar assembly and 10.2 kN for the alumina-stainless steel assembly, contrary to tests results which show stronger bounds for alumina-stainless steel. If the strength magnitude is correctly reproduced (around 11 kN), these slight differences between experiments and predictions could be explained by uncertainties on some materials properties given by literature data, especially for the strength of alumina and metallization. Moreover, such brittle materials often exhibit an important scatter in their properties that can be attributed due to a heterogeneous initial defect distribution. In this sense, the direct characterization of materials considered here and the coupling of the approach with reliability methods [START_REF]Structural reliability analyses[END_REF] would surely allow a more realistic estimation of the ceramic metal strength.

Finally, note in Fig. 9 that finite-element simulations lead to a non-linear response of the assembly before failure, contrary to the almost linear result in experiments (Fig. 4). It seems that such discrepancy comes from the difference in the application of the mechanical loading in both cases. In the numerical model, the Fig. 9 Reaction force according to the applied displacement solicitation is directly applied to the assembly and induces the development of some plastic deformation within metallic materials. On the contrary, it is difficult to avoid additional displacements in testing conditions due to the experimental device (especially due to the elastomer washer located between grip and specimen), as shown by the difference in displacement magnitude in Fig. 4 (expressed in millimetres) and Fig. 9 (expressed in micrometres); in this way, plastic deformation during experimental tests may be hidden by such supplementary components.

CONCLUSION AND PERSPECTIVES

In view of the design of joints in engineering structures, the numerical model developed in this work represents an essential tool to study the mechanical behaviour and the reliability of ceramic metal brazed assemblies. Indeed, the brazing process model allows a consistent estimation of the residual stresses developed within such structures during the cooling phase, which remains crucial to get a realistic representation of their in-use behaviour.

Results obtained in this work are really promising since the numerical tool is able to correctly predict the experimental failure mode of various assemblies whether the coefficient of thermal expansion of metal and ceramic is matched or not. Nevertheless, if the tensile strength magnitude is reasonably reproduced, the quantitative agreement could be improved in the future by an experimental characterization of the properties of brittle materials considered here (alumina and metallization) including notably their potential scattering. Moreover, it may be interesting to complete this work by simulating the damage growth within the ceramic material and to compare this to the crack pattern generated during experimental tests.

Fig. 2

 2 Fig. 2 Tensile specimen geometry of ceramic metal seal (distances in millimetre)

Fig. 7

 7 Fig.[START_REF] Yu | Residual stresses in ceramic-interlayer-metal joints[END_REF] Evolution of the maximum value of the maximum principal stress during the brazing process for each type of specimen

Table 1

 1 Mechanical and thermal properties of materi-

		als according to temperature: alumina (A), filler
		metal (FM), stainless steel (SS), and Kovar (K)
				Temperature, T ( • C)
		20	100	200	300	400	500	600
	Young's modulus (GPa)
	A	330	330	330	330	330	330	330
	FM [9] 79	79	79	79	79	79	79
	SS	193	191	183	-	168	-	148
	K	138	138	138	138	138	138	138
	Poisson ratio
	A	0.26	0.26	0.26	0.26	0.26	0.26	0.26
	FM [9] 0.33	0.33	0.33	0.33	0.33	0.33	0.33
	SS	0.22	0.27	0.27	-	0.31	-	0.37
	K	0.32	-	-	0.32	-	0.32	-
	Yield stress (MPa)
	FM [9] 247	183	87	40	27	13	6
	SS	247	243	169	148	136	133	125
	K	345	-	-	223	-	217	-
	Tensile strength (MPa)
	SS	973	774	667	635	633	589	517
	K	517	-	-	462	-	385	-
	Tensile failure strain
	SS	0.48	0.44	0.38	0.34	0.35	0.34	0.34
	K	0.34	-	-	0.42	-	0.36	-
	Coefficient of thermal expansion (×10 -6 / • C)
	A	-	6.2	7.4	7.8	8	8.1	8.2
	FM	-	16.7	19.2	20.2	20.4	20.7	20.5
	SS	-	18.4	20.3	20.9	21.3	21.4	21.7
	K	-	5.9	5.2	5.1	5.1	6.2	7.8