T Boukhobza
email: taha.boukhobza@cran.uhp-nancy.fr

Sensor location for discrete mode observability of switching linear systems with unknown inputs

Keywords: Switching linear systems, mode observability, graph theory, sensor placement

This paper deals with the problem of additional sensor placement in order to recover the discrete mode observability of switching structured linear systems with unknown inputs. Such property is quite important for designing control laws, observers, fault detection and isolation schemes (when the fault occurrence implies a commutation between two modes) and so on. The proposed method, based on a graph-theoretic approach, assumes only the knowledge of the system's structure. We express, in graphical terms new necessary and sufficient conditions for the discrete mode generic observability. When these conditions are not satisfied, we propose a sensor placement procedure which allows us to recover the mode observability. Our approach can be implemented by classical and quite simple graph-theory algorithms.

1 Introduction A subclass of hybrid systems, called switching systems, is suitable to represent systems that can switch between different behaviours according to a discrete mode variable value. Many engineering systems are designed to be switching systems in order to simplify their functional description and maintain the flexibility in operation. Thus, switching systems have become ubiquitous in modern technology and the subject of significant research activities in automatic control area. For such systems, estimating the active mode is of great interest in many control areas and studies presented in [START_REF] Babaali | Observability of switched linear systems in continuous time[END_REF][START_REF] De Santis | Observability of internal variables in interconnected switching systems[END_REF][START_REF] Vidal | Observability of linear hybrid systems[END_REF] are among the most significant works which deal with this problem. In this context, our study concerns the discrete mode observability for switching linear systems (SLS) using only continuous variable measurements. More precisely two questions are addressed: establish necessary and sufficient conditions to characterise structurally the discrete mode observability and when this property is not satisfied, provide additional sensor locations in order to recover it. One motivation among others to this study lies in the fact that in many practical cases, the faults, when they occur in a system, may change completely the system dynamics. Thus, in these cases, it may be more pertinent to consider a switching linear system, where the nominal system corresponds to a specific mode and to each fault occurrence is associated a discrete mode value. Fault detectability and isolability is in this case equivalent to the discrete mode observability. From the point of view of the class of systems under study our paper is close to [START_REF] Babaali | Observability of switched linear systems in continuous time[END_REF][START_REF] De Santis | Observability of internal variables in interconnected switching systems[END_REF][START_REF] Vidal | Observability of linear hybrid systems[END_REF][START_REF] De Santis | Observability of internal variables in interconnected switching systems[END_REF]. Indeed, the presented work deals with the discrete mode observability of switching continuous time linear systems with unknown inputs, where the discrete mode signal is assumed to be exogenous to the continuous part and unobserved by a discrete measure. However, comparatively with the previous published results in the field, our proposed method and the obtained results present three main originalities. Firstly, few works deal with the observation of hybrid systems with unknown inputs, representing distur-bances, unmodeled dynamics or faults for example. Some studies ( (Pina and Botto, 2006) and the references therein), aim at designing unknown input observers for SLS, but they do not study the mode observability conditions. Otherwise, [START_REF] Vu | On invertibility of switched linear systems[END_REF] addresses the problem of recovering the discrete mode variable and the input but assumes that the initial state is known. Secondly, most part of observability studies use algebraic or geometric approaches and so requires the exact knowledge of the state space matrices characterizing the systems' model. In many modelling problems or during conception stage, these matrices have a number of fixed zero entries determined by the physical laws while the remaining entries are not precisely known. In these cases, to study the structural properties, like observability, the idea is that we only keep the zero/non-zero entries in the state space matrices. Thus, we consider models where the fixed zeros are conserved while the non-zero entries are replaced by free parameters. Many interesting works on these models, called structured models, are related to the graph-theoretic approach and aim to analyse properties such as controllability, observability . . . [START_REF] Dion | Generic properties and control of linear structured systems: A survey[END_REF]. It results from these works that the graph-theoretic approach provides quite simple and elegant analysis tool. Finally, our approach allows us to address the problem of sensors' location to recover the discrete mode observability. This problem has not been resolved yet and is interesting from a conception point of view. Comparatively with [START_REF] Boukhobza | Observability of switching structured linear systems with unknown input: a graph-theoretic approach[END_REF] where the authors express in graphic terms necessary and sufficient conditions for the discrete mode observability of SLS, the novelties, as we highlight this after Proposition 1, lie in the formulation of the simpler and more computationally efficient conditions. These improvements offer the possibility to propose a solution to the sensors' location problem when the discrete mode is not observable. All the proposed results are based on classical combinatorial algorithms with polynomial order complexity. This is quite important when we deal with large scale systems. Moreover, since we consider structured systems, our ap-proach can be used during a conception stage. The paper is organised as follows: after Section 2, which is devoted to the problem formulation, some definitions related to the graph-theoretic approach are given in Section 3. The main results are stated in Section 4 before a brief conclusion.

Problem statement

Consider the following SLS Σ :

{ ẋ = A(q)x + B(q)u y = C(q)x + D(q)u (1)
where x ∈ R n , u ∈ R m and y ∈ R p are respectively the state vector, the unknown input vector and the output (measurement) vector and where A(•), B(•), C(•) and D(•) are real matrices of compatible dimensions. For the exogenous and unobserved discrete mode variable (or switching signal) taking its values in Q def = {1, . . . , N }, we assume, as in [START_REF] Babaali | Observability of switched linear systems in continuous time[END_REF], that only a finite number of jumps can occur in any finite time interval. This implies that all the system trajectories are well defined and infinitely right-differentiable. Therefore, Zeno behaviour can not occur. Without loss of generality and for the sake of homogeneity, all the inputs are assumed to be unknown. In fact, the control input signals, whose values are known, are considered to be measured i.e. we associate a virtual output equation of the form y k ′ = u k to these inputs. Our aim is to address, in the same framework discrete mode observability of autonomous SLS and SLS with known or unknown inputs. In [START_REF] De Santis | Observability of internal variables in interconnected switching systems[END_REF], discrete mode observability is defined as the ability to reconstruct the mode starting from the knowledge of the input and the output, for any non-zero input value and for all initial conditions. It results that, the discrete mode observability as it is defined in [START_REF] De Santis | Observability of internal variables in interconnected switching systems[END_REF], cannot be used in our context, since it can never be satisfied for SLS with unknown inputs or autonomous SLS. Thus, we choose here to have a more general framework by imposing mode distinguishability not for all but for generic initial conditions x 0 and unknown inputs u: Definition 1 (Mode distinguishability) Two modes q ∈ Q and q ′ ∈ Q with (q ̸ = q ′ ) are distinguishable if at least one of the two following conditions holds: -there exist an integer s ≥ 0 and an expression Ψ q (y, ẏ, . . . , y (s) ) = 0 satisfied for mode q but not for mode q ′ for almost all initial conditions x 0 and inputs u.

-there exist an integer s ′ ≥ 0 and an expression Ψ q ′ (y, ẏ, . . . , y (s ′ ) ) = 0 satisfied for mode q ′ but not for mode q for almost all initial conditions x 0 and inputs u.

Here, " for almost all initial conditions x 0 and inputs u(t) " is to be understood as " for all (x T 0 , u T ) T ∈ R n+m except for the zero set of some polynomials with real coefficients in the n + m initial state and input components. The zero set of these polynomial forms a proper algebraic variety of R n+m which has Lebesgue measure zero. The interpretation of Definition 1 is that q is distinguishable from q ′ if, for generic initial states x 0 and unknown input u, we can rule out q or q ′ when observing the output over [0, T ]. Definition 2 (Discrete mode observability) SLS (Σ) is discrete mode observable if its modes are all distinguishable two-by-two i.e. ∀q ∈ Q, ∀q ′ ∈ Q, with q ̸ = q ′ , q and q ′ are distinguishable. Discrete mode observability analysis can be reduced to the study of the distinguishability of each pair of modes. Thus, for the sake of simplicity, there is no loss of generality in considering that we have only two modes. Since we study a structural property as the observability, it is pertinent to deal with structured systems, for which we assume that only the sparsity pattern of matrices A(q), B(q), C(q) and D(q) is known for q ∈ {1, 2}. So, to each entry of these matrices, we only know whether its value is fixed to zero, or that it has an unknown value represented by a real parameter λ i . We denote by A λ (q), B λ (q), C λ (q) and D λ (q) respectively the matrices obtained by replacing the non-zeros in A(q), B(q), C(q) and D(q), for q ∈ {1, 2} by the corresponding parameters λ i and the studied systems have the form

Σ Λ : { ẋ(t) = A λ (q)x(t) + B λ (q)u(t) y(t) = C λ (q)x(t) + D λ (q)u(t) (2)
If all parameters λ i are numerically fixed, we obtain a socalled admissible realization of structured switching linear systems (SSLS) (Σ Λ ). We say that a property is true generically for SSLS (Σ Λ ) if it is true for almost all parameters λ i . Discrete mode observability analysis is based on the distinguishability of the structured models associated to the two discrete modes. So, it is pertinent and necessary to highlight the similarities and the differences between the models associated to these modes. Thus, we decompose each structured matrix into two parts: the first one is common to the two modes and the second one is specific to each mode i.e. for q ∈ {1, 2},

A λ (q) = A λ 0 + A λ q , B λ (q) = B λ 0 + B λ q , C λ (q) = C λ 0 + C λ q and D λ (q) = D λ 0 + D λ q .
When we write the matrices under this form, we assume that the entries of all these matrices constituting vector Λ = (λ 1 , λ 2 , . . . , λ h )

T ∈ R h can take any value in R h or equivalently that parameters λ i are free. Note that there is no third mode 0 but we use q = 0 as subscript to represent the common part between the two modes, in order to exhibit structurally their differences and their similarities. All the results exposed in the paper lean on the hypothesis that all the entries of matrices A λ 0 , A λ q , B λ 0 , B λ q , C λ 0 , C λ q , D λ 0 and D λ q are free. This can be obtained when dealing with a model close to the physical laws governing the system. On the one hand, for any other decomposition of A λ (q), B λ (q), C λ (q) and D λ (q) which does not satisfy this hypothesis, the method presented here is not guaranteed. On the other hand, when applying a coordinate change, if we break the parameter independence or in other terms if we obtain matrices having entries linked by fixed algebraic relations, then the results of the paper are not guaranteed. Note that by not guaranteed, we mean that the results may be correct may be not. It depends if the relation between the parameters introduce a singularity in the different structural properties used in the proposed method (observability subspace dimensions for example). To summarize, our aim is to provide necessary and sufficient conditions to check if a SSLS is generically discrete mode observable i.e. is discrete mode observable for almost all its realizations or equivalently for almost all parameters λ i . Next, when these conditions are not satisfied, we provide a procedure for adding sensors in order to recover one of them and so the discrete mode observability property. Note that we do not address, as in [START_REF] Boukhobza | Observability of switching structured linear systems with unknown input: a graph-theoretic approach[END_REF], the problem of state estimation of SLS in the presence of unknown inputs.

Graphical representation and definitions

The digraph associated to (Σ Λ ) is noted G(Σ Λ ). It is constituted by a vertex set V and an edge set E i.e. G(Σ Λ ) = (V, E). The vertices are associated to the continuous state, the input and the output components of (Σ Λ ) and the directed edges represent links between these variables. More precisely, V = X ∪ U ∪ Y, where X = {x 1 , . . . , x n }, U = {u 1 , . . . , u m } and Y = {y 1 , . . . , y p } are respectively the set of state, input and output vertices. For q ∈ {0, 1, 2}, we define E q = A q -edges∪B q -edges∪C q -edges∪D q -edges, where, for q ∈ {0, 1, 2},

A q -edges = {(x j , x i ) | A q (i, j) ̸ = 0}, B q -edges = {(u j , x i ) | B q (i, j) ̸ = 0}, C q -edges = {(x j , y i ) | C q (i, j) ̸ = 0} and D q -edges = {(u j , y i ) | D q (i, j) ̸ = 0}. Finally the edge set is E = 2 ∪ q=0 E q .
Here, M (i, j) is the (i, j)th element of matrix M and (v 1 , v 2 ) denotes a directed edge from vertex v 1 ∈ V to vertex v 2 ∈ V. The edges included in E 0 represent the common part of the two modes' models, while E q , for q ∈ {1, 2} is related to the specific part of each mode's model. For the sake of clarity, the vertices are written in bold fonts to differentiate them from the corresponding variables. Each edge is associated to a free non-zero parameter of the system's model called the weight of the edge. Note that number q is written under each E q -edge and represents its index. An edge belonging to E q , q ̸ = 0, is said specific to mode q. Note that even if it is conceptually feasible to have an edge with index 0 having the same end vertices than an edge with index q ̸ = 0, we will, for a sake of coherence and readability, avoid this case and prefer two edges with indices 1 and 2. The index 0 will, then be reserved to the common part of the two modes. Example 1 To the system defined by the matrices below, we associate the digraph in Figure 1. A realisation of this system could be the numerical specified system having the following dynamics:

       λ 7 0 0 0 0 0 0 0 0 0 0 0 λ 8 0 0 0 0 0 0 λ 9 0 0 0 0 0 0 0 0 0 0 0 λ 10        ,all the entries of A λ 1 are zero except A λ 1 (1, 2) = λ 11 , A λ 1 (4, 4) = λ 12 , A λ 1 (4, 5) = λ 13 , A λ 1 (5, 2) =
Mode 1 :

                                                             ẋ1 = x 2 ẋ2 = -x 3 ẋ3 = 2x 2 ẋ4 = -3x 4 + x 5 -x 8 ẋ5 = 2x 2 ẋ6 = u 1 ẋ7 = -2x 5 + x 6 ẋ8 = 2x 7 y 1 = x 1 y 2 = x 5 y 3 = x 4 y 4 = x 8
Mode 2 :

                                                             ẋ1 = 0 ẋ2 = -x 3 ẋ3 = 2x 2 ẋ4 = 2x 5 -x 8 ẋ5 = -x 2 ẋ6 = u 1 ẋ7 = -2x 5 + 3x 6 ẋ8 = 2x 7 y 1 = x 1 y 2 = x 5 y 3 = x 4 y 4 = x 8 (3)
The digraph representing the SSLS is built from the superposition of the digraphs related to each mode. In order to study the properties of the system associated to a specific mode q, we have to restrict the edge set to E 0 ∪ E q . In this context, many of the functions and specific vertex subsets, defined below, present an index q related to the considered mode.

Let us now give some useful definitions and notations.

• A path is simple when every vertex occurs only once in this path.

• A cycle is a path of the form v s0 → v s1 → . . . → v s i → v s0
, where v s0 , v s1 , . . . , v s i are distinct.

• For q ∈ {1, 2}, we say that path P is included in E 0 ∪ E q if all its edges are included in E 0 ∪ E q .

• A path is a Y-topped path if its end vertex belongs to Y.

In the sequel, V 1 and V 2 represent two subsets of V. We denote by card(•) the cardinality function and

V 1 \ V 2 is the set of elements in V 1 which are not in V 2 .
• Denote by P red q (V 1 ) the set of all the predecessors of vertices included in V 1 in mode q.

• A path P = v s0 → v s1 → . . . → v s i is said a V 1 → V 2 path if v s0 ∈ V 1 and v s i ∈ V 2 .
Moreover, if the only vertex of P which belongs to V 1 is v s0 and the only vertex of

P which belongs to V 2 is v s i , P is called a direct V 1 → V 2 path. • For q = {1, 2}, ρ q [ V 1 , V 2 ] is the maximal number of disjoint V 1 → V 2 paths included in E 0 ∪E q . Moreover, a set of ρ q [ V 1 , V 2 ] disjoint V 1 → V 2 paths included in E 0 ∪ E q is called maximum V 1 → V 2 linking in E 0 ∪ E q . • For q ∈ {1, 2}, µ q [ V 1 , V 2 ] denotes the minimal num- ber of vertices of U ∪ X ∪ Y belonging to a maximum V 1 → V 2 linking included in E 0 ∪ E q . • For q ∈ {1, 2}, V ess,q [ V 1 , V 2 ] is the vertex subset in- cluding the vertices present in all the maximum V 1 → V 2 linkings included in E 0 ∪ E q .
• For q ∈ {1, 2}, there exists a unique vertex subset noted

S o q [ V 1 , V 2
] and called minimum output separator which is the set of begin vertices of all direct

V ess,q [ V 1 , V 2 ] → V 2 paths included in E 0 ∪ E q .
• For q ∈ {1, 2}, there exists a unique vertex subset noted

S i q [ V 1 , V 2 ] and called minimum input separator which is the set of end vertices of all direct V 1 → V ess,q [ V 1 , V 2 ] paths included in E 0 ∪ E q . In Example 1, ρ 1 [ U, Y ] = ρ 2 [ U, Y ] = 1 and ρ 1 [ {x 1 , x 2 , x 3 }, Y ] = ρ 2 [ {x 1 , x 2 , x 3 }, Y ] = 2. Moreover, V ess,2 [ {x 3 }, Y ] = {x 3 , x 2 , x 5 } but V ess,1 [ {x 3 }, Y ] = {x 3 , x 2 } because there exists in E 0 ∪ E 1 a x 3 → Y path which does not cover x 5 : x 3 → x 2 → x 1 → y 1 . Finally, S o 1 [ U, Y ] = {x 8 }, S o 2 [ {x 3 }, Y ] = {x 5 }, S o 1 [ {x 2 , x 3 }, Y ] = {x 2 }, S o 2 [ {x 2 , x 3 }, Y ] = {x 5 }, S i 1 [ {x 2 , x 3 }, Y ] = {x 2 } = S i 2 [ {x 2 , x 3 }, Y ] .
4 Main result

Preliminaries

Let V Y ⊆ Y denotes a vertex subset playing the role of the output vertex subset instead of Y. Our aim is to characterize the generic dimension of the observability subspace when only measurements related to V Y are available. To do so, we subdivide first the system into two parts from the observability point of view: the maximal over-determined part and the rest related to the square part and the underdetermined part. The discrete mode observability can be achieved only in the over-determined part. This subdivision is provided in (van der [START_REF] Trentelman | The generic number of invariant zeros of a structured linear system[END_REF] and is equivalent to the Dulmage-Mendelsohn decomposition when representing the system by a bipartite graph. Definition 3 [START_REF] Boukhobza | State and input observability for structured linear systems: A graph-theoretic approach[END_REF] In digraph

G(Σ Λ ), for each vertex subset V Y ⊆ Y and q ∈ {1, 2}, we define: • X 1,q (V Y ) def = { x i | ρ q [ U ∪ {x i }, V Y ] > ρ q [ U, V Y ]} ; • Y 0,q (V Y ) def = V Y ∩ V ess,q [ U, V Y ] ; • Y 1,q (V Y ) def = V Y \ Y 0,q (V Y ); • U 0,q (V Y ) def = {u i | there is no edge from u i to X 1,q (V Y ) ∪ Y 1,q (V Y )}; • U 1,q (V Y ) def = U \ U 0,q (V Y ); • X s,q (V Y ) def = S o q [ U 0,q (V Y ), V Y ] ∩ X; • X 0,q (V Y ) def = X \ ( X 1,q (V Y ) ∪ X s,q (V Y )
) . We can uniquely decompose for each mode the system into two subsystems: -(Σ 1,q ), representing, from the observability point of view, the over-determined part of the system. Indeed, it is proven in [START_REF] Boukhobza | State and input observability for structured linear systems: A graph-theoretic approach[END_REF]van der Woude, 2000) that this part remains left invertible even if we remove any one of its measurements. This is due to the fact that, in this part, by definition no output vertex is essential in an input-output linking and all the state vertices constituting this subsystem can reach an output vertex independently from the input-output paths. It has necessarily more outputs (measurements) than inputs and is defined by input U 1,q (V Y ) and X s,q (V Y ), state X 1,q (V Y ) and output Y 1,q (V Y ).

-(Σ 0,q ) defined by input U 0,q (V Y ), state X 0,q (V Y ) and output X s,q (V Y ) ∪ Y 0,q (V Y ). This subsystem represents the square and under-determined parts of the system.

U 1,q (V Y ) X s,q (V Y ) Y 0,q (V Y ) X 1 (V Y ) Y 1,q (V Y ) X 0,q (V Y ) X s,q (V Y ) Σ Σ Σ Σ 0,q (V Y ) Σ Σ Σ Σ 1,q (V Y ) V Y U 0,q (V Y ) Figure 2. System subdivision In Example 1, X 1,1 (Y) = {x 1 , x 2 , x 3 , x 4 , x 5 }, Y 1,1 (Y) = Y, Y 0,1 (Y) = ∅ = U 1,1 (Y), U 0,1 (Y) = {u 1 }, X s,1 (Y) = {x 8 }, X 0,1 (Y) = {x 6 , x 7 }.
The decomposition is the same for q = 2. For V Y = Y and for each q ∈ {1, 2}, X 0,q , X s,q , X 1,q , U 0,q , U 1,q , Y 0,q and Y 1,q denote the state, the unknown input and the output associated to vertex subsets X 0,q (Y), X s,q (Y), X 1,q (Y), U 0,q (Y), U 1,q (Y), Y 0,q (Y) and Y 1,q (Y) vectors respectively.

Remark 1 When all the inputs are known, or equivalently measured, then subsystem Σ 0,q , q ∈ 1, 2 contains only the state vertices which are not connected to the measurements.

Important remarks are summarized in the following lemmas:

Lemma 1 There cannot exist a relation linking only the output components of square subsystem Σ 0,q , Y 0,q and X s,q .

Proof: Using results shown in (van der Woude, 2000) (Theorem 5.1) concerning the case of square systems, we have that the generic number of invariant zeros of the pencil matrix of system Σ 0,q is equal to card(X 0,q ) + card(X s,q ) + card(U 0,q ) -

( µ q [ U 0,q (Y), X s,q (Y) ∪ Y 0,q (Y) ] - ρ q [ U 0,q (Y), X s,q (Y) ∪ Y 0,q (Y) ] + card(X s,q )
) . Thus, the dimension of the strongly observable subspace and so the number of possible independent observation equations is equal to µ

q [ U 0,q (Y), X s,q (Y) ∪ Y 0,q (Y) ] -ρ q [ U 0,q (Y), X s,q (Y) ∪ Y 0,q (Y) ]
. Moreover, the cardinality of U 0,q (Y) ∪ X 0,q (Y) representing the unknown variables for this system, is at least equal to µ q

[ U 0,q (Y), X s,q (Y) ∪ Y 0,q (Y) ] - ρ q [ U 0,q (Y), X s,q (Y) ∪ Y 0,q (Y)
] because all the linkings between U 0,q (Y) and X s,q (Y) ∪ Y 0,q (Y) cover vertices included in U 0,q (Y) ∪ X 0,q (Y). Since, in the best case, the number of independent observation equations is equal to the number of unknown variables, we cannot eliminate unknown variables from some equations to obtain a relation linking only Y 0,q and X s,q and their derivatives.

△ Lemma 2 For subsystem (Σ 1,q ), for each Y u = {y i1,q , y i2 , . . . ,

y i k } ⊂ Y 1,q (Y) such that ρ q [ U 1,q (Y)∪ X s,q (Y), Y u ] = card(U 1,q (Y) ∪ X s,q (Y))
, there exist generically a matrix G, a function φ and an integer ν ≤ n 1,q such that ( X T s,q , U T 1,q

) T = φ(Y u , Ẏu , . . . , Y (ν) u ) + GX 1,q , where Y u (ν) = (y (ν) i1,q , y (ν) i2 , . . . , y (ν)
i k ) T . Proof: According to [START_REF] Dion | Generic properties and control of linear structured systems: A survey[END_REF] (Theorem 4), for subsystem (Σ 1,q ), the existence of Y u = {y i1,q , y i2 , . . . ,

y i k } ⊂ Y 1,q (Y) such that ρ q [ U 1,q (Y)∪ X s,q (Y), Y u ] = card(U 1,q (Y) ∪ X s,q (Y))
, implies the invertibility of (Σ 1,q ) using only Y u . Thus, using the inversion algorithm of [START_REF] Silverman | Inversion of multivariable linear systems[END_REF], we can express the input of such system i.e. U 1,q and X s,q in function of its state X 1,q and its output components Y u and their derivatives. Thus, there exist a matrix generically G, a function φ and an integer ν ≤ n 1,q such that (

X T s,q , U T 1,q , ) T = φ(Y u , Ẏu , . . . , Y (ν) u ) + GX 1,q , where Y u (ν) = (y (ν) i1,q , y (ν) i2 , . . . , y (ν) i k ) T .
△ As a corollary of this lemma, we have that: Lemma 3 For each y i ∈ Y 1,q (Y), there exists an output algebraic relation linking it to other measurements of Y 1,q (Y). Proof:

y i / ∈ V ess,q [ U 1,q (Y) ∪ X s,q (Y), Y 1,q (Y) ] im- plies that there exists necessarily a subset Y u ⊂ Y 1,q (Y) such that ρ q [ U 1,q (Y)∪X s,q (Y), Y u ] = card(U 1,q (Y)∪ X s,q (Y)).
Using the previous lemma, there exist a function φ and an integer ν ≥ 0 such that for mode q, ( X T s,q , U T 1,q

) T = φ(Y u , Ẏu , . . . , Y (ν)
u ) + GX 1,q . Substituting this in subsystem (Σ 1,q ), we have:

         Ẋ1,q = ( A 1,1 + (A 1,s , B 1,1 )G ) X 1,q + φx(Yu, Ẏu, . . . , Y (ν) u ) def = ÃX 1,q + φx(Yu, Ẏu, . . . , Y (ν) u ) Y 1,q = ( C 1,1 + (C 1,s , D 1,1 )G ) X 1,q + φy(Yu, Ẏu, . . . , Y (ν) u ) def = CX 1,q + φy(Yu, Ẏu, . . . , Y (ν) u )
(4) Consider then that the characteristic equation of matrix à has the form Ãn1 + . . . + a k Ãk + . . . + a 0 à = 0, where n 1 = card(X 1,q (Y)). Multiplying this equation by Ci , where Ci is the line of matrix C related to output y i in mode q in (6), we obtain an equation on the form:

( C i Ãn1 + . . . + a k Ci Ãk + . . . + a 0 Ci Ã) X 1,q = 0 (5)
Since, y i is an output of Σ 1,q , at least a term Ci Ãk of the latter equation is not zero and depends on y i . Then, (7) leads to an output algebraic equation where some derivatives of y i intervene.

△

Definition 4 Consider SSLS (Σ Λ ) associated to di- graph G(Σ Λ ). To each vertex subset V Y ⊆ Y and q ∈ {1, 2}, we associate the integer β q (V Y ) defined as µ q [ U 0,q (V Y ), X s,q (V Y ) ∪ Y 0,q (V Y ) ] - ρ q [ U 0,q (V Y ), X s,q (V Y ) ∪ Y 0,q (V Y ) ] plus the maximal number of vertices of X 1,q (V Y )∪U 1,q (V Y )∪X s,q (V Y ) covered by a disjoint union of -a ( U 1,q (V Y ) ∪ X s,q (V Y ) ) → Y 1,q (V Y ) linking of maximal size; -Y 1,q (V Y )-topped paths ; -cycles covering only elements of X 1,q (V Y ). Lemma 4 Consider SSLS (Σ Λ ) represented by digraph G(Σ Λ,lin ), β q (V Y
) is equal to the generic dimension of the observable subspace in the extended state and input space (x T (t), u T (t)) T for mode q and restricting the measurements to the output components associated to V Y . Proof: Using results proved in (van der Woude, 2000) (Theorems 5.1 and 5.2 ) and in (van der [START_REF] Dion | Generic properties and control of linear structured systems: A survey[END_REF] (Theorem 3.6) and considering that the measurements of the system are reduced to the components associated to the elements of V Y , for each mode q, number β q (V Y ) is equal to n + m -g_n q,inv,z (P q (s)), where g_n q,inv,z (P q (s)) denotes the generic number of invariant zeros of the pencil matrix P q (s) associated to mode q. Moreover, from [START_REF] Trentelman | The generic number of invariant zeros of a structured linear system[END_REF]) (Lemma 7.7), we have that n + m -g_n q,inv,z (P q (s)) is also equal to the dimension of the observable subspace, i.e. the number of observable directions or independent state and input combinations, in the extended state and input space. △ To achieve the discrete mode observability analysis, it is necessary to have an output algebraic relation linking some outputs and their derivatives and specific to only one of the two modes. When several output components appear in this equation, that means that for some order k i , y (ki) i can be written as a linear combination of y (k) i , k < k i , and of other output components with their derivatives. In this case, the intersection of observability subspace generated by y i and the other components is not empty. This can be characterized by the generic dimensions of such subspaces and so by β q (•).

Definition 5 For each output y i ∈ Y 1,q (Y), and each mode q, we define integer d q (y i ) = β q (Y ) -β q (Y \ {y i }).

Obviously, for all k > d q (y i ), in mode q, y (k) i can be written as a linear combination of y (j) i , j ≤ d q (y i ), and other output components with their derivatives.

Discrete mode observability analysis

Using the previous settings and definitions, we analyse, hereafter, the existence of an output algebraic relation depending on the mode to achieve mode distinguishability. Proposition 1 SSLS (Σ Λ ), with two possible modes q ∈ {1, 2}, associated to digraph G(Σ Λ ) is generically discrete mode observable iff one of the following conditions is satisfied: 1.) For some q ∈ {1, 2}, there exists a cycle C included in E 0 ∪ E q , covering only elements of X 1,q (Y) and containing at least one edge of E q ; 2.) For some q ∈ {1, 2}, there exists a direct X s,q (Y) ∪ U 1,q (Y) → Y 1,q (Y) path included in E 0 ∪E q and containing at least one edge of E q ; 3.) For q ̸ = q ′ , Y 1,q ̸ = Y 1,q ′ ; 4.) For some q ∈ {1, 2}, there exist a vertex subset

V Y ⊆ Y 1,q (Y) such that V Y ⊆ Y 1,q (V Y ) and either V Y Y 1,q ′ (V Y ) or β q (V Y ) > β q ′ (V Y ); 5
.) For some q ∈ {1, 2}, ∃ an edge e κ ∈ E q ending by x ℓ ∈ X 1,q (Y), there exists a v P → y i path of length strictly greater than d q (y i ) which covers e κ and such that x ℓ belongs to a direct S o q [{v P }, Y 1,q ] → y i path included in E 0 ∪ E q .

Proof: Sufficiency: We will prove hereafter that each of the five conditions of Proposition 1 leads to the existence of an output algebraic relation which is satisfied in only one mode and so, according to Definition 1 ensures the mode distinguishability. Condition 1: Let us denote by x i one of the vertices of cycle C verifying Condition 1. Since this vertex belongs to X 1,q (Y), there exist a x i → y j path in E 0 ∪ E q , where

y j ∈ Y 1,q (Y) and a subset Y u ⊂ Y 1,q (Y) \ {y i } with ρ q [ U 1,q (Y) ∪ X s,q (Y), Y u ] = card(U 1,q (Y) ∪ X s,q (Y)) < ρ q [ U 1,q (Y) ∪ X s,q (Y) ∪ {x i }, Y u ∪ {y j } ] .
From Lemma 2, there exist a matrix G, a function φ and an integer ν ≥ 0 such that for mode q, ( X T s,q , U T 1,q

) T = φ(Y u , Ẏu , . . . , Y (ν) u ) + GX 1,q .
Substituting this in subsystem (Σ 1,q ) and using notations of (4):

         Ẋ1,q = ( A 1,1 + (A 1,s , B 1,1 )G ) X 1,q + φx(Yu, Ẏu, . . . , Y (ν) u ) def = ÃX 1,q + φx(Yu, Ẏu, . . . , Y (ν) u ) Y 1,q = ( C 1,1 + (C 1,s , D 1,1 )G ) X 1,q + φy(Yu, Ẏu, . . . , Y (ν) u ) def = CX 1,q + φy(Yu, Ẏu, . . . , Y (ν) u )
(6) Since the elements of G are represented by the edges belonging to the Y u -topped paths, they are independent from the elements of à associated to cycle C. Thus, the digraph representation of (6) contains also C. Therefore, from [START_REF] Reinschke | Multivariable Control. A Graph Theoretic Approach[END_REF]) (Theorem 21.1), the characteristic equation of matrix Ã, has the form Ãn1 + . . . + a k Ãk + . . . + a 0 à = 0, where n 1 = card(X 1,q (Y)) and contains a term a n1-k Ãn1-k 1,1 , where k is the length of C and a n1-k depends on the product of the weights of the edges constituting C and so, on a specific entry of A q according to the existence of an edge belonging to E q in C. Thus,

( C j Ãn1 + . . . + a k Cj Ãk + . . . + a 0 Cj Ã) X 1,q = 0 (7)
where Cj is the line of matrix C related to output y j in mode q in (6) i.e. y j = Cj X 1,q + φ y,j (Y u , Ẏu , . . . , Y (ν) u ). On the one hand, since x i belongs to a cycle and as there exists a x i → y j path if we denote by ℓ > 0 the length of this path, then ∀k ≥ ℓ -1, Cj Ãk ̸ = 0. Furthermore, since ℓ + k ≤ n 1 , then term a n1-k Cj Ãn1-k ̸ = 0 and also term Cj Ãk . On the other hand, due to dynamics (6) for all

k ≥ 0 Cj Ãk = y (k) j + φ u (Y u , Ẏu , . . . , Y (ν+k) u
). Thus, (7) leads to an equation in the form:

y (n1+1) j +. . .+a k y (k) j +. . .+a 0 y j +υ(Y u , . . . , Y (2n1) u ) = 0
(8) Therefore, as a n1-k is specific to mode q, equation (7) leads to an output algebraic relation of the kind Ψ q (Y 1,q , Ẏ1,q , . . . , Y (s) 1,q ) = 0 parametrized by at least an element of A q specific to mode q.

Condition 2: Let denote by v i the vertex of U 1,q (Y) ∪ X s,q (Y) from which there is a direct X s,q (Y) ∪ U 1,q (Y) → Y 1,q (Y) path P included in E 0 ∪ E q and containing at least an edge e κ of E q . Let us denote also by λ κ the non-zero parameter (or weight) associated to e κ , y j the end of P and ℓ its length. Since

v i belongs to U 1,q (Y) ∪ X s,q (Y), it exists a subset Y u = {y i1 , y i2 , . . . , y i k } ⊂ Y 1,q (Y) \ {y i } such that ρ q [ U 1,q (Y) ∪ X s,q (Y), Y u ] = card(U 1,q (
Y) ∪ X s,q (Y)) without using edges and vertices of P . From Lemma 2, there exist a matrix G, a function φ and an integer ν ≤ n 1 such that the dynamics equation of subsystem (Σ 1,q ) is in the form (6). Since the elements of G are represented by edges belonging to the Y u -topped paths, they are independent from the elements of Ã. From the characteristic equation of matrix Ã, we can write an equation as (7), where also Cj is the line of matrix C related to output y j in mode q in (6) i.e.

y j = Cj X 1,q + φ y,j (Y u , Ẏu , . . . , Y (ν) u ). Using equations of 6), we have Cj Ãk X 1,q = y (k) j -Cj ( φ (k-1) x (Y u , Ẏu , . . . , Y u (ν) ) + Ãφ (k-2) x (Y u , Ẏu , . . . , Y u (ν) ) + . . . + Ãk-2 φx (Y u , Ẏu , . . . , Y u (ν) ) ) -φ (k) y,j (Y u , Ẏu , . . . , Y (ν) u ).
Since there exists a path of length ℓ from v i to y j containing an edge e κ , term C j Ãℓ-1 , for all k ≥ ℓ, is not zero and depends on the element of A q denoted previously λ κ specific to mode q. On the other hand, term φ

(k-ℓ) x (Y u , Ẏu , . . . , Y u (ν)
) is not zero for all k ≥ ℓ, because there exists a path between v i and an element of Y u as by definition of

Y u ρ q [ U 1,q (Y) ∪ X s,q (Y), Y u ] = card(U 1,q (Y) ∪ X s,q (Y)). This implies that, C j Ãℓ-1 φ (k-ℓ) x (Y u , Ẏu , . . . , Y u (ν)
) is not zero for all k ≥ ℓ and depends on λ κ . In particular, this implies that term Cj Ãn1 , present in relation ( 7) is not zero and depends on λ κ . Thus, substituting terms Cj Ãk X 1,q by y

(k) j -Cj ( φ (k-1) x (Y u , Ẏu , . . . , Y u (ν) ) + Ãφ (k-2) x (Y u , Ẏu , . . . , Y u (ν) ) + . . . + Ãk-2 φx (Y u , Ẏu , . . . , Y u (ν) ) ) -φ (k) y,j (Y u , Ẏu , . . . , Y (ν)
u ), in equation ( 7), we obtain an output algebraic relation and knowing that term

C j Ãℓ-1 φ (k-ℓ) x (Y u , Ẏu , . . . , Y u (ν)
) is specific to mode q, the obtained output algebraic relation is parametrized by at least a specific element of A q and so allows to achieve the distiguishability.

Condition 3: This condition implies that there exists y i ∈ Y 1,q (Y) \ Y 1,q ′ (Y). That means, using Lemma 3, that all the possible output algebraic relations in mode q ′ do not include this y i or its derivatives but there is an output algebraic relation including y i and other outputs of Y 1,q (Y) and their derivatives in mode q . Therefore, the latter relation is satisfied only in modeq and so achieves discrete mode observability. Condition 4: If V Y Y 1,q ′ (Y), that means from Lemma 3 that there exists a relation linking the components of V Y = {y i1 , y i2 , . . . , y i ℓ } and their derivatives in mode q but not in mode q ′ . This achieves the discrete mode observability. Otherwise, if β q (V Y ) > β q ′ (V Y ), then the generic dimension of the observability subspace associated to V Y is greater in mode q than in mode q ′ . In this case, let us denote by k i1 , k i2 , . . . , k i ℓ an ℓ-uple of integers such that the observability subspace associated to V Y by in mode q is generated by Y e = (y i1 , ẏi1 , . . . , y (ki 1 ) i1 , y i2 , ẏi2 . . . , y i ℓ , . . . , y

(ki ℓ ) i ℓ ) T where ℓ ∑ j=1 (k ij + 1) = β q (V Y ). Since, β q ′ (V Y ) < β q (V Y )
at least one output derivative y (ki j ) ij can be expressed in mode q ′ using lower derivatives of y ij and other elements of Y e . This expression is valid in mode q ′ but not in mode q and this achieves discrete mode observability.

Condition 5: The fact that, for some q, y i belongs to Y 1,q (Y) implies that there exists a vertex subset

Y u ⊆ Y 1,q (Y) \ {y i } such that ρ q [ U 1,q (Y) ∪ X s,q (Y), Y u ] = card(U 1,q (Y) ∪ X s,q (Y)
). This implies, from Lemma 2, that there exist a matrix G, a function φ and an integer ν ≤ n 1 such that the dynamics equation of subsystem (Σ 1,q ) can be put on form (6). Moreover, by definition of d q (y i ), we have that, the dimension of the observability subspace is equal to integer d q (y i ) plus the dimension of the observability subspace without y i . This means that, the d th q (y i ) derivative of y i can be expressed using the other output components and the lower derivatives of y i . Thus, ∀k ≥ d q (y i ), there exist a minimal subset

Ỹ ⊆ Y 1,q (Y) \ (Y u ∪ {y i }), such that y (k) i
is a linear combination of the first derivatives of y (s) i , s = 0, . . . , k-1 and the all the possible derivatives (until the n th 1 the dimension of the under-determined part) of components y l of Ỹ. Thus, there exist some constant real parameters denoted α i,s and α l,s such that

y (k) i = ∑ s< ki α i,s y (s) i + ∑ l | y l ∈ Ỹ n 1 ∑ s=0 α l,s y (s) l + υ(Yu, . . . , Y (2n 1 ) u ) (9)
where n 1 = card(X 1,q ). The aim of the rest of the proof is to show that if Condition 5 is satisfied, some of these parameters are specific to mode q and this relation allows to achieve the mode distinguishability.

Using the system's equations described in (6), we can deduce that for each s and l, y

(s) i = Ci Ãs X 1,q + f ct i,s (Y u , . . . , Y (n1+s) u
) and y

(s) l = Cl Ãs X 1,q + f ct l,s (Y u , . . . , Y (n1+s) u
), where Ci (resp. Cl ) is the row of matrix C corresponding to output y i (resp. y l ). Substituting the latter equalities in (9) , all functions of Y u and its derivatives will simplify as they are independent from the considered elements of X 1,q and (9) can be written as 10), all the components of Ỹ intervene. Let us denote by v P = x j the begin vertex of the so-called path P satisfying Condition 5 (i.e. P is a y i -topped path of length k + 1 strictly greater than d q (y i ) and covers e s ) and e j the j th Euclidean vector. As this relation is valid whatever the state vector X 1,q , we can remove it and by multiplying relation (10) in the right by e j , we obtain: 11) where each non-zero component of Cl Ãs is associated to the paths arriving to y l ∈ Ỹ of length s + 1. Since all the {x j } → Ỹ ∪ {y i } paths starting from x j cover, by definition, S o q [ {x j }, Ỹ ∪ {y i } ] def = {x r }, then there exist k r and k ′ such that k r + k ′ = k and Ci Ãk e j = Ci A kr ∆ r Ãk ′ e j where ∆ r is a diagonal matrix having only one nonzero element ∆ r (r, r) = 1. We can do the same reasoning for each term Cl Ãs e j and so there exist s r and s ′ such that s ′ r + s ′ = s and Cl Ãs e j = Cl Ãsr ∆ r Ãs ′ e j . The fact that end vertex of e κ , x ℓ belongs to a direct S o q [{x j }, Y 1,q ] → y i path implies that specific edge e κ ∈ E q belongs to a S o q

Ci A k X 1,q = ∑ s<dq (y i ) α i,s Ci A s X 1,q + ∑ y l ∈ Ỹ n 1 ∑ s=0 α l,s Cl Ãs X 1,q (10) Since subset Ỹ is minimal i.e. ∀y j ∈ Ỹ, β q ( Ỹ ∪ {y i } ∪ Y u ) \ {y j } ) -β q (Y u ∪ Ỹ \ {y j }) > k i , then in relation (
Ci A k e j = ( ∑ s<dq (y i ) α i,s Ci A s + ∑ y l ∈ Ỹ n 1 ∑ s=0 α l,s Cl Ãs ) e j (
[ v P , Ỹ ∪ {y i } ] → Ỹ ∪ {y i } path.
This means that edge e κ appears in only some of

S o q [ v P , Ỹ ∪ {y i } ]
→ Ỹ ∪ {y i } paths. Thus, some, but not all, terms Ci A kr and Cl Ãsr contain the non-zero parameter corresponding to edge e κ , specific to mode q. Denoting by C r = e T r , where e r is the r th Euclidean vector, we have that Ci Ãk e j = Ci

A kr ∆ r Ãk ′ e j = α ′ C r Ãk ′ e j
and Cl Ãs e j = Cl Ãsr ∆ r Ãs ′ e j = α ′ l,s C r Ãs ′ e j , where

α ′ l,s def = ( Cl Ãsr ∆ r )(C r C T r ) -1 = ( Cl Ãsr ∆ r ).
Thus, after substitution of the previous terms in relation (11),

α ′ Cr A k ′ e j = ( ∑ sr ≤s<dq (y i ) α ′ i,s α i,s Cr A s-sr + ∑ l | y l ∈ Ỹ n 1 ∑ s=sr α ′ l,s α l,s Cr Ãs-sr ) e j ( 12
)
where some, but not all coefficients α ′ and α ′ l,s depend on the weight of e κ . This weight cannot be factorized and simplified because all the coefficients do not depend on it (some S o q [ v P , Ỹ ∪ {y i } ] → Ỹ ∪ {y i } paths do not contain edge e κ ). Therefore, equality ( 12) is valid only if some of the coefficients α, α i,s and α l,s depend also on the weight λ κ of e κ . Thus, by means of equation ( 9) containing coefficients α i,s and α l,s , we obtain an output algebraic relation depending on λ κ and satisfied only when the discrete mode variable is equal to q. Necessity: Firstly, from Lemma 1, there cannot exist an output algebraic relation linking the output components of the under-determined subsystem (Σ 0,q ). So, only the edges in subsystem (Σ 1,q ) can lead to an output algebraic relation (between outputs Y 1,q (Y) and their derivatives) useful to deduce the discrete mode. In this case, some outputs have to be used to invert (Σ 1,q ) in order to eliminate from the output equation all the unknown inputs and so to use all the possible derivatives of the other outputs in an output algebraic equation. It is necessary then to consider a subset

Y u ⊆ Y 1,q (Y) such that ρ q [ U 1,q (Y) ∪ X s,q (Y), Y u ] = card(U 1,q (Y)∪X s,q (Y))
. Indeed, from Lemma 2 and Theorem 4 of [START_REF] Dion | Generic properties and control of linear structured systems: A survey[END_REF], only in such a case the system is left-invertible and so there exist integer ν ≥ 0, matrix G and function φ such that ( X T s,q , U T 1,q

) T = φ(Y u , Ẏu , . . . , Y u (ν) ) + GX 1,q
. This quantity can then be substituted in subsystem (Σ 1,q ) for mode q, to obtain an equation of form (6). When Conditions 1 and 2 are not satisfied, it is not possible to obtain an output algebraic relation achieving mode distinguishability from the characteristic equation of any matrix à of (6) related to any choice of Y u , when this equation has the same degree for the two modes. It is the case when Condition 4 is not satisfied. Moreover, when Condition 2 is not satisfied, for any possible choice of

Y u , φ y (Y u , Ẏu , . . . , Y (ν) u ) and φ x (Y u , Ẏu , . . . , Y (ν)
u ) do not depend on any specific element of A q , B q , C q or D q , q ∈ {1, 2}. Furthermore, any output relation can be put on the form (9), for some integer k, output component y i and some set of output components Y which do not include y i and are disjoint from Y u . Moreover, when Conditions 3 and 4 are not satisfied, the output components implicated in any of these expressions and their derivatives degrees are the same for modes 1 and 2. Thus, the only way to have a difference between the two output algebraic equations respectively is that there exists at least one coefficient α i,s or α l,s of Equation ( 9) characterizing specifically a parameter specific only to one of the two modes' model. Without loss of generality, we can assume that Y is minimal i.e. in the output algebraic relations and so all the components of Y appear. If it is not the case, we remove them from Y. Relation (9) implies that the observable subspace obtained using outputs {y i } ∪ Y u and the one generated by Ỹ ∪ Y u have non zero intersection. Defining

k i = β q ( Ỹ ∪ {y i } ∪ Y u ) -β q ( Ỹ ∪ Y u ) ≥ d q (y i ), relation (9) cannot be satisfied for some k ≤ k i -1. Thus, in out- put algebraic relation (9), k ≥ k i . Furthermore, since Y is minimal, we cannot express y (ki) i
using only a part of Ỹ i.e. ∀y j ∈ Ỹ. To guarantee discrete mode observability, there must exist at least a y i -topped path P , of length greater or equal to k i + 1 and so of length strictly greater than d q (y i ), associated to this relation. Let us denote by x j the begin vertex of P (it cannot be a U 1,q (Y) ∪ X s,q (Y) → Y 1,q (Y) path when Condition 2 is not satisfied) and e j the j th Euclidean vector. When Condition 2 is not satisfied, in Equation (9), υ(Y u , Ẏu , . . . , Y (n) u ) does not depend on any element specific of A q , B q , C q or D q (q ∈ {1, 2}). Then, equation ( 9), can be written as relation (11), where Cl Ãs

is associated y l -topped paths, y l ∈ Ỹ of length s + 1. All the {x j } → Ỹ ∪ {y i } paths cover x r = S o q [ x j , Ỹ ∪ {y i } ]
. So, if for each x ℓ end vertex of a specific edge belonging to P , x ℓ does not belong to any direct S o q [{v P }, Y 1,q ] → y i path, then all the paths starting from x r to Y contain only edges of E 0 . Then according to the same arguments as previously (in the part of the proof devoted to the sufficiency of Condition 5), we can write an equation similar to (12) with the same notations, where α ′ and α ′ l,s ′ depend only on the entries related to the edges of E 0 when Condition 5 is not satisfied. Equation ( 12) implies that there exists an output algebraic relation with the use of only virtual output Cr x in addition to Y u . This relation can be due to the characteristic equation of matrix A which includes only the terms related to cycles in the digraph. When Condition 1 is not satisfied, there is no edge of E q , q ∈ {1, 2} in any cycle. Thus, as all α ′ and α ′ l,s ′ depend only on the entries related to the edges of E 0 , it is also the case for coefficients α i,s and α l,s in equation ( 9). So, all the existing relations of the form (9) do not contain terms of A q , B q , C q or D q (q ∈ {1, 2}). We can do the same reasoning

for all Y u ⊆ Y 1,q (Y) s.t. ρ q [ U 1,q (Y) ∪ X s,q (Y), Y u ] = card(U 1,q (Y) ∪ X s,q (Y)). △
Comments and interpretation: To establish the discrete mode observability, we first search in the graph the edges specific to one mode and we interest to their position w.r.t. the output vertices. If there is a specific edge belonging to any cycle in (Σ 1,q ) then the distinguishability is possible (first condition). If a specific edge belong to any input → output path in (Σ 1,q ) then the distinguishability is possible (second condition). If a specific edge allows to modify the output subdivision Y 0 /Y 1 (third condition) or to modify observability subspace of any output measurements set (fourth condition), then the distinguishability is possible. Finally, if a specific edge belongs to an output rooted path including a state vertex which can be linked independently to other outputs, then the distinguishability is possible (fifth condition). If no condition is satisfied, then the two modes are sufficiently similar or their differences are not observable from the measurements and so they are not distinguishable.

An intuitive structural interpretation of the conditions and what they entail can be also provided, knowing that an output algebraic relation can exist only in the over-determined part of a system and that any output belonging to this overdetermined part belongs to at least one output algebraic equation:

• Condition 1 means that there exists an output combination with a dynamics depending on the discrete mode variable but not from the unknown input.

• Condition 2 means that an observable input-output behaviour depends on the discrete mode variable. This dependence is structural when the relative degree of the considered input changes else it is parametric.

• Condition 3 means that the under-determined part of the system changes according to the discrete mode variable. Thus, there exists an output algebraic relation including a particular output y i in a mode while all the possible output algebraic equations in the other mode do not include this output component.

• Condition 4 means that the dimension of the observability subspace associated to some output components depends on the discrete mode variable. This is equivalent to say that the observability index of some output components changes and so the modes are distinguishable from their observability characteristics. This implies that there exists an output algebraic relation between several outputs of the system and whose order depends on the discrete mode.

• Condition 5 means that there exists an output algebraic relation between several outputs of the system and that this relation contains some parameters depending on the discrete mode.

To summarize, the conditions provided in Proposition 1 can be classified into two categories. The first one includes Conditions 3, 4, 2 when the subdivision of the system for the two modes are different and a part of Condition 1. In this case, the two modes can be said structurally distinguishable. The distinguishability here depends on the structure of the systems associated to each mode and not on the parameters' values. In the case of Condition 1, when the cycle exists in a mode and not in the other one, there is then at least an additional relation differentiating the two modes and so the distinguishability is here also structural. When only Condition 5, the distinguishability is purely parametric. The main differences with the formulation given in [START_REF] Boukhobza | Observability of switching structured linear systems with unknown input: a graph-theoretic approach[END_REF] of the discrete mode observability conditions lie in three points:

• Condition 3 of [START_REF] Boukhobza | Observability of switching structured linear systems with unknown input: a graph-theoretic approach[END_REF] has been split in two conditions here Condition 3 and Condition 4. This is done for computational reasons. Indeed, Condition 3 of [START_REF] Boukhobza | Observability of switching structured linear systems with unknown input: a graph-theoretic approach[END_REF] is stated as "For some q ∈ {1, 2}, there exist a vertex y i ∈ Y 1,q (Y) and disjoint vertex subsets Ỹ ⊆ Y 1,q (Y) \ {y i } (this subset can be empty), Y u ⊆ Y 1,q (Y) \ {y i } such that:... " three conditions must be verified. This implies a combinatorial search to check a property on all the output vertices of Y 1,q (Y) but also on all the possible subsets Ỹ and Y u . In the present paper, Condition 3 does not necessitate any computation and for Condition 4, we check a test on only one output vertex subset V Y . On the one hand, this is computationally easier and on the other hand, globally only two conditions must be verified on V Y , knowing that one of them (V Y ⊆ Y 1,q (V Y )) can be expressed easily using a linking condition.

• Condition 4 of [START_REF] Boukhobza | Observability of switching structured linear systems with unknown input: a graph-theoretic approach[END_REF]) is substituted in Proposition 1 by Condition 5. In [START_REF] Boukhobza | Observability of switching structured linear systems with unknown input: a graph-theoretic approach[END_REF], checking Condition 4 implies, as previously, to test 5 properties, on all possible groups constituted by an output vertex, a path and two disjoint output vertices.

In the present paper, checking Condition 5 is equivalent to check a condition on the position of a specific edge w.r.t. its distance to an output vertex. We do that by working on a simplified graph where are removed all the edges which do not belong to a sufficiently long path. Condition 5 necessitates the computation of, at most, n output separators.

• Due to the changes in the formulation of the discrete mode conditions, as we will see below, the sensor location study to recover the discrete mode observability is possible because it can be centred on the position of specific edges w.r.t. output vertices in the digraph. With conditions given in [START_REF] Boukhobza | Observability of switching structured linear systems with unknown input: a graph-theoretic approach[END_REF], this is not possible, because the two last ones depend on a group constituted of at least three parts: a single output vertex and two subsets of output vertices. Thus, when adding a sensor, we must consider all the cases where each of these groups changes.

In the conditions given in Proposition 1, we must check a property for at most one output vertex or one output vertex subset.

Remark 2 When all the inputs are known, according to Remark 1, Σ 1,q , q ∈ 1, 2 will contain all the input vertices and all the state vertices which are connected to the outputs. That means that, the subdivision of the system do not give an under-determined part. Consequently, the specific edges belonging to any Y -topped path may be useful to establish the discrete mode observability. This can be illustrated in Example 1 where edge (x 6 , x 7 ) is specific but as it belongs to Σ 0,q part, q ∈ 1, 2, it cannot be used to generate an output algebraic relation specific to mode 1 or mode 2. But, if input u 2 is known (or measured), this edge will belong to an input-output path in Σ 1,q , q ∈ 1, 2, Condition 2 will be satisfied and the weight of this edge participates to an output algebraic relation specific to mode 1 or mode 2. Note that, when all the input are measured, the Proposition is applicable, by adding in the system output measurements on all the inputs i.e. output vertices and edges linking each known input vertex to its associated measurement vertex.

Example 1 Continued It is easy to see that Condition 1 is satisfied while Condition 2 is not. Indeed, cycle x 4 → x 4 contains a specific edge related to mode 1 and x 4 ∈ X 1,1 , then Condition 1 is satisfied and the system is discrete mode observable. Moreover, U 1,1 = U 1,2 = ∅ and X s,1 = X s,2 = {x 8 } but there is no direct path including specific edges between X s,1 or X s,2 and Y, so Condition 2 cannot be satisfied. Let us check Condition 5. There is only one edge (x 2 , x 1 ) specific to mode 1 and this edge can belong only to y 1topped paths. So, let us compute d 1 (y 1 ):

β 1 (Y) = 9 and β 1 (Y \ {y 1 }) = 8. Thus, d 1 (y 1 ) = 1. Pred 1 ({x 1 }) = {x 2 , x 3 } and we have S o 1 (x 2 , Y 1,1 ) = S o 1 (x 3 , Y 1,1 ) = {x 2 }. x 1 belongs to a direct {x 2 } → y 1 path included in E 0 ∪ E 1 .
Let us search now a y 1 -topped path P whose length is greater or equal to d 1 (y 1 ) + 1 = 2 and it includes edge (x 2 , x 1 ). We can choose P = x 2 → x 1 → y 1 with v P = x 2 and so, Condition 5 is then satisfied. This graphic condition characterizes output algebraic relation ẏ1 λ7λ11 -ẏ2 λ8λ14 = 0 depending on λ 11 and λ 14 and specific to mode 1 only. This relation is not satisfied for mode 2 because, in this case ẏ1 is identically zero. Relation ẏ1 = 0 is specific to mode 2 even if there is no specific coefficient to mode 2 appearing because here Condition 4 is satisfied: or in other terms, β 1 ({y 1 }) = 3 ̸ = β 2 ({y 1 }) = 1. This traduces that in mode 2, ẏ1 = 0 while it is not the case in mode 1.

To handle a ically specified tnumero a specific realization of the structured system with numerically specified parameters, we only apply the previous generic results. In the case of the specific system given at the end of Example 1, we can conclude to the discrete mode observability for almost any unknown input and initial state because the following output relations allow to distinguish the mode: ẏ1 = 0 which is valid in mode in mode 2, ẏ1 -ẏ2 2 = 0 and ẏ3 = -3y 3 + y 2 -y 4 which are valid only in mode 1. Note that sometimes (it is not the case for this example), the generic algebraic relation which allows the distinguishability, have some singularities w.r.t. the parameter values. In this case, for the specific realizations presenting these singularities, the generic result is not applicable.

Additional sensor location to recover the mode observability

When none of the conditions of Proposition 1 is satisfied, we propose hereafter a strategy in order to precise as finely as possible the additional sensors' locations allowing to distinguish the two discrete modes. This problem is more complicated than a classical observability problem [START_REF] Commault | Observability preservation under sensor failure[END_REF][START_REF] Boukhobza | State and input observability recovering by additional sensor implementation: a graph-theoretic approach[END_REF] or fault diagnosability one [START_REF] Commault | Sensor location for diagnosis in linear Systems: a structural analysis[END_REF] because of the objective is not to detect or to observe a state variable or an input variable but to detect a change in a dynamics. Knowing that this information is supported only by the edges, it is pertinent to concentrate our reasoning particularly on the ones which are not common to the two modes. Choosing one of these specific edges, denoted e s = (v s , v ′ s ) ∈ E q , we will treat all the situations, according to its position in the digraph. To obtain all the additional sensors' locations ensuring the discrete mode observability, it is then sufficient to roll out the following procedure for all the edges which are not common to the two modes. In the sequel, Y add denotes the set of additional sensors. Obviously, the added sensors are considered to be common to the two modes. Indeed, it is not realistic to add sensors only active during a certain mode, especially when the system is not discrete mode observable. 1) e s = (v s , v ′ s ) belongs to a cycle but included Σ 0,q : In this case, to satisfy to Condition 1 it is necessary to add a sensor in order to make v ′ s belonging to Σ 1,q . It does not matter if v s (when different from v ′ s ) still belongs to Σ 0,q after the additional sensor placement, because in this case, v s will belong to X s,q (Y) and Condition 2 will be satisfied. Thus, the objective is to have v ′ s ∈ X 1,q (Y ∪ Y add ). Therefore, let us denote by S u,q = S i q (U, Y), it is necessary (but not always sufficient) to place a sensor to measure any vertex covered by any direct S u,q ∪ {v ℓ } → S i q (S u,q ∪ {v ℓ }, Y) path and strictly before S i q (S u,q ∪{v ℓ }, Y), where v ℓ is any vertex belonging to the cycle. If, after adding a sensor, the cycle is not in Σ 1,q , we must repeat the placement above considering the new sensor set until obtaining v ′ s ∈ X 1,q (Y ∪ Y add ). The maximal number of additional sensors to make this inclusion true is card(S u,q ∪ {v ′ s }) -ρ q [S u,q ∪ {v ′ s }, Y]. Example 2 Consider now the SSLS represented by the digraph depicted in Figure 3. For this example, the differences between the two modes lie in two edges: (x 5 , x 6 ) and (x 3 , x 7 ). All the state, input and output vertices belong to Σ 0,q for the two modes be-

cause of ρ 1 [U, Y] = ρ 2 [U, Y] = 2 = card(Y)
. Thus, cycle x 5 → x 6 → x 5 containing a specific edge to mode 1 is included also in Σ 0,1 . To satisfy Condition 1, we must add a sensor. Here, S u,1 = {x 4 , u 2 } and S i 1 (S u,1 ∪{x 6 }, Y) = S i 1 (S u,1 ∪ {x 5 }, Y) = {y 1 , y 2 }. Therefore, an additional sensor measuring any vertex on any {x 4 , u 2 , x 5 } → {y 1 , y 2 } path or {x 4 , u 2 , x 6 } → {y 1 , y 2 } path i.e. any component in {x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , u 2 } is necessary and here sufficient to ensure the discrete mode observability. Note that, we find u 2 in the previous subset. Indeed, as it is highlighted in Remark 2 when input u 2 is known, then the under-determined part of the system includes more vertices and edges and this makes the discrete mode-observability possible.

2) e s = (v s , v ′ s ) belongs to an U -rooted simple path but included Σ 0,q : In this case, to satisfy to Condition 2, it is necessary and sufficient to add a sensor in order to make v s either in X s,q (Y ∪ Y add ) or in X 1,q (Y ∪ Y add ). Therefore, let us denote by S u,q = S i q (U, Y), it is necessary and sufficient to place a sensor to measure v s or any vertex covered by any direct S u,q ∪ {v ′ s } → S i q (S u,q ∪ {v ′ s }, Y) path and strictly before S i q (S u,q ∪ {v ′ s }, Y). Example 3 Consider again the SSLS represented by the digraph depicted in Figure 3. Specific edge to mode 2 (x 3 , x 7 ) is included in Σ 0,1 . To satisfy Condition 2, we must add a sensor in order to make

x 3 ei- ther in X s,2 (Y ∪ Y add ) or in X 1,2 (Y ∪ Y add ). Since x 3 ∈ V ess,2 [U, Y], we can make it included as well in X s,2 (Y ∪ Y add ) or in X 1,2 (Y ∪ Y add ). Here, S u,2 = {x 4 , u 2 } and S i 1 (S u,2 ∪ {x 7 }, Y) = {x 2 , x 4 }.
Therefore, to satisfy Condition 2, it is necessary and sufficient to measure {u 2 , x 3 , x 7 }. Note that, we find u 2 in the previous subset. Indeed, as it is highlighted in Remark 2 when input u 2 is known, then the under-determined part of the system includes more vertices and edges and this makes the mode-observability possible. Note that, by this additional sensor location, dedicated to recovering Condition 2, we can satisfy in the same time Condition 3 since these conditions are intricately linked.

3) e s = (v s , v ′ s ) is exclusive to one mode, not covered neither by a cycle nor by an U -rooted path: In this case, the predecessors of v s do not include an input vertex. In addition to this, as the added sensor corresponds to a vertex and a common edge, it is not possible to satisfy, with any additional sensor located anywhere, Condition 3. Thus, we interest here only to recovering Condition 4 and this can be possible only if (v s , v ′ s ) does not belong to E 0 (is exclusive to one mode). Let us denote by V q,Obs the set of all observable input and state components, which is equal to {v ∈ X ∪ U, β q (Y ∪ {v}) = β q (Y)} [START_REF] Boukhobza | Partial state and input observability recovering by additional sensor implementation: a graph-theoretic approach[END_REF]. To satisfy condition 4, it is necessary and sufficient to place a sensor to take measures on the following set {v ′ s } ∪ { x j successor of v ′ s such that the maximal length path arriving to x j covers necessarily e s } . This set can be unfortunately empty. If it is not the case, the new output allows to construct a set of outputs V Y , which has not the same observability subspace dimension in mode q and in mode q ′ . Indeed, the added measurement is sensitive only in mode q to v s representing the most distant unmeasurable state vertex for the added sensor. For this example, without unknown inputs, all the state and output vertices belong obviously to Σ 1,q . But there cannot exist an output algebraic relation since there is only one output (Conditions 3 and 5 cannot be satisfied), no cycle (Condition 1 cannot be satisfied), no input (Condition 2 cannot be satisfied) and Condition 4 is not satisfied . Since there exists an exclusive edge (x 3 , x 1 ) related to mode 1, we can satisfy Condition 4 by adding a sensor. The set of successors of x 1 are {x 2 , x 6 , y 1 } and V 1,Obs = ∅. We can place a sensor on x 1 or on x 6 since the maximal length path arriving to these vertices covers necessarily (x 3 , x 1 ). It is not the case for x 2 since there is a maximal path arriving to x 2 and starting in x 5 and dos not cover (x 3 , x 1 ). Thus, adding a sensor on x 1 or on x 6 allows to satisfy Condition 4 and to recover the discrete mode observability. 4) e s = (v s , v ′ s ) does not belong neither to a cycle nor to an U -rooted simple path and the previous step does not succeed: In this last case, the aim is to add some sensors in order to recover Condition 5. To do so, we proceed in two steps. The objective of the first step is to ensure v s belongs to a direct S o q [Pred q ({v s }), Y 1,q ] → Y 1,q path included in E 0 ∪ E q . The objective of the second step is to ensure that, for at least one output y i successor of v ′ s , we have that d q (y i ) is less or equal than the maximal distance from v ′ s to y i . • Step 1 : If v s does not belong to a direct S o q [Pred q ({v s }), Y 1,q ] → y i path included in E 0 ∪ E q , then it is necessary and sufficient to place a sensor to measure v s or any vertex covered by any direct Pred q ({v s }) → S i q (Pred q ({v s }), Y) path. • Step 2 : If after Step 1 above, Condition 5. is not satisfied, then we propose an iterative additional sensor location procedure: While Condition 5. is not satisfied choose y i one of the output components successors of v ′ s place an additional sensor on any predecessor of y i which is not in V q,Obs and from which the maximal distance to y i is less or equal than the one from v ′ s to y i . The algorithm above leads, when we consider always the same output component successor of v ′ s , y i to the decrease of d q (y i ) until attaining a value less than the distance from v ′ s to y i . In this case, Condition 5. will be satisfied.

Example 5 Consider the SSLS represented by the digraph depicted in Figure 5. We have that ∀q ∈ {1, 2}, X 0,q (Y) = U 0,q (Y) = X s,q (Y) = Y 0,q (Y) = ∅, U 1,q (Y) = {u 1 }, X 1,q (Y) = X and Y 1,q (Y) = Y. Furthermore, ∀q ∈ {1, 2}, β q (Y) = 6, β q ({y 2 , y 3 }) = 3, β q ({y 1 , y 3 }) = 5 and β q ({y 1 , y 2 }) = 5. It is easy to see that the three first conditions of Proposition 1 are not satisfied. Moreover, since there is no exclusive edge, Condition 4 cannot be satisfied. The only specific edge is (x 4 , y 1 ) and has as output successor only y 1 . When we take y i = y 1 we have for both q = 1 and q = 2, d q (y 1 ) = 3 and there is no y 1 -topped path with a length greater than d q (y 1 ) + 1 = 4. Thus, Condition 5 is also not satisfied and this system is not discrete mode observable. In order to recover Condition 5, we must add at least a sensor. First, note that here v P of Proposition 1, can be only x 4 . S 0 1 [{x 4 }, Y 1,1 ] = {x 4 } and we have that y 1 belongs to a direct {x 4 } → y 1 ] path. Therefore, the first step is not necessary. We go directly to Step 2: here y i = y 1 and V 1,Obs = {x 2 , x 3 , x 6 , u 1 }. Vertices x 1 and x 4 are the only ones in te previous list from which there exists a path having a length less or equal to the ones from x 4 to y 1 . Thus, we must measure x 1 or x 4 . Assume that we measure x 1 . In this case, the system will not be discrete mode observable since d 1 (y 1 ) = 1 and as the path between x 4 and y 1 is also of length 1. Thus, we must measure also x 4 . In this case, d 1 (y 1 ) = 0, Condition 5 will be satisfied. Note that instead measuring x 4 , we can measure also x 5 even if this vertex is not related to y 1 , it allows also to obtain d 1 (y 1 ) = 1. In fact, our procedure leads always to a solution but is not exhaustive for this last procedure stage.

Note that, when we deal with a specific realization of a SSLS with numerically specified parameters, we can obviously apply the proposed method above in order to add sensors for recovering the discrete mode observability. After choosing the additional sensors, we must verify, as it is done for Example 1 that the obtained algebraic relations useful to establish the distinguishability presents no singularities. If it is the case, then we must eliminate the useless sensor(s) and add a new other one using the procedure above. We do that until we recover the discrete mode observability of the considered specific realization. If there exist a difference between the two modes in the system's dynamics, after a finite number of iterations the provided procedure will always allow to recover the discrete mode observability.

Conclusion

In this paper, we propose a graph-theoretic tool to analyse generic mode discrete mode observability for switching structured linear systems with unknown inputs. More precisely, we provide new necessary and sufficient conditions to ensure the generic discrete mode observability of switching structured linear systems. Next, when these observability conditions are not satisfied, we study the location of additional sensors in order to recover the generic discrete mode observability. Our approach uses classical programming techniques and is free from numerical difficulties. This makes our proposed method well-suited to treat large scale systems.

Figure 1 .

 1 Figure 1. Digraph associated to system of Example 1

Figure 3 .

 3 Figure 3. Digraph associated to the system of Example 2

Example 4

 4 Consider the SSLS represented by the digraph depicted in Figure 4.

Figure 4 .

 4 Figure 4. Digraph of the system of Example 4

Figure 5 .

 5 Figure 5. Digraph of the system of Example 5

Preprint submitted to Automatica December 4, 2011


Vincent Monchiet
email: vincent.monchiet@univ-mlv.fr

Djimedo Kondo
email: djimedo.kondo@upmc.fr

Semi-analytical investigation of stress interfacial effects in ductile media with nanosized spheroidal cavities

Keywords: Ductile Materials, Yield Criterion, Size effect, Spheroidal Voids, Interfacial Stress

In this paper we investigate the interfacial stress effects on the macroscopic yield strength of plastic porous media containing nanosized spheroidal cavities. The solid matrix is assumed to obey the von Mises criterion with associated flow rule. Analysis of a rigid-ideal plastic spheroidal unit cell, containing a confocal spheroidal cavity, and subjected to arbitrary mechanical loadings is made. Void size effects are captured by considering at the interface between the matrix and the cavity a surface stress model, which relates the jump of the traction vector to the interfacial residual stress and to interfacial plastic strain. The resulting macroscopic criterion for the nanoporous material exhibits unusual features such as (i) an increase of the yield stress when the void size is decreased, (ii) asymmetry between the yield stress in uniaxial tension and compression.

Introduction

The elastic properties of a solid are significantly affected by the presence of surfaces and interfaces (see [START_REF] Wong | Nanobeam mechanics: elasticity, strength, and toughening of nanorods and nanotubes[END_REF][START_REF] Cuenot | Surface tension effect on the mechanical properties of nanomaterials measured by atomic force microscopy[END_REF] for experimental evidence and [START_REF] Miller | Size-dependent elastic properties of nanosized structural elements[END_REF][START_REF] Diao | Atomistic simulation of the structure and elastic properties of gold nanowires[END_REF] for numerical results). Surface effects are attributed to the presence of few layers of atoms which experience a different local environment than atoms in the bulk and have a different equilibrium positions and energy. The classic threedimensional linear-elasticity theory generally neglects these effects since the considered objects and structural elements are, at less, microsized but never of the size of one or few nanometers. For such nanosized objects, the interfacial effects become predominant since the area of surface per unit of volume is very high. Surface effects in standard continuum theories are treated within the framework of the Gurtin and Murdoch stress interface model [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF] which assumes a jump of the traction vector while the displacement field is considered continuous across the surface. The jump condition may comply with a generalized Young-Laplace equation which extends to solid-solid interface the well known equation that describes capillarity effects in fluid mechanics. Thus, the discontinuity of the traction vector consists of two parts, the first one is attributed to the presence of interfacial residual stresses, independent of the deformation, and the second one being related to elastic deformation of the interface. Stress surface model have been used to model the inclusions size dependency of elastic properties of nanocomposites [START_REF] Duan | Karihaloo Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress[END_REF][START_REF] Sharma | Size-dependent Eshelby's tensor for embedded nanoinclusions incorporating surface/interface energies[END_REF][START_REF] Le Quang | Size-dependent effective thermoelastic properties of nanocomposites with spherically anisotropic phases[END_REF][START_REF] Brisard | Hashin-Shtrikman bounds on the bulk modulus of a nanocomposite with spherical inclusions and interface effects[END_REF].

However, surface effects on the yield strength and plastic behavior of nano objects and nanostructured materials have also been observed (see for instance [START_REF] Marszalek | Atomic force microscopy captures quantized plastic deformation in gold nanowires[END_REF][START_REF] Wu | Mechanical properties of ultrahighstrength gold nanowires[END_REF] for experimental evidence and [START_REF] Diao | Yield Strength Asymmetry in Metal Nanowires[END_REF][START_REF] Diao | Atomistic simulations of the yielding of gold nanowires[END_REF][START_REF] Gall | The Strength of Gold Nanowires[END_REF][START_REF] Yang | Atomistic simulation on size-dependent yield strength and defects evolution of metal nanowires[END_REF] for numerical simulations). Indeed, atomistic-based simulations performed by [START_REF] Gall | The Strength of Gold Nanowires[END_REF] have shown that the yield stress of gold nanowires strongly increases as the cross-section decreases. Note also that [START_REF] Diao | Yield Strength Asymmetry in Metal Nanowires[END_REF] reports that the magnitude of the yield stress is larger in tension than in compression for very small nanowires, the authors attributing this asymmetry mainly to surface effects. Few recent studies consider stress interfaces effects in the context of non-linear composites and nanoporous plastic materials [START_REF] Zhang | Effect of surface energy on the yield strength of nanoporous materials[END_REF][START_REF] Zhang | Effect of surface stress on the asymmetric yield strength of nanowires[END_REF][START_REF] Zhang | Effect of surface/interface stress on the plastic deformation of nanoporous materials and nanocomposites[END_REF][START_REF] Dormieux | An extension of Gurson model incorporating stresses effects[END_REF][START_REF] Goudarzi | Surface energy effects on the yield strength of nanoporous materials containing nanoscale cylindrical voids[END_REF]. In particular, by performing a limit analysis of a hollow sphere, [START_REF] Dormieux | An extension of Gurson model incorporating stresses effects[END_REF] generalize the Gurson model [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I. -Yield criterion and flow rules for porous ductile media[END_REF] in order to predict void size effects. To this end, they make use of a plastic version of the Gurtin stress interface model (see for instance [START_REF] Monchiet | Interfacial models in viscoplastic composites materials[END_REF]) which relates the interfacial stress to the plastic deformation at the cavity surface. The resulting model shows a void size dependency of the macroscopic yield strength of nanoporous media: for nanosized cavities, the strength domain appears to be significantly larger than that predicted by the Gurson model. Alternatively, non-local plasticity theories have been used in [START_REF] Wen | The modified Gurson model accounting for the void size effect[END_REF][START_REF] Huang | Coupling effects of void size and void shape on the growth of prolate ellipsoidal microvoid[END_REF][START_REF] Li | Combined effects of void shape and void size Ű oblate spheroidal microvoid embedded in infinite non-linear solid[END_REF][START_REF] Li | RVE-based studies on the coupled effects of void size and void shape on yield behavior and void growth at micron scales[END_REF] instead of the von Mises criterion for describing the plastic behavior of the solid matrix. Yet, a major advantage of the analysis based on stress interface over non-local plasticity theories, is hat it is possible to derive closed-form expressions of the overall plastic dissipation of porous solids containing nanosized cavities. Nevertheless, in [START_REF] Dormieux | An extension of Gurson model incorporating stresses effects[END_REF], the authors restrict their analysis to the case of spherical cavities. In this contribution, we provide an extension of the model presented in [START_REF] Dormieux | An extension of Gurson model incorporating stresses effects[END_REF] to the case of spheroidal cavities. By doing so, we also extend the work of Gologanu et al. [START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids Ű case of axisymmetric prolate ellipsoidal cavities[END_REF][START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids -case of axisymmetric oblate ellipsoidal cavities[END_REF] by incorporating both void shape and void size effects.

Definition of the unit cell

Let us consider a spheroidal cavity of semi-axes a 1 and b 1 embedded in a spheroid of semi-axes a 2 and b 2 . The axis of the spheroids are aligned with 0x 3 , where (0, x 1 , x 2 , x 3 ) is a cartesian coordinate system of orthonormal basis (e 1 , e 2 , e 3 ). The volume of the cavity is V 1 = 4πa 1 b 2 1 /3 while the total volume of the unit-cell is V 2 = 4πa 2 b 2 2 /3. The shape of the cavity is described by the aspect ratio a 1 /b 1 , with a 1 > b 1 corresponding to a prolate cavity while b 1 > a 1 to an oblate one. Let us denote by c the focal distance and by e 1 the void eccentricity, defined by:

c = a 2 1 -b 2 1 = a 2 2 -b 2 2 e 1 = c a 1 e 2 = c a 2 (prolate) c = b 2 1 -a 2 1 = b 2 2 -a 2 2 e 1 = c b 1 e 2 = c b 2 (oblate) (1)
We will use both cylindrical coordinates ρ, θ, z with (e ρ , e θ , e z ) the associated orthonormal basis and the classical spheroidal coordinates λ, ϕ, θ (associated orthogonal basis (e λ , e φ , e θ )) defined by:

             x 1 = b sin(ϕ) cos(θ)
x 2 = b sin(ϕ) sin(θ) ). The matrix of the spheroidal unit cell is made of a rigid-plastic material obeying to the von Mises yield criterion, σ eq ≤ σ 0 , (σ 0 being the yield stress in tension and σ eq the von Mises equivalent stress σ eq =3 2 σ : σ) and the associated flow rule:

x 3 = a cos(ϕ)                    e λ = 1 L λ a sin(ϕ) e ρ + b cos(ϕ)
σ = 2σ 0 3 d d eq , with d eq = 2
The surface between the void and the solid matrix, denoted Γ, is assumed to be described by a stress-type interface model, which was introduced by [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF] in the context of elasticity and extended by [START_REF] Dormieux | An extension of Gurson model incorporating stresses effects[END_REF][START_REF] Monchiet | Interfacial models in viscoplastic composites materials[END_REF] to plasticity. Stress interfaces ensure the continuity of the velocity field while the traction vector, t = σ.n, undergoes a jump which is governed by the Young-Laplace equation:

[t] Γ = -div s (τ ) (4)
In the above equation, div s (τ ) denotes the surface divergence of the tensor τ such that vector div s (τ ) = (τ ⊗ ∇) : P (n), with P (n) = I -n ⊗ n, I being the second order identity tensor and n the normal unit vector taken on the interface Γ and oriented from the void to the solid matrix. In ( 4), the notation [t] Γ represents the jump of t across Γ defined as follows: [t] Γ = t + -t -, where t + and t -are the values of the traction vector calculated on both sides of Γ.

When the interface Γ is the bounday between a void and a solid, t -= 0 and relation ( 4) reduces to t + = -div s (τ ).

In equation ( 4), the interfacial stress, τ , is given by:

τ = τ r P (n) + 2τ 0 3d s eq (d s + tr(d s )P (n)) (5)
where τ r is the interfacial residual stress, τ 0 is a material parameter and d s is the interfacial plastic strain rate which is defined as the restriction of the total strain rate to the tangent plane of normal unit vector n. In (5), d s eq is the surface equivalent strain rate. The expressions of d s and d s eq are:

d s = P (n).d.P (n) d s eq = 2 3 d s : d s + tr(d s ) 2 1/2 (6)
Note that for a spheroidal surface n = e λ , P (n) = e θ ⊗ e θ + e ϕ ⊗ e ϕ and the quantities d s and d s eq read:

d s = d θθ e θ ⊗ e θ + d ϕϕ e ϕ ⊗ e ϕ + d θϕ (e θ ⊗ e ϕ + e ϕ ⊗ e θ ) d s eq = 4 3 (d 2 θθ + d 2 ϕϕ + d 2 θϕ + d θθ d ϕϕ ) 1/2 (7)
3 Limit analysis taking into consideration interfacial stresses

As already mentioned, Hill-Mandel kinematic homogenization approach will be used to derive the overall plastic potential of the porous solid. Thus, uniform strain rate boundary conditions are considered on external surface of the hollow spheroid:

v(λ = λ 2 ) = D.x ( 8
)
where D is the macroscopic strain rate tensor. The limit stress at the macroscopic scale, Σ, is:

Σ = ∂Π ∂D (D) (9)
where Π(D), is the macroscopic dissipation given by:

Π(D) = inf v∈K 1 V 2 Ω-ω σ 0 d eq dV + Γ [t] Γ .vdS ( 10
)
where K is the space of admissible velocity fields, i.e. of continuous and differentiable velocity fields that comply with condition [START_REF] Gall | The Strength of Gold Nanowires[END_REF]. The first integral in [START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids -case of axisymmetric oblate ellipsoidal cavities[END_REF] is computed over the solid matrix, defined by Ω -ω, where Ω is the domain corresponding to the unit cell, while ω is the domain occupied by the void. The second integral in [START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids -case of axisymmetric oblate ellipsoidal cavities[END_REF] is performed over the surface Γ of the void. It is recalled that

V 2 = 4πa 2 b 2 Γ [t] Γ .vdS = - Γ div s (τ ).vdS (11)
Moreover, for any continuously differentiable second order tensor τ and vector v,

div s (τ .v) = div s (τ ).v + τ : d s (12)
The integral over Γ of the quantity div s (τ .v) being null over any closed surface, it follows that:

Γ div s (τ ).vdS = - Γ τ : d s dS (13)
Finally, replacing τ by its expression (5), one obtains:

Π(D) = inf 1 V 2 Ω-ω σ 0 d eq dV + 1 V 2 Γ τ 0 d s eq dS + 1 V 2 Γ τ r tr(d s )dS (14)
Note that this expression of Π(D) contains three terms: the first is classic in the context of Limit Analysis of ductile porous media; the last two terms are associated to interfacial plastic dissipations. Note that, in the spheroidal coordinate system, the surface integral is defined by:

Γ • dS = ϕ=π ϕ=0 θ=2π θ=0 • b 1 L λ 1 sin(ϕ)dϕdθ ( 15
)
where L λ 1 is the value of L λ for λ = λ 1 , i.e. L λ 1 = a 2 1 sin 2 (ϕ) + b 2 1 cos 2 (ϕ); for the definition of the volume integral in the expression of Π(D, the reader is referred to [START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids Ű case of axisymmetric prolate ellipsoidal cavities[END_REF][START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids -case of axisymmetric oblate ellipsoidal cavities[END_REF].

The trial velocity field

A crucial step in the derivation of an approximate closed form expression of the macroscopic potential of the ductile porous material lies in the choice of the trial velocity field. Generally, this field is composed in two parts: one involving a constant traceless tensor A, one, denoted v E , which is heterogeneous, i.e.

v = A.x + Bv E (16)
Due to additional difficulties related to the consideration of interfacial stress effects, we propose here to adopt, for v E , the velocity field considered by Gologanu et al. [START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids Ű case of axisymmetric prolate ellipsoidal cavities[END_REF][START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids -case of axisymmetric oblate ellipsoidal cavities[END_REF] and Monchiet et al. [START_REF] Monchiet | Approximate criteria for anisotropic metals containing non spherical voids[END_REF]. This field has the property to comply with uniform strain rate conditions on any iso-λ = cst spheroid. Its expression is:

                   v E λ = c 3 bL λ 1 + 1 2 (1 -3α)(1 -3 cos 2 (ϕ)) v E ϕ = 3c 3 4ab 2 L λ [(1 -α)b 2 -2αa 2 ] sin(2ϕ) v R θ = 0 ( 17
)
where α depends on λ or equivalently on e and is given by:

α =          ab 2 c 3 arctanh c a - b 2 c 2 (prolate void) - ab 2 c 3 arctan c a + b 2 c 2 (oblate void) (18)
For the explicit dependence of a and b on the coordinates λ and e, the reader is referred to section 2.

The verification of uniform strain rate boundary conditions at the outer spheroid λ = λ 2 leads to:

A = D -D m T ; B = a 2 b 2 2 c 3 D m ( 19
)
where the second order tensor T is given by:

T = 3 2 (1 -α)(e 1 ⊗ e 1 + e 2 ⊗ e 2 ) + 3αe 3 ⊗ e 3 (20)
From equation [START_REF] Li | RVE-based studies on the coupled effects of void size and void shape on yield behavior and void growth at micron scales[END_REF], it is readily seen that the strain rate field in the matrix is the sum of a homogenous deviatoric field A and a non-homogeneous field d E :

d = A + Bd E (21)
with d E given by:

                                 d E λλ = - 3c 3 ab 2 (1 -α) + 3c 3 a 2b 2 L 2 λ (1 -3α) sin 2 (ϕ) d E ϕϕ = 3c 3 2ab 2 (1 -α) - 3c 3 a 2b 2 L 2 λ (1 -3α) sin 2 (ϕ) d E θθ = 3c 3 2ab 2 (1 -α) d E λϕ = 3c 3 4bL 2 λ (1 -3α) sin(2ϕ) (22)
For application purpose, we consider axisymmetric mechanical loadings. Thus, the non-null components of the macroscopic strain rate tensor, D, are D 11 = D 22 and D 33 . Let us denote D q = 2(D 33 -D 11 )/3. It can then be shown that the equivalent strain rates d eq and d s eq , which appear in the expression of the macroscopic dissipation Π(D) (see ( 14)) read:

d eq = D 2 q + 2D q D m F 1 (λ, ϕ) + D 2 m F 2 (λ, ϕ) d s eq = D 2 q G 1 (ϕ) + 2D q D m G 2 (ϕ) + D 2 m G 3 (ϕ) (23)
In which, functions F i (λ, ϕ) and G i (λ, ϕ) being given by:

                                       F 1 (λ, ϕ) = -(z 2 + 3u(α -w 2 )), F 2 (λ, ϕ) = z 2 2 + 6z 2 u(α -w 2 ) + 3u 2 1 + 3α 2 + 2(1 -3α)w 2 , G 1 (ϕ) = 1 -3w 2 + 3w 4 , G 2 (ϕ) = 1 f (1 -3w 2 ) -(1 -3w 2 + 3w 4 )∆ , G 3 (ϕ) = 1 f 2 4 -2(1 -3w 2 )∆ + (1 -3w 2 + 3w 4 )∆ 2 (24)
with:

u = a 2 b 2 2 ab 2 , w = b cos(ϕ) L λ ∆ = z 1 -f z 2 , z 1 = 1 -3α 1 , z 2 = 1 -3α 2 (25)
The invariants of the macroscopic stresses at yielding are:

           Σ m = 1 3 ∂Π ∂D m Σ q = ∂Π ∂D q (26)
where Σ m = tr(Σ/3 = (2σ 11 + Σ 33 )/3 and Σ q = Σ 33 -Σ 11 (see equation [START_REF] Huang | Coupling effects of void size and void shape on the growth of prolate ellipsoidal microvoid[END_REF] for the expression of Π(D) and ( 23) to [START_REF] Wu | Mechanical properties of ultrahighstrength gold nanowires[END_REF] for d e q and d seq ). Thus, the macroscopic criterion is obtained by computing numerically the three integrals in equation ( 14).

Results

In this section, we propose to illustrate the salient features of the new model, namely the effects of the void size on the macroscopic yield strength of the ductile material containing spheroidal nanocavities, in this section we present projections of the developed analytic macroscopic criterion in the deviatoric plane. As mentioned above, axisymmetric loadings are considered; the stress invariants being Σ q = Σ 33 -Σ 11 and Σ m = (2Σ 11 + Σ 33 )/3. We first consider the limiting case of spherical nano-cavities. In figure 2, we compare the predictions obtained by numerically evaluating the integral [START_REF] Huang | Coupling effects of void size and void shape on the growth of prolate ellipsoidal microvoid[END_REF] with that obtained with the approximate criterion [START_REF] Dormieux | An extension of Gurson model incorporating stresses effects[END_REF] for various values of the non dimensional parameter k = τ 0 /(b 1 σ 0 ) and fixed porosity f = 0.1 (τ r , being considered null). Clearly, the results shows an important effect of the the cavity size on the macroscopic yield locus, the yield surface being larger when the parameter k = τ 0 /(aσ 0 ) increases, this corresponding to a decrease of the size of the void. Note the very good agreement between the approximate criterion of Dormieux-Kondo [START_REF] Dormieux | An extension of Gurson model incorporating stresses effects[END_REF] and the numerical results. One reason is that the trial velocity field used by [START_REF] Dormieux | An extension of Gurson model incorporating stresses effects[END_REF] is the one already used by Gurson [START_REF] Gurson | Continuum theory of ductile rupture by void nucleation and growth: Part I. -Yield criterion and flow rules for porous ductile media[END_REF] to which the trial velocity field (Eq. ( 16) with ( 17)) reduces in the case of spherical cavities. Thus, the numerical results validate the approximations used in [START_REF] Dormieux | An extension of Gurson model incorporating stresses effects[END_REF] by these authors. In the particular case k = 0, the criterion [START_REF] Dormieux | An extension of Gurson model incorporating stresses effects[END_REF] reduces to Gurson's one. This is physically sound, since for large values of the cavity radius the term related to the interfacial stress vanishes. Σ q /σ 0 Σ q /σ 0 Σ m /σ 0 Σ m /σ 0 Fig. 4. Yield loci for an oblate cavity with an aspect ratio a 1 /b 1 = 1/2 (at the left) a 1 /b 1 = 1/5 (at the right) and a porosity f = 0.1.

0 0.5 1 1.5 2 2.5 3 -1.5 -1 -0.
Σ q /σ 0 Σ q /σ 0 Σ m /σ 0 Σ m /σ 0
On figure 3 is represented the yield surface for the case of a prolate cavity. On the left, the macroscopic yield surface corresponds to cavities having an aspect ratio a 1 /b 1 = 2 while at the right, similar results are provided for the aspect ratio a 1 /b 1 = 5. The porosity f = 0.1 is considered and various values of the non dimensional parameter k = τ 0 /(b 1 σ 0 ) are taken (the interfacial residual stress τ r is still considered as zero). Figure 4 displays similar results for the case of an oblate cavity. At the right, the aspect ratio is a 1 /b 1 = 1/2 while at the left, the aspect ratio is a 1 /b 1 = 1/5. As for the case of a spherical void, the surface effect induce an increase of the resistance surface of the ductile porous medium with nano spheroidal cavities. It must be also noted that the size effect seems to be more significant in the case of an oblate cavity than for a prolate one (particularly for the aspect ratio a 1 /b 1 = 1/5). The reason is that, for the same volume, the area of the void surface is greater for an oblate cavity than for a prolate one. Σ q /σ 0 Σ q /σ 0 Σ m /σ 0 Σ m /σ 0 Fig. 5. Yield loci for a prolate cavity with an aspect ratio a 1 /b 1 = 5 (at the left) for an oblate cavity with a 1 /b 1 = 1/5 (at the right) and a porosity f = 0.1 and for two values of the parameter k r = τ r /(b 1 σ 0 ).

For completeness, we now propose to evaluate the effect of the interfacial residual stresses on the macroscopic yield surface of nanoporous materials. On figure 5, at the left, are represented the surfaces of resistance for a prolate cavity with the aspect ratio a 1 /b 1 = 1/2, a porosity f = 0.1 and two values for the non dimensional parameter k r = τ r /(b 1 σ 0 ). At the right, are represented similar results for the case of an oblate cavity having the aspect ratio a 1 /b 1 = 1/5. A significant influence of the residual stress on the yield strength of ductile porous media with nanosized spheroidal cavities is observed. More precisely, that residual stresses induce a translation of the center of the macroscopic yield surface along both axis related to the macroscopic stress invariants Σ m and Σ q . This imply an asymmetry of the macroscopic yield strength between tension and compression. Again, it is observed that the interfacial stress effects more pronounced for an oblate cavities than for a prolate one (and for the same reasons that one already mentioned above).

Conclusion

In this study the combined effects of void shape and size on yielding of the porous aggregate were investigated. To this end we have performed the limit analysis of a spheroidal unit cell made up of a von Mises solid matrix containing a confocal spheroidal cavity and a stress interface which induce a jump of the traction vector across the surface of the void. That surface effects are considered with the plastic version of the Gurtin model [START_REF] Monchiet | Interfacial models in viscoplastic composites materials[END_REF]. In the latter, the jump of the traction vector at the void-matrix interface results from two contributions: : surface residual stresses and plastic strain rates that occurs at the surface of the void. Numerical calculations, based on the trial velocity field introduced by Gologanu et al. [START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids Ű case of axisymmetric prolate ellipsoidal cavities[END_REF][START_REF] Gologanu | Approximate models for ductile metals containing non-spherical voids -case of axisymmetric oblate ellipsoidal cavities[END_REF], has been performed in order to derive an upper bound for the macroscopic yield criterion. Illustrations has been provided for various shape of the cavity and various values of the material parameters which enter into the stress interface model. That applications clearly show a significant effect of the stress interface on the macroscopic yield surface of plastic materials containing spheroidal nanocavities. First, the interfacial stress effect due to the surface plastic deformation is characterized, at the macroscopic level, by an increase of the yield strength. The interfacial residual stress induces a change in position of the macroscopic plastic surface which implies an asymmetry between tension and compression. For completeness, it must be mentioned that the surface effects on the macroscopic yield strength of nanoporous media are more significant for oblate cavities than for a prolate one.

e 3 e ϕ = 1 3 e θ = e θ ( 2 )

 132 L λ b cos(ϕ) e ρ -a sin(ϕ) e with L λ = a 2 sin 2 (ϕ) + b 2 cos 2 (ϕ), λ ∈ [0, +∞[, ϕ ∈ [0, π] and e ρ = cos(θ)e 1 + sin(θ)e 2 , θ ∈ [0, 2π]. In the above equations: a = c cosh(λ) and b = c sinh(λ) for a prolate void, while for the case of an oblate void a = c sinh(λ) and b = c cosh(λ). The iso-λ surfaces define confocal spheroids with foci c = |a 2 -b 2 | and eccentricity e = c/a, for a prolate void while e = c/b for an oblate one. The porosity f is defined by: f = (a 1 b 2 1 )/(a 2 b 2 2

Fig. 1 .

 1 Fig. 1. Schematic diagram showing the hollow spheroid with a stress interface.

5 Σ q /σ 0 Σ m /σ 0 Fig. 2 .

 5002 Fig. 2. Yield loci for spherical nanovoids. Comparison between the upper bound of [6] (dashed line) and the exact, numerical two-field criteria (full line with circles) for a porosity f = 0.1 and for various values of k = τ 0 /(b 1 σ 0 ).

Fig. 3 .

 3 Fig. 3. Yield loci for a prolate cavity with an aspect ratio a 1 /b 1 = 2 (at the left) a 1 /b 1 = 5 (at the right) and a porosity f = 0.1.

d : d, d = 1 2 (∇ ⊗ v + v ⊗ ∇)(3)where v is the velocity field, d the strain rate tensor and d eq the von Mises equivalent strain rate.

/3 is the volume of Ω. Using the generalized Young-Laplace equation (4), it is readily seen that the second integral is related to the interfacial residual stress and interfacial plastic strain. It then follows that: