F Morelli

X Bombois

H Hjalmarsson

L Bako

K Colin

Least costly identification experiment for the identification of one module in a dynamic network

In this paper we consider the design of the least costly experiment for the identification of one module in a given network of locally controlled systems. The identification experiment will be designed in such a way that we obtain a sufficiently accurate model of the to-be-identified module with the smallest identification cost i.e. with the least perturbation of the network.

Introduction

This paper contributes to the efforts of developing techniques for the identification of large-scale or interconnected systems when the topology of the network is known. In many papers, the problem is seen as a multivariable identification problem and structural properties of the system are then used to simplify this complex problem (see e.g. [START_REF] Haber | Moving horizon estimation for large-scale interconnected systems[END_REF]). The identifiability of the multivariable structure is studied in a prediction error context in [START_REF] Weerts | Identifiability in dynamic network identification[END_REF] while this multivariable structure is exploited in other papers to reduce the variance of a given module in the network (see [START_REF] Hägg | On identification of parallel cascade serial systems[END_REF][START_REF] Gunes | A variance reduction technique for identification in dynamic networks[END_REF][START_REF] Everitt | On the variance analysis of identified linear MIMO models[END_REF]). In other contributions, conditions are derived for the consistent estimation of a given module in a dynamic network (see e.g. [START_REF] Dankers | Identification of dynamic models in complex networks with prediction error methods -predictor input selection[END_REF][START_REF] Gevers | Identification in dynamic networks: identifiability and experiment design issues[END_REF][START_REF] Hendrickx | Identifiability of dynamical networks with partial node measurements[END_REF]).

While many different problems have thus been extensively studied in the dynamical network context, this is not the case for optimal experiment design (i.e. the problem of designing the excitation signal of an identification experiment to guarantee a certain model accuracy under some constraints on this excitation signal). In our previous contribution [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF], we made the first steps towards optimal experiment design in a dynamical network context. In [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF], we considered the case of a network made up of locally controlled systems, i.e. modules, whose interconnection is realized by exchanging their measured output between neighbouring modules (this type of networks is usual in the literature on multi-agent systems (see e.g. [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF][START_REF] Korniienko | Performance control for interconnection of identical systems: Application to pll network design[END_REF])). For this particular type of dynamical networks, we showed how to design the excitation signals that have to be added to each module in order to identify models of these different modules that are sufficiently accurate to enhance the network performance by a redesign of the local controllers. The accuracy of each model can be measured by the inverse of the covariance matrix of the identified parameter vector of each module. In [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF], we have derived an expression for the inverse of this covariance matrix as an affine function of the excitation signal spectra. It is important to note that the inverse of the covariance matrix of a given module l is obviously a function of the excitation signal applied to this particular module, but also, though in a lesser extent, a function of the excitation signals applied to all modules k 1 having a path to l. Consequently, the excitation signal applied to such a module k contributes to the accuracy of the model of l. In other words, the propagation of the excitation signals due to the interconnection is a positive feature when we want to obtain sufficiently accurate estimates of every module with the smallest excitation power (see also [START_REF] Wahlberg | Variance results for identification of cascade systems[END_REF][START_REF] Everitt | A geometric approach to variance analysis of cascaded systems[END_REF][START_REF] Everitt | Variance results for parallel cascade serial systems[END_REF]).

In some cases, we may only be interested in the accurate identification of one specific module l via the application of an excitation signal to l. Since the other modules have not to be identified, the propagation of this excitation signal to the other modules due to the interconnection is a negative feature that have to be limited as much as possible 1 . For this purpose, we extend the least costly identification framework (see [START_REF] Bombois | Least costly identification experiment for control[END_REF]) to this particular dynamic network identification problem. In particular, we design the spectrum of the excitation signal applied to l in such a way that the accuracy of the identified model (measured via the inverse of the covariance matrix) is larger than a given threshold while entailing the smallest perturbation on the network. The perturbation (i.e. the cost of the identification) will be measured by the sum of the effects of the excitation signal on the input and output of each system in the network.

With respect to the least costly framework introduced in [START_REF] Bombois | Least costly identification experiment for control[END_REF] for a single closed loop, the cost of the identification experiment in the network context thus not only contains the perturbation induced by the excitation signal in the closed loop where the system has to be identified, but also the perturbation induced in other loops by this excitation signal. This propagation of the effect of the excitation signal is due to the fact that the output signal of the to-be-identified loop (which is perturbed by the excitation signal) is transmitted to neighbouring modules. In this paper, in order to reduce this propagation, we propose an approach where the signal transmitted to the neighbouring modules is no longer the actual output signal, but a sanitized version of this output signal where the contribution of the excitation signal has been (partially) removed. Indeed, using an initial estimate of the to-be-identified system, we derive an estimate of this contribution and we subtract this estimate from the measured output signal before the transmission to the neighbouring modules.

This new configuration is inspired by the concept of stealth identification that we introduced in [START_REF] Potters | Optimal experiment design in closed loop with unknown, nonlinear and implicit controllers using stealth identification[END_REF] for a single closed loop and that we here extend to the network case. The use of the stealth identification in this paper is also a new application of this concept since, in [START_REF] Potters | Optimal experiment design in closed loop with unknown, nonlinear and implicit controllers using stealth identification[END_REF], it was introduced as a tool to enable classical optimal experiment design in a loop where the controller is not Linear Time Invariant (LTI). With respect to [START_REF] Potters | Optimal experiment design in closed loop with unknown, nonlinear and implicit controllers using stealth identification[END_REF] 2 , we also analyze which accuracy condition the initial estimate used to compute the sanitized version of the output signal must respect for the stealth configuration to be effective (i.e., to yield a smaller identification cost). Moreover, another contribution of the present paper is to robustify the stealth approach by considering the uncertainty of this initial estimate and its influence on the cost of the identification. For this purpose, as we will see in the sequel, we will need to consider an optimal experiment design where the cost constraint is robustified with respect to the initial uncertainty of the initial estimate. In [START_REF] Bombois | Robust optimal identification experiment design for multisine excitation[END_REF], we have recentlty proposed an approach to tackle such a robustified cost constraint. With respect to the earlier approaches for this problem [START_REF] Rojas | Robust optimal experiment design for system identification[END_REF][START_REF] Kumar | Robust plant friendly optimal input design[END_REF][START_REF] Forgione | Data-driven model improvement for model-based control[END_REF], the approach in [START_REF] Bombois | Robust optimal identification experiment design for multisine excitation[END_REF] does not entail any kind of approximation. However, the possible high dimension of the network can imply an excessive computational complexity for the approach in [START_REF] Bombois | Robust optimal identification experiment design for multisine excitation[END_REF]. To avoid this problem, we propose in this paper an alternative approach which is more appropriate to the network situation (and its possible high dimension) and which, while being (slightly) more conservative, has the same property as the one in [START_REF] Bombois | Robust optimal identification experiment design for multisine excitation[END_REF] i.e., it is not based on any approximation. This absence of approximation is also the main difference with the approach proposed in the conference paper [START_REF] Morelli | Optimal experiment design for the identification of one module in the interconnection of locally controlled systems[END_REF] on which the present paper is based.

Note finally that the framework considered here is much different than the frameworks of [START_REF] Vincent | Input design for structured nonlinear system identification[END_REF][START_REF] Hägg | On optimal input design for networked systems[END_REF], which are, to our knowledge, the only other papers treating the optimal experiment design problem in a network. In [START_REF] Vincent | Input design for structured nonlinear system identification[END_REF], the authors consider input design for nonparametric identification of static nonlinearities embedded in a network. The main purpose of [START_REF] Hägg | On optimal input design for networked systems[END_REF] lies in the use of measurable disturbances in optimal experiment design.

Notations:

The matrix I n denotes the identity matrix of dimension n. The matrix

   X1 0 0 0 . . . 0 0 0 XN   
will be denoted diag(X 1 , ..., X N ) when the elements X i (i = 1, ..., N ) are scalar quantities, while it will be denoted bdiag(X 1 , ..., X N ) when the elements X i (i = 1, ..., N ) are matrices. For a matrix A, A T denotes the transpose of A and A * its conjugate transpose. For a vector of transfer function R(z), R i (z) denotes the i th entry of R(z). Finally, ⊗ denotes the Kronecker product and the Redheffer star product [START_REF] Zhou | Essentials of Robust Control[END_REF].

Description of the network configuration

We consider a network made up of N mod single-input single-output (SISO) systems S i (i = 1, . . . , N mod ) operated in closed loop with a SISO decentralized controller K i (i = 1, . . . , N mod ):

S i : y i (t) = G i (z, θ 0,i )u i (t) + v i (t) (1)
u i (t) = K i (z)(y ref,i (t) -y i (t)) (2)
where the signal u i is the input applied to the system S i and y i is the measured output. This output is made up of a contribution of the input u i and of a disturbance term v i (t) = H i (z, θ 0,i )e i (t) that represents both process and measurement noises. The different systems S i (i = 1, . . . , N mod ) are thus described by two stable transfer functions G 0,i (z) = G i (z, θ 0,i ) and H 0,i (z) = H i (z, θ 0,i ), the later being also minimum-phase and monic. These transfer functions are parametrized by an unknown true parameter vector θ 0,i ∈ R n θ i in a known model structure. For each i (i = 1, . . . , N mod ), the signal e i (i = 1, . . . , N mod ) defining v i is a zero mean white noise signal of variance σ 2 e i and these white noise signals e i are for simplicity supposed to be mutually independent. In (2), y ref,i is a reference signal that will be computed based on the measured outputs of neighbouring modules (see later). We can rewrite the above equations as follows:

ȳ(t) = Ḡ(z, θ 0 )ū(t) + H(z, θ 0 )ē(t) (3)
ū(t) = K(z)(ȳ ref (t) -ȳ(t)) (4)
where ȳ = (y 1 , . . . , y N mod ) T and ū, ē, ȳref are defined in a similar way and where θ 0 = (θ T 0,1 , . . . , θ T 0,N mod ) T ∈ R n θ concatenates the true parameter vectors θ 0,i (i = 1, ..., N mod ). In these equations, we also use the notation Ḡ = diag(G 1 , . . . , G N mod ) ( H and K are defined in a similar way).

The closed-loop systems described in (3)-(4) will be interconnected via the following equation:

ȳref (t) = A ȳ(t) + B ref ext (t) (5)
where the matrix A and the vector B represent the flow of information in the network and ref ext is a (scalar) external reference signal that should be followed by all outputs y i and that is generally only available at one node of the network. This type of interconnections is typical in formation control or multi-agent systems (see e.g. [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF][START_REF] Korniienko | Performance control for interconnection of identical systems: Application to pll network design[END_REF]).

To illustrate [START_REF] Dankers | Identification of dynamic models in complex networks with prediction error methods -predictor input selection[END_REF], let us consider the network represented in Figure 1. In this network, we have N mod = 6 systems/modules, all of the form (1) and all operated as in (2) with a decentralized controller K i (see Figure 2). These local closed loops are represented by a circle/node in Figure 1. The objective of this network is that the outputs y i of all modules follow the external reference ref ext even though this reference is only available at Node 1. For this purpose, a number of nodes are allowed to exchange information (i.e. their measured output) with some other neighbouring nodes. The arrows between the nodes in Figure 1 indicate the flow of information.

K i (z) G 0;i (z) y i (t) v i (t) - + + y ref;i (t)
To the network From the network For example, Node 5 receives the output of two nodes (i.e. Nodes 3 and 4) and sends its output (i.e. y 5 ) to three nodes (Nodes 3, 4 and 6). The reference signal y ref,i of Node i will be computed as a linear combination of the received information at Node i. For Node 5, y ref,5 will thus be a linear combination of y 3 and y 4 . More precisely, for all outputs y i to be able to follow the external reference ref ext , A and B in (5) are generally chosen as [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF][START_REF] Korniienko | Performance control for interconnection of identical systems: Application to pll network design[END_REF]:

u i (t)
A =         0 0 0 0 0 0 1/3 0 1/3 1
/3 0 0 0 0.5 0 0 0.5 0 0 0.5 0 0 0.5 0 0 0 0.5 0.5 0 0 0 0 0 0 1 0

        B = (1, 0, ..., 0) T .
The matrix A is called the normalized adjacency matrix in the literature [START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF]. Using (5), we, e.g., see that the tracking error signals y [START_REF] Korniienko | Performance control for interconnection of identical systems: Application to pll network design[END_REF][START_REF] Fax | Information flow and cooperative control of vehicle formations[END_REF]. A normalized adjacency matrix can be defined for any information flow using the following rules. Row i of A is zero if no output is sent to Node i. If y i is sent to Node j, entry (j, i) of A will be nonzero. Finally, all nonzero entries in a row are equal and sum up to one. We also need to introduce the notion of (directed) path between two nodes. There exists a path from Node i to Node j if A ji = 0 or we can find a set of ζ intermediary nodes described by the indexes {n 1 , ..., n ζ } such that A n 1 i = 0, A n 2 n 1 = 0, ... , A jn ζ = 0. We moreover introduce the following definitions: Definition 1 Consider an arbitrary node of a network containing N mod nodes, say Node j (j = 1, ..., N mod ). For this node, we define the set P j as a set of indexes of nodes. A certain index i = j belongs to P j if there exists a path from Node j to Node i. Similarly, for the same Node j, we also define the set L j . A certain index i = j belongs to L j if there exists a path from Node i to Node j.

As an example, P 5 = {2, 3, 4, 6} and L 5 = {1, 2, 3, 4} for the network of Figure 1. For the sequel, it is important to note the following facts. If an external signal (e.g., the disturbance v i or an excitation signal r) is added to Node j, this external signal will also influence all nodes i with i ∈ P j . Conversely, Node j will be influenced by all external signals added in nodes i with i ∈ L j .

In the sequel, we will suppose that an identification procedure has delivered initial estimates θ init,i of θ 0,i (i = 1, ..., N mod ) and that all these estimates are normally distributed around θ 0,i with a covariance matrix P init,i . This can e.g. be done using an open-loop experiment on each system S i disconnected from the network or via the identification procedure presented in [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF] (the experiment is then done in the network configuration). We can then say that the concatenated vector θ init = (θ T init,1 , . . . , θ T init,N mod ) T is normally distributed around θ 0 = (θ T 0,1 , . . . , θ T 0,N mod ) T with a covariance matrix P init = bdiag(P init,1 , ..., P init,N mod ). Based on this statistical property, the following ellipsoid U init is a β%-confidence region for the unknown parameter vector θ 0 :

U init := θ ∈ R n θ | (θ -θ init ) T P -1 init (θ -θ init ) < χ β (6)
where P r(χ 2 (n θ ) < χ β ) = β and θ = (θ T 1 , ..., θ T N mod ) T . This ellipsoid U init can be considered as an uncertainty set for the unknown true parameter vector θ 0 . From now onwards, we will therefore assume that θ 0 ∈ U init .

In the sequel, we will suppose that the model accuracy obtained after this initial experiment is satisfactory for all, but one node, say Node l (l = 1, ..., N mod ). We will therefore have to perform a new identification experiment to obtain a better estimate of the parameter vector θ 0,l describing this node. This experiment is described in the next section.

3 Identification of one module in the network and cost of the experiment 3.1 Cost of an experiment in the stealth and non-stealth configurations

K l (z) G 0;l (z) y l (t) v l (t) r(t) + - + + y ref;l (t)
To the network From the network u l (t)

x(t) - + ỹl (t)
Figure 3: To-be-identified node (i.e. Node l) during the identification experiment. In the stealth setting, x(t) is given by [START_REF] Everitt | A geometric approach to variance analysis of cascaded systems[END_REF]. In the non-stealth setting, x(t) = 0.

During the identification experiment, we will apply for a duration N (i.e. from t = 1 till t = N ) an excitation signal r(t) of spectrum Φ r at the output of the controller K l of Node l (see Figure 3). This excitation will allow to obtain sufficiently informative data u l (t) and y l (t) (t = 1, . . . , N ) for an accurate identification of S l (see Section 3.2). In Figure 3, we also observe the signal x(t). In this paper, we will consider two choices for x(t) corresponding to two configurations/settings: the stealth and the non-stealth configurations. In the non-stealth setting, the signal x(t) will be chosen equal to zero. This choice corresponds to the classical setting for an identification experiment in a closed-loop/network context (see e.g. [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF]). In the stealth setting, x(t) will be chosen as the following estimate of the contribution of r in y l :

x(t) = G l (z, θ init,l ) 1 + K l (z)G l (z, θ init,l ) =T init,l(z) r(t) (t = 1, . . . , N ) (7)
where θ init,l is the subvector of θ init corrresponding to S l . Note that (7) can be easily computed since both r and T init,l (z) are known. As shown in Figure 3, the signal x(t) is subtracted from the measured output y l (t) to give ỹl (t) = y l (t) -x(t) which will be the signal that will be transmitted to compute ȳref . Consequently, during the identification experiment, the equations ( 4)-( 5) become:

ū(t) = ml r(t) + K(z) (ȳ ref (t) -ȳ(t)) (8) ȳref (t) = A (ȳ(t) -ml x(t)) + B ref ext (t) (9)
where mi (i = 1, . . . , N mod ) denotes a unit (column) vector of dimension N mod for which the i th entry is equal to 1 and the other entries are equal to zero. We will show, in the sequel, the advantage of the stealth setting in order to reduce the cost of the identification experiment. For this purpose, we need to define the latter. Let us first consider the stealth setting. In this case, the output vector ȳ and the input vector ū in the network configuration (3)-( 8)-( 9) can be rewritten as follows as a function of the external signals r, ref ext and ē:

ȳ(t) = R y (z, θ 0 ) r(t) + R y ext (z, θ 0 ) ref ext (t) + S y (z, θ 0 ) ē(t) (10) ū(t) = R u (z, θ 0 ) r(t) + R u ext (z, θ 0 ) ref ext (t) + S u (z, θ 0 ) ē(t) (11)
for some vectors of transfer functions R y ext , R u ext , some matrices of transfer functions S y , S u and

R u (z, θ 0 ) = ml S 0,l (z) + N (z, θ 0 ) (T 0,l (z) -T init,l (z)) (12) R y (z, θ 0 ) = Ḡ(z, θ 0 ) R u (z, θ 0 ) ( 13
)
where

S 0,l (z) = 1/(1 + K l (z)G l (z, θ 0,l )) and T 0,l (z) = G l (z, θ 0,l )/(1 + K l (z)G l (z, θ 0,l
)) are scalar transfer functions and N (z, θ 0 ) is a vector of transfer functions:

N (z, θ 0 ) = K(z) S(z, θ 0 ) I N mod -A S(z, θ 0 ) Ḡ(z, θ 0 ) K(z) -1 A ml S(z, θ 0 ) = I N mod + Ḡ(z, θ 0 ) K(z) -1 (14)
For the sequel, it is important to note that T init,l (z) (see [START_REF] Everitt | A geometric approach to variance analysis of cascaded systems[END_REF]) is the initial model of T 0,l (z) that corresponds to the parameter θ init,l . Consequently, in ( 12), T 0,l (z) -T init,l (z) is the modeling error of this initial model T init,l (z).

With respect to the normal operations ( 3)-( 4)-( 5), ȳ and ū are thus perturbed during the identification experiment by R y (z)r(t) and R u (z)r(t), respectively. Consequently, it makes sense to define the cost of the identification experiment as:

J(Φ r , θ 0 ) = 1 2π π -π R y (e jω , θ 0 ) * R y (e jω , θ 0 ) + η R u (e jω , θ 0 ) * R u (e jω , θ 0 ) Φ r (ω) dω J(Φ r , θ 0 ) = 1 2π π -π R * (e jω , θ 0 )R(e jω , θ 0 ) Φ r (ω) dω ( 15
)
where η is an user chosen weighting factor and where R(z, θ 0 ) is the following vector of transfer functions of dimension 2N mod :

R(z, θ 0 ) = R y (z, θ 0 ) √ η R u (z, θ 0 ) (16)
The cost J(Φ r , θ 0 ) can be rewritten as the sum of the individual costs J i (Φ r , θ 0 ) (i = 1, ..., N mod ) in each module: [START_REF] Ghosh | Optimal identification experiment design for lpv systems using the local approach[END_REF], we can also rewrite (18) as follows:

J(Φ r , θ 0 ) = N mod i=1 J i (Φ r , θ 0 ) with (17)
J i (Φ r , θ 0 ) = 1 2π π -π R * i (e jω , θ 0 )R i (e jω , θ 0 )Φ r (ω) dω (18) R i (z, θ) = R y i (z, θ) √ η R u i (z, θ) where R y i (resp. R u i ) is the i th entry of R y (resp. R u ). Due to
J i (Φ r , θ 0 ) = 1 2π π -π |G i (e jω , θ 0,i )| 2 + η |R u i (e jω , θ 0 )| 2 Φ r (ω) dω (19)
In the non-stealth setting, the cost J(Φ r , θ 0 ) of an identification experiment and the individual costs J i (Φ r , θ 0 ) can be defined in a very similar way. However, the expressions for R y and R u that are used in [START_REF] Haber | Moving horizon estimation for large-scale interconnected systems[END_REF] and ( 18) have a different expression:

R u,N S (z, θ 0 ) = ml S 0,l (z) + N (z, θ 0 ) T 0,l (z) (20) R y,N S (z, θ 0 ) = Ḡ(z, θ 0 ) R u,N S (z, θ 0 ) (21)
By comparing ( 20) and ( 12), we observe that the modeling error T 0,l (z) -T init,l (z) is replaced by T 0,l (z) in the expression of R u,N S . For both the stealth and non-stealth settings, using the definition of P l (see Definition 1), the individual costs J i (Φ r , θ 0 ) for i ∈ P l will be equal to zero. However, for i ∈ P l , these costs J i (Φ r , θ 0 ) will be nonzero. Consequently, the excitation signal r will not have an influence only on Node l (where it is applied and where it is necessary for the identification of S l ), but also on all nodes i with i ∈ P l . For the network in Figure 1, if the excitation signal r is applied in Node 5, besides J 5 (Φ r , θ 0 ) , the individual costs J i (Φ r , θ 0 ) for i = 2, 3, 4 and 6 will also be non-zero. This result is equivalent to the fact that N i (z, θ 0 ) (i.e. the i th entry of N (z, θ 0 )) is a nonzero transfer function for all i ∈ P l and is equal to zero for all i = l, i ∈ P l .

The role of the stealth compensation x(t) is to reduce as much as possible this propagation of the influence of the excitation r (applied in Node l) towards the nodes i ∈ P l . Before explaining this in more details, let us make the following assumptions on Node l and the considered network: Assumption 1 Consider the set P l (see Definition 1) corresponding to the to-be-identified Node l. We assume that P l is a non-empty set. Assumption 2 Consider the set P l (see Definition 1) corresponding to the to-be-identified Node l and the vector of transfer functions N (z, θ 0 ) (see [START_REF] Gunes | A variance reduction technique for identification in dynamic networks[END_REF]). We will assume that, for all i ∈ P l , the

i th entry N i (z, θ 0 ) of N (z, θ 0 ) is such that N i (e jω , θ 0 ) = 0 for (almost) all frequencies.
If P l would be empty, there is of course no need for the stealth setting since the signal r will not be propagated to other nodes. Assumption 2 will in fact always hold, except in pathological cases that we here want to formally exclude.

We can now explain the role of the stealth compensation in reducing the propagation of the influence of the excitation r towards the nodes i ∈ P l . Let us first consider the ideal case i.e. when T init,l = T 0,l . This choice does not change the situation in the non-stealth setting i.e. J i (Φ r , θ 0 ) remains nonzero for all i ∈ P l since R u,N S i and R y,N S i remains the same nonzero transfer functions for all these i (they are not function of T init,l ). However, in the stealth setting, for all i = l, the transfer functions R u i and R y i are identically zero when T init,l = T 0,l . Consequently, in this ideal case, the effect of the excitation r(t) will only be felt in the to-be-identified module i.e. J i (Φ r , θ 0 ) = 0 for all i = l.

In practice, T init,l will of course always be different from T 0,l , but, as shown in the following proposition, the stealth configuration will remain beneficial if T init,l satisfies a certain accuracy constraint that will hold in the vast majority of the cases.

Proposition 1 Consider that, following the procedure described in this section, an excitation signal r(t) of spectrum Φ r is applied to Node l of a network like the one described in Section 2 and satisfying Assumptions 1 and 2. Let us for this spectrum Φ r compute the individual costs J i (Φ r , θ 0 ) (i = 1, ..., N mod ) in the stealth setting and in the non-stealth setting using the respective expression for these costs in the two settings (see Section 3.1). Suppose finally that the initial model T init,l (z) of T 0,l (z) satisfies the following accuracy constraint at the frequencies ω where Φ r (ω) = 0:

|T 0,l (e jω ) -T init,l (e jω )| |T 0,l (e jω )| < 1. ( 22
)
Then, for all i ∈ P l (see Definition 1), the individual cost J i (Φ r , θ 0 ) in the stealth configuration is strictly smaller than the one in the non-stealth configuration. Recall also that, for the nodes i with i = l and i ∈ P l , J i (Φ r , θ 0 ) = 0 in both configurations.

Proof. See Appendix A.

The property ( 22) is equivalent to say that, at ω, T init,l is a better model of T 0,l than the zero transfer function (i.e. |T 0,l (e jω ) -T init,l (e jω )| < |T 0,l (e jω ) -0|). As already mentioned, this property will generally be met in practice. The advantage of the stealth configuration will be further discussed in the next section.

Identification of one given module

In Section 2, we assumed that we have obtained an initial estimate θ init = (θ T init,1 , . . . , θ T init,N mod ) T of the true parameter vector θ 0 = (θ T 0,1 , . . . , θ T 0,N mod ) T . As already mentioned at the end of Section 2, we will suppose that we want to increase the accuracy of the estimate3 θ init,l of the true parameter vector θ 0,l corresponding to Node l. The accuracy of θ init,l can be measured with P -1 init,l where P init,l is the covariance matrix of θ init,l . The accuracy of θ init,l can be improved by combining it with a new estimate of θ 0,l obtained using a data set {y l (t), u l (t)|t = 1, . . . , N } collected as shown in Figure 3 in the stealth or the non-stealth setting.

We consider for this purpose a full-order model structure

M = {G l (z, θ l ), H l (z, θ l ) | θ l ∈ R n l } for S l .
We suppose that M is globally identifiable at θ 0,l i.e., θ l = θ 0,l is the only parameter vector for which G l (z, θ l ) and H l (z, θ l ) corresponds to S l . We will also suppose that the excitation signal r(t) (see Figure 3) and the white noise vector ē (see [START_REF] Bombois | Robust optimal identification experiment design for multisine excitation[END_REF]) are uncorrelated and that ref ext (t) is a stationary signal uncorrelated with r(t) and ē(t). Then, using the data set {y l (t), u l (t)|t = 1, . . . , N }, an estimate θN,l of θ 0,l can be obtained via prediction error identification [START_REF] Ljung | System Identification-Theory for the User 2nd edition PTR[END_REF]:

θN,l = arg min θ l 1 N N t=1 2 (t, θ l ) with: ( 23
) (t, θ l ) = H -1 l (z, θ l ) (y l (t) -G l (z, θ l )u l (t))
Using a similar reasoning4 as in [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF], we will show that θN,l is a consistent estimate of θ 0,l i.e., θ 0,l is the unique minimum of the asymptotic criterion Ē 2 (t, θ l ) = lim N →∞ 1 N N t=1 E 2 (t, θ l ) (with E the expectation operator). Let us for this purpose make a similar assumption as Assumption 2 to exclude pathological cases for the matrix S u in [START_REF] Gevers | Informative data: how to get just sufficiently rich?[END_REF]. Assumption 3 Consider the set L l (see Definition 1) corresponding to the to-be-identified Node l and the matrix of transfer functions S u (z, θ 0 ) (see [START_REF] Gevers | Informative data: how to get just sufficiently rich?[END_REF]). We will assume that, for all i ∈ L l , the (l, i) entry S u li (z, θ 0 ) of S u (z, θ 0 ) is such that S u li (e jω , θ 0 ) = 0 for (almost) all frequencies ω.

Proposition 2 Consider the network setup described in Section 2 and the identification procedure described above yielding the estimate θN,l of θ 0,l using data collected as shown in Figure 3 (using the stealth or the non-stealth settings). Consider also the set L l corresponding to Node l (see Definition 1) and suppose that Assumption 3 holds. Then, the estimate θN,l is a consistent estimate of θ 0,l if the product K l (z)G l (z, θ 0,l ) contains (at least) one delay and if one of the following two conditions is satisfied

• the set L l is non-empty. In this case, θN,l is consistent even if the external excitation r(t) and the external reference signal ref ext are equal to zero.

• The order of excitation of r (i.e. the number of frequencies at which its spectrum Φ r (ω) is nonzero in ] -π π]) and the order of excitation of the external reference signal ref ext are sufficient for [START_REF] Ljung | System Identification-Theory for the User 2nd edition PTR[END_REF] to be consistent when the data set {y l (t), u l (t)|t = 1, . . . , N } is collected in the following simple closed-loop system:

y l (t) = G l (z, θ 0,l )u l (t) + H l (z, θ 0,l )e l (t) (24)
u l (t) = r(t) + K l (z)(ref ext (t) -y l (t)) (25)
Proof. See Appendix B.1.

Let us interpret this proposition for the example of the network of Figure 1. In this network, L l is non-empty for all nodes l = 1. For all these nodes, as shown in the proof of Proposition 2, the perturbations v i (t) = H i (z, θ i,0 )e i (t) for i ∈ L l will be sufficient to yield informative data {y l (t), u l (t)|t = 1, . . . , N } and thus a consistent θN,l (even if r(t) = ref ext (t) = 0). It is however to be noted that, unless we have the luxury of performing an arbitrary long experiment, a nonzero excitation r(t) will generally be required to obtain the desired accuracy for θN,l (see later). As far as Node 1 is concerned, we see that this node can indeed be described as in ( 24)- [START_REF] Potters | Experiment design for identification of structured linear systems[END_REF]. Consequently, the consistency of θN,l can be checked using the data informativity result for direct closed loop identification (see e.g. [START_REF] Gevers | Informative data: how to get just sufficiently rich?[END_REF]). In this paper, the consistency of θN,l can be checked by verifying a condition on the excitation order of the excitation signal, which uniquely depends on the order of the controller K l (z) and on the respective parametrization and the orders of G l (z, θ l ) and H l (z, θ l ). Note that, since ref ext is assumed quasi-stationary and independent of r, the orders of excitation of r and ref ext just add up.

We will from now on suppose that the estimate θN,l is consistent. As shown in Appendix B.2, the estimate θN,l is then also (asymptotically) normally distributed around θ 0,l with a covariance matrix P θ l that can be estimated from the data and whose inverse has the following expression:

P -1 θ l (Φ r , θ 0 ) = M ē(θ 0 )+ N 2πσ 2 l π -π F l (e jω , θ 0,l )F * l (e jω , θ 0,l ) |R u l (e jω , θ 0 )| 2 Φ r (ω) + |R u ext,l (e jω , θ 0 )| 2 Φ refext (ω) dω (26)
with R u l and R u ext,l the l th entry of R u and of R u ext , respectively, and with Φ refext the power spectrum

of ref ext , F l (z, θ l ) = H -1 l (z, θ l ) ∂G l (z,θ l )
∂θ l and M ē(θ 0 ) the contribution of ē to the accuracy of the estimate (see Appendix B.2 for more details). We observe that P -1 θ l (Φ r , θ 0 ) is an affine function of the power spectrum Φ r of the excitation signal r and of the power spectrum Φ refext of the external reference ref ext (and a more complex function of θ 0 ). Equation [START_REF] Potters | Optimal experiment design in closed loop with unknown, nonlinear and implicit controllers using stealth identification[END_REF] pertains to the stealth configuration. In the non-stealth configuration, we can use the same expression for P θ l (Φ r , θ 0 ), but we have to replace R u l by R u,N S l (see [START_REF] Kumar | Robust plant friendly optimal input design[END_REF]). We can combine the information on θ 0,l contained in the estimates θN,l and θ init,l using the following estimator θfinal,l = (P -1 θ l + P -1 init,l ) -1 P -1 θ l θN,l + P -1 init,l θ init,l whose covariance matrix is given by (P -1

θ l + P -1 init,l ) -1 [22, page 464].
The accuracy of the estimate θfinal,l can thus be measured with P -1 θ l (Φ r , θ 0 ) + P -1 init,l . In this paper, we will suppose that this accuracy will be deemed satisfactory if the following accuracy constraint P -1 θ l (Φ r , θ 0 )+P -1 init,l > R adm is satisfied. The matrix R adm is a given strictly positive-definite and symmetric matrix that reflects the desired accuracy [START_REF] Potters | Experiment design for identification of structured linear systems[END_REF][START_REF] Ghosh | Optimal identification experiment design for lpv systems using the local approach[END_REF].

Optimal experiment design problem

As mentioned in the introduction, we will design the spectrum Φ r of the excitation signal r of the identification experiment described in the previous section in such a way that the accuracy constraint P -1 θ l (Φ r , θ 0 ) + P -1 init,l > R adm is satisfied with the smallest cost J(Φ r , θ 0 ) (see [START_REF] Haber | Moving horizon estimation for large-scale interconnected systems[END_REF]). This optimization problem can thus be formulated as follows:

min Φr J(Φ r , θ 0 ) (27)
P -1 θ l (Φ r , θ 0 ) + P -1 init,l ≥ R adm (28)
This optimization problem can be considered both in the stealth and in the non-stealth setting by using the respective expressions for J(Φ r , θ 0 ) and for P -1 θ l (Φ r , θ 0 ) in both cases. Before discussing how this optimization problem can be solved in practice, let us formulate the following result that illustrates the advantage of the stealth configuration.

Proposition 3 Consider an identification experiment in Node l of a network satisfying Assumptions 1 and 2. Consider, for this identification experiment, the optimal experiment design problem ( 27)-( 28) in the stealth and in the non-stealth setting (i.e. using the respective expressions for J(Φ r , θ 0 ) and for P -1 θ l (Φ r , θ 0 ) in both cases) and let us denote by Φ S r,opt and Φ N S r,opt the optimal spectra obtained in these two settings. Then, we have that the optimal cost J(Φ S r,opt , θ 0 ) in the stealth setting is strictly smaller than the cost J(Φ N S r,opt , θ 0 ) in the non-stealth setting if the model T init,l of T 0,l used in the stealth compensation [START_REF] Everitt | A geometric approach to variance analysis of cascaded systems[END_REF] has the following property for all ω where Φ N S r,opt (ω) = 0:

|T 0,l (e jω ) -T init,l (e jω )| |T 0,l (e jω )| < |R u l (e jω , θ 0 )| |R u,N S l (e jω , θ 0 )| ( 29
)
where R u l (resp. R u,N S l ) is the l th entry of R u (resp. R u,N S ) defined in (12) (resp. (20)).

Proof. See Appendix C.

As shown in Proposition 3, we thus see that the stealth configuration, which can be very easily implemented in a multi-agent network, will be, in many cases, advantageous 5 to obtain the required accuracy for the model of S l with the smallest possible identification cost.

Let us now turn to the problem of solving the optimal experiment design problem ( 27)- [START_REF] Safonov | Propagation of conic model uncertainty in hierarchical systems[END_REF]. We observe that, like many optimal experiment design problems, this optimization problem is dependent on the unknown vector θ 0 = (θ T 0,1 , . . . , θ T 0,N mod ) T . Since θ 0 is unknown, the optimization problem ( 27)-( 28) cannot be tackled as such. A commonly used approach to circumvent this problem is to replace θ 0 by an initial estimate. If we use the initial estimate θ init for this purpose (see Section 2), this would yield an optimization problem consisting in minimizing J(Φ r , θ init ) under the constraint P -1 θ l (Φ r , θ init ) + P -1 init,l ≥ R adm . In the stealth configuration, this approach will have a major disadvantage since T init,l in ( 7) is also computed based on θ init . The latter has as consequence that J(Φ r , θ init ) = J l (Φ r , θ init ) (because J i (Φ r , θ init ) = 0 for all i = l). In other words, the propagation of the signal r(t) towards the nodes i with i ∈ P l will not be taken into account in the optimal experiment design problem if we replace θ 0 by θ init . We will therefore instead consider the following formulation where the cost constraint has been robustified using the initial uncertainty region U init (see ( 6)):

min Φr,γ γ (30)
J(Φ r , θ) ≤ γ ∀θ ∈ U init (31)
P -1 θ l (Φ r , θ init ) + P -1 init,l ≥ R adm (32)
This optimization problem can be considered both in the stealth and in the non-stealth settings by using the respective expressions for J(Φ r , θ) and for P -1 θ l (Φ r , θ init ) in both cases. As opposed to the case where θ 0 is replaced by θ init , the above formulation will also take into account the propagation of the excitation signal in the stealth setting. In this setting, we will also observe a robustification of the stealth compensation. Indeed, the robustified formulation will favour spectra Φ r yielding, for all θ ∈ U init , small perturbations R u i (z, θ)r(t) and R y i (z, θ)r(t) for i = l and for i ∈ P l (see ( 11)-( 10)). For nodes i ∈ P l , this e.g. means that the power of the following signal has to be made small for all θ ∈ U init :

R u i (z, θ)r(t) = Ni (z, θ) G l (z, θ) 1 + K l (z)G l (z, θ) -T init,l (z) r(t)
Consequently, the robustified optimal experiment design problem will generally and among other considerations favour spectra Φ r (ω) with more contributions in the frequency ranges where the stealth compensation will be more effective due to a small uncertainty of T init,l (z) (thereby robustifying the stealth configuration).

In both the stealth and non-stealth settings, the robustified optimal experiment design problem ( 30)- [START_REF] Weerts | Identifiability in dynamic network identification[END_REF], has also the following properties. If we denote by Φ r,opt and γ opt the solution of this optimization problem, we have that γ opt = sup θ∈U init J(Φ r,opt , θ). Since we assume that θ 0 ∈ U init , this robustified formulation ensures that the a-priori unknown cost J(Φ r,opt , θ 0 ) is guaranteed to be smaller than γ opt . This would have not been the case if, instead of the robustified constraint (31), we would have used the non-robustified constraint J(Φ r , θ init ) ≤ γ.

Since the optimization problem ( 30)-( 32) can be considered for the stealth and the non-stealth configurations, the solution of the optimization problem in both settings can be compared to verify whether the stealth configuration indeed yields a smaller cost. Note that this will generally be the case since, as explained above, the robust formulation will imply a robustification of the stealth compensation. However, in the case of ( 30)-(32), we do not have a condition such as [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF] to guarantee this property.

Tackling the robust cost constraint in a convex way

Since [START_REF] Weerts | Identifiability in dynamic network identification[END_REF] is affine in the decision variable Φ r , the optimization problem ( 30)-(32) will be convex if [START_REF] Wahlberg | Variance results for identification of cascade systems[END_REF] can be transformed into a constraint linear in the decision variables Φ r and γ. However, as very often in robustness analysis theory, we cannot find a tractable linear constraint that is equivalent to [START_REF] Wahlberg | Variance results for identification of cascade systems[END_REF], but we will show in this section that we can find one that implies [START_REF] Wahlberg | Variance results for identification of cascade systems[END_REF]. Consequently, if we solve the optimization problem with this alternative constraint, its solution γ opt will be an upper bound for sup θ∈U init J(Φ r,opt , θ). This entails a certain conservatism, but this γ opt remains an upper bound for the (unknown) cost J(Φ r,opt , θ 0 ) of an identification experiment with spectrum Φ r,opt .

In order to derive a tractable constraint implying [START_REF] Wahlberg | Variance results for identification of cascade systems[END_REF], we have to take into account that we are in the network context and thus that the vector θ can be of high dimension. Consequently, instead of working directly on the uncertain vector θ as in [START_REF] Bombois | Robust optimal identification experiment design for multisine excitation[END_REF], it is preferable to consider the so-called hierarchical approach. The hierarchical approach has indeed been introduced in [START_REF] Safonov | Propagation of conic model uncertainty in hierarchical systems[END_REF] to analyze the robustness of large-scale (interconnected) systems. Consequently, our objective here will be to determine a tractable constraint implying [START_REF] Wahlberg | Variance results for identification of cascade systems[END_REF] and that can be used in the context of the hierarchical approach.

To derive such a tractable linear constraint implying (31), we will assume that the to-bedesigned excitation signal r is a multisine i.e., r(t) = L m=1 A m sin(ω m t) where the frequencies ω m (m = 1, . . . , L) are fixed by the user (as e.g. a fine grid of the frequency range [0 π]) and where the amplitudes A m (m = 1, . . . , L) will be optimally determined. Such parametrization of the excitation signal is classically used in optimal experiment design and corresponds to the following spectrum:

Φ r (ω) = π L m=1 c m (δ(ω -ω m ) + δ(ω + ω m )) ≥ 0 ∀ω (33)
where c m = A 2 m 2 (m = 1, . . . , L) will be the decision variables of the optimization problem. The positivity of Φ r (ω) for all ω can be imposed by the constraints c m ≥ 0 (m = 1, . . . , L) on these decision variables.

Using ( 33) and ( 15), the robust cost constraint (31) can be rewritten as follows:

L m=1 c m R * (e jωm , θ)R(e jωm , θ) ≤ γ ∀θ ∈ U init (34)
In the next section, we will show that the hierarchical approach can be used to deduce, for each ω, an accurate upper bound α(ω) for

J wc (ω) = sup θ∈U init R * (e jω , θ)R(e jω , θ) (35)
i.e. J wc (ω) ≤ α(ω). This computable upper bound α(ω) for J wc (ω) is important since it is a necessary ingredient to derive a tractable linear constraint implying the robust cost constraint (34) as shown in the following proposition.

Proposition 4 Consider the robust cost constraint (34) corresponding to a spectrum of the type [START_REF] Zhou | Essentials of Robust Control[END_REF].

Then, the constraint (34) holds for a given γ if the following inequality linear in the decision variables c m (m = 1, . . . , L) holds:

L m=1 c m α(ω m ) ≤ γ ( 36
)
where α(ω) (m = 1, ..., L) is an upper bound for J wc (ω) (see(35))

Proof. For any frequency ω m , we have that:

R * (e jωm , θ)R(e jωm , θ) ≤ α(ω m ) ∀θ ∈ U init

The latter relation confirms that (34) holds if (36) holds.

Let us observe that (36) is an inequality constraint linear in the coefficients c m (m = 1, . . . , L). Moreover, since Φ r is affine in c m (m = 1, . . . , L), the accuracy constraint ( 32) is also linear in these coefficients. Consequently, the following Linear Matrix Inequality (LMI) optimization problem is a convex formulation for the original robust optimal experiment design problem (30)- [START_REF] Weerts | Identifiability in dynamic network identification[END_REF]. LMI formulation. Consider the parametrization [START_REF] Zhou | Essentials of Robust Control[END_REF] for the to-be-designed spectrum Φ r . For this parametrization, the LMI optimization problem has as decision variables a scalar γ > 0 and coefficients c m ≥ 0 (m = 1, . . . , L) and consists in determining the smallest value of γ for which both the LMI constraint (32) and the constraint (36) hold for some c m ≥ 0 (m = 1, . . . , L).

Let us denote by c m,opt (m = 1, . . . , L) and by γ opt the solution of this LMI optimization problem and let us also denote by Φ r,opt the spectrum corresponding to the coefficients c m,opt . Then, due to Proposition 4, we have that γ opt is an upper bound of sup θ∈U init J(Φ r,opt , θ) and thus of J(Φ r,opt , θ 0 ). Moreover, by construction, the spectrum Φ r,opt is also the one yielding the smallest value of (the upper bound of) sup θ∈U init J(Φ r,opt , θ) for which the accuracy constraint ( 32) is met.

Remark. The solution γ opt = L m=1 c m,opt α(ω m ) of the optimization problem gives an upper bound J ub for the total cost J(Φ r,opt , θ 0 ). We can also compute an upper bound J ub i for the individual costs J i (Φ r,opt , θ 0 ) defined in [START_REF] Hendrickx | Identifiability of dynamical networks with partial node measurements[END_REF]. This upper bound J ub i (i = 1, ..., N mod ) is given by:

J ub i = L m=1 c m,opt α i (ω m ) (37)
where, for any frequency ω, α i (ω) (i = 1, ..., N mod ) is an upper bound for

J wc,i (ω) = sup θ∈U init R * i (e jω , θ)R i (e jω , θ) (38)
Such an upper bound can also be computed using the tools that will be presented in the next section.

Remark. With respect to our earlier contribution [START_REF] Morelli | Optimal experiment design for the identification of one module in the interconnection of locally controlled systems[END_REF], the approach that we propose in this paper to robustify the cost constraint is not based on any approximation.

6 Computation of α(ω) using the hierarchical approach

LFT representation of R(z, θ)

To be able to use the LMI formulation of the optimal experiment design problem ( 30)- [START_REF] Weerts | Identifiability in dynamic network identification[END_REF] given in the previous section, it is necessary to determine a way to compute an (accurate) upper bound α(ω) for J wc (ω) at a given 6 frequency ω. As mentioned before, the hierarchical approach will be here used for this purpose. In order to use the hierarchical approach for the problem at stake (see e.g. [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF]), we first need to rewrite s(t) = R(z, θ)r(t) in a Linear Fractional Transformation (LFT) representation having a special form i.e., we have to determine vectors of signals p and q such that: p(t) = T (z, θ)q(t) and q(t)

s(t) = M (z) p(t) r(t) (39)
where M (z) is not a function of θ and where T (z, θ) is a diagonal matrix of dimension N mod × N mod for which the (i, i) entry is a closed-loop transfer function related to an isolated 7 loop made up of the controller K i and a model G i (z, θ i ) of G i (z, θ i,0 ). In our case, this upper part will be chosen as:

T (z, θ) = diag(T 1 (z, θ 1 ), . . . , T N mod (z, θ N mod )) ( 40
)
where

T i (z, θ i ) = G i (z, θ i )/(1 + K i (z)G i (z, θ i ))
. Note that, in the sequel, we will often use the shorthand notation R(z, θ) = F(M (z), T (z, θ)) for the LFT (39). We will determine this LFT both in the stealth setting and in the non-stealth setting. Let us start with the stealth setting and let us pose ē = 0 and ref ext = 0 in ( 3)-( 8)-( 9). If we express these equations for an arbitrary θ, we obtain

ȳ(t) = Ḡ(z, θ)ū(t) (41) ū(t) = ml r(t) + K(z) (ȳ ref (t) -ȳ(t)) (42) ȳref (t) = A   ȳ(t) -ml T init,l (z)r(t) =x(t)    (43)
Inserting ( 42) in (41) yields:

ȳ(t) = T (z, θ) K(z)ȳ ref (t) + ml r(t) (44)
with the definition of T (z, θ) given in (40). Let us now define p(t) := ȳ(t) and q(t) := K ȳref (t) + ml r(t) and let us notice that, using (43), q(t) is equal to:

q(t) = K(z)A    ȳ(t) p(t) -ml T init,l (z)r(t)    + ml r(t)
Using ( 10) and ( 11), let us also notice that s(t) = (s T y (t), sT u (t)) T with sy (t) = R y (z, θ)r(t) = ȳ(t) = p(t) and su (t) = √ ηR u (z, θ)r(t) = √ ηū(t) (when ē = 0 and ref ext = 0). Inserting (43) in (42) and using the fact that ȳ(t) = p(t), we obtain:

ū(t) = K(z) (A -I N mod ) p(t) + ml -K(z)A ml T init,l (z) r(t)
This therefore yields the following expression for M (z):

M (z) =   K(z)A ml -K(z)A ml T init,l (z) I N mod 0 √ η K(z) (A -I N mod ) √ η ml -K(z)A ml T init,l (z)   ( 45
)
7 By isolated, we means that the considered loop is not connected to other loops via a network connection.

Let us now consider the non-stealth configuration. In this configuration, R y and R u (see ( 12)-( 13)) are replaced by R y,N S and R u,N S (see ( 20)-( 21)) in the above reasoning. By doing this, R(z, θ) can also be written as an LFT F(M (z), T (z, θ)) with

M (z) =   K(z)A ml I N mod 0 √ η K(z) (A -I N mod ) √ η ml  

Hierarchical approach

Based on the LFT representation R(z, θ) = F(M (z), T (z, θ)) deduced in the previous subsection (for both the stealth and non-stealth settings), we have now all the elements to derive the hierarchical approach to determine an upper bound α(ω) for J wc (ω) (see ( 35)). This approach is an adaptation of the procedure proposed in Section 3 of our previous contribution [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF]. The main difference with [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF] is that R(z, θ) is here a vector of transfer function while, in [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF], R(z, θ) was a scalar transfer function.

Let us first notice that, at each frequency ω, the frequency response T (e jω , θ) of every transfer matrix T (z, θ) with θ ∈ U init lies in the following multiplicative uncertainty region [START_REF] Zhou | Essentials of Robust Control[END_REF]:

T (ω) = { T (e jω ) | T (e jω ) = (1 + ∆(e jω )) T (e jω , θ init ) with ∆(e jω ) ∈ ∆(ω)} (46) ∆(ω) = {∆(e jω ) = diag(∆ 1 (e jω ), ..., ∆ N mod (e jω )) | |∆ i (e jω ) -ci (ω)| < ρ i (ω) i = 1, ..., N mod } (47)
where, for each i = 1, ..., N mod , ci (ω) is a complex scalar and ρ i (ω) is a real scalar determined as follows: min

ρ i (ω), ci (ω) ρ i (ω) s.t. | Ti (e jω , θ i ) -ci (ω)| < ρ i (ω) ∀θ i ∈ U init,i (48)
with Ti (e jω , θ i ) =

T i (e jω ,θ i )-T i (e jω ,θ init,i ) T i (e jω ,θ init,i )

and

U init,i = {θ i | θ ∈ U init } = θ i |(θ i -θ init,i ) T P -1 init,i (θ -θ init,i ) ≤ χ β
By virtue of (48), T (ω) is in fact the smallest multiplicative uncertainty region that contains T (e jω , θ) for all θ ∈ U init . As shown in Appendix D, the quantities ci (ω) and ρ i (ω) can be exactly computed for each i and for each ω. Consequently, T (ω) can thus be entirely defined.

Using the LFT R(z, θ) = F(M (z), T (z, θ)), the quantity J wc (ω) at a given ω can be rewritten as:

J wc (ω) = sup θ∈U init F * (M (e jω ), T (e jω , θ)) F(M (e jω ), T (e jω , θ))) (49)
Using the multiplicative uncertainty region T (ω) defined above, we have the following property:

J wc (ω) ≤ sup T (e jω )∈T (ω)
F * (M (e jω ), T (e jω )) F(M (e jω ), T (e jω )))

= α(ω) (50)
Using simple transformation, the LFT F(M (e jω ), T (e jω )) in T (e jω ) can be transformed in an LFT in ∆(e jω ) i.e., F M (e jω ), T (e jω ) = F M ∆ (e jω ), ∆(e jω ) with

M ∆ (e jω ) = 0 T (e jω , θ init ) I N mod T (e jω , θ init ) M (e jω )
Using this new LFT description, α(ω) in (50) can be rewritten equivalently as:

α(ω) = sup ∆(e jω )∈∆(ω) F * M ∆ (e jω ), ∆(e jω ) F M ∆ (e jω ), ∆(e jω ) (51)
An accurate upper bound for this quantity α(ω) can be determined using the LMI optimization problem given in the following proposition. This computable upper bound for α(ω) is thus also an upper bound for J wc (ω) and will therefore be chosen as the quantity α(ω) that is used in the LMI formulation of Section 5.

Proposition 5 Consider a given frequency ω and the quantities ρ i (ω) and ci (ω) (i = 1, ..., N mod ) defining the set T (ω) (see (46)). Define R ω (resp. C ω ) as a diagonal matrix of dimension N mod whose elements are ρ 2 i (ω) (resp. ci (ω)) (i = 1..., N mod ). Then, an upper bound for α(ω) (see ( 51)) is given by γopt (ω) where γopt (ω) is the solution of the following LMI optimization problem. This LMI optimization problem has as decision variables a real scalar γ(ω) > 0 and a strictly positive definite diagonal matrix

T ω ∈ R N mod ×N mod . min γ(ω), Tω γ(ω) s.t. M ∆ (e jω ) I * N (γ(ω)) M ∆ (e jω ) I < 0 (52) with N (γ(ω)) ∆ =     T ω (R ω -C * ω C ω ) 0 0 I 2N mod T ω C * ω 0 0 0 T ω C ω 0 0 0 -T ω 0 0 -γ(ω)     (53)
Proof. See Appendix E.

Remark. From a computational point-of-view, the hierarchical approach is advantageous for large networks. Indeed, due to the embedding in a multiplicative uncertainty, the complexity of the multiplier T ω in the above proposition remains very limited (i.e., N mod elements in T ω ). Moreover, the quantities ρ i (ω) and ci (ω) are computed locally in each node. In other words, the complexity of the network does not play a role in the computation of ρ i (ω) and ci (ω).

Remark. The quantity α i (ω) (i = 1, ..., N mod ) that can be used to determine the upper bound J ub i on the individual cost J i (see (37)) can also be determined using the hierarchical approach presented above. For this purpose, we just have to replace, in the above reasoning, the LFT representation for R(z, θ) by an LFT representation for R i (z, θ) (this LFT representation can be easily deduced from the one of R(z, θ)).

7 Numerical illustrations

First numerical illustration

In this first numerical illustration, we consider the network of Figure 1 made up of N mod = 6 homogenous nodes. In other words, the true systems S i (i = 1, ..., 6) are all identical and given by the following ARX system [START_REF] Id Landau | A flexible transmission system as a benchmark for robust digital control[END_REF] with two resonance peaks:

y i (t) = (z -3 B 0 (z))/(A 0 (z))u i (t) + (1)/(A 0 (z))e i (t) with B 0 (z) = 0.10276 + 0.18123z -1 , A 0 (z) = 1 -1.99185z -1 + 2.20265z -2 - 1.84083z -3 + 0.89413z -4
. The variances of the white noises e i (t) are all equal to 0.5. We further suppose that these true systems are all controlled by the same local controller

K(z) = K B (z) K A (z) with K B (z) = 0.03742-0.06719z -1 +0.06995z -2 -0.03814z -3 -0.02546z -4 +0.06323z -5 -0.04707z -6 + 0.03222z -7 and K A (z) = 1 -3.348z -1 + 5.953z -2 -7.163z -3 + 6.143z -4 -3.705z -5 + 1.368z -6 - 0.2482z -7 .
For simplicity, the initial estimate θ init and its covariance matrix P init is determined from a single open-loop experiment on the ARX system (disconnected from the network) with a white input signal of variance 19 and of duration N = 1000. This open-loop experiment yields the following identified polynomials B id = 0.09535z -3 +0.1769z -4 and A id = 1-1.989z -1 +2.187z -2 -1.822z -3 +0.887z -4 corresponding to the identified parameter vector θ module,ol of dimension 6 with covariance matrix P module,ol . Based on this open-loop experiment, the initial estimates θ init,i of θ 0,i (i = 1, ..., 6)) can all be chosen equal to θ module,ol i.e. θ init = (θ T module,ol , θ T module,ol , ..., θ T module,ol ) T and P init is chosen equal to I N mod ⊗ P module,ol . Using these quantities, we can define the uncertainty region U init corresponding to a confidence level β of 95% (χ β = 50.9985).

Our objective will be to design the spectrum Φ r (ω) of the excitation signal r that has to be added to Node 5 (i.e. l = 5) during an identification experiment of duration N = 1000 to improve the accuracy of the model of S 5 in such a way that the following accuracy constraint is satisfied P -1 θ 5 (Φ r , θ 0 ) + P -1 init,5 > R adm where R adm is chosen as the inverse of the diagonal matrix diag((0.019) 2 , (0.022) 2 , (0.018) 2 , (0.009) 2 , (0.005) 2 , (0.009) 2 ). We will furthermore suppose that ref ext (t) = 0 during the identification experiment.

To determine the spectrum Φ r (ω) satisfying this accuracy constraint with the smallest identification cost, we consider the optimization problem ( 30)- [START_REF] Weerts | Identifiability in dynamic network identification[END_REF]. We will consider this optimization problem both in the stealth setting and in the non-stealth setting. For both settings, we define the cost of the identification experiment as in [START_REF] Haber | Moving horizon estimation for large-scale interconnected systems[END_REF] with η = 1 and we parametrize Φ r (ω) as in [START_REF] Zhou | Essentials of Robust Control[END_REF] with L = 20 frequencies distributed in the frequency range 10 -2 , π . In the stealth setting, the stealth compensation is implemented using [START_REF] Everitt | A geometric approach to variance analysis of cascaded systems[END_REF] with the transfer function T init,5 that can be constructed based on θ init,5 = θ module,ol .

To solve the optimization problem ( 30)-( 32) in both settings, we compute, for each setting, the corresponding quantity α(ω) for each frequencies ω m (m = 1, ..., L) present in Φ r (ω) (see Section 6). The optimal spectrum Φ r,opt and the optimal value γ opt can then be deduced8 using the LMI optimization problem given at the end of Section 5.

Let us first consider the stealth setting. In this case, the optimal spectrum Φ r,opt corresponds to a multisine for which all the amplitudes are negligible except at two frequencies ω = 1.35 and ω = 1.65 and γ opt is equal to 34.9466. This value is an upper bound J ub for sup θ∈U init J(Φ r,opt , θ) as discussed in Section 4 and thus also an upper bound for the a-priori unknown cost J(Φ r,opt , θ 0 ). As mentioned in the beginning of Section 3, our LMI approach entails a certain conservatism. Let us thus check the extent of this conservatism in this example. For this purpose, we compute a lower bound J lb for sup θ∈U init J(Φ r,opt , θ) by considering the maximal value of J(Φ r,opt , θ i ) over a set of 1000 grid points θ i ∈ U init . This procedure yields J lb = 33.92. Consequently, the conservatism of our procedure remains limited in this example since there is only 3% of difference between J lb and J ub . In this simulation example, we can also compute J(Φ r,opt , θ 0 ) i.e. J(Φ r,opt , θ 0 ) = 31.39. Consequently, γ opt is indeed an upper bound for J(Φ r,opt , θ 0 ). Note also that, as opposed to this, J(Φ r,opt , θ init ) = 30.79 would underestimate the actual cost of the experiment.

In order to have a better idea of how this cost is distributed in each of the six modules, let us consider the upper bound J ub i (see (37)) of the individual costs J i (Φ r,opt , θ 0 ) for i = 5 and for all i ∈ P 5 = {2, 3, 4, 6}. These upper bounds are given in Table 1. We observe that J ub 5 is almost equal to the total cost (for i = 5, J ub i ≤ 3.5 10 -4 ). Let us now compare these results with the ones obtained in the non-stealth setting. The optimal multisine in this setting is very similar to the one in the stealth setting and γ opt = 34.9557. We observe that this value is (slightly) larger than in the stealth case where γ opt = 34.9466. Moreover, as shown in Table 1, the costs J ub i for i ∈ P 5 are all at least 16 times larger than when the stealth configuration is implemented. This shows the advantage of the stealth configuration. The maximum of these individual costs for i = 5 (i.e. J ub 6 = 6 10 -3 ) remains though small with respect to J ub 5 = 34.9470. This is due to the large attenuation of this particular network. In other networks, this attenuation could be much smaller and would even more justify the use of the stealth configuration. This will be illustrated in the second numerical illustration.

Table 1: Upper bounds J ub i for the individual costs J i (i = 2, ..., 6) obtained using the optimal spectrum in the stealth and non-stealth settings (first illustration) In this second numerical illustration, we consider the network of Figure 4 made up of N mod = 2 nodes in cascade for which we wish to identify the first node. In this network, the matrices A and B in the interconnection (9) are given by

J ub 2 J ub 3 J ub
A = 0 0 1 0 B = 1 0
The true system S 1 is equal to the ARX system used in the previous section while the true system S 2 is given by the following first-order ARX system: y 2 (t) = (3.6 z -1 )/(1 -0.7 z -1 )u 2 (t) + 1/(1 -0.7 z -1 )e 2 (t) (i.e. θ 0,2 = (-0.7 3.6) T ). The controller K 1 is also the same as the one in the previous subsection while K 2 (z) = 10. The variance of the white noises e 1 (t) and e 2 (t) are here also both equal to 0.5. Note that the controller K 2 ensures a large closed-loop bandwidth. Consequently, in the non-stealth configuration, any perturbation due to r(t) in y ref,2 (t) = y 1 (t) will have a strong influence on u 2 (t) and y 2 (t). A stealth configuration may thus have a large impact in this example. For simplicity, the initial estimate θ init,1 and its covariance matrix P init,1 will be chosen as in the previous section (i.e. the estimate obtained with an open-loop experiment with input variance of [START_REF] Korniienko | Performance control for interconnection of identical systems: Application to pll network design[END_REF]. The initial estimate θ init,2 and its covariance matrix P init,2 are determined in a similar way (open-loop experiment with input variance of [START_REF] Korniienko | Performance control for interconnection of identical systems: Application to pll network design[END_REF]. This yields θ init,2 = (-0.69 3.57) T and a covariance matrix P init,2 . Based on this, the initial estimate θ init of θ 0 is chosen as (θ T init,1 , θ T init,2 ) T and P init is chosen equal to bdiag(P init,1 , P init,2 ). Using these quantities, we can define the uncertainty region U init corresponding to a confidence level β of 95% (χ β = 15.5).

Our objective will be to design the spectrum Φ r (ω) of the excitation signal r that has to be added to Node 1 (l = 1) during an identification experiment of duration N = 1000 to improve the accuracy of the model of S 1 in such a way that the following accuracy constraint is satisfied P -1 θ 1 (Φ r , θ 0 ) + P -1 init,1 > R adm where R adm is chosen as in the previous subsection. We will here also suppose that ref ext (t) = 0 during the identification experiment.

To determine the spectrum Φ r (ω) satisfying this accuracy constraint with the smallest identification cost, we consider the optimization problem ( 30)- [START_REF] Weerts | Identifiability in dynamic network identification[END_REF]. We will consider this optimization problem here also both in the stealth setting and in the non-stealth setting. For both settings, we here also define the cost of the identification experiment as in [START_REF] Haber | Moving horizon estimation for large-scale interconnected systems[END_REF] with η = 1 and we parametrize Φ r (ω) as in [START_REF] Zhou | Essentials of Robust Control[END_REF] with the same L = 20 frequencies as in the previous subsection. In the stealth setting, the stealth compensation is implemented using [START_REF] Everitt | A geometric approach to variance analysis of cascaded systems[END_REF] with the transfer function T init,1 that can be constructed based on θ init,1 (T init,1 is here equal to the transfer function T init,5 in the previous subsection)

Let us compare the value γ opt obtained when solving ( 30)- [START_REF] Weerts | Identifiability in dynamic network identification[END_REF] in the stealth and in the nonstealth settings. Like in the previous subsection, the optimal cost γ opt in the stealth setting is smaller than in the non-stealth setting: the obtained γ opt are respectively 33.87 and 42.38. Note that the advantage of the stealth configuration is here very clear since the stealth configuration allows to reduce the predicted cost by 25 %.

In order to have a better idea of how this cost is distributed in each of the two modules in both settings, we can consider Table 2 giving the upper bound J ub i of the individual costs J i (Φ r,opt , θ 0 ) (i = 1, 2). By comparing J ub 2 = 0.17 (in the stealth setting) and J ub 2 = 8.61 (in the non-stealth setting), we observe that the stealth configuration plays its role entirely by reducing, by a factor of 50, the propagation of the excitation signal to Node 2. This strong reduction is also observed for the actual individual cost J 2 (Φ r,opt , θ 0 ) which is equal to 0.03 in the stealth setting and to 7.04 in the non-stealth setting.

Table 2: Upper bounds J ub i for the individual costs J i (i = 1, 2) obtained using the optimal spectrum in the stealth and non-stealth settings in the second numerical illustration

J ub 1 J ub 2 STEALTH
33.70 0.17 NON-STEALTH 33.77 8.61

Case of a larger U init

To show the effectiveness of the stealth configuration even in the case of a larger uncertainty U init (and thus a worse model T init,l ), we repeat the above comparisons when θ init is identified with an input signal with a variance 10 times smaller than previously. In the first numerical illustration, this larger U init yields a value of γ opt = 131.66 in the stealth setting and a value of γ opt = 131.71 in the non-stealth one. In the second numerical illustration, we obtain γ opt = 119 in the stealth setting and γ opt = 157.84 in the non-stealth setting. Consequently, with this larger U init and this worse model T init,l , the stealth configuration also yields a smaller identification cost than the non-stealth configuration. Even though the obtained reduction is smaller due to the worse model T init,l , the effectiveness of the stealth setting in reducing the propagation of the influence of the excitation signal remains confirmed when looking at Tables 3 and4 giving, for this larger U init and this worse T init,l , the upper bounds J ub i of the individual costs J i (Φ r,opt , θ 0 ) for the first and second numerical illustrations, respectively. In Table 3, we observe that J ub i for i ∈ P 5 remain 4.5 times smaller (instead of 16 times smaller in Table 1) when the stealth configuration is implemented. In Table 4, J ub 2 remains 7 times smaller (instead of 50 times smaller in Table 2) when the stealth configuration is implemented. Table 3: Upper bounds J ub i for the individual costs J i (i = 2, ..., 6) obtained using the optimal spectrum in the stealth and non-stealth settings for a larger U init (first numerical illustration) i for the individual costs J i (i = 1, 2) obtained using the optimal spectrum in the stealth and non-stealth settings for a larger U init (second numerical illustration)

J ub 2 J ub 3 J ub 4 J ub 5 J ub 6 STEALTH 3
J ub 1 J ub 2 STEALTH
113.57 5.43 NON-STEALTH 118.12 39.72 It is to be noted that, for both numerical illustrations, the condition ( 29) is verified at all frequencies even with the worse model T init,l considered in this section. Consequently, by virtue of Proposition 3, the result presented above could have been expected. However, note that this condition (29) cannot be verified in practice since it requires the knowledge of the unknown true system and that Proposition 3 only pertains to the ideal optimal experiment design problem ( 27)- [START_REF] Safonov | Propagation of conic model uncertainty in hierarchical systems[END_REF] and not to (30)- [START_REF] Weerts | Identifiability in dynamic network identification[END_REF].

Conclusions

This paper extends the least costly identification experiment design framework to the case of the identification of one module in a network of locally controlled systems. The cost of the identification experiment (that is minimized under a certain accuracy constraint) is here defined as a function of the perturbations induced by the excitation signal on the input and output signals of each module. The propagation of the influence of the excitation signal is further reduced by an extension of the stealth identification paradigm. Note that the results of this paper can easily be extended to network situation where the cost of the experiment is not related to the perturbation induced by the excitation signal on the output of each module, but is more related to the perturbation induced by the excitation signal on the differences between the outputs of neighbouring modules. This situation can occur when the different agents in the network must remain as much as possible in a given formation configuration [START_REF] Seiler | Disturbance propagation in vehicle strings[END_REF].

where ∆G l (z, θ l ) = G l (z, θ 0,l ) -G l (z, θ l ) and ∆H l (z, θ l ) = H l (z, θ 0,l ) -H l (z, θ l ). Using now the fact that there is at least a delay in the product K l G l (and thus in the product ∆G l (z, θ l )S u ll ), that H l is monic and that r, ref ext and the white noises e j (j = 1, ..., N mod ) are mutually independent, the power Ē 2 (t, θ l ) of (t, θ l ) is given by: Ē 2 (t, θ l ) = σ 2 e l + Ēs

with s r = ∆G l (z,θ l ) H l (z,θ l ) R u l r, s refext = ∆G l (z,θ l ) H l (z,θ l ) R u ext,l ref ext , s e l = ∆H l (z,θ l ) H l (z,θ l ) + ∆G l (z,θ l )
H l (z,θ l ) S u ll e l and, for i ∈ L l , s e i = ∆G l (z,θ l ) H l (z,θ l ) S u li e i . Since Ē 2 (t, θ 0,l ) = σ 2 e l , we thus need to have Ē 2 (t, θ * l ) = σ 2 e l for all minimizers θ * l of the asymptotic criterion (56). Since we assume that the model structure M is globally identifiable at θ 0,l , the consistency will thus be proven if we show that Ēs 2 r (t, θ l ) = Ēs 2 refext (t, θ l ) = Ēs 2 e j (t, θ l ) = 0 (j = l and j ∈ L l ) implies ∆G l (z, θ l ) = ∆H l (z, θ l ) = 0.

Let us first consider the case where L l is non-empty i.e. there is at least one path from a node i = l to Node l. Due to the existence of this path, S u li is a nonzero transfer function (see Assumption 3). Consequently, Ēs 2 e i (t, θ l ) = 0 implies ∆G l (z, θ l ) = 0. Since ∆G l (z, θ l ) = 0, we have that Ēs 2 e l (t, θ l ) = 0 implies that ∆H l (z, θ l ) = 0 and we have therefore proven the consistency of θN,l for this case (independently of the value of r and ref ext ).

Let us now consider the case where L l is empty (it is e.g. the case of Node 1 in the network of Figure 1). In that particular case, y ref,l must9 be equal to ref ext . Consequently, the data {y l (t), u l (t) | t = 1, ..., N } are generated by the closed-loop system described by ( 24)-( 25) and the consistency property have to be evaluated on this loop. As shown in e.g. [START_REF] Gevers | Informative data: how to get just sufficiently rich?[END_REF], when {y l (t), u l (t) | t = 1, ..., N } are generated by the closed-loop system described by ( 24)-( 25), the orders of excitation of r and/or ref ext will need to be sufficient to yield a consistent θN,l .

Finally, let us note that the above reasoning remains exactly the same for the non-stealth setting i.e. when R u l is replaced by R u,N S l in the expression of s r (t, θ) in (59).

B.2 Accuracy

Since (t, θ 0,l ) = e l (t) and since θN,l is a consistent estimate of θ 0,l , θN,l is asymptotically normally distributed around θ 0,l [START_REF] Ljung | System Identification: Theory for the User, 2nd Edition[END_REF] and the inverse of its covariance matrix is given by:

P -1 θ l = N σ 2 e l
Ēψ l (t, θ 0,l )ψ T l (t, θ 0,l ) with ψ(t, θ l ) = -∂ (t,θ l ) ∂θ l [START_REF] Ljung | System Identification: Theory for the User, 2nd Edition[END_REF]. Let us first show that the above expression is equivalent to [START_REF] Potters | Optimal experiment design in closed loop with unknown, nonlinear and implicit controllers using stealth identification[END_REF] in the stealth setting. It is easy to show (see e.g. [START_REF] Bombois | Least costly identification experiment for control[END_REF]) that ψ(t, θ 0,l ) = F l (z, θ 0,l )u l (t) + L l (z, θ 0,l )e l (t) with F l (θ l ) as defined below [START_REF] Potters | Optimal experiment design in closed loop with unknown, nonlinear and implicit controllers using stealth identification[END_REF] and with L l (θ l ) = H -1 l (θ l ) ∂H l (θ l ) ∂θ l . Using [START_REF] Gevers | Informative data: how to get just sufficiently rich?[END_REF] and recalling that r and ref ext and the white noises in ē are all mutually independent, we obtain the expression (26) with [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF]: )Z * l (e jω )dω with σ 2 e i (i = 1, ..., N mod ) the variance of e i and with Z l (z) a matrix of transfer functions of dimension n θ l × N mod whose l th column is L l + F l S u ll and whose k th -column (k = l) is equal to F l S u lk (S u lk is the entry l × k of S u in ( 11)).

M ē = N 2πσ 2
D Computation of the quantities ci (ω) and ρ i (ω)

This appendix shows how the quantities ci (ω) and ρ i (ω) (see (48)) can be computed for a given i and a given ω. For this purpose, let us first give an expression of Ti (e jω , θ i ) as a function of θ i using the following notation for G i (e jω , θ i ) = Z 1,i (e jω )θ i 1+Z 2,i (e jω )θ i . In the last expression, Z 1,i (z) and Z 2,i (z) are row vectors containing only delays or zeros (see [START_REF] Bombois | Robustness analysis tools for an uncertainty set obtained by prediction error identification[END_REF]). This yields Ti (e jω , θ i ) = -1 + Z N,i (e jω )θ i 1 + Z D,i (e jω )θ i (61)

with Z D,i = Z 2,i + K i Z 1,i and Z N,i = Z 1,i

T i (e jω ,θ init,i ) -Z D,i . Note that, compared to Section 3 in [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF], Z N,i and Z D,i have here different expression since T i is defined differently. Based on (61), the quantities ci (ω) and ρ i (ω) can be exactly computed using the LMI optimization problem given in the following proposition whose proof can be found in our previous contribution [START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF].

Proposition 6 Consider the notation Ti (e jω , θ i ) = -1+Z N,i (e jω )θ i 1+Z D,i (e jω )θ i given in (61). The optimization problem (48) at a given ω and at a given i is equivalent to the following LMI optimization problem having as decision variables a positive real scalar µ i (ω), a complex scalar ci (ω), a positive real scalar ξ i (ω) and a skew-symmetric matrix X i (ω) ∈ R (n θ i +1)×(n θ i +1) (n θ i is the dimension of θ i ):

min µ i (ω) subject to -µ i (ω) λ i (ω) λ * i (ω) -A i (ω) -ξ i (ω)B i + jX i (ω)
< 0 (62)

with j = √ -1, λ i (ω) = Z N,i -Z D,i ci -1 -ci and A i (ω) = Z * D,i Z D,i Z * D,i Z D,i 1 B i = P -1
init,i -P -1 init,i θ init,i -θ T init,i P -1 init,i θ T init,i P -1 init,i θ init,i -χ β

The above optimization problem is not explicitly function of ρ i (ω). However, the optimal ρ i (ω) can be obtained by taking the square root of the optimal µ i (ω).

E Proof of Proposition 5

The result of Proposition 5 will be proven if we can show that the LMI (52) implies:

F * M ∆ (e jω ), ∆(e jω ) F M ∆ (e jω ), ∆(e jω ) < γ(ω) ∀∆(e jω ) ∈ ∆(ω)

To show this, let us expand the LFT F M ∆ (e jω ), ∆(e jω ) into p = ∆(e jω )q and q s = M ∆ (e jω ) p 1 (64) where q, p and s are complex vectors and where s = F M ∆ (e jω ), ∆(e jω ) . Let us now consider this LFT (64) for a given ∆(e jω ) ∈ ∆(ω) and let us also consider the corresponding signals p, q and s. Let us then pre-and post-multiply the LMI constraint (52) with (p * , 1) and (p T , 1) T , respectively. Using (64), this yields:

q p * T ω (R ω -C * ω C ω ) T ω C * ω T ω C ω -T ω =A q p + s * s < γ(ω) (65)
27

Since p = ∆(e jω )q, we can rewrite the first term of the left hand side of (65) as follows:

q p * A q p = q * I N mod ∆(e jω ) * A I N mod ∆(e jω ) q (66)

The above reasoning can be done for any value of ∆(e jω ) ∈ ∆(ω). In other words, for the matrix T ω found by the optimization problem, (65) holds true for all ∆(e jω ) ∈ ∆(ω). Let us also observe that, when T ω is a strictly positive definite diagonal matrix, (66) is a positive quantity for all ∆(e jω ) ∈ ∆(ω). Recalling that s = F M ∆ (e jω ), ∆(e jω ) , we have therefore also that (63) holds, which is the desired result.

Figure 1 :

 1 Figure 1: Example of graph representation of the network, each circle represents a node i and the edges represent the communication link between the nodes

Figure 2 :

 2 Figure 2: Representation of a single module/node i

3 7

 3 -5 8.8 10 -5 34.9461 3.5 10 -4 NON-STEALTH 1.2 10 -7 1.5 10 -3 1.5 10 -3 34.94706 10 -

Figure 4 :

 4 Figure 4: Graph representation of the network used in the second numerical illustration

10 -7 2 .

 102 1 10 -3 2.1 10 -3 131.649 8.3 10 -3 NON-STEALTH 1.3 10 -6 9.4 10 -3 9.5 10 -3 131.656 3.8 10 -2

  ref,1 -y 1 and y ref,2 -y 2 of Nodes 1 and 2 are respectively given by ref ext -y 1 and 1/3 ((y 1 -y 2 ) + (y 3 -y 2 ) + (y 4 -y 2 )). Similar relations can be found for all the other nodes. If the different loops [K i G i ] are designed to make the tracking error y ref,i -y i as small as possible, it can be proven that such an interconnection allows good tracking of ref ext at all nodes

Table 4 :

 4 Upper bounds J ub

  2 r (t, θ l ) + Ēs 2 refext (t, θ l ) + Ēs 2 e l (t, θ l ) +

	Ēs 2 e i (t, θ l )	(59)
	i∈L l

This is especially the case when the experiment is performed in a network where all modules have to track a given reference. In this case, the excitation signal introduces an undesired perturbation on the tracking performance.

and with respect to the conference paper[START_REF] Morelli | Optimal experiment design for the identification of one module in the interconnection of locally controlled systems[END_REF] on which the present paper is based

This estimate θ init,l is also the one with which the transfer function T init,l used for the stealth compensation is constructed.

The main differences of Proposition 2 in this paper with respect to Theorem 1 in[START_REF] Bombois | Optimal identification experiment design for the interconnection of locally controlled systems[END_REF] are that we here consider the influence of refext and that we also derive the consistency for the stealth setting.

The condition (29) on T init,l is more complex than[START_REF] Ljung | System Identification: Theory for the User, 2nd Edition[END_REF]. However, (29) will be respected if T init,l is not a too poor estimate of T 0,l .

We will indeed need to determine α(ω) at the L frequencies present in[START_REF] Zhou | Essentials of Robust Control[END_REF].

Note that the whole procedure yielding Φr,opt and γopt takes 11 seconds using Matlab.

If it was not the case, Node l could not track refext while we have stated that the objective of the network configuration is that each node follows refext.

A Proof of Proposition 1

Let us consider the expression [START_REF] Korniienko | Performance control for interconnection of identical systems: Application to pll network design[END_REF] for the individual costs J i (Φ r , θ 0 ) (i = 1, ..., N mod ) and introduce the following notations to distinguish them in the stealth (superscript S) and non-stealth case (superscript N S):

where R u is given by ( 12) and R u,N S is given by [START_REF] Kumar | Robust plant friendly optimal input design[END_REF]. Let us now observe that, for i = l, we have that:

Let us first consider the indexes i ∈ P l and recall Assumption 2. In this case, if T init,l satisfies [START_REF] Ljung | System Identification: Theory for the User, 2nd Edition[END_REF] for all frequencies ω where Φ r (ω) = 0, we have for all these frequencies that:

Consequently, for all i ∈ P l , we have that J S i (Φ r , θ 0 ) < J N S i (Φ r , θ 0 ). Finally, for the indexes i = l, i ∈ P l , J S i (Φ r , θ 0 ) = J N S i (Φ r , θ 0 ) = 0 since Ni (z, θ 0 ) = 0 in this case.

B Consistency and accuracy of (23)

B.1 Proof of Proposition 2

Let us first consider the stealth setting. To show the consistency of ( 23), we have to show that θ 0,l is the unique minimum of the following asymptotic identification criterion:

The prediction error (t, θ l ) in ( 23) can be rewritten as follows using ( 1) and ( 11):

In the non-stealth setting, the above reasoning and the expression (26) remain valid, but we have to replace R u l by R u,N S l (see [START_REF] Kumar | Robust plant friendly optimal input design[END_REF]).

C Proof of Proposition 3

Let us introduce the following notations to distinguish the covariance matrice P θ l (Φ r , θ 0 ) in the stealth and non-stealth case and let us also consider the notations introduced in (54)-(55) to distinguish the individual costs in those two cases.

Using the optimal spectrum Φ N S r,opt in the non-stealth case, let us define the following spectrum:

If an excitation signal r having that spectrum is used during an experiment where the stealth configuration is implemented, it is clear that the obtained covariance matrix P θ l ,S (Φ S r , θ 0 ) will be equal to the one obtained in the non-stealth case with Φ N S r,opt (i.e. P θ l ,N S (Φ N S r,opt , θ 0 )). Consequently, an experiment with this spectrum Φ S r will satisfy the accuracy constraint [START_REF] Safonov | Propagation of conic model uncertainty in hierarchical systems[END_REF]. Due to (60), (54) and (55), we have also the following relation for the individual cost at Node l: J S l (Φ S r , θ 0 ) = J N S l (Φ N S r,opt , θ 0 ) Moreover, for i = l, we have that:

Consequently, due to Assumption 1, for i ∈ P l , we have that J S i (Φ S r , θ 0 ) < J N S i (Φ N S r,opt , θ 0 ) if T init,l satisfies (29) for all frequencies ω where Φ S r (ω) = 0 (and thus for all frequencies ω where Φ N S r,opt (ω) = 0). Moreover, for i = l and i ∈ P l , we have that J S i (Φ S r , θ 0 ) = J N S i (Φ N S r,opt , θ 0 ) = 0. Due to [START_REF] Hägg | On optimal input design for networked systems[END_REF], we have thus shown that, in the stealth case, we can find a spectrum Φ S r leading to the same accuracy as with Φ N S r,opt , but with a strictly smaller cost. The result of the proposition is therefore proven since, by definition, J S (Φ S r , θ 0 ) ≥ J S (Φ S r,opt , θ 0 )