Nawel Khenak

Jean-Marc Vézien

David Thery

Patrick Bourdot

Spatial Presence in Real and Remote Immersive Environments

Keywords: Telepresence, Spatial Presence, Immersion, User Evaluation, Reality-Test H.5.1 [Information Interfaces and Presentation]: Multimedia Information System-Virtual reality, Telepresence System; [Information Interfaces and Presentation]: Multimedia Information System-Evaluation

This paper presents an experiment assessing the feeling of spatial presence in both real and remote environments (respectively the socalled "natural presence" and "telepresence"). Twenty-eight (28) participants performed a 3D-pointing task while being located in a real office and the same office remotely rendered over HMD. The spatial presence was evaluated by means of the ITC-SOPI questionnaire and users' behaviour analysis (trajectories of head during the task). The analysis also included the effect of different levels of immersion of the system -visual-only versus visual and audio -rendering in such environments. The results show a higher sense of spatial presence for the remote condition, regardless of the degree of immersion, and for the "visual and audio" condition regardless of the environment. Additionally, trajectory analysis of users' heads reveals that participants behaved similarly in both environments.

INTRODUCTION AND RATIONALE

Spatial presence defined as the sense of "being there" [START_REF] Biocca | The cyborg's dilemma: Progressive embodiment in virtual environments[END_REF] encompasses the user's ability to experience a feeling of presence [START_REF] Zhao | Reconceptualizing presence: Differentiating between mode of presence and sense of presence[END_REF]. If the environment is real and non-mediated, the user experiences a natural presence; if the environment is real and mediated, the user experiences a remote presence; and if the environment is computer generated, the user experiences a virtual presence [START_REF] Sheridan | Musings on telepresence and virtual presence[END_REF]. While studies have focused on the evaluation of the sense of presence across real and virtual environments, no study was conducted to compare between real and remote environments within the specific context of spatial presence. This was mainly due to the low immersive quality of technologies at that time. Indeed, many scholars emphasized the importance of high-quality natural sensory channels in the emergence of remote presence [START_REF] Sheridan | Musings on telepresence and virtual presence[END_REF]. Recent technological breakthroughs, including HMDs visual quality, sound spatialization and overall system latency reduction, allow for more sophisticated user experiments in terms of realistic sensation and higher level of immersion, permitting better comparison of the sense of spatial presence between real and remote environments. The main goal of the conducted study is thus to evaluate the user's sense of spatial presence within real and remote immersive environments in almost similar visual and auditory conditions.

(3) firstname.lastname@limsi.fr So far, assessing presence was mostly performed through subjective questionnaires. Lessiter et al.'s ITC Sense of Presence Inventory (ITC-SOPI [START_REF] Lessiter | A cross-media presence questionnaire: The ITC-Sense of Presence Inventory[END_REF]) was chosen to measure the sense of spatial presence. This popular questionnaire has the advantage to be quite easy to administer and score. In parallel, an approach based on behavioural indicators sought to establish reliable and validated measures for presence [START_REF] Mania | The effects of levels of immersion on memory and presence in virtual environments: A reality centered approach[END_REF]. Then, objective metrics related to user's behaviour were logged to study their relation with the feeling of spatial presence and whether they are conceivable tools for its objective assessment.

EXPERIMENTAL DESIGN AND METHOD

Two rectangular rooms with a similar office setting were used for the experiment. Subjects were either located in an "operating room", where they performed the task directly, or in a "teleoperating room", from which they performed the task through a teleoperation system. The task consisted in pointing a sequence of images that were displayed sequentially on 12 different tablets in a random but fixed order, one at a time (as soon as a tablet was pointed at, the image disappeared, and another one appeared in another location). The tablets were attached to the 4 walls of the operating room using a uniform sampling procedure. Sounds were played from the tablets. Thus, the subjects were seated in the middle of the operating or the teleoperating room and had to perform a sequence of pointing actions for a fixed period of time (3 minutes).

Hadware and Software set-up

Remote visual capture was obtained with a Ricoh Theta-V 360° panoramic camera placed at the centre of the operating room. Images were streamed at a resolution of 3840 x 1920@30fps to a real-time rendering engine visualized in a HTC Vive VR headset, with a 90Hz refresh rate. Because the field of view of the headset was limited to about 110°, subjects in the operating room wore a headset mock-up with an identical field of view. An infrared tracking system was set up in the operating room consisting of a six-camera ARTrack5 tracking network. Subjects wore infrared markers on both their pointing hand and their head. In the teleoperation setting, head tracking was achieved using the Vive lighthouse system. Remote action was made possible by tracking user gesture with a Leap Motion sensor attached to the headset.

In the operating room, the subjects naturally experienced 3D sound perception coming from the tablets around them. To reproduce a similar auditory perception in the remote setting, a Tetramic First-Order ambisonic microphone was used to capture the sound, which was rendered binaurally over headphones (Sennheiser HD 280) in the teleoperating room. A patch designed in the Max MSP software managed all the audio processing pipeline. A reverberation delay of 1.2 second was added to simulate the acoustics of the operating room while the source aperture was fixed to 90°. Finally, the RMS level was set equal in both real and remote conditions to 75 dBA.

USER EVALUATION

A mixed-design analysis was run with two independent variables: the type of environment with 2 modalities labelled "real" and "remote", representing respectively the operating room and the teleoperating room, and the level of immersion with 2 modalities labelled "with sound" and "no sound", representing respectively the multisensory condition -visual with audio rendering -and the visual-only, silent condition. The order of the factors was counterbalanced across the 28 participants of the study (19 males, 9 females; Mean = 27,4 years, Sd = 4,4 years), using a Latin Square.

Once the pointing task was completed, participants were administered the ITC-SOPI questionnaire [START_REF] Lessiter | A cross-media presence questionnaire: The ITC-Sense of Presence Inventory[END_REF]. The participants performed the task and filled the questionnaire twice, for the "with sound" and "no-sound" condition, but only in one of the two environments. Finally, personal demographic variables (age, gender, experience with VR systems) were also collected.

In total, 54 trials were registered. The study comprised 2 different dependent variables, namely the perceived spatial presence deducted from responses to the questionnaire, and the user behaviour described by the head-related hesitation movement, i.e. the number of subject's turnarounds to locate the targets during the evaluation.

RESULTS AND DISCUSSION

The analyses were divided into a between-subject study between the environments and a within-subject study between the levels of immersion. All the analyses were performed using RStudio.

The perceived spatial presence was assessed using a two-way repeated measures ANOVA between both environments and levels of immersion. Statistically significant results were found between "real" and "remote" conditions [𝐹(52) = 16.7, 𝑝 < .001, 𝑟 = 0.2] and between "no-sound" and "with sound" conditions [𝐹(52) = 11.5, 𝑝 < .01, 𝑟 = 0.2]. Thus, a Wilcoxon signed rank dependent and independent tests were performed respectively between levels of immersion and environments. A statistically significant higher sense of presence was reported in the "with sound" condition in both real and remote environments ([𝑉 = 1.5, 𝑝 < .01, 𝑟 = 0.8]; [𝑉 = 7, 𝑝 < .01, 𝑟 = 0.9]). A higher sense of presence was also found in the "remote" condition compared to the "real" condition for both "with sound" and "no-sound" conditions ([𝑊 = 153.5, 𝑝 < .01, 𝑟 = 1.1]; [𝑊 = 158.5, 𝑝 < .01, 𝑟 = 1.0]).

The higher sense of presence with audio shows the usefulness of a high degree of immersion to improve the feeling of presence [START_REF] Sheridan | Musings on telepresence and virtual presence[END_REF]. More surprising, the higher sense of presence reported in the remote environment show that subjects experienced an illusion of more "real" presence in the remote environment. This hyper-presence [START_REF] Biocca | The cyborg's dilemma: Progressive embodiment in virtual environments[END_REF] could be resulting from the lack of familiarity of participants with VR headsets (66,7% were beginners), especially in such a telepresence configuration.

Trajectory analysis was performed with the help of an ad-hoc hesitation indicator, which measured how often subjects would "change their mind" about the location of the pointing target.

A two-way repeated measures ANOVA was conducted to compare between the environments and levels of immersion. The results showed that for both real and remote environments, the hesitation of participants was reduced significantly when audio renderings were provided. Thus, participants had similar reaction whereas in the real and remote condition. However, results of comparison between the environments showed that participants hesitated more in the "remote" condition, especially in the "with sound" condition. A possible explanation is that localization was probably degraded by the low spatial resolution of the First-order Ambisonic microphone used during the evaluation. This highlights the importance of accurate reproduction of the auditory component to reach renderings comparable to real situations [START_REF] Postma | Perceptive and objective evaluation of calibrated room acoustic simulation auralizations[END_REF].

Finally, these results contradict those from the spatial presence scale of the questionnaire. Thus, it would be interesting not only to analyze the trajectory in its totality (research phase and pointing phase) but analyze each phase separately to provide more reliable tools based on behavioral indicators.

CONCLUSION

This paper presented a user evaluation of the sense of spatial presence between a real and a remote environment captured through telepresence technologies, with different levels of immersion. Such comparative analyses between real-world and telepresence conditions will benefit the presence research community as well as the designers of future teleoperation systems.

Figure 1 :

 1 Figure 1: (Top) General setting of participants (Left) In the operating room. (Right) In the teleoperating room. (Bottom) First person view of participants.

ACKNOWLEDGEMENTS

This research was supported by EquipEx DIGISCOPE (ANR-10-EQPX-26-01) operated by the French National Research Agency (ANR), as part of the program Investissements d'Avenir.


Kévin Colin
email: kevin.colin@ec-lyon.fr

Xavier Bombois
email: xavier.bombois@ec-lyon.fr

Laurent Bako
email: laurent.bako@ec-lyon.fr

Federico Morelli
email: federico.morelli@ec-lyon.fr

Closed-loop Identification of MIMO Systems in the Prediction Error Framework: Data Informativity Analysis

Keywords: System Identification, Closed-loop Identification, MIMO Systems, Prediction Error Methods, Consistency

In the Prediction Error Identification framework, it is essential that the experiment yields informative data with respect to the chosen model structure to get a consistent estimate. In this work, we focus on the data informativity property for the identification of Multi-Inputs Multi-Outputs system in closed-loop and we derive conditions to verify if a given external excitation combined with the feedback introduced by the controller yields informative data with respect to the model structure. This study covers the case of the classical model structures used in prediction-error identification and the classical types of external excitation vectors, i.e., vectors whose elements are either multisine or filtered white noises.

Introduction

For an identification within the Prediction Error framework, it is crucial to choose an excitation that yields informative data. Indeed, the prediction error estimate is then guaranteed to be consistent (provided the chosen model structure is globally identifiable at the true parameter vector) [START_REF] Ljung | System identification: Theory for the user. Prentice Hall information and system sciences series[END_REF][START_REF] Soderstrom | System Identification[END_REF]. In this work, we study the data informativity with respect to (w.r.t.) Multi-Input Multi-Output (MIMO) model structures for the identification of MIMO systems in the closed-loop configuration (direct closedloop identification).

Data informativity is obtained when the input excitation is sufficiently rich to guarantee that the prediction error is distinctive for different models in the considered model structure. For Single-Input Single-Output (SISO) systems, this property has been largely studied. In this case, the data are informative if the input signal is sufficiently rich of an order that depends on the type and the complexity of the considered model structure. More precisely, an input signal is sufficiently rich of an order η if and only if its input power spectrum has a non-zero amplitude in η different frequencies in the interval ] -π, π] (see e.g. [START_REF] Ljung | System identification: Theory for the user. Prentice Hall information and system sciences series[END_REF]). Due to renewed interest in optimal exper-iment design (see e.g. [START_REF] Bombois | Least costly identification experiment for control[END_REF][START_REF] Jansson | Optimal experiment design in closed loop[END_REF]), where the covariance matrix of the identified model is involved, there has been a lot of works to connect the positive definiteness of the covariance matrix to the data informativity [START_REF] Sanfelice Bazanella | Necessary and sufficient conditions for uniqueness of the minimum in prediction error identification[END_REF][START_REF] Gevers | Identification and the information matrix: How to get just sufficiently rich[END_REF]. In addition, necessary and sufficient conditions have been derived for the data informativity in both the open-loop and closedloop case in [START_REF] Gevers | Informative data: How to get just sufficiently rich?[END_REF][START_REF] Gevers | Identification and the information matrix: How to get just sufficiently rich[END_REF]. In particular, these papers derive the minimal order of signal richness that the excitation signal must have to ensure data informativity and this is done for all classical model structures (BJ, OE, ARX, ARMAX, FIR). In the closed-loop case, this minimal order is related to the complexity of the controller present in the loop during the identification. In [START_REF] Gevers | Informative data: How to get just sufficiently rich?[END_REF][START_REF] Gevers | Identification and the information matrix: How to get just sufficiently rich[END_REF], it is also shown that, if the controller is sufficiently complex, the data can be informative even if the excitation signal is equal to zero (costless identification).

While the data informativity seems to be a grown-up research area in the SISO case, this cannot be said for the MIMO case. In [START_REF] Ng | Identifiability of MIMO linear dynamic systems operating in closed loop[END_REF], and more recently in [START_REF] Sanfelice Bazanella | Closed-loop identification of MIMO systems: a new look at identifiability and experiment design[END_REF], attention has been given to determine the minimal order that the controller must have to ensure that informative data are obtained when the external excitation is zero. Two conditions have been derived, one being sufficient and the other one necessary. In the majority of the cases, though, a (nonzero) external excitation will be required to yield informative data. Up to our knowledge, in the MIMO case, there is (almost) no result about the minimal richness this external excitation must have to yield informative data and about how this minimal signal richness relates to the complexity of the controller. Perhaps, the only result in that matter is given in [START_REF] Sanfelice Bazanella | Closed-loop identification of MIMO systems: a new look at identifiability and experiment design[END_REF]. In [START_REF] Sanfelice Bazanella | Closed-loop identification of MIMO systems: a new look at identifiability and experiment design[END_REF], it is indeed said that an external excitation signal r (t ) with a strictly positive definite power spectrum matrix Φ r (ω) at all ω always yields informative data for direct closed-loop identification. This condition is of course only sufficient and is moreover very restrictive. As an example, a multisine excitation will never respect this condition.

In this paper, we will therefore derive a condition allowing to verify whether, for a given MIMO controller, an arbitrary external excitation r (t ) yields informative data for the direct closed-loop identification of a given MIMO system in a full-order model structure. We will do that for the classical model structures used in the Prediction Error framework (FIR, ARX, ARMAX, OE, BJ) and for both multisine external excitations and filtered white noise external excitations.

As we will see in this paper, data informativity will be guaranteed in the MIMO closed-loop case if, for each channel/output, a certain matrix is full row rank. This matrix depends on the model structure complexity, on the controller coefficients and on the external excitation parametrization (i.e., amplitude, phase-shift and frequencies for multisine and filter coefficients for filtered white noise) . We will also observe that this matrix clearly separates the contributions of the controller and of the external excitation to the informativity of the collected inputoutput data. This paper builds upon our previous contributions where we consider the data informativity problem for the open-loop identification of MIMO systems [START_REF] Colin | Data informativity for the identification of MISO FIR system with filtered white noise excitation[END_REF][START_REF] Colin | Data Informativity for the Open-Loop Identification of Multivariate System in the Prediction Error Framework[END_REF][START_REF] Colin | Informativity: how to get just sufficiently rich in the MISO FIR case?[END_REF].

Notations

For a complex-valued matrix A, A T denotes its transpose and A * its conjugate transpose. A positive semidefinite (resp. definite) matrix A is denoted A 0 (resp. A 0). We will denote A i k the (i , k)-entry of the matrix A, A i : the i -th row of A and A :k the k-th column of A. The identity matrix of size n ×n is denoted I n and 0 n×p is the n × p matrix full of zeros. For the sake of simplicity, we will often drop the index n × p and just write 0. The notation diag(a 1 , • • • , a n ) refers to the n × n diagonal matrix whose elements in its diagonal are the scalars a 1 , • • • , a n . For a vector x ∈ R n , the notation ||x|| refers to the Eu-

clidean norm, i.e. ||x|| = x 2 1 + • • • + x 2 n .
For two integers n ≤ p, the set n, p is the set of consecutive integers between n and p.

For quasi-stationary signals x(t ) [START_REF] Ljung | System identification: Theory for the user. Prentice Hall information and system sciences series[END_REF], we define the operator Ē [x(t )] = lim

N →+∞ 1 N N t =1 E [x(t )
] where E is the expectation operator.

For discrete-time model, z is the forward-shift operator. The degree of a polynomial P (X ) is denoted deg(P (X )). When X = z -1 , we say that ρ is the delay of P (z -1 ) when the first non-zero coefficient is linked to z -ρ , i.e. P (z -1 ) = p ρ z -ρ + p ρ+1 z -(ρ+1) + • • • + p n z -n with p ρ = 0.

Prediction Error Framework in closed-loop

Consider a MIMO system S with an input vector u ∈ R n u and an output vector y ∈ R n y , described by

S : y(t ) = G 0 (z)u(t ) + H 0 (z)e(t ) (1)
where G 0 (z) is a stable matrix of transfer functions of dimension n y × n u , H 0 (z) a stable, inversely stable and monic 1 matrix of transfer functions of dimension n y ×n y and e ∈ R n y is a vector made up of zero-mean white noise signals such that Ē e(t )e T (t ) = Σ 0 0. We will make the following assumption for the sake of simplicity.

(z) = diag(H 0,1 (z), • • • , H 0,n y (z))
where each scalar transfer function H 0,i (z

) (i = 1, • • • , n y )
is stable, inversely stable and monic.

As shown in Fig. 1, the system S is under feedback control with a stabilizing controller described by a matrix of (rational) transfer functions K (z) of dimension n u ×n y . The reference signal is set to 0. For identification purpose, a quasi-stationary external excitation r ∈ R n u can be added to the control effort such that the input u is given by

u(t ) = -K (z)y(t ) + r (t ) (2)
Fig. 1. Closed-loop configuration.

We will assume that the excitation vector r is uncorrelated with the vector of white noises e and that there is no algebraic loop in the closed loop made up of K (z) and G 0 (z), i.e., there is at least a delay in all entries of the matrix K (z)G 0 (z). In this work, we focus on developing conditions to get a consistent estimate of (G 0 (z), H 0 (z)) when considering the direct closed-loop identification approach, i.e., by using the data x(t ) = (y T (t ), u T (t )) T for the identification.

We will consider two types of quasi-stationary external excitation r . The first type is a multisine where each entry

r k of r (k = 1, • • • , n u ) is a multisine made up of sinusoids at s different frequencies ω l (l = 1, • • • , s), i.e., r k (t ) = s l =1 Λ kl cos(ω l t + Ψ kl ) k = 1, • • • , n u (3)
where Λ kl and Ψ kl are respectively the amplitude and phase shift of the sinusoid at the frequency ω l . Note that Λ kl and Ψ kl can be zero for some k

(k = 1, • • • , n u ) but, for each l = 1, • • • , s,
there exists (at least) a value of k for which Λ kl = 0. In the second type, r is generated as r = M v via a stable transfer matrix

M (z) = (M kq (z)) (k,q)∈ 1,n u × 1, f and a vector v = (v 1 , • • • , v f ) T containing f independent white noises signals v q (q = 1, • • • , f ) of covariance Σ v 0.
In other words, each entry r k of r is given by:

r k (t ) = f q=1 M kq (z)v q (t ) k = 1, • • • , n u (4)
Note that some M kq (z) can be identically zero and that f is not necessarily equal to n u but, for each q = 1, • • • , f , there exists (at least) a value of k for which M kq (z) = 0 .

The system S is identified within a full-order model structure M = (G(z, θ), H (z, θ)) | θ ∈ D θ where θ ∈ R n is a parameter vector and the set D θ restricts the parameter vector θ to those values for which G(z, θ) is stable and H (z, θ) is stable and inversely stable. The model structure is said to be full-order if ∃θ 0 ∈ D θ such that (G(z, θ 0 ), H (z, θ 0 )) = (G 0 (z), H 0 (z)). We will suppose that M is globally identifiable at the true parameter vector θ 0 , i.e. (G(z, θ), H (z, θ)) = (G 0 (z), H 0 (z)) ⇒ θ = θ 0 [START_REF] Sanfelice Bazanella | Necessary and sufficient conditions for uniqueness of the minimum in prediction error identification[END_REF][START_REF] Sanfelice Bazanella | Closed-loop identification of MIMO systems: a new look at identifiability and experiment design[END_REF].

Assume that we have collected a set of N input-output data:

Z N = x(t ) = y T (t ), u T (t ) T | t = 1, • • • , N
For each (G(z, θ), H (z, θ)) ∈ M , we can define the one-step ahead predictor ŷ(t , θ) for the output y(t ) using Z N :

ŷ(t , θ) = W y (z, θ)y(t ) + W u (z, θ)u(t ) = W (z, θ)x(t ) (5) W u (z, θ) = H -1 (z, θ)G(z, θ) (6) W y (z, θ) = I n y -H -1 (z, θ) (7)
where

W (z, θ) = (W y (z, θ),W u (z, θ)).
As will be illustrated in the sequel, the data x(t ) = (y T (t ), u T (t )) T must be informative with respect to M to get the consistency of the estimate.

Definition 2 (Data Informativity [2])

Consider the framework defined above with the data x(t ) = (y T (t ), u T (t )) T collected on the true system S operated in closed-loop with a controller K (z) and a quasi-stationary external excitation r (see [START_REF] Sanfelice Bazanella | Closed-loop identification of MIMO systems: a new look at identifiability and experiment design[END_REF]). Consider also a model structure

M yielding the predictor ŷ(t , θ) = W (z, θ)x(t ). Define the set ∆ W = ∆W (z) = W (z, θ ) -W (z, θ ) | θ and θ in D θ .
The data x(t ) are said to be informative w.r.t. the model structure M if, for all ∆W (z) ∈ ∆ W , we have

Ē ||∆W (z)x(t )|| 2 = 0 =⇒ ∆W (z) ≡ 0 (8)
where the notation ∆W (z) ≡ 0 means that ∆W (e j ω ) = 0 at all or almost all ω ∈] -π, π].

Data informativity combined with global identifiability at θ 0 are important properties since they imply that the prediction error criterion [START_REF] Ljung | System identification: Theory for the user. Prentice Hall information and system sciences series[END_REF][START_REF] Soderstrom | System Identification[END_REF] defined below yields a consistent estimate θN for θ 0 :

θN = arg min θ∈D θ V N θ, Z N (9) V N θ, Z N = 1 N N t =1 T (t , θ)Σ -1 0 (t , θ) (10)
where (t , θ) = y(t ) -ŷ(t , θ) and where Σ 0 is assumed known for simplicity 2 . The estimate θN is consistent if it converges to θ 0 with probability equal to 1 when N → +∞.

In [START_REF] Colin | Data informativity for the identification of MISO FIR system with filtered white noise excitation[END_REF][START_REF] Colin | Data Informativity for the Open-Loop Identification of Multivariate System in the Prediction Error Framework[END_REF][START_REF] Colin | Informativity: how to get just sufficiently rich in the MISO FIR case?[END_REF], we derived conditions to ensure the data informativity in open-loop (K (z) ≡ 0) for input vectors u = r of the types (3)-( 4) and for the classical types of model structures M in open-loop i.e. FIR, ARX, OE and BJ model structures. In the sequel, we will study the closed-loop case. In the next section, we present the model structures considered in this study.

Considered model structures

For the sake of simplicity, we will here also restrict attention to the classical types of model structures i.e. FIR, ARX, ARMAX, OE and BJ. For this purpose, let us decompose θ as follows

θ =   θ η   ( 11
)
where θ is made up by the parameters uniquely found in G(z, θ) and η contains the rest of the parameters.

The ARMAX model structure can be described as follows

   G(z, θ) = A(z, η) -1 B (z, θ) H (z, θ) = A(z, η) -1 C (z, η) (12)
where A(z, η) and C (z, η) are diagonal and monic polynomial matrices of dimension n y × n y and B (z, θ) is a polynomial matrix of dimension n y × n u . Note that the ARX and the FIR model structures are special cases of the ARMAX one with C (z, η) = I n y in the ARX case and C (z, η) = A(z, η) = I n y in the FIR case.

For BJ model structures, G(z, θ) and H (z, θ) do not share common parameters:

   G(z, θ) = G(z, θ) H (z, θ) = H (z, η) (13) with H (z, η) = diag(H 1 (z, η), • • • , H n y (z, η)).
Let us introduce some further notations for these MIMO model structures.

ARMAX/ARX/FIR: For the MIMO ARMAX/ARX/FIR model structure in [START_REF] Ng | Identifiability of MIMO linear dynamic systems operating in closed loop[END_REF], B (z, θ) is a matrix made up of n y n u independently parametrized polynomials

B i k (z, θik ) = z -ρ i k Bik (z, θik ) (i = 1, • • • , n y , k = 1, • • • , n u )
where the delays ρ i k can be all different and Bik (z, θik ) is a polynomial. All

H i (z, η) = C i (z, η)/A i (z, η) are parametrized independently with a parameter vector η i , i.e, H i (z, η) = H i (z, η i ) = C i (z, η i )/A i (z, η i ) where C i (z, η i ) and A i (z, η i ) are monic polynomials.
The coefficients of the polynomial Bik are the parameters in θik while the ones in the polynomials A i (z, η i ) and C i (z, η i ) are the ones in η i . Consequently, we have that:

Bik (z, θik ) = θik,1 + deg( Bik ) m=1 θik,(m+1) z -m (14)
C i (z, η i ) = 1 + deg(C i ) m=1 η i ,m z -m (15) A i (z, η i ) = 1 + deg(A i ) m=1 η i ,(m+deg(C i )) z -m (16)
where θik,m denotes the m-th entry of θik and η i ,m the m-th entry of η i . The number of parameters to identify in Bik , C i and A i is thus equal to deg( Bik )

+ 1, deg(C i ) and deg(A i ) respectively.
BJ/OE: For the MIMO BJ/OE model structure in [START_REF] Skogestad | Multivariable feedback control: analysis and design[END_REF], G(z, θ) is a matrix made up of n y n u independently parametrized transfer functions

G i k (z, θik ) = z -ρ i k Bik (z, θik )/F i k (z, θik ) where Bik (z, θik ) and F i k (z, θik ) are polynomials. The diagonal matrix H (z, θ) is com- posed by n y independently parametrized transfer func- tions H i (z, η i ) with H i (z, η i ) = C i (z, η i )/D i (z, η i ) where C i (z, η i ) and D i (z, η i ) are monic polynomials.
The coefficients of the polynomials Bik and F i k are the parameters in θik while the coefficients of the polynomials C i (z, η i ) and D i (z, η i ) are the ones in η i . Consequently, we have that:

Bik (z, θik ) = θik,1 + deg( Bik ) m=1 θik,(m+1) z -m (17) F i k (z, θik ) = 1 + deg( Fik ) m=1 θik,(m+deg( Bik )+1) z -m (18) C i (z, η i ) = 1 + deg(C i ) m=1 η i ,m z -m (19) D i (z, η i ) = 1 + deg(D i ) m=1 η i ,(m+deg(C i )) z -m (20)
where θik,m denotes the m-th entry of θik and η i ,m the m-th entry of η i . The number of parameters to identify in Bik ,

F i k , D i and C i is thus equal to deg( Bik )+1, deg(F i k ), deg(D i ) and deg(C i ) respectively.
For all classical model structures, the parameter vector θ ∈ R ñ is the concatenation of θik

(i = 1, • • • , n y , k = 1, • • • , n u ) i.e. θ = ( θT 11 , θT 12 , • • • , θT 1n u , • • • , • • • , θT n y 1 , θT n y 2 , • • • , θT n y n u ) T . The parameter vector η ∈ R n η is also the con- catenation of η i (i = 1, • • • , n y ) i.e. η = (η T 1 , η T 2 , • • • , η T n y ) T .
These classical model structures are globally identifiable at the true parameter vector θ 0 if θ 0 does not lead to a pole/zero cancellation.

Index notations:

In the sequel, we will use the index i ∈ 1, n y to specify the entries y i of y while k ∈ 1, n u will be used to specify the entries u k of u.

Data informativity for MIMO systems in closed-loop

Simplification of the study

We will first see in the next theorem how the controller K (z) and the external excitation r contribute to the data informativity in the closed-loop MIMO case. Section 4. Recall that r and e are independent. For each ∆W (z) ∈ ∆ W , we define similarly ∆W y (z) and ∆W u (z). Then, the data x(t ) are informative w.r.t. the model structure M if and only if, for all

Theorem 3 Consider Definition 2 and one of the model structures M defined in

∆W (z) = (∆W y (z), ∆W u (z)) ∈ ∆ W ,    ∆W y (z) -∆W u (z)K (z) ≡ 0 Ē ||∆W u (z)r (t )|| 2 = 0 =⇒ (∆W y , ∆W u ) ≡ (0, 0) (21)
PROOF. See Appendix A.

Remark 4

As mentioned in the introduction, a sufficient condition for the informativity is to choose r such that its power spectrum matrix is strictly positive definite at all frequencies, i.e. Φ r (ω) 0 ∀ω. Indeed, by using Parseval theorem

Ē [||∆W u (z)r (t )|| 2 ] = 0 is equivalent to tr 1 2π π -π ∆W u (e j ω )Φ r (ω)∆W * u (e j ω )d ω = 0 where tr is the trace operator. Consequently, if Φ r (ω) 0 for all ω, then Ē [||∆W u (z)r (t )|| 2 ] = 0 implies ∆W u ≡ 0. When ∆W u ≡ 0,
the first equation of the left hand side of (21) is equivalent to ∆W y ≡ 0. Consequently, we have informativity when Φ r (ω) 0 ∀ω (or at almost all ω). However, this sufficient condition for data informativity is in fact very restrictive. It will indeed never be verified for multisine excitation such as [START_REF] Bombois | Least costly identification experiment for control[END_REF] or for filtered white noise excitation of the type (4) when f < n u , while such excitation signals can of course yield informative data (see, e.g., Section 9).

In the next theorem, we will show that we can simplify the result of Theorem 3.

Theorem 5 Consider Definition 2, Theorem 3 and one of the model structures M defined in Section 4. Define the sets

∆ W,i = {∆W i : (z) | ∆W i : (z) is the ith row of ∆W (z) ∈ ∆ W } (i = 1, • • • , n y ). For each ∆W i : (z) ∈ ∆ W,i ,
i : (z) = (∆W y,i : (z), ∆W u,i : (z)) ∈ ∆ W,i :    ∆W y,i : (z) -∆W u,i : (z)K (z) ≡ 0 Ē ||∆W u,i : (z)r (t )|| 2 = 0 =⇒ (∆W y,i : , ∆W u,i : ) ≡ (0, 0) (22)
PROOF. See Appendix B.

Theorem 5 allows to simplify the data informativity verification: it can be done channel-by-channel (or outputby-output). Hence, in the sequel we will consider an arbitrary i and restrict attention to (22) for that particular i .

In its actual form, ( 22) is function of the rational transfer functions matrix (∆W y,i : (z), ∆W u,i : (z)) ∈ ∆ W,i . We need to transform (22) into a polynomial matrix form for the development of the conditions on data informativity. Let us for this purpose first observe that the first equation of the left hand side of (22) can be rewritten as follows

∆W y,i : (z), -∆W u,i : (z)   I n y K (z)   ≡ 0 (23)
By considering a right-factorization of K (z) and a leftfactorization of (∆W y,i : (z), -∆W u,i : (z)), we will prove that we can transform (23) into a polynomial matrix form. Moreover, we will see that this left-factorization of (∆W y,i : (z), -∆W u,i : (z)) will allow us to transform the second equation of the left hand side of (22) into a polynomial matrix form.

To simplify the presentation in the sequel, we will often use shorthand notations for each transfer of the form M (z), M (z, θ ) and M (z, θ ) by dropping the argument and we will denote them by M , M and M respectively.

Factorization of (22) into a polynomial matrix form

We are going to consider a right-factorization for K (z) and a left-factorization for the rational block matrix (∆W y,i : (z), -∆W u,i : (z)) which should be valid for all ∆W i : (z) = (∆W y,i : (z), ∆W u,i : (z)) ∈ ∆ W,i . The one that we will choose is obtained by putting all entries of (∆W y,i : (z), -∆W u,i : (z)) on the same denominator. It is given in the next lemma.

Lemma 6 Consider the model structures defined in Section 4. For these model structures, the rational blockmatrix (∆W y,i : (z), -∆W u,i : (z)) can be left-factorized into

(∆W y,i : (z), -∆W u,i : (z)) = Q -1 i (z)(Υ y,i (z), Υ u,i (z))
where the row polynomial vector Υ y,i (z) of dimension n y has all its entries equal to 0 except possibly the i -th one denoted Υ y,i i (z). The row polynomial vector Υ u,i (z) is of dimension n u . The scalar polynomials Q i (z) and Υ y,i i (z) and the row polynomial vector Υ u,i (z) are given by Theorem 5), let us define the set Y i made up of all polynomial matrices (Υ y,i i (z), Υ u,i (z)) obtained by considering the leftfactorization of ∆W i : (z) ∈ ∆ W,i (see Lemma 6).

• for FIR: Q i = 1, Υ y,i i = 0, Υ u,i = B i : -B i : . • for ARX: Q i = 1, Υ y,i i = A i -A i , Υ u,i = B i : -B i : . • for ARMAX: Q i = C i C i , Υ y,i i = C i A i -C i A i , Υ u,i = C i B i : -C i B i : . • for OE: Q i = n u k=1 F i k F i k , Υ y,i i = 0 and Υ u,i = (Υ u,i k ) k∈× 1,n u with Υ u,i k = (B i k F i k -B i k F i k ) n u l =1,l =k F i l F i l • for BJ: Q i = C i C i n u k=1 F i k F i k , Υ y,i i = D i C i n u k=1 F i k F i k - D i C i n u k=1 F i k F i k and Υ u,i = (Υ u,i k ) k∈ 1,n u with Υ u,i k = D i C i B i k F i k n u l =1,l =k F i l F i l -D i C i B i k F i k n u l =1,l =k F i l F i l PROOF. See Appendix C. Based on the set ∆ W,i (i = 1, • • • , n y ) (see
For the controller K (z), we will consider the rightfactorization consisting in putting all entries of K (z) on the least common multiple J (z) of the denominator of the entries of K (z) (J (z) is a polynomial). Therefore, K (z) can be rewritten as

K (z) = N (z)V -1 (z) (24)
where N (z) is a FIR matrix of dimension n u × n y and

V (z) = diag(J (z), • • • , J (z) n y times
).

Based on the factorization of K (z) in (24) and of ∆W i : (z) ∈ ∆ W,i in Lemma 6, we can transform (22) into a polynomial matrix form.

For all i = 1, • • • , n y , denote Ξ i (z) =   V i : (z) N (z)   ( 25
)
where V i : (z) is the i -th row of V (z).

For each ∆W i : (z) = (∆W y,i : (z), ∆W u,i : (z)) ∈ ∆ W,i , consider the left-factorization of (∆W y,i

: (z), -∆W u,i : (z)) = Q -1 i (z)(Υ y,i (z), Υ u,i (z))
given in Lemma 6 and denote Υ y,i i the i -th element of the row polynomial vector Υ y,i . Then, the data x(t ) are informative w.r.

t. the model structure M if and only if, for all

i = 1, • • • , n y ,    (Υ y,i i (z), Υ u,i (z))Ξ i (z) ≡ 0 Ē ||Υ u,i (z)r (t )|| 2 = 0 =⇒ (Υ y,i i , Υ u,i ) ≡ (0, 0) (26) for all (Υ y,i i , Υ u,i ) ∈ Y i .
PROOF. See Appendix D.

Main result for data informativity

In this section, we derive the main result of this paper. This result will allow us to check data informativity by verifying for each channel i = 1, • • • , n y whether a given matrix is full row rank. As we will see in the sequel, this matrix will depend on the complexity of the model structure, on the controller coefficients and on the external excitation parametrization (amplitudes, phase-shifts, frequencies for multisine excitation and filter coefficients for filtered white noise excitation).

For this purpose, a first step is to give a formal expression of the polynomial Υ y,i i and the row vector of polynomials

Υ u,i = (Υ u,i 1 , • • • , Υ u,i n u ) in Y i .
Using Lemma 6 and the notations introduced in Section 4, we can determine the scalars η y,i , η u,i k [START_REF] Mišković | Closed-loop identification of multivariable systems: With or without excitation of all references?[END_REF] and the entries of Υ u,i in Y i can be expressed as follows:

(i = 1, • • • , n y , k = 1, • • • , n u ) such that all Υ y,
Υ y,i i (z) = δT y,i Z y,i (z) (27) Υ u,i k (z) = δT u,i k Z u,i k (z) (28)
where Using ( 27) and (28) and defining δu,i = ( δT

Z y,i (z) = (z -1 , • • • , z -η y,i ) T , Z u,i k (z) = (z -ρ i k , • • • , z -η u,i k ) T and
(i = 1, • • • , n y , k = 1, • • • , n u )
u,i 1 , • • • , δT u,i n u
) T , the vectors of polynomials (Υ y,i i , Υ u,i ) in the set Y i can be rewritten as follows:

(Υ y,i i (z), Υ u,i (z)) = δT i bdiag(Z y,i (z), Z u,i (z)) (29)
where δi = ( δT

y,i , δT u,i ) T and Z u,i = bdiag(Z u,i 1 , • • • , Z u,i n u ) (with bdiag(X 1 , • • • , X n ) a block diagonal matrix whose blocks are given by X i (i = 1, • • • , n))
. We will denote by ζ i (resp. ζ u,i ) the dimension of δi (resp. δu,i ). These dimensions can be easily deduced based on η y,i , η u,i k

(i = 1, • • • , n y , k = 1, • • • , n u ). Let us also introduce the set D i = { δi | (29) ∈ Y i }.
Note that, except in the FIR case where

D i = R ζ i , D i is a subset of R ζ i .
In Sections 6 and 7, we will show that, using ( 27)-(29), we can determine matrices A i , B i and C i such that the left hand side of (26) for a given (Υ y,i i , Υ u,i ) ∈ Y i can be expressed as a function of the parameter vector δi ∈ D i defining (Υ y,i i , Υ u,i ) (see ( 29)). In particular, the first equation of the left hand side of ( 26) is equivalent to:

           δT y,i δT u,i δT i   A i B i   = 0 0 (BJ/ARX/ARMAX case) δT u,i B i = 0 (OE/FIR case) (30)
The difference in the above equation follows from the fact that Υ y,i i = 0 for OE/FIR model structures (see Lemma 6).

The second equation of the left hand side of ( 26) is equivalent to:

δT u,i C i = 0 (31)
As will be shown in Sections 6 and 7, the dimension and the elements of the matrices A i , B i and C i will be function of the complexity of the model structure, the controller coefficients and the external excitation parametrization.

Theorem 8 Consider the data x(t ) generated as in Section 3 with an external excitation r (t ). Consider also Definition 2, Theorem 7 and one of the model structures M

defined in Section 4. Consider finally the notations introduced in Section 5.3 and, in particular, the equations (30) and (31) that are respectively equivalent to the first and second equations of the left hand side of (26). Then, the data x(t ) are informative with respect to M if, for each i = 1, • • • , n y , at least one of the matrices P Moreover, when r = 0, the data x(t ) are informative with respect to M if, for each i = 1, • • • , n y , the matrix P (a) i defined above is full row rank.

PROOF. We will prove the theorem in the BJ/ARX/ARMAX case. The proof for the OE/FIR case can be derived using the same argumentation. Note first that δi =( δy,i , δu,i ) in ( 30)-( 31) is constrained to lie in D i . When δi ∈ D i , having a full row rank P (a) i is a sufficient condition 3 for δi = 0 to be the unique solution of (30). Note also that δi = 0 is equivalent to (Υ y,i i Υ u,i ) = (0, 0) (see ( 29)). Consequently, using the equivalence recalled in the statement of the theorem, when P (a) i is full row rank, we have also that

(Υ y,i i (z), Υ u,i (z))Ξ i (z) ≡ 0 =⇒ (Υ y,i i , Υ u,i ) ≡ (0, 0) for all (Υ y,i i , Υ u,i ) ∈ Y i .
It is clear by Theorem 7 that the latter implies that x(t ) is informative (in particular when r = 0).

Note now that (30) and (31) can be combined into δT

i P (c) i = 0. If P (c)
i is full row rank, then δT i P (c) i = 0 implies δi = 0. Consequently, using the equivalence recalled in the statement of the theorem, when P (c) i is full row rank, we have also that (26) holds for all (Υ y,i i , Υ u,i ) ∈ Y i . It is clear by Theorem 7 that the latter implies that x(t ) is informative.

Finally, if P (b)

i is full row rank, the equation (31) implies δu,i = 0. Consequently, using the equivalence recalled in the statement of the theorem, when P (b) i is full row rank, we have also that: Ē ||Υ u,i (z)r (t )|| 2 = 0 =⇒ Υ u,i ≡ 0 for all Υ u,i (z) ∈ Y i . Since Υ u,i ≡ 0, the first equation of the left hand side of (26) reduces to Υ y,i i (z)J (z) ≡ 0 where J (z) is a given polynomial defined before in (24). The latter always implies that Υ y,i i ≡ 0. Consequently, by virtue of Theorem 7, we have also data informativity when P (b) i is full row rank.

Let us now show how we can rewrite the left hand side of (26) into (30)-(31).

6 Rewriting of (Υ y,i i , Υ u,i )Ξ i (z) ≡ 0 into (30)

We will consider the BJ/ARX/ARMAX case since it is the more general. Using (29) and (25), we have that:

(Υ y,i i (z), Υ u,i (z))Ξ i (z) = ( δT y,i , δT u,i )L(z) (32)
with L(z) a polynomial matrix of dimension ζ i × n y :

L(z) =   Z y,i (z)V i : (z) Z u,i (z)N (z)   Each entry L m j (z) of L(z) (m = 1, • • • , ζ i , j = 1, • • • , n y ) is a polynomial in z -1 and can therefore be rewritten as λ T m j Z L (z) with Z L (z) = (z -β mi n , z -β mi n+1 , • • • , z -β max ) T
where λ m j is a vector containing the coefficients 4 of the polynomial L m j (z) and z -β mi n , z -β max are the smallest and largest value of the monomials z -β among all entries L m j of L. This yields to the following expression for L(z):

L(z) =   Z y,i (z)V i : (z) Z u,i (z)N (z)   =   A i B i   (I n y ⊗ Z L (z)) (33)
where ⊗ represents the Kronecker product and the matrix

(A T i , B T i ) T is such that its m-th row (m = 1, • • • , ζ i ) is given by (λ T m1 , • • • , λ T mn y ).
Using now (32) and (33), we see that (Υ y,i i (z), Υ u,i (z))Ξ i (z) ≡ 0 is equivalent to (30). The matrix (A T i , B T i ) T has ζ i rows and a number of columns equal to n y (β max -β mi n +1). This number of columns therefore depends both on the complexity of the controller and on the complexity of the model structures (via β mi n , β max , η u,i k , η y,i ). The more complex the model structure and the controller, the larger this number of columns is.

Example 9

Consider the following ARX model structure M with n u = 2 inputs and n y = 1 output:

G(z, θ) = θ11,1 z -1 1 + η 1,1 z -1 , θ12,1 z -1 + θ12,2 z -2 1 + η 1,1 z -1 H (z, θ) = 1 1 + η 1,1 z -1
where θ = ( θ11,1 , θ12,1 , θ12,2 , η 1,1 ) T ∈ D θ . In this case, we have the left-factorization of (∆W y,1: (z), -∆W u,1: (z)) in Lemma 6 of the form (∆W y,1: (z), -∆W u,1:

(z)) = Q -1 1 (z)(Υ y,1 (z), Υ u,1 (z)) with Q 1 (z) = 1.
From Appendix E, the polynomial Υ y,1 (z) = Υ y,11 (z) has an order of η y,1 = 1 and is given by

Υ y,11 (z) = δ y,1 T Z y,1 (z) with δ y,1 = η 1,1 -η 1,1 and Z y,1 (z) = z -1 .
From Appendix E, the polynomial entries Υ u,11 (z) and Υ u,12 (z) of Υ u,1 (z) have a degree of η u,11 = 1 and η u,12 = 2 respectively and are successively given by

Υ u,11 (z) = δ T u,11 Z u,11 (z) Υ u,12 (z) = δ T u,12 Z u,12 (z)
where

• δ u,11 = θ 11,1 -θ 11,1 and Z u,11 = z -1 . • δ u,12 = ( θ 12,1 -θ 12,1 , θ 12,2 -θ 12,2 ) T and Z u,12 = (z -1 , z -2 ) T .
Then, we have that

(Υ y,11 (z), Υ u,1 (z)) = ( δT y,1 , δT u,1 ) δT 1 bdiag(Z y,1 , Z u,1 ) ( 34
)
where Z u,1 = bdiag(Z u,11 , Z u, 12) and δ u,1 = (δ T u,11 , δ T u,12 ) T .

The true system S is put under feedback control with the following stabilizing controller K (z):

K (z) =   0.4 + 0.29z -1 -0.07z -2 0.04 + 0.01z -1 -0.3z -2   N (z) 1 1 -0.62z -1 + 0.07z -2 V -1 (z)
In this case, the least common multiple of the denominators of K (z) is directly J (z) = 1 -0.62z -1 + 0.07z -2 . Consequently, the matrix Ξ 1 (z) in Theorem 7 is equal to Ξ 1 (z) = (J T (z), N T (z)) T . Therefore, (Υ y,11 (z), Υ u,1 (z))Ξ 1 (z) can be rewritten as

( δT y,1 , δT u,1 )   Z y,1 (z)J (z) Z u,1 (z)N (z)   L(z) = ( δT y,1 , δT u,1 )         z -1 -0.62z -2 + 0.07z -3 0.4z -1 + 0.29z -2 -0.07z -3 0.04z -1 + 0.01z -2 -0.3z -3 0.04z -2 + 0.01z -3 -0.3z -4         L(z)
The smallest and largest value of the monomials z -β among the entries of L(z) are β mi n = 1 and β max = 4 respectively. Therefore, the latter can be rewritten as follows

( δT y,1 , δT u,1 )   A 1 B 1         z -β mi n . . . z -β max       = ( δT y,1 , δT u,1 )         1
                z -1 z -2 z -3 z -4        
The entries in the matrices A 1 and B 1 are the polynomial coefficients of the numerators in N (z) and the least common multiple denominator J (z) of the controller K (z). The number of rows of A 1 and B 1 are respectively equal to the dimension of δy,1 and δu,1 , which are 1 and 3 respectively.

The matrix P (a) 1 = (A T 1 , B T 1 ) T has a rank of 4: it is fullrow rank. Therefore, from Theorem 8, the data x(t ) are informative with respect to M in the costless framework (r = 0). The paper [START_REF] Sanfelice Bazanella | Closed-loop identification of MIMO systems: a new look at identifiability and experiment design[END_REF] proposes another (sufficient) condition to verify whether r = 0 can yield informative data. Here, this condition is not satisfied. Consequently, the sufficient condition of Theorem 8 is in this example less restrictive than the one in [START_REF] Sanfelice Bazanella | Closed-loop identification of MIMO systems: a new look at identifiability and experiment design[END_REF].

7 Rewriting of Ē ||Υ u,i (z)r (t )|| 2 = 0 into (31) 7.1 Main idea of the rewriting of Ē ||Υ u,i (z)r (t )|| 2 = 0
This equation has already been studied in the open-loop case [START_REF] Colin | Data Informativity for the Open-Loop Identification of Multivariate System in the Prediction Error Framework[END_REF]. The idea for the rewriting of Ē ||Υ u,i (z)r (t )|| 2 = 0 is based on the introduction of the regressor concept [START_REF] Colin | Data Informativity for the Open-Loop Identification of Multivariate System in the Prediction Error Framework[END_REF][START_REF] Gevers | Informative data: How to get just sufficiently rich?[END_REF]. For this purpose, let us observe that the n u entries Υ u,i k of the polynomial vector Υ u,i are the FIR filters given in (28) with the orders η u,i k given in appendix E. Hence, we have that

Υ u,i (z)r (t ) = n u k=1 Υ u,i k (z)r k (t ) = n u k=1 δT u,i k φ r k ,i (t ) = δT u,i 1 δT u,i 2 • • • δT u,i n u δT u,i φ r,i (t ) ( 35
)
where

φ r,i (t ) =         φ r 1 ,i (t ) φ r 2 ,i (t ) . . . φ r nu ,i (t )         with φ r k ,i (t ) =         r k (t -ρ i k ) r k (t -ρ i k -1) . . . r k (t -η u,i k )         (36)
The signal vector φ r,i (t ) is is generally called regressor or regressor vector in the literature [START_REF] Gevers | Informative data: How to get just sufficiently rich?[END_REF]. We have

that Ē ||Υ u,i (z)r (t )|| 2 = 0 ⇔ δT u,i Ē φ r,i (t )φ *
r,i (t ) δu,i = 0. In [START_REF] Colin | Data Informativity for the Open-Loop Identification of Multivariate System in the Prediction Error Framework[END_REF], we have shown that, when r (t ) is given either by (3) or (4), the regressor vector φ r,i can be rewritten as φ r,i (t ) = C i ϕ i (t ) where ϕ i (t ) is a complex or real-valued signal vector of dimension d such that Ē ϕ i (t )ϕ * i (t ) 0 and C i a complex or real-valued time independent deterministic matrix of dimension

ζ u,i × d . Consequently, we have that Ē ||Υ u,i (z)r (t )|| 2 = 0 ⇔ δT u,i C i Ē ϕ i (t )ϕ * i (t ) C * i δu,i = 0 ⇔ δT u,i C i = 0 1×ζ u,i . There- fore, Ē ||Υ u,i (z)r (t )|| 2 = 0 is rewritten as δT u,i C i = 0 1×ζ u,i (37)
As shown in [START_REF] Colin | Data Informativity for the Open-Loop Identification of Multivariate System in the Prediction Error Framework[END_REF], when the external signal is given by (3), we have that φ r,i = C i ϕ i with ϕ i (t ) = 1 2 (e j ω 1 t , e -j ω 1 t , • • • , e j ω s t , e -j ω s t ) and

C i = (C T i ,1 , • • • , C T i ,n u
) T a column block matrix of dimension ζ u,i × 2s such that:

C T i ,k =             Λ k1 e -j ρ i k ω 1 • • • Λ k1 e -j η u,i k ω 1 Λ * k1 e j ρ i k ω 1 • • • Λ k1 e j η u,i k ω 1 . . . • • • . . . Λ ks e -j ρ i k ω s • • • Λ ks e -j η u,i k ω s Λ * ks e j ρ i k ω s • • • Λ * ks e j η u,i k ω s            
where the phasors Λkl are given by Λkl = Λ kl e j

Ψ kl (k = 1, • • • , n u , l = 1, • • • , s).
Note that the larger the number s of sinusoids in r , the larger the number of columns in C i .

As shown in [START_REF] Colin | Data Informativity for the Open-Loop Identification of Multivariate System in the Prediction Error Framework[END_REF], when r is given by ( 4), we can also derive an expression for the matrix C i 5 . In this case, the larger the complexity of the M (z) and the number f of white noises generating r are, the larger the number of columns of C i is.

Interpretation of Theorem for data informativity

Theorem 8 shows that we can verify whether a given external excitation r (t ) yields informative data by checking, for each channel, if one of the three matrices P (a) i ,

P (b) i or P (c) i is full row rank.
In particular, if P (a) i is full row rank for each channel i , then the data x(t ) generated as in Section 3 will yield informative data even if the external excitation r (t ) is equal to zero.

To have that P (a) i is full row rank, the number of its columns should be larger than ζ i . If the complexity of the controller and of the model structure (which determines the number of columns of P (a) i ) is not sufficient, P (a) i will not be full row rank.

Even if P (a) i is not full row rank, we can of course obtain informative data by adding a nonzero excitation r (t ) of the type (3) or (4). In this case, the data informativity can be checked by verifying whether P (c) i is full row rank. 5 In [START_REF] Colin | Data Informativity for the Open-Loop Identification of Multivariate System in the Prediction Error Framework[END_REF], this matrix is denoted L .

We observe that P (a) i and P (c) i have the same number of rows, but the number of columns of P (c) i is larger than the one of P (a) i (due to the matrix C i linked to the external excitation). Consequently, even if P (a) i has too few columns, the addition of the external excitation can allow P (c) i to have more columns than rows (and thus to imply (in the vast majority of the cases) that the data x(t ) are informative.

To do that, we can choose an excitation signal yielding a full row rank matrix C i , but it is as such not necessary since it is in theory sufficient to have that the sum of the number of columns in C i and the number of columns in P (a) i is larger than ζ i . A similar phenomenon was observed in the SISO case [START_REF] Gevers | Informative data: How to get just sufficiently rich?[END_REF][START_REF] Gevers | Identification and the information matrix: How to get just sufficiently rich[END_REF] with the external excitation.

As already mentioned, in the vast majority of the cases, the matrices P (a) i , P (b) i , P (c) i will be full row rank when the number of rows is smaller than the number of columns. However, for some badly chosen external excitations, controllers, ..., a rank deficiency can occur (as was also observed in in the open loop case [START_REF] Colin | Data Informativity for the Open-Loop Identification of Multivariate System in the Prediction Error Framework[END_REF]) and it is thus important to formally verify the rank of these matrices.

Numerical example

True system to be identified

Consider the following BJ system S with n u = 2 inputs and n y = 1 output given by

y 1 (t ) = -z -1 1-0.4z -1 2z -1 =G 0 (z) u(t ) + 1 1 + 0.5z -1 e 1 (t )
where u(t ) = (u 1 (t ), u 2 (t )) T . The system S is put under feedback control with a stabilizing controller K (z) given by

K (z) =   N 11 (z) N 21 (z)   1 J (z) V -1 (z) J (z) = 1 -0.3z -1 -0.49z -2 + 0.155z -3 + 0.06z -4 -0.02z -5 N 11 (z) = 1 + 0.1z -1 -0.24z -2 -0.004z -3 + 0.008z -4
N 21 (z) = 0.7 + 0.13z -1 -0.033z -2 + 0.0116z -3 -0.004z -4

We will identify S within a full-order model structure M as in [START_REF] Skogestad | Multivariable feedback control: analysis and design[END_REF] with

ρ 11 = ρ 12 = 1, B11 = θ11,1 B12 = θ12,1 F 11 = 1 + θ11,2 z -1 F 12 = 1 C 1 = 1 D 1 = 1 + η 1,1 z -1
where θ = ( θ11,1 , θ11,2 , θ12,1 , η 1,1 ) T . With this model structure, from Lemma 6, the vector δ1 = ( δT y,1 , δT u,1 ) T ∈ D 1 contains 10 polynomial coefficients with 3 in δy,1 and 7 in δu,1 since η y,1 = 3, η u,11 = 3 and η u,12 = 4 (see Appendix E).

Costless identification

Let us study if we can get informative data with a costless experiment, i.e., without external excitation (r = 0). For that, we should calculate the rank of the matrix P (a) 1 in Theorem 8 since r = 0. But, first, let us observe that the controller has been chosen such that

H -1 (z, θ ) -H -1 0 (z) ∆W y (z) = (H -1 0 (z)G 0 (z) -H -1 (z, θ )G(z, θ )) ∆W u (z) K (z)
with θ = (0.2, 0.7, 0, 0.7) T . From Theorem 3, a zero excitation (i.e, r = 0) will therefore not yield informative data with respect to M . Let us verify that the condition in Theorem 8 allows one to check this fact. By following the steps given in Section 5.3, we obtain the following matrix

P (a) 1 =   A 1 B 1   : P (a) 1 =                             1
                           
The matrix P (a) 1 has 8 columns for 10 rows. Consequently, it cannot be full row rank. In this example, the (sufficient) condition for data informativity given in Theorem 8 allows to verify without any problem that r = 0 cannot yield informative data.

Identification with external excitation (r = 0)

As the costless identification cannot yield informative data, we need to add an external excitation r . In this paragraph, we will propose 2 cases.

Case 1: Let us consider a multisine excitation. To make P (b) 1 full row rank, we need at least four sinusoids in r (t ) since ζ u,1 = 7. However, to make P (c) 1 full row rank, one sinusoid should be sufficient (two columns in C 1 ) since P (a) 1 has already 8 columns. Let us verify this property. Let us therefore choose a signal (r 1 (t ), r 2 (t )) T with one sinusoid at ω 1 = 0.1rad/s and with the following phasors Λ kl (k = 1, 2, l = 1)

Λ 11 = 2e j 0.2 Λ 21 = 2e j 0.2
i.e., r 1 (t ) = r 2 (t ) = 2 cos(ω 1 t + 0.2) ∀t . Here, the matrix C 1 is given by

C 1 =                  Λ 11 e -j ω 1 Λ * 11 e j ω 1 Λ 11 e -2 j ω 1 Λ * 11 e 2 j ω 1 Λ 11 e -3 j ω 1 Λ * 11 e 3 j ω 1 Λ 21 e -j ω 1 Λ
The rank of the corresponding matrix P (c) 1 is equal to 10: the data x(t ) generated with this excitation r are thus also informative with respect to M . It is important to see that we can generate informative data by only exciting one signal in r (as was observed in [START_REF] Mišković | Closed-loop identification of multivariable systems: With or without excitation of all references?[END_REF]) or, as in the previous case, by putting r 1 = r 2 .

Monte-Carlo simulations

In order to confirm that the input choices in Cases 1 and 2 yield informative data, we have applied each of these external excitation vectors to the true system in 1000 identification experiments (with different realizations of the white noise e, assumed to be Gaussian with of variance of σ 2 0 = 0.01) and we have identified θN (see ( 9)) for each experiment. For each input vector, we have computed the mean of these 1000 estimates and we have in each case observed that this mean is almost equal to θ 0 , given in Table 1, illustrating the consistency.

Conclusion

In this work, we have developed a condition that allows one to check if a closed-loop identification experiment could yield informative data with respect to MIMO model structures. We have addressed this study for classical model structures and classical external excitations (multisine or filtered white noise). This condition can be easily verified and depends on the controller complexity, the external excitation parametrization and the complexity of the model structure. Based on this condition, we give also hints to design the experiment to yield informative data.

We are going to prove that (A.2) is equivalent to the left hand side of (21).

Let us first prove that the first equation of (A.2) is equivalent to ∆W y (z) -∆W u (z)K (z) ≡ 0. Since Ē e(t )e T (t ) 0 and H 0 (z) is stable and inversely stable, we have that the power spectrum matrix Φ ṽ of ṽ(t ) = H 0 (z)e(t ) is strictly positive definite at all frequencies. Consequently, the first equation of (A.2) is equivalent to

∆W y (z)S y (z) -∆W u (z)S u (z)K (z) ≡ 0 (A.3)
From the push-through rule [13, Chapter 3], S u (z)K (z) = K (z)S y (z). Consequently, (A.3) is equivalent to

∆W y (z) -∆W u (z)K (z) S y (z) ≡ 0 (A.4)
Finally, by post-multiplying by S -1 y (z), we obtain

∆W y (z) -∆W u (z)K (z) ≡ 0 (A.5)
which is the first equation of the left hand side of (21).

Let us now prove that the second equation of (A.2) is equivalent to Ē ||∆W u (z)r (t )|| 2 = 0. Combining the second equation of (A.2) with (A.5) leads to

Ē ||∆W u (z) S u (z) + K (z)S y (z)G 0 (z) r (t )|| 2 = 0 (A.6) Again, from the push-through rule [13, Chapter 3], S y (z)G 0 (z) = G 0 (z)S u (z). Consequently, (A.6) is equivalent to Ē ||∆W u (z) I n u + K (z)G 0 (z) S u (z)r (t )|| 2 = 0 (A.7)
Finally, by observing 6 that I n u + K (z)G 0 (z) = S -1 u (z), we obtain that the second equation of (A.2) is equivalent to Ē ||∆W u (z)r (t )|| 2 = 0. This concludes the proof.

B Proof of Theorem 5

We are going to prove that the property (21) for all ∆W (z) = (∆W y (z), ∆W u (z)) ∈ ∆ W is equivalent to the fact that the property (22) holds for all ∆W i : (z) = (∆W y,i : (z), ∆W u,i : (z)) ∈ ∆ W,i and for all

i = 1, • • • , n y .
Let us first observe that ∆W (z) ≡ 0 is equivalent to ∆W i : (z) ≡ 0 ∀i ∈ 1, n y . Secondly, it is also straightforward to see that ∆W y (z) -∆W u (z)K (z) ≡ 0 is equivalent to ∆W y,i : (z) -∆W u,i : (z)K (z) ≡ 0 ∀i ∈ 1, n y . Finally, let us prove that Ē ||∆W u (z)r (t )|| 2 = 0 is equivalent to Ē ||∆W u,i : (z)r (t )|| 2 = 0 ∀i ∈ 1, n y . For this purpose, observe that the term Ē ||∆W u (z)r (t )|| 2 = 0 can be recast 6 And by assuming that S u (z) does not filter out any part of r . as follows

Ē ||∆W u (z)r (t )|| 2 = n y i =1 Ē ||∆W u,i : (z)r (t )|| 2 Since the term Ē ||∆W u,i : (z)r (t )|| 2 is non-negative (i = 1, • • • , n y ), we have indeed that Ē ||∆W u (z)r (t )|| 2 = 0 is equivalent to Ē ||∆W u,i : (z)r (t )|| 2 = 0 (i = 1, • • • , n y ).
We have thus proven that the property (21) for all ∆W (z) ∈ ∆ W is equivalent to

   ∆W y,i : (z) -∆W u,i : (z)K (z) ≡ 0 Ē ||∆W u,i : (z)u(t )|| 2 = 0 (i = 1, • • • , n y ) (B.1) =⇒ ∆W i : (z) = (∆W y,i : (z), ∆W u,i : (z)) = 0 (i = 1, • • • , n y ) for all ∆W i : (z) = (∆W y,i : (z), ∆W u,i : (z)) ∈ ∆ W,i (i = 1, • • • , n y ).
Using the parametrization introduced in Section 4, we observe that there are no common parameters in ∆W i : (z) = (∆W y,i : (z), ∆W u,i : (z)) and ∆W j : (z) = (∆W u, j : (z), ∆W y, j : (z)) ( j = i ). Therefore, (B.1) is equivalent to the fact that property (22) holds for all ∆W i : (z) = (∆W y,i : (z), ∆W u,i : (z)) ∈ ∆ W,i and for all i = 1, • • • , n y , completing the proof.

C Proof of Lemma 6

For the right factorization of (∆W y,i : (z), -∆W u,i : (z)) ∈ ∆ W,i , we will consider the one that consists on putting all entries of (∆W y,i : (z), -∆W u,i : (z)) on the same denominators.

First, the matrix H (z, θ) is diagonal for all θ ∈ D θ and so only the i -th entry of the 1×n y row-matrix ∆W y,i : is possibly non-zero. Let us denote ∆W y,i i the i -th entry of ∆W y,i : . Observe first that ∆W y,i i (z) = W y,i i (z, θ ) -W y,i i (z, θ

) = 1 -H -1 i (z, θ ) -1 -H -1 i (z, θ ) = H -1 i (z, θ ) -H -1 i (z, θ ).
Let us first study the case of the ARMAX model structure (see [START_REF] Ng | Identifiability of MIMO linear dynamic systems operating in closed loop[END_REF]). For all ∆W i : (z) = (∆W y,i : (z), ∆W u,i : (z)) ∈ ∆ W,i , we have that

   ∆W y,i i (z) = C -1 i (z, θ )A i (z, θ ) -C -1 i (z, θ )A i (z, θ ) -∆W u,i : (z) = C -1
i (z, θ )B i : (z, θ ) -C -1 i (z, θ )B i : (z, θ ) (C.1) Therefore, by posing Q i = C i C i , we obtain the factorization proposed in the lemma. Since ARX and FIR model structures are special cases of ARMAX model structures with C i (z, η) = 1 and C i (z, η) = A i (z, η) = 1 (i = 1, • • • , n y ) respectively, we also obtain the factorization proposed in the lemma for these two model structures.

Let us now consider the BJ model structures in [START_REF] Skogestad | Multivariable feedback control: analysis and design[END_REF]. For all ∆W i : (z) = (∆W y,i : (z), ∆W u,i : (z)) ∈ ∆ W,i , we have that

   ∆W y,i i (z) = C -1 i D i -C -1 i D i ∆W u,i : (z) = C -1 i D i G i : -C -1 i D i G i : (C.2)
where G i : and G i : are respectively the i -th row of G and G . Since C i , C i , D i and D i are scalars, we have that (∆W y,i i (z), -∆W u,i : (z))

= C -1 i C -1 i (C i D i -C i D i , C i D i G i : -C i D i G i : ) (C.3)
Let us put each entry of G i : (z, θ ) and G i : (z, θ ) on the same denominator as follows

G i : (z, θ ) = F -1 i (z, θ )G i (z, θ ) (C.4) G i : (z, θ ) = F -1 i (z, θ )G i (z, θ ) (C.5)
where F i (z, θ ) = n u k=1 F i k and G i (z, θ ) = (G i ,k ) k∈ 1,n u is a row vector of polynomials with G i ,k (z, θ ) = B i k (z, θ ) n u l =1,l =k F i l (z, θ ). The matrices F i (z, θ ) and G i (z, θ ) are defined similarly.

Since F i (z, θ ) and F i (z, θ ) are scalars, (C.3) can be recast as follows

∆W y,i i = Q -1 i Υ y,i i = Q -1 i (F i F i C i D i -F i F i C i D i ) -∆W u,i : = Q -1 i Υ u,i = Q -1 i (C i D i F i G i -C i D i F i G i )
where

Q i = C i C i F i F i . For OE, C i = C i = D i = D i = 1.
Hence, for OE model structures, Υ y,i i ≡ 0 and Υ u,i is the one defined in the statement. This concludes the proof.

D Proof of Theorem 7

We will prove that the property ( 22) is equivalent to (26). First, (Υ y,i i , Υ u,i ) ≡ (0, 0) is equivalent to (∆W y,i : , ∆W u,i : ) = (0, 0) since Q i is invertible. Hence the right hand sides of ( 22) and (26) are equivalent.

Secondly, the equation ∆W y,i : (z) -∆W u,i : (z)K (z) = 0 can be rewritten as follows By observing that all entries Υ y,i j ( j = i ) of the row vector Υ y,i are equal to 0, the latter is equivalent to

Theorem 7

 7 Consider Definition 2, Theorem 5 and one of the model structures M defined in Section 4. Consider the right-factorization of K (z) in (24).

  as a function of the model structure type and of the model structure complexity are given in Appendix E.



  are full row rank.• (a) The matrix P (a) for the BJ/ARX/ARMAX case and to B i for the OE/FIR case.• (b) The matrix P (b) i is equal to C i .• (c) The matrix P (c) /ARMAX case and to B i C i for the OE/FIR case.

  0.01 -0.3

  -0.3 0.49 -0.155 0.06 -0

  ∆W y,i : (z), -∆W u,i : (z) By post-multiplying by V (z) and by pre-multiplying by Q i (z), the latter is equivalent to Υ y,i (z), Υ u,i (z)

Υ

  y,i i (z), Υ u,i (z) first equation of the left hand side of (26).Finally, Ē [||∆W u,i : (z)r (t )|| 2 ] = 0 is equivalent to Ē [||Q -1 i (z)Υ u,i (z)r (t )|| 2 ] = 0. The latter is equivalent to Ē [||Q i (z)Q -1 i (z)Υ u,i (z)r (t )|| 2 ] = 0 which is in turn equivalent to Ē [||Υ u,i (z)r (t )|| 2 ] = 0, which is the desired result.E Degrees of the polynomials Υ y,i i and Υ u,i kWhen non-zero (i.e., for the BJ/ARX/ARMAX case), Υ y,i i (z) ca be written as in (27) with η y,i (see Lemma 6):• η y,i = deg(A i ) for the ARX case. • η y,i = deg(A i ) + deg(C i ) for the ARMAX case. • η y,i = deg(C i )+deg(D i )+2 n u k=1 deg(F i k ) for the BJ case.The polynomials Υ u,i k (z) can be written as in (28) with ρ i k as defined in Section 4 and with η u,i k (see Lemma 6):• η u,i k = deg(B i k ) for the FIR/ARX case. • η u,i k = deg(B i k ) + deg(C i ) for the ARMAX case. • η u,i k = deg(B i k ) + deg(F i k ) + 2 n u l =1,l =k deg(F i l ) for the OE case. • η u,i k = deg(C i ) + deg(D i ) + deg(B i k ) + deg(F i k ) + 2 n ul =1,l =k deg(F i l ) for the BJ case.

  we define similarly ∆W u,i : (z) and ∆W y,i : (z) which are the i -th row of ∆W u (z) and ∆W y (z) respectively. Then, the data x(t ) are informative w.r.t. the model structure M if and only if, for all i = 1, • • • , n y , the following property holds for all ∆W

  δy,i , δu,ik are vectors of coefficients. The values of η y,i , η u,i k

Table 1

 1 Mean of the identified parameter vector with N = 10000 for different cases over 1000 experiments.

	θN	θ11,1	θ11,1	θ12,1	η 1,1
	θ 0	-1	-0.4	2	0.5
	Case 1 -1.0010 -0.3999 1.9996 0.4994
	Case 2 -1.0045 -0.3987 2.0034 0.5033

Preprint submitted to Automatica

This means that H 0 and H -1 0 are stable and H 0 (z = ∞) = I n y .

It can however be estimated together with θN (see e.g. [10, Chapter 15])

If D i would be equal to R ζ i , this would be a necessary and sufficient condition.

These coefficients correspond to the ones in N (z) and J (z) of the right-factorization of the controller K (z) in (24).

A Proof of Theorem 3

The proof is adapted from the one in [START_REF] Sanfelice Bazanella | Closed-loop identification of MIMO systems: a new look at identifiability and experiment design[END_REF]. The idea is to prove that the left hand sides of ( 8) and ( 21) are equivalent and that the same holds for the right hand sides. Note that the latter is straightforward since ∆W (z) ≡ 0 ⇔ (∆W y (z), ∆W u (z)) ≡ (0, 0).

First, combining (1) and (2) leads to

where S u and S y are respectively the input and output sensitivity transfer functions of the closed-loop depicted in Fig. 1, defined respectively by S u (z)