Joachim Schöpfel

Sylvain Vanacker

Eric Kergosien

Bernard Jacquemin

Master's theses and open scholarship: a case study

Keywords: Digital libraries, Open repositories, Open access, Electronic theses and dissertations, Master's theses, Open scholarship, Cloud

come   L'archive ouverte pluridisciplinaire

Introduction

Open scholarship has become the new paradigm of scientific communication. The term refers to "teaching and research practices that espouse openness (i.e.) a collection of emergent scholarly practices that espouse openness and sharing", including open access, open publishing and open education (Veletsianos and Kimmons, 2012, p. 167). An ever-increasing number of books, journals and repositories provide free and unrestricted access to scientific content from all disciplines. Two third of the open institutional repositories contain more or less important collections of electronic theses and dissertations (ETDs), mostly PhD theses, considered as the most useful kinds of invisible scholarship and the most invisible kinds of useful scholarship [START_REF] Suber | Open access. Essential Knowledge series[END_REF].

The situation is quite different for Master's theses. Even if their number is several times higher 1 , they are much less well represented in open repositories than PhD theses. Less than 1 per cent of all repositories clearly indicate that they contain Master's theses, and freely available Master's theses represent less than 0.5 per cent of the scientific items retrieved by the Bielefeld Academic Search Engine. However, Master's theses can be an excellent showcase for the academic Master programs and research. They are original work of the mind, and at least for the best of them, they contribute to scientific and professional knowledge. The following paper provides an overview on the nature, publishing an interest of Master's theses and, based on our project at the University of Lille (France), describes some essential aspects for the processing and valorization of these documents in the academic cloud, as a contribution to open scholarship.

Master's theses -an overview

The Master's degree dates back to the origin of European universities. Today, it is usually a second-cycle academic degree requiring previous study at the Bachelor's level (or equivalent). It is awarded by universities or colleges upon completion of a one or two years program demonstrating advanced disciplinary knowledge or professional skills. Master's degree names and the structure and duration of Master programs vary according to the country and even the university; diversity is the rule.

In general, the granting of a Master's degree requires some kind of thesis or dissertation on a research or a professional project (internship), where the students have to demonstrate their ability of independent and rigorous thinking and of complex problem solving, along with field-specific methodology and knowledge. A recent report issued by the US National Academies of Sciences, Engineering, and Medicine (2017) highlights the potential scientific quality of Master's theses based on substantial undergraduate research experiences defined as "an inquiry or investigation conducted by an undergraduate student that makes an original intellectual or creative contribution to the discipline"2 .

Alongside research capacities and originality, quality is assessed in terms of innovation and discovery, refinement of experimental design, new research techniques, teamwork and writing skills. "As Master's theses aim to provide evidence of the skills learned, both practical and theoretical skills are exemplified in theses" (Seliger, 2015, p. 131). Their quality is impacted by the Master program and in particular, by the importance of research. Master's theses are a "complex genre", considering both the set of problems related to its configuration (structure, language, norms of reference) and the factors that constrain its production (methodological procedures, student/supervisor relationship, time management etc.) [START_REF] Carvalho | Writing a master dissertation -students' difficulties and coping strategies[END_REF].

As mentioned above, few repositories contain Master's theses. Institutional repositories: Some institutions disseminate their undergraduate students' theses and dissertations in their own institutional repository, as part of its intellectual output and along with articles, communications, working papers etc. For instance, in the University of Amsterdam, Scripties repository Master's theses represent 70 per cent of the nearly 50,000 undergraduate works3 .

General aggregating repositories: These multi-institutional or consortial repositories aggregate files and metadata from different document types and institutions. The Latin America portal LA Referencia for instance provides access to nearly 1.5 million publications, one third being Master's theses, mostly from Brazil4 .

ETD repositories: As a specific variant of the multi-institutional repositories mentioned above, they only contain ETDs from different institutions, like the OhioLINK Electronic Thesis and Dissertation Center with over 90,000 undergraduate, Master's and doctoral theses and dissertations from 32 universities and colleges5 and the Scandinavian DiVA portal for academic publications and student papers produced at 44 universities, colleges of higher education, research institutes and museums, with more than 250,000 student papers from BA, Master and other diplomas of which 72 per cent are in open access 6 .

Because of their number, diversity and quality, Master's theses prove their usefulness for studies on the evolution and distribution of research subjects and methodologies. A synthesis of published research with samples of Master's theses reveals three different approaches (for more details and further reading, see annex).

Content analysis: Nearly 20 recent papers from Iran, Slovenia, Brazil, China, Zambia, Turkey and other countries share an exploratory-descriptive approach and aim at assessing research paradigms, the diversity and trends of research topics, writing skills and information literacy, the students' background and the choice of research methods like qualitative or action research. They also share the basic assumption that Master's theses generally display sufficient quality and relevance for content analysis and scientometrics.

TDM: Some studies, especially from China, apply TDM techniques to Master's theses, such as quantitative analysis of the frequency of key words, co-word matrixes, co-occurrence network analysis, factorial or cluster analysis, strategy coordinate analysis and multidimensional scaling analysis. Their objective is the assessment of research topics ("hot issues"), methodologies or funding trends through the automatic exploitation of Master's theses.

Citation analysis: Citation analysis constitutes a third approach to Master's theses. These studies are based on the references produced by the Master students and assess the use of scientific literature in terms of document types, preferred journals, geographic origin of cited sources, and so on.

Along with the potential for scientometrics and TDM, there are other reasons for open access to Master's theses, such as enhancing accessibility, increasing the impact in scientific journals and conferences [START_REF] Gillet | Measuring the international spreading of the knowledge produced by French dental master theses[END_REF], preventing plagiarism, strengthening visibility for Master's institutions [START_REF] Xia | Current Trends in Institutional Repositories for Institutions Offering Master's and Baccalaureate Degrees[END_REF], contributing to the curriculum quality and fostering the potential use of outcomes especially from less known programs and disciplines [START_REF] Martens | Enhanced Accessibility of Master Theses Related to Real Estate[END_REF].

Quality is essential for the interest and potential of Master's theses. Not all Master's theses' reflect substantial undergraduate research experiences or professional projects. Poor quality of abstracts can be a limit to TDM. Evaluation can be helpful to control the quality and to select the best papers. Even if the criteria are neither homogeneous nor applied in the same way everywhere, with a large diversity of written guidelines and evaluation grids, of the supervisors' role and of oral exams to adjust the final mark [START_REF] Bøhn | The assessment of master's theses: findings of a survey, spring 2011[END_REF], the evaluation generally includes some similar features, such as background knowledge, description of methods, data and findings, discussion of the findings and the use of references. Sometimes, the Master assessment includes research skills and the relation with the supervising research team. And even if their quality may vary and should be controlled, their diversity and richness make them interesting and useful resources for citation analyses, for content mining and scientometric assessment. But this potential requires open accessibility and reusability. The Lille project MemorySID provides some insight how to proceed.

The Lille project MemorySID

The following section describes the project MemorySID at the University of Lille (France), with nearly 70,000 students the largest French University and one the largest French-speaking universities worldwide. The project started in 2008, to contribute to open scholarship through open access publishing of Master's theses in library and information sciences, via a national infrastructure in the public academic cloud. With the years, the project rationale shifted from self-archiving to a two-layer approach with two digital libraries, one in intranet and the other in the cloud.

Phase one (2008-2015): self-archiving

Ten years ago, from 2008 on, the Department of Information and Documentation Sciences (SID) at the University of Lille (France) started encouraging the Master students to deposit their theses on a national repository for Master's theses library and information sciences called mémSIC7 . The objective was to publish a selection of good and non-confidential works in open access and to guarantee long-term findability, accessibility and preservation through a national OA infrastructure, to valorize the students' work and to increase the Master programs' visibility [START_REF] Chauvin | Les mémoires de Master dans les archives ouvertes[END_REF][START_REF] Mann | La valorisation des mémoires d'étudiants sur une archive ouverte institutionnelle : un fonds et un contexte spécifiques[END_REF]. The three guiding principles for this first project phase were:

• validation by the jury (faculty);

• double authorization by the student and by the project tutor (intership); and

• self-archiving by the students.

The academic cloud was preferred to a local, institutional solution to optimize visibility and usage. The success was mixed. Between 2008 and 2015, 70 theses were deposited on mémSIC, with scores equivalent to the US academic A and B grades (for French readers: 14/20 or more). The impact in terms of usage was more than satisfying, with statistics going up to several thousand downloads per paper (see below). However, the main problem was the lack of motivation for self-archiving and, in some cases, the lack of quality of metadata and files. Another problem was that mémSIC does not allow the creation of an institutional collection or portal, thus reducing the impact for the Master program.

Phase two (from 2016 on): digital library

To resolve these issues, a follow-up project was launched in 2016 called MemorySID [START_REF] Vanacker | Processus de diffusion des mémoires de Master[END_REF]. For this second phase, the priorities were different:

• retrodigitization of all print Master's theses still in the academic library and department holdings;

• integration of all digitized Master's theses into the campus-wide Nuxeo document management system (DMS), accessible for students and scholars via their intranet workspace; and

• creation of a new collection on the national repository DUMAS, with a selection of the best theses ("showcase").

In other words, we privileged a two-layered approach with two digital libraries, one as a folder in the academic intranet and the other as a portal in the cloud, on the national ETD repository DUMAS (Figure 1). DUMAS is hosted by the French Center for Direct Scientific Communication8 which operates also the French national open repository HAL, and it contains more than 19,000 Master's theses. Only library and faculty can deposit files and create metadata. There is no self-archiving. Each deposit needs validation by the repository's administrator. Insofar, DUMAS is a kind of crossover between an open repository and a digital library. After removing duplicates, 765 print theses were digitized, 583 from the library holdings (1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004) and 227 from the SID department (2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016). The text files (PDF) were uploaded in the DMS and indexed in a Dublin Core (DC) compliant metadata format. To the 765 print works from 1996-2016, 9 native digital Master's theses were added in 2017. From these 774 Master's theses, 715 are searchable and available for all students and scholars on the campus, via the academic virtual workspace, while the others remain in the "dark" part of the intranet because of their confidential content. The digital library (folder) is administered by the faculty (SID Department), while the content of the DMS is maintained and backed-up by the university IT Department.

From the 715 Master's theses in the public DMS, 87 per cent have a sufficient score (grade A or B) and are authorized for online dissemination, according to the administrative records (i.e. defense minutes). So far (July 2018), 523 theses from the DMS have been deposited on the DUMAS repository and are online. Additionally, the metadata from 64 self-archived theses on mémSIC (first project phase) were transferred to DUMAS. This means that at present, the new Lille LIS Master's theses collection on the DUMAS repository provides unrestricted access to 587 theses from 1996 to 2017.

Usage of the DUMAS collection

The following usage statistics are from July 19, 2018. They assess the download figures for all 587 Master's theses available on DUMAS, for the period from January 2015 to June 2018.

Cumulative figures: up to now, the 587 items have been downloaded 296,074 times (Figure 2). The median download statistics for the last 12 months are 3,855 per month, and they are continuously increasing.

Usage statistics per item: the analysis of the usage statistics reveals that all deposited Master's theses, without exception, have been accessed and downloaded (Figure 3).

Figure 3 shows that the differences of usage among the Master's theses are very important, ranging from 10 to more than 60,000 downloads, with a median of 43. At least two factors may explain these differences: 1. The date of deposit and publishing: the items with more than 10,000 downloads have all been deposited between 2009 and 2011, while the items with low download figures are all from 2017 and 2018.

2. The topics: probably, theses with "hot issues" will be more downloaded. But the real impact of the content on usage statistics is difficult to assess and surely affected by other variables like the quality of indexing and referencing.

Information of interesting and recent deposits is published via social media, e.g. Twitter, Facebook and LinkedIn; metadata, links and the full text can be discovered on sites like Scoop.it, CiteULike or portals for students. However, without Digital Object Identifier minting and a more sophisticated altmetrics tool it is impossible to assess the real impact of the deposits inside and outside of the academic communities.

Where do the successful download requests come from? The DUMAS repository identifies the country of nearly 80 per cent downloads.

In total, 30 per cent successful download requests come from France, another 13 per cent are from the three French-speaking Maghreb countries Morocco, Algeria and Tunisia. The USA represents 3 per cent, Germany and Canada 2 per cent each, the UK, South Africa and China 1 per cent each. The member states of the so-called "Francophonia" represent 60 per cent of the download requests, i.e. 75 per cent of all geographically identified downloads. However, the collection's outreach is global and not limited to the French-speaking world: the complete download list contains 186

Discussion

MemorySID is a faculty-based project in social sciences and humanities, launched and undertaken by the SID department staff to increase the outreach and impact of the Lille LIS Master program. From the beginning, it raised questions and problems that had to be fixed by the faculty, in partnership with the academic library, the IT department and the host of the national open repository. Four issues appear crucial for the future development of open and unrestricted dissemination of Master's theses.

IP clearance

Intellectual property (IP) is an important issue. Open access to Master's theses via open repositories means uploading and publishing, and both need authorization from the rights holder and, at least in France where dissertations are considered as administrative proof for the diploma, also from the institution.

Once the student has left the Master program, this IP clearance is difficult to obtain and the risk is high that time produces an increasing number of "orphan theses". The best moment is at the defense (viva), when the student presents the results of her or his research work and internship. This is the opportunity for the student and the jury to sign the authorization for open access publishing, as part of the minutes of the thesis defense. The signed minutes are archived by the department and serve as proof for the dissemination via the DMS and the open repository.

But this is not enough. Some theses contain sensitive information or personal data while others include material protected by third party rights (photos, maps, etc.). Also it may be appropriate to request a third authorization from the professional tutor of the internship. If the student cannot guarantee full rights clearance for included material, it will be prudent either to drop this material or to exclude the thesis from open access.

At present, of all theses preserved in the DMS, four out of five are freely shared on the internet, while 12 per cent are restricted to on-campus access via the intranet and 8 per cent are in the DMS but not available at all (Figure 4).

Open access 80%

Confidential 8%

Intranet 12% 2. Guidelines about how to write a Master's thesis, how to use protected thirdparty material, how to handle confidential and sensitive information, personal data (privacy issues), etc.

Sometimes, fears are expressed that unrestricted dissemination of Master's theses may increase the risk of plagiarism. Indeed, this "collateral damage" cannot be excluded but on the other hand, open access to Master's theses will improve the performance of anti-plagiarism software.

Quality, metadata and identifiers

The second issue is quality. While all Master's theses should be preserved in the intranet DMS, only the best ones should be disseminated in open access, for two reasons:

1. Insofar, as the collection provides an excellent showcase and high visibility for the Master program, the dissemination of poor papers should be avoided.

2. To foster the potential for TDM, particular attention should be paid to the quality of metadata.

The quality issue has a paradoxical effect: if the deposit is limited to the best ("outstanding") Master's theses, only a small part will be available, reducing their diversity and representativeness. On the other hand, a selective "institutional quality label" will increase their value for the human reader and the machine.

At Lille, the threshold for online dissemination is set at 14 marks out of 20, which is similar to the US A and B grades. This score is the result of collegiate evaluation by at least two scholars, applying the usual criteria, e.g. quality of the analysis, independent thinking, presentation, structure, language and style and references. Nevertheless, the score is only one criteria and the final decision is taken at the defense (viva), by the jury.

The format and quality of metadata remains a challenge. Standard metadata improve findability and interoperability. Both systems, the DMS (Nuxeo) as well as DUMAS, make use of the Dublin Core (DC) elements. To limit inconsistent indexing, some elements were refined, such as rights (level of availability), source (Master program, level, date of the viva), description (author's abstract) and filename.

The DUMAS metadata schema follows the Text Encoding Initiative format of the French national HAL repository with some specific elements for Master's theses, e.g. the name of the Master program with its subdivisions, specialties and options. The problem is that the DMS and DUMAS follow different schemas, with different refinements and values, and that their metadata silos are not connected. This means the creation of two different metadata sets for the same document, one for the DMS and the other for the repository. The lack of interconnection and interoperability is a source of inconsistency and errors, causes double workload and requires more post-deposit curating than necessary. A three-level routine reduces the metadata heterogeneity and error level:

1. Inconsistencies and anomalies are rejected by the DUMAS administrator during the validation.

2. The index of authors and directors' names is periodically controlled by the local administrator.

3. The tag cloud is checked from time to time for synonymy, variants etc.

A last comment on unique identifiers: DUMAS systematically attributes a specific HAL identifier to each new deposit. To track impact on social media (altmetrics), it may be interesting to consider additionally digital object identifier minting.

Text and data mining

TDM tools and methods are already applied to Master's theses, especially to assess research topics, methodologies and funding trends. In Lille, a research team started to mine PhD and Master's theses in exploratory research projects, to identify specific topics in different disciplines like Agronomics, Cultural Studies or Information Sciences, and geographical information related to the concept of territory [START_REF] Kergosien | Automatic Identification of Research Fields in Scientific Papers[END_REF].

These studies confirm that the diversity and volume of Master's theses bear interesting potential for TDM and can produce relevant new knowledge about research topics, methods, people, etc. As the output of TDM is conditioned by the quality of the corpus, this means that special attention should be paid to the quality of titles, abstracts, key words and the structuration, annotation, indexing and open formats, whenever possible.

In an uncertain legal environment of TDM like in France, it would be appropriate and helpful to publish Master's theses under a liberal open license (CC-BY or CC-BY-SA) to foster their reuse and exploitation; however, this remains an exception up to now.

Preservation

Master's theses are part of what is called gray literature, not easy to identify, not always well controlled by catalogs and databases and often at risk of disappearance [START_REF] Schöpfel | Grey Literature[END_REF]. In France, Master's theses must be conserved for at least five years, as a proof of graduation. MemorySID adopts a three-level strategy to guarantee longterm preservation at least for the best Master's theses:

• Remaining print copies are preserved by the faculty administration during the obligatory five-year period, in the facilities of the SID Department.

• All digital copies are preserved in the university DMS and maintained by the IT Department for at least 5-10 years, the usual lifespan of documents in intranet.

• As for the selection of best and non-confidential Master's theses, they are preserved in the DUMAS open repository. DUMAS is part of the French national open scholarship infrastructure HAL which is backed up by the French National Computing Center for Higher Education9 specialized in long-term preservation of digital library resources, research data etc.

Thus, the DUMAS solution includes outsourcing of the long-term availability of Master's theses, in a dark archive in the public academic cloud which can be used as a failsafe during disaster recovery.

Conclusion

The environment of open scholarship provides excellent opportunities for a new approach to the management of Master's theses. Based on the Lille project and other studies, the following matrix sums up some key factors for the contribution of Master's theses to open scholarship:

• Strength: their diversity, richness and representativeness make Master's theses interesting for reading, content mining, citation analysis, etc. which is an important feature of the emerging landscape of open science. Also, they contribute to the visibility and promotion of the institution and the Master program.

• Weakness: the main problem is related to their unequal quality. Sensitive content (personal data, corporate information etc.) is another problem.

• Opportunity: the European and French open science policy creates a favorable environment for the dissemination of Master's theses on open repositories; this is emphasized by the institutions' interest to promote and valorize their Master programs.

• Threat: similar to PhD dissertations [START_REF] Schöpfel | Degrees of Secrecy in an Open Environment. The Case of Electronic Theses and Dissertations[END_REF], the major threat is access restriction because of intellectual property, i.e. no authorization by students. Restrictive policies of institutional repositories ("scientists only") and lack of interest for undergraduate work are other barriers.

A collection of Master's theses in the public academic cloud is a unique showcase for the academic excellence of Master programs, of teaching and research. The download statistics show that its impact and visibility are real and significant. It can also produce a rich and representative corpus for TDM and scientometrics. However, this potential is conditioned by a couple of key variables, such as a selective approach, long-term preservation and free dissemination on open repositories. It is only when these conditions are met, that Master's theses fully contribute to open scholarship.

Figure 1 :

 1 Figure 1: Two-layered approach of MemorySID

Figure 2 :

 2 Figure 2: Cumulative download statistics (N = 587)

Figure 3 :

 3 Figure 3: Download statistics per thesis (N = 587, logarithmic scale)

Figure 4 :

 4 Figure 4: Status of Master's theses in the DMS (N = 774) The key factor for the IP-related process is the faculty support, i.e. the personal and professional commitment to open scholarship which implies two "upstream actions": A clear and explicit communication about open repositories, open scholarship and sharing.

  Out of the 3,519 repositories of the OpenDOAR directory, 57 per cent contain ETDs but only 0.5 per cent clearly indicate that this means also Master's theses. The international Registry of Open Access Repository Mandates and Policies (ROARMAP) with 941 open access policies keeps a record of 37 per cent mandates and instructions regarding ETDs but nothing about Master's theses. Many institutions do not consider undergraduate work as research output worth open scholarship. Those which do have different options. Three vectors of open access dissemination via repositories can be distinguished.

Council of Undergraduate Research https://www.cur.org/

Scripties www.scriptiesonline.uba.uva.nl

LA Referencia www.lareferencia.info

OhioLINK ETD Center https://etd.ohiolink.edu/

DiVA www.diva-portal.org

Mémoires de master en sciences de l'information et de la communication https://memsic.ccsd.cnrs. fr

CCSD https://www.ccsd.cnrs.fr/

CINES https://www.cines.fr/en/

Statistics from 2017, covering all disciplines including medicine, produced by the French Ministry for Higher Education and Research


Guillaume Mercère
email: guillaume.mercere@univ-poitiers.fr

Marco Lovera
email: lovera@elet.polimi.it

Convergence analysis of instrumental variable recursive subspace identification algorithm *

Keywords: Subspace methods, recursive algorithms, linear systems, instrumental variable, convergence analysis

The convergence properties of recently developed recursive subspace identification methods are investigated in this paper. The algorithms operate on the basis of instrumental variable (IV) versions of the propagator method for signal subspace estimation. It is proved that, under suitable conditions on the input signal and the system, the considered recursive subspace identification algorithms converge to a consistent estimate of the propagator and, by extension, to the state-space system matrices.

Introduction

The problem of recursive subspace model identification (RSMI) has been an active area of research in recent years (see, e.g., [START_REF] Verhaegen | A fast recursive MIMO state space model identification algorithm[END_REF][START_REF] Cho | Fast recursive identification of state space models via exploitation displacement structure[END_REF][START_REF] Gustafsson | Instrumental variable subspace tracking using projection approximation[END_REF][START_REF] Lovera | Recursive subspace identification of linear and non linear Wiener state space models[END_REF][START_REF] Oku | Recursive 4SID algorithms using gradient type subspace tracking[END_REF][START_REF] Mercère | Recursive subspace identification based on instrumental variable unconstrained quadratic optimization[END_REF]). Most RSMI algorithms are inspired by offline versions of subspace model identification (SMI) techniques and therefore rely on the availability of efficient updating methods for the numerical linear algebra algorithms used in batch SMI.

So far, two main approaches to the RSMI problem have been considered. First, some works have proposed adaptations of SMI algorithms in order to update the singular value decomposition (SVD) [START_REF] Verhaegen | A fast recursive MIMO state space model identification algorithm[END_REF][START_REF] Cho | Fast recursive identification of state space models via exploitation displacement structure[END_REF]. Unfortunately, these techniques have the drawback of requiring the disturbances acting on the system output to be spatially and temporally white, which is restrictive in practice. The other approach [START_REF] Gustafsson | Instrumental variable subspace tracking using projection approximation[END_REF][START_REF] Lovera | Recursive subspace identification of linear and non linear Wiener state space models[END_REF][START_REF] Oku | Recursive 4SID algorithms using gradient type subspace tracking[END_REF][START_REF] Mercère | Recursive subspace identification based on instrumental variable unconstrained quadratic optimization[END_REF] relies on the strong analogies between RSMI and signal processing techniques dedicated to direction of arrival (DOA) estimation [START_REF] Krim | Two decades of array signal processing research: the parametric approach[END_REF]. More precisely, two points of view are suggested to find alternatives to the SVD in a recursive framework:

• The first one consists in adapting the so-called Yang's criterion [START_REF] Yang | Projection approximation subspace tracking[END_REF] to the recursive update of the observability matrix [START_REF] Lovera | Recursive subspace identification of linear and non linear Wiener state space models[END_REF][START_REF] Oku | Recursive 4SID algorithms using gradient type subspace tracking[END_REF]. In particular, DOA estimation algorithms have been adjusted in order to deal with more general types of perturbations thanks to the use of instrumental variables (IV).

• The second one rests on the adaptation of another array signal processing technique: the propagator method [START_REF] Munier | Spatial analysis using new properties of the cross spectral matrix[END_REF]. The advantage of this approach over the previous conception lies in the use of a linear operator and quadratic criteria which lead to recursive least squares implementations for the algorithms [START_REF] Mercère | A new recursive method for subspace identification of noisy systems: EIVPM[END_REF][START_REF] Mercère | Recursive subspace identification based on instrumental variable unconstrained quadratic optimization[END_REF][START_REF] Mercère | Sequential correlation based propagator algorithm for recursive subspace identification[END_REF].

While a significant level of maturity has now been reached on the algorithmic side, very limited attention has been dedicated to the analysis of the convergence properties of the developed RSMI techniques. In [START_REF] Oku | Recursive 4SID algorithms using gradient type subspace tracking[END_REF], the proposed gradient-based RSMI method was analysed and conditions on the gain of the gradient iteration were derived. However, the convergence study is based on assumptions on the signal-to-noise ratio which limit the validity of the results.

In the light of the above discussion, the aim of this paper is to analyse the convergence properties of two recursive implementations of the MOESP class [START_REF] Verhaegen | Identification of the deterministic part of MIMO state space models given in innovations form from input output data[END_REF] of subspace identification algorithms. More precisely, the Instrumental Variable Propagator Method (IVPM) and the Extended Instrumental Variable Propagator Method (EIVPM) RSMI techniques, which operate on the basis of the IV version of the propagator method for signal subspace estimation, have been considered and the following results have been derived:

• for the EIVPM algorithm it can be shown that asymptotic convergence can be characterised in terms of the so-called critical relation for the consistency of IV subspace identification algorithms, first derived in the classical paper [START_REF] Jansson | On consistency of subspace methods for system identification[END_REF]. As a consequence, convergence can be guaranteed under suitable persistency of excitation conditions in the absence of process noise;

in a number of special cases (e.g., single input systems, white noise or ARMA input signal) whenever process noise is present.

• for the IVPM algorithm, which can be considered as a special case of EIVPM, a detailed analysis is proposed for single input systems. More precisely, it is possible to show that in this case IVPM converges to consistent estimates under less restrictive persistency of excitation conditions than for EIVPM.

The paper is organised as follows. In Section 2, the system model is introduced, the main notations are defined and the general assumptions are stated. In Section 3 an overview of the considered RSMI algorithms is provided. In particular, the stages necessary to the recursive estimation of the statespace matrices are developed. The analysis of the convergence properties of the considered algorithms is presented in Section 4. Conditions on the input signal and the identified system are more precisely given, which ensure the consistency of the propagator estimates and, by extension, of the extended observability matrix. These convergence properties of the propagator are completed by the expressions of the asymptotic distribution of the estimates. Some simulations are presented in Section 5 to illustrate these theoretical results. Finally, concluding remarks are provided in Section 6.

Problem formulation and notations

Assume that the true system can be described by the discrete-time linear time-invariant state-space model in innovation form

x(t + 1) = Ax(t) + Bu(t) + Ke(t) y(t) = Cx(t) + Du(t) + e(t) (1)
with n y outputs y, n u inputs u, n x states x. e is a white process noise. Furthermore, the following assumptions hold throughout the paper:

1. the innovation process e is a stationary zero mean white process noise with second moments

E e(t)e T (τ ) = R e δ tτ (2)
where δ tτ is the Kronecker delta; 2. the system (1) is asymptotically stable; 2 3. the pair {A, C} is observable and the pair {A, [B K]} is reachable;

4. the input u is a quasi stationary deterministic sequence uncorrelated with the noise e.

To deal with deterministic and stochastic signals in a compact manner, the following operator is defined

Ē [•] = lim N →∞ 1 N N t=1 E [•] ( 3
)
where E is the expectation operator. For two signals a(t) and b(t), the cross covariance matrix will be denoted as R ab = Ē a(t)b T (t) while estimates of signal correlations will be denoted by Rab (t) = t k=1 λ t-k a(k)b T (k), where 0 < λ ≤ 1 is a forgetting factor.

Overview of RSMI algorithms

The algorithms considered in this paper recursively estimate the {A, B, C, D} state space matrices at each new data acquisition. The proposed methods are based on the estimation of a basis for the observability subspace from the input-output (I/O) relation [START_REF] Mercère | Recursive subspace identification based on instrumental variable unconstrained quadratic optimization[END_REF]

y f (t) = Γ f x(t) + H f u f (t) + G f e f (t) = = Γ f x(t) + H f u f (t) + b f (t) (4)
where the stacked vectors y f , u f and e f are defined as

y f (t) = y T (t) • • • y T (t + f -1) T ∈ R nyf ×1 (5)
with f > n x , Γ f is the observability matrix

Γ f = C T (CA) T • • • CA f -1 T T , (6)
H f is the block Toeplitz matrix of the impulse responses from u to y [START_REF] Viberg | Subspace based methods for the identification of linear time invariant systems[END_REF] and b f = G f e f with G f the block Toeplitz matrix of the impulse responses from e to y. The class of techniques considered herein is based on the application of the so-called propagator method [START_REF] Munier | Spatial analysis using new properties of the cross spectral matrix[END_REF] to the recursive estimation of Γ f . To this purpose, note that letting

Y f ∈ R nyf ×N , U f ∈ R nuf ×N and B f ∈ R nyf ×N be the Hankel I/O data matrices defined as Y f ( t) = y f (t) • • • y f (t + N -1) , (7)
with N >> f > n and t = t + N -1, equation ( 4) can be written in matrix form as

Y f ( t) = Γ f X( t) + H f U f ( t) + B f ( t), (8)
where

X( t) = x( t) • • • x( t + N -1) .
As is well known from the offline subspace identification literature, a quantity directly related with the observability subspace can be obtained by computing the projection

Z f of Y f on the kernel of U f Z f ( t) = Y f ( t)Π U ⊥ f ( t) = = (Γ f X( t) + B f ( t)) Π U ⊥ f ( t). ( 9
)
Considering now the time update of

Z f Z f ( t) = Z f ( t -1) z f ( t) , (10)
it is clear that the observation vector z f ( t) will carry all the relevant information for the estimation of the observability subspace contained in the data at time t. Therefore, a two-step procedure for the recursive estimation of the system matrices can be devised:

1. the update of the observation vector z f from the I/O measurements by considering a recursive formulation of the orthogonal projection performed in equation ( 9) (see Subsection 3.1);

2. the estimation of a basis of Γ f from this observation vector by adapting the propagator method [START_REF] Munier | Spatial analysis using new properties of the cross spectral matrix[END_REF] (see Subsection 3.2).

Both stages are considered in the following subsections.

Recursive estimation of the observation vector

The problem of updating the observation vector can be solved by recursively updating the projection (9) at each iteration. Several techniques for the computation of this update have been developed in the literature (see, e.g., [START_REF] Lovera | Recursive subspace identification of linear and non linear Wiener state space models[END_REF][START_REF] Oku | Recursive 4SID algorithms using gradient type subspace tracking[END_REF]). In this paper, an approach based on the matrix inversion lemma [START_REF] Golub | Matrix computations[END_REF] is used. It has indeed the advantage of providing an explicit expression of the observation vector in terms of the I/O data. The idea is to recursively update the quantity

Z f ( t) = Y f ( t)Π U ⊥ f ( t) Z f ( t) = Y f ( t) I -U T f ( t) U f ( t)U T f ( t) -1 U f ( t)
at each new data acquisition, knowing that

U f ( t) = U f ( t -1) u f ( t) , Y f ( t) = Y f ( t -1) y f ( t)
by applying the matrix inversion lemma to

U f U T f -1
. It can be shown that the observation vector z f ( t) can be recursively computed with the following algorithm:

Algorithm 1 Assume that, at time t -1, W f ( t -1) = U f ( t -1)U T f ( t -1) -1 and V f ( t -1) = Y f ( t -1)U T f ( t -1
) have been estimated. Then, when a new I/O data sequence {u f ( t), y f ( t)} is acquired, the observation vector is updated by means of the recursion

β f ( t) = W f ( t -1)u f ( t) (11a) δ f ( t) = u T f ( t)β f ( t) (11b) α f ( t) = 1 1 + δ f ( t) (11c) z f ( t) = α f ( t) (y f ( t) -V f ( t -1)β f ( t)) (11d) V f ( t) = V f ( t -1) + y f ( t)u T f ( t) (11e) W f ( t) = W f ( t -1) -α f ( t)β f ( t)β T f ( t). ( 11f
)
Remark 2 It is easy to show that the observation vector z f ( t) can be equivalently written as

z f ( t) = α f ( t) Γ f x( t) + bf ( t) , (12)
where

bf ( t) = b f ( t) -B f ( t -1)U T f ( t -1)β f ( t) (13) x( t) = x( t) -X( t -1)U T f ( t -1)β f ( t). ( 14
)
Therefore, it is possible to apply the propagator method to the subspace identification problem by exploiting the analogy between [START_REF] Mercère | Sequential correlation based propagator algorithm for recursive subspace identification[END_REF] and the data generation model used in array signal processing problems [START_REF] Krim | Two decades of array signal processing research: the parametric approach[END_REF]

z(t) = Γ(θ)s(t) + b(t) ( 15
)
where z is the output of the n z sensors of the antenna array, Γ(θ) the steering matrix, s the vector of the n s signal waveforms and b the additive noise.

Remark 3 In the following, the scaling factor α f (see [START_REF] Mercère | Sequential correlation based propagator algorithm for recursive subspace identification[END_REF]) will be neglected for simplicity

z f ( t) = Γ f x( t) + bf ( t). ( 16
)
Note that this simplification does not affect the properties of the algorithm since [START_REF] Verhaegen | A fast recursive MIMO state space model identification algorithm[END_REF] provides all the information needed to estimate span col {Γ f }.

Recursive update of the observability matrix

Once the observation vector is estimated, the second step of the recursive subspace identification procedure consists in the online update of the observability matrix. A number of methods have been developed in the literature in order to avoid the computation of a complete SVD in this step of recursive subspace identification [START_REF] Lovera | Recursive subspace identification of linear and non linear Wiener state space models[END_REF][START_REF] Oku | Recursive 4SID algorithms using gradient type subspace tracking[END_REF][START_REF] Mercère | Recursive subspace identification based on instrumental variable unconstrained quadratic optimization[END_REF]. In this paper, the focus is on algorithms based on the propagator concept [START_REF] Munier | Spatial analysis using new properties of the cross spectral matrix[END_REF].

Under assumption 3, since Γ f ∈ R nyf ×nx with n y f > n x , the extended observability matrix has at least n x linearly independent rows, which can be gathered in a submatrix Γ f1 . Then, the complement Γ f2 of Γ f1 can be expressed as a linear combination of these n x rows. So, there is a unique linear operator P f ∈ R nx×(nyf -nx) , named propagator [START_REF] Munier | Spatial analysis using new properties of the cross spectral matrix[END_REF], such that

Γ f2 = P T f Γ f1 . (17)
Furthermore, it is easy to verify that

Γ f = Γ f1 Γ f2 = Γ f1 P T f Γ f1 = I nx P T f Γ f1 . (18)
Thus, since rank

{Γ f1 } = n x , span col {Γ f } = span col I nx P T f . (19)
Equation ( 19) implies that it is possible to estimate the observability matrix (in a particular basis) by estimating the propagator. This operator can be estimated from [START_REF] Verhaegen | A fast recursive MIMO state space model identification algorithm[END_REF]. Indeed, applying a data reorganization to the observation vector so that the first n x rows of Γ f are linearly independent, ( 16) can be partitioned as

z f (t) = z f1 (t) z f2 (t) = I nx P T f Γ f1 x(t) + bf1 (t) bf2 (t) (20)
where z f1 ∈ R nx×1 and z f2 ∈ R (nyf -nx)×1 are the components of z f respectively corresponding to Γ f1 and Γ f2 . In the noise free case, it is easy to show that

z f2 = P T f z f1 . (21)
In the presence of noise, this relation no longer holds. However, by assuming we have collected (or are about to collect) N I/O measurements, an unbiased estimate of the propagator P f can be obtained by introducing an instrumental variable ξ ∈ R n ξ ×1 , assumed to be uncorrelated with the noise but sufficiently correlated with the state vector x, in the sense that

R e f ξ = 0 and rank {R xξ } = n x , (22)
and by defining the cost function (see [START_REF] Gustafsson | Instrumental variable subspace tracking using projection approximation[END_REF])

J IV (P f ) = Rz f 2 ξ (N ) -P T f Rz f 1 ξ (N ) 2 F . ( 23
)
Four algorithms (IVPM, EIVPM, EIVsqrtPM and COIVPM [START_REF] Mercère | A new recursive method for subspace identification of noisy systems: EIVPM[END_REF][START_REF] Mercère | Sequential correlation based propagator algorithm for recursive subspace identification[END_REF]) have been developed to minimise this criterion according to the number of instruments in ξ. In the following the IVPM and EIVPM algorithms will be described in detail. However it should be noticed that, along the same lines, the asymptotic properties of the other two algorithms can be worked out (the details are omitted for brevity). In fact, the EIVsqrtPM and the COIVPM algorithms differ from the IVPM and EIVPM only in an algorithmic way and are based on the same least squares estimate [START_REF] Mercère | Sequential correlation based propagator algorithm for recursive subspace identification[END_REF].

The IVPM algorithm

Assuming that it is possible to build an instrumental variable such that n ξ = n x , the least squares estimate of the propagator is given by

PT f (N ) = Rz f 2 ξ (N ) R-1 z f 1 ξ (N ) (24)
provided the indicated inverse exists (see Section 4 for invertibility conditions). Then, an RLS recursive version of (24), named IVPM [START_REF] Mercère | Recursive subspace identification based on instrumental variable unconstrained quadratic optimization[END_REF], can be obtained by applying the matrix inversion lemma [START_REF] Golub | Matrix computations[END_REF] to

R z f 1 ξ K f (t) = ξ T (t)L f (t -1) λ + ξ T (t)L f (t -1)z f1 (t) (25a) L f (t) = 1 λ (L f (t -1) -L f (t -1)z f1 (t)K f (t)) (25b) PT f (t) = PT f (t -1)+ + z f2 (t) -PT f (t -1)z f1 (t) K f (t), (25c)
where L f (t) = R-1 z f 1 ξ (t) and 0 < λ ≤ 1 is a forgetting factor.

The EIVPM algorithm

In many cases, it is difficult or impossible to construct an instrumental variable such as n ξ = n x . Thus, it is necessary to increase the number of instruments. By assuming that the input is sufficiently "rich" (see Section 4) so that the (rectangular) matrix Rz f 1 ξ (N ) is full rank, the least squares estimate of the propagator is given by PT

f (N ) = Rz f 2 ξ (N ) R † z f 1 ξ (N ) (26) with R † z f 1 ξ (N ) = RT z f 1 ξ (N ) Rz f 1 ξ (N ) RT z f 1 ξ (N ) -1
. Then, a recursive version of (26), named EIVPM [START_REF] Mercère | Recursive subspace identification based on instrumental variable unconstrained quadratic optimization[END_REF], can be obtained by adapting the overdetermined instrumental variable technique first proposed in [START_REF] Friedlander | The overdetermined recursive instrumental variable method[END_REF]

g f (t) = Rz f 2 ξ (t)ξ(t) z f2 (t) (27a)
Λ(t) = -ξ T (t)ξ(t) λ λ 0 (27b) Ψ f (t) = Rz f 1 ξ (t -1)ξ(t) z f1 (t) (27c) K f (t) = Λ(t) + Ψ T f (t)L f (t -1)Ψ f (t) -1 Ψ T f (t)L f (t -1) (27d)
P T f (t) = P T f (t -1)+ + g f (t) -P T f (t -1)Ψ f (t) K f (t) (27e) Rz f 1 ξ (t) = λ Rz f 1 ξ (t -1) + z f1 (t)ξ T (t) (27f) Rz f 2 ξ (t) = λ Rz f 2 ξ (t -1) + z f2 (t)ξ T (t) (27g) L f (t) = 1 λ 2 (L f (t -1) -L f (t -1)Ψ(t)K f (t)) (27h) with L f (t) = Rz f 1 ξ (t) RT z f 1 ξ (t) -1
.

Remark 4 The computational complexity of EIVPM and IVPM has been analysed in [START_REF] Mercère | Recursive subspace identification based on instrumental variable unconstrained quadratic optimization[END_REF], where a comparison with other RSMI algorithms based on PAST [START_REF] Yang | Projection approximation subspace tracking[END_REF] and EIVPAST [START_REF] Gustafsson | Instrumental variable subspace tracking using projection approximation[END_REF] has been presented. It was more precisely shown that the overall computational cost of PM approaches is comparable to the one of PAST and its by-products.

It is interesting to note that, from the second iteration on, the update of the estimated subspace is always in the form Γf = I nx Pf T . This means that, after a short transient period, the recursive estimation is made in the same state-space basis. This loss of flexibility can be firstly considered as a drawback, more particularly from a numerical point of view. Indeed, the fact that the first n x rows of Γf are equal to the identity matrix could lead to conditioning problems when the proposed algorithms are applied to ill conditioned systems. However, we have not encountered any numerical problems when simulating the algorithm in Matlab. Furthermore, contrary to the PAST approach where the signal subspace is obtained by minimising a modified function obtained after an approximation, the PM cost functions are always computed without approximations nor constraints. More particularly, in the PAST framework, the minimization of approximated criterion leads to a matrix having columns that are not exactly orthonormal. This property evolves during the recursive minimization since, under some conditions, the minimizer converges to a matrix with orthonormal columns [START_REF] Yang | Projection approximation subspace tracking[END_REF]. This evolution can be interpreted as a slow change of basis, which implies that it is not possible to guarantee that Γ f (t) and Γ f (t -1) are expressed in the same state-space coordinates. This might represent a problem in the estimation of the state-space realization Â, B, Ĉ and D, and particularly so whenever RSMI techniques are used for change detection purposesThe PM algorithms do not suffer from this drawback.

Convergence analysis

In this Section, the convergence conditions and the asymptotic distribution of the estimation error of the EIVPM and IVPM algorithms are derived. More precisely, conditions will be established by which the considered recursive algorithms converge to consistent estimates of the propagator P f .

Convergence analysis of EIVPM

Consider the extended instrumental variable estimate (26) and assume that λ = 1. Then

PT f (t)R(t) = 1 t Rz f 2 ξ (t) 1 t RT z f 1 ξ (t) (28)
with

R(t) = 1 t Rz f 1 ξ (t) 1 t RT z f 1 ξ (t)
. Note that the right hand side of (28) can be equivalently written as

PT f (t)R(t) = 1 t t τ =1 z f2 (τ )ξ T (τ ) 1 t RT z f 1 ξ . (29) Now, since z f1 (t) z f2 (t) = I nx P T f Γ f1 x(t) + bf1 (t) bf2 (t) , (30)
it is possible to write z f2 in terms of the true

P T f z f2 (t) = P T f z f1 (t) + bf2 (t) -P T f bf1 (t) . (31)
Letting η(t) = bf2 (t) -P T f bf1 (t) and introducing equation (31) in the relation (29), we get

PT f (t)R(t) = P T f 1 t Rz f 1 ξ (t) 1 t RT z f 1 ξ (t) + 1 t Rηξ (t) 1 t RT z f 1 ξ (t) (32)
and finally, from (28) and (32) we have

PT f (t) -P T f R(t) = 1 t Rηξ (t) 1 t RT z f 1 ξ (t). ( 33
)
Along the lines of [8, Chapter 4], the convergence analysis of EIVPM is based on the analysis of (33), i.e.,

1. proving that PT f (t) -P T f R(t) → 0 with probability 1 (w.p.1) as t → ∞;

2. deriving conditions under which R(t) converges to a constant matrix R as t → ∞, with R full rank.

These two steps are considered in the following subsections.

Convergence of PT

f (t) -P T f R(t)
From (33), it is easy to establish the following proposition:

Proposition 5 Under assumptions 1-4, consider algorithm (27) and further assume that ξ ∈ R n ξ ×1 (n ξ ≥ n x ) is uncorrelated with the noise but sufficiently correlated with the state vector x, i.e., (22) holds.

Then PT f (t) -P T f R(t) → 0 w.p.1 as t → ∞.

Proof 6 This result can be proved by showing that the quantity 1 t Rηξ (t), which appears in the right hand side of equation (33), converges to 0. To this purpose, consider the definition of bf in equation ( 13) and compute the correlation

1 t t τ =1 bf (τ )ξ T (τ ) = 1 t t τ =1 G f e f (τ ) -E f (τ -1)U T f (τ -1)β f (τ ) ξ T (τ ) ( 34
)
and note that the above quantity tends to (see, e.g., [START_REF] Liu | Identification of multi input and multi output systems by observability range space extraction[END_REF])

G f R e f ξ -R e f u f R -1 u f R u f ξ (35)
w.p.1 when t → ∞. Furthermore, according to the assumptions on ξ and u, it is straightforward that R e f u f = 0 and R e f ξ = 0. Thus, we have

1 t t τ =1 bf (τ )ξ T (τ ) → 0 w.p.1 as t → ∞ (36)
and recalling that η(t) = bf2 (t) -P T f bf1 (t) we have

1 t t τ =1 η(τ )ξ T (τ ) → 0 w.p.1 as t → ∞, (37)
from which, in view of (33), the thesis follows.

Convergence of R(t)

In order to complete the convergence analysis, we need to study under which conditions R(t) converges to a full rank matrix in order to conclude that PT f (t) tends to P T f w.p.1 as t tends to infinity. To this purpose, it is sufficient to analyse the rank of Rz f 1 ξ (t) as t tends to infinity by construction of R(t). Indeed, if conditions are fulfilled such as Rz f 1 ξ (t) is full rank, R(t) will satisfy the same property and will be invertible. For that, note first of all that in the propagator basis (see (30)) the first n x rows of (16) (i.e., the z f1 part) are given by

z f1 (t) = x(t) + bf1 (t). ( 38
)
Then, recalling the definition of x given in ( 14), we have that

1 t Rz f 1 ξ (t) = 1 t t τ =1 bf1 (τ )ξ T (τ ) + 1 t t τ =1 x(τ ) -X(τ -1)U T f (τ -1)β f (τ ) ξ T (τ ) . ( 39
)
Since (36) holds, the first term of the right hand side of (39) can be neglected. Therefore, we have that

1 t Rz f 1 ξ (t) → Rz f 1 ξ = R xξ -R xu f R -1 u f R u f ξ (40)
w.p.1 when t tends to infinity. Noting that matrix

R xξ -R xu f R -1 u f R u f ξ is the Schur complement of the block R u f in matrix R xξ R xu f R u f ξ R u f , a sufficient condition that guarantees that Rz f 1 ξ is full rank is given by rank Ē x(t) u f (t) ξ(t) u f (t) T = n x + f n u . ( 41
)
Note that the above condition does not depend on the choice of the instrumental variable. Thus, the invertibility of R is ensured for any choice of instrumental variable such that (41) holds. Therefore, (41) is a sufficient condition that guarantees that R(t) converges to a constant, full rank matrix R, as t → ∞.

If the instrumental variable vector is constructed as

ξ(t) = y T p (t) u T p (t) T = [ y T (t-p) ••• y T (t-1) u T (t-p) ••• u T (t-1) ] T , ( 42
)
where parameter p defines the horizon over which past input and output data are taken to form the IV vector, then equation ( 41) corresponds to the so-called critical relation for the consistency of IV subspace identification algorithms, first derived in the classical paper [START_REF] Jansson | On consistency of subspace methods for system identification[END_REF]. In particular, conditions under which (41) holds have been derived in the cited paper and lead to the following result for the convergence of the EIVPM algorithm.

Proposition 7 Under assumptions 1-4 (see Section 2), consider the EIVPM algorithm (27) and further assume that:

• K = 0 ( i.e., no process noise);

• the instruments are composed by past input and output data with p ≥ n x (see (42));

• the input u is persistently exciting of order f + p + n x ;

• the forgetting factor λ is chosen equal to 1.

Then PT f (t) → P T f w.p.1 as t → ∞

In addition, convergence can also be guaranteed if K = 0, for example1

Proposition 8 Under assumptions 1-4 (see Section 2), consider the EIVPM algorithm (27) and further assume that:

• the instruments are composed by past input and output samples with p ≥ n x (see (42));

• the input u is a zero mean white sequence;

• rank {C p {A, B} C p {A, G}} = n x with C p {L, M} = L p-1 M • • • LM M (43) G = Ē x(t + 1)y T (t) ; (44)
• the forgetting factor λ is chosen equal to 1.

Then PT f (t) → P T f w.p.1 as t → ∞ Other special cases (e.g., single input systems, ARMA input signal) can be analysed by exploiting the results in [START_REF] Jansson | On consistency of subspace methods for system identification[END_REF], and are omitted for brevity.

Convergence analysis of IVPM

In the special case of the IVPM algorithm, the previous analysis can be further developed in order to derive more stringent convergence conditions. For that, note first of all that, assuming λ = 1, the quadratic criterion (23) is minimised by PT

f (t) such that PT f (t)R(t) = 1 t Rz f 2 ξ (t) (45)
where now R(t) = 1 t Rz f 1 ξ (t). By following the same steps as in paragraph 4.1.1, it is straightforward to prove that PT

f (t) -P T f R(t) = 1 t Rηξ (t) (46)
and that 1 t Rηξ (t) → 0 w.p.1 as t → ∞ when the input u is uncorrelated with the innovation e (system in open loop) and when the instrumental variable ξ ∈ R nx×1 is uncorrelated with the noise but sufficiently correlated with the state vector x. Thus, the following proposition is verified: Proposition 9 Under assumptions 1-4 (see Section 2), consider algorithm (25) and further assume that ξ ∈ R nx×1 is uncorrelated with the noise but sufficiently correlated with the state vector x.

Then PT f (t) -P T f R(t) → 0 w.p.1 as t → ∞.
The second step of the convergence analysis of IVPM is, just as before, the derivation of conditions under which R(t) converges to a full rank matrix. More precisely, it is possible to exploit the conclusions obtained in analysing EIVPM in order to conclude that

R(t) → R = R xξ -R xu f R -1 u f R u f ξ (47)
w.p.1 when t tends to infinity. Indeed, the sufficient condition (41) insures, as previously, that R(t) converges to a full rank matrix. However, in the IVPM case, a constraint has to be satisfied:

n ξ = n x .
To study the influence of this condition in a widespread situation, consider that the instruments are chosen as past inputs, i.e.,

ξ(t) = u p (t) = u T (t -p) • • • u T (t -1) T (48)
with n u p = n x . Thus, we get

R(t) → R = R xup -R xu f R -1 u f R u f up (49)
w.p.1 when t tends to infinity. As in paragraph 4.1.2, the application of the Schur complement leads to a sufficient condition for the invertibility of R:

R xup R xu f R u f up R u f = Ē x(t) u f (t) u p (t) u f (t) T > 0. (50)
In order to check if (50) holds, we proceed along the lines of the analysis of the IV-4SID algorithm performed in [START_REF] Jansson | On consistency of subspace methods for system identification[END_REF]. To this purpose, the state vector is decomposed into its deterministic and its stochastic parts

x(t) = x d (t) + x s (t) (51)
where x d is due to the observed inputs u

x d (t) = F u (q -1 )u(t) (52)
and x s is caused by the innovation e x s (t) = F e (q -1 )e(t). ( 53)

Substituting ( 51)-( 53) in (50) we have

Ē x(t) u f (t) u p (t) u f (t) T = Ē x d (t) u f (t) u p (t) u f (t) T ( 54
)
since u is assumed to be uncorrelated with the innovation. Now, let

F u (q -1 ) = N F u (q -1 ) D F u (q -1 ) (55) with N F u (q -1 ) = N F u 1 q -1 + • • • + N F u nx q -nx and D F u (q -1 ) = D F u 0 + • • • + D F u nx q -nx .
Then, the deterministic state x d can be easily related to the input and the state-space matrices as follows

x d (t) = 1 D F u (q -1 ) N F u nx • • • N F u 1    u(t -n x ) . . . u(t -1)    . (56)
In the single input case, equation ( 54) can be equivalently written as

Ē x(t) u f (t) u p (t) u f (t) T = M 1 D F u (q -1 ) Ē ū(t)ū T (t) (57) with ū(t) = u(t -n x ) • • • u(t + f -1) T and M =      N F u nx • • • N F u 1 0 • • • 0 D F u nx • • • D F u 1 D F u 0 0 . . . . . . 0 D F u nx • • • D F u 1 D F u 0      . ( 58
)
It is possible to show that M is full rank if the system is reachable (see [5, Appendix A] for details). By assuming this condition is fulfilled, the matrix in (57) is positive definite if u is persistently exciting of order f + n x . Therefore R in ( 49) is also nonsingular and PT f (t) → P T f w.p.1 as t → ∞. This result is summed up in the following proposition: Proposition 10 Under assumptions 1-4 (see Section 2), consider the IVPM algorithm (25) and further assume that:

• the system is single input;

• the input is persistently exciting of order f + n x ;

• the forgetting factor λ is chosen equal to 1.

Then PT f (t) → P T f w.p.1 as t → ∞.

Asymptotic distribution

Once it is proved that the IVPM and EIVPM algorithms lead to consistent estimates, the following phase consists in expressing the asymptotic distribution of the projector estimates, which can provide a useful tool for the assessment of the accuracy of the computed estimates. For that, we first analyse the distribution of the EIVPM estimate. IVPM will follow as a special case.

The probability distribution of the EIVPM estimate is characterised by the following proposition:

Proposition 11 Consider the EIVPM algorithm (27). Then, under the assumptions of Proposition 7, the propagator estimate is asymptotically normal:

√ tvec PT f (t) -P T f ∈ AsN   0, RT z f 1 ξ R-1 ⊗ I    Q 1 0 . . . 0 Q n ξ       (59) with Rz f 1 ξ = R xξ -R xu f R -1 u f R u f ξ , R = Rz f 1 ξ RT z f 1 ξ and Q i = lim t→∞ 1 t E   t j=1 η(j)ξ T i (j) t k=1 ξ i (k)η T (k)   . ( 60
)
Proof 12 From (33), it follows that

PT f (t) -P T f 1 t Rz f 1 ξ (t) 1 t RT z f 1 ξ (t) = 1 t Rηξ (t) 1 t RT z f 1 ξ (t). ( 61
)
According to paragraph 4.1.2, we have (see (40))

1 t Rz f 1 ξ (t) → Rz f 1 ξ = R xξ -R xu f R -1 u f R u f ξ w.p.1 as t → ∞ (62)
Thus, by vectorizing PT f (t) -P T f , knowing that [START_REF] Golub | Matrix computations[END_REF] vec {ABC} = C T ⊗ A vec {B} where ⊗ is the Kronecker product, we get

√ tvec PT f (t) -P T f = RT z f 1 ξ R-1 ⊗ I vec 1 √ t t τ =1 η(τ )ξ T (τ ) (63)
Now, from the central limit theorem we know that, by writing the i -th component of ξ as ξ i , for all i

1 √ t t τ =1 η(τ )ξ T i (τ ) ∈ AsN(0, Q i ), (64)
with Q i defined as in (60). Thus, it follows that

vec 1 √ t t τ =1 η(τ )ξ T (τ ) ∈ AsN   0,    Q 1 0 . . . 0 Q n ξ       . (65)
(59) follows by combining (63) and (65).

In the same way, it is straightforward to prove that:

Proposition 13 Consider the IVPM algorithm (25); then, under the assumptions of Proposition 10, the propagator estimate is asymptotically normal:

√ tvec PT f (t) -P T f ∈ AsN   0, R-1 ⊗ I    Q 1 0 . . . 0 Q nx       (66) with R = R xup -R xu f R -1 u f R u f up .

Simulation example

Consider the following fourth-order system:

x(t + 1) =

y(t) = [ -0.5749 1.0751 -0.5225 0.183 ] x(t) (68)
-0.7139 u(t) + 0.9706 e(t).

This SISO process has been chosen in order to apply at the same time the IVPM and EIVPM algorithms.

The input u and the innovation e are white Gaussian noises with zero mean and variance 1 and 9 respectively. This leads to a signal to noise ratio (of variances) at the output (cov[y]/cov[e]) of 4.4. The initial estimates of the system matrices are randomly generated under the constraint that the absolute value of the maximum eigenvalue of Â(0) is less than 1 (stability requirement). The forgetting factor is fixed at 1 in order to meet the assumptions of the convergence study. The instrumental variable vector is constructed using past values of the input u. All these conditions allow to satisfy the constraints of Propositions 8 and 10. The results presented in this Section have been obtained by repeating the identification procedure 100 times using different noise realizations for e.

The first result to be verified is the asymptotic convergence of the matrix R(t) to a full rank matrix; for the considered example, R(t) converges to a matrix with a condition number of about 3 (σ max ≃ 30 and σ min ≃ 3).

In order to underline the efficiency of the instrumental variable propagator based methods, it is next proposed to compare the accuracy of the estimated poles obtained with IVPM and EIVPM and two other subspace identification techniques: the recursive EIVPAST algorithm [START_REF] Gustafsson | Instrumental variable subspace tracking using projection approximation[END_REF][START_REF] Lovera | Recursive subspace identification of linear and non linear Wiener state space models[END_REF] and the batch PI MOESP algorithm [START_REF] Verhaegen | Identification of the deterministic part of MIMO state space models given in innovations form from input output data[END_REF]. More precisely, the eigenvalues of the estimated A matrices during the Monte Carlo simulation are plotted and compared with the true ones (i.e., ±0.67 ± 0.67). 300 samples are used for each realization. The final model, i.e., Â(300), B(300), Ĉ(300), D(300) is considered for the recursive methods. The parameters are chosen as follows: f = p = 4 for IVPM and f = p = 8 for the other algorithms. Figure 1 shows that IVPM and EIVPM produce eigenvalue estimates which are equally accurate as the ones obtained with the batch PI MOESP algorithm. On the contrary, the EIVPAST method yields several poorly accurate estimates. It can be noted that the variance of the eigenvalue estimates of IVPM is slightly larger than the EIVPM one. Furthermore, in practice, it seems to be more convenient to use EIVPM instead of IVPM since the first one only needs an upper bound of the system order for the instrumental variable construction, which is not the case with IVPM.

Finally, the averaged time histories of the real parts of the estimated eigenvalues obtained with EIVPM and EIVPAST techniques have been analysed. These averaging results (omitted for brevity) illustrate the fact that the transient behaviour of EIVPAST method is relatively more chaotic than the EIVPM one. This could be explained by the fact that the projection approximation which characterises PAST-based algorithms might lead to a loss of accuracy during transients.

Concluding remarks

In this paper, the convergence properties of a class of recursive subspace identification algorithms have been investigated. More precisely, online implementations of the MOESP methods based on instrumental variable versions of the propagator technique for signal subspace estimation have been analysed, convergence proofs for the IVPM and EIVPM algorithms have been derived and conditions on the input signal have been given which guarantee consistent estimates of the observability matrix. A simulation example has been used to illustrate the validity of the underlying assumptions and to assess the performance of the algorithms.

Figure 1 :

 1 Figure 1: Estimated poles obtained with IVPM, EIVPM, EIVPAST and PI MOESP algorithms for a Monte Carlo simulation of size 100.

This particular case will be considered in the numerical experiments.