Siún Nic Mhuirí
email: siun.nicmhuiri@dcu.ie

A lens on two classrooms: Implications for research on teaching

Keywords: Research frameworks, culture, international, mathematics teaching

 as an analytic lens on episodes of mathematics teaching from two different countries. This lens highlights differences in teaching approaches across the two settings and draws attention to the need for further interrogation of how culture, curriculum and values inform teaching practices. It also has implications for research practices as it shows that the theoretical frameworks and methodological tools that are used in research are not value free or culturally neutral.

Introduction

My motivation in writing this paper is to address some of the questions that were raised in TWG19 at CERME10. I hope to contribute to discussions on methods used to conduct research on mathematics teaching. I have a particular interest in how research frameworks are used by different communities. At CERME10, I presented a view of research frameworks as theoretical frameworks or methodological tools which 'frame' or structure a coherent set of understandings about a theme [START_REF] Nic Mhuirí | Considering research frameworks as a tool for reflection on practices: Grainsize and levels of change[END_REF]. In this paper, I use the Teaching for Robust Understanding (TRU) framework [START_REF] Schoenfeld | Classroom observations in theory and practice[END_REF] to analyze segments of mathematics lessons. I raise questions about how culture, curriculum and values inform teaching practices and how research frameworks reflect this (or not). I conclude by considering the implications for research practice more generally.

Theoretical Framework

I situate my work in the sociocultural perspective where learning is conceived as transformation of participation in social practices (e.g., [START_REF] Lerman | Cultural, discursive psychology: A sociocultural approach to studying the teaching and learning of mathematics[END_REF]. From this perspective, a classroom community can be understood as a community of practice (CoP) [START_REF] Wenger | Communities of practice: Learning, meaning, and identity[END_REF]. The CoP theory developed from research on apprenticeship contexts. Critics argue that it does not offer a theoretical base for formal teaching where teachers are accountable for learning outcomes (e.g., [START_REF] Goos | Developing a communal identity as beginning teachers of mathematics: Emergence of an online community of practice[END_REF]. Attention to teacher as agent for educational and social change [START_REF] Cochran-Smith | Inquiry as stance: Practitioner research for the next generation[END_REF] is necessary to envisage how new practices might come to be established within the community. Notwithstanding student agency and the influence of the wider sociocultural context, the teacher is assumed to have agency, and generally some authority, in choosing actions which shape the practices of classrooms, i.e., "the repeated actions in which students and teachers engage as they learn" (Boaler, 2002, p. 113). However, teaching is more than simply a collection of practices. Following [START_REF] Biesta | Thinking philosophically about teaching[END_REF], I recognize teaching as relational, intentional and purposeful. Firstly, teaching implies a relationship between the person teaching and the one being taught. The teacher also has a role to play in 'relationally bridging' student and subject [START_REF] Grootenboer | Identity as a lens to understand learning mathematics: Developing a model[END_REF]. While learning may occur, in the absence of teaching, teaching is considered to be intentional as teachers deliberately aim to teach their students. Furthermore, education is a "teleological practice where the question of what education is for can never be evaded" (Biesta & Stengel, 2016, p. 33, original emphasis).

Biesta and Stengel contrast 'purpose' with the more concrete 'aims' which a teacher might endeavor to achieve. They describe the purpose of education as concerned with justification for engaging in teaching and consider 'purpose' to be normative and indicative of what is 'educationally desirable ' (2016, p. 31). They identify three important domains of educational purpose: qualification; socialization; and subjectification. Qualification is understood as connected with "the transmission and acquisition of knowledge and skills" and socialization is understood as "the way in which through education we become part of existing cultures and traditions and form our identity" (Biesta & Stengel, 2016, p. 26). Subjectification is "an educational orientation concerned with the ways in which human beings can be subjects in their own right, rather than objects of the actions and activities of others (2016, p. 21). It is difficult to argue with [START_REF] Mosvold | Mathematical knowledge for teaching and the teaching of mathematics[END_REF] contention that "while there are other important aims of education, teaching is centrally about supporting the learning of subject matter" (p. 3111). However, Biesta and Stengel maintain that such aims should be articulated in relation to the domains of educational purpose.

The view of education proposed by Biesta and Stengel centralizes teacher agency as it highlights the role of teacher judgment. In every activity, teachers make (tacit) judgments about the balance between the three domains of educational purpose. This has connections with [START_REF] Schoenfeld | How we think: A theory of human decision-making, with a focus on teaching[END_REF] theory of teachers' decision-making. He maintains that for a well-practiced activity like teaching, decision making is a function of teachers' knowledge, resources, goals, beliefs and orientations. His model of in-the-moment decision making offers a fine-grained lens on teacher judgments. By situating such judgments in the broader field of the three domains of educational purpose, Biesta and Stengel's philosophy facilitates consideration of the bigger picture. It raises questions about how society, culture and curriculum shape teachers' orientations and how beliefs about educational purpose influence everyday decision-making.

Methodology

The data was collected by international researchers for different purposes. It was shared amongst TWG19 members who had expressed an interest in engaging in the analysis of common data at CERME10. Four pieces of data were shared, three of which contained video clips. I focused only on the video data which consisted of episodes from an American, a Greek, and a Norwegian classroom. I will use the TRU Framework [START_REF] Schoenfeld | Classroom observations in theory and practice[END_REF] for analysis. This framework consists of five dimensions: the mathematics; cognitive demand; access to content; agency, authority and identity; uses of assessment. The mathematics involves the disciplinary concepts and practices made available for learning. Cognitive demand aims to capture the extent to which students have opportunities to engage in 'productive struggle'. Access to content addresses the extent to which activity structures support the active engagement of all students. Agency, authority and identity refers to the extent to which students have opportunities to instigate and contribute to discussions in ways that contribute to their agency, mathematical authority and to the development of positive identities. Uses of assessment relates to how classroom activities elicit student thinking and subsequent interactions respond to those ideas. [START_REF] Schoenfeld | Classroom observations in theory and practice[END_REF] describes his goals in the creation of the framework as being concerned with identifying a relatively small but 'complete' number of categories of classroom activities for observation, i.e., no other categories or dimensions are thought to be necessary for analysis. This claim of 'completeness' was the main reason for choosing to use the framework as a methodological tool. The TRU framework can be used for evaluation purposes but I aim only to highlight important issues and to interrogate the process of analysis itself.

The TRU approach to classroom observations separates or parses lesson by the nature of the activity structure that occurs: Whole class activities (including topic launch, teacher exposition, and whole class discussion); small group work; student presentations; and individual student work. Each episode should be relatively short but 'phenomenologically coherent ' (2013, p. 617). The TRU framework contains detailed rubrics for all five dimensions across each of these activity structures. It might be considered that these activity structures are relatively unambiguous and should be easily recognized across international classrooms, e.g., individual work or small group work. However, issues of curriculum and culture should not be ignored [START_REF] Andrews | Finnish mathematics teaching: a case of uniquely implicit didactics[END_REF]. For example, the activity structure 'topic launch' would seem to have strong connections to the 'Launch-Explore-Summarize Teaching Model' used in the US based Connected Mathematics Project Curriculum (https://connectedmath.msu.edu/). The extent to which this model of instruction is recommended by, or embedded in, other jurisdictions internationally is questionable. It is also possible that other important activity structures exist locally that are not captured in the TRU framework (c.f., [START_REF] Clarke | Addressing the challenge of legitimate international comparisons of classroom practice[END_REF]. Despite these concerns, it was possible to categorize the data using the TRU activity structures so it was decided that it was appropriate to proceed with analysis.

To some extent the data was 'pre-parsed' as only selected elements of the lessons were shared. There were a variety of activity structures in evidence across the different classroom. The activities in the Greek classroom were conducted in a whole class setting with some elements of teacher exposition and whole class discussion. The activity structures in the Norwegian classroom involved a whole class topic launch, small group work/individual work and a series of student presentations. The activity in the US classroom centered on a single student presentation. Given the constraints of this paper, I decided to focus on the student presentations across the US and the Norwegian data. An overview of the data is shown in table 1.

US Data

Norwegian Data Summer program for 5 th graders.

Majority low SES participants.

Experienced teacher recognized as expert (Professor Deborah Ball, University of Michigan)

Video clip (c. 3 minutes) and contextual information.

Small group (5 students) primary teaching.

Experienced teacher recognized locally as expert

Video clip (c. 21 minutes) with English subtitles and transcription.

Overview of video content

In the US video, a student presents a solution to a task concerning what fraction is marked on a number line. She gives the incorrect answer of 1/7. Her justification is that there are seven equal parts shown on the number line. The teacher invites students to question her reasoning.

In the Norwegian example, the teacher gives the students the task of figuring out what year the King was born. In effect, students have to compute 2017 -80. After about 5 minutes, the teacher asks students to present their solution methods. These are summarized below.

 Presentation 1: 80 -17 = 63, 2000 -63 = 1937. The calculations were carried out using the standard algorithm even though the procedure has not yet been taught for higher number ranges. The teacher models the students' thinking on an empty number line and also models the algorithm.  Presentation 2: 2000 -80 = 1920; 1920 + 17 = 1937. The calculations were done mentally but recorded in vertical format at the board.  Presentation 3: 2017 -80 = 1937 using standard algorithm.  Presentation 4: 1900 Presentation 4: + 20 = 1920;;1920+ 17 = 1937. Teacher questioning appears to aim to expose reasoning.  Presentation 5: The first student complains that she did not have a chance to write her method (standard algorithm) on the board. Teacher invites her to do this.

Results

Due to space limitations the full TRU framework rubric for student presentations is not presented here. Instead, under each heading, I give a brief overview of all levels with a full description of the most relevant levels (in italics). I then present an analysis of relevant classroom events.

The Mathematics

Level 1 (answers without reasons) and Level 2 (procedural mathematics with no expectation of reasoning) do not apply in either case. Level 3 is described as follows: The Mathematics presented is relatively clear and correct AND either includes justifications and explanations OR the teacher encourages students to focus on central mathematical ideas and explaining and justifying them.

In the US classroom, explanations were given by the presenter Aniyah but the mathematics was incorrect. The teacher encouraged other students to ask questions but not to evaluate (i.e., agree or disagree with) her solution. Toni questioned why she chose one-seventh. Lakeya questioned her choice of 1 for the numerator. Dante's question is unclear but may also be targeting the numerator. The focus on justification and key mathematical ideas would place this at level 3 of the rubric.

In the Norwegian classroom, the mathematical reasoning was correct and the explanations were generally clear (level 3). Where this was not clear (e.g., presentation 4), the teacher asked clarifying questions. In presentation 5, the student inverted the numbers when writing the standard subtraction algorithm. She appeared to recognize her own error and the teacher said he understood her thinking. He appeared to value reasoning above procedures. All presentations can be described by level 3.

Cognitive Demand

Level 1 (familiar facts and procedures) does not apply to either case. Level 2: Presentation offers possibilities of conceptual richness or problem solving challenge, but teaching interactions tend to 'scaffold away' these possibilities, resulting in a straightforward or familiar focus on facts and procedures. Level 3: The teacher's hints or scaffolds support presenters and/or class in 'productive struggle' in building understanding and engaging in mathematical practices.

In the US classroom, the mathematics was cognitively demanding for the presenter Aniyah and perhaps for others, e.g., Dante. Other students were invited to ask questions and in this way possibly provide support for Aniyah. The extent to which these interactions were 'productive' for her and/or other students is not obvious on completion of the clip. Level 3 most closely describes this extract but it was the classroom community, rather than the teacher, that was providing the scaffolding.

The task in the Norwegian classroom was of appropriate challenge. One student solved it quickly but others needed more time and one student did not come to a correct solution. In presentation 1, the student was clear in her ideas but this transitioned quickly to teacher explanation. The second student presented a mental method. Again, the teacher took responsibility for explaining this. No scaffolding occurred in presentation 3. It might be considered that a more challenging example of the standard algorithm had been addressed in presentation 1. This might explain why the teacher did not dwell on this example. The teacher's questions in presentation 4 appeared to attempt to scaffold the presenter and clarify her ideas. Across all presentations, the focus remained on student thinking rather than procedures. However, the teacher's actions are closer to level 2 than to level 3. While presenters did not often need the teacher's support, his input may have the effect of 'scaffolding away' the opportunities for students to build their own understanding of each other's ideas.

Access to Mathematical Content

The descriptions for this dimension refer to teacher-presenter conversations (C) and whole class discussions (W). Level 1 (no support (C) or significant disengagement (W)) and level 2 (ineffective scaffolding (C) or uneven participation without teacher action (W) do not apply. Level 3: Teacher supports presenters if needed (C) or the presentation evolves into whole class activity in which the teacher actively supports broad participation and/or what appear to be established participation structures result in such participation (W).

The US classroom was very much orientated to the whole class situation (W). Not all students contributed to this discussion but the teacher deliberately orchestrated whole-class consideration of Aniyah's idea. This is indicative of level 3 of the framework.

In the Norwegian classroom, the interaction was between the teacher and each presenter in turn (C).

The teacher asked clarifying questions. Some questions might be considered to be dual-purpose in that the other students might have benefitted from them. However, the discussion never 'evolves into whole class activity' (W) and the students were not explicitly asked to comment on each other's ideas or to make connections across suggestions. It is hard to match this with TRU framework descriptions as for teacher-presenter conversations (C) level 3 still refers to active supporting of presenters (which was only necessary in presentation 4).

Agency, authority and identity

Level 1 (presentation constrained by teacher questions) does not apply. Level 2: Presenters have the opportunity to demonstrate individual proficiency but the discussions do not build on student's ideas. Level 3: Student presentations result in further discussions of relevant mathematics or students make meaningful reference to other students' ideas in their presentations. (To qualify as an idea what is referred to must extend beyond the tasks, diagrams etc. that is referred to)

In the US classroom, Aniyah's presentation appeared to launch some classroom discussion of meaningful mathematics. This would place it at level 3 of the framework. It can be argued that students such as Toni take on an evaluative role. She acts as a mathematical authority and demonstrates agency. Such activities are envisaged to contribute to positive mathematical identity. The nature of Aniyah's experience is less clear and at the end she chooses to sit down rather than continue presenting/defending her idea. It is necessary to track participation over a longer period before one can make any claims about identity or agency [START_REF] Nic Mhuirí | Investigating student participation trajectories in a mathematical discourse community[END_REF].

The Norwegian classroom can technically be considered at a level 3 but while students are the source of ideas, they are not the source of discussion. Talk is teacher-led at all times. Student contributions are reformulated and explained by the teacher presumably for the purposes of ensuring others understand. Consequently the teacher retains mathematical authority. He also 'rates' the solution strategies of two students. After presentation 3, he first praises the girl who used the standard algorithm. Then he compares it to the previous presentation, saying "But his way to do it, I'd say, uses a method that's easy to calculate in your head. Really smart." By implication, the solution using the standard algorithm in a number range students have not officially been taught yet, is positioned as not as 'smart.' This hierarchical positioning of solutions, and by implication students, does not occur anywhere else but it does speak to issues of identity and authority.

Use of Assessment

Level 1 (reasoning not pursued) and level 2 (specific student ideas not utilized) do not apply. Level 3: In presentation and discussion, the teacher solicits student thinking and responds to student ideas by building on productive beginnings or addressing emerging misunderstandings.

The US classroom episode is aligned with the level 3 description and might be considered formative assessment in action. The teacher is activating the other students to respond to Aniyah's misunderstanding. It would be necessary to see how this plays out to judge whether the strategy is effective for Aniyah and other students.

In the Norwegian classroom, the teacher emphasizes student thinking. In presentation 1, the teacher uses the student's ideas as a launch to model her solution on an empty-number line and to model the standard subtraction algorithm. This episode might be considered to 'build on productive beginnings' (level 3). Levels of understanding were not generally made explicit in the classroom dialogue. The teacher posed a question which explicitly sought to assess students' understanding just once. It is possible that in this small group the teacher could closely observe indications of student (mis)understanding. Late in the episode, (c. 18 minutes) a student explained that he had attempted to solve the task using the standard algorithm but had gotten an incorrect answer. It remains unclear whether he learned how to complete this correctly from the class dialogue.

Discussion

On an evaluative level, the short US video scored higher on the dimensions of the TRU framework than the Norwegian example. The analysis showed differences in the extent to which agency was devolved to students. This devolution was carefully orchestrated by the teacher in the US classroom. Student presentations never evolved into whole-class discussion in the Norwegian setting. All interactions were funneled through the teacher and it was never explicitly stated that the students should attempt to understand each other's reasoning though this may have been an implicit teacher expectation. Such norms were possibly well established in comparison to the US summer school where the teacher was working to establish norms. Indeed, when the Norwegian students were working on developing solutions, they displayed some annoyance that one of the participants indicated the answer before all had completed working. In one of his only explicit directions, the teacher gave the following instruction: "A good tip right now is not to trust that one sitting beside you […] you can only trust yourself. Think for yourself and trust yourself." This appears to emphasize individual effort and indicates an expectation that all students should be able to devise a solution independently. Individual thinking, including errors, was valued in the US classroom but the teacher also seemed to be trying to set an expectation that the community should support the individual in making sense of mathematics. It might be argued that particular forms of socialization and subjectification [START_REF] Biesta | Thinking philosophically about teaching[END_REF] were being actively pursued by the US teacher.

The extent to which certain forms of socialization and subjectification are interwoven into the TRU framework warrants further attention. The framework presents a leveled or hierarchical positioning of various teacher practices that is not value-free. For example, in the Access to Mathematical Content dimension, two different participation structures are recognized: teacher-presenter conversations and whole-class involvement. In the Norwegian classroom, it appears that teacherpresenter conversations occur for the benefit, but without the involvement, of the whole-class. This participation structure is not recognized by the TRU framework but has some similarities to Andrew's (2011) discussion of the 'implicit didactics' of Finnish classrooms where teachers' extended conversations with a competent child in a whole-class setting appeared to be a common feature. These embedded, but unspoken, expectations raise particular challenges for researchers.

Any research lens is informed by the values of the researcher and the research tradition from which the lens is drawn. Often what is valued by a lens remains implicit and unexplored. In this case, the disconnection between the TRU framework and the Norwegian classroom highlights something about the lens itself. This disconnection also raises questions about culture, and whether it is suitable to use a framework developed in one environment to analyze teaching in a different context where conceptions of expert practice may be quite different [START_REF] Clarke | Addressing the challenge of legitimate international comparisons of classroom practice[END_REF]. The aim of this paper however was not to compare teachers (or contexts) but to explore some of the challenges of conducting research in mathematics education. Limitations to this research include the length of the US video and the outsider-status of the researcher in relation to both contexts. The analysis of the Norwegian data was conducted with an English transcript and it is likely that particular nuances of language and meaning have been lost in translation. However, this brief analysis does draw attention to the need for further interrogation of how culture and values inform teaching practices and research frameworks.

Table 1 : Overview of data

 1


Anna Östman
email: anna.ostman@mdh.se

Teachers' probing questions in mathematical classrooms connected to their practice of encouraging students to explain their thinking

Keywords: probing questions, teachers' practice, talk moves, elementary school

Introduction

Asking questions is a common used teaching activity in mathematical classrooms and by arguing that there is a relationship between teaching and learning [START_REF] Smith | Five practices for orchestrating productive mathematical discourse[END_REF], the conclusion is that questions play an important role in teachers' practice in the mathematical classroom [START_REF] Boaler | The importance, nature and impact of teacher questions[END_REF]. [START_REF] Boaler | The importance, nature and impact of teacher questions[END_REF] studied teachers' questions to find the nuances and differences of teachers' questions in mathematical classrooms. They found nine different types of questions (e.g. recall questions, questions supporting students in developing mathematical knowledge, questions which explore meanings and relationships, probing questions to get students to explain their thinking and questions generating discussions). The probing questions are considered critical in encouraging students to explain, clarify and reason. Connected to the probing questions, talk moves are considered useful for teachers in having a fruitful discussion with students [START_REF] Smith | Five practices for orchestrating productive mathematical discourse[END_REF][START_REF] Kazemi | Intentional talk: How to structure and lead productive mathematical discussions[END_REF]. The talk moves that can be seen in mathematical classrooms are for example revoicing, repeating and wait time [START_REF] Kazemi | Intentional talk: How to structure and lead productive mathematical discussions[END_REF]. Asking questions seems to be a teaching activity that teachers do not plan ahead of the lesson even though teachers make explicit that questions are important in the teaching [START_REF] Boaler | The importance, nature and impact of teacher questions[END_REF]. Asking probing questions also make students aware of their own responses and make students answer in the way they think the teacher expects an answer [START_REF] Smith | Five practices for orchestrating productive mathematical discourse[END_REF]. Therefore, in this study I have chosen to focus on the specific questions where teachers are probing students to make their thinking explicit, e.g. "How did you think?", "How did you get your answer?"

The aim of the study is to categorize the different types of responses teachers give to students when students have answered teachers' probing questions. The prospective results will be analysed with the attempt to present different types of interaction patterns emerging from data, which are connected to teachers' responses and teachers' talk moves when students have made their thinking available and explicit through oral explanations. The results will be presented both qualitatively and quantitatively.

Theoretical framework and methodology

The data in this study consists of videotaped mathematics lessons from 16 teachers, one lesson from each teacher, with six to nine years old students. In the data analysis, all episodes where teachers ask probing questions [START_REF] Boaler | The importance, nature and impact of teacher questions[END_REF] were first identified together with the lesson phase in which each probing question appeared. In these episodes teachers' talk moves [START_REF] Kazemi | Intentional talk: How to structure and lead productive mathematical discussions[END_REF] were used to find categories of different interaction patterns among teachers' responses. The interaction patterns are grown from the data and the patterns are used to explore the different ways teachers follow up students' answers after the probing question.

Findings

Preliminary findings from the 16 teachers indicate that probing questions are used in all phases of the mathematical lessons and that students' responses to the probing questions vary between different teachers. The probing questions are most common during the lesson phase whole class discussion and student work. In the preliminary results, I have found that teachers use revoicing in 50% of the interactions following probing questions. Revoicing seems to help students extend their thinking and the results show that when a student's first answer is short without an explicit thought and teachers follow up with revoicing, students often express a developed thought (see Example 1). In contrast, when a student first answers with an explicitly developed thought and the teacher uses the talk move revoicing the next answer from the student is often short (see Example 2). The pattern starts over here [pointing to the pearls] and we were thinking small large small large Teacher: You did think small large small large

Student: Yes

The preliminary results also show that teachers who regularly use the talk move wait time have students answering the probing question with developed thoughts. However, the talk move wait time is not commonly used by the teachers in this study.

Example 1 :

 1 Short student answer and teacher revoicingstudents express a developed thought Teacher: Tell me, how do you think? first, I counted those [showing cards with number 6, 1] and then those [showing cards with numbers 3, 4, 2] Example 2: Developed student answer and teacher revoicing -students stop deepening their answer Teacher: Please, explain your pattern, how do you think? Student:


Karin Rudsberg
email: karin.rudsberg@oru.se

Marcus Sundhäll
email: marcus.sundhall@oru.se

Per Nilsson
email: per.nilsson@oru.se

Investigating the relation between teachers' actions and students' meaning making of mathematics

Keywords: Mathematics teaching, meaning making, pragmatism, epistemological move analysis

In this paper we present and illustrate a framework for analyzing the relation between teachers' actions and students' meaning making in mathematics. We adopt a pragmatic perspective on learning, and a methodological approach using already analyzed material, to see whether and how the framework of epistemological move analysis can contribute when analyzing the relation between teachers' actions and students' meaning making. The results suggest that epistemological move analysis can be used to identify teachers' purpose in students meaning making in mathematics, by analyzing students' responses. Further, it makes it possible to identify what earlier knowledge students use, and how they use it, to re-actualize mathematical objects, relations and concepts.

Introduction

The teacher plays a major role in making classroom talk the arena for learning mathematics and teachers' interactive strategies are central to how students become engaged in math-talks in a classroom practice [START_REF] Walshaw | The teacher's role in classroom discourse: A review of recent research into mathematics classrooms[END_REF]. To succeed in eliciting and extending students' mathematics, a teacher should encourage the mathematical activities of analyzing, making connections, and generalizing [START_REF] Fraivillig | Advancing children's mathematical thinking in everyday mathematics classrooms[END_REF]. It has been proven important that a teacher stimulates students to struggle with important mathematics by challenging them with non-routine tasks and trying to find out more efficient solution methods [START_REF] Hiebert | The effects of classroom mathematics teaching on students' learning[END_REF]. However, providing successful math-talks in practice can be challenging. Teachers tend to funneling [START_REF] Bauersfeld | Interaction, construction, and knowledge: Alternative perspectives for mathematics education[END_REF] students' responses towards the answer they want, blocking them from being fully engaged in math-talk [START_REF] Brodie | Working with learners' mathematical thinking: Towards a language of description for changing pedagogy[END_REF]. Also, rather than addressing problems in terms of the underlying concepts, students address problems mainly in terms of what they think that the teacher is expecting [START_REF] Millar | Improving Science Education: The Contribution of Research[END_REF].

Talk-based teaching of mathematics is a complex teaching practice, calling for the need of theoretical constructs that take serious the social and interactive nature of teaching and learning [START_REF] Lerman | The social turn in mathematics education research[END_REF]. However, concerns have been raised that a focus on social and interactive issues in mathematics teaching may steer away learning from content [START_REF] Lerman | Cultural psychology, anthropology and sociology: The developing 'strong' social turn[END_REF]. For instance, analyses of practices or activity systems common in sociocultural traditions tend to focus on what is done in a general objective [START_REF] Bakker | An inferentialist perspective on the coordination of actions and reasons involved in making a statistical inference[END_REF], which leaves little room for fine-grained analysis of how students make sense of mathematical content.

In the present paper we use a pragmatic perspective on learning [START_REF] Dewey | Experience and nature[END_REF][START_REF] Dewey | Experience and nature[END_REF]) as a starting point, where meaning making is considered to emerge "in the process of doing and undergoing the consequences of action" (Rudsberg, Öhman & Östman, 2013, p. 600). Based on this perspective the aim of this paper is to present and illustrate a framework that enables detailed investigations of teachers' role in directing students' meaning making processes of mathematics in talk-based teaching. The methodological approach in this paper is to use already analyzed material to see whether and how the framework can contribute when analyzing the relation between teachers' actions and students' meaning making. We aim to clarify and investigate both social and individual aspects of the meaning making process in mathematics, in a descriptive manner.

Theoretical background

A pragmatic perspective on meaning making: The first question to be addressed is how to understand learning and meaning making in classroom practice. Often the visibility of such processes is seen as a central problem (Östman & Öhman, 2010), for example the problem of not knowing what the students have in mind. The problem of visibility is often connected to a view of learning as something mental, something that takes place in the student's head hidden from the observer [START_REF] Rudsberg | Analysing Students' Learning in Classroom Discussions about Socio-scientific Issues[END_REF]. However, here we take Dewey´s theory of action [START_REF] Dewey | Experience and nature[END_REF][START_REF] Dewey | Experience and nature[END_REF] 1 as a starting point in which meaning making and learning are understood in terms of people's actions; something they do, in relation to the environment and the specific situation at hand. This can be seen as a way to dissolve dualisms, such as, for example, between the concepts of inner mind and reality [START_REF] Garrison | An introduction to Dewey's theory of functional "trans-action": An alternative paradigm for activity theory[END_REF]. [START_REF] Dewey | Experience and nature[END_REF][START_REF] Dewey | Experience and nature[END_REF] describes life as a process of constant change that contains an active phase, doing, and a passive phase, undergoing the consequences of action. Accordingly, meanings are seen as the way that humans respond to the environment and they are practical in the sense that "we use them as means to consequences" (Garrison, 2001, p. 284). [START_REF] Dewey | Experience and nature[END_REF][START_REF] Dewey | Experience and nature[END_REF] states that meaning "is not indeed a psychic existence; it is primarily a property of behaviour" (p. 179). Communication is seen as a coordinating activity in which the participants make something in common [START_REF] Garrison | An introduction to Dewey's theory of functional "trans-action": An alternative paradigm for activity theory[END_REF]. In this way, "meaning making emerges in the process of doing and undergoing the consequences of action" (Rudsberg et al., 2013, p 600) and in this process experiences continually changes. Continuity can be seen as the process where students re-actualise earlier knowledge in order to make meaning in a new situation. In the process of re-actualization earlier knowledge also transforms i.e., earlier knowledge is not perceived as fixed entities. Meaning making can thus be understood as a practical social process [START_REF] Garrison | Deweyan pragmatism and the epistemology of contemporary social constructivism[END_REF]. Hence, in this pragmatic perspective meaning making and learning is possible to observe in students' coordinating activities. Learning takes place when students develop a new or more developed and specific repertoire of coordinating activities [START_REF] Semetsky | On the creative logic of education, or: re-reading Dewey through the lens of complexity science[END_REF].

Epistemological move analysis: Epistemological move analysis (EMA) is based on a pragmatic understanding of epistemology, where it is seen as both a part and a result of human practices [START_REF] Wickman | Learning as discourse change: a sociocultural mechanism[END_REF]. This means that what counts as relevant knowledge is different in different practices and the knowledge that is considered to be true and valid is a discursive construction (ibid.). EMA focuses on the relation between the epistemological moves that the teachers make and the students' meaning making (Lidar, Lundqvist & Östman, 2006). An epistemological move can be defined as the way a teacher's utterance affects a student's meaning making in an educational situation, that is, the way a teacher's actions facilitate a certain meaning making. In other words, how a teacher's answer to, and/or evaluation of, a student's utterance affects the student's meaning making. EMA has been used in educational practices of, for example, science education [START_REF] Lidar | Teaching and Learning in the Science Classroom[END_REF]. From previous research, we can delineate at least five different types of epistemological moves: confirming, re-constructing, instructional, generative, reorienting (ibid.)2 .

As described above EMA enables investigations of the relation between the teacher's actions and students' meaning making in mathematics. In order to investigate students' meaning making EMA includes Practical Epistemology Analysis (PEA) (Lidar, Lundqcist & Östman, 2006). That is, EMA can be seen as a development of PEA with a specific focus on the role of teachers' actions. EMA aims at giving account of both the content and the processes of students' meaning making in relation to the teacher's moves. In the analysis of students' meaning making four central components are used; encounter, gap, stand fast and relation [START_REF] Wickman | Learning as discourse change: a sociocultural mechanism[END_REF]. Teaching takes place in situations where students encounter their environment and interact with, for example, other students and the teacher. The interaction is made in relation to the purpose of the activity. That is, the first step of the analysis is to clarify what the student encounter and the purpose of the activity. In education, students sometimes do not immediately know how to proceed, i.e., the students cannot proceed by habit and a gap occurs. In teaching situations, a gap becomes visible when for example the teacher asks a question or when the students hesitate. In order for the interaction to proceed the gap needs to be filled, by, for example, responding to a question. This is done in the process of re-actualization. In the process of re-actualization the students bridges the gap by creating relations between their earlier knowledge, what stands fast, and the problem at hand. That is, the students' earlier knowledge becomes actualized in a new situation. The earlier knowledge becomes visible in the educational practise as the students use knowledge without hesitation and no explanations are needed. The relations created by the students constitute the content of the individual's meaning making.

In our analysis we focus on the relation between the teacher's actions and students' meaning making. The teachers' actions, for example, giving instructions, asking questions or making comments, can be seen as guidelines that direct the students' meaning making in a certain way. These actions made by the teachers indicate both what counts as knowledge and appropriate ways of acquiring this knowledge in the specific educational practice. Here it is important to notice that we categorize teachers' epistemological moves in terms of the function they have for the students' meaning making. On this account, we have analyzed whole events in order to clarify the differences between students' utterances before and after the teacher's actions. An event consists of a sequence consisting of three turns, including (1) a student's actions, (2) teachers' actions that encourage a student's action, and (3) a student's actions after the teacher's actions [START_REF] Klaar | Children's meaning making in nature in an outdoor-oriented and democratic Swedish preschool practice[END_REF].

Illustration of EMA in mathematics education

To illustrate epistemological move analysis we use two examples from Blanton's and Kaput's paper focusing on classroom practices in algebraic reasoning [START_REF] Blanton | Characterizing clasroom practice that promotes algebraic reasoning[END_REF]. These examples are rich in mathematical content, given that the characterization is built on features of algebraic reasoning. Blanton and Kaput aim to "understand those types of instructional practice that indicated a generative and self-sustained capacity to build on students' algebraic reasoning" (ibid, p. 416).

From this study we have chosen examples from audio-taped classroom talks. To identify the teacher's purpose of these classroom talks, the authors have studied notes by and interviews with the teacher. By using epistemological move analysis we can study the relationship between the purpose that the teacher expresses and the purpose visible in how the student responds to his/her moves. Blanton's and Kaput's paper includes several examples of principles for task-design and indications on what kind of question the teacher should ask to promote algebraic reasoning. However, their intention is "not to address how June's practice affected student achievement; this would require a more detailed look at how her actions played out in the classroom and how students were involved" (ibid, p. 435). In this paper we aim to employ such a detailed look by analyzing how the teacher's epistemological moves direct and support students' meaning making. This way we can clarify students' meaning making concerning both process and knowledge content in the actual situation. Important to notice, however, is that we do not investigate whether or not this change is of long term.

Algebraic treatment of number:

In reviewing homework of addition tasks, the teacher shifts focus to properties of even and odd numbers and, during this sequence, there is an encounter established between the teacher (June) and two students (Tony and Jenna). In the first part of this encounter a scheme of even and odd numbers is used. The teacher's purpose is that the students should find an algebraic principle of parity in adding odd and even numbers. This aim becomes visible when the teacher uses large numbers to guide Tony to use an algebraic principle instead of computation that specifies even and odd numbers. The transcript (ibid.) used in this illustration is the following: "June challenged a student's use of arithmetic strategy to deduce that 5+7 was even:

1

June: How did you get that? 2 Tony: I added 5 and 7 and then I looked over there [pointing to a visible list of even and odd numbers on the wall] and saw that it was even. 3

June: What about 45678+85631? Odd or even? 4 Jenna: Odd. 5

June: Why? 6 Jenna: Because 8 and 1 is even and odd, and even and odd is odd." (p. 422).

Teacher "spontaneously shifted the focus from computing sums to determining if the sum of two numbers would be even or odd. When students responded by first computing the sum to determine if it was even or odd, June began to use numbers that were sufficiently large so that students could not compute. Instead, they were forced to attend to the structure in the inscriptions themselves." (ibid, p. 417). The gap in this encounter is between the use of addition in order to determine if the sum is even or odd and the ability to use an algebraic principle in order to determine if the sum will be even or odd. The gap becomes visible when the teacher (line 3) uses large numbers.

The students are able to make arguments on the parity of sums of odd and even numbers without using computations. The stand fast can be regarded as computation of specific numbers and using the list on the wall to determine odd and even numbers. This is re-actualized into the ability to use numbers as placeholders, or variables, for any odd or even number. This change is made because of the teacher's epistemological moves in line 3 and 5. Together these moves enabled the student to change the way of determining if a sum of two positive integers will be even or odd. Hence, the student has made new mathematical meaning about the structure of the inscription itself in terms of an algebraic principle of even and odd numbers. This way the teacher's epistemological moves in line 3 and 5 are generative move as they have the function of generate students learning of algebraic principles and relations. The teacher's purpose becomes visible in line 5 since the response by the student in line 6 is built on an algebraic principle rather than on computation.

Finding a mathematical formula:

In working with the concept of area, the teacher introduced an activity in which students should determine how many two-colored counters were needed to cover a large purple square. In this sequence we can conclude that it is an encounter between the teacher and four students (June, Mari, Zolan, Stephanie, Kevin) and also one more student (unidentified in the audio-taped recording), who may or may not be one of the other four students. The teacher's purpose is that the students reach an understanding of how to use length and width to determine the area of rectangular planar objects, i.e., finding a formula for the area. This purpose becomes visible in line 23 as it generates the concepts of length and width in line 24. The transcript (ibid.) used in this illustration is as follows: "June had the following conversation with her students: What do you call "this way" and "that way"? 26 Student:

Length and width!" (p. 428-429).

The purpose in this encounter is located between the use of square tiles for covering rectangular planar objects and finding a formula for area of such objects. In line 13 the teacher asks the students if they know the size of the large purple square after the students have covered it with tiles. However, the teacher's first question (line 13) does not have any function in terms of meaning making, since the student cannot bridge the gap that the question creates. Indeed, the teacher's question (line 13) is an action, but it is not an action that qualifies as an epistemological move. A gap between square tiles and the size of a planar object becomes visible when the teacher in line 15 asks the students to look at the square and explain what they see. This is an instructional move since the student's answer is describing what he (Zolan) can see and not mentioning anything on area. This means that the gap lingers. In the third question (line 17) the teacher explicitly asks the students to find the area of the desk. This is a re-orienting move since the students now are focusing on the total amount of squares rather than numbers of columns and rows. Here it stands fast for the student how to cover a square object with square tiles (line 18). This earlier knowledge is reactualized in order to make new meaning to view area as a grid of square tiles. Hereby the students are able to fill the gap.

When the teacher explicitly asks the students to determine the area of another planar object (line 19), the gap changes. The new gap is between using columns and rows and finding the formula for area. Related to the change of gap the student also actualizes other earlier knowledge that becomes visible when the teacher asks the students to determine the area of yet a new object (line 19). Here the knowledge that stands fast can be regarded as knowing how to use columns and rows to cover a rectangular planar object. This is re-actualized when the student starts to investigate the grid in a new way by counting the numbers of squares in the bottom row (line 20). In this way the teacher's action functions as a re-orienting move.

Finally, the teacher makes two re-constructing moves (line 21 and 23). The first becomes visible when the student (line 22) shows awareness of the importance in comparing the total number of square tiles with the numbers of columns and rows. Thereafter the student (line 24) is able to relate number of columns and rows to create a formula for area. Finally, the students are able to fill the gap by finding a formula, in terms of length and width, for determining area of planar objects.

Concluding remarks

The analysis shows how teacher's epistemological moves during math-talk is crucial for students' meaning making processes, both concerning the procedure (how) and the mathematical content (what). For example, in the first illustration the generative move deepens the student's mathematical knowledge in terms of an ability to build on algebraic principle rather than computation when determining even and odd numbers. We can conclude that the teacher enables the students to generalize [START_REF] Fraivillig | Advancing children's mathematical thinking in everyday mathematics classrooms[END_REF], which is done by challenging them to find a more general and efficient solution to the problem at hand [START_REF] Hiebert | The effects of classroom mathematics teaching on students' learning[END_REF]. By using EMA it was possible to identify how the students' earlier knowledge in this specific situation was re-actualized into the ability to use numbers as placeholders for any odd or even number. More explicitly, this reactualization is made because of the teacher's epistemological move (line 3), made visible by the student's answer in line 6, where the latter uses numbers 1 and 8 as placeholders for odd and even numbers. By this re-actualization the student fills the gap that becomes visible by the teacher's question in line 3. In this way the student has made new meaning about an algebraic principle of odd and even numbers. Here we stress that the contribution in using EMA is to point out what actually takes the meaning making a step forward, rather than analyzing certain approaches in forehand intended to promote specific motives. The strength in EMA is in the situational, rather than long term. However, in combining EMA with well-known strategies promoting algebraic reasoning, such as the framework used by Blanton & Kaput, sustainable learning perspectives may be closely related to situational and contextual perspectives.

This can be compared with our second illustration. In the first part of the transcript the teacher poses different questions with the purpose of enabling for the students to understand a formula for the area of rectangular objects. However, in the first part of the sequence the students answer the teacher's questions without re-actualizing any earlier knowledge about area. Instead they answer the question more in terms of what they think that the teacher are expecting [START_REF] Millar | Improving Science Education: The Contribution of Research[END_REF]. Important to notice here is also that the teacher's purpose, to understand how to determine area, isn't visible in the students' actions until later in the sequence (line 19-20). At the end of the sequence (line 24-26), after the re-constructing move, which also clarifies the teacher's purpose, the students re-actualize earlier knowledge that enables them to establish a formula for area. By using EMA we can identify that the teacher's purpose is not initially clear to the students. The students are not able to see how the general properties for area correlate to the teacher's purpose. The students try to answer the question (line 13) but are not able to fill the gap until the teacher changes character of the questions (line 23). This illustration can be seen as an example of funneling [START_REF] Bauersfeld | Interaction, construction, and knowledge: Alternative perspectives for mathematics education[END_REF], with the consequence of students being partly blocked from being fully engaged in the talk [START_REF] Brodie | Working with learners' mathematical thinking: Towards a language of description for changing pedagogy[END_REF].

To summarize, using EMA, we can analyze the relation between teachers' actions and students' meaning making concerning process (how) and content (what) in mathematics. This way, EMA can contribute to the research on social and interactive processes in mathematics [START_REF] Lerman | Cultural psychology, anthropology and sociology: The developing 'strong' social turn[END_REF]. Further, EMA makes it possible to identify the role of teachers' purpose in students' meaning making in mathematics. However, it should be stressed that meaning making is seen as situational.

We believe that the concept of re-actualization should get more attention in studying the role of teachers' actions for students' meaning making in mathematics. Re-actualization should not be misinterpreted as transfer or application, but rather as the process describing what stands fast in a certain encounter and how the students use knowledge that stands fast to create new meaning of mathematical objects, principles or concepts. Based on the present analysis, we claim that EMA offers analytical tools to enhance our understanding of the relationship between content and process in student's meaning-making in mathematics and the roles teachers' actions can have in providing directions for meaning-making.

For a more thorough elaboration of Dewey' theory of action and its use, see, for example,Östman and Öhman (2010).

The moves identified in this study will be described in the analysis; for a description of earlier identified epistemological moves see, for example,[START_REF] Lidar | Teaching and Learning in the Science Classroom[END_REF] andRudsberg and Öhman (2010).


Ye Wang
email: yewang@hrbeu.edu.cn

Sorin Olaru
email: sorin.olaru@l2s.centralesupelec.fr

Giorgio Valmorbida
email: giorgio.valmorbida@l2s.centralesupelec.fr

Vicenç Puig
email: vicenc.puig@upc.edu

Gabriela Cembrano
email: gabriela.cembrano@upc.edu

Set-invariance characterizations of discrete-time descriptor systems with application to active mode detection

Keywords: Robust invariant sets, Active mode detection, Input design, Set separation, Discrete-time descriptor systems

In this paper, we study robust invariant set characterizations of discrete-time descriptor systems and propose an active mode detection mechanism for discrete-time descriptor systems considering multiple modes of operation. The considered class of descriptor systems assumes regularity, stability and is affected by unknown-but-bounded disturbances. As a first theoretical result, we establish a general framework for robust invariant sets for discrete-time descriptor systems in both causal and noncausal cases. Particular transformations are subsequently proposed for handling causal and non-causal descriptor systems and will be used to characterize the effects of disturbances. Based on these set-theoretic notions and a designed input signal for active set separations, we propose an active mode detection mechanism by exploiting the strong invariance properties.

Introduction

Set-invariance theory has played an essential role in automatic control with a variety of applications to control systems, since it is widely used for guar-anteeing the stability and achieving desired performance [START_REF] Blanchini | Set invariance in control[END_REF][START_REF] Kolmanovsky | Theory and computation of disturbance invariant sets for discrete-time linear systems[END_REF]. For systems affected by disturbances, different techniques in set-invariance theory are used for the computation of robust invariant (RI) sets. These techniques have been applied to linear dynamical systems [START_REF] Raković | Invariant approximations of the minimal robust positively invariant set[END_REF], linear parametervarying systems [START_REF] Seron | On robust stability and set invariance of switched linear parameter varying systems[END_REF], and nonlinear systems [START_REF] Alamo | Convex invariant sets for discrete-time Lur'e systems[END_REF]. In particular, ultimate boundedness methods are used to compute RI sets with relative low complexity [START_REF] Kofman | A systematic method to obtain ultimate bounds for perturbed systems[END_REF][START_REF] Olaru | Positive invariant sets for fault tolerant multisensor control schemes[END_REF]. Furthermore, set-invariance characterizations are instrumental for control strategies, such as fault detection and isolation [START_REF] Blanchini | Active fault isolation: A dualitybased approach via convex programming[END_REF][START_REF] Xu | Actuator-fault detection and isolation based on settheoretic approaches[END_REF], faulttolerant control [START_REF] Olaru | Positive invariant sets for fault tolerant multisensor control schemes[END_REF][START_REF] Seron | Fault tolerant control allowing sensor healthy-to-faulty and faulty-tohealthy transitions[END_REF][START_REF] Stoican | Set-theoretic Fault-tolerant Control in Multisensor Systems[END_REF]) and robust model predictive control [START_REF] Mayne | Robust model predictive control of constrained linear systems with bounded disturbances[END_REF]. A remarkable application of RI sets is on mode detection of systems subject to multiple modes of operation. Indeed, since different operating modes lead to different RI sets, the distance between these sets can be used for monitoring and mode detection. Due to the fact that the RI sets of different modes may overlap, an additive input signal can be conveniently designed to separate a parametrization of the RI sets, represented by tubes of trajectories. In this case, the set-theoretic mode detection mechanism is called active. In the literature, this mechanism is also called active fault diagnosis [START_REF] Raimondo | Closed-loop input design for guaranteed fault diagnosis using set-valued observers[END_REF]. A set of additive inputs are designed to guarantee fault diagnosis outputs that are only consistent with one faulty scenario. These additive inputs can be obtained from the solution to a mixedinteger quadratic program or using a multi-parametric approach [START_REF] Marseglia | Active fault diagnosis: A multi-parametric approach[END_REF].

The above described methods have been proposed for standard dynamical systems modeled only with differential/difference equations. Due to mass, volume or energy conservation laws, difference equations describing a dynamical system can be coupled with a set of algebraic equations. This class of systems described by differential/difference and algebraic equations is called descriptor, singular, or differential/difference-algebraic systems [START_REF] Dai | Singular Control Systems[END_REF][START_REF] Duan | Analysis and Design of Descriptor Linear Systems[END_REF]. For this class of systems, set-invariance characterizations as well as their mode detection are still not widely developed. Instances of such systems can be found in water distribution networks (Wang, Puig & Cembrano 2017), chemical processes [START_REF] Biegler | Control and Optimization with Differential-Algebraic Constraints[END_REF], electrical circuits [START_REF] Riaza | Differential-algebraic systems: Analytical aspects and circuit applications[END_REF]) and economic models [START_REF] Dai | Singular Control Systems[END_REF]. From a theoretical point of view, descriptor systems satisfying a well-posed property, for which a solution exists and is unique, are called regular [START_REF] Dai | Singular Control Systems[END_REF][START_REF] Oarȃ | Numerical solution to a descriptor discrete-time algebraic Riccati equation[END_REF]. Regularity, however, does not imply causality and models of interest in economy are noncausal, see e.g. the Leontief model [START_REF] Dai | Singular Control Systems[END_REF]. In terms of a control system, stability [START_REF] Halanay | Stability and stable oscillations in discrete time systems[END_REF] is an important property for the analysis of boundedness and convergence of the closed-loop trajectory. In particular, in terms of descriptor systems, admissibility guarantees the properties of regularity, causality and stability. The present paper aims to revisit all these properties of descriptor systems and exploit the underlying structural properties in a set-theoretic framework.

Systems modeled in a descriptor framework can be affected by uncertainties, such as modeling errors and disturbances. Also, faults from actuator and sensor malfunctions may change the dynamics and constraints of the system and therefore the system can evolve or switch to different modes of operation. For instance, in cyberphysical systems, the system model can be changed by faults of different nature, such as process/system faults, actuator and sensor faults, as well as communication faults. Thus, a suitable mode detection mechanism is required to identify whether the actual cyber-physical system matches with the prediction by checking the feedback information.

The main contribution of this paper is to present a general framework for set-invariance characterization of discrete-time descriptor systems as well as an application to active mode detection. The proposed computation of invariant sets relies on partitioning the state space for both causal and non-causal descriptor systems under standard notations. Besides, we propose an active mode detection mechanism based on positive set-invariance characterizations for discrete-time descriptor systems.

The preliminary results presented in this paper have been reported in [START_REF] Wang | Robust invariant sets and active mode detection for discrete-time uncertain descriptor systems[END_REF]. Additional improvements and new contribution are summarized as follows:

• (A detailed review of definitions and properties of discrete-time descriptor systems) These properties prove to be useful for the computation of RI sets. • (A general framework for set-invariance characterizations of discrete-time descriptor systems) This framework completes the preliminary results in [START_REF] Wang | Robust invariant sets and active mode detection for discrete-time uncertain descriptor systems[END_REF]. In addition, the convergence time for each RI set and the results of checking the compatible initial states are provided. • (A novel active mode detection mechanism) The strong invariance properties for detecting mode of operation are formulated.

2 Background and preliminaries

Discrete-time descriptor systems

Consider the discrete-time linear time-invariant (LTI) descriptor system with additive disturbances

Ex(k + 1) = Ax(k) + B w w(k), (1)
where x ∈ R n and w ∈ R q denote the state vector and the disturbance vector, respectively,

k ∈ N. A ∈ R n×n , B w ∈ R n×q and E ∈ R n×n with rank(E) = r ≤ n.
Definition 1 (Regularity) A descriptor system (1) is said to be regular if it has a unique solution defined as an application x : N → R n which satisfies (1) for any disturbance realization w : N → R q and a compatible initial state x(0).

From the above definition, if the system (1) is regular, then it has a unique solution for the disturbance-free case (w ≡ 0). We also say the matrix pair (E, A) is regular.

Definition 2 (Causality) A regular descriptor system (1) is said to be causal if x(k), ∀k ∈ N is determined completely by the initial condition x(0) and w(j), for j = 0, . . . , k. Otherwise, it is said to be non-causal.

Definition 3 (Asymptotic Stability) A regular descriptor system (1) is said to be asymptotically stable for the disturbances-free case (w ≡ 0) if lim k→∞ x(k) = 0.

Definition 4 (Admissibility) A descriptor system (1) for the disturbances-free case (w ≡ 0) is said to be admissible if it is regular, causal and asymptotically stable.

Lemma 1 ( [START_REF] Dai | Singular Control Systems[END_REF])) For the matrix pair (E, A) of the descriptor system (1), the following properties hold

• (Regularity) the pair (E, A) is regular if det(zE -A) is not identically zero. • (Causality) the pair (E, A) is causal if deg(det(zE - A)) = rank(E). • (Asymptotic stability) the pair (E, A) is asymptoti- cally stable if |ν| < 1, ∀ν ∈ λ (E, A), where λ (E, A)
denotes the generalized eigenvalues of E and A.

In the following, admissibility is not part of the assumption, i.e. the study concerns both causal and non-causal descriptor systems.

Assumption 1 The descriptor system (1) (the matrix pair (E, A)) is regular and asymptotically stable in the disturbance-free case (w ≡ 0).

We now establish suitable transformations that decompose the descriptor system (1) into subsystems for setinvariance characterizations and active mode detection.

Definition 5 (Equivalence of descriptor systems) Consider two descriptor systems respectively defined by the triplets (E, A, B w ) and ( Ẽ, Ã, Bw ). If there exists a pair of non-singular matrices Q ∈ R n×n and P ∈ R n×n satisfying QEP = Ẽ, QAP = Ã, QB w = Bw , then these two systems are called restricted equivalent under the transformation (Q, P ).

For the descriptor system (1), we now present two standard restricted equivalent forms that are of interest (Duan 2010, Chapter 2).

Consider the descriptor system (1) with rank(E) = r.

There always exists a transformation (Q, P ) yielding

QEP = I r 0 0 0 , QAP = A 1 A 2 A 3 A 4 , QB w = B w1 B w2 , (2)
with

A 1 ∈ R r×r , A 2 ∈ R r×(n-r) , A 3 ∈ R (n-r)×r , A 4 ∈ R (n-r)×(n-r) , B w1 ∈ R r×q and B w2 ∈ R (n-r)×q .
Lemma 2 (Dynamics decomposition form [START_REF] Duan | Analysis and Design of Descriptor Linear Systems[END_REF])) The descriptor system (1) is causal if and only if there exists a transformation (Q, P ) yielding (2) with a non-singular block matrix A 4 .

Based on the above lemma, an equivalent causal descriptor system in a standard dynamical form is presented in the following.

Lemma 3 (Equivalent causal descriptor system) A causal descriptor system (1) with rank(E) = r can be transformed into the following form

x(k + 1) = Ãx(k) + Bw w(k), (3)
where

à = A 1 -A 2 A -1 4 A 3 0 -A -1 4 A 3 A 1 -A 2 A -1 4 A 3 0 , ( 4a
) Bw = B w1 -A 2 A -1 4 B w2 0 -A -1 4 A 3 B w1 -A 2 A -1 4 B w2 -A -1 4 B w2 . ( 4b
)
and

A 1 , A 2 , A 3 , A 4 , B w1 , B w2 are defined in (2) and x(k) = x1 (k) x2 (k) = P -1 x(k), w(k) = w(k) w(k + 1)
, ( 5)

with x1 (k) ∈ R r , x2 (k) ∈ R (n-r) .
PROOF. See the proof of (Wang, Olaru, Valmorbida, Puig & Cembrano 2017, Lemma 3). 2

Remark 1 In Lemma 3, w(k + 1) only appears in the algebraic equation of the descriptor system (1), which is used to compute x2 (k + 1).

The regular descriptor system (1) also allows the transformation in the so-called Kronecker canonical form.

Lemma 4 (Kronecker canonical form [START_REF] Dai | Singular Control Systems[END_REF]))

The descriptor system (1) is regular if and only if there exists a transformation ( Q, P ) yielding

QE P = I p 0 0 N , QA P = Ā 0 0 I , QB w = Bw1 Bw2 , (6)
with -p) is a nilpotent matrix (that is there exists a scalar s > 0 such that N s = 0 and N s-1 = 0, s ≤ n -p) and p ≤ r = rank(E).

Ā ∈ R p×p , Bw1 ∈ R p×q , Bw2 ∈ R (n-p)×q . More- over, N ∈ R (n-p)×(n
Computationally efficient and numerically stable methods exist to obtain these transformations as reported in [START_REF] Gerdin | Computation of a canonical form for linear differential-algebraic equations[END_REF][START_REF] Varga | Solving Fault Diagnosis Problems[END_REF].

Lemma 5 (Causality [START_REF] Dai | Singular Control Systems[END_REF])) The descriptor system (1) transformed in the Kronecker canonical form (6) is causal if and only if N = 0.

Background of set-invariance theory

We now introduce set-theoretic notions for discrete-time descriptor systems. For a regular and stable descriptor system (1), we consider that the additive disturbances are unknown but bounded in a known set

w(k) ∈ W = {w ∈ R q : |w| ≤ w} , ∀k ∈ N, (7)
with w ∈ R q and elementwise inequality.

As a consequence of boundedness of the disturbances and the stability of the dynamics, the system trajectories eventually converge to a bounded region of the state space [START_REF] Kolmanovsky | Theory and computation of disturbance invariant sets for discrete-time linear systems[END_REF] for the forward trajectories. Given an initial state x(0) and the unique solution to (1) (note that the discrete-time domain of the solution may include negative values for backward propagations), the following definitions are introduced in terms of the set-theoretic analysis.

Definition 6 (RI set) A set Ω ∈ R n is said to be robust invariant (RI) with respect to the system (1) if x(0) ∈ Ω implies x(k) ∈ Ω, ∀w(k) ∈ W and ∀k ∈ Z. Definition 7 (RPI set) A set Ω ∈ R n is said to be ro- bust positively invariant (RPI) with respect to the sys- tem (1) if x(0) ∈ Ω implies x(k) ∈ Ω, ∀w(k) ∈ W and ∀k ∈ N.
Definition 8 (mRPI set) An RPI set Ω ∞ ∈ R n is said to be minimal RPI (mRPI) with respect to the system (1) if it is contained in every closed RPI set.

Definition 9 (L-steps RNI set) A set Ω ∈ R n is L- steps robust negatively invariant (RNI) with respect to the system (1) if x(L) ∈ Ω implies x(L+k) ∈ Ω, ∀w(k) ∈ W and ∀k ∈ Z [-L,0] .

Robust invariant set characterizations of discrete-time descriptor systems

In this section, we formulate explicit expressions of several RI sets and approximations of minimal RI sets for discrete-time descriptor systems in both causal and noncausal cases. Furthermore, the convergence time for each RI set is provided.

For an admissible descriptor system (1), the set analysis will be performed using the dynamics decomposition form. From Lemma 3, there exists a transformation (Q, P ) leading to (2) and ( 3)-( 5). We consider a partition of the matrix P as P = [P 1 , P 2 ] with P 1 ∈ R n×r and P 2 ∈ R n×(n-r) . The structure of the mRPI set of the admissible descriptor system (1) is characterized in the following theorem1 .

Theorem 1 (mRPI set of admissible descriptor systems) Consider an admissible descriptor system (1) with the dynamics decomposition form in (2) and w(k) ∈ W, ∀k ∈ N. The mRPI set Ω c is given by

Ω c = P 1 Φ 1 ⊕ P 2 Φ 2 ,
where

Φ 1 = ∞ i=0 Ã1 Bw1 W, (8a)
Φ 2 = -A -1 4 A 3 Φ 1 ⊕ -A -1 4 B w2 W , ( 8b
)
with Ã1 = A 1 -A 2 A -1 4 A 3 and Bw1 = B w1 -A 2 A -1 4 B w2 .
PROOF. With the transformation (Q, P ), the descriptor system (1) is equivalent to a dynamical system including two subsystems as in ( 4). On the one hand, from (4) we have

x1 (k + 1) = Ã1 x1 (k) + Bw1 w(k). (9)
The admissibility of (1) implies the matrix Ã1 is Schur. Then, the characterization of the mRPI set for x1 can be obtained as in (8a) using the standard LTI notions [START_REF] Kolmanovsky | Theory and computation of disturbance invariant sets for discrete-time linear systems[END_REF].

On the other hand, from (2) we obtain x2

(k) = -A -1 4 A 3 x1 (k) -A -1 4 B w2 w(k)
, which is an algebraic equation. Thus, we obtain the mRPI set Φ 2 by a linear projection image of the set Φ 1 in (8a), which leads to (8b). By definition in (5) and using the Minkowski addition of the sets obtained via the linear mapping defined by the matrices P 1 and P 2 , we can determine the mRPI set

Ω c = P 1 Φ 1 ⊕ P 2 Φ 2 .
2

Remark 2 For invariant approximations of the mRPI set Ω c in Theorem 1, several approaches can be applied to the LTI part of the dynamics leading to an approximation of Φ 1 (see e.g. [START_REF] Olaru | Positive invariant sets for fault tolerant multisensor control schemes[END_REF], Raković et al. 2005)) while the approximation of Φ 2 is a projection involving the approximation of Φ 1 and W. Furthermore, by applying the iterative approximation approach in [START_REF] Olaru | Positive invariant sets for fault tolerant multisensor control schemes[END_REF], arbitrarily close approximations can be obtained

Ω c 0 ⊇ • • • ⊇ Ω c .
Based on the above results, we present a practical condition of the compatibility check for any initial state x(0).

Corollary 1 Consider an initial state x(0) for the admissible descriptor system (1) in (3)-( 5). If

x2 (0) / ∈ -A -1 4 A 3 ζ * Φ 1 ⊕ -A -1 4 B w2 W , (10)
where x(0) = P 1 x1 (0) + P 2 x2 (0) and ζ * = min{ζ ∈ R : x1 (0) ∈ ζΦ 1 }, then x(0) is not a compatible initial state for (1) and it is independent of any disturbance realization w(0) ∈ W.

PROOF. Based on the mRPI set Ω c , if x(0) is compatible, then it holds x2 (0) ∈ Φ 2 , ∀w(0) ∈ W. On the other hand, for a scalar ζ * , if the condition (10) does not hold, then x(0) is not compatible. Note that the set in ( 10) is not an RPI set but it represents a constraint for the descriptor part of states whenever this constraint is violated, leading to the incompatibility of the algebraic equations. 2Remark 3 By Definition 6 and its characterization in Theorem 1, the consistency in terms of initial state x(0) with the descriptor model (1) can be tested. In presence of the disturbance w(0) ∈ W, x(0) may not be a compatible initial state. This shows that x(0) should be understood as an implicit function of w(0), i.e. x(w(0)), by means of the solution of algebraic equations.

To complete the study of admissible descriptor systems, the computation result of the convergence time for discrete-time admissible descriptor systems is provided based on the result in (Seron et al. 2012, Appendix A). This is equivalent to an upper bound for the total number of steps necessary for the system trajectories to reach the set Ω c from a given initial state.

Theorem 2 (Convergence time for admissible descriptor systems) Consider an admissible descriptor system (1), w(k) ∈ W, ∀k ∈ N and an approximation of Ω c 0 ⊇ Ω c with Ω c 0 = P 1 Φ1,0 ⊕ P 2 Φ 2 . For a compatible initial state x(0), the system trajectory x(k) belongs to Ω c 0 , that is, x1 (k) defined in (5) belongs to Φ1,0 , for k ≥ T ca , where T ca is the convergence time corresponding to (1) and depends on x(0) and ε.

PROOF. Based on Lemma 3, x2 (k) has no dynamics and is a linear mapping of x1 (k) and w(k). By directly applying the result in [START_REF] Seron | Fault tolerant control allowing sensor healthy-to-faulty and faulty-tohealthy transitions[END_REF], Appendix A) to x1 (k) with its dynamics x1 (k + 1) = Ã1 x1 (k) + Bw1 w(k), we can obtain the convergence time T ca . 2

In case that the descriptor system (1) is regular and stable but not causal, there might exist a unique solution at each time [START_REF] Dai | Singular Control Systems[END_REF]. We now consider a non-causal and stable descriptor system (1) and use the Kronecker canonical form in (6) for the RPI characterization.

From Lemma 4, a non-causal descriptor system (1) can be transformed in (6) with a nilpotent matrix N satisfying N = 0. As introduced in (Dai 1989, Chapter 8), for a regular matrix pair (E, A), there exists a suitable transformation ( Q, P ) with P = P1 P2 , P1 ∈ R n×p , P2 ∈ R n×(n-p) yielding to (6).

For the transformed system in the Kronecker form, we use the following partitioning form

x(k) = x1 (k) x2 (k) = P -1 x(k), QB w = Bw1 Bw2 , (11)
with x1 (k) ∈ R p , x2 (k) ∈ R (n-p) .
Based on the Kronecker canonical form in Lemma 4, we have that

x1 (k + 1) = Āx 1 (k) + Bw1 w(k), (12a) N x2 (k + 1) = x2 (k) + Bw2 w(k). ( 12b
)
The structure in ( 12) highlights the fact that the noncausal descriptor system (1) is stable if and only if the matrix Ā is Schur. We now formulate the mRPI set of discrete-time non-causal descriptor systems.

Theorem 3 (mRPI set of non-causal descriptor systems) Consider a non-causal descriptor system (1) with the Kronecker canonical form in (6) and w(k) ∈ W, ∀k ∈ N. The mRPI set Ω n is given by

Ω n = P1 Θ 1 ⊕ P2 Θ 2 ,
where

Θ 1 = ∞ i=0 Āi Bw1 W, (13a)
Θ 2 = n-p-1 i=0 -N i Bw2 W . ( 13b
)
PROOF. The non-causal descriptor system can be decomposed in two subsystems, where (12a) is an ordinary difference equation. Hence, the mRPI set of x1 can be constructed as in (13a). On the other hand, from (12b), the anti-causal state x2 (k) can be propagated as follows:

x2 (k) = N x2 (k + 1) -Bw2 w(k), x2 (k + 1) = N x2 (k + 2) -Bw2 w(k + 1)
, and after the (n -p)-step iterations, we can obtain

x2 (k) = N (n-p) x2 (k + n -p) - n-p-1 i=0 N i Bw2 w(k + i). ( 14
)
Since N is a nilpotent matrix with N n-p = 0, we know that for k > n -p, N k = 0. Therefore, (14

) becomes x2 (k) = - n-p-1 i=0 N i Bw2 w(k + i). With w(k) ∈ W, ∀k ∈ N, the set for x2 can be com- puted by Θ 2 = n-p-1 i=0 -N i Bw2 W = -Bw2 W ⊕ -N Bw2 W ⊕ • • • ⊕ -N n-p-1 Bw2 W . Finally, we derive the mRPI set Ω n = P1 Θ 1 ⊕ P2 Θ 2 . 2
Remark 4 Theorem 3 builds on the assumption that the time domain of solution to the system (1) is N. The existence of this infinite-time trajectory leads to a positive invariance property although the system is not causal. Theorem 3 should be reconsidered in case that the trajectories are defined only for a finite-time window.

For a non-causal descriptor system (1), we also present the results of the compatibility check for any initial state x(0) and convergence time. The proofs are similar to the ones of Corollary 1 and Theorem 2.

Corollary 2 Consider an initial state x(0) of a noncausal descriptor system (1). If x2 (0) / ∈ Θ 2 where x(0) = P1 x1 (0) + P2 x2 (0), then x(0) is compatible for (1) irrespective of any disturbance realization w(0) ∈ W.

Theorem 4 (Convergence time for non-causal descriptor systems) Consider a non-causal descriptor system (1) affected by disturbances w(k) ∈ W, ∀k ∈ N and let the set Ω n 0 ⊇ Ω n with Ω n 0 = P 1 Θ1,0 ⊕ P 2 Θ 2 . For a compatible initial state x(0), the system trajectory x(k) converges to Ω n 0 in T cn iterations, that is, x1 (k) defined in (11) belongs to Θ1,0 , for k ≥ T cn , where T cn is the convergence time corresponding to (1) and depends on x(0) and ε.

As an extension for a non-causal descriptor system (1), we now focus on trajectories defined only on a finitetime window, that is x(k), k ∈ Z [0,L] with L > 0. The dynamics of a non-causal descriptor system (1) obey the equivalent subsystems in (12) but the set-invariance characterization need to be relaxed in order to consider the finite number of dynamical constraints as well as the structural particularities (algebraic equations) related to anti-causality.

The difficulties are related to a combination of causal and anti-causal dynamics in (12a) and (12b). For (12a), the positive invariance will be the appropriate concept while for (12b), the negative invariance offers the suitable framework in a predefined finite-time window L.

Theorem 5 (L-steps RNI set) Consider the anticausal subsystem (12b).

A set Υ is L-steps RNI if Υ ⊇ N Υ ⊕ -Bw2 W ⊇ • • • ⊇ N L Υ L-1 i=0 -N i Bw2 W . (15)
PROOF. From (12b), we have x2 (k) = N x2 (k + 1) -Bw2 w(k). For a finite time window L > 0, x2 (L) ∈ Υ .

By the backward propagations of x2 (k + L) ∈ Υ for any k ∈ Z [-L,0] , we can derive (15). 2

Corollary 3 Given L 1 -and L 2 -step RNI sets Υ 1 and Υ 2 with L 1 ≥ L 2 ≥ n -p satisfying Υ 1 ⊇ Υ 2 , then N l Υ 1 l-1 i=0 -N i Bw2 W ⊇ N l Υ 2 l-1 i=0 -N i Bw2 W , ∀l ≥ 0.
PROOF. The relationship (15) holds for l = 0

as

Υ 1 ⊇ Υ 2 . Suppose N l Υ 1 l-1 i=0 -N i Bw2 W ⊇ N l Υ 2 l-1 i=0 -N i
Bw2 W holds for some l ≥ 0. Then, by pre-multiplying with N and Minkowski summing the set -Bw2 W on both sides, we obtain

N l+1 Υ 1 l i=0 -N i Bw2 W ⊇ N l+1 Υ 2 l i=0 -N i Bw2 W .
The proof is completed by induction. 2

Remark 5 The set Θ 2 in (13b) is L-steps RNI with respect to (12b), ∀L > 0.

Remark 6 Consider the set Θ 2 as in (13b). An L-steps RNI set with respect to (12b) can be constructed iteratively starting from Υ 0 = Θ 2 and for i ∈ Z [1,L] , the recursive construction is given by

Υ i = x ∈ X 2 : ∃w ∈ W, N x -Bw2 w ∈ Υ i-1 , ( 16
)
and X 2 ⊆ R (n-p) is a pre-defined set of state constraints for x2 .

Theorem 6 (L-steps RNI set of non-causal descriptor systems) Consider a non-causal descriptor system (1) in with the Kronecker form in (6). The set

Ω = P1 Θ 1 ⊕ P2 Υ, ( 17
)
guarantees that x(k) ∈ Ω, ∀k ∈ Z [0,L] if x1 (0) ∈ Θ 1 and x2 (L) ∈ Υ . PROOF. From (13a), the set Θ 1 is RPI for the dynam- ics of x1 (k). If x1 (0) ∈ Θ 1 , then it follows x1 (k) ∈ Θ 1 , ∀k ∈ Z [0,L] . Meanwhile, the set Υ is L-steps RNI for x2 (k) as discussed in Theorem 5. If x2 (L) ∈ Υ , then it follows x2 (k) ∈ Υ , ∀k ∈ Z [0,L]
. Thus, we obtain Ω by a linear mapping of Θ 1 and Υ as in ( 17). 2

Proposition 1 Consider a non-causal descriptor system (1) in the restricted equivalent form (6) and define a finite-time trajectory

x(k) for k ∈ Z [0,L] with L > 0. If x(0) ∈ Ω 0 for L > s = n -p with N s = 0 and N s-1 = 0, then x(k) ∈ Ω 0 for k ∈ Z [0,L-s] and x(k) ∈ Ω k-(L-s) for k ∈ Z [L-s,L] ,
where

Ω i = P1 Θ 1 ⊕ P2 Υ i with Υ i in (16).
PROOF. For k ∈ Z [0,L-s] , from ( 14), x(k) is contained in the RI set Ω 0 = Θ 2 as defined in ( 17). On the other hand, for k ∈ Z [L-s,L] , the anti-causal component is contained in Υ i , which can be propagated by using ( 16) leading to the confinement of the finite time trajectories for L -s < k < L. 2

4 Active mode detection for discrete-time descriptor systems

In this section, we propose an active mode detection mechanism based on the RPI set characterizations for systems with multiple modes of operation and no switch between different modes. The objective is the identification of the current operating mode in a finite time with any initial state x(0). This operating mode will be detected from a (finite) predefined set of modes of operation. The algorithmic procedure is able to detect the current operating mode based on the offline design of active detection input and online monitoring.

Problem formulation

Consider a family of discrete-time descriptor systems corresponding to multiple modes of operation as

E σ x(k + 1) = A σ x(k) + B σ u(k) + B σ w w(k), (18)
where

E σ ∈ R n×n with rank(E σ ) ≤ n, A σ ∈ R n×n , B σ ∈ R n×m , B σ w ∈ R n×q , and σ ∈ Σ d = {1, .
. . , d} denotes the constant mode index and u(k) ∈ R m denotes an additive input vector at time instant k. It is assumed that the descriptor system (18) is regular and stable for any σ ∈ Σ d , then it follows that matrices (E σ -A σ ) are non-singular.

In order to check the compatibility of any state x(k), based on the Kronecker canonical form in Lemma 4, let us denote the partitioning form x = [x 1 x 2 ] , where x 1 ∈ R p is the dynamical part corresponding to the dynamics (12a) and x 2 ∈ R (n-p) is the algebraic part corresponding to the algebraic equation (12b). Based on this notation, we also denote

B σ = [B σ 1 B σ 2 ] and B σ w = [B σ w1 B σ w2 ] .
Let us also denote the transformation Qi , P i for the descriptor system (18) at mode i ∈ Σ d such that Qi E i P i and Qi A i P i satisfy the Kronecker canonical form in (6).

From the RPI set characterizations in Section 3, the RPI set Pi composed by Pi = P

i 1 Φ i 1 ⊕ P i 2 Φ i 2 with P i = [ P i 1 P i 2 ]. Theorem 7 A state x(k) = [x 1 (k) x 2 (k)
] is compatible with respect to the descriptor system (18) in an

operating mode i ∈ Σ d only if x 2 (k) satisfies x 2 (k) ∈ P i 2 Φ i 2 . ( 19
)
PROOF. Based on the Kronecker canonical form in (6), with the transformation Qi , P i in mode i ∈ Σ d , for a compatible state x(k), the corresponding algebraic equation (12b) should be satisfied. Thus, the condition (19) could be used for checking the operating mode i ∈ Σ d . 2

Based on the above theorem, we state the following corollary without proof.

Corollary 4 For an initial state

x(0) = x 1 (0) x 2 (0) , if x 2 (0) ∈ P i 2 Φ i 2 , then the initial operating mode set Σ d (0) = Σ d \ {i}.
The objective of the mode detection is to decide which mode σ ∈ Σ d is active in (18) by monitoring the current state x(k) and without prior knowledge on w(k) ∈ W. The initial state x(0) is assumed to be known and we make use of the RPI sets of (18) of each mode σ ∈ Σ d as Pσ when u ≡ 0. For a state x(k) of ( 18), ∀k ∈ N, in the mode i ∈ Σ d , we split x(k) = xi (k) + xi (k) with the nominal and perturbed dynamics

E i xi (k + 1) = A i xi (k), (20a)
E i xi (k + 1) = A i xi (k) + B i w w(k), (20b)
where xi ∈ R n and xi ∈ R n .

The basic passive mode detection mechanism (u ≡ 0) can be summarized as follows:

Proposition 2 Consider the compatible initial state x(0) = xi (0) + xi (0) satisfying x(0) -xi (0) ∈ Pi , and let the set of viable modes be initialized as

Σ(0) = Σ d . Given the state measured at time k, if x(k) / ∈ xi (k) ⊕ Pi , then the mode i is not the current operating mode, that is, Σ(k) = Σ(k) \ {i}.
PROOF. The error dynamics xi (k) = x(k)-xi (k) satisfy (20b) and the initialization ensures xi (k) ∈ Pi . If the system (18) is operating in mode i, then the positive invariance of Pi is guaranteed using (20b). Whenever x(k) / ∈ xi (k) ⊕ Pi , the positive invariance is violated and the mode i cannot represent the current operating mode.

2

Remark 7 Assuming Σ d (k) = Σ d (k -1) \ {i}, ∀i ∈ Σ d such that x(k) -xi (k) ∈ Pi , then Card (Σ d (k))
is monotonically decreasing as time k increases. However, one cannot guarantee Card (Σ d (k)) → 1. Example. Consider three modes of operation in (18). As shown in Fig. 1, from an initial state x(0), the mode shown in blue sets is detected after several steps. As time k increases, the modes in red and green sets are discarded. Note that the system state trajectory x(k) may always stay in the intersection of three sets during propagations. Thus, we cannot discard any mode.

This passive mode detection does not guarantee the mode identifiability regardless of the initial conditions. Indeed,

σ∈Σ d
Pσ = ∅ and thus there exists at least a realization w(k), ∀k ∈ N, which does not allow to decrease the cardinality of Σ d (k) and eventually identify the current mode of operation. The active mode detection is intended to enhance the monitoring process by the injection of an excitation signal.

Design of active detection input

For any two different modes i, j ∈ Σ d , the active detection input denoted by u(k) is designed to guarantee P i (k) ∩ P j (k) = ∅ for some k ∈ N, where P i (k) and P j (k) denote the tube of trajectories parameterized by u(k). From (18), the system (18) in modes i and j can be formulated as

E i x(k + 1) = A i x(k) + B i u(k) + B i w w(k), (21a)
E j x(k + 1) = A j x(k) + B j u(k) + B j w w(k). (21b)
Recall that for u(k) = 0 in (21), it follows P i (k) = Pi and P j (k) = Pj .

Similar to (20), assuming the system (18) in mode i ∈ Σ d , we split x(k) = xi (k) + xi (k) with

E i xi (k + 1) = A i xi (k) + B i u(k), (22a)
E i xi (k + 1) = A i xi (k) + B i w w(k). (22b)
With an active detection input u(k), ∀k ∈ N, the state x(0) has to be decomposed as x(0) = xi (0) + xi (0) (for instance in mode i ∈ Σ d ) to satisfy the algebraic equations in the descriptor model ( 18). Based on this observation, we introduce the following proposition to check whether the initial state x(0) is compatible by testing the satisfaction of algebraic equations in (18) for different modes.

Proposition 3 Given the set of modes Σ d . For any i ∈ Σ d such that rank(E i ) < n, if B i 2 = 0, then ∃u(0) such that x(0) ∈ Pi . 2

The result in Proposition 3 shows that descriptor systems have structural advantages in view of mode detection, that is, the algebraic equations in a descriptor systems must hold. When an additional detection input signal is applied, by checking ( 23), some modes can be discarded.

We now present the procedure to design a constant active detection input ū = 0 that can be applied to the system (18) with a finite detection time N T as

u(k) = ū, if k ≤ N T -1, 0, otherwise.
With this constant input ū, ( 22) becomes

E i xi (k + 1) = A i xi (k) + B i ū, (24a)
E i xi (k + 1) = A i xi (k) + B i w w(k). (24b)
Recall x(0) = x 1 (0) , x 2 (0) . The initial condition is given by xi 1 (0) = x 2 (0) and xi 2 (0) satisfies (24a) with ū.

By definition of the RPI set, we denote xi (k + 1) ∈ Pi , ∀x i (k) ∈ Pi , ∀w(k) ∈ W, ∀k ∈ N. The system trajectory in mode i belongs to the parameterized RPI set, that is, x(k) ∈ P i (k) = xi (k) ⊕ Pi , with xi (k) obtained from (24a) and ∀w(k) ∈ W, ∀k ∈ N.

From the nominal dynamics (24a), the stability is guaranteed when the system evolves towards the equilibrium point

xi ∞ = E i -A i -1 B i ū. (25)
In the following theorem, we present the set separation condition for the design of ū.

Theorem 8 For any two modes i, j ∈ Σ d , the sets

P i ∞ = xi ∞ ⊕ Pi , P j ∞ = xj ∞ ⊕ Pj (26)
satisfy P i ∞ ∩ P j ∞ = ∅ if and only if there exists an active detection input ū such that

(E i -A i ) -1 B i -(E j -A j ) -1 B j ū / ∈ S ij . ( 27
)
PROOF. From ( 26),

P i ∞ ∩ P j ∞ = ∅ is equivalent to xi ∞ ⊕ Pi ∩ xj ∞ ⊕ Pj = ∅. ( 28
)
By adding -x ∞ j to the above both sets in (28), we obtain xi

∞ ⊕ -x j ∞ ⊕ Pi ∩ xj ∞ ⊕ -x j ∞ ⊕ Pj = ∅, which can be simplified as xi ∞ -xj ∞ ⊕ Pi ∩ Pj = ∅ leading to (27) based on (25). 2
Let us denote the half-space representation of the set S ij as

S ij = {x ∈ R n : H ij x ≤ b ij } , ∀i, j ∈ Σ d ,
where

H ij ∈ R pij ×n , b ij ∈ R pij
, and p ij is the total number of the linear constraints corresponding to S ij .

Based on the set separation condition in ( 27), the constant active detection input u ∈ [u min , u max ] can be obtained by solving offline the following mixed-integer optimization problem.

Problem 1 (Constant active detection input)

minimize u u 2 , (29a) subject to xi ∞ = E i -A i -1 B i u, ( 29b
) xj ∞ = E j -A j -1 B j u, (29c) u min ≤ u ≤ u max , (29d)
H ij xi ∞ -xj ∞ ≥ b ij -M r ∆ ij + r , (29e) ∆ ij = δ 1 , . . . , δ pij ∈ {0, 1} , (29f) pij l=1 δ l = p ij -1, ∀i, j ∈ Σ d , i = j (29g)
with an arbitrary large positive scalar M r and an arbitrary small positive scalar r .

The optimal solution of Problem 1 defines the constant active detection input ū = u.

Remark 8 The computational complexity of Problem 1 relies on the total amount of decision variables that includes the vector u ∈ R m and the binary variables in ∆ ij . The worst-case complexity of ∆ ij is 2 pij , where p ij is the number of linear constraints for the set S ij .

Example. Consider the same three modes of operation in (18). By solving Problem 1, a constant active detection input ū can be obtained. As shown in Fig. 2, mRPI sets for three modes can be separated by ū. With the constant detection input obtained from solving Problem 1, the guaranteed mode detection result is presented in the following theorem.

Theorem 9 If ū is a feasible solution of Problem 1, then for any initial state x(0), there exists a finite time N T (x(0)) such that the detection Card(Σ d (k)) = 1 is achieved in k ≤ N T (x(0)). Moreover, the convergence time from x(0) to the set P i ∞ denoted as T i c can be computed explicitly for any i ∈ Σ d . Then, the upper bound for the detection time is

N T (x(0)) = max i T i c . (30)
PROOF. By the design of ū, it is guaranteed P i ∞ ∩ P j ∞ = ∅ for any two modes i, j ∈ Σ d . For a given initial state x(0) compatible with the mode i in (24), one has x(T i c ) ∈ P i ∞ independent of the operating mode. But P i ∞ ∩ P j ∞ = ∅ for all i = j and x N T (x(0)) ∈ P i ∞ only holds for the current operating mode. 2

Active mode detection algorithm

Based on the above results, we now propose an algorithm to achieve the mode detection by updating online the active input according to the monitoring of the compatible modes. Overall, this leads to a piecewise constant signal and a detection time upper-bounded by N T (x(0)). Example. By applying the above algorithm to the same example, the simulation result is shown in Fig. 3. From an initial state x(0), the operating mode can be detected at time k = 3 and the system (18) is in th Mode 2 since the state trajectory only stays in the blue set at time k = 3.

Remark 9 Active mode detection can also achieved via online redesign of the separation signals. A receding sequence of detection inputs can be designed with a given detection window through a corresponding mixed-integer optimization structurally similar to Problem 1. Then, online monitoring procedure can be carried out with these time-varying detection inputs. As a result, the operating mode can be detected within a predefined window.

Conclusion

In this paper, we have studied robust invariant set characterizations of discrete-time descriptor systems in both causal and non-causal cases. Based on two restricted equivalent forms of descriptor systems, the explicit results on robust invariant set characterizations are provided. Besides, we have also proposed an active mode detection mechanism based on RPI set-invariance characterizations for discrete-time descriptor system with multiple modes of operation. Based on the separation of RPI sets of descriptor systems, we have proposed a method to design an active detection input and an active mode detection algorithm.

Fig. 1 .

 1 Fig. 1. A passive mode detection example.

  From (22b), we know x(0) = xi (0) + xi (0) and xi (0) ∈ Pi . Based on the nominal descriptor dynamics (22a), xi (0) is also constrained by u(0) at time k = 0. If B i 2 = 0, then xi (0) = 0. Considering the boundedness of Pi and the fact that x(0) = xi (0)+ xi (0), there exists u(0) acting on xi (0) that satisfies (23).

  Fig. 2. Three mRPI sets and separated sets.

Fig. 3 .

 3 Fig. 3. Active model detection with ū. Offline procedure: For any Σ ⊆ Σ d with Card(Σ) ≥ 2, compute ūΣ as the solution of Problem 1.Online procedure: Input an initial state x(0);(i) Initialize Σ d (0) = Σ d ; (ii) Compute the compatible state xi (0) with u(0) = ūΣ d (0) and x(0); (iii) k = 0; (iv) while (Card(Σ d (k)) > 1) 1) for i ∈ Σ d (k), if x(k) ∈ xi (k) ⊕ Pi , then Σ d (k) = Σ d (k) \ {i}; 2) u(k) = ūΣ d (k) ; 3) Update the nominal state xi (k + 1) by (24a); 4) k = k + 1; (v) Obtain Card(Σ d (k)) = 1and the operating mode is detected.

the Minkowski sum is denoted by ⊕.

This work was partially funded by the Spanish State Research Agency (AEI) and the European Regional Development Fund (ERFD) through the project DEOCS (ref. DPI2016-76493-C3-3-R), the FPI grant (ref. BES-2014-068319), the FPI mobility grants (ref. EEBB-I-17-12257 and EEBB-I-18-12900), and by a grant of Ministry of Research and Innovation, CNCS -UEFISCDI, project number PN-III-P1-1.1-TE-2016-0862, MOSCBIOS, within PNCDI III. The material in this paper was partially presented at the 56th