Laurent Lesnard
email: laurent.lesnard@sciences-po.fr

Man Yee Kan

Two-Stage Optimal Matching Analysis of Workdays and Workweeks

Keywords: Cost, Optimal Matching, Time Use, Two-Stage Optimal Matching, Work Schedule, Work Time

We apply Optimal Matching (OM) at two stages for the analysis of workdays and workweeks using data from the UK 2000 Time Use Survey. We only employ substitutions but no insertion or deletion when calculating the distance matrix between sequences. The costs are defined according to the transitional frequencies of events at a given time. Our study demonstrates how OM can be adapted to the number of periodicities and theoretical concerns of the topic by adjusting its costs and parameters. There are 7 main types of workweeks in the UK and standard workweeks account for only 1 in 4 workweeks.

Introduction

Work time is an important dimension of quality of life and social stratification. Previous studies showed that the average work time has been on the decline whereas that on leisure has increased since the 1960s in industrialised countries [START_REF] Dumazedier | Toward a society of leisure?[END_REF][START_REF] Robinson | Time For Life. The Surprising Ways Americans Use Their Time[END_REF][START_REF] Gershuny | Changing Times: Work and Leisure in Postindustrial Society[END_REF]. Yet recent research has shown that the decrease in work time has been the most prevalent among low-earning and low-status workers but has increased in the case of higher-grade professionals and managers. The two opposite trends in work time have progressively reversed the social class-work time gradient so that at the beginning of the 2000s, high skilled workers have longer work hours than unskilled ones [START_REF] Gershuny | Changing Times: Work and Leisure in Postindustrial Society[END_REF]. Relatively few studies, however, have focused on the scheduling of work time, although it is obvious that not only the total amount of work time but also its scheduling are very significant to one's social and family life. In this article, we introduce a statistical technique that is particularly useful for analysing the scheduling of time use. We call this technique "two-stage optimal matching" (2SOM). We employ it for the analysis of 7-day diary data from the UK 2000 Time Use Survey to define a typology of workweeks.

A handful of studies have explored the scheduling of everyday activities using time diary data. Time diary data are usually collected by respondents' records of their activities at every 10 minute or 15 minute time slots. Unlike questionnaire stylised data, these data not only provide information on the amount of time spent on daily activities but also the scheduling of these activities. However, before the advent of Optimal Matching Analysis (OM), these rich data, especially their scheduling dimension, were not thoroughly analysed. For example, the first graphical representations of the timing of daily activities can be found in [START_REF] Szalai | The Use of Time. Daily Activities of Urban and Suburban Populations in Twelve Countries Mouton[END_REF]. [START_REF] Wilson | Activity pattern analysis by means of sequence-alignment methods[END_REF] first applied OM to time diary data to explore the timing of daily activities. [START_REF] Lesnard | Schedules as sequences: A new method to analyze the use of time based on collective rhythm with an application to the work arrangements of french dual-earner couples[END_REF] introduced an advanced version of OM, Dynamic Hamming Matching, DHM, which is particularly adapted to the analysis of time use data. Using this sequence analysis variant, researchers have identified a variety of workdays in France (Lesnard 2006a) and in Belgium [START_REF] Glorieux | The coming of the 24-hour economy? Changing work schedules in belgium between 1966 and 1999[END_REF]. The workdays include "shifted" (morning, evening or night shifts), "fragmented" (two short work spells where there is a long break between them), "short" (a short spell of work episode), standard (9-to-5 work hours), and "long" (a long spell of work episode). In France, standard workdays increasingly gave way to non-standard work schedules between the 1980s and the 1990s. [START_REF] Lesnard | Off-scheduling within dual-earner couples: An unequal and negative externality for family time[END_REF] suggests that the expansion of non standard workdays is one of the key factors to explain the increase in desynchronization of work time for dual-earner couples.

Most studies on the scheduling of work have been confined to one day diary data despite the fact that work is very likely organized according to longer time framesone week to the least. Average workweek time is regularly measured and reported in official labour statistics. The number of weekly work hours is also a topic of frequent debates and negotiations among policy makers, trade unions and academics. Nevertheless, relatively little attention has been given to scheduling of those work hours over the week. Admittedly, the analysis of workweek schedules has been restricted by the lack of suitable data (most time use surveys only collected day-long rather than week-long diaries). Furthermore, analysing the scheduling of work over the week is methodologically more challenging than focusing on workdays because it requires taking into account both of the scheduling of work hours within the day and that of workdays over the week.

Following the recent Eurostat guidelines on collecting time use data, some recent national time use surveys (e.g. in UK 2000, France 1999, Belgium 1999and Finland 2000) have collected 7-day working time data using the "workweek grid" method [START_REF] Robinson | Measuring the complexity of hours at work: The weekly work grid[END_REF]. The week-long time use data provide great opportunities for researchers to investigate patterns of work schedules, but so far very few studies have fully exploited the strength of these using advanced statistical techniques. For example, when examining individuals' workweeks and the synchronicity of work time of dual earner couples, [START_REF] Chenu | Synchronicity in the work schedules of working couples[END_REF] adopted a series of indicators and numeric indexes such as the length of workweek and the amount of work during weekends to analyse individual workweeks rather than a systematic tool of estimating the distance among different types of workweeks and that between partners' work time.

Optimal Matching and cost setting

Methods for describing sequential data (e.g. data concerning lifecycle events, and career trajectories) have been available to social scientists for more than three decades. Among these methods, the most popular one is certainly Optimal Matching Analysis (OM), introduced to the social sciences by Andrew Abbott and his colleagues in the 1980s 1 . OM is basically a distance measure adapted to sequence data in which dissimilarity between two sequences is given by the minimal total cost to match them [START_REF] Kruskal | An overview of sequence comparison. Time warps, string edits, and macromolecules: the theory and practice of sequence comparison[END_REF][START_REF] Durbin | Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids[END_REF]. Three transformations are allowed to transform one sequence into the other one: insertion, deletion, and substitution. The total cost of any matching is the sum of the weighted number of transformations required. The lowest matching cost is used as the measure of the dissimilarity between two sequences. As a result, the kind and the number of transformations used depend on the relative cost of insertion-deletion and substitution. Insertion and deletion, commonly called indel, are completely symmetrical in OM and therefore are given the same cost.

Despite OM has been used for more than three decades, users are often uncertain about how to set costs and how this might affect results [START_REF] Wu | Some comments on "sequences analysis and optimal matching methods in sociology: Review and prospects[END_REF]. As [START_REF] Stovel | Ascription into achievement: Models of career systems at lloyds bank, 1890-1970[END_REF] put it, "The assignment of transformation costs haunts all optimal matching analyses". Statistical software in the early 1980s were not yet well adapted for analysing sequence data, and empirical data analyses based on them usually took hours or even longer to complete. Hence it was difficult to test how different values of cost might lead to different results. Considerable progress has been made on this issue with the developments in relevant statistical packages and programs. Furthermore, social scientists have increasingly been employing, experimenting with, and reflecting on optimal matching.

In social science research, sequence data are usually concerned with events and time, which determine how the two kinds of operations (i.e. indel and substitutions) are used in OM. Indels warp time in order to match identically coded but remote events while substitutions focus on the comparison of contemporaneous but distinct events [START_REF] Lesnard | Introduction aux méthodes d'appariement optimal (optimal matching analysis)[END_REF]. The succession of indels and substitutions for matching two sequences can be seen as a series of acceleration and deceleration to match identically coded events, as well as a couple of replacements of states by one another. The ratio of indel cost to substitution one determines whether it is preferable to simplify time or events when comparing pairs of sequences. In fact, what is now commonly known as Optimal Matching was originally a refinement suggested by Vladimir [START_REF] Levenshtein | Binary codes capable of correcting deletions, insertions, and reversals[END_REF] to improve the similarity measure introduced by Richard [START_REF] Hamming | Error-detecting and error-correcting codes[END_REF], who measured similarity by the number of identical contemporaneous tokens (see Table 1). When no substitution operations are used, whether because they are not allowed as in the Levenshtein II distance or because their cost is greater than twice the indel one,2 then OM is equivalent to finding the longest common subsequence [START_REF] Kruskal | An overview of sequence comparison. Time warps, string edits, and macromolecules: the theory and practice of sequence comparison[END_REF] 3 , Time warps, string edits, and macromolecules: the theory and practice of sequence comparison, 1-44}. When no indel operations are used (because they are not allowed by definition, e.g. Hamming distance or their cost is greater than the substitution one), OM amounts to counting the number of dissimilar contemporaneous events4 . As a result, the ratio of indel cost to substitution one determines the kind of similarity that OM will be most sensitive to (see Figure 1). If the timing of events is not too important, then users should favor costs close to Levenshtein II. On the contrary, high indel costs should be used when timing is important for the analysis. Users should set the same cost for indel and substitution if they want to use both kinds of operations in a more or less balanced way (Lesnard 2009a).

Figure 1 -Patterns corresponding to costs in OM

Unlike the three historical OM variants described in Table 1, it is common, especially in social research, to define a substitution cost for each pair of states5 . This approach makes it possible to have some substitution costs higher as well as some lower than the indel cost. When it is higher, the two different events will not be substituted but the algorithm will try to shift the two sequences so as to find identical but shifted subsequences. When it is lower, the two different events will be substituted and be kept locally in synchronization. However it should be noted that defining a substitution matrix amounts to deciding a priori that some pairs of states are closer to one another than others. This a priori knowledge can be informed by theory (for instance, see [START_REF] Halpin | Class careers as sequences: An optimal matching analysis of work-life histories[END_REF], hypotheses (for an illustrative example, see [START_REF] Stovel | Ascription into achievement: Models of career systems at lloyds bank, 1890-1970[END_REF], previous findings or even the sequences themselves6 .

When adapting OM to various kinds of data and research questions, the flexibility offered by substitution costs is even greater than in the case of defining a pairwise substitution matrix. For example, [START_REF] Lesnard | Schedules as sequences: A new method to analyze the use of time based on collective rhythm with an application to the work arrangements of french dual-earner couples[END_REF]Lesnard ( , 2009a) ) recommends a variant of Hamming matching called Dynamic Hamming Matching, in which only substitution operations with time-varying costs inversely proportional to the series of transition matrices are used, to be applied to the case where sequences are of equal length and that preserving the timing of event is crucial. Indel costs are usually set relatively to substitution ones, but a notable exception is the OMv dissimilarity measure (introduced by [START_REF] Halpin | Optimal matching analysis and life course data: The importance of duration[END_REF]) in which the indel cost is inversely proportional to the spell length so as to take the duration of an event spell into account.

The versatility of OM is also illustrated when it is applied to multiple domain sequences, i.e. sequences unfolding in several distinct, but theoretically interdependent, domains such as family, employment and housing careers [START_REF] Pollock | Holistic trajectories: A study of combined employment, housing and family careers by using multiple-sequence analysis[END_REF]. Although it is possible to conduct sequences analyses for each of the domains, a simpler way is to define substitution matrices for each and then combine them [START_REF] Stovel | Ascription into achievement: Models of career systems at lloyds bank, 1890-1970[END_REF][START_REF] Pollock | Holistic trajectories: A study of combined employment, housing and family careers by using multiple-sequence analysis[END_REF], a method which is called Multiple Sequence Analysis (MSA). In Gary [START_REF] Pollock | Holistic trajectories: A study of combined employment, housing and family careers by using multiple-sequence analysis[END_REF] study, the substitution cost concerning employment statuses (e.g. employed or self-employed) as well as housing tenure statuses (e.g. owning with a mortgage or owning outright) is considered to be equal to the sum of the substitution costs concerning respectively employment statuses and housing tenure statuses. Hence the multiple domain issue can be dealt with by multiple OM analyses or by combining multiple substitution matrices for a single OM analysis 7 .

As suggested by Andrew [START_REF] Abbott | Reply to levine and wu[END_REF], the hypotheses on which OM rests are not about how data are generated 8 but on the kinds of patterns that users expect to see prior to the analysis. In standard OM, the ratio of indel cost to substitution one is a kind of slider with which users can choose among infinite combinations of patterns, ranging from the number of similar contemporaneous events to the longest common subsequence. The importance attached to the timing of sequences decreases as the cursor moves along. Focusing the analysis on a certain type of pattern does not imply, as unfairly criticized by [START_REF] Levine | But what have you done for us lately?: Commentary on abbot and tsay[END_REF] and [START_REF] Wu | Some comments on "sequences analysis and optimal matching methods in sociology: Review and prospects[END_REF], that OM will create ex nihilo this particular pattern. It just implies that the pattern being looked for will be easier to be identified from the data if it exists. Using multiple substitution costs enables researchers to tune OM finely for particularly nature of data and research questions. But, as illustrated by Multiple Sequence Analysis, the flexibility offered by OM is actually greater than that by setting a matrix of pairwise substitution costs. In what follows, we use workweeks as an example to demonstrate how OM can be adapted to cope with long sequences and multiple periodicities.

Analysing workdays and workweeks

The data come from the UK Time Use Survey (UKTUS) 2000-01, carried out by the Office for National Statistics from June 2000 to September 2001 [START_REF] Ipsos-Rsl | United kingdom time use survey[END_REF]. The sample is nationally representative of about 6,400 households in the UK. The response rate was 60%. All individuals aged 8 or above in the households were requested to fill in individual questionnaires and diaries. In addition to the traditional 2 single day diaries (one in the week, one in the weekend), the UKTUS also collected 7-day workweek grid diaries. In each day of the workweek grid diaries, the time is divided into 96 15-minute slots. Respondents were requested to indicate their work or study episodes by drawing a line across the start and the end of each of their work or study episode. They were also instructed to exclude travelling time and meal breaks in their work or study time.

The design of the diaries, however, does not enable users to distinguish between work and study spells. In order to build a typology of workweeks for the present study, only respondents in employment (part-time or full-time), as reported in the household questionnaires, were selected. Of the 9,823 respondents who filled in the week grid diaries, 4,944 were in employment and recorded work time on at least one of the 7 7 In the case of multiple OM analyses, domains are assumed to be independent (correlations may only be identified from the results). When combining multiple substitution matrices for a single OM analysis, domains are assumed to be interdependent. 8 That is, the aim of the analysis is not to provide a model of underlying processes, but rather to describe the data.

days. In the workweeks, there are 21,122 workdays in which respondents recorded at least one work episode9 .

Investigating workweek patterns involves analysing two nested periodicities: days within weeks and hours within days. At the level of the day, the focus should be placed on the scheduling of work hours. At the level of the week, it is rather which days are scheduled for work that is of interest. Although it is possible to apply OM directly to the 672 15-minute time slots of the workweek grids, it will be more appropriate to take account of these two nested periodicities in the analysis as workers are likely to schedule their work time at two stages in real life. The issue of intra-day work time variations over the week is similar to that of seasonality in time-series analysis. As the main goal of time-series analysis is to model trends (e.g. trends of the unemployment rate), seasonality is often considered not directly relevant and therefore only controlled by modelling it separately. In the case of scheduling of work hours within a day, intra-day variation is a much more important issue than the case of seasonality in time-series analysis, so it should be analysed separately.

Our approach is to apply OM in two steps, a method we call two-stage optimal matching (2SOM). At the first stage, OM is applied to the 96 15-minute time slots to define typologies of workdays. The sample is consisted of day-long diaries recording at least 15 minutes of work time (there are repeated records from respondents who had more than one work day in the week). 21,122 workdays are derived from the original 4,944 workweeks. These sequences are made of two states: work and non work. Cluster analysis is applied to the resulting dissimilarity matrix to produce a typology of workdays. At the second stage, OM is employed to analyse 7-day weeks, which were made of the types of workdays identified in the first stage. The states include the types of workdays (e.g. standard, long and so on) and "rest", a category to take into account the days with no work at all.

As mentioned in the previous section, the costs at the two stages should be set according to the importance of timing for the analysis. Focussing on timing is certainly crucial for the first stage, which is concerned with the scheduling of work time during the day. The parameters used should be those on the Hamming distance pole in Figure 1. We will use a variant of Hamming distance, Dynamic Hamming Matching, DHM [START_REF] Lesnard | Schedules as sequences: A new method to analyze the use of time based on collective rhythm with an application to the work arrangements of french dual-earner couples[END_REF], which has been applied to the analysis of work schedules in recent studies [START_REF] Glorieux | The coming of the 24-hour economy? Changing work schedules in belgium between 1966 and 1999[END_REF], Lesnard 2006a, Lesnard 2006b[START_REF] Lesnard | Off-scheduling within dual-earner couples: An unequal and negative externality for family time[END_REF]). In DHM, only substitution operations are used. In order to make the costs sensitive to the timing of sequences, their values are defined to be varying with time and inversely proportional to transition frequencies between pairs of states at a particular time. The rationale for DHM is that transition frequencies between states reveal their relative distances at a given t. A high frequency of transitions between any two states at t indicates that many individuals switch states at the time and therefore the likelihood that these two states belong to the same type of trajectory at that time is high. As a result, the distance between these two states is considered to be short. On the contrary, a low frequency of transitions suggests that the two states belong to two different types of trajectory and hence their distance is considered to be long at t. DHM fits well with the requirements of the first stage of the analysis. At the second stage, timing is crucial too. For example, working on an evening shift is likely to have different implications for social life on different days of the week (e.g. Saturday vs. Monday). Thus DHM is also an appropriate parameterization for the second stage. We hence analyse workweeks with two-stage Dynamic Hamming matching. However, it should be noted that different OM parameterizations could have been used at the different stages of the analyses. Figure 2 summarizes the analyses we have conducted. At each of the two stages of the analysis, we employ the method beta-flexible [START_REF] Belbin | A comparison of two approaches to beta-flexible clustering[END_REF][START_REF] Milligan | A study of the beta-flexible clustering method[END_REF] to calculate the distance between groups (as opposed to the original elements that is called linkage in cluster analysis). Beta-flexible has proved more robust to recover structure in presence of outliers and noise than Ward's or other classical linkages [START_REF] Milligan | An examination of the effect of six types of error perturbation on fifteen clustering algorithms[END_REF][START_REF] Milligan | A monte carlo study of thirty internal criterion measures for cluster analysis[END_REF]. We conduct the sequence analysis by using the seqcomp plugin in Stata10 . We used SAS for the cluster analysis because the beta-flexible linkage is not implemented in Stata11 . Results

First stage

At the first stage of analysis, we focused on the 34, 608 days in the 4,944 weeks from the sample 12 . We applied Dynamic Hamming Matching (DHM) to the 21,122 days with at least one work spell (61% of the days). Nevertheless the sample was too big for the 4 Gb memory limit imposed by 32-bit systems of our computers. We thus split these 21,122 days into two sub-samples and conducted two analyses separately 13 . Agglomerative hierarchical clustering (linkage: beta-flexible with ß = -0.3) is applied to the two distance matrices produced by DHM. There are no definitive criteria to determine the number of clusters, but the "elbow criterion" usually gives interesting starting points 14 . In both samples, the first significant spike in the intergroup distance occurs for the seven-group partition, suggesting that very dissimilar groups have just been joined and hence there are at least eight types of workdays in the data. Another smaller spike is observed in the 9-cluster solution of the second sample. We examined and compared visual representations of the clusters and the summary statistics (including average total worktime, medians of the start, the middle and the end time of the workday) of the different partitions between 11 and 8 groups.

12 1,078 out of the 34,608 days (3.11%) have missing values. Visual inspection reveals that these missing values appear to be coding errors caused by a single data coder, who coded "missing" instead of "zeroes" for work spells during work days and for all values during non-work days. Once these false missing values were replaced with zeroes, no further missing value is found. 13 Clustering a 21,122 x 21,122 matrix with SAS 9.1 on a MS Windows 32-bit computer was not possible. The sample was randomized before being split. 14 An elbow, or a spike, in the intergroup distance indicates that the two very dissimilar clusters have been merged. In such a case, the cluster solution just before this merging should be considered rather than the one just after.

Figure 3 -Typology of workdays

Finally, we adopted the 10-cluster solution as it is the most succinct one that incorporates all major types of part-time workdays, which are important characteristics of the UK workdays, and all the other major categories. We moved on to match identical clusters in the typologies from the two samples based on tempograms15 and summary statistics. This step has been straightforward as the different types of workdays identified from the two randomized sub-samples are highly similar to each other, as well as to typologies found from previous studies [START_REF] Glorieux | The coming of the 24-hour economy? Changing work schedules in belgium between 1966 and 1999[END_REF], Lesnard 2006a, Lesnard 2009b). The final typology is summarized in Table 2 and represented graphically in Figure 3. The first two types are standard workdays (Type 1, Clusters 1 and 2). The first one, which is also the most common type of workday in the UK, is the traditional 9-to-5 workday. Workdays of this group start at around 9 am and end at around 5 pm. The second type of standard workdays is a variant of the 9-to-5 one, but the starting time and the end time are both one hour earlier -we hence call it the 8-to-4 workday. Although being the most common type of work schedules, standard workdays (9-to-5 and 8-to-4) account for just over half of the workdays in the UK (52.1%). Other types of workdays deviate from the standard workdays in two main ways: length and schedule. Type 2 of workdays, long workdays (Clusters 3 and 4), have distinctly long total work time (over 10 hours). There are minor differences between them: the former is a longer version of the 9-to-5 workdays and the latter is characterized by evening work in the workplace or at home. Long workdays constitute 15.5% of total workdays. There are three groups of shorter workdays. Two of them are part-time workdays (Type 4, Clusters 8 and 9), which make up 14.3% of the total workdays. They have the major work spell in the morning or in the evening. The third one, short workdays (Type 5, Cluster 10) contains very short total work time and is characterized by multiple, short, and staggered work spells. Short workdays represent 4.6% of total workdays. The final type, shift workdays (Type 3, Clusters 5, 6 and 7), depart from the standard ones in their schedules. There are three types of shifts: morning, evening, and night, which add up to 13.4% of the total workdays. The total work time of these shift workdays is more or less the same as the standard ones. However, most of the work on these days is carried out before 9 am or after 5 pm. Interestingly, morning shift is the most common among the three types of shift workdays, whereas night shift is the least common form.

Figure 4 -Simplified Workweeks

To build simplified workweeks, we employ a 5-category typology as described in the above paragraph and in Table 2 16 . Furthermore, we add the category rest days to take account of the days that contain no work spell. Therefore every week contains 7 episodes (Monday to Sunday) and 6 states (5 types of workdays and 1 type of rest days). The visual representation of these simplified workweeks is given in Figure 4.

The proportion of each of the 5 types of workdays remain more or less stable during weekdays. Standard workdays make up about 60% of the weekdays. As expected, the results are very different during weekends. Work is uncommon on Saturdays and Sundays. It is worth mentioning that the proportion of non standard workdays (shift, part-time and short) is much higher on weekends than on weekdays. That is, weekend work is atypical, and the types of workdays on weekends are more likely to be atypical as well.

Second stage

To build a typology of workweeks, we run DHM on these simplified workweeks. In Figure 5, the intergroup distances for the series of nested partitions indicate that there are at least five types of workweeks. Two other, smaller, spikes occur at the 10-cluster and 18-cluster solutions. The results suggest that we should examine the series of partitions ranging from 18 to 5.

Figure 5 -Cluster solutions and DHM distance (beta-flexible linkage)

We first tried to reduce the number of groups starting from the 18-group solution by the algorithm results. However, some of the groupings suggested did not seem appropriate 17 . We therefore decided to reduce the number of groups manually based on descriptive statistics and visual representations of the cluster solutions. Cluster analysis is basically an algorithm used to produce nested sets of clusters. As an algorithmic method, it is based on the repetition of a finite sequence of simple and intuitive instructions. In the beginning, each element is a separate cluster and on each step the two "closer" clusters are merged 18 . This sequence of instructions could be done manually, though it is much more efficient to run it by computers. However, efficiency comes at the expense of rigid rules which may not be entirely satisfactory for the last steps of the grouping. For the last few steps of the agglomerative clustering, successive groupings suggested by the algorithm are not theoretically or empirically satisfactory. We therefore conduct the grouping by linking clusters that are similar in theoretically important characteristics (e.g. number of workdays, proportion of work done on weekends, and total work hours). To adopt results

17 For instance, the first grouping suggested by the algorithm is to combine clusters 5 and 16. 18 At first, each element forms a separate cluster. The original dissimilarity matrix determines the distances among the clusters at this stage. However, when two elements are combined to form a new cluster, the original matrix cannot be referenced to directly because it does not provide information about the distance between the newly formed cluster and the rest of the original elements. At this stage, we conduct the linkage manually.

different from those suggested by the algorithm, however, one has to provide convincing arguments and justifications. Figure 6 shows the eighteen-cluster solution. Table 3 displays descriptive statistics on the eighteen clusters and how we have reorganised them. Cluster 2 represents the standard 9-to-5 Monday-to-Friday workweek (standard workweek). We combine Clusters 8, 10, 13, 14, and 16 into a single category, long workweek, because they are all characterized by long work hours over the week (the average workweek time is longer than 45 hours, and in three of them it is over 48 hours, i.e., the maximum limit suggested by the European Working Time Directives). We then group Clusters 6, 7, and 15 together because they all consist of shift hours on workdays (shift workweeks). Clusters 9 and 12 form another category, alternate workweeks, in the new typology. Unlike other types of workweeks, they are not composed of a uniform type of workdays. Instead the types of workdays vary over the week (e.g. standard hours on Monday, shift hours on Tuesday, part-time work on Wednesday and so on). [START_REF] Gershuny | Changing Times: Work and Leisure in Postindustrial Society[END_REF]. Our typology goes further to identify the distribution of workdays over the week, and that working on weekends is a key characteristic of the work time trends. Finally, Type G, short workweeks, which represent 15% of the workweeks, are composed of only three work days. Nevertheless, standard work hours are usually involved during workdays. The average weekly work hours are short (21.6 hours). In fact, short workweeks can also be defined as a variant of part-time work in the UK.

Similarly with Types E and F, there is a high proportion of weekend work (9.6%).

Figure 8 -Tempograms of the final typology of workweeks (15-minute time slots)

To assess the effectiveness of two-stage optimal matching, we represent the typology using the 672 15-minute episodes in Figure 8. As can been seen, the seven types are very distinct from one another, suggesting that the first step of 2SOM is effective. The strength of 2SOM lies in making the interpretation of results easier. The results would have been much more difficult to interpret if we had not obtained the first stage results. It should be noted that, however, applying OM directly on the 672 15-minute episodes yields slightly different results from 2SOM19 .

Advantages of analysing both workdays and workweeks

As can be seen from Figure 7, workweeks are usually dominated by one type of workdays: for example, long workweeks are composed of mostly long workdays, and shift workdays are common in shift workweeks. This suggests that work is not randomly scheduled over days and weeks, but is instead highly temporally structured. Previous research demonstrated that work schedules mostly reflect the preferences of the employers rather than those of the employees [START_REF] Lesnard | Off-scheduling within dual-earner couples: An unequal and negative externality for family time[END_REF][START_REF] Golden | Flexible work schedules: Which workers get them?[END_REF].

This lack of variation in the types of workdays within a workweek gives some confidence to researchers who only have day-long time use data: that analysing how work is organised at the level of the day is likely to give good insights into how work is scheduled over a longer period. Nevertheless, our findings also show that there is one major drawback from this approach: the overall proportion of atypical or nonstandard workweeks will be underestimated if the figures are generalized from the analysis of workdays alone. It is because standard workdays, though not being the dominant types of workdays, occur also in long, shift and all types of part-time workweeks. In other words, observing a standard workday in a sample is not a good predictor of whether or not the rest of the week will be made of only standard workdays. In contrast, a non standard workday is an excellent predictor of atypical workweeks.

Thus, researchers will overestimate the proportion of standard workweeks based on the number and proportion of standard workdays in their samples. In this study, 52% of wordays are standard (31% when both workdays and rest days are taken into account), but standard workweeks only account for 26% of the total workweeks. On the other hand, long workdays represent 16% of the workdays but 20% of the workweeks are long. In sum, the proportion of workweeks will be more accurately represented should the analyses be conducted at both the day-and the week-levels rather than solely at the day-level.

Conclusion

We have demonstrated that two-stage optimal matching (2SOM) can be usefully applied to the analysis of workweeks. Our study surpasses past ones by providing important insights into the schedule of work hours within workdays and the structure of work days over the week. We have identified 7 types of workweeks and more varieties of part-time work in the UK.

Methodologically, this study has contributed to the on-going reflections and discussions on how costs should be set in OM. We suggest that costs should be defined in accordance with the kind of patterns that researchers expect to see or consider theoretically interesting. For example, we have defined costs based on transitional frequencies of two states at a given time in this study. Setting the costs based on theoretically concerns will help to emphasize and capture certain patterns effectively. But it does not necessarily imply that the results will be completely changed should a different cost is chosen. In cluster analysis, there are often stable clusters, which exist in the outputs however the parameters are set20 . But the proportion of the stable clusters may vary with the cost. That is, non core cases of a cluster are prone to move to another cluster if the algorithm changes. In this regard, to stress a certain kind of patterns in OM by defining the ratio of indel cost to substitution ones is tantamount to adjusting the sensitivity of border cases to the stable clusters. When indel costs are low compared to substitution ones, the distance will favour the number of identical events regardless of their location in the sequences. When they are high, similarity of the border cases will be estimated according to their existing positions in the sequences.

In the case of the scheduling of work, it is essential to compare sequences based on their local similarity (otherwise the schedule of events itself will be altered). Furthermore, we employ time-varying substitution costs because transition frequencies provides significant empirical and theoretical information on sequence proximity. To apply OM on other research topics, we recommend users define the costs based on theories and previous findings. When no previous reference is available, researchers may adopt neutral costs (i.e. close to Levenshtein I) so as to favour neither local nor remote similarity.

OM is a versatile technique that can easily take into account two periodicities in its analysis. Although it is possible to apply OM directly to the 672 episodes, it is more appropriate to focus on each of the nested periodicities so that the patterns found are clearer and easier to be identified at each stage. The two-stage optimal matching (2SOM) for cases of nested periodicities is analogous to noise filtering or seasonal adjustment in time series analysis. The first stage of OM, in which analyses are guided by theories and previous findings, acts as a form of noise filtering.

Finally, we have a suggestion for researchers who intend to apply 2SOM on other workweek data. The UK 2000 time use data used in this study, unlike other survey data such as the France Time Use Study 1998-1999, do not contain information about whether the respondents considered the workweeks filled in to be a "normal" week. Consequently our sample is likely to contain respondents who were on vacation or had some usual work patterns. In future research, where possible, researchers should restrict their analyses to workweeks that respondents indicated as normal.

Figure 2 -

 2 Figure 2 -Synoptic representation of Two-Stage Dynamic Hamming Matching applied to workweeks.

Figure 3 -

 3 Figure 3 -Tempograms of the typology of workdays

Figure 6 -

 6 Figure 6 -The eighteen-cluster solution produced by flexible-beta clustering

Figure 7 -

 7 Figure 7 -Tempograms of the final typology of workweeks

Table 1 -The three historical Optimal Matching distances

 1

		Operations used and costs
		Substitution	Insertion and deletion
	Hamming	Yes (cost=1)	No
	Levenshtein I	Yes (cost=1)	Yes (cost=1)
	Levenshtein II	No	Yes (cost=1)

Table 2 -Types of workdays

 2

	Name		Size (%)
	1 Standard	9 to 5	34.92	52.13
		8 to 4	17.21
	2 Long	long	10.76	15.54
		long day and evening	4.78
	3 Shift	morning shift	6.65	13.42
		evening shift	3.74
		night shift	3.03
	4 Part-time	part-time morning	9.66	14.33
		part-time afternoon	4.67
	5 Short	short atypical	4.56	4.56

Table 3 -Descriptive statistics of the eighteen-cluster solution produced by beta- flexible clustering

 3

	Origi nal clust er id	New cluster id	Size	Work time	Num ber of days off	Num ber of work days	Proport ion of work on Sat.	Proport ion of work on Sun.	Proportio n of work on weekend	Proportion of full Saturdays off	Proportion of full Sundays off
	2	1 26.10	42.24	1.76	5.24	4.97	3.19	4.08	80.43	86.90
	8	2	4.39	49.18	1.64	5.36	10.37	8.23	9.30	59.49	72.31
	10	2	3.62	60.53	1.21	5.79	40.81	30.21	35.51	6.83	25.47
	13	2	5.40	46.37	1.95	5.05	7.63	5.13	6.38	72.50	80.00
	14	2	4.57	57.20	1.59	5.41	10.00	6.06	8.03	62.56	81.28
	16	2	2.00	46.84	1.55	5.45	8.82	5.27	7.05	56.18	74.16
	6	3	4.03	31.78	2.23	4.77	13.86	9.60	11.73	53.63	65.92
	7	3	5.67	42.94	1.51	5.49	15.89	14.87	15.38	45.63	51.98
	15	3	2.41	35.08	2.11	4.89	3.32	4.53	3.92	88.79	84.11
	9	4	4.90	35.14	1.61	5.39	16.26	13.10	14.68	38.99	51.38
	12	4	2.05	23.34	1.98	5.02	6.81	6.42	6.62	57.14	73.63
	11	5	6.86	20.39	2.30	4.70	5.93	3.87	4.90	73.77	81.31
	17	5	2.27	23.12	3.03	3.97	5.13	3.19	4.16	75.25	88.12
	4	6	5.83	36.76	2.32	4.68	12.56	6.99	9.78	59.07	77.61
	5	6	4.54	31.15	3.08	3.92	10.10	3.44	6.77	66.83	87.13
	1	7	7.24	13.06	5.11	1.89	9.15	7.08	8.12	67.70	74.53
	3	7	6.41	30.89	2.85	4.15	11.40	8.39	9.89	63.16	70.53
	18	7	1.71	23.08	3.78	3.22	14.39	15.75	15.07	57.89	52.63

Table 4 -Descriptive statistics of the final typology of workweeks

 4 Type C, shift workweeks, constitute 12% of the total workweeks. Like long workweeks, they have a high proportion of weekend work (12%). The average work time is shorter than the standard workweek (37.7 hours). From the cluster (Cluster 4, Figure7), we see that in the case where Saturday or Sunday is a workday, a rest day will take place between Monday and Friday. In other words, shift workweeks are not only characterized by shift hours of work, but also a shift of the days off from weekends to weekdays.Alternate workweeks, Type D, are made of more than one type of workdays. There is a high proportion of part-time workdays and hence the average weekly work time is considerably short (31.7 hours). Weekend work is also common in this type of workweeks (12.3%). Accordingly, days off tend to shift towards Monday to Friday. It is well documented that part time work rate is relatively high in the UK compared with other developed countries. In the present study, we have identified two main types of part-time workweeks, which together form about one fifth of the total workweeks. Type E, the standard workday part time, is similar with the standard workweek but have about one more day off (2.5 c.f. 1.8). Type F, part workday parttime, is characterized by part workweek (2.7 days off on average) and part-time work hours during workdays. Both Types E and F of part-time workweeks have shorter total work hours than the standard workweeks (the figures being 21.1 and 34.3 respectively). Furthermore, weekend work is more common than the standard workweek, especially working on Saturdays. These results are consistent with previous studies on work time trends, which show that long and short workdays are both increasingly common in economically advanced societies

	Name	Size	Work	Num	Num	Proport	Proport	Proportion	Proporti	Proporti
			time	ber of	ber of	ion of	ion of	of work	on	of	on	of
				days	work	work	work	on	full		full
				off	days	on Sat.	on Sun.	weekend	Saturday s off	Sunday s off
	A Standard	26.1	42.2	1.8	5.2	5	3.2	4.1	80.4		86.9
	B Long	20	52.1	1.6	5.4	14.9	10.6	12.7	53.8		68.1
	C Shift	12.1	37.7	1.9	5.1	12.7	11.1	11.9	56.9		63
	D Alternate	7	31.7	1.7	5.3	13.5	11.1	12.3	44.3		57.9
	E Part-time I	9.1	21.1	2.5	4.5	5.7	3.7	4.7	74.1		83
	F Part-time II	10.4	34.3	2.7	4.3	11.5	5.4	8.5	62.5		81.8
	G Short	15.4	21.6	4	3	10.7	8.6	9.6	64.7		70.4

If the cost for substituting two events is higher than that for one insertion and one deletion, then substitutions are never used[START_REF] Kruskal | An overview of sequence comparison. Time warps, string edits, and macromolecules: the theory and practice of sequence comparison[END_REF].

The dissimilarity measure suggested by[START_REF] Elzinga | Sequence similarity: A nonaligning technique[END_REF] does not belong to the Optimal Matching family. However, if it had to be located on this scale, it would be on the far right of it.

When indel operations are allowed but are penalized by a very high cost, they are used only when sequences are not of equal length. In such cases, the resulting dissimilarity is likely to be defined above all by the length of sequences.

A recent exception is[START_REF] Brzinsky-Fay | Lost in transition? Labour market entry sequences of school leavers in europe[END_REF], who set indel cost to 1 and substitution cost to 2.

Frequencies of transitions between states have often been used as an inverse measure of distance between states (e.g.[START_REF] Abbott | Measuring resemblance in sequence analysis: An optimal matching analysis of musicians careers[END_REF].

As a consequence, full-time students, some of whom might have a part-time job, were not included in the analysis. This is likely to decrease the number of part-time workweeks.

This plugin is available free of charge at http://laurent.lesnard.free.fr.

Now it is possible to conduct all the analyses with the TraMineR library in R[START_REF] Gabadinho | Mining sequence data in r with the traminer package: A user's guide[END_REF], which implements Dynamic Hamming Matching since the version 1.4 released on August 6, 2009.

A tempogram is a graphical representation of the state distribution for each time-slot.

Previous studies on French workweeks also supported the simplified groupings for different types of standard workdays, shift workdays and part-time workdays respectively. The simplification can be justified by the proximity in workers' backgrounds and characteristics among the subgroups (For more detail, seeLesnard 2006a, Lesnard 2009b).

Given the fact that the typologies are difficult to interpret, it is not possible to describe precisely the difference in the results of one-stage OM and 2SOM. Overall, the difference should not be significant because the first-stage 2SOM is guided by theories and empirical findings of previous studies. The additional figure is available upon request to the authors.

In OM, stable clusters are composed of identical or almost identical sequences, which will end up being grouped together in no regard to the costs, since costs are, by definition, only used when sequences are dissimilar.

This project was supported by a Postdoctoral Fellowship from the British Academy and a British Academy/CNRS Joint Project Grant. Earlier versions of this paper were presented at the IATUR 2008 conference, the ISA Research Committee 33 2008 conference, and seminars in Oxford and Surrey.


Keywords: Contrôle, boucle fermée, écoulement compressible, réduction du bruit Control, feedback law, cavity, compressible flow, noise reduction

Les perturbations acoustiques présentes dans un écoulement de cavité bidimensionnel et compressible sont analysées et contrôlées au moyen de simulations numériques directes (DNS) et d'une loi de rétroaction en retour de sortie liant l'actionneur de type jet synthétique à des prises de pression pariétales. La loi de contrôle est obtenue en bâtissant un modèle réduit dynamique non linéaire de l'écoulement non contrôlé et forcé par une excitation à large spectre puis en menant une étude de contrôle actif de type "Linear Quadratic Regulator". La loi de contrle implémentée dans les DNS conduit à une réduction significative et globale des émissions acoustiques (jusqu'à 10 dB). Le contrôle reste efficace bien en dehors de la fenêtre temporelle d'observation qui a servi à l'obtention du modèle réduit. L'approche continue d'amortir les perturbations pour un contrôle de moyenne amplitude induisant des effets non linéaires.

List of Figures

10 a 1 (t) coefficient versus time. Comparisons of the integration of the ROM, in unactuatted and actuated situation, the reference is the a 1 (t) given by the POD . . . . . . . . . . . .

Préambule

Le travail inclus dans ce rapport fait suite à l'appel d'offre "Mathématique" de la FNRAE et présente une réponse au thème 4 "Identification, détection, contrôle en propagation d'ondes".

Le projet ECOSEA se propose de traiter l'Estimation, le COntrôle et la Stabilisation d'Ecoulements Aérodynamiques. L'équipe de l'IMFT s'est concentrée sur un écoulement classique mais riche par son contenu : une couche limite bidimensionnelle compressible, laminaire affleurant une cavité rectangulaire. Un mécanisme de rétroaction génère une instabilité auto-entrenue qui est la source d'ondes acoustiques qui se propagent dans l'écoulement. Il s'agit donc de chercher des stratégies de contrôle actif en boucle fermée qui aboutiront à une réduction significative des émissions acoustiques. Ce cas de contrôle aéroacoustique s'avère plus difficile et exigeant numériquement qu'un contrôle d'un décollement dans un fluide incompressible à relativement bas nombre de Reynolds, Nous allons ici décrire une méthodologie qui pourra être étendue à d'autres types d'écoulements, et pour d'autres objectifs tels que la réduction de traînée ou l'augmentation de la portance. Il s'agirait alors de contrôler des ondes d'instabilités dans les couches limites ou de réduire ou supprimer les décollements de couche limite ou les recirculations de culot de corps épais.

La démarche de contrôle d'écoulements proposé est potentiellement susceptible d'apporter des améliorations dans la conception et les performances de véhicules tant terrestre, que navals et aériens.

La projet ECOSEA se proposait d'améliorer quelques points durs existants associé au contrôle en boucle fermée :

1. Les simulations numériques ou les expériences produisent des champs de vitesse, de pression ou de température de dimension N , de N = 10 4 à 10 8 . Cette dimension ou degrés de liberté est énorme par rapport aux dimensions des problèmes rencontrés en automatique (science des systèmes), .

Pour appliquer les théories des systèmes il faut donc diminuer la dimension de nos problèmes en utilisant la technique de réduction de modèle.

2. Pour améliorer un dispositif de contrôle il faut placer, autant que possible, les actionneurs et les capteurs optimalement, or leurs positions sont souvent choisies en fonction de contraintes de conception ou technologiques. Il serait donc très intéressant de connaître les positions optimales au sens mathématique du terme.

3. L'estimation de l'état en fonction de mesures de capteurs est un problème à lui seul et de nombreuses approches existent. En mécanique des fluides, on utilise principalement des estimations stochastiques linéaires ou l'approche basée sur le filtre de Kalman. Des approches plus complexes existent en automatique ou dans les problèmes d'assimilation de données qui devraient être testées pour des écoulements.

4. Le principe de séparation définie en théorie du contrôle propose de traiter séparément la partie estimation de la partie contrôle . Mais le couplage existe malgré tout. Peu de travaux portent sur une meilleure prise en compte de ce couplage, en particulier en présence de nonlinéarités. Une partie du travail a été ou sera présentée au Congrès Franc ¸ais de Mécanique en 2011, au congrès internationale MUSAF, à Toulouse en septembre 2013 et au workshop du GDR " contrôle du décollement " à Lille en novembre 2013. Un article dans un livre va être publié et un article est en cours de rédaction pour une revue internationale, ce qui explique que la suite du rapport est en langue anglaise.

Chapter 1

Methodology and applications

In this chapter the methodology which includes numerical tools and theories are presented. The testcase is introduced and the key point of the actuation mode determination is discussed. Some intermediaite results are displayed and analysed.

Introduction

In most of the flow configurations of practical interest, the range of spatial and temporal scales in strong interaction is so large that an accurate analysis and computation require a huge number of degrees of freedom. This issue becomes critical when the objective is to develop a model-based flow control strategy. Indeed, passive flow control is directly linked to shape optimization that is in general an iterative procedure, and active flow control is typically an unsteady approach for which we would like to act in real-time. One way of reducing the number of degrees of freedom by several orders of magnitude is to use Reduced-Order Models (ROMs) based on projection methods. Different approaches can be employed. For instance, the linear global or local stability eigenmodes of the flow can be exploited for deriving such model. However the eigenvalue problem is so large for realistic flows that this approach is very often limited by the computer resources and accuracy. But coupled with a projection method and sometimes using a Proper Orthogonal Decomposition (POD) it is possible to derive a linear balanced reduced order model which can be controlled by the linear control theory. Many applications and developments can be found in the literature [START_REF] Ahuja | Feedback control of unstable steady states of flow past a flate plate using reduced-order estimators[END_REF][START_REF] Barbagallo | Closed-loop control of an open cavity flow using reduced-order model[END_REF]5]. Another approaches will define some heuristic or simple linear models from the flow physical analysis (using POD for instance) [2]. Here, we will use reduced-order model based on Proper Orthogonal Decomposition (POD) and consider for configuration a compressible cavity flow computed by Direct Numerical Simulation (DNS). In the objective to reduce flow instabilities and the level of noise emissions, we design a feedback control law between few wall pressure sensors located at strategic locations in the cavity and an actuator. A major difficulty is to introduce explicitly the control law in the ROM model obtained by Galerkin projection of the Navier-Stokes equations onto the POD modes. For that, we first determine an actuation mode from a given forced simulation. After linearization of the reduced-order model around an equilibrium state, we will determine a feedback control law from a Linear Quadratic Regulator (LQR) approach. Its efficiency and robustness are finally tested in the DNS code. The general methodology is presented on picture 1.1. The initial sensitivity study based on adjoint of the direct numerical simulations is included.

Compared to our previous studies on the subject [START_REF] Kasnako Glu | Reduced Order Modeling, nonlinear analysis and control methods for flow control[END_REF][START_REF] Nagarajan | Development and Application of a Reduced Order Model for the Control of Self-Sustained Instabilities in Cavity Flows[END_REF], some improvement have been implemented. Usually, a feedback law is a direct relation between the state of the system and the actuation as shown in figure 1.2. Here, we overpass the knowledge of the state to determine an output feedback law directly from some sensors' values (red dashed arrow of the figure 1.2 ). Moreover, we have changed the inner product used in the POD, modified the governing equations in the Galerkin projection and simplified the linear control approach. In addition, a new identification model of the output was also introduced.

In Section 1.2, the configuration and the parameters of the direct numerical simulations are given, and the chirp actuation is described. The POD reduced-order model and the LQR strategy are discussed in Section 1.3. The feedback law is numerically tested on the DNS in Section 2. Finally, the paper ends with some perspectives.

Direct Numerical Simulations 1.2.1 Governing equations, configuration

The governing equations are the two-dimensional compressible Navier-Stokes equations written with the conservative formulation. The static pressure p is given from the ideal gas state equation. No slip boundary conditions are imposed at the wall except when actuation is on. The present reference test case is very close to one of those found in [START_REF] Rowley | On Self-sustained Oscillations in twodimensional Compressible Flow over Rectangular Cavities[END_REF]. We consider a laminar boundary layer flow over a rectangular cavity of aspect ratio length/depth (L/D) equal to 2 (Figs. 1.3 and 1.4). At the upstream cavity corner, the Reynolds number based on the momentum boundary layer thickness is 188 and the Reynolds number based on L is 2982. The air upstream conditions are Mach number M ∞ = 0.6 and a temperature T ∞ = 298 • K [START_REF] Moret | Aeroacoustic investigation and adjoint analysis of subsonic cavity flows[END_REF][START_REF] Nagarajan | Development and Application of a Reduced Order Model for the Control of Self-Sustained Instabilities in Cavity Flows[END_REF]. This flow is well known to produce self-sustained instabilities. The impingement of the shear layer on the downstream edge of the cavity induces a pressure wave which excites the shear layer instability by a feedback mechanism. It is therefore characterized by a strong coupling between hydrodynamic instability and an acoustic wave propagation [17]. Generally, studies aim to reduce the noise emitted outside the cavity at some specific frequency approximated by the Rossiter formula. Several acoustic modes can be found depending on the state and properties of the boundary conditions and of the aspect ratio of the cavity flow. In this 2D test case, the fundamental noise frequency (see Fig. Six unsteady pressure sensors are chosen at the cavity walls for deriving the control law (Fig. 1.3) and 15 additional probes are introduced to record the temporal evolutions of flow fields for spectral analysis (Fig. 1.4). The reference simulations are run over N t = 25000 uniform time steps ∆t from 0 to 22.52 where the time is nondimensionalized by the upstream sound velocity a ∞ and the cavity length L. Over this time horizon, approximately 10 Rossiter's cycles are displayed (Fig. 2.2). The DNS code (named ASIA) has been validated in various configurations. Numerical schemes, non reflecting boundary conditions and other informations are found in [START_REF] Moret | Aeroacoustic investigation and adjoint analysis of subsonic cavity flows[END_REF].

The simulation code (ASIA) has been validated in various configurations a [START_REF] Moret | Aeroacoustic investigation and adjoint analysis of subsonic cavity flows[END_REF]. An alternate fourth order Runge-Kutta scheme is used for temporal discretisation and sixth order compact schemes are used for spatial derivative discretisation [START_REF] Kloker | A robust high-resolution split-type compact FD scheme for spatial direct numerical simulation of boundary-layer transition[END_REF]11]. A non regular cartesian multi-block grid is designed to capture high gradients close to the wall. The characteristic boundary conditions are implemented to avoid acoustic wave reflections on open boundaries of the computational domain [START_REF] Poinsot | Boundary conditions for direct simulations of compressible viscous flows[END_REF]22] and a buffer region is added at the outlet of the domain to damp any residual numerical waves for x/D > 9.96 (Fig. 1.4). Some ghost cells are introduced into the mesh to avoid any wall point to be a computed grid node.

Chirp actuation

For introducing explicitly the control term in the POD model, at least one actuation mode must be determined from a forced simulation. The results will strongly depend on the actuation law imposed in this simulation. The location of the forcing and its type (direction, physical quantity) have been set after a numerical sensitivity analysis of the pressure field to different types of forcing using adjoint DNS of the Navier-Stokes equations [3, [START_REF] Moret | Aeroacoustic investigation and adjoint analysis of subsonic cavity flows[END_REF]21]. The actuator is finally centered close to the upstream edge of the cavity at x/D ≈ 3.6 that corresponds to the highest sensitivity zone, and the action is on the normal velocity at the wall simulating a blowing/suction effect. From the sensitivity analysis, we impose a spatial Gaussian distribution actuation law on a small wall slot of width 15 streamwise steps ∆x, centered on a given location x 0 :

f w (x, t) = γ(t) exp(- ||x -x 0 || 2 σ 2 ), σ = f (∆x)
To excite a large bandwidth of flow frequencies, we impose for forcing term γ(t) a slowly varying amplitude and frequency called chirp actuation and given by:

γ(t) = A 1 sin(2πSt 1 t) × sin (2πSt 2 t -A 2 sin(2πSt 3 t)) .
(1.1)

The non-dimensional amplitude A 1 governs the linear effect of the perturbation with respect to a given time mean field. The other parameters control the large time variation of the phase, the frequency and the amplitude. We set A 2 = 27, St 1 = 1/30, St 2 = 2/15 and St 3 = 8/3 (reference time L/a ∞ ). An analytical forcing as equation (1.1) is preferred, even if not necessary, to a white noise because this equation is used for the identification model of outputs and in the calculus of the actuation mode. As displayed in Fig. 1.6, the range of Strouhal numbers covered by the chirp excitation is large and includes the second Rossiter mode (St ≈ 0.7) and its harmonics. The more the amplitude A 1 increases, the more the non-linear effects become dominant in the flow. In the following, A 1 is hence set to 0.01 of the upstream velocity U ∞ to keep approximately linearity and to produce an observable and significant effect.

When actuation is one at the wall, the velocity, pressure and energy boundary conditions are modified. Since the actuation is localized in a small wall length, no nonreflective boundary conditions are required, and no numerical wave reflection are observed as shown in section 2. 1.3 Reduced Order Modelling

Proper Orthogonal Decomposition and actuation mode

Let q(x, t) be the state variables, and x denotes the location vector (x, y). Proper Orthogonal Decomposition [START_REF] Noack | Reduced-Order Modelling for Flow Control[END_REF] is performed over a reduced space domain centered on the cavity (see Fig. 1.4). For that, 250 snapshots sampled every 100 time steps of the numerical simulations are considered for the snapshot POD algorithm. Here, we choose an inner product weighted by the inverse of the variance of each state variable component as in [6] (cf. Appendix A)

For an un-actuated flow (variables with superscript u), the POD expansion reads

q u (x, t) = qu (x) + N i=1 a u i (t)φ u i (x) (1.2)
where qu (x) is the mean time flow, φ u i (x) are the POD modes and a u i (t) the mode coefficients. For representing the actuated flows (superscript a), an actuation mode ψ(x) is added to the previous unactuated expansion leading to

q a (x, t) = qa (x) + N i=1 a a i (t)φ u i (x) + γ(t)ψ(x), (1.3)
where γ(t) is the control parameter. The actuation mode is determined from the snapshots obtained in section 1.2.2 for the chirp actuation (see [START_REF] Kasnako Glu | Reduced Order Modeling, nonlinear analysis and control methods for flow control[END_REF][START_REF] Nagarajan | Development and Application of a Reduced Order Model for the Control of Self-Sustained Instabilities in Cavity Flows[END_REF]). By construction, ψ(x) is orthogonal to the un-actuated POD modes φ u i (x) following the simple algorithm :

1. a a i (t) = q a (x, t) -qa (x), φ u i (x) Ω 2. ψ(x) = q a (x, t) -qa (x) - N i=1 a a i (t)φ u i (x), γ(t) t / γ(t), γ(t) t
The index Ω and t in the inner products denotes respectively the spatial and temporal inner products.

The expansion (1.3) is based on the assumption that the mean flow is approximately not changed by the actuation i.e. qa (x) ≈ qu (x). In our case, 6 POD modes are seen to be sufficient to represent the energetic content of the flow.

The figure 1.7 show the streamwise velocity component of the actuation mode φ u when the reference forcing is a sine fonction. The mode is localized in space in the wall boundary layers and in the shear layer. The dashed red lines show the same actuation mode when another spatial inner product based on the total energy to get the correlation matrix is used for the POD determination. It indicates that the inner product definition does not play an important role. The figure 1.8 shows the same actuation with a chirp forcing. We guest the location of the actuation at the wall, upstream the cavity. We also clearly see some spatial localized structures around the downstream cavity corner, which is the source of the noise emission related to the Rossiter feedback mechanism. The actuation mode seems to have well captured the intrinsic physic of the flow and the place where to modify the flow characteristics. The response of the flow to a large bandwith frequency signal appears therefore more convenient from a control view point than a simple one frequency forcing. Because of its large bandwith frequency, more flow modes are excited and finaly the actuation mode is spatially localized close to the origin of the noise emission, at the downstream corner of the cavity. We can recognize as well the location of the forcing upstream the cavity.

Galerkin projection and ROM

The Reduced Order Model (ROM) of the cavity flow is obtained by a Galerkin projection of the compressible Navier-Stokes equations onto the spatial POD modes. To avoid cubic terms in the dynamical model, we have used the same formulation of the governing equations as in [6, 23], i.e.

ζ t = ζ(u x + v y ) -uζ x -vζ y u t = -uu x -vu y -ζp x + 1 Re ζ 4 3 u x - 2 3 v y x + (v x + u y ) y v t = -uv x -vv y -ζp y + 1 Re ζ 4 3 v y - 2 3 u x y + (v x + u y ) x p t = -up x -vp y -γp(u x + v y ) + γ ReP r [(pζ) xx + (pζ) yy ] + γ -1 Re u x 4 3 u x - 2 3 v y + v y 4 3 v y - 2 3 u x + (v x + u y ) 2 (1.4)
where ζ = 1/ρ, and u, v and p are the velocity components and pressure, respectively. In (1.4), P r and Re denote the reference Prandtl and Reynolds numbers.

By introduction of (1.3) into (1.4), and by projection on the POD modes, a non linear dynamical system of size N is determined for a(t):

ȧ = C + La + a T Qa + γ Ĉ + γ La + γ 2 Q, (1.5)
where the superscript T denotes the transpose. The coefficients of this reduced-order model depend on the mean flow, on the POD modes and on the actuation mode. The analytical expressions of the coefficients are given in an appendix B. It is now well known that reduced-order models derived by the traditional POD Galerkin approach may be fragile when used for control design [START_REF] Cordier | Identification strategies for model-based control[END_REF]. For this reason, we calibrate the coefficients C, L, Q of the un-actuated model (1.5) using similar numerical methods as those described in [START_REF] Nagarajan | Development and Application of a Reduced Order Model for the Control of Self-Sustained Instabilities in Cavity Flows[END_REF]. Moreover, we checked for different values of the control parameter that the dynamical description of (1.5) remains satisfactory when the forcing was introduced. Some comparisons are shown in an appendix C. The first step in the design of a feedback control law by the LQR analysis is to linearize the reducedorder model (1.5) around an equilibrium point, a e , the steady solution of (1.5). The corresponding equilibrium state q e (x) is computed for the un-actuated system (1.5) with a Newton iterative algorithm. The resulting reduced-order model in the new set of coordinates ã = aa e is:

ȧ = Lã + ãT Qa + a T Qã + ( Ĉ + La)γ = Ãã + Bγ.
(1.6)

Identification model of outputs

When the mode coefficients a of (1.6) are not available, an estimation based directly on real-time measurements of surface pressure at a small number N s = 6 of locations is derived. Let y(t) be the pressure measurements, the objective is to determine the model ŷ(t) = Ca(t) + Dγ(t) where ŷ(t) = y(t) -ȳ.

The matrix C and the vector D are determined by linear stochastic estimation (LSE) and results are simply obtained by exploiting the orthogonality property of the augmented basis ({φ i } N i=1 , ψ) [START_REF] Nagarajan | Development and Application of a Reduced Order Model for the Control of Self-Sustained Instabilities in Cavity Flows[END_REF]. By considering that each sensor y i is located on the POD mesh at a position x k it can be demonstrated that Cij = φ u j (x k ) and Di = ψ p (x k ), where ψ p is the pressure component of the actuation mode. If the sensors are not located on the POD grid, a projection gives (1.7)

Cij = ỹu i , a u j t a u j , a u j t and Di = y a i -y u i -Cij (a a j -a u j ), γ t γ, γ t

Linear Quadratic Regulator and output feedback control

The LQR design of the system (1.6) and (1.7) computes the gain matrix K c , such that the state-feedback law

γ(t) = -K c ã(t) (1.8)
minimizes the quadratic cost function given by J = T h 0 ãT ã + 2 γ 2 dt where is a penalization term corresponding to the control cost. When the value of is high, the best possible stabilization is obtained but for an expensive cost. For an infinite time horizon (T h → ∞), the control gain is given by

K c = 1 2 BT X, where X is solution of the continuous time algebraic Riccati equation ÃT X + X Ã - 1 2 X B BT X + Id = 0.
By manipulating (1.7) and (1.8), we can obtain an expression for γ. If the number of Galerkin POD modes N is equal to the number of sensors N s then

γ(t) = -K c ( C -DK c ) -1 (y -ȳ -Ca e ).

First conclusions

Otherwise, we use the pseudoinverse and find

γ(t) = -K c [( C -DK c ) T ( C -DK c )] -1 ( C -DK c ) T (y -ȳ -Ca e ).
In summary, the direct output feedback control law reads:

γ(t) = α T (y(t) -ȳ) + β, (1.9)
where α is a vector of dimension N s and β = -α T Ca e is a real number. From equation (1.9) which is a linear relationship between the actuation and the outputs, β governs the level of the steady part of the actuation while the α vector governes the damping rate of the unsteady part of the actuation.

First conclusions

On the based of a reduced order model and of a Linear Quadratic Regulator control approach, it is possible to design an actuation feedback law from sensors signal suitly located in the flow. An important issue is to generate some flow fields in known actuated conditions in order to well capture and to model an actuated mode which is a RHS forcing term of a nonlinear dynamical system.

Another important point is that the actuation low is linear with respect to the fluctuated pressure and the sum of a damping term and a constante velocity. Because of its simple shape the actuation law can be easily implemented in the DNS code. The design of the feedback finaly is quite cheap considering that only two DNS and some computations on a ROM of low dimension (lower than 10) are required.

Let recall that the complexity of the numerical suite to get the actuation laws has not been realy described here, but numerous options and parameters need to be set carefully.

Chapter 2

Feedback control of acoustic waves with DNS and POD analysis

This chapter is concentrated on the analysis of the actuation law and of the actuated flow when the feedback control law is implemented in the Direct Numerical Simulations.

Actuation level and efficiency

The direct output feedback law is implemented in the full DNS code. Equation (1.9) shows that the control amplitude max |γ| depends on β which is a velocity. β is a function of the equilibrium state a e which in general is not unique. It is also related to the output identification matrix C : the sensor positions (Fig. 1.3) must lead to a good observability obtained from a Lyapunov equation [START_REF] Lewis | Optimal control[END_REF]. The D and K c vector dependence means that the actuation mode must be representative of the physics, the state space system must therefore have a good controllability verified by a Lyapunov equation as well. Finally, the value of β is maximum when the control parameter goes to 0. When all conditions are fulfilled the control law produces a flow stabilization in the direct numerical simulations, otherwise exponential divergence occurs introduced by the components of α. With the chosen parameters, we find β=5.16 m/s (case A). From Fig. 2.1, we see that the amplitude of the actuation slowly decreases in time and the pressure fluctuations do the same. However the damping rate is quite small and a long time simulation is required to get a strong noise reduction of several decibels (a maximum of 3 dB after 45000 time steps). It is quite complex and time consuming to control the values of β and α by changing sensors' positions, actuation modes or by playing with the other parameters. So to improve the efficiency of actuation in the DNS, we propose to progressively increase the value of β in the control law (1.9) to 10.16 m/s (case B) and 17 m/s (case C). Here, only results of case C are discussed. For lower values of β, the conclusions are similar but the control effect is lower. For case C, the maximun actuation level in the flow field is around 8 % of the upstream velocity and generates weak nonlinearities. The damping of the control law in Fig. 2.1 indicates that efficiency is higher when γ(t) is larger. The actuation seems to converge towards a constante value respectively of 3.56, 8.16 and 14.5 m/s for cases A to C. It indicates that the flow outputs y(t) -ȳ converges towards a non zero constante value and the flow goes towards another stable state different from equilibrium state of the unactuated flow: the non zero wall steady asymptotic actuation is now a part of this new actuated equilibrium state.

The control law has been designed for a time period of 25 000 ∆t but actuation remains efficient for longer time, far from the initial time window T h (at least up to 80 000 ∆t). The control law is quite robust but the quasi-perfect periodicity of the initial unactuated flow contributes to this result.

To go deeper into the flow analysis we have compared for the probes displayed in Fig. 1.4, the nondi- mensional pressure p/p ∞ obtained in case C with the unforced flow results. Some general trends are discussed now. In the main part of the computational domain, and especially in the emission zone and inside the directivity cone of the acoustic waves, the mean time pressure value has greatly decreased. In zones far upstream and far downstream the cavity, very close to the wall (probes 1, 2 and 8) the sound waves originated from the cavity are quasi unobservable and the mean pressure is greatly increased by the actuation perturbation. The Sound Pressure Level (SPL) has decreased globally except in some small localized area discussed later. Figure 2.2 displays signals obtained for some typical probes, one located in the vicinity of the downstream corner and of the sound source location (probe 7), and another one located in the far field (probe 10) that is more representative of the sound propagation.

Figure 2.3 demonstrates the control efficiency. Fast Fourier Transforms (FFT) of pressure's signals indicate that the control damps the fundamental frequency and its harmonics (a maximum of 1/8 approximately). A subharmonic has been generated by the actuation, a footprint of the weakly nonlinear effect.

Noise reduction, Sound Pressure Level

The global noise reduction is displayed in Fig. 2.4 through the difference of SPL between the actuated flow (case C) (left of fig. 2.5) and the unactuated flow (right of fig. 2.5). A damping of approximately 10 dB is reached close to the maximum of the SPL of the unactuated flow, in the shear layer and close to the downstream corner. On some localized and small zone located on the top of the cavity and downstream, the SPL has increased of several dB (up to 6 or more). Actually, the SPL level of the unactuated flow was really low and even with this amplification, the SPL remains low comparing to the mean SPL level as demonstrated on figure 2.5. The weakly nonlinear effect of the actuation on the flow is emphasized by the appearance of waves on some contours around -7 dB on figure 2.4 and which are not seen on figure 2.5. It could generated a secondary subharmonic instability (see Figure 2.3 as well) as it is well known in boundary layers or other flows but further investigations are necessary.

POD analysis of the actuated flow

It could be interested to carry out a Proper Orthogonal Decomposion following equation (1.2) of the actuated flow and to compare its new dynamics to those of the unactuated flow.

The figure 2.6 shows the Relative Information Content of the both flows. To get 99.99 % of the full energy, four modes are required in the unactuated flow while ten modes are not sufficient with actuation. The flow dynamics has been drastically complexified.

The mean perturbated pressure fields before and after actuation (the zeroth POD modes) ploted on figure 2.4 recall that the level of actuation is high enough to introduce a variation and therefore a nonlinearity of the mean flow. It is therefore interesting to remind that the design of the actuation mode and the design of the actuation law was assuming a negligible nonlinear effect on the mean flow. It is another indicator of a certain robustness of the actuation law.

The phase portraits of the first five temporal coefficients are shown on figure 2.8. The curve for the unactuated flow are all closed (in red) because of the perfect time periodicy of the flow. However, for the unactuated flow, again the new flow dynamics is exhibited by large oscillations. For all phase portraits, we can guess that the curves will converge towards a fixed point. It is at least obvious for the coefficients a 2 versus a 1 . It confirms that for infinite horizon, all flow fluctuations will be damped and the the flow will converge towards a steady new equilibrium state which can be determined asymptoticaly from the location of the fixed points in the phase portrait. This part requires more investigations however.

Summary

The efficient and in some way the robustness of the actuation law have been numerically proved. The linear property of the actuation law has allowed to use the constante β as a paremeter to improve the efficency of the control. As result the actuated flow seems to converge towards a new equilibrium state while the actuation law becomes steady for infinite time. Some weakly nonlinear effects are observed without perturbating the control efficiency.

Summary and future works

A global noise reduction, up to 10 dB, has been reached by implementing a direct output feedback actuation law into Direct Numerical Simulations, using a LQR strategy based on Reduced Order Modelling.

The actuation law remains really efficient far after the time window used to design the law or by tuning the actuation by a constant value, which leads to some weak nonlinearities, proving its robustness.

Concerning computational resources, the proposed approach requires only two DNS for determining the actuation mode, and then rests only on reduced-order models. Compared to an optimal open-loop control approach or to an optimization strategy for which many DNS simulations have to be done, our approach is really cheaper.

As current works, we no longer use a LQR control approach. We improve the former Linear Quadratic Gaussian control implementation in the ROM by proposing a feedback law given by a convolution product :

γ(t) = t t-tc Ns i=1
G i (t -τ )y i (τ ) dτ t c is a delayed time which allows to have the history of the flow in the actuation law. This more complex actuation law can take into account of the noise or uncertainties associated to sensors and the actuators.

As the perspectives, more investigations are necessary to take into account of the actuation mode in the ROM. For instance we could add the actuation time derivative as it is done by some authors. Another possibibility is to activate simultaneoustly several actuators at several positions.

The robusness can be as well enhance by using H 2 and H ∞ control in the ROM but as well by designing and testing actuation law in some given range of Reyndols numbers and Mach numbers.

Moreover, the general control efficiency could be enhanced by going further towards nonlinear flow control.

The last but probably most important issue would be to define a wind tunnel experiment to control for instance separation at relative low Mach number and Reynolds number and to use the DNS and this methodology to propose and to test various feedback law depending on actuators positions and types and on sensors positions.

1

  Travaux sur le contrôle des écoulements à l'IMFT . . . . . . . . . . . . . . . . . . . . . 1.1 General methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Direct feedback output control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Sensors' locations in non dimensional variables. Numbers 7 and 8 are off. . . . . . . . . 1.4 POD domain and probes' locations in non dimensional variables. . . . . . . . . . . . . . 1.5 Spectrum of the uncontrolled flow at x/D = 3.33. FFT done over 75000 time steps. . . 1.6 FFT over 100 000 time steps of the chirp actuation. . . . . . . . . . . . . . . . . . . . . 1.7 u velocity of the actuation mode with a sine forcing . . . . . . . . . . . . . . . . . . . . 1.8 u velocity of the actuation mode with a chirp forcing (A 1 = 0.001) . . . . . . . . . . . 2.1 Forcing γ(t) for three values of β and mean values. . . . . . . . . . . . . . . . . . . . . 2.2 Time evolutions of pressure for the unactuated flow and case C. . . . . . . . . . . . . . . 2.3 FFT of pressure signals. Comparisons between the unactuated flow and the controlled one (case C). The values of probe 10 are scaled by 5. . . . . . . . . . . . . . . . . . . . 2.4 Difference of SPL between the unactuated flow and the controlled one (case C). Values in decibels (dB). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Sound Pressure Level of the unactuated and actuated flows . . . . . . . . . . . . . . . . 2.6 Relative Information Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Mean pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Phase portraits, in black: closed curves for unactuated flow, in red, convergente curves for actuated flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 a k (t), k = 1..4. Comparisons of the time integration coefficients of the unactuated ROM obtained by two inner products, the one associated on the isentropic model (Isen) and the variance inner product associated to the Navier-Stokes equation in Vigo's formulation . .

Figure 1 -

 1 Figure 1 -Travaux sur le contrôle des écoulements à l'IMFT

Figure 1 . 1 -Figure 1 . 3 -Figure 1 . 4 -

 111314 Figure 1.1 -General methodology Figure 1.2 -Direct feedback output control

  1.5) is found at a Strouhal number St L = f × L/U ∞ = 0.703, close to the so called 1.2 Direct Numerical Simulations second Rossiter mode given by St L = 0.74.

Figure 1 . 5 -Figure 1 . 6 -

 1516 Figure 1.5 -Spectrum of the uncontrolled flow at x/D = 3.33. FFT done over 75000 time steps.

Figure 1 . 7 -Figure 1 . 8 -

 1718 Figure 1.7 -u velocity of the actuation mode with a sine forcing

  where a, b t = T 0 a(t)b(t) dt. Finally, introducing ỹ = ŷ -Ca e we find the output identification model: ỹ = C ã + Dγ.

2. 2

 2 Noise reduction, Sound Pressure Level γ(t) t/∆t

Figure 2 . 1 -

 21 Figure 2.1 -Forcing γ(t) for three values of β and mean values.

Figure 2 . 2 -

 22 Figure 2.2 -Time evolutions of pressure for the unactuated flow and case C.

FFigure 2 Figure 2 . 4 -

 224 Figure 2.3 -FFT of pressure signals. Comparisons between the unactuated flow and the controlled one (case C). The values of probe 10 are scaled by 5.

Figure 2 . 5 -

 25 Figure 2.5 -Sound Pressure Level of the unactuated and actuated flows

Figure 2 . 8 -

 28 Figure 2.6 -Relative Information Contents Figure 2.7 -Mean pressure

  5. Enfin, le mouvement des fluides répond au modèle des équations de Navier-Stokes qui sont fortement non linéaires. La physique même des écoulement est fortement non linéaire. La conception de la loi de con dans les problèmes non linéaires se fait très souvent autour d'un état tangent suite à une linéarisation locale. Des travaux à l'Institut de Mathématique de Toulouse ont démarré sur la conception de lois de contrôle nonlinéaires. Les résultats sont encore mitigés, mais clairement des recherches appronfondies doivent être poursuivies dans ce sens.

	L'équipe de l'IMFT s'est attachée à améliorer les points 1, 2 et 3 et un peu 4. Les ressources dédiées
	au projet n'ont pas permis d'avancer sur le point 5.

Acknowledgements

Appendix

Bibliography

Acknowledgments

This work has been supported by the Fondation de la Recherche pour l'Aéronautique et l'Espace (FRAE) under the ECOSEA project. We would also like to thank the CALMIP center in Toulouse for providing the computational resources. This work is another brick inside a complex association of various numerical tools and various theories. So I would like to thanks all people (colleagues, students) and financial supports who helped me during at least the last decade.

Appendix A-Inner product a) Variance inner product

On introducing q = (ζ, u, v, p) with ζ = 1/ρ and for non-dimensional variables we have:

In the inner product, the weight α µ are the inverse of the square of the variance of the variable µ :

μ is the temporal mean value of µ on the time interval [0 T ].

b) Energy based inner product

For the case of compressible flows, the velocity variables are dynamically coupled with the thermodynamic variables. The inner product which defined the correlation tensor adds the flow variables and the thermodynamic variables. A question arises when we add two variables of different dimensions to make sense when we use the usual L 2 inner product. The scalar inner product can be computed for each variable as it has been used by [START_REF] Rowley | Simulation, and Control of Cavity Flow Oscillations[END_REF].

For vector valued variables one choice could be to non-dimensionalize the variables, but the meaning of non-dimensionalization on the optimality of the projections is not clear. [START_REF] Rowley | Simulation, and Control of Cavity Flow Oscillations[END_REF] seeks an inner product, for the isentropic flow variables can be interpreted as an energy. This inner product has been used widely to study cold isentropic flows at low Mach numbers by [START_REF] Rowley | Model reduction of compressible flows using POD and Galerkin projection[END_REF], [START_REF] Gloerfelt | Compressible Proper Orthogonal Decomposition/Galerkin reduced order model of self sustained oscillations in a cavity[END_REF].The vector variable is defined with q = (u, v, c), where u and v are flow variables and c the local speed of sound. The inner product is then written.

where γ is the ratio of specific heats and α is a parameter to be chosen. α = 1, respectively α = 1/γ corresponds to the integration of the stagnation enthalpy and respectively of the stagnation energy [START_REF] Gloerfelt | Compressible Proper Orthogonal Decomposition/Galerkin reduced order model of self sustained oscillations in a cavity[END_REF].

.

B-ROM with the 1/ρ formulation Galerkin projection

The low-order model of the full Navier-Stokes equation has cubical implicit terms,Vigo suggests the use of a formulation by using the primitive variables (u, v, p) and the specific volume ζ = 1 ρ . The fully compressible Navier-Stokes equation in terms of these variables can be written as:

On introducing q = (ζ, u, v, p) the above equation can be re-casted as

with

Actuated nonlinear dynamical system

By taking into account of the actuation mode ψ * in the Galerkin projection of the Navier-Stokes equation on the POD modes,an actuated nonlinear dynamical system can be written as :

where

C-Unactuated and actuated ROM -Calibration

The first inner product used to get POD spatial modes and time coefficients was the one based on the energy (eq. 2). To get the reduced order model, we used the isentropic equation in the Galerkin projection similarly to [START_REF] Gloerfelt | Compressible Proper Orthogonal Decomposition/Galerkin reduced order model of self sustained oscillations in a cavity[END_REF]. In the figures 9 and 10 this case is referred as "Isen". Then we used the inner product based on the variance (eq. 1) and the equations 3 was used in the Galerkin projection. This case is referred as "Vigo" in the figures. The figure 10a) shows that the POD coefficient a 1 (t) are very similar and so far inner-product independent, except that the amplitude level are not equal because of the POD spatial mode normalization. The conclusion is the same for the other time coefficients of interest. The next figures 10b) to 10c) show the comparison of the three first time coefficients. As expected, the coefficients obtained from the time integration of the ROM are unstable, but they seems more accurate in comparison with the POD reference coefficients in the Vigo's case. That let us think that it was necessary to choose this later inner product and the Navier-Stokes equations in the Galerkin projection for control purpose. Naturally, after the calibration process, the time coefficients given from the ROM time integration perfectly match with the reference.

In the figure 10 are compared the a 1 (t) coefficients before the calibration process, in four case : unactuated POD, unactuated ROM with Vigo's formulation, with the chirp actuation and the isentropic model and with the chirp actuation and the Vigo's formulation. The conclusion is quite obvious, the use of the Vigo's formulation is a great improvement to better take into account of the actuation in the nonlinear dynamical system.

D-ANR Cormored / Ecosea project tasks

The Ecosea project lead to large significant improvement from the ANR Cormored project.


ehralam-ehralam. O lo sole, o lo si'a ma olo sanga modi la ek nia la'a mine? Sirua penge lahresuk-hresuk ratam inoiron. Ma pema kne nau tam pepiak nia reroi nau am'ma ratusana nia ni. Pe ma noi nau, hmei ni perane Pautamu kekngauk la Le-keu re sirua pema sole hea. Pe ma du'a lade sia-sian nei re hmei ek nrei, tasi nrei ma nusa lau pluak ne si lu si'a lau. O lo nge si'a memouk. O lo sanga modi ita kua pa hresuk rin, I noi tam ron. sira, bainhira iha bailoro naruk, susar ba imi atu hetan bee, uza Ratusana ne'e hodi husu no bolu bee husi fatin-fatin atu bee bele mai iha fatin ne'ebé imi hela ba.

Múzika husi siklu rituál husi bee midar iha Ataúro

Iha illa Ataúro, depois fulan balu husi tempu maran ne'ebé naruk, bee-moos ladún iha. Iha aldeia oin-oin, ema organiza rituál ida atu hetan bee midar (ek, er, eg), husi rai (ek matan Hresuk, eg matan Raklungu) no husi lalehan (uran «rain «). Rituál ne'e hala'o durante semana balu iha fulansetembru to'o fulan-outubru antes tempu udan hahú. Hodi hetan ''favór'' ne'e sira kanta knananuk, ho tambór iha fatin sira tuirmai ne'e: iha bee matan husi aldeia, iha rate sira nia família nian no iha fatin husi estátua sira. Naran nelu tolan, eventu ne'e hanesan tempu atu kanta siklu naruk husi knananuk souk sosé iha Makili / sour sésé iha Makadade. Rituál ne'e lakon ona dezde tinan 1973 husi parte hotu-hotu husi illa exetu iha Makili.

Rituál bee nian hahú iha suku ida-idak nia bee-matan (Bitau ba grupu Manroni, Er hlerin ba Adadi, Epua ba Humangili). Uma-hun hotu ba hamutuk iha bee-matan atu halo fatin moos, atu troka pipa au tolu no atu husu ba maromak ba bee liu husi knananuk serimoniál lubun ida (er hleri «knananuk bee nian»-hresuk/raklungu). Istória ida husi Makili konta kona-ba oinsá ema halai sai husi bee laran, entaun maromak bee-matan nian, husu ba feto ida naran Katisuk atu mai iha bee-matan nia oin kada tinan atu hananu bainhira kos hena (kngohi) iha bandeja (ho betel) ho tudik espesiál hanaran Ratusana, hodi bolu bee.

Iha Ataúro, uma-hun balun mak protetór ba bee-matan (Ruma Lari, Tua Bere Doe iha Makadade, Hnelak no Mau Tuda iha Makili). Uma-hun sira-ne'e mak sei kanta no baku tambór durante rituál sira. Iha Makili, knaar sira-ne'e fahe ba malu: uma-hun Hnelak baku tambór bainhira uma-hun Mau Tuda kanta. Knananuk serimónia nian ba bee midar sira hanaran souk iha hresuk, sour iha raklungu. Iha faze daruak, uma-hun ida-idak ba iha ninia rate família rasik hodi «bolu sira nia bei'ala sira» (roi sa pohé), kanta knananuk sita, hakoi mate nian. Sira halo oferta (hnaro, Hresuk, rahnia Raklungu), no bolu ema ne'ebé mate ona nia klamar liu husi dere topipi atu mai hasoru ema ne'ebé moris hodi ajuda sira atu buka bee midar.

Tambór Tihak no estátua Sa 'et, Makili, 2022© Bluemilk Studio Tambór Tihak, Makili, 2022 © Bluemilk Studio


Bénédicte Lebret
email: benedicte.lebret@inrae.fr

Armelle Prunier

Armelle Prunier -Inrae

Rearing non-castrated male pigs in

come

This project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement No 816172