Romain Serizel

Nicolas Turpault

Hamid Eghbal-Zadeh

Ankit Parag

Large-Scale Weakly Labeled Semi-Supervised Sound Event Detection in Domestic Environments

Keywords: Sound event detection, Large scale, Weakly labeled data, Semi-supervised learning

This paper presents DCASE 2018 task 4. The task evaluates systems for the large-scale detection of sound events using weakly labeled data (without time boundaries). The target of the systems is to provide not only the event class but also the event time boundaries given that multiple events can be present in an audio recording. Another challenge of the task is to explore the possibility to exploit a large amount of unbalanced and unlabeled training data together with a small weakly labeled training set to improve system performance. The data are Youtube video excerpts from domestic context which have many applications such as ambient assisted living. The domain was chosen due to the scientific challenges (wide variety of sounds, time-localized events. . . ) and potential industrial applications.

INTRODUCTION

We are constantly surrounded by sounds and we rely heavily on these sounds to obtain important information about what is happening around us. Ambient sound analysis aims at automatically extracting information from these sounds. It encompasses disciplines such as sound scene classification (in which context does this happen?), sound event detection (SED) and classification (what happens during this recording?) [START_REF] Virtanen | Computational analysis of sound scenes and events[END_REF]. It has been attracting a continuously growing attention during the past years as it can have a great impact in many applications including smart cities, autonomous cars or ambient assisted living.

In this task, we focus on SED with time boundaries in domestic applications. The system then has to detect when an sound event occurs and what is the class of the event (as opposed to audio tagging where only the presence of a sound event is important regardless of when it happened). Current systems heavily rely on a supervised training phase and usually require a large set of sound recordings labeled in terms of event with time boundaries. Obtaining such annotations is tedious and it is hardly feasible to gather a sufficient amount of data to train state-of-the-art systems that are often relying on complex deep networks architectures [START_REF] Parascandolo | Recurrent neural networks for polyphonic sound event detection in real life recordings[END_REF][START_REF] Takahashi | Deep convolutional neural networks and data augmentation for acoustic event detection[END_REF][START_REF] Adavanne | Sound event detection in multichannel audio using spatial and harmonic features[END_REF][START_REF] Cakir | Convolutional recurrent neural networks for polyphonic sound event detection[END_REF][START_REF] Xu | Large-scale weakly supervised audio classification using gated convolutional neural network[END_REF].

We propose to follow-up on DCASE2017 task 4 [START_REF] Mesaros | Dcase 2017 challenge setup: Tasks, datasets and baseline system[END_REF] and investigate the scenario where a large scale corpus is available but only a small amount of the data is labeled. We propose to use a subset of the Audioset corpus [START_REF] Gemmeke | Audio set: An ontology and human-labeled dataset for audio events[END_REF] targeting classes of sound events related to domestic applications. The labels are provided at clip level (an event is present or not within a sound clip) but without the time boundaries (weak labels, that can also be refereed to as tags) in order to further decrease the annotation time. These constraints indeed corresponds to constraints faced in many real applications where the budget allocated to annotation is limited.

In order to fully exploit this dataset, the proposed systems will have to tackle two different problems. The first problem is related to the exploitation of the unlabeled part of the dataset either in unsupervised approaches [START_REF] Salamon | Unsupervised feature learning for urban sound classification[END_REF][START_REF] Jansen | Unsupervised learning of semantic audio representations[END_REF] or together with the labeled subset in semi-supervised approaches [START_REF] Zhang | Semi-supervised learning helps in sound event classification[END_REF][START_REF] Komatsu | Acoustic event detection method using semi-supervised non-negative matrix factorization with a mixture of local dictionaries[END_REF][START_REF] Elizalde | An approach for self-training audio event detectors using web data[END_REF]. The second problem is related to the detection of the time boundaries and how to train a system that can detect these boundaries from weakly labeled data. Currently, most of the state-of-the-art approaches rely mainly on smoothing techniques to ensure time consistency [START_REF] Xu | Large-scale weakly supervised audio classification using gated convolutional neural network[END_REF][START_REF] Jeong | Audio event detection using multiple-input convolutional neural network[END_REF][START_REF] Lee | Combining multi-scale features using sample-level deep convolutional neural networks for weakly supervised sound event detection[END_REF] which is not sufficient to estimate the time boundaries accurately. The evaluation metric chosen is penalizing these boundary estimation errors heavily in order to emphasize this latter aspect.

This manuscript describes DCASE2018 task 4 and is organized as follows. Section 2 presents the dataset used in task 4. The task evaluation procedure is described in Section 3. The baseline system is described and evaluated in 4. Conclusions and perspectives are presented in Section 5

AUDIO DATASET

The task employs a subset of Audioset [START_REF] Gemmeke | Audio set: An ontology and human-labeled dataset for audio events[END_REF]. Audioset consists of an expanding ontology of 632 classes of sound events and a collection of 2 million human-labeled 10-seconds sound clips (less than 21% are shorter than 10 seconds) drawn from 2 million YouTube videos. The ontology is specified as a hierarchical graph of event categories, covering a wide range of human and animal sounds, musical instruments and genres, and common everyday environmental sounds.

Audioset provides labels at clip level (without time boundaries for the events) also known as weak labels. Audio clips are collected from Youtube videos uploaded by independent users so the number of clips per class varies dramatically and the dataset is not balanced 1 . Google researchers conducted a quality assessment task where experts were exposed to 10 randomly selected clips for each class of sound events and discovered that in most of the cases not all the clips contains the event related to the given label. More information about the initial annotation process can be found in Gemmeke et al. [START_REF] Gemmeke | Audio set: An ontology and human-labeled dataset for audio events[END_REF].

We will focus on a subset of Audioset that consists of 10 classes of sound events (Table 1). The development set provided for task 4 is split into a training set and a test set2 .

Training set

Labeled training set

This set contains 1,578 clips (2,244 class occurrences) for which weak labels provided in Audioset have been verified and corrected by human annotators. The weak labels are provided in a tabseparated csv file under the following format:

[filename (str)][tab][class label (str)]

The first column is the name of the sound clip downloaded from YouTube composed of the YouTube ID of the video and the time boundaries of the 10-seconds audio clip extracted from the video. The last column corresponds to the sound events that are present in the clip, each separated by a semi-colon.

The amount of clips per class of sound events is presented in Table 1 and the number of classes observed per clip is presented in Table 2. One-third of the clips in this set contain at least two different classes of sound events.

Unlabeled in domain training set

This set contains 14,412 clips. The clips are selected such that the distribution per class of sound event (based on Audioset labels) is close to the distribution in the labeled set. Note however that given the uncertainty on Audioset labels this distribution might not be exactly similar. Audioset labels have not been verified for this subset and should not be used during the systems development.

Unlabeled out of domain training set

This set is composed of 39,999 clips extracted from classes of sound events that are not considered in the task. Note that these clips are chosen based on the Audioset labels which are not verified for this subset and therefore might be noisy. Additionally, as the speech class is present in about half of the clips in Audioset, the unlabeled out of domain set also contains almost 20000 clips with speech. This was the only way to have a set which is somehow representative of Audioset. Indeed, discarding speech would have also meant discarding many other classes of sound events and the variability of the set would have been penalized. Audioset labels have not been verified for this subset and should not be used during the systems development.

Test set

The test set is designed such that the distribution in term of clips per class of sound event is similar to that of the weakly labeled training set. The size of the test set is such that it represents about 20% of the size of the weakly labeled training set, it contains 288 clips (906 events). The test set is annotated with strong labels, with time boundaries (obtained by human annotators).

The minimum length for an event is 250 ms. The minimum duration of the pause between two events from the same class is 150 ms. When the silence between two consecutive events from the same class was less than 150 ms the events have been merged to a single event. The strong labels are provided in a tab-separated csv file under the following format:

[filename (str)][tab][onset (float)][tab][offset (float)][tab][class label (str)]
The first column, is the name of the audio file downloaded from YouTube, the second column is the onset time in seconds, the third column is the offset time in seconds and the last column corresponds to the class of the sound event.

The amount of events per class is presented in Table 1. This table also presents the mean and median duration of the events for each class. From these durations it is possible to categorize the events into three classes: short events (Alarm, cat, dishes, dog and speech), events with variable duration (blender and running water) and long events that span almost over the whole clip (electric shaver, frying and vacuum cleaner). This classification is confirmed by the observation of the duration distribution presented on Figure 1.

One of the focus of this task is the development of approaches that can provide fine time-level segmentation while learning on weakly labeled data. The observation of the event duration distribution confirms that in order to perform well it is essential to design approaches that are efficient at detecting both short events and events that have a longer duration. Table 3 presents the time proportion of overlapping events in the test set. With overlapping events occurring about 12% of the time it is important to design approaches that are able to detect and properly classify the overlapping events. However, this proportion is lower than the proportion of the clips containing multiple sound events (about 64%). This tends to confirm that being able to discriminate between events that occur at different times within the clip is of a high importance as well. This aspect reinforces the focus on efficient time segmentation.

Evaluation set

The evaluation set contains 880 10-seconds audio clips. The process to select the clips was similar to the process applied to select clips in the training set and the test set, in order to obtain a set with comparable classes distribution. Labels with time boundaries (obtained by human annotators) will be released after the DCASE 2018 challenge is concluded.

TASK DESCRIPTION

The task consists of detecting sound events within web videos using weakly labeled training data. The detection within a 10-seconds clip should be performed with start and end timestamps. Task rules are detailed on the challenge webpage 3 .

Task evaluation

Submissions will be evaluated with event-based measures for which the system output is compared to the reference labels event by event [START_REF] Mesaros | Metrics for polyphonic sound event detection[END_REF]. True positives are the occurrences when an event present 3 http://dcase.community/challenge2018/ in the system output corresponds to an event in the reference annotations. The correspondence between event boundaries are estimated with a 200ms collar tolerance on onsets and a tolerance on offsets that is the maximum of 200ms and 20% of the event length. False positives are obtained when an event is present in the system output but not in the reference annotations (or not within the tolerance on the onset or the offset). False negatives are obtained when an event is present in the reference annotations but not in the system output (or not within the tolerance).

Submissions will be ranked according to the event-based F1score. The F1-score is first computed class-wise over the whole evaluation set:

F 1c = 2T Pc 2T Pc + F Pc + F Nc , (1)
where T Pc, F Pc and F Nc are the number of true positives, false positives and false negative for class of sound event c over the whole evaluation set, respectively. The final score is the F1-score average over class regardless of the number of events per class (macro-average):

F 1macro = c∈C F 1c nC , (2)
where C is the classes ensemble and nC. Evaluation is done using sed eval toolbox [START_REF] Mesaros | Metrics for polyphonic sound event detection[END_REF]. The choice of the macro averaging allows for according the same importance to each class of sound event in an unbalanced scenario as in task 4.

BASELINE

System description

The baseline system is based on convolutional recurrent neural networks (CRNN). The number of 64 log mel-band magnitudes are Figure 2: Baseline system description extracted from 40 ms frames with 50% overlap. Using these features, we train a first CRNN with three convolution layers (64 filters (3x3), max pooling (4) along the frequency axis and 30% dropout), one recurrent layer (64 Gated Recurrent Units with 30% dropout on the input), a dense layer (10 units sigmoid activation) and global average pooling across frames (Figure 2).

The system is trained for 100 epochs (early stopping after 15 epochs patience) on weak labels ('train/weak', 1,578 clips). 20% of this set is used for validation. This model is trained at clip level, the annotations only indicate if the event is present or not during the clip. The inputs are 500 frames long for a single output frame. This first model is used to predict labels of unlabeled files ('train/unlabel in domain', 14,412 clips).

A second model based on the same architecture (Figure 2) is trained on predictions of the first model on the unlabeled subset ('train/unlabel in domain'). The files with manual annotations ('train/weak') are used for the validation of the model. The main difference with the first pass model is that the output dense layer is time-distributed in order to be able to predict events at the frame level. The inputs are 500 frames long, each of them labeled identically following clip labels. The model outputs a decision for each of these 500 frames. Median filtering over 51 frames (≈ 1 s) is applied to the output of the network, in order to obtain the events onset and offset for each file. The full process is described in Algorithm 1.

System performance

The F1-score performance of the baseline system after the first pass (training on 'train/weak' data) and after the second pass (training on 'train/unlabel in domain' with labels obtained with the first pass system) are presented in Table 4. The second pass improves performance compared to the first pass in macro-average and for all classes expect for 'Dishes'. Therefore, the baseline system takes advantage of the unlabeled data even though this could probably be done more efficiently by the systems submitted to the task.

The F1-score is close to zero for all the short event classes. It is slightly higher for event with variable duration (running water Predict class for unlabeled data ('train/unlabel in domain') 6: end procedure 7: procedure SECOND PASS (at frame level) 8:

Train a CRNN on labels predicted for the unlabeled data during the first pass ('train/unlabel in domain')

⊲ weak data ('train/weak') is used for validation ⊲ Labels are used at frames level. If a class is present in the clip, all frames contain the label. Evaluate the prediction on the test set with respect to manual annotations and blender) and performs the best for the long event class. This tends to indicate that the baseline performs very poorly in timesegmentation which is one of the focus of this task. Errors in segmentation are heavily penalized by the metric as they result in both a false positive and a false negative. Therefore, in order to improve performance, submitted systems should propose efficient time-segmentation approaches.

CONCLUSION

This paper presents DCASE 2018 task 4 on large-scale weakly labeled semi-supervised SED in domestic environments. The goal is to exploit a small dataset of weakly labeled sound clips (without time boundaries) together with a larger unlabeled dataset to perform SED (with time boundaries). This indeed corresponds to a realistic scenario as obtaining time coded annotations is time consuming and is generally considered as one of the principal bottlenecks in training SED systems. The design of the dataset with a wide variability in event lengths and the choice of a metric that is heavily penalizing segmentation errors put a strong focus on the problem of localizing the events in time.

  In order to reflect what could possibly happen in a real-world scenario, we provide 3 different splits of training data in task 4 training set: a labeled training set, an unlabeled in domain training set and an unlabeled out of domain training set.

Figure 1 :

 1 Figure 1: Duration distribution by class of sound events.

Algorithm 1 4 :

 14 Script description 1: procedure DOWNLOAD THE DATA (ONLY THE FIRST TIME) 2: end procedure 3: procedure FIRST PASS (at clip level) Train a CRNN on weakly labeled data ('train/weak') ⊲ 20% of data used for validation 5:

9 :

 9 Predict strong test labels ('test/')⊲ Predict an event with an onset and an offset 10: end procedure 11: procedure EVALUATION (event based) 12:

Table 1 :

 1 Class-wise statistics

			Count		Duration (in s)
	Class	Training	Test		Test
		clips	clips events	mean median
	Alarm/bell/ringing	205	45	112	1.53	0.58
	Blender	134	30	40	5.35	4.59
	Cat	173	32	97	0.81	0.71
	Dishes	184	35	122	0.56	0.42
	Dog	214	29	127	1.03	0.66
	Electric shaver/toothbrush	103	25	28	7.41	8.78
	Frying	171	24	24	9.34	10.00
	Running water	343	63	76	5.61	5.53
	Speech	550	105	261	1.51	1.20
	Vacuum cleaner	167	35	36	8.66	9.99
	Total	2244	288	906	2.41	1.03
	Number of classes	1	2	3 and more
	Clip proportion	62.36% 32.89%	4.75%

Table 2 :

 2 Proportion of clips with multiple classes of sound events (training set)

Table 3 :

 3 Proportion of overlapping events (test set)

	Number of events	1	2	3 and more
	Time proportion	86.86% 10.40%	2.74%
	Clip proportion	33.68% 17.71%	48.61%

Table 4 :

 4 13: end procedure Event based F1-scores

	Class	1 st pass 2 nd pass
	Alarm/bell/ringing	3.2%	3.9%
	Blender	10.1%	15.4%
	Cat	0.0%	0.0%
	Dishes	1.9%	0.0%
	Dog	0.0%	0.0%
	Electric shaver/toothbrush	18.2%	32.4%
	Frying	9.4%	31.0%
	Running water	7.6%	11.4%
	Speech	0.0%	0.0%
	Vacuum cleaner	24.8%	46.5%
	Macro average	7.51%	14.06%

see also https://research.google.com/Audioset/ /dataset/index.html

The annotations files and the script to download the audio files is available on the git repository for task 4 https: //github.com/DCASE-REPO/dcase2018_baseline/tree/ master/task4/dataset

This work has been funded by the French region Grand-Est. † This work has been funded by the Austrian Ministries BMVIT and BMWFW, and the Province of Upper Austria, via the COMET Center SCCH.


Jean-Pierre Kahane

Yitzhak Katznelson

Restrictions of continuous functions

published or not. The documents may come

Introduction

Given a continuous real-valued function on [0, 1], and a closed subset E ⊂ [0, 1] we denote by f E the restriction of f to E, that is, the function defined only on E that takes the same values as f at every point of E. The restriction f E will typically be "better behaved" than f . It may have bounded variation when f doesn't, it may have a better modulus of continuity than f , it may be monotone when f is not, etc. All this clearly depends on f and on E, and the questions that we discuss here are about the existence, for every f , or every f in some class, of "substantial" sets E such that f E has bounded total variation, is monotone, or satisfies a given modulus of continuity. The notion of "substantial" that we use is that of either Hausdorff or Minkowski dimensions, both are defined below.

Here is an outline of the paper. We refer to theorems by the subsection in which they are stated.

Section 2 deals with restrictions of bounded variation. Theorem 2.1, part I states that every continuous real-valued function on [0, 1] has bounded variation on some set of Hausdorff dimension 1/2. Part II of the theorem shows that this is optimal by constructing an appropriate lacunary series whose sum has unbounded variation on every closed set of Minkowski dimension bigger than 1/2 (and hence on every set of Hausdorff dimension bigger than 1/2). Analogous results for R dvalued functions are proved in subsection 2.6.

Section 3 deals with restrictions that satisfy a Hölder condition with parameter α ∈ (0, 1). It was known, though never stated in this form, that for every continuous function f on [0, 1] and every α ∈ (0, 1) there exists sets E of Hausdorff dimension 1 -α such that f E satisfies a Hölder α condition (see subsection 3.1). Extending the methods used in the proof of theorem 2.1, we give an elementary proof of the result (theorem 3.1 part I) and show, in part II, that it is optimal by constructing, as in the proof of part II of theorem 2.1, an approriate lacunary series whose sum is a function for which nothing better can be done.

In section 4, theorem 4.1, we construct continuos functions f that satisfy a Hölder-α condition for all α < 1 and yet if f E is Lipschitz or monotone, then E is "arbitrarily thin". Theorem 4.2 deals with monotone restrictions of continuous functions.

In section 5 we consider the relative advantage of restrictions of functions that satisfy various Hölder smoothness conditions, give partial results and point out some open problems.

By including the short section 1, we try to make the exposition elementary and self-contained, requiring no background material beyond what should be "commonly known".

Notations and terminology.

A modulus of continuity is a monotone increasing continuous concave function ω(t) on [0, 1], such that ω(0) = 0.

Given a real-valued function f on [0, 1], a closed set E, and a modulus of continuity ω, we write

f E ∈ C ω if for all t ∈ E there exist δ = δ (t) > 0 and C = C(t) such that if τ ∈ E and |t -τ| ≤ δ (t) then | f (t) -f (τ)| ≤ C(t)ω(t -τ).
For ω(t) = t α , 0 < α ≤ 1 we write Lip α instead of C ω . Lip 1 is usually referred to as the Lipschitz class, while Lip α , 0 < α < 1, as the Hölder α class. [START_REF] Kahane | Some random series of functions[END_REF] The (total) variation, var(E, f ), of a function f on a closed set E, is defined by

var(E, f ) = sup ∑ | f (x j+1 ) -f (x j )|,
the supremum is for all finite monotone increasing sequences {x j } ⊂ E. We write

f ∈ BV (E) if var(E, f ) < ∞. The oscillation of g on a set E is (1) osc(g, E) = max x∈E g(x) -min x∈E g(x).
Finally, if E ⊂ [0, 1] is closed, we denote by |E| the (Lebesgue) measure of E.

Dimensions

1.1 (Lower) Minkowski dimension.

DEFINITION. Let s > 0. An s-separated set of length m is a set J = {x j } m j=1 in [0, 1] such that |x k -x j | > s for j = k.
For a subset E ⊂ [0, 1], denote by L n (E) the smallest number of intervals of length n -1 needed to cover E. Denote by L * n the largest number L such that E contains some n -1 -separated sequence of length L.

Lemma.

(2)

L n (E) ≤ L * 2n (E) ≤ L 2n (E). PROOF: A pair of points whose distance is > (2n) -1 cannot belong to the same interval of length (2n) -1 . Conversely, if {x j } L * n j=1 is a maximal (2n) -1 separated subset of E, then the intervals of length n -1 centered at x j cover E. ◭
The Minkowski dimension, M-dim(E) of E is defined as the limit, if it exists,

(3)

M-dim(E) = lim n→∞ log L n (E) log n = lim n→∞ log L * n (E) log n .
The lower Minkowski dimension L M-dim(E) of E is well defined for all sets by

(4) L M-dim(E) = lim inf log L n (E) log n = lim inf log L * n (E) log n . Example. If E = { 1 j } ∞ j=1 , the subset { 1 j } n j=1 is n -2 separated and L * n 2 (E) ≥ n.
On the other hand the intervals [ jn -2 , ( j + 1)n -2 ], j = 1, . . . , n cover { 1 j } ∞ j=n , and n additional intervals of the same size cover { 1 j } n j=1 , so that L n 2 (E) ≤ 2n. By (2) L n 2 (E) ∼ n, the limit in (3) exists, and M-dim(E) = 1 2 .

Hausdorff dimension.

The Hausdorff dimension H -dim(E) of a set E ⊂ R is the infimum of the numbers c for which there is a constant C such that, for every ε > 0, there exists a covering of E by intervals I n satisfying:

(5) sup

n |I n | < ε and ∑ |I n | c < C.
Since covering by intervals of arbitrary lengths ≤ ε can be more efficient than covering by intervals of a fixed length, (6)

H -dim(E) ≤ L M-dim(E);
the Hausdorff dimension of a set E is bounded above by its lower Minkowski dimension. The inequality can be strict: for example, if E is countable then H -dim(E) = 0, while L M-dim(E) can be as high as 1.

A useful criterion for a lower bound on the Hausdorff dimension of a closed set E is the following: Lemma. Assume that E carries a probability measure µ such that µ(I) ≤ C|I| δ for every interval I then H -dim(E) ≥ δ . PROOF: If c < δ , and I n are intervals such that |I n | < ε and ∪I n ⊃ E, then (7)

1 ≤ ∑ µ(I n ) ≤ C ∑ |I n | δ ≤ Cε δ -c ∑ |I n | c .
That means ∑|In| c > C -1 ε c-δ which is unbounded as ε → 0. ◭

Determining functions.

A Hausdorff determining function is a continuous nondecreasing function h on [0, 1] satisfying h(0) = 0. The Hausdorff dimension introduced in the previous subsection uses explicitly, in (5), the functions h c (t) = t c , with 0 < c ≤ 1 as does (implicitly) the definition of the Minkowski dimension.

A set E ⊂ [0, 1] has zero h-meassure if, for every ε > 0, there exist intervals

I n such that ∑ h(|I n |) < ε and E ⊂ ∪I n . A set E ⊂ [0, 1] is Minkowski-h-null if lim inf L n h(1/n) = 0. A set that is Minkowski h-null has zero h-measure.
The converse is false.

Restrictions of Bounded Variation

2.1 The total variation of restrictions. Given a function f on R and a closed set E, we denotes the total variation of the restriction f E of f to E by var(E, f ), and write f ∈ BV (E) if var(E, f ) < ∞.

Theorem. I:

For every real-valued f ∈ C([0, 1]), there are closed sets G ⊂ [0, 1], such that H -dim(G) ≥ 1 2 and f ∈ BV (G). II: There exists real-valued functions F ∈ C([0, 1]) such that var(E, F) = ∞ for every closed set E ⊂ [0, 1] such that L M-dim(E) > 1
2 , (and, in particular, for closed sets E such that H -dim(E) > 1 2 ).

2.2

The proof of part I of the theorem uses the following lemma.

Lemma. Let I be an interval and E

⊂ I a closed set, ϕ ∈ C(E) and osc(ϕ , E) = a.
Then there are subsets E j ⊂ E, j = 1, 2, carried by disjoint intervals I j , such that

|E j | ≥ 1 4 |E| and osc(ϕ , E j ) ≤ a 2 . PROOF: If I = [t 1 ,t 2 ] let t 3 be such that |E ∩ [t 1 ,t 3 ]| = 1 2 |E|. Set I 1 = [t 1 ,t 3 ] and I 2 = [t 3 ,t 2 ]. Define E 1 ⊂ I 1 as follows: Let [c, c + a] be the smallest interval containing ϕ(E ∩ I 1 ). Write G 1 = E ∩ ϕ -1 ([c, c + 1 2 a]) and G 2 = E ∩ ϕ -1 ([c + 1 2 a, c + a]
), and observe that either

|G 1 | ≥ 1 2 |E| or |G 2 | ≥ 1 2 |E| (or both).
Set E 1 as G 1 in the first case, and as G 2 otherwise. Define E 2 ⊂ I 2 in the same way.

◭

We call the sets E j descendants of E, and refer to the replacement of each E by its two descendants as the standard procedure. We sometime use the alternate procedure in which we replace each E by only one of the two descendants.

PROOF OF THE THEOREM, PART I : Let f ∈ C([0, 1]) be real-valued. We apply the lemma, with ϕ = f , repeatedly. We use the standard procedure most steps and the alternate procedure occasionally, c(k) ∼ 2 log 2 k times out of k. After k iterations we have a set

E k which is the union of 2 k-c(k) ∼ 2 k k -2 sets E k,α , each of Lebesgue measure ≥ 2 -2k
, carried by disjoint intervals I k,α , and such that

osc(g, E k,α ) ≤ 2 -k . Write G = k E k . For x, y ∈ G let k(x, y) be the last k such that x and y are in the same component E k,α . Remember that | f (x) -f (y)| ≤ 2 -k .
In a monotone sequence

{x j } N j=1 ⊂ G and any k ∈ N, there are at most 2 k-c(k) ∼ 2 k k -2 values of j for which k(x j , x j+1 ) = k; so that (8) ∑ | f (x j+1 ) -f (x j )| ≤ ∑ 2 k-c(k) 2 -k ∼ ∑ 2 k k -2 2 -k = ∑ k -2 .
It follows that the total variation of f G is bounded by ∑ k -2 .

Let µ k a probability measure carried by E k that puts the same mass 2 c(k)-k on every E k,α . Observe that, for all l ∈ N, µ k+l (E k,α ) = µ k (E k,α ).

Let µ be a weak-star limit of µ k as k → ∞. Since every interval I of length 2 -2k intersects at most two sets of the form E k,α we have µ(I)

≤ C|I| k-c(k) 2k
and, by lemma 1.2 H -dim G ≥ 1/2. ◭

2.3

The proof of part II of the theorem is a construction that uses as a building block the 2-periodic function ϕ, defined by:

(9) ϕ(2m + x) = 1 -|x| for |x| ≤ 1 and m ∈ Z.
We write ϕ n (x) = ϕ(2nx).

Lemma. Let J = {x j } ⊂ [0, 1] be an s-separated monotone sequence of length m. If m > 2n, then, for a > 0, ( 10
)
var(J, aϕ n ) = ∑ |aϕ n (x j+1 ) -aϕ n (x j )| ≥ (m -2n)2nas.
PROOF: There are at most 2n values of j for which x j and x j+1 are separated by some ℓ 2n , (ℓ = 1, . . . , 2n). For all other j we have aϕ n linear and

|aϕ ′ n | = 2an in [x j , x j+1 ] so that (11) |aϕ n (x j+1 ) -aϕ n (x j )| = 2na(x j+1 -x j ) ≥ 2nas,
and there are at least m -2n such values of j. ◭

2.4

We can modify aϕ n somewhat without affecting (10) materially. PROOF: For the values of j for which x j and x j+1 are not separated by some ℓ 2n we have

Lemma. Let g ∈ C([0, 1]), g ∞ < nsa/
|aϕ n (x j+1 ) -aϕ n (x j )| = 2na(x j+1 -x j ), |G(x j+1 ) -G(x j )| ≤ na 10 (x j+1 -x j ), |g(x j+1 ) -g(x j )| ≤ nas 5 ≤ na 5 (x j+1 -x j ), (13)
so that

|G + aϕ n + g)(x j+1 ) -(G + aϕ n + g)(x j )| ≥ (2na - na 10 )(x j+1 -x j ) - nsa 5 > nsa which implies (12) ◭
We use the lemma with m = 20n and the right-hand sides of (10) and (12) will be (wastefully) written simply as n 2 as.

2.5

PROOF OF THEOREM 2.1, PART II : For sequences {a l }, a l > 0, and {n l } ⊂ N write: m l = 20n l , s l = n -2 l log n l , and

(14) F = ∞ ∑ l=1 a l ϕ n l , G k = k-1 ∑ l=1 a l ϕ n l , g k = ∞ ∑ l=k+1 a l ϕ n l ,
The sequences {a l }, a l > 0 and {n l } ⊂ N are chosen (below) so that

a. a k log n k > k, b. ∑ k-1 l=1 a l n l < 1 10 a k n k c. ∑ l>k a l < 1 10 n k a k s k .
These conditions guarantee that the lemma applies with

n = n k , G = G k and g = g k so that if J is s k separated of length m k , then (15) var(J, F) ≥ n 2 k a k s k = a k log n k > k. If L M-dim(E) > 1/2 then, for all k > k(E), E contains s k -separated se- quences J E (n k ) of length m k , so that (16) var(E, F) ≥ var(J E (n k ), F) > k,
and the function F = ∑ ∞ l=1 a l ϕ n l has infinite variation on every closed E such that L M-dim(E) > 1 2 . The sequences {a l } and {n l } are defined recursively: Take a 1 = 1/2 and n 1 = 100. If a l and n l defined for l ≤ k, set a k+1 = 1 20 a k n -1 k , and observe that this rule guarantees that ∑ j>k a j < 2a k , so that c. is satisfied. Now take n k+1 big enough to satisfy conditions a. and b. ◭

E). II:

There exists continuous R d -valued functions F such that if E ⊂ [0, 1] is closed and L M-dim(E) > 1 d+1 then var(E, F) = ∞.
The proofs of both parts are the obvious variations on the proofs for d = 1. The proof of part I differs from that of the corresponding part of Theorem 2.1 only in the estimate of the measures of the sets E k,α defined at the k'th stage, carried, as before, by disjoint intervals I k,α , and such that osc(g, E k,α ) ≤ 2 -k , but now of Lebesgue measure ≥ 2 -(d+1)k . This guarantees that the Hausdorff dimension of the set, constructed as before, is ≥ 1 d+1 .

For part II we replace the function ϕ n by ψ n = ψ n,d (mx) where m = [n 1/d ] (the integer part of n 1/d ) and ψ n,d is an even 2-periodic R d -valued function satisfying: ψ n,d ≤ 1 and, for x, y such that [x] = [y] and |x -y| ≥ 1/n:

(17) ψ n,d (x) -ψ n,d (y) ≥ n -1 d so that (18) ψ n (x) -ψ n (y) ≥ n -1 d if [mx] = [my] and |x -y| ≥ n -d+1 d .
A set

E such that L M-dim(E) > 1 d+1 , E contains, when n is large, n -d+1 d -separated sequences {x j } L 1 of length L >> n 1 d
and for all, but at most m ∼ n 1 d values of j, we have ψ n (x j+1 ) -ψ n (x j ) ≥ n -1 d so that the variation of ψ n on E is large. One can construct the functions ψ n,d as follows. Let A m = A m,d be the set of (m + 1)

d points v l = (k 1 , . . . k d ) satisfying 0 ≤ k j ≤ m in N d ,
enumerated in a way that v l+1v l = 1, i.e., v l and v l+1 have the same entries except for one, on which they differ by 1. The function ψ n,d is defined on [-1, 1] by stipulating that it is 2-periodic, even, and it maps

[ l (m+1) d , l+1 (m+1) d ] linearly onto [ v l m , v l+1 m ].
3 Hölder restrictions 3.1 Theorem. I: Assume 0 < α < 1. Given a continuous function f , there exists a closed set E such that H -dim E = 1 -α, and f E ∈ Lip α . II: For 0 < α < 1 there exist continuous functions f such that if f E ∈ Lip α for a closed set E, then H -dim E ≤ 1 -α.

Part I of the theorem derives easily from properties of Gaussian stationary processes on the circle, established in [1]. The proof reads:

"Take a Gaussian stationary process X on the circle (Fourier series with independent Gaussian coefficients) such that X ∈ Lip α and H -dim X -1 (0) = α a.s. Then write E = (Xf ) -1 (0) and apply remark 2 in Chapter 14, section 5, page 206 of [1]."

Part II of the theorem shows that part I is optimal. We give here an elementary proof of both parts.

3.2

We prove part I of the theorem by the method used in the proof of part I of theorem 2.1. The following is an extension of the procedures introduced in 2.2. Then

Lemma. Let E

⊂ I ⊂ [0, 1] be a closed set, f ∈ C R (E) and osc( f , E) = a. Given ε > 0, integers k ≥ 2
| Ẽm | ≥ |E| 1-ε k . Let J = [min x∈E f (x), max x∈E f (x)] (so that |J| = a). Divide J into l equal intervals, J s , s = 1, . . . , l, and write E m,s = Ẽm ∩ f -1 J s . For every m let s(m) be such that |E m,s(m) | ≥ |E| 1-ε
kl , and set E m = Ẽm,s(m) . ◭

We refer to this as the k, l, ε procedure on (I; E), call the pairs (I m ; E m ) the (first generation) descendants of (I; E) and rename them as (I 1,m ; E 1,m ).

We rename the parameters k, l, ε as k 1 , l 1 , ε 1 , and repeat the procedure on each (I 1,m ; E 1,m ) with parameters k 2 , l 2 , ε 2 . We have the second generation, with k 1 k 2 descendants named (I 2,m ; E 2,m ), m = 1, . . . , k 1 k 2 .

We iterate the procedure repeatedly with parameters k j , l j , ε j for the j'th round, and denote

(19) K n = n ∏ j=1 k j , L n = n ∏ j=1 l j ηn = n ∏ 1 (1 -ε j ).
After n iterations we have

K n intervals I n,m , each carrying a subset E n,m of E such that |E n,m | ≥ ηn K -1 n L -1
n |E|, and any two are separated by intervals of length ≥ ε n ηn-1 K -1 n L -1 n-1 |E|. Given α ∈ (0, 1), we choose the parameters k j , l j uniformly bounded, and ε j → 0 so that (20)

α n = log L n log K n + log L n -log(ε n ηn ) > α, β n = log K n log K n + logL n -log ηn < 1 -α, and α n → α, β n → 1 -α. Denote E * n = ∪ K n m=1 E n,m , observe that E * n ⊂ E * n-1 , and set E * = ∩E * n .
We claim that E * satisfies the requirements of part I of the theorem. To prove the claim we need to show a.

H -dim E * ≥ (1 -α). b. f E * ∈ Lip α .
PROOF: For claim a. we construct a probability measure µ * on E * , such that for every α ′ > α, there exists a constant C = C(α ′ ) such that µ * (I) ≤ C|I| α ′ for all intervals I. By lemma 1.2 this proves H -dim E * ≥ (1 -α).

Denote by µ n the probability measure obtained by normalizing the Lebesgue measure on E * n by multiplying it, on each E n,m , by a constant

c n,m = K -1 n |E n,m | -1 , so that µ n (E n,m ) = K -1
n . The sequence {µ n } converges in the weak-star topology to a measure µ * carried by E * . Observe that µ * (E n,m ) = µ n (E n,m ) = K -1 n . We evaluate the modulus of continuity of the primitive of µ * by estimating the size of intervals A such that µ * (A) ≥ 2K -1 n . Such interval must contain an interval I n,m , and hence E n,m , and it follows that

(21) |A| ≥ |I n,m | ≥ |E n,m | ≥ ηn K -1 n L -1 n |E|
which means that for every α ′ > α we have for n large enough and every interval

I n,m (22) µ * (I n,m ) ≤ |I n,m | log Kn log Kn+log Ln-log ηn = |I n,m | β n ≤ |I n,m | 1-α ′
and it follows that for arbitrary intervals I and any α ′ > α, as |I| → 0

(23) µ * (I) = O |I| 1-α ′
which means that the Hausdorff dimension of E * is at least 1 -α.

The modulus of continuity ϑ of f E * is determined by: "Let x, y ∈ E * . Let n be the smallest index such that x, y are not in the same

E n,m . Then |x -y| ≥ ε n ηn K -1 n L -1 n |E| and | f (x) -f (y)| ≤ L -1 n-1 ." which translates to ϑ (ε n ηn K -1 n L -1 n-1 |E|) ≤ L -1 n-1 , or, for t in this range ϑ (t) = O (t α n )
, and for all t

(24) ϑ (t) = O (t α ) ◭
Remark: Reversing the inequalities in (20) by an appropriate choice of the parameters we obtain a set E * that has positive measure in dimension 1 -α, such that the modulus of continuity of f E * is bounded by t α |log t| α+ε as t → 0.

Proof of theorem 3.1, part II.

As in section 2, we write

(25) f (x) = ∞ ∑ 1 a j ϕ(λ j x), and f n (x) = n ∑ 1 a j ϕ(λ j x)
where ϕ is the 2-periodic function defined by (9), a j is fast decreasing, λ j fast increasing. Both a j and λ j depend on α, and will be defined inductively. Choose (arbitrarily) a 1 = 1 2 , and λ 1 = 10. Assuming a j and λ j have been chosen for j ≤ n, we shall choose a n+1 small (see below) and then λ n+1 a large enough integral multiple of λ n so that:

(26) λ n | λ n+1 , and a n+1 λ n+1 ≥ 2 n ∑ 1 a j λ j ,
The divisibility guarantees that that f n is linear in each of the intervals ( j λ n , j+1 λ n ) and the successive inequalities in (26

) that | d dt f n | ≥ 1 2 a n λ n > 2 n . Let E be closed, and assume that f E ∈ Lip α . Denote E n = {x : x ∈ E, | f (x) -f (y)| ≤ n|x -y| α for all y ∈ E such that |x -y| ≤ λ -1 n }.
Clearly E n ⊂ E n+1 , and E * = lim E n ⊃ E. It suffices, therefore, to show that E n can be covered by intervals I j,n such that

∑ j |I j,n | β < ε n,β , with ε n,β → 0 for every β > 1 -α. Write E n, j = E n ∩ [ j λ n , j+1 λ n ].
For x, y ∈ E n, j , and in particular the pair x, y such that E n, j ⊂ [t, y] we have

(27) n|x -y| α ≥ | f (x) -f (y)| ≥ 1 2 a n λ n |x -y| -2a n+1 .
If a n+1 is small enough, this implies |x -y| 1-α ≤ 2n a n λ n , and E n can be covered by λ n intervals I j,n of length |I j,n | ≤ 2n

a n λ n 1 1-α .
For any β , (28)

|I j,n | β ≤ 2n a n λ n β 1-α , and ∑ |I j,n | β ≤ 2n a n β 1-α λ 1-β 1-α n .
For β > 1 -α the exponent of λ n is negative, and we take λ n big enough (after choosing a n ). This concludes the proof of theorem 3. ◭ 4 Lipschitz and monotone restrictions 4.1 Lipschitz restrictions. Part II of theorem 3 indicates that there are continuous functions f such that if f E ∈ Lip 1 then H -dim E = 0. The following refinement shows that even if f is "almost" Lip 1 , the set E can be "arbitrarily" thin.

Theorem. Given a Hausdorff determining function h, and a modulus of continuity ω such that lim s→0 ω(s)/s = ∞, there exist functions f

∈ C ω such that if f E ∈ Lip 1 ,
then E has zero h-measure.

Notice that the assumption lim s→0 ω(s)/s = ∞, allows ω(s) = O (s α ) for all α < 1. The corresponding f ∈ C ω belongs to Lip α for all α < 1.

PROOF: We use again the series (25), namely

f = ∞ ∑ 1 a j ϕ(λ j x),
and adapt the parameters a n and λ n to the current context. Both a j and λ j will be defined inductively, a j will be fast decreasing, λ j fast increasing.

Denote by ω n (s) = max x, |τ|≤s a n |ϕ(λ n (x+τ))-ϕ(λ n (x))|, the modulus of con- tinuity of a n ϕ(λ n x). The condition ∑ n ω n (s) = O (ω(s)), as s → 0, guarantees that f ∈ C ω . Observe that

(29) ω n (s) = min(a n , a n λ n s) = a n if s > λ -1 n a n λ n s if 0 ≤ s ≤ λ -1 n .
i. The first condition we impose on a n , λ n is: a n ≤ ω(1/λ n ). It implies that ω n (s) ≤ min(a n , ω(s)) for all s. As ω(1/λ ) >> 1/λ , the condition is consistent with having a n λ n arbitrarily large.

ii. Given a n and λ n , define c n by the condition ω(c n ) = 2 n a n λ n c n = 2 n ω n (c n ). This implies that for s ≤ c n we have ω(s) ≥ 2 n a n λ n s and (30)

ω n (s) ≤ a n if s > c n 2 -n ω(s) if s ≤ c n .
so that for c n+1 ≤ s ≤ c n we have ∑ ω j (s) ≤ ω(s) + ∑ ∞ j=n+1 a j . It follows that if a n decreases fast enough (while λ n increases, allowing a n λ n to be as large as is needed), we have indeed f ∈ C ω .

iii. Assuming a j and λ j have been chosen for j ≤ n, we shall choose a n+1 small (see below) and then λ n+1 a large enough integral multiple of λ n so that:

(31) λ n | λ n+1 , and a n+1 λ n+1 ≥ 2 n ∑ 1 a j λ j ,
The divisibility guarantees that that f n is linear in each of the intervals ( j λ n , j+1 λ n ) and the successive inequalities in (31

) that | d dt f n | ≥ 1 2 a n λ n >> 2 n . Let E be closed, and assume that f E ∈ Lip 1 . Denote E n = {x : x ∈ E, | f (x) -f (y)| ≤ n|x -y| for all y ∈ E such that |x -y| ≤ λ -1 n }.
Clearly E n ⊂ E n+1 , and E * = lim E n ⊃ E. It suffices, therefore, to show that E n can be covered by intervals I j,n such that

∑ j h(|I j,n |) < ε n , with ε n → 0. Write E n, j = E n ∩ [ j λ n , j+1 λ n ]. If x, y ∈ E n, j then (32) n|x -y| ≥ | f (x) -f (y)| ≥ 1 2 a n λ n |x -y| -2a n+1 which implies |x -y| ≤ 4a n+1 /(a n λ n -2n
). It follows that E n can be covered by λ n arcs of length bounded by l n = 4a n+1 /(a n λ n -2n) < 5a n+1 /a n λ n .

Choose a n+1 small enough so that λ n h(l n ) < n -n , and then λ n+1 appropriate to guarantee (31). PROOF: Now we have to give up the building block ϕ defined in (9) and the corresponding functions ϕ n . Let us denote by ψ m (x) the 1-periodic function satisfying: ψ m (0) = ψ(1) = 0, ψ m (m -1 ) = 1 and ψ m (x) linear on [0, m -1 ] and on [m -1 , 1].

Remark

Write f = ∑ ∞ 1 a j ψ m j ((-1) j λ j x) and f n = ∑ n 1 a j ψ m j ((-1) j λ j x), where a j , m j , and λ j will be defined inductively.

The first conditions are (33) m j-1 λ j-1 | m j λ j , and a n λ n ≥ 2 n-1 ∑ 1 a j m j λ j , so that f n is linear in each of the intervals (n-intervals) ( j m n λ n , j+1 m n λ n ). Each such interval is divided in the next generation into one "fast" interval on which | d dt f n+1 | ∼ a n+1 m n+1 λ n+1 and the union of the remaining "slow" intervals on which | d dt f n+1 | ∼ a n+1 λ n+1 .

For even n (resp. odd n) f n is increasing (resp. decreasing) on the fast intervals and decreasing (resp. increasing) on the unions of the slow ones contained in an (n -1)-interval.

Let E be closed, f E monotone increasing. Let n be even. Then, if J is the slow part of an n-interval, the diameter of J ∩ E is bounded by a n+1 /a n λ n . The number of such J's is λ n . Choose a n+1 such that λ n h(a n+1 /a n λ n ) → 0. E \ J is covered by the union of the fast n-intervals that is λ n intervals of length m -1 n . Choose m n (after choosing λ n ) so that λ n h(m -1 n ) → 0.

2. 6

 6 R d -valued functions. The generalization of Theorem 2.1 to R d -valued functions is the following statement: Theorem. I: For every continuous R d -valued function g, there are closed sets E ⊂ [0, 1], such that H -dim(E) ≥ 1 d+1 and g ∈ BV (

  and l ≥ 2, there are subsets E m ⊂ E, m = 1, 2, . . . , k, carried by disjoint intervals I m , such that a. The distance between any two E m ′ s is at least |E|ε/k; b. |E m | ≥ 1-ε kl |E|; c. osc( f , E m ) ≤ a l . PROOF: Choose the increasing sequence {x m }, m = 0, . . . , k so that |E ∩ [0, x m ]| = |E| m k , and let y m = x m + |E|ε/k. Write I m = [y m , x m+1 ] and Ẽm = E ∩ I m .

  Assume that 0 < β < α < 1. There exist functions f ∈ Lip β such that if f E ∈ Lip α , then E has Hausdorff dimension bounded by 1-α 1-β . PROOF: We keep the notations used in the proof of theorem 4.1. As observed there, the condition f ∈ Lip β is equivalent to a n = O λ -β n (if λ n grows fast enough). Now a Is the following statement valid? Assume 0 < β < α < 1. If f ∈ Lip β there exists a set E such that H -dim E = 1-α 1-β , and f E ∈ Lip α . 5.2 Bounded variation. For α ∈ (0, 1), denote by α the Lip α norm. It is easy to see that aϕ n α ∼ an α and if n k increases fast enough, say n k+1 > 2n k , then ∑ a k ϕ n k ∈ Lip α if, and only if,a k = O n -α k .Theorem. There exists real-valued functions F ∈ Lipα such that if E ⊂ [0, 1] is closed and L M-dim(E) > 1 2-α then var(E, F) = ∞.PROOF: As in the example above defineF = ∑ a k ϕ n k where now n k = a -1/α k . If L M-dim(E) > 1 2-α ,and we set s k = n α-2 k log n k , then E contains s k -separated sequences J ′ k of length m k > 20n k , and var(E, F) = ∞ since for every k, (34) var(E, F) ≥ var(J ′ k , F) ≥ n 2 k a k s k = log n k . ◭ Question: Is the result best possible: does every f ∈ Lip α have bounded variation on some set of dimension c = 1 2-α ?

:

  The proof shows, in fact, that E is Minkowski h-null.

◭ 4.2 Monotone restrictions. Does there exist a function f ∈ C([0, 1) such that if f E is monotone then E has Hausdorff dimension 0?

Theorem. Given a Hausdorff determining function h, there exists f ∈ C([0, 1]) such that if f E is monotone, then E has zero h-measure.

Some classics refer to the Hölder classes as the Lipschitz α classes -hence the notation.


Josette Badet

Roger W Jeanloz

come

Isolation and partial purification of lipid-linked oligosaccharides from calf pancreas


Keywords: Data-driven control, Model reference control, Loewner interpolation

In this article, a direct data-driven design method, based on frequency-domain data, is proposed. The identification of the plant is skipped and the controller is designed directly from the measurements. The identification task is reported on a fixed-order controller using for the first time the Loewner approach, known for model approximation and reduction. The method is validated on two numerical examples including the control of an industrial hydroelectric generation plant, modelled by irrational equations.

INTRODUCTION

In many control engineering applications, no mathematical description of the plant is available or easily accessible.

Given some input-output data collected on a system, one can either identify a model of the plant and then, design a controller using any kind of model based technique (indirect methods), or directly use the experimental data to design a controller (direct methods).

In this paper, we are interested in the latter case. Indeed, direct methods are particularly interesting when a model of the system would be too time-consuming, too complex or too costly to obtain. Another application is for industrial processes which tend to exhibit various parametric variations. Modelling and identifying these behaviors and taking them into account in the controller design is a difficult task which is avoided using a direct data-based method. Moreover, it is worth noting that a model is always a compromise between complexity and reliability, and a major difficulty is to determine which parts of the dynamics have to be considered. Furthermore, the identification can result in a complex structure for the plant, and consequently for the controller, and a reduction step (for the controller) might be needed, which is a complex task (see [START_REF] Antoulas | A tutorial introduction to the Loewner framework for model reduction[END_REF]). Of course, authors are conscious that, with direct methods, the identification problem is shifted to the control one, and the problem is then to know which parts of the dynamics are to be controlled. However, since the selection of the controller is done directly from the experimental data, direct datadriven methods seem less conservative, and not sensitive to modelling errors. Moreover, they are less time-consuming since the modelling and/or identification steps are skipped and the resulting control law is tailored to the actual system. However, authors are conscious that an advantage of indirect methods over direct ones is that the specifications can be more advanced and that the model can be used for other purpose (stability and robustness analysis, simulation, etc...).

Numerous direct methods, also referred to as "model free" control, have been proposed to improve the results and try to achieve the best possible performance without using any plant model. One of the first direct methods relies the unfalsified control concept, see [START_REF] Safonov | The unfalsified control concept and learning[END_REF].

Another one is the Iterative Feedback Tuning (IFT, [START_REF] Hjalmarsson | Iterative feedback tuningan overview[END_REF]) which finds the controller parameters thanks to an adaptative and iterative control algorithm based on explicit criterion minimization. Note that Kammer introduced in [START_REF] Kammer | Direct iterative tuning via spectral analysis[END_REF] a frequencydomain variant of the IFT. Among all these techniques, the Virtual Reference Feedback Tuning (VRFT) is a direct method that has been introduced in [START_REF] Campi | Virtual reference feedback tuning: a direct method for the design of feedback controllers[END_REF] in its original version. It consists in minimizing a performance criterion, which is a convex function for an a priori fixed poles selection. It finds the controller (in a class a priori selected) so that the difference between the closed-loop and an objective transfer is minimized. See also [START_REF] Formentin | Data-driven design of braking control systems[END_REF] for an application of the VRFT and Formentin et al. (2014) for a comparison with indirect methods. In an applicative set-up, direct methods such as the VRFT are really useful when a quick and low-cost controller has to be synthesized. One of the main problem is that selecting a class of controllers, as it is required in the methods mentioned above, can be particularly difficult.

In this article, a one shot direct method using frequencydomain data is proposed. In this approach, the user does not have to choose a structure for the controller, which makes it really easy to use. It can be applied to MIMO plants and it is suitable for large scale problems. As the VRFT, it is a model reference-based control technique: the specifications are expressed as a reference transfer function which represents the desired behavior in closed-loop. The problem considered (see Figure 1) is to design a controller for an unknown plant P on the basis of n u input-n y output, The goal is to identify a controller K that gives a closedloop as close as possible to the reference transfer M if inserted in the closed-loop. The measurements can be given as a data-set {u(t it ), y(t it )}, i t = 1 . . . N t , where u(t it ) ∈ R nu and y(t it ) ∈ R ny , or directly as samples of the frequency response of the plant

{ω i f , Φ i f }, i f = 1 . . . N f , where Φ i f ∈ C ny×nu and ω i f ∈ R + .
The closed-loop objective transfer M and the open-loop experimental data of the plant are used to get the frequency response of the "ideal" controller denoted K(ıω i f ) for a limited frozen set of frequency values. This controller is "ideal" in sense that it would give exactly the objective transfer function M if inserted in the closed-loop. Then, the "ideal" controller, known over a limited data samples, will be approximated by K, a linear time-invariant system defined by the following generalized state-space form:

K : E ẋ(t) = Ax(t) + Be(t) u(t) = Cx(t) + De(t) , (1)
where x(t) ∈ R n the state vector, e(t) = r * (t)-y(t) ∈ R ny the input vector of the controller, which is the tracking error, u(t) ∈ R nu the output vector (i.e. the control signal) and

E, A ∈ R n×n , B ∈ R n×ny , C ∈ R nu×n and D ∈ R nu×ny . Its associated transfer function is K(s) = C(sE - A) -1 B + D.
The direct data-driven control design technique proposed in this paper relies on two steps: (i) the obtention of an "ideal" frequency response of the controller K(ıω i f ) and (ii) its identification K through the Loewner framework in the frequency domain. As said earlier, this method, called L-DDC (Loewner Data-Driven Control), does not require any a priori selection of a class of controllers, the user does not have to determine a priori the poles which is a long and difficult task that do not guarantee the selected poles optimality. Moreover, it is applicable to MIMO systems, a single set of input-output measurements is needed and, as shown in the rest of the paper, is easy to apply.

Let us denote ı the complex variable and † the Moore-Penrose inverse. s ∈ C denotes the complex conjugate of s ∈ C. The (A, E) pencil denotes the generalized eigenvalues of A -λE. F(.) denotes the Fourier operator.

This article is organized in five sections. Section 2 introduces the Loewner framework which is the frequencybased interpolation technique used in this work and which plays a pivotal role in the proposed approach. Then, the proposed method to design a controller on the basis of frequency-domain data is exposed in Section 3. Finally, two applications are considered in Section 4. The first one is an academical example concerning a flexible transmis-sion. The second one consists in designing a controller for an open-channel hydraulic system described by an irrational transfer function. Conclusions and outlooks are finally given in Section 5.

PRELIMINARY RESULTS: LOEWNER-BASED IDENTIFICATION

The Loewner approach, exposed in [START_REF] Mayo | A framework for the solution of the generalized realization problem[END_REF], is an interpolatory method initially used for model approximation and reduction. In its original version, it constructs a descriptor model in state-space form directly from the frequency-domain data so that the model performs a barycentric Lagrange interpolation (see [START_REF] Ionita | Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems[END_REF] for further details).

This model can be reduced to balance between its complexity and its accuracy. In order to construct such a realization, the following inputs are required: (i) left interpolation point (µ j ) j=1...q ∈ C and left tangential directions (l j ) j=1...q ∈ C ny , and (ii) right interpolation points

(λ i ) i=1...k and right tangential directions (r i ) i=1...k ∈ C nu .
The following vectors are then defined from the input data:

v T j = l T j K(µ j ) ∀j = 1 . . . q w i = K(λ i )r i ∀i = 1 . . . k .
The interpolation points correspond to the data of the model to be identified, which are, in our approach, the samples of the frequency response of the "ideal" controller

{ω i f , K(ıω i f )}, i f = 1 . . . N f .
The computation of the samples K(ıω i f ) and the separation of the data between left and right interpolation points, respectively (µ j ) j=1...q and (λ i ) i=1...k , are explained in Section 3. Note that in case of a SISO system, the tangential directions (l j ) j=1...q and (r i ) i=1...k are useless and can be fixed to 1. In the Loewner approach, one seek for a model K that interpolates the data as follows:

l T j K(µ j ) = l T j K(µ j ) = v T j ∀j = 1 . . . q K(λ i )r i = K(λ i )r i = w i ∀i = 1 . . . k ,
where K represents the ideal controller (Section 3 explains how to obtain samples of its frequency response for a given set of frequency values).

Based on the (µ j , l T j , v T j ) and (λ i , r i , w i ) data, one can construct the Loewner and shifted Loewner matrices L and L σ as follows for all j = 1 . . . q and i = 1 . . . k:

[L] j,i = v T j r i -l T j w i µ j -λ i , [L σ ] j,i = µ j v T j r i -λ i l T j w i µ i -λ j . (2)
As explained in [START_REF] Mayo | A framework for the solution of the generalized realization problem[END_REF], one of the main advantages of the Loewner framework is that the minimal Mc Millan order of the interpolating model K can be obtained by evaluating r = rank[L, L σ ]. The matrices of the model K are then computed as follows :

E = -Y * LX , A = -Y * L σ X , B = Y * V , C = W X , (3)
where X ∈ C k×r and Y ∈ C q×r are two matrices determined by the singular value decomposition of the Loewner pencil (L σ , L) as follows:

[L, L σ ] = Y Σ l X * , L L σ = Ỹ Σ r X * , (4)
where Σ l , Σ r ∈ R r×r .

Computing ν = rank(L) allows to know whether the model is strictly proper rational or not. If not, a D-term or a polynomial term is added. More details are available in [START_REF] Antoulas | A tutorial introduction to the Loewner framework for model reduction[END_REF].

In addition to determining the smallest exact interpolating model, the Loewner framework allows to control the complexity of the identified model: by keeping the n largest singular values of the decomposition of the Loewner pencil only (4), the obtained realization is a n-th order one. In our case, n is the objective order for the controller (n ≤ r).

Usually, the Loewner framework presented here is used for model order reduction, see [START_REF] Dalmas | From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity[END_REF] for an industrial application. In the context of the proposed data-driven controller design, the Loewner framework will be used for the first time to identify directly a controller of a limited complexity. The proposed procedure is detailed in the next section.

MAIN CONTRIBUTIONS: LOEWNER BASED DATA-DRIVEN CONTROL DESIGN

As introduced above, we propose a model reference control technique using frequency-domain data. The proposed procedure is indicated in the following L-DDC algorithm (Algorithm 1).

The idea is to use the interconnection of Figure 1 and to exploit experimental data

{ω i f , Φ i f }, i f = 1 . . . N f to
determine the frequency response of the "ideal" controller K(ıω i f ). The "ideal" controller is the one which would allow to get exactly the desired closed-loop transfer materialized by the function M in Figure 1. Then, we propose to identify the controller using the Loewner framework which principles are recalled in Section 2.

Our approach does not require to select a class of controllers a priori, contrary to the VRFT which allows to tune controllers with a prescribed structure. This is actually, from a practical viewpoint, one of the main strengths of our approach and is due to the interpolatory property of the Loewner framework.

As in the classic VRFT method, the performance objective is fixed by a transfer function M which represents the dynamic one wants to achieve in closed-loop.

If a time data-set {u(t it ), y(t it )}, i t = 1 . . . N t , of inputoutput measurements of the plant is available, a Fourier transform of the input and output signals is done and samples of the plant frequency response are estimated:

Φ i f = Y (ıωi f ) U (ıωi f ) = F (y(ti t )) F (u(ti t )) , ∀i f = 1 . . . N f .
Obviously, the signal u should sufficiently excite the system to get richenough information in terms of frequencies.

With reference to Algorithm 1, first step (1) aims at obtaining the frequency response of the "ideal" controller K(ıω i f ) at given frequencies (ıω i f ) i f =1...N f . Then, steps (2) to (6) consist in identifying the controller from the "ideal" frequency response computed above. This is achieved through the Loewner framework. The data are equally separated between left and right interpolation points. In [START_REF] Ionita | Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems[END_REF], the author recommends to alternate between left and right to avoid rank loss in the Cauchy-like Loewner matrix L. Let us consider the following shift selection:

ıω = µ 1 λ 1 . . . µ N f 2 λ N f 2
. Finally, to deal with real arithmetic only, the complex conjugate values are considered, ensuring L realness (see [START_REF] Ionita | Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems[END_REF]), so the left and right interpolation points are given by: µ

= µ 1 µ 1 . . . µ N f 2 µ N f 2 , λ = λ 1 λ 1 . . . λ N f 2 λ N f 2 ,
where q = k = N f . Note that K(-ıω) = K(ıω) (in order to compute the vectors v T j ∀j = 1 . . . q = N f and

w i ∀i = 1 . . . k = N f ).
Concerning the tangential directions, if the system is SISO, the tangential directions are fixed to 1. In a multivariable case, the choice of the tangential directions is still an open problem and it could be a way to improve our method in the future. If not choosing them randomly, an idea is to spread the directions on the unit sphere. See [START_REF] Ionita | Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems[END_REF] for further details on these problems. Data:

• Samples of the frequency response of the plant

{ω i f , Φ i f }, i f = 1 . . . N f . Note that q = k = N f .
• Objective order n for the controller • Reference transfer function M Solution:

(1) Compute the samples of the frequency response of the "ideal" controller as follows:

∀ i f = 1 . . . N f , K(ıω i f ) = (Φ i f -Φ i f M (ıω i f )) † M (ıω i f ).
(2) Divide the samples between left and right interpolation points, respectively (µ j ) j=1...q and (λ i ) i=1...k .

Choose the left and right tangential directions, respectively (l j ) j=1...q ∈ C ny and (r i ) i=1...k ∈ C nu . See the comments above for the choice of the repartition and the directions. Algorithm 1. L-DDC algorithm Remark 1. An optional filtering of the frequency-domain data ω i f , i f = 1 . . . N f , can be applied before or after computing the ideal frequency response at step (1). We have not proposed any filtering option yet, this would be an interesting outlook for further research.

Remark 2. Note that the complexity of the L-DDC algorithm resides in the Singular Value Decomposition computed at step (5) of the algorithm. Consequently, it is well tailored to large scale problems (i.e. with a large number of data), and there is no limitations to MIMO cases, see [START_REF] Demourant | A new frequency-domain subspace algorithm with restricted poles location through lmi regions and its application to a wind tunnel test[END_REF] for a multivariable example.

NUMERICAL APPLICATIONS

In order to illustrate the main features of the L-DDC algorithm, simulation examples are presented. The first one is a simple academical example while the second one is an industrial simulation one (it consists in an application to an open channel flow described by an irrational transfer function).

Flexible transmission system

Here, the considered plant is the well-known flexible transmission system. This example has been used in [START_REF] Campi | Virtual reference feedback tuning: a direct method for the design of feedback controllers[END_REF] and in [START_REF] Hjalmarsson | Model-free tuning of a robust regulator for a flexible transmission system[END_REF] as a benchmark for the presentation of the VRFT using time-domain data. The system consists of three horizontal pulleys connected by two elastic belts (see [START_REF] Campi | Virtual reference feedback tuning: a direct method for the design of feedback controllers[END_REF]).

The system input is the angular position of the first pulley and its output is the angular position of the third pulley.

The control objective is to pilot the angular position of the third pulley. The behavior of the plant is described by the following discrete-time linear transfer function (using a sampling time T s = 0.05s):

P (z) = z -3 B(z) A(z)
, where B(z) = 0.28261 + 0.50666z -1 and A(z) = 1 -1.41833z -1 + 1.58939z -2 -1.31608z -3 + 0.88642z -4 . The control objective is defined, as in [START_REF] Campi | Virtual reference feedback tuning: a direct method for the design of feedback controllers[END_REF], by the following closed-loop reference transfer function:

M (z) = z -3 (1-α) 2
(1-αz -1 ) 2 , with α = e -Tsω and ω = 10 the desired bandwidth.

For the VRFT technique, the selected class of controllers proposed in [START_REF] Campi | Virtual reference feedback tuning: a direct method for the design of feedback controllers[END_REF] is defined as

K V RF T (z, ϑ) = ϑ0+ϑ1z -1 +ϑ2z -2 +ϑ3z -3 +ϑ4z -4 +ϑ5z -5 1-z -1
, where ϑ i , ı = 1 . . . 5 are the decision variables. In the noise free simulation case, the parameters found by the VRFT are (see [START_REF] Campi | Virtual reference feedback tuning: a direct method for the design of feedback controllers[END_REF]) ϑ 2 = [0.32905, -0.59771, 0.70728, -0.64010, 0.46499, -0.11763]

T .

The input signal used for the simulation was a linear swept-frequency signal from f = 1Hz to f = 500Hz (N f = 750). Now, to stick to the L-DDC approach, a Fourier transform of the input-output signals allows us to compute samples of the frequency response of the plant:

Φ i f = Y (ıωi f ) U (ıωi f ) , ∀i f = 1 . . . N f .
Then, following the L-DDC algorithm, the frequency response of the ideal controller is computed and data are shown as blue dots in Figure 2

: ∀i f = 1 . . . N f , K(ıω i f ) = M (ıωi f ) Φi f -Φi f (ıωi f ) .
The model of the 5-th order controller obtained by the L-DDC algorithm is visible in Figure 2. First, note that the frequency response of the ideal controller is really similar to the VRFT controller one, which allows to get quite exactly the reference transfer in closed-loop.

The L-DDC controller, obtained by applying the algorithm of Section 3, fits the data of the ideal controller with a 5-th order (same order as the VRFT controller). Note that the minimal order giving an exact interpolation of the data is r = 72. However, the singular values of the Loewner matrix L, given in Figure 3, show that the five first singular value are enough to describe the controller. The Bode magnitude plots of the plant P and the reference model M are shown in Figure 4 (solid red and solid blue respectively). The closed-loops obtained by the two different controllers are also visible in Figure 4. The VRFT controller allows to get a similar closed-loop to the reference transfer. However the L-DDC controller gives The fact that the L-DDC controller allows a better fit of the reference transfer can be explained by the fact that the user does not have to select a structure a priori, it is directly identified from the "ideal" frequency response. The absence of a user-defined structure adds degrees of freedom to the problem that are used toward the realization of the objective transfer function M . Not choosing a structure for the controller is a strength of our method and makes it really appealing in practice.

Application to an open-channel flow for hydroelectricity (MISO case)

Now let us move to an industrial problem provided by the French power producer EDF (Electricité de France).

EDF uses water resources to generate green energy with run-of-the-river power plants. They rely on open-channel hydraulic systems that are non-linear and which dynamic depends on the operating point. Here, for simplicity we will consider one single operating point only.

Their physical model requires partial differential equations (namely Saint-Venant equations). In [START_REF] Dalmas | From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity[END_REF], a new irrational transfer function is proposed for open channels to represent the level-to-flow variations for any operating point. It is the solution of Saint-Venant equations under many assumptions. The system has two inputs, the entering and the outgoing flows q e and q s , and one output, the water depth h. The transfer is given by: h

(x, s, Q 0 ) = G e (x, s, Q 0 )q e (s) + G s (x, s, Q 0 )q s (s) = P (x, s, Q 0 ) q e q s
, where G e (x, s, Q 0 )= λ 1 (s)e λ2(s)L+λ1(s)x -λ 2 (s)e λ1(s)L+λ2(s)x B 0 s(e λ1(s)L -e λ2(s)L )

G s (x, s, Q 0 )= λ 1 (s)e λ1(s)x -λ 2 (s)e λ2(s)x B 0 s(e λ1(s)L -e λ2(s)L ) where x is the position of the measurement point on the channel, Q 0 the nominal flow, L the length of the open channel. B 0 , λ 1 (s) and λ 2 (s) depend on the canal configuration and the nominal flow (see [START_REF] Dalmas | From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity[END_REF]). The system, which dynamic is visible in Figure 5, is extremely slow, has a delay behavior and a pole in limit of stability. Moreover, it has an infinite number of poles since the transfer function is irrational.

The system have been approximated in [START_REF] Dalmas | From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity[END_REF] by a 21-st order rational transfer function depending on the nominal flow with input time delays: τ e 500s and τ s 1500s, on q e and q s respectively. This approximation is shown in Figure 5 (dashed red).

The objective is to stabilize the system and to obtain a faster dynamic. The reference closed-loop M is chosen to be a second order continuous transfer function:

M (s) = 1 1+ 2ξ ω 0 s+ s 2 ω 2 0
, with ω 0 = 10 -5 rad.s -1 and ξ = 1. In this example, the frequency approach is interesting since the system is represented by an irrational transfer function. Therefore, one cannot have a time-domain simulation. However we still can estimate samples of the frequency response of the system {ω i f , Φ i f }, i f = 1 . . . N f , from which the ideal controller's frequency response can be deduced. To qs Fig. 6. L-DDC controller synthesized of order 2 (dashed blue), full order r = 215 (solid black) compared with the ideal frequency response (red dots).

The "ideal" controller K(ıω i f ), which exactly provides the desired closed-loop behavior dictated by M when placed in the closed-loop, is obtained as follows (step (1) of the L-DDC algorithm):

∀i f = 1 . . . N f , K(ıω i f ) = (Φ i f -M (ıω i f )Φ i f ) † M (ıω i f )
The samples of the frequency response Φ i f = P (ıω i f ), i f = 1 . . . N f = 500 are extracted from the irrational transfer function P . Note that the Moore-Penrose inverse † is used because it is no longer a SISO case. Therefore, the pseudoinverse should be employed.

The result of the identification step is given in Figure 6. The mallest order giving an exact interpolating model for the considered data is r = rank[L, L σ ] = 215. Even if a controller with an higher order would be more accurate in terms of fitting the ideal frequency response, the second order controller K, n = 2, shown in Figure 6, which is an approximation of the ideal controller, should be sufficient to obtain good performances. Step response of the closed loop using the 2nd order L-DDC controller (dashed blue) and of the objective transfer (solid red).

Since we had no access to the EDF simulator, the 21-st order rational transfer function of [START_REF] Dalmas | From infinite dimensional modelling to parametric reduced order approximation: Application to open-channel flow for hydroelectricity[END_REF] is used to simulate the closed-loop with the 2-nd order controller obtained by the L-DDC. The results are shown in Figure 7. The closed-loop dynamic is similar to the objective behaviour. Note that the command signal (not shown here for space limitations) is reasonable, the maximum flow variation is around 3m 3 .s -1 , which is in the acceptable range for this application with a controller of order 2. In this case, the L-DDC method is appealing since it does not require to simulate the complex system described by an irrational transfer function to obtain timedomain data. Only samples of the frequency response of the plant are needed, which can be estimated directly from the irrational transfer function. Moreover, one should notice that controlling such an infinite order model is also quite challenging even for model-based methods. A perspective is to try this controller on the EDF simulator instead of using the approximate model to validate the performances.

CONCLUSION AND DISCUSSION

In this paper, the L-DDC (Loewner Data-Driven Control), which is a direct data-driven method involving frequency-domain data, has been proposed. It is a model reference technique which specifications are imposed easily through a reference transfer function representing the desired closed-loop behavior. Despite its simplicity, the L-DDC approach seems to provide good performances and, thanks to the Loewner framework, a set of interesting properties.

First, the frequency response of the "ideal" controller is computed thanks to frequency-domain data from the plant and the reference transfer. This controller is called "ideal" because it would give exactly the objective if inserted in the closed-loop. Then, the Loewner framework is used to identify the controller on the basis of this "ideal" frequency response. The Loewner framework provides a nice bundle of properties we pretend to exploit in the future of this work (such as the rank, the interpolary features or the potential structuration of the controller for example).

The main advantage of the L-DDC method relies on its simplicity, the user does not have to choose a structure for the controller, only the order, which becomes a tunable parameter allowing to find a compromise between complexity and reliability. This technique is appealing for engineers for applications when a controller should be synthesized quickly and for which it would be too costly or too complex to identify a model. It is a one shot method and the obtained controller is tailored to the actual system. It can be applied to MIMO plants and it is suitable for large scale problems since the computational cost is low. The use of more properties of the Loewner framework is promising. However, the L-DDC technique, as other direct methods, does not allow the use of model-based analysis of stability and robustness and further research will address this point.

Frequency

  -domain data-driven control design in the Loewner framework P. Kergus * C. Poussot-Vassal * * F. Demourant * * S. Formentin * * *

Fig. 1 .

 1 Fig. 1. Problem formulation: M is the reference transfer, P is the plant and K the controller to be identified respectively denoted u and y, measurement data without identifying a dynamical model of the plant.

  (3) Compute the Loewner pencil (L, L σ ) following the definitions of (2). (4) Compute the dimension of the minimal representation r = rank[L, L σ ]. (5) Perform the Singular Value Decomposition of the Loewner pencil as in (4). If n ≤ r, only the n largest singular values are kept. Otherwise, the controller will be of order r. (6) Compute the descriptor controller model as indicated by (3) to obtain the matrices E ∈ R n×n , A ∈ R n×n , B ∈ R n×ny , C ∈ R nu×n and D ∈ R nu×ny . The controller K is then given as in (1).

Fig. 2 .Fig. 3 .Fig. 4 .

 234 Fig. 2. Identification of the controller by the L-DDC algorithm (solid clear blue) and VRFT controller (solid black). The blue dots corresponds to the frequency response of the ideal controller computed from the data and the objective transfer function M .

Fig. 5 .

 5 Fig. 5. System dynamic for nominal flow Q 0 = 1400 m 3 .s -1 : original plant in solid blue and approximation obtained by Dalmas et al. (2016) in dashed red.

  Fig. 7. Step response of the closed loop using the 2nd order L-DDC controller (dashed blue) and of the objective transfer (solid red).


domain: sdv sdv.bbm.bs chim

text:
R Suffix

P Suffix

H Meier

We present solid-state NMR assignments of the N-terminal domain of the DnaB helicase from Helicobacter pylori (153 residues) in its microcrystalline form. We use a sequential resonance assignment strategy based on three-dimensional NMR experiments. The resonance assignments obtained are compared with automated resonance assignments computed with the ssFLYA algorithm. An analysis of the 13 C secondary chemical shifts determines the position of the secondary structure elements in this α-helical protein.

9 Abstract We present solid-state NMR assignments of the 10 N-terminal domain of the DnaB helicase from Helicobac-11 ter pylori (153 residues) in its microcrystalline form. We 12 use a sequential resonance assignment strategy based on 13 three-dimensional NMR experiments. The resonance 14 assignments obtained are compared with automated reso-15 nance assignments computed with the ssFLYA algorithm. 16 An analysis of the 13 C secondary chemical shifts deter-17 mines the position of the secondary structure elements in 18 this a-helical protein. 19 20 Keywords HpDnaB Á Assignments Á Solid-state NMR Á 21 Secondary chemical shifts Á ssFLYA

Biological context

DnaB helicases are bacterial ATP-driven enzymes which unwind double-stranded DNA in the presence of ATP during the fork movement in 5 0 -3 0 direction in DNA replication (LeBowitz and McMacken 1986). Structurally, DnaB is a two-domain helicase with an amino-terminal domain and a carboxy-terminal domain separated by a linker region. The full-length protein forms ring-shaped hexameric assemblies which encircle single-stranded DNA (LeBowitz and McMacken 1986). The C-terminal domain supports the ATPase activity and is involved in the ring formation, while the N-terminal domain is forming an a-helical globule which has the function to activate the helicase. A fundamental step in DNA replication is the strand synthesis which is initiated by an interaction between the N-terminal domain of the helicase DnaB and the primase DnaG [START_REF] Corn | Regulation of bacterial priming and daughter strand synthesis through helicase-primase interactions[END_REF].

This work focuses on the investigation of the N-terminus of DnaB extracted from Helicobacter pylori (H. pylori), a gram-negative microaerophilic spiral shaped bacterium which is, with a worldwide prevalence of approximately 50 %, responsible for the most common chronic bacterial infections, such as gastric ulcer diseases and gastric adenocarcinoma (Parsonnet et al. 1991;Peterson 1991). The replication system in H. pylori exhibits significant differences compared to other microorganisms which were investigated in detail, e.g. Escherichia coli. The most relevant differences are the absence of the recF gene, the presence of the dnaA gene *600 kb away from the dnaN-gyrB genes and most importantly the absence of the dnaC gene (Soni et al. 2003). In E. coli, DnaC is essential for loading DnaB helicase at oriC, which is the origin of the chromosomal DNA replication, whereas in the case of H. pylori HpDnaB itself is able to take over the (Soni et al. 2005). The crystal structure of the C-terminal domain [START_REF] Stelter | [END_REF], as well as of the 121 N-terminal residues of HpDnaB were determined (Kashav et al. 2009). The latter one was found to consist of a dimer and two further degradation peptide fragments in the asymmetric unit (Kashav et al. 2009). This manuscript presents the solid-state NMR spectroscopic investigation of the N-terminus of HpDnaB as a step towards an integrated structural biology approach aiming at a detailed structural description of the full-length protein. In this context, solidstate NMR studies on a sedimented sample of the fulllength protein have been described (Gardiennet et al. 2012). Previous biochemical investigations suggest that the N-terminal domain and the linker region play an important role in multimerisation, quaternary state transition and activity of HpDnaB (Kashav et al. 2009;Nitharwal et al. 2007).

U N C O R R E C T E D P R O O F DnaC function

Methods and experiments

Protein expression and purification, sample preparation

The DNA fragment corresponding to the N-terminal domain of the H. pylori helicase DnaB (strain 26695) was amplified by PCR (forward 5 0 -caccatggatcatttaaagcatttgcag-3 0 and reverse 5 0 -gcaccatagaaggctttaggaattag-3 0 ) from genomic DNA and inserted into the plasmid pET151/ DTopo (Invitrogen TM ). The resulting vector was introduced into E. coli BL21(DE3) cells (One Shot Ò BL21 Star TM (DE3) Chemically Competent E. coli, Invitrogen TM ) and protein overexpression was performed in minimal M9 medium (Studier 2005) containing D-[U-13- C]glucose 2 g L -1 (Cambridge Isotope Laboratories, Inc. CLM-1396-PK) and 15 NH 4 C l 2gL -1 (Sigma-Aldrich Ò 299251) as the only nitrogen and carbon sources. After cell lysis by a microfluidization process, 13 C-15 N-DnaB-Nter was purified by Ni 2? -agarose affinity chromatography (Qiagen TM ). The pseudo-affinity tag was subsequently cleaved with the TEV (Tobacco Etch Virus) protease by dialysis. Six additional residues of the tag remain in the sequence (see Fig. 1).

For crystallization, 13 C-15 N-DnaB-Nter was concentrated to 23.5 mg/ml using a centrifugal concentrator with a 10 kDa cut-off (Vivaspin Ò 20 VS2001 Sartorius), and the buffer was exchanged during the concentration step with the final buffer (50 mM Tris-HCl pH 6.5, 100 mM NaCl). Crystallization of the protein was performed by mixing an equal volume of protein and crystallization buffer (100 mM HEPES pH 7.0, 0.1 % (m/v) sodium azide, 10 % (v/v) polyethylene glycol 10000) in a nine-well glass plate with 2.3 M NaCl solution in the reservoir. Microcrystals 105 were obtained after 1 week at 20 °C and were harvested 106 and centrifuged into the NMR rotor (25,0009g during 1 h 107 at 4 °C) using a homemade device [START_REF] Bo ¨ckmann | Characterization of different water pools in solid-state NMR protein samples[END_REF].

108 Solid-state NMR spectroscopy 109 Solid-state NMR spectra were acquired at 18.8 and 20.0 T 110 static magnetic field strengths using 3.2 mm Bruker Bios-111 pin ''E-free'' probes (Gor'kov et al. 2007)

U N C O R R E C T E D P R O O F
U N C O R R E C T E D P R O O F
comparable spectra in all cases. Spectra were also recorded on samples with the His-tag, and showed differences large enough to discard the further use of this sample. TALOS? calculations were performed using version 3. 9) conclusion. The Ca and Cb chemical shifts of these two threonine residues already reveal that Thr116 is located in an a-helix, whereas Thr147 is most probably located in a loop in agreement with the observed flexible character. The chemical shifts have been deposited in the BMRB database under the accession number 26548.

U N C O R R E C T E D P R O O F 162 spectra shown in
As indicated in Fig. 6, for some resonances in the amino-acid region Thr116-Lys127 a peak doubling is observed which is exemplarily illustrated by analyzing the unique serine-alanine amino acid pair (residues Ser124 and Ala125). The sequential backbone walk for these residues using NCACB, NcoCACB and CANcoCA spectra is illustrated in Fig. 7 and clearly demonstrates the presence of two sets of resonances. Pronounced spectroscopic differences are mainly observed for the 15 N frequencies. The observed peak doubling might indicate crystallographically distinct molecules in the asymmetric unit, which would agree with the previously published crystal structure for the residues 1-121 consisting of a dimer in the asymmetric unit (Kashav et al. 2009).

The results of the manual assignment procedure are validated by automated peak assignments as implemented in the solid-state FLYA algorithm (Schmidt et al. 2013). Figure 8 illustrates the good agreement between the manually assigned residues and the assignments obtained by FLYA calculations based on the peak lists from the manual assignment procedure. Only a few significant differences (e.g. for the two prolines Pro43 and Pro44, as well as for Thr147, and the C 0 resonances of Ser89 and Val135) were observed and in those cases the manually assignment was carefully rechecked and its resonances were taken for the final assignment as they unambiguously result from the performed sequential walk.

Secondary structure

27

  Solid-state NMR sequential assignments of the N-terminal domain 3 of HpDnaB helicase 4 Received: 29 March 2015 / Accepted: 11 August 2015 8 Ó Springer Science+Business Media Dordrecht 2015

Fig. 1

 1 Fig.1Amino-acid sequence of the N-terminus of HpDnaB as extracted from the uniprot database (The UniProt Consortium-Activities at the Universal Protein Resource (UniProt) 2014). At the N-terminal domain a part of the tag (amino acid sequence GIDPFT) is still present after the cleavage of the tag with TEV protease (shown in grey). Residues highlighted in red are located in a-helices as determined from NMR secondary chemical shifts (see Fig.9)

  8 (Shen et al. 2009). The DSSP algorithm (Kabsch and Sander 1983) was applied using the corresponding web interface (http://www.cmbi.ru.nl/dssp.html) with the 3D atomic coordinates extracted from the pdb file 3GXV (Kashav et al. 2009). Solid-state FLYA calculations (Schmidt et al. 2013; Schmidt and Gu ¨ntert 2012) were performed with CYANA version 3.97. The tolerance value for chemical shift 153 matching was set to 0.55 ppm for 13 C and 15 N. The cal-154 culations are based on experimental peak lists as obtained 155 from the manual assignment procedure. 156 Assignment and data deposition 157 The solid-state NMR spectra of the N-terminal domain of 158 HpDnaB (residues 1-153) reveal significant spectral 159 overlap as expected for a protein of 153 amino acids, even 160 though also many isolated, well resolved signals are 161 detected as can be seen in the 2D dipolar correlation NMR

Fig. 3 Fig. 4 Fig. 5 Fig. 6

 3456 Fig.2a 2D 13 C,13 C DARR spectrum of the N-terminus of HpDnaB measured at 20.0 T with a spinning frequency of 17.0 kHz and 10 ms DARR mixing. The spectrum includes the labels for the Ca-Cb peaks as predicted from the manually created shift list using the CcpNmr software (black chain A, green chain B). In the Ca/Cb region seven isolated peaks could not be assigned, most probably because the corresponding residues are located in flexible parts of the protein. b 2D NCA spectrum of the N-terminus of HpDnaB acquired at 20.0 T with a spinning frequency of 17.0 kHz. The spectrum includes the labels for the peaks as predicted from the manually created shift list using the CcpNmr software (black chain A, green chain B) assuming that only intraresidual peaks with a through-space limit corresponding to one bond are visible (x), peaks labeled with plus indicate N-Cb resonances

  Secondary chemical shifts were obtained by subtracting the random-coil shifts (Wang and Jardetzky 2002) from the observed solid-state NMR chemical shifts (Wishart et al. 1992) and are visualized in Fig. 9. These data clearly illustrate the dominant a-helical character of the N-terminus of HpDnaB. Seven a-helices are identified in total, the longest one is reaching from residue 95 to 117. Our results indicate an interruption after the sixth a-helix and a succeeding seventh a-helix with positive secondary chemical shifts observed for residues 120-135. It is reasonable to assume that the C-terminal part of the protein is mainly flexible, since most resonances are not detected in the performed NMR experiments. The visible residue Thr147 shows Ca/Cb chemical shifts more typical for a loop or extended conformation than an a-helical arrangement (Wang and Jardetzky 2002). The secondary chemical shifts

Fig. 7 Fig. 8 Fig. 9 a

 789 Fig. 7 Illustration of peak doubling for a unique Ser-Ala amino acid pair (124Ser-125Ala). a The sequential walk along the protein backbone based on NcoCACB (negative peaks are illustrated in turquoise), CANcoCA (positive peaks are shown in pink) and

Table 1 . 13 C and 15 N resonance assignments were obtained 120 by using a previously established assignment strategy 121 based on a sequential walk applying 3D NMR spectra 122 (Habenstein et al. 2011; Schuetz et al. 2010) which were 123 analyzed with the software CcpNmr (Fogh et al. 2002; 124 Stevens et al. 2011; Vranken et al. 2005). Many resonances 125 were assigned applying conventional experiments, such as

 1

	. The MAS
	spinning frequency was set to 17.5 and 17.0 kHz for the	112
	two fields, respectively. The 2D and 3D spectra were	113
	processed with the software TOPSPIN (version 3.2, Bruker	114
	Biospin) with a shifted (2.0-2.8) squared cosine apodiza-	115
	tion function and automated baseline correction in the	116
	direct dimension. The sample temperature was set to	117
	278 K, for more details of the conducted experiments see	118
		119
	NCACB, NCACX, NCOCX and CANCO. In case of	126
	spectral overlap, it was essential to complement the	127
	assignment strategy with relayed NMR experiments, such	128
	as NcoCACB, CANcoCA and NcaCBCX. Although those	129
	experiments are less sensitive due to four polarization	130
	transfer steps, they strongly benefit from a larger spectral	131
	dispersion (Schuetz et al. 2010), and the signal/noise ratio	132
	achieved under the conditions used here (high field, full	133
	rotor, moderate-sized protein) is good. A selective C 0 -Ca	134
	polarization transfer in those experiments was achieved by	135
	a modified band-selective homonuclear cross-polarization	136
	step (Chevelkov et al. 2013). The spectra used for assign-	137
	ment were all recorded on a single sample, whereas	138
	reproducibility was carefully checked by 2D measurements	139
	on samples from different preparations which yield	140

Table 1

 1 Overview about experimental parameters of the performed solid-state NMR experiments

	Experiment	DARR	NCA	NCACB	NCACX	NCOCX
	(a)
	MAS frequency/kHz	17.0	17.0	17.5	17.0	17.0
	Field/T	20.0	20.0	18.8	20.0	20.0
	Transfer I	HC-CP	HN-CP	HN-CP	HN-CP	HN-CP
	1 H field/kHz	59.4	60.8	58.5	60.8	60.8
	X field/kHz	40.9	43.1	44.0	43.1	43.1
	Shape	Tangent 1 H	Tangent 1 H	Tangent 1 H	Tangent 1 H	Tangent H
	13 C carrier/ppm	95	-	-	-	-
	Time/ms	0.5	0.6	0.6	0.6	0.6
	Transfer II	DARR	NC-CP	NC-CP	NC-CP	NC-CP
	1 H field/kHz	17.0	-	-	-	-
	13 C field/kHz	-	6.1	10.0	6.1	6.1
	15 N field/kHz	-	10.2	27.0	10.2	10.6
	Shape	-	Tangent 13 C	Tangent 13 C	Tangent 13 C	Tangent C
	Carrier/ppm	95	60	56	60	178
	Time/ms	10	6	4	6	4
	Transfer 3	-	-	DREAM	DARR	DARR
	1 H field/kHz	-	-	-	25.1	25.1
	13 C field/kHz	-	-	7.9	-	-
	N field/kHz	-	-	-	-	-
	Shape	-	-	Tangent 13 C-	-
	Carrier/ppm	-	-	56	178	178
	Time/ms	-	-	4	60	30
	t1 increments	2000	2000	80	108	108
	Sweep width (t1)/kHz	100	66.7	5.7	6.0	6.0
	Acquisition time (t1)/ms	10	15	7.1	9.0	9.0
	t2 increments	2988	3072	116	136	116
	Sweep width (t2)/kHz	100	100	10.9	8.6	6.4
	Acquisition time (t2)/ms	14.9	15.4	5.3	8.0	9.0
	t3 increments	-	-	1988	3072	3072
	Sweep width (t3)/kHz	-	-	100	100	100
	Acquisition time (t3)/ms	-	-	9.9	15.4	15.4
	1 H Spinal64 (Fung et al. 2000)	89	89	90	89	89
	decoupling power/kHz
	Interscan delay/s	2.1	3	2.5	2.6	2.6
	Number of scans	12	8	24	8	8
	Measurement time/h	14	13	157	87	74
	Experiment	CANCO	NcoCACB	CANcoCA	NcaCBCX	CCC
	(b)
	MAS frequency/kHZ	17.0	17.0	17.0	17.0	17.5
	Field/T	20.0	20.0	20.0	20.0	18.8
	Transfer I	HC-CP	HN-CP	HC-CP	HN-CP	HC-CP
	1 H field/kHz	59.4	60.8	58.4	58.4	66.0
	X field/kHz	40.9	43.1	40.8	45.2	50.0
	Shape	Tangent 1 H	Tangent 1 H	Tangent 1 H	Tangent 1 H	Tangent H
	13 C carrier/ppm	95	-	103	-	58.5
	Time/ms	0.5	0.6	0.5	0.6	0.7
	Transfer II	CN-CP	NC-CP	CN-CP	NC-CP	DREAM

Table 2

 2 Statistics of the manually performed peak assignments

	Category	Assigned/%
	C	69.3
	CA	70.6
	CB	69.4
	CG	61.7
	CD	54.5
	CE	34.5
	CZ	17.6
	N	69.9
	Residue Ala	91.7
	Residue Arg	50.0
	Residue Asn	55.6
	Residue Asp	62.5
	Residue Cys	100.0
	Residue Gln	42.9
	Residue Glu	72.2
	Residue Gly	66.7
	Residue His	60.0
	Residue Ile	81.3
	Residue Leu	68.8
	Residue Lys	72.7
	Residue Met	50.0
	Residue Phe	85.7
	Residue Pro	66.7
	Residue Ser	88.9
	Residue Thr	100.0
	Residue Tyr	75.0
	Residue Val	75.0
	Data created with the CcpNmr
	software

in vivo.

Stevens T et al (2011) A software framework for analysing solid-state

MAS NMR data. J Biomol NMR 51:437-447

Studier FW (2005) Protein production by auto-induction in high-

31:1647-1651Solid-state NMR sequential assignments of the N-terminal domain of HpDnaB helicase

Acknowledgments This work was supported by the Agence Nationale de la Recherche (ANR-11-BSV8-021-01, ANR-12-BS08-0013-01), the ETH Zurich, the Swiss National Science Foundation (Grant 200020_159707 and 200020_146757). PG is supported by the Lichtenberg Program of the Volkswagen Foundation, AB and LT by the CIBLE program 2011 from the Re ´gion Rho ˆne-Alpes.

268

A complete comparison of the secondary structure ele-269 ments obtained from solid-state NMR with those found in 270 the published crystal structure is hampered by the fact that 271 the latter one consists only of residues 1-121 (Kashav et al. 272 2009) and might crystallize differently than the full-length 273 N-terminal domain (no diffracting crystals could be 274 obtained for the full-length N-terminal domain). Never-275 theless, the structure of the globular N-terminal domain 276 seems to be roughly similar between the 1-121 and 1-153 277 residues samples, since in general a good agreement 278 between NMR data [secondary chemicals shifts and TALOS? backbone torsion angle calculations (Shen et al. 2009)] and the X-Ray structure is observed (see Table 3).

Conclusions

We describe the sequential resonance assignment of the N-terminal domain of HpDnaB (residues 1-153) based on 3D solid-state NMR experiments leading to a site-specific chemical shift assignment of approximately 70 % of the backbone resonances. Those manually obtained peak assignments are validated by solid-state FLYA calculations. Seven a-helices where identified by a secondary chemical shift analysis which mostly agrees with the single crystal structure published for the residues 1-121 (PDB 3GXV). The observed peak doubling might point to crystallographically distinct molecules in the asymmetric unit.

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