MD Hania Szajewska

MD Andrea Horvath

MD Anna Piwowarczyk

eradication rates and side effects during treatment

Keywords: H. pylori < Topics, Meta-analyses < Topics, Probiotics/prebiotics < Topics, Diarrhoea < Topics RCT, H pylori, probiotics, microbiota

INTRODUCTION

The most commonly prescribed triple therapy, consisting of use of a proton pump inhibitor with clarithromycin and amoxicillin, remains the recommended first-choice treatment for Helicobacter pylori infection. 1 2 3 One major problem with this therapy, as well as with other H pylori eradication regimens, is unsatisfactory eradication rates largely due to the increased resistance to antibiotics, primarily to clarithromycin. 4 5 6 In addition, adverse effects are experienced by about 5% to 30% of patients receiving H pylori eradication therapy and further contribute to treatment failure. [START_REF] Tong | Meta-analysis: the effect of supplementation with probiotics on eradication rates and side effects during Helicobacter pylori eradication therapy[END_REF] Measures to overcome these problems include the use of probiotics, which are live microbial food ingredients that are beneficial to health. [START_REF]WHO Working Group Report on Drafting Guidelines for the Evaluation of Probiotics in Food[END_REF] The rationale for the use of probiotics as adjunctive treatment for H pylori infection is based on the results of studies that have shown that various lactobacilli (eg, Lactobacillus johnsonii La1, L acidophilus CRL 639, L casei), or their metabolic products, can inhibit or kill H pylori in vitro. 9 10 A recent systematic review [START_REF] Tong | Meta-analysis: the effect of supplementation with probiotics on eradication rates and side effects during Helicobacter pylori eradication therapy[END_REF] evaluated the effects of supplementation with probiotics on H pylori eradication rates and side effects of anti-H pylori treatment. Fourteen randomised controlled trials (RCTs) of varying methodological quality involving 1671 patients were identified. In patients with H pylori infection, probiotic supplementation improved eradication rates. In two RCTs that evaluated patients with eradication failure, probiotic supplementation also improved eradication rates.

Probiotics reduced therapy-related side effects overall and individual symptoms of diarrhoea, epigastric pain, nausea, and taste disturbance.

Opponents of using a meta-analytical approach to assess the efficacy of probiotics argue that the beneficial effects of probiotics seem to be strain specific, thus, pooling data on different strains may result in misleading conclusions. A more favourable approach is to perform a meta-analysis that evaluates the effect of administering a clearly defined probiotic preparation (single or in combination). Given these considerations, the aim of the current review was to update and synthesise the available clinical trial evidence of the likely effects of S boulardii given in addition to standard eradication therapy on major clinical outcomes related to H pylori

METHODS

The guidelines from the Cochrane Collaboration for undertaking and reporting the results of a systematic review and meta-analysis [START_REF] Higgins | Cochrane Handbook for Systematic Reviews of Interventions 4[END_REF] and the PRISMA statement [START_REF] Liberati | The PRISMA statement for reporting systematic reviews and meta-analyses of studies that evaluate healthcare interventions: explanation and elaboration[END_REF] were followed for this systematic review and meta-analysis.

Criteria for considering studies for this review

All relevant RCTs that compared use of S boulardii alone or during H pylori eradication therapy with use of placebo or no treatment were eligible for inclusion.

Participants of any age had to be H pylori-infected subjects, as assessed by generally accepted methods (ie, the [START_REF]Assessing risk of bias in included studies[END_REF] C-urea breath test [UBT], histopathology, or the rapid urease test). The primary outcome measure was the rate of H pylori eradication, which had to be confirmed by a negative [START_REF]Assessing risk of bias in included studies[END_REF] C-UBT or other generally accepted method at least 4 weeks after treatment. The secondary outcome measures were the frequencies of adverse effects (overall and specific). The adverse effects of interest were any common gastrointestinal adverse effects that occurred during anti-H pylori therapy, including diarrhoea, taste disturbance, nausea, vomiting, bloating, loss of appetite, abdominal pain, constipation, and the need for discontinuation of the H pylori therapy.

Search methods for identification of studies

The Cochrane Central Register of Controlled Trials (CENTRAL, the Cochrane Library), MEDLINE, and EMBASE databases were searched for relevant studies in July 2010. The principal search text word terms and MESH headings used were as follows: probiotic*, Saccharomyces boulardii and S boulardii, Helicobacter pylori and H pylori. Two (AH, AP) reviewers independently carried out the search, and they did not impose any language restrictions. The reference lists from identified studies and key review articles were also searched to identify any other relevant studies. The

Data collection and analysis

Three reviewers using a standardised approach independently undertook the literature search, data extraction, and quality assessment. The data sought included baseline characteristics of the patients, details of the H pylori eradication therapy, and details related to the use of experimental and control interventions (including dose and duration), type of outcome measure (primary vs secondary), methods of checking H pylori status, and/or assessment of side effects. Minor disagreements were resolved by discussion.

Assessment of risk of bias in included studies

The reviewers independently, but without being blinded to the authors or journal, assessed the risk of bias in the studies that met the inclusion criteria. The Cochrane Collaboration's tool for assessing risk of bias was used, which includes the following criteria: adequacy of sequence generation, allocation concealment, and blinding of participants, personnel and outcome assessors; and extent of loss to follow-up, ie, the proportion of patients in whom the investigators were not able to determine outcomes (incomplete outcome data). In all cases, an answer of 'yes' indicates a low risk of bias, and an answer of 'no' indicates a high risk of bias. [START_REF]Assessing risk of bias in included studies[END_REF]

Measures of treatment effect

The dichotomous outcomes, the results for individual studies, and pooled statistics are reported as the risk ratio (RR) between the experimental and control groups with 95% confidence intervals (95% CI). The weighted mean difference (WMD) between the treatment and control groups was selected to represent the difference in continuous outcomes (with 95% CI).

Dealing with missing data

We assessed pooled data using available case analysis, ie, an analysis in which data are analysed for every participant for whom the outcome was obtained, rather than intention-to-treat analysis with imputation. [START_REF]Special topics in statistics[END_REF]

Assessment of heterogeneity

Heterogeneity was quantified by [START_REF] Chey | American College of Gastroenterology guideline on the management of Helicobacter pylori infection[END_REF] and I 2 , which can be interpreted as the percentage of the total variation between studies that is attributable to heterogeneity rather than to chance. A value of 0% indicates no observed heterogeneity, and larger values show increasing heterogeneity. If heterogeneity was not revealed, we present results of only the fixed effects model. If there was substantial heterogeneity (over 50%), all analyses were based on the random effects model if it was still considered appropriate to pool the data.

Assessment of reporting biases

To test for publication bias, we planned to use a test for asymmetry of the funnel plot proposed by Egger et al. [START_REF] Egger | Bias in meta-analysis detected by a simple, graphical test[END_REF] This test detects funnel plot asymmetry by determining whether the intercept deviates significantly from zero in a regression of the normalized effect estimate (estimate divided by its standard error) against precision (reciprocal of the standard error of the estimate) weighted by the reciprocal of the variance of the estimate (on StatsDirect, version 2.3.8). However, the publication bias was not formally assessed using a funnel plot due to the small number of studies (<10) included in the analyses of the primary and secondary outcome measures.

Data synthesis (Statistical methods)

The data were analysed using Review Manager (RevMan) [Computer program.

Subgroup analysis and sensitivity analyses

For the primary outcome, pre-planned subgroup analysis based on age (adults vs children) was performed. Additionally, when there was statistically significant heterogeneity in the primary outcome across studies, sensitivity analyses were planned to determine the impacts of allocation concealment (adequate versus inadequate and/or unclear) and attrition (<20% versus ≥20%). The latter were not performed, as there was no heterogeneity in the primary outcome.

RESULTS

The literature search yielded 894 articles, of which six were reviewed in full text (figure 1). 16 17 18 19 20 21 Of these studies, five RCTs 17 18 19 20 21 met the inclusion criteria.

All were published in English. These trials randomised a total of 1307 patients, of which 1227 were followed up. Table 1 summarises the characteristics of the included studies. The characteristics of the excluded trials, with reasons for exclusion, are available upon request. Four studies enrolled only adults, 17 18 19 21 and one RCT [START_REF] Hurduc | A randomized, open trial evaluating the effect of Saccharomyces boulardii on the eradication rate of Helicobacter pylori infection in children[END_REF] (n=90) was undertaken exclusively in children (age range: 3 to 18 years). The sample size ranged from 43 to 661 participants. In all studies, S boulardii was used in addition to standard triple therapy consisting of a proton pump inhibitor and two antibiotics.

In all included trials, clarithromycin was one of the antibiotics used. The daily dose of S boulardii ranged from 500 mg 18 20 to 750 mg [START_REF] Song | The effect of probiotics and mucoprotective agents on PPI-based triple therapy for eradication of Helicobacter pylori[END_REF] to 1000 mg. 17 19 Two RCTs 17 18 were placebo controlled; in the remaining three trials, 19 20 21 there was no additional intervention in the control group. Except for one multi-centre trial, 19 the included studies were single-centre trials. The studies were undertaken in countries such as Italy (one RCT [START_REF] Cremonini | Effect of different probiotic preparations on anti-helicobacter pylori therapy-related side effects: a parallel group, triple blind, placebo-controlled study[END_REF] ), Korea (one RCT [START_REF] Song | The effect of probiotics and mucoprotective agents on PPI-based triple therapy for eradication of Helicobacter pylori[END_REF] ), Romania (one RCT [START_REF] Hurduc | A randomized, open trial evaluating the effect of Saccharomyces boulardii on the eradication rate of Helicobacter pylori infection in children[END_REF] ), and Turkey (two RCTs 17 19 ).

Risk of bias in included studies

With the exception of one RCT by Cremonini et al., [START_REF] Cremonini | Effect of different probiotic preparations on anti-helicobacter pylori therapy-related side effects: a parallel group, triple blind, placebo-controlled study[END_REF]

Heterogeneity

Significant heterogeneity (I 2 ≥50%) was found for the overall incidence of adverse effects (chi 2 =9.75, p=0.04, I 2 =59%) and epigastric pain (chi 2 =4.75, p=0.09, I 2 =58%). In all cases, the observed statistical heterogeneity was not judged to be clinically relevant (ie, studies consistently reported results in the same direction with clinically insignificant differences between the studies). However, there were too few studies to adequately determine heterogeneity. [START_REF]Analysing data and undertaking metaanalyses[END_REF]

Effects of interventions

Primary outcome: H pylori eradication rates

Data regarding the effects of S boulardii supplementation on H pylori eradication rates were available from four trials, 17 18 20 21 which reported data from 915 participants (825 adults and 90 children) (figure 2). In two RCTs, the eradication rate was a primary outcome; 20 21 in the remaining two RCTs, 17 18 it was a secondary outcome.

We found a significant difference between the S boulardii-supplemented group and the control group with respect to H pylori eradication rates (four RCTs, n=915, RR 1.13, 95% CI 1.05 to 1.21). Of the 460 patients in the S boulardii group, 370 (80%, 95% CI 77% to 84%) experienced eradication compared with 324 of the 455 patients (71%, 95% CI 67% to 75%) in the control group. Thus, the administration of S boulardii along with the standard therapy resulted in a 9% higher absolute eradication rate (ARR 9%, 95% CI 3.6% to 14%). The number needed to treat (NNT) was 11 (95% CI 7 to 28). The pooled results of the three RCTs conducted in adults 17 18 21 showed a statistically significant increase in the eradication rate in favour of S boulardii compared with placebo or no treatment (3 RCTs, n=825, RR 1.12, 95% CI 1.04 to 1.22).

Secondary end points: adverse effects and compliance

Data regarding therapy-related adverse effects were available from all five of the included trials (figure 3). We found a significant difference between the S boulardiisupplemented group and the control group with respect to the risk of overall adverse effects (five RCTs, n=1305, RR 0.46, 95% CI 0. In regard to specific adverse effects, the risk of therapy-related diarrhoea was statistically lower in the S boulardii group compared with the control group (four RCTs, n=1215, 5.6% vs 12.2%, respectively, RR 0.47, 95% CI 0.32 to 0.69, NNT 16, 95% CI 11 to 30). However, we found no significant difference between the study groups with respect to epigastric pain, taste disturbance/dry mouth, nausea, or abdominal gas/bloating (see figure 2). Additionally, there was no significant difference between the groups in the frequency of vomiting, constipation, or other nonspecific reactions such as urticaria/skin reactions, palpitations, aphthous lesions in the mouth, belching, loss of appetite, blurred vision, or the presence of Clostridium difficile toxin.

The forest plots for these outcomes are not presented, as these outcomes have been reported in only one or two trials. The need for discontinuation of the eradication treatment was not reported in any trial.

DISCUSSION

Summary of evidence

This meta-analysis of RCTs showed that in patients with H pylori infection, addition of S boulardii to triple therapy compared with placebo or no intervention improved eradication rates, reduced overall therapy-related adverse effects, and decreased some individual symptoms such as diarrhoea. As the majority of included patients were adults, our results may be applicable primarily to such a population.

Quality of the evidence

In our analysis, the studies seemed methodologically sound with regard to sequence generation, >80% follow-up, and intention-to-treat analysis. Potential limitations included unclear or inadequate allocation concealment and no blinding in some trials. This can overestimate the effect and skew the results in favour of either treatment, depending on the biases of the investigators. Reassuringly, the direction of the effect for the primary outcome (eradication rate), as well as that for adverse effects, was similar and the benefit was reproducible, regardless of the methodological concerns. Study limitations also included a small sample size in some trials. In only two RCTs, 17 18 sample size calculations were available. However, to increase power is one of the reasons why a meta-analysis is performed within a systematic review. [START_REF] Higgins | Cochrane Handbook for Systematic Reviews of Interventions Version 5[END_REF]

How the intervention might work

The exact mechanisms by which S boulardii might exert its actions in increasing the eradication rates are unclear. One possible explanation is that this beneficial effect is due to a reduction in therapy-related side effects and, consequently, better compliance with treatment. Additional mechanisms, discussed in detail elsewhere, [START_REF] Mcfarland | Systematic review and meta-analysis of Saccharomyces boulardii in adult patients[END_REF] include interference with pathogenic toxins, preservation of cellular physiology, interference with pathogen attachment, interaction with normal microbiota, or contribution to the reestablishment of short chain fatty acid levels. In addition, stimulation or modulation of immune responses, both within the lumen and systemically, although not clearly linked to H pylori infection, may also contribute.

Agreement and disagreement with other studies or reviews

With regard to the eradication rate, overall gastrointestinal side effects, and risk of diarrhoea, the results of our review are in line with conclusions of the previous review by Tong et al. [START_REF] Tong | Meta-analysis: the effect of supplementation with probiotics on eradication rates and side effects during Helicobacter pylori eradication therapy[END_REF] Our findings with regard to therapy-related diarrhoea are in line with and add to a previously published meta-analysis on the effects of S boulardii in preventing antibiotic-associated diarrhoea in children and adults. [START_REF] Szajewska | Meta-analysis: non-pathogenic yeast Saccharomyces boulardii in the prevention of antibiotic-associated diarrhoea[END_REF] This meta-analysis documented that treatment with S boulardii compared with placebo reduced the risk of antibiotic-associated diarrhoea from 17.2% to 6.7% (RR 0.43, 95% CI 0.23 to 0.78).

Of note, the effect size with respect to diarrhoea was similar in the current and previous meta-analyses (reduction of 53% vs 57%, respectively). Collectively, these data support the use of S boulardii for the prevention of diarrhoea associated with antibiotic treatment, regardless of the reason for which the antibiotics were used.

A number of studies suggest that the dose of probiotic is important. 26 27 28 29 The daily doses of S boulardii ranged from 500 mg to 1000 mg. The largest effect on the eradication rate was observed in the largest, but open-label, RCT by Song et al., [START_REF] Song | The effect of probiotics and mucoprotective agents on PPI-based triple therapy for eradication of Helicobacter pylori[END_REF] which used the daily dose of S boulardii 750 mg (corresponding to ≈22.5 x 10 9 CFU).

Whether or not this dose contributed to the beneficial effect of S boulardii on the eradication rate is not clear, but it could not be excluded.

Can we be satisfied with the eradication rate?

In 2007, Graham et al. [START_REF] Graham | A report card to grade Helicobacter pylori therapy[END_REF] proposed that one judge the effectiveness of H pylori eradication therapy against an established target, such as a "report card." According to the proposed classification system, only therapies that score excellent, ie, those that achieve ≥95% eradication success in the local populations, should be prescribed.

In our review, the H pylori eradication rate in the triple therapy group was 71% and increased to 80% with S boulardii supplementation. Thus, even when supplemented with S boulardii, this treatment did not achieve the desired level of success.

Nevertheless, when making clinical decisions, it seems reasonable to consider the mode of therapy with higher efficacy. Recently, it has been documented that 12 sequential therapy compared with standard triple therapy may be more effective for H pylori eradication. [START_REF] Gatta | Sequential therapy or triple therapy for Helicobacter pylori infection: systematic review and meta-analysis of randomized controlled trials in adults and children[END_REF] Considering the beneficial effect of S boulardii documented in our analysis, one could speculate that the addition of S boulardii to the sequential therapy may result in even higher eradication rates. Further trials are needed to confirm this assumption.

Safety

Whereas no adverse effects other than those attributed to H pylori eradication therapy were observed in any of the included trials, the administration of S boulardii is not without risk. A recent systematic review [START_REF] Whelan | Safety of probiotics in patients receiving nutritional support: a systematic review of case reports, randomized controlled trials, and nonrandomized trials[END_REF] documented that some probiotic products, particularly S boulardii and Lactobacillus GG, have been shown to increase the risk of complications in specific patient groups. Of note, most complications have occurred in immunocompromised subjects or in patients with other life-threatening illnesses managed in intensive care units. It was also stated that all case reports that detailed infections caused by certain probiotics (ie, S boulardii or Lactobacillus GG) are likely to reflect their wider use in the clinical setting rather than their increased virulence.

Overall, probiotics are safe for use in otherwise healthy populations, but caution should be taken in patients with risk factors for adverse events (eg, patients with central venous catheters or increased bacterial translocation).

CONCLUSIONS

There is evidence to recommend the use of S boulardii as a safe option for increasing H pylori eradication rates, although only moderately, and decreasing overall therapyrelated side effects, particularly diarrhoea, in settings where standard triple therapy is recommended and in non-risk populations. While caution is advised in view of the methodological concerns regarding some of the included studies, it is reassuring that there was consistency of the effect across studies with regard to these outcomes. As the majority of included patients were adults, studies in children are needed.

F o r P e e r R e v i e w

DISCUSSION

Summary of evidence

24 Summarize the main findings including the strength of evidence for each main outcome; consider their relevance to key groups (e.g., healthcare providers, users, and policy makers).

  choice of the probiotic S boulardii was determined by the fact that it is widely available and commonly used in many countries.

5 boulardii

 5 principal pharmaceutical company Biocodex (Gentilli, France) that manufactures S was contacted to help identify published and unpublished data. The ClinicalTrials.gov website was also searched for RCTs that were registered but not yet published. Certain publication types (ie, letters to the editor, abstracts, proceedings from scientific meetings) were excluded, unless a full set of data was obtained from the authors.

Version 5 .

 5 0. Copenhagen: The Nordic Cochrane Centre, The Cochrane Collaboration, 2008]. Absolute risk reduction (ARR) and number needed to treat (NNT), all with a 95% CI, were calculated using StatsDirect statistical software (version 2,7,8 [2010-03-15]).

  discussed in the Introduction. The major question with regard to the meta-analysis by Tong et al. is whether it was appropriate to pool data on different probiotic microorganisms. The risk is that pooling data from different genera, species, strains, and doses of probiotics obtained in different settings and/or populations, presumably with variations in their native intestinal microbiota, may result in misleading conclusions. The results could be erroneously extrapolated to other probiotics, including those that have not been adequately studied. Given these considerations, our work focused on one type of a clearly defined, single-organism, specifically S boulardii. Thus, our results precisely define the effects of S boulardii supplementation on the rates of H pylori eradication, adverse effects, and patient compliance.

13 CONTRIBUTORS

 13 HS initially conceptualised this study. AH and HS contributed to the initial protocol of the study. All authors were responsible for data collection, data analysis, data interpretation, and preparation of the report. HS assumed the main responsibility for the writing of this manuscript. All authors contributed to (and agreed upon) the final version.ACKNOWLEDGMENTS, COMPETING INTERESTS, FUNDINGAuthors' declaration of personal interest: H. Szajewska served as a speaker for Biocodex, the manufacturer of S boulardii. Two remaining authors declare no conflict of interest.Declaration of funding interests: This study was funded in full by The Medical University of Warsaw.

Figure 1 .Figure 2 .Figure 3 .

 123 FIGURE LEGENDFigure 1. Identification process for eligible trials Figure 2. Primary outcome: Effect of S boulardii (SB) on H pylori eradication rates Figure 3. Secondary outcomes: Effect of S boulardii (SB) on H pylori eradication therapy-related adverse effects

20 Figure 3 . 1 . 4 . 4

 203144 Figure 3. Secondary outcomes: Effect of S boulardii (SB) on H pylori eradication therapy-related adverse effects

9 Limitations 25 - 10 Conclusions 26 11 FUNDINGFunding 27 13 From:

 9251026112713 Discuss limitations at study and outcome level (e.g., risk of bias), and at review-level (e.g., incomplete retrieval of identified research, reporting bias). 9Provide a general interpretation of the results in the context of other evidence, and implications for future research. 10-Describe sources of funding for the systematic review and other support (e.g., supply of data); role of funders for the systematic review. Moher D, Liberati A, Tetzlaff J, Altman DG, The PRISMA Group (2009). Preferred Reporting Items for Systematic Reviews and Meta-Analyses: The PRISMA Statement. PLoS Med 6(6): e1000097. doi:10.1371/journal.pmed1000097 For more information, visit: www.prisma-statement.org. Page 2 of 2

  all included trials had a number of methodological limitations (see table2).

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  3 to 0.7). Of the 665 patients in the S boulardii group, 86 (12.9%, 95% CI 10.4% to 15.7%) experienced any adverse effect compared with 156 of the 640 patients (24.3%, 95% CI 21% to 27.8%) in the control group. Thus, the coadministration of S boulardii with the standard eradication therapy resulted in an 11.4% lower absolute adverse effects rate (ARR 11.4%, 95% CI 7.3% to 15.6%). The number needed to treat was 9 (95% CI 7 to 14).

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Table 1 . Characteristics of included studies S boulardii group (daily dose)

 1

	Study ID	Patients	Cont/Exp	Eradication regimen			Control	Primary/secondary	H. pylori	Follow-	Score system
	(Country)		(Follow-	(daily dose)			group	outcomes	infection.	up	for assessing
			up)		mg	CFU			Initial		side effects
									diagnosis/re-
									checking
	Cindoruk	H pylori-	62/62	Lansoprazole (30 mg	1000 mg	≈20 x 10 9	Placebo	Side-	Histology/UB	6 wk	Questionnaire
	et al.	positive	(FU	twice daily)	(in 2	CFU *		effects/eradication	T		by De Boer
	(Turkey)	symptomatic	62/62)	Amoxicillin (1 g twice	doses, for			success
	(17)	adults		daily)	2 wk)
				Clarithromycin (500
				mg twice daily)
				14 days
	Cremonini	H pylori-	21/22	Rabeprazole (20 mg	≈500 mg	10 x 10 9	Placebo	Side-	UBT/UBT	5 to 7	Questionnaire
	et al. (Italy)	positive	(FU	twice daily)	(in 2	CFU		effects/eradication		wk	by De Boer
	(18)	asymptomatic	20/21)	Clarithromycin (500	doses, for			rate			Deleted: 0
		adults		mg twice daily)	2 wk)*
				Tinidazole (500 mg
				twice daily)
				1 wk
	Duman et	H pylori-	185/204	Omeprazole (20 mg	1000 mg	≈20 x 10 9	No	Incidence of	UBT,	2 to 4	Interview
	al.	positive	(FU	twice daily)	(in 2	CFU*	treatme	diarrhoea during	histology/not	wk
	(Turkey)	symptomatic	172/196)	Clarithromycin (500	doses, for		nt	and following the	applicable	(14 -45
	(19)	adults		mg twice daily)	2 wk)			antibiotic		days)
				Amoxicillin (1 g twice				treatment/duration
				daily)				of diarrhoea and
				2 wk				frequency of bowel
								movements during
								a diarrhoeal episode
	Hurduc et	H pylori-	42/48	Omeprazole or	500 mg	≈10 x 10 9	No	Eradication	Rapid urease	4 to 6	Recorded in
	al.	positive	(FU	Esomeprazole (1	(in 2	CFU	treatme	rate/Adverse events	test,	wk	the
	(Romania)	symptomatic	42/48)	mg/kg/day twice	doses, for		nt		histology/rapi		questionnaire
	(20)	children		daily) for 3 weeks	4 wk)				d urease test,
				Amoxicillin (50					histology
				mg/kg/day twice
				daily) for 7-10 days
				Clarithromycin (15
				mg/kg/day twice
				daily) for 7-10 days

Table 2 . Methodological quality summary: review authors' judgements about each methodological quality item for each included study

 2

	Study ID	Adequate	Allocation	Blinding?	Incomplete data
		sequence	concealment?		addressed?
		generation?
	Cindoruk et al. 2007	Yes	Unclear	Yes	Yes
	(17)
	Cremonini et al. 2002	Yes	Yes	Yes	Yes
	(18)
	Duman et al. 2005 (19)	Unclear	Unclear	No	Yes
	Hurduc et al. 2008 (20)	Yes	Unclear	No	Yes
	Song et al. 2010 (21)	Yes	Unclear	No	Yes
	In all cases, an answer of 'yes' indicates a low risk of bias, and an answer of 'no' indicates a high risk of
	bias

Primary outcome: Effect of S boulardii (SB) on H pylori eradication rates

		Experimental	Control		Risk Ratio	Risk Ratio
	Study or Subgroup	Events	Total	Events	Total	Weight	M-H, Fixed, 95% CI	M-H, Fixed, 95% CI
	1.6.1 Eradication in children
	Hurduc (SB 500 mg)	45	48	34	42	11.1%	1.16 [0.98, 1.36]
	Subtotal (95% CI)		48		42	11.1%	1.16 [0.98, 1.36]
	Total events	45		34
	Heterogeneity: Not applicable
	Test for overall effect: Z = 1.76 (P = 0.08)
	1.6.2 Eradication in adults
	Cremonini (SB 500 mg)	17	20	16	20	4.9%	1.06 [0.80, 1.41]
	Song (SB 750 mg)	264	330	237	331	72.6%	1.12 [1.02, 1.22]
	Cindoruk (SB 1000 mg)	44	62	37	62	11.4%	1.19 [0.92, 1.54]
	Subtotal (95% CI)		412		413	88.9%	1.12 [1.04, 1.22]
	Total events	325		290
	Heterogeneity: Chi² = 0.35, df = 2 (P = 0.84); I² = 0%
	Test for overall effect: Z = 2.86 (P = 0.004)
	Total (95% CI)		460		455	100.0%	1.13 [1.05, 1.21]
	Total events	370		324
	Heterogeneity: Chi² = 0.47, df = 3 (P = 0.92); I² = 0% Test for overall effect: Z = 3.21 (P = 0.001) Test for subgroup differences: Not applicable			0.5 Favours experimental Favours control 0.7 1 1.5	2

.94 [0.53, 1.65]

  Specify any assessment of risk of bias that may affect the cumulative evidence (e.g., publication bias, selective reporting within studies).Risk of bias within studies19 Present data on risk of bias of each study and, if available, any outcome level assessment (seeitem 12). For all outcomes considered (benefits or harms), present, for each study: (a) simple summary data for each intervention group (b) effect estimates and confidence intervals, ideally with a forest plot. Present results of any assessment of risk of bias across studies (seeItem 15). 16Additional analysis 23 Give results of additional analyses, if done (e.g., sensitivity or subgroup analyses, meta-regression [seeItem 16]).

			16
	Section/topic individual studies Results of	on page # Reported 20 8-9 # Checklist item
	TITLE Title Synthesis of INTRODUCTION Rationale Objectives METHODS Protocol and registration Eligibility criteria Information sources Search Study selection Data collection process Data items Risk of bias in individual studies Summary measures results studies Risk of bias across	18 1 Identify the report as a systematic review, meta-analysis, or both. Total 48 21 330 62 204 665 21 330 62 204 617 21 330 62 413 21 330 62 204 617 21 330 62 413 21 62 83 Events 13 12 63 37 31 156 6 20 19 28 73 3 11 27 41 8 19 9 0 36 3 7 13 23 4 15 19 Total 42 20 331 62 185 640 20 331 62 185 598 20 331 62 413 20 331 62 185 598 20 331 62 413 20 62 82 Weight 8.6% 7.7% 36.6% 25.0% 22.1% 100.0% 4.7% 29.0% 29.6% 36.7% 100.0% 13.2% 36.6% 50.2% 100.0% 8.0% 52.4% 36.4% 3.2% 100.0% 10.2% 34.0% 55.8% 100.0% 25.3% 74.7% 100.0% M-H, Random, 95% CI 0.27 [0.10, 0.76] 0.24 [0.08, 0.72] 0.76 [0.54, 1.08] 0.38 [0.23, 0.63] 0.50 [0.28, 0.87] 0.46 [0.30, 0.70] 0.16 [0.02, 1.20] 0.55 [0.27, 1.13] 0.47 [0.23, 0.96] 0.45 [0.25, 0.83] 0.47 [0.32, 0.69] 0.95 [0.22, 4.18] 1.00 [0.44, 2.28] 0.33 [0.17, 0.65] 0.62 [0.27, 1.41] 0.12 [0.02, 0.87] 0.74 [0.38, 1.45] 1.00 [0.43, 2.35] 2.72 [0.11, 66.41] 0.72 [0.35, 1.48] 0.32 [0.04, 2.80] 0.72 [0.23, 2.23] 0.54 [0.23, 1.26] 0.56 [0.29, 1.08] 0.95 [0.27, 3.30] 0.93 [0.49, 1.77] Control Risk Ratio Risk Ratio F o r P e e r R e v i e w F o r P e e r R e v i e w F o r P e e r R e v i e w F o r P e e r R e v i e w F o r P e e r R e v i e w 21 Present results of each meta-analysis done, including confidence intervals and 3 Describe the rationale for the review in the context of what is already known. 4 Provide an explicit statement of questions being addressed with reference to participants, interventions, comparisons, outcomes, and study design (PICOS). 5 Indicate if a review protocol exists, if and where it can be accessed (e.g., Web address), and, if available, provide registration information including registration number. 6 Specify study characteristics (e.g., PICOS, length of follow-up) and report characteristics (e.g., years considered, language, publication status) used as criteria for eligibility, giving rationale. 7 Describe all information sources (e.g., databases with dates of coverage, contact with study authors to identify additional studies) in the search and date last searched. 8 Present full electronic search strategy for at least one database, including any limits used, such that it could be repeated. 9 State the process for selecting studies (i.e., screening, eligibility, included in systematic review, and, if applicable, included in the meta-analysis). 10 Describe method of data extraction from reports (e.g., piloted forms, independently, in duplicate) and any processes for obtaining and confirming data from investigators. 11 List and define all variables for which data were sought (e.g., PICOS, funding sources) and any assumptions and simplifications made. 12 Describe methods used for assessing risk of bias of individual studies (including specification of whether this was done at the study or outcome level), and how this information is to be used in any data synthesis. 13 State the principal summary measures (e.g., risk ratio, difference in means). F o r P e e r R e v i e w measures of consistency. 0Experimental M-H, Random, 95% CI 22	19-20 1 3 3-4 No 4 4-5 4 5 5 5 6 6
	Synthesis of results	0.01 14 Describe the methods of handling data and combining results of studies, if done, 0.1 1 10 100 including measures of consistency (e.g., I ) for each meta-analysis. 2	6-7
		Favours experimental Favours control
		Page 1 of 2
	Section/topic	# Checklist item	Reported on page #
	Risk of bias across studies	15 14-15

7

Additional analyses

16

Describe methods of additional analyses (e.g., sensitivity or subgroup analyses, meta-regression), if done, indicating which were pre-specified.

7

RESULTS

Study selection 17

Give numbers of studies screened, assessed for eligibility, and included in the review, with reasons for exclusions at each stage, ideally with a flow diagram.

7

Study characteristics

18 For each study, present characteristics for which data were extracted (e.g., study size, PICOS, follow-up period) and provide the citations.

ABSTRACT

Structured summary 2 Provide a structured summary including, as applicable: background; objectives; data sources; study eligibility criteria, participants, and interventions; study appraisal and synthesis methods; results; limitations; conclusions and implications of key findings; systematic review registration number. 2


Alessandro De Gregorio
email: alessan-dro.degregorio@uniroma1.it

Parametric estimation for planar random flights

Keywords: asymptotically efficient, discretely observed process, likelihood function, distance process

The aim of this paper is to consider a parametric estimation problem for two-dimensional random motions at finite speed with an infinite number of directions (planar random flight). In particular we are interested in estimating the unknown value of the parameter λ, the underlying rate of the Poisson process, when the process is observed at n + 1 equidistant discrete times. We introduce three different estimators, based on the distance between two consecutive observed points. Furthermore, their empirical performance is analyzed by means of a Monte Carlo analysis for small sample size n.

Introduction

We consider the motion in the plane of a particle starting at arbitrary point (x 0 , y 0 ), moving with constant velocity c and taking directions uniformly distributed in (0, 2π]. The changes of direction are governed by a homogenous Poisson process with parameter λ > 0. Let N (t) be the number of Poisson events in the interval [0, t], the position at time t > 0 of a particle performing a random flight is

X(t) = x 0 + c N (t)+1
j=1 (s js j-1 ) cos θ j , Y (t) = y 0 + c N (t)+1 j=1 (s js j-1 ) sin θ j , where θ j are independent random variables uniformly distributed in (0, 2π], while s j , j = 1, ..., n are the instants at which Poisson events occur (s 0 = 0 and s N (t)+1 = t). Random flights in R 2 have been studied in [START_REF] Stadje | The exact probability distribution of a two-dimensional random walk[END_REF], [START_REF] Masoliver | Some two and three-dimensional persistent random walk[END_REF], [START_REF] Kolesnik | A planar random motion with an infinite number of directions controlled by the damped wave equation[END_REF]. De Gregorio and [START_REF] De Gregorio | Some results on random flights[END_REF] and [START_REF] Orsingher | Random flights in higher spaces[END_REF] analyze the random flights in higher spaces and derive their explicit distribution in R 4 .

The only references about the statistical inference of random motion at finite velocity consider estimation problem for the telegraph process which describes the motion of a particle on the real line. [START_REF] Yao | Estimation of Noisy Telegraph Process: Nonlinear Filtering Versus nonlinear Smoothing[END_REF] estimates the state of the telegraph process under white noise perturbation and studies performance of nonlinear filters. [START_REF] Iacus | Statistical analysis of the inhomogeneous telegrapher's process[END_REF] considers the estimation of the parameter θ of a non-constant rate λ θ (t). More recently, De Gregorio and Iacus (2008) introduce a pseudo-maximum likelihood estimator and a moment estimator for the parameter λ when the sample paths of the telegraph process are observed only at equidistant discrete times. The authors also analyze the same statistical problem for a geometric telegraph process particularly interesting in view of financial applications. For a telegraph process observed at discrete times, [START_REF] Iacus | Estimation for discretely observed telegraph process[END_REF] study the asymptotic (i.e. the mesh decreases to zero and the horizon interval tends to infinity) properties of two moment estimators and propose also an estimator consistent, asymptotically normal and asymptotically efficient.

The aim of this paper is the estimation of the parameter λ which governs the rate of change of direction of a random flight {(X(t), Y (t)), t > 0}. If the whole path of the process in observed, the parameter λ is estimated with N (T )/T (see [START_REF] Yu | Statistical inference for spatial Poisson processes[END_REF] and the estimation does not present problem. We assume that the process {(X(t), Y (t)), 0 < t < T } is observed only at n + 1 equidistant times 0 = t 0 < t 1 < .. < t n = T , where t i = i∆ n = i∆, i = 0, 1, ..., n. Since {(X(t), Y (t)), 0 < t ≤ T } is not markovian, it is not possible to explicit the likelihood function of the points observed as product of the transition densities. This fact implies that we can not use the tools developed for the diffusion processes (see [START_REF] Sørensen | Estimating functions for discretely observed diffusions: A review[END_REF][START_REF] Sørensen | Parametric inference for diffusion processes observed at discrete points time: a survey[END_REF]and Iacus, 2008, for an account of these estimation methods). The main idea of this paper is to consider the distance between two consecutive observed points which provides a sequence of independent and identically distributed (i.i.d.) random variables. In this way, we are able to propose a maximum likelihood estimator, based on the likelihood function of the distances, and moment type estimator as well. Furthermore, it is also possible to consider a statistic which counts the number of times in which the process doesn't change direction in the intervals [t i , t i+1 ), i = 0, 1, ..., n.

We believe that the statistical problems treated in this paper are interesting because the planar random flights seem to be useful, for example, in ecology and biology. In fact, [START_REF] Holmes | Are diffusion models too simple? A comparison with telegraph models of invasion[END_REF] and [START_REF] Holmes | Partial differential equations in ecology: spatial interactions and population dynamics[END_REF] consider these models to represent the displacements of the animals and microorganisms on a surface.

The paper is organized as follows. In Section 2 we describe the random motion considered here and calculate the exact value of the moments of the process R(t) = X 2 (t) + Y 2 (t). In Section 3 we introduce and study the estimators for the parameter λ. Their empirical performance is analyzed by means of a Monte Carlo analysis in Section 4.

Planar random flights: description and distance process

Let us consider a particle starting at the arbitrary point (x 0 , y 0 ) of the plane R 2 , moving with constant finite speed c. The initial direction is a random vector v = (cos θ, sin θ) uniformly distributed on the unit circumference, i.e. the random variable θ is uniformly distributed in (0, 2π]. The changes of direction are governed by a homogeneous Poisson process with parameter λ > 0. Therefore, when a Poisson event occurs the particle takes a new direction uniformly distributed on the unit circle, independently from the previous one.

We indicate the position of the particle at time t > 0 with the stochastic process (X(t), Y (t)), which is called random flight in the plane. At time t the particle is located in the disc

S 2 ct = {(x, y) : (x -x 0 ) 2 + (y -y 0 ) 2 ≤ c 2 t 2 }, (2.1)
with probability 1. If no Poisson event occurs the particle is located on the circle

∂S 2 ct = {(x, y) : (x -x 0 ) 2 + (y -y 0 ) 2 = c 2 t 2 }, with probability P{(X(t), Y (t)) ∈ ∂S 2 ct } = e -λt .
The remaining part of the distribution lies in the interior of (2.1) and represents the absolute continuous component of the distribution P {X(t) ∈ dx, Y (t) ∈ dy} .

We note that the random flights have trajectories which assume the form of broken lines where the single steps have random length and are uniformly oriented in (0, 2π]. However, the total length for any sample paths at time t is ct. The density law of (X(t), Y (t)) (see [START_REF] Kolesnik | A planar random motion with an infinite number of directions controlled by the damped wave equation[END_REF]) is equal to

p(x, y; t) = λ 2πc e -λt+ λ c √ c 2 t 2 -(x-x0) 2 -(y-y0) 2 c 2 t 2 -(x -x 0 ) 2 -(y -y 0 ) 2 1 {(x-x0) 2 +(y-y0) 2 <c 2 t 2 } + e -λt 2πct δ(c 2 t 2 -(x -x 0 ) 2 -(y -y 0 ) 2 ), (2.2)
with (x, y) ∈ S 2 ct and δ(•), 1(•) representing respectively the Dirac's delta function and the indicator function. Now, we put (x 0 , y 0 ) = (0, 0) and introduce the following distance process

R(t) = X 2 (t) + Y 2 (t), (2.3)
i.e. the euclidean distance from the origin of the position reached by the moving particle at time t. It's clear that for 0 r < ct

P {R(t) r} = P (X(t), Y (t)) ∈ S 2 r = λ 2πc S 2 r exp -λt + λ c c 2 t 2 -x 2 -y 2 c 2 t 2 -x 2 -y 2 dxdy = 1 -exp -λt + λ c c 2 t 2 -r 2
where S 2 r is a disk with radius r. Then, the complete distribution of R(t) becomes

p R (r, t) = λ c r exp{-λt + λ c √ c 2 t 2 -r 2 } √ c 2 t 2 -r 2 1 {0<r<ct} (2.4) +e -λt δ(c 2 t 2 -r 2 ).
We note that the density (2.4) coincides with formula [START_REF] Iacus | Estimation for discretely observed telegraph process[END_REF] in [START_REF] Kolesnik | A planar random motion with an infinite number of directions controlled by the damped wave equation[END_REF], when we ignore the angular component. By taking into account the probability law (2.4), we are able to derive the moments of R(t).

Theorem 2.1. Let (x 0 , y 0 ) = (0, 0) and p ≥ 1, we have that

ER p (t) = (ct) p e -λt √ π 2 λt p-1 2 Γ p 2 + 1 I p+1 2 (λt) + L p+1 2 (λt) + 1 (2.5)
with modified Bessel function

I ν (x) = ∞ k=0 x 2 2k+ν 1 Γ(k + 1)Γ(k + ν + 1)
and modified Struve function

L ν (x) = ∞ k=0 x 2 2k+ν+1 1 Γ(k + 3 2 )Γ(k + ν + 3 2 )
Proof. In view of (2.4), we can write

ER p (t) = λ c e -λt ct 0 r p+1 e λ c √ c 2 t 2 -r 2 √ c 2 t 2 -r 2 dr + (ct) p e -λt . (2.6)
Now, we work out the integral in (2.6). Hence

ct 0 r p+1 e λ c √ c 2 t 2 -r 2 √ c 2 t 2 -r 2 dr = ∞ k=0 1 k! λ c k ct 0 r p+1 (c 2 t 2 -r 2 ) k-1 2 dr = ∞ k=0 1 k! λ c k (ct) p+k+1 2 Γ p 2 + 1 Γ k+1 2 Γ k+1 2 + p 2 + 1 = √ πΓ p 2 + 1 ∞ k=0 1 k! λt 2 k (ct) p+1 Γ(k) Γ k+1 2 + p 2 + 1 Γ k 2 = √ πΓ p 2 + 1 (ct) p+1 ∞ k=0 λt 2 2k 1 2kΓ(k)Γ k + p+1 2 + 1 + ∞ k=0 λt 2 2k+1 1 (2k + 1)Γ k + p+1 2 + 3 2 Γ k + 1 2 = √ π 2 Γ p 2 + 1 (ct) p+1 2 λt p+1 2 ∞ k=0 1 k! λt 2 2k+ p+1 2 1 Γ k + p+1 2 + 1 + 2 λt p+1 2 ∞ k=0 λt 2
2k+ p+1 2 +1 1 Γ k + p+1 2 + 3 2 Γ k + 3 2 = √ π 2 Γ p 2 + 1 (ct) p+1 2 λt p+1 2 I p+1 2 (λt) + L p+1 2 (λt) . (2.7)
By inserting (2.7) into (2.6) we obtain the result (2.5).

Remark 2.1. We observe that

lim λ→∞ ER p (t) = 0, lim λ→0 ER p (t) = (ct) p .
In other words, if λ grows to infinity the changes of direction increase and consequently the distance from the origin decreases. Viceversa for λ → 0 the particle tends to achieve the edge of the disc S 2 ct .

Estimation for discretely observed planar random flights

We assume that the planar random flight {(X(t), Y (t)), 0 < t ≤ T }, with (X(0), Y (0)) = (0, 0), is observed only at n + 1 equidistant discrete times 0 = t 0 < t 1 < ... < t n = T, where t i = i∆ n = i∆, i = 0, 1, ..., n. We use the following notation to simplify the formulas:

(X(t i ), Y (t i )) = (X(i∆ n ), Y (i∆ n )) = (X i , Y i ).
The interest is the estimation of the parameter λ whilst the velocity c is assumed to be known. However, if in the interval (i∆ n-1 , i∆ n ] there are not changes of direction, then

(X i -X i-1 ) 2 +(Y i -Y i-1 ) 2 = c 2 ∆ 2 n .
If ∆ n is suitably small, there is high probability of observing N (t i ) -N (t i-1 ) = 0 and c can be calculated without error. Furthermore, by setting

R i = (X i -X i-1 ) 2 + (Y i -Y i-1 ) 2
we are able to estimate consistently the velocity c by means of the following estimator

ĉn = M n ∆ n , (3.1)
where M n = max{R i ; i = 1, ..., n}. By noting that R i ≤ c∆ n with the equality holding if and only if N (t i ) -N (t i-1 ) = 0, we have that P {ĉ n = c} = (1e -λ∆n ) n → 0 as n → ∞. Therefore, if c is unknown the estimators below should be considered substituting c with ĉn . Analogously to the telegraph process, the random flights are not markovian. For this reason we cannot write the explicit likelihood function of the process in the form of product of transition densities as well as for diffusion processes. Nevertheless,

(X i -X i-1 , Y i -Y i-1 , θ i ), i = 1, ..., n
, where θ i is the angular direction (unobservable) of the particle at time t i , is a special hidden Markov model. However, to face the estimation problem we prefer to follow alternative arguments explained below.

The contrast function approach

In order to estimate λ, we introduce a suitable contrast function as follows. We observe that

R 2 i = c 2    N (ti)+1 j=N (ti-1)+2 (s j -s j-1 ) 2 + j =k (s j -s j-1 )(s k -s k-1 ) cos(θ j -θ k )    therefore, while (X i , Y i ) are not independent, R i , i = 1, ...,
L n (λ) = L n (λ|R 1 , ..., R n ) (3.2) = n i=1 p R (R i , ∆ n ) = n i=1 λ c R i exp{-λ∆ n + λ c √ u n,i } √ u n,i
1 {un,i>0} + e -λ∆n δ(u n,i = 0) ,

where u n,i = c 2 ∆ 2 n -R 2 i . The densities p R (R i , ∆ n ) appearing in (3.
2) represent the distribution of the random distance between the position of a planar random flight at time t i-1 , namely (X i-1 , Y i-1 ), and one reaches at the instant t i , namely (X i , Y i ). The likelihood function (3.2) is equivalent to

L n (λ) = e -λ∆n(n-n + ) n + i=1 λ c R i exp{-λ∆ n + λ c √ u n,i } √ u n,i (3.3)
where n + is the number of the planar random flights with at least one change of direction.

Remark 3.1. In the expression (3.3), the factor e -λ∆n(n-n + ) concerns the singular part of the densities p R (R i , ∆ n ), while the product contains the absolutely continuous components of the distributions of R i 's. Note that for increasing values of λ, the absolutely continuous component of (3.3) has a bigger weight than the singular component; viceversa for small values of λ.

At this point, it is quite natural to consider as estimator for λ, the maximum likelihood estimator (MLE) based on L n (λ) (contrast function). Formula (3.3) yields

F n (λ) = d dλ log L n (λ) = -n∆ n + 1 c n + i=1 √ u n,i + n + λ , (3.4)
and the estimator is obtained by solving F n (λ) = 0. By taking into account the function (3.4) we derive the following MLE for the parameter λ

λn = arg max λ>0 L n (λ) = cn + cn∆ n - n + i=1 √ u n,i . (3.5)
It is easy to see that d 2 dλ 2 log L n (λ) < 0 and the uniqueness of the estimator (3.5) holds. The following Theorem on the asymptotic properties of λn holds.

Theorem 3.1. The estimator (3.5) is consistent, asymptotically normal and asymptotically effcient

n∆ n ( λn -λ) d → N (0, λ) (3.6) for ∆ n → 0, n∆ → ∞ as n → ∞. Proof. Since n + = n i=1 1 {Ri<c∆n} , d 2 dλ 2 logL n (λ) = -n + λ 2 , we get that 1 n∆ n E d 2 dλ 2 logL n (λ) = - n n∆ n λ 2 (1 -e -λ∆n ) = - 1 λ 2 (λ + o(1)) → - 1 λ .
Then, the statements of Theorem follow immediately by the standard asymptotical theory for the MLE in i.i.d case.

A moment-type estimator

The random flights and the related process R(t) are not stationary. However, it's possible to use Theorem 2.1 to introduce a moment-type estimator based on the sample moments. Therefore, we consider the following estimator λn,p = arg min

λ>0 {m p -ER p (∆ n )} 2 , (3.7)
where m p = 1 n n i=1 R p i and ER p (∆ n ) is given by (2.5). The estimator λn,p is unique. In fact the the function ER p (∆ n ) is monotonically decreasing respect to λ and the Remark 2.1 permits us to claim that lim λ→0 ER p (∆ n ) = (c∆ n ) p and lim λ→∞ ER p (∆ n ) = 0. Being m p in [0, (c∆ n ) p ] because 0 R i c∆ n , then λn,p is the unique solution to (3.7).

Estimators based on a counting function

As in [START_REF] Iacus | Estimation for discretely observed telegraph process[END_REF], we consider the following counting statistic

G n = 1 n∆ n n i=1 1 {Ri<c∆n} = 1 n∆ n n i=1
1 {N ([ti-1,ti))≥1} , which provides the average number of the intervals in which the direction has changed at least once (up the constants 1/∆ n ). It's easy to see that

E(G n ) = 1 -e -λ∆n
∆ n , then we introduce the following statistic

λn = - 1 ∆ n log (1 -∆ n G n ) (3.8)
which represents an unbiased estimator for the rate λ.

  n, is a sequence of i.i.d. random variables. By taking into account the distribution (2.4), we consider the following likelihood function

Acknowledegments

I am grateful to the referees for their helpful comments and suggestions, which led to improvements to the first draft of the paper.

ACCEPTED MANUSCRIPT

Theorem 3.2. For n∆ n = T → ∞ and ∆ n → 0 as n → ∞ the estimator (3.8) is consistent, asymptotically normal and asymptotically efficient n∆ n ( λnλ) d → N (0, λ).

(3.9)

Proof. The result can be proved following the same steps of the proof of Theorem 3.1 in [START_REF] Iacus | Estimation for discretely observed telegraph process[END_REF].

Remark 3.2. Another advantage of the estimator λn is that it doesn't consider the probability law of (X(t), Y (t)) as (3.5). This fact permits us to estimate the parameter λ for a random flight performing its motion in R

Monte Carlo analysis

We analyze the empirical performance of the estimators (3.5), (3.7) and (3.8) by means of a Monte Carlo analysis with n < ∞ fixed (small samples). In order to evaluate the estimator (3.7) we fixed p = 1 and considered the mean

We simulate 10000 sample paths of the planar random flights in the interval [0, T ], with T = 500, for different values of λ and c = 1. For any trajectories we have sampled n = 200, 300, 500, 1000 values subsequently used to estimate the unknown parameter λ. The results for the estimators λn , λn,1 and λn have been reported in the Table 1. Furthermore in the tables there is a column Bias representing the bias of the estimator and another one SE, which represents the values of the standard error. For example, for λn , these values are calculated as follows

where N = 10000 is the number of simulations. It emerges, as expected, that the bias and the standard error tend to zero when the sample size increases. It is clear that the true value of the parameter λ and the bias are correlated as well as the standard error. In fact, for fixed n, as the more λ increases the more Poisson events remain hidden to the observer. Furthermore, it is possible to compare the performance of the three estimators by means of the values obtained for SE. For each λ and n, from the Table 1 emerges that the standard error for λn is smaller than one for λn,1 and λn . Then we can conclude that the maximum psuedo-likelihood estimator seems to be the best estimator with respect to the variability of the estimates.

A C C E P T E D M A N U S C R I P T


Y Ren

Nikolaos Halidias

N Halidias-Y

Nikolaos Ren

Halidias

Remarks and corrections on "An existence theorem for stochastic functional differential equations with delays under weak assumptions, Statistics and Probability Letters, 78, 2008." by

Keywords: AMS Subject Classification: 60H10, 60H20, 60H30 stochastic functional differential equations, comparison theorem

In this note we would like to give some remarks and open problems concerning comparison results for stochastic functional differential equations. Our starting point was the paper of Halidias and Ren 2008 where one can find some wrong arguments in the comparison result. However, as we shall prove in this note, the results in that paper are true under some more restrictive conditions on the data.

Introduction

Let t 0 be a positive constant and (Ω, F, P ) be a complete probability space with a filtration {F t } t≥t0 satisfying the usual conditions. Assume that B(t) is a Brownian motion defined on (Ω, F, P ). Let C([-τ, 0]; R) denote the family of bounded continuous R-valued functions ϕ defined on [-τ, 0] with norm ||ϕ|| = sup -τ ≤θ≤0 |ϕ(θ)|. We denote by M 2 ([-τ, 0]; R) the family of all F t0 -measurable, R-valued process ψ(t) = ψ(t, ω), t ∈ [-τ, 0] such that

E 0 -τ |ψ(t)| 2 dt < ∞.
In this note we will study stochastic functional differential equations like the following,

d[X(t) -D(X(t))] = f (t, X t ) dt + g(t, X(t)) dB(t), t 0 ≤ t ≤ T, (1)
where X t = {X(t + θ) : -τ ≤ θ ≤ 0} could be considered as a C([-τ, 0]; R)-value stochastic process. We define the initial value of (1) as follows:

X t0 = ξ = {ξ(θ) : -τ ≤ θ ≤ 0} is F t0 -measurable, C([-τ, 0]; R) -value random variable such that ξ ∈ M 2 ([-τ, 0]; R). (2)
Here, f : [t 0 , T ] × C([-τ, 0], R) → R, g : [t 0 , T ] × R → R and D : R → R and assume that are Borel measurable. We will study also the following stochastic functional differential equation of neutral type,

d[X(t) -D(X t )] = f (t, X t ) dt + dB(t), t 0 ≤ t ≤ T, (3)
with the initial value defined as before. Here,

D : C([-τ, 0]; R) → R, f (t, x) : [t 0 , T ] × C([-τ, 0]; R), are Borel measurable.
In the following two sections we prove comparison theorems for problems (1) and ( 3) with initial conditions like [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]. In section 4 we give some remarks and conclusions.

Comparison theorem for sfdes

Our goal here is to give a comparison result for problem (1), [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF]. So, consider the following problems,

X(t) -D(X(t)) = ξ 1 (0) -D(X(t 0 )) + t t0 f 1 (s, X s )ds + t t0 g(s, X(s))dB(s), t 0 ≤ t ≤ T, (4) X(t) -D(X((t)) = ξ 2 (0) -D(X(t 0 )) + t t0 f 2 (s, X s )ds + t t0 g(s, X(s))dB(s), t 0 ≤ t ≤ T. (5)
Theorem 1 Suppose that problems (4) and ( 5) have unique strong solutions,

X(t), Y (t) re- spectively. Suppose that f 1 (t, x) ≤ f 2 (t, x) for all x ∈ C([-τ, 0]; R), ξ 1 (0) ≤ ξ 2 (0) and that x → f 1 (t, x) or x → f 2 (t, x) is increasing, in the sense that if x ≤ y then f (t, x) ≤ f (t, y) for x, y ∈ C([-τ, 0]; R). Then, we have that X(t) ≤ Y (t) a.s. for all t > 0.

Proof.

For a discussion about the existence and uniqueness of strong solutions to problem like (3) one can see [START_REF] Mao | Stochastic Differential Equations and Applications[END_REF].

We define now the following operator, p(•,

•) : C([-τ, 0]; R)×C([-τ, 0]; R) → C([-τ, 0]; R), with p(x, y)(θ) = max θ∈[-τ,0] {x(θ), y(θ)}
It is clear that y → p(x, y) satisfies growth and Lipschitz conditions with constant one.

Suppose that f 2 is increasing. Consider the following problem,

Z(t) -D(p(X(t), Z(t))) = ξ(0) -D(p(X(t 0 ), Z(t 0 )) + t t0 f 2 (s, p(X s , Z s )) + t t0 g(s, p(X(s), Z(s)))dB(s), t 0 ≤ t ≤ T.
It's clear that this problem has a unique solution noting that y → p(x, y) satisfies growth and Lipschitz conditions. We shall prove that Z(t) ≥ X(t) a.s. Suppose that this is not true. Then, the following stopping time, τ 1

τ 1 = inf{t ∈ [t 0 , T ] : X(t) -Z(t) > 0},
is such that τ 1 < T . In this case we define another stoping time τ 2 defined as

τ 2 = inf{t ∈ [τ 1 , T ] : Z(t) -X(t) > 0},
Thus, we have produced the stochastic interval (τ 1 , τ 2 ) in which X(τ 1 ) = Z(τ 1 ) and X(t) ≥ Z(t) on (τ 1 , τ 2 ). For a discussion about stopping times and stochastic intervals one can see [START_REF] Mao | Stochastic Differential Equations and Applications[END_REF] and [START_REF] Karatzas | Brownian Motion and Stochastic Calculus[END_REF].

Then we have

X(t) -Z(t) = X(τ 1 ) -Z(τ 1 ) -D(X(τ 1 )) + D(p(X(τ 1 ), Z(τ 1 ))) + D(X(t)) -D(p(X(t), Z(t))) + t τ1 (f 1 (s, X s ) -f 2 (s, p(X s , Z s )))ds + t τ1 (g(s, X(s)) -g(s, p(X(s), Z(s))))dB(s), t ∈ (τ 1 , τ 2 ).
It is obvious that,

D(X(τ 1 )) -D(p(X(τ 1 ), Z(τ 1 ))) = 0 because X(τ 1 ) = Z(τ 1 ), D(X(t)) -D(p(X(t), Z(t))) = 0, because X(t) ≥ Z(t) on t ∈ (τ 1 , τ 2 ) t τ1 (f 1 (s, X s ) -f 2 (s, p(X s , Z s )))ds ≤ t τ1 (f 1 (s, X s ) -f 2 (s, X s ))ds ≤ 0, t ∈ (τ 1 , τ 2 ) t τ1 g(s, X(s)) -g(s, p(X(s), Z(s))))dB(s) = 0, t ∈ (τ 1 , τ 2 ),
noting that always p(X(t), Z(t)) ≥ X(t).

Thus in fact we have equality, i.e. Z(t) = X(t). So, always Z(t) ≥ X(t) and thus Z(t) = Y (t) the unique solution of (4).

Comparison theorem for neutral sfdes with additive noise

In this section we consider the following stochastic functional differential equation of neutral type,

d[X(t) -D(X t )] = f (t, X t ) dt + dB(t), t 0 ≤ t ≤ T,
with the initial value defined as before. Here, D : C([-τ, 0]; R) → R, f (t, x) : [t 0 , T ] × C([-τ, 0]; R), are Borel measurable. Our goal is to obtain a comparison theorem for this kind of problem.

Let the following problems,

X(t) -D(X t ) = ξ(0) -D(X t0 ) + t t0 f 1 (s, X s )ds + B t -B t0 , t 0 ≤ t ≤ T, (6)
X(t) -D(X t ) = ξ(0) -D(X t0 ) + t t0 f 2 (s, X s )ds + B t -B t0 , t 0 ≤ t ≤ T. (7)
Theorem 2 Suppose that problems (6) and (7) have unique strong solutions. Suppose further that x → f 2 (t, x), x → D(x) are increasing operators in sense that we have described before. If,

f 1 (t, x) ≤ f 2 (t, x) for all x ∈ C([-τ, 0]; R) then X(t) ≤ Z(t) a.s. for all t.
Proof.

As before we consider the operator p(•, We consider the following problem, Z(t) -D(p(X t ), Z t )) = ξ(0) -D(p(X(t 0 ), Z(t 0 )) + t t0 f 2 (s, p(X s , Z s ))) + B t -B t0 , t 0 ≤ t ≤ T.

•) : C([-τ, 0]; R) × C([-τ, 0]; R) → C([-τ, 0]; R), with p(x, y)(θ) = max
It's clear that this problem has a unique strong solution. We are going to show that Z(t) ≥ X(t) a.s. for all t > 0.

Suppose that this is not true. Then as before we can construct a stochastic interval (τ 1 , τ 2 ) in which X(τ 1 ) = Z(τ 1 ) and X(t) ≥ Z(t) on (τ 1 , τ 2 ).

  θ∈[-τ,0] {x(θ), y(θ)}

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

Thus we have,

Arguing as before and noting that f 2 , D are increasing operators we have that X(t) ≤ Z(t) a.s. for all t. But Z(t) = Y (t) the solution of problem (7), so we have done.

The proof of Theorem 2 is essential the same with that of Theorem 1. But the key difference is that we have integrate over (t 0 , t) with t ∈ (τ 1 , τ 2 ) in Theorem 2 while in Theorem 1 we have integrate over (τ 1 , t) with t ∈ (τ 1 , τ 2 ).

We can have another comparison theorem with different D. Assume that ξ = x 0 ∈ R and suppose that D 1 (x 0 ) = D 2 (x 0 ).

Consider the following problems,

Theorem 3 Suppose that problems (8) and ( 9) have unique strong solutions. Suppose further that x → f 2 (t, x), x → D 2 (x) are increasing operators in sense that we have described before. If, f 1 (t, x) ≤ f 2 (t, x) and D 1 (x) ≤ D 2 (x) for all x ∈ C([-τ, 0]; R) then X(t) ≤ Z(t) a.s. for all t.

We omit the proof because is the same as Theorem 2.

Remarks

There exists some open problems, for example what can we say if both D, g are functionals depending on X t = {X(t + θ) : θ ∈ [-τ, 0]}. We would like also to point out that the results in Halidias and Ren 2008 does not hold as they are but are true if we assume further that d = 1, x → f 2 (t, x) is increasing and that g is not a functional depending on X t .


Duro Basic
email: duro.basic@schneider-electric.com

François Malrait
email: francois.malrait@schneider-electric.com

Pierre Rouchon
email: pierre.rouchon@mines-paristech.fr

Modeling magnetic saturation and saliency effects via Euler-Lagrange

Keywords: Lagrangian with complex coordinates, magnetic saturation, saliency, permanent-magnet machine, high frequency injection

come

Modeling magnetic saturation and saliency effects via Euler-Lagrange models with complex currents for three-phase permanent magnet machines D. Basic * F. Malrait * * P. Rouchon * * *

INTRODUCTION

Modeling permanent magnet machines with magnitude saturation is not a straightforward task and could lead to complicated developments when a detailed physical description is included (see, e.g., [START_REF] Chiasson | Modeling and High Performance Control of Electric Machines[END_REF], [START_REF] Boldea | The Induction Machine Handbook[END_REF]). Even if such effects are not dominant they play an important at low speed and/or high torque and stator current. The contributions of this communication are as follows :

• simples models (see ( 10)), including simultaneously saturation and saliency and extending directly usual models used in the literature. • A first experimental validation of such magnetic saturation models for a 1.2 kW machines. It relies on high frequency voltage injection and on the measures of the resulting ripples on the stator current.

We exploit here [START_REF] Basic | Euler-lagrange models with complex currents of three-phase electrical machines and observability issues[END_REF] that proposes an extension to complex electrical variable of Lagrangian modeling of electrical machines.

In section 2 we recall the simplest model of a permanent magnet machine and its Euler-Lagrange formulation based on the two scalar components of the complex stator current. We recall the complexification procedure and explain how to derive the Euler-Lagrange equations directly with complex stator current. Then we provide the general form of physically consistent models (equation ( 5)). Finally we obtain, just by simple modification of the magnetic Lagrangian, physically consistent models with magnetic saturation and saliency effects (equation ( 10)). We also derive the associated magnetic energies of such non-linear magnetic models. In section 3, we propose a simple experimental validation of the nonlinear magnetic model introduced in previous section (equation (10) with lambda given by ( 12)). The validation is based on high frequency voltage injection and measurement of the associated currentripples. Perturbations techniques provide analytic expressions of these ripples, expressions depending on the level of permanent currents. Experimental data on a 1.2 kW machine confirm the fact that the current ripples are an increasing function of the current static offset. These experimental data also confirm that the injected frequencies are not too high (less than a few kHz) and that the main flux distribution is not altered by skin effects in magnets and laminations and that magnetic core losses could be neglected here. Simulations confirm that such dependance is typical of such magnetic saturations.

EULER-LAGRANGE MODELLING WITH COMPLEX CURRENTS AND VOLTAGES

The usual model and its Euler-Lagrange formulation

In the ( , ) frame (total power invariant transformation), the dynamic equations read (see, e.g., [START_REF] Chiasson | Modeling and High Performance Control of Electric Machines[END_REF], [START_REF] Leonhard | Control of Electrical Drives[END_REF]):

⎧  ⎨  ⎩ ( ˙ ) = ℑ ( ( ¯ ) * ) - ( + ¯ ) = - (1)
where

• * stands for complex-conjugation, = √ -1 and is the number of pairs of poles.

• is the rotor mechanical angle, and are the inertia and load torque, respectively. • ∈ ℂ is the stator current, ∈ ℂ the stator voltage. • = ( + )/2 with inductances = > 0 (no saliency here).

• The stator flux is = + ¯ with the constant ¯ > 0 representing to the rotor flux due to permanent magnets.

It is well known that (1) derives from a variational principle (see, e.g., [START_REF] Ortega | Passivity-Based Control of Euler-Lagrange Systems[END_REF]) and thus can be written as Euler-Lagrange equations with source terms corresponding to energy exchange with the environment. Consider the additional complex variable ∈ ℂ defined by = . The Lagrangian associated to this system is the sum of the mechanical kinetic Lagrangian ℒ and magnetic one ℒ defined as follows:

ℒ = 2 ˙ 2 , ℒ = 2 + 2 (2)
where = ¯ / > 0 is the permanent magnetizing current.

Take the complete Lagrangian ℒ = ℒ + ℒ as a real function of the generalized coordinates = ( , , ) and generalized velocities ˙ = ( ˙ , , ):

ℒ( , ˙ ) = 2 ˙ 2 + 2 ( ( + cos ) 2 + ( + sin ) 2 ) (3)
with = + , ( and real) and ˙ = = + , ( and real). Then the mechanical equation in (1) reads

( ∂ℒ ∂ ˙ ) - ∂ℒ ∂ = -
where -corresponds to the energy exchange through the mechanical load torque. Similarly, the real part of complex and electrical equation in (1) reads

( ∂ℒ ∂ ˙ ) - ∂ℒ ∂ = -
and its imaginary part

( ∂ℒ ∂ ˙ ) - ∂ℒ ∂ = - since ∂ℒ ∂ = ∂ℒ ∂ = 0 and ˙ = .
The energy exchanges here are due to the power supply through the voltage and also to dissipation and irreversible phenomena due to stator resistance represented by the Ohm law -.

Euler-Lagrange equation with complex current

The drawback of such Lagrangian formulation is that we have to split into real and imaginary parts the generalized coordinates associated to and ˙ = . We do not preserve the elegant formulation of the electrical part through complex variables and equations. We will show here that it is still possible to extend such complex formulation to the Euler-Lagrange equations. It seems that it has never been used for electrical machines. We recall here below the principle of such complexification (usual in quantum electro-dynamics) and then applied it to the above Euler-Lagrange formulation. Consider a Lagrangian system with two generalized coordinates 1 and 2 corresponding to a point = 1 + 2 in the complex plane ( = √ -1). The Lagrangian ℒ( 1 , 2 , ˙ 1 , ˙ 2 ) is a real function and the Euler-Lagrange equations are

( ∂ℒ ∂ ˙ 1 ) - ∂ℒ ∂ 1 = 0, ( ∂ℒ ∂ ˙ 2 ) - ∂ℒ ∂ 2 = 0.
Using the complex notation , we have 1 = + * 2 and 2 = - * 2 , thus ℒ is also a function of , * , and

* : L( , * , ˙ , ˙ * ) ≡ ℒ ( + * 2 , - * 2 , ˙ + ˙ * 2 , ˙ -˙ * 2
) .

The above identity defines L as a function of the 4 complex independent variables ( , * , ˙ , ˙ * ). Simple computations show that

2 ∂ L ∂ = ∂ℒ ∂ 1 - ∂ℒ ∂ 2 , 2 ∂ L ∂ * = ∂ℒ ∂ 1 + ∂ℒ ∂ 2 and similarly 2 ∂ L ∂ ˙ = ∂ℒ ∂ ˙ 1 - ∂ℒ ∂ ˙ 2 , 2 ∂ L ∂ ˙ * = ∂ℒ ∂ ˙ 1 + ∂ℒ ∂ ˙ 2 .
Thus with this complex notation, we can gather the two real Euler-Lagrange equations into a single complex one

( ∂ℒ ∂ ˙ 1 + ∂ℒ ∂ ˙ 2 ) = ∂ℒ ∂ ˙ 1 + ∂ℒ ∂ ˙ 2 that reads now simply ( 2 ∂ L ∂ ˙ * ) -2 ∂ L ∂ * = 0.
Let us apply this complexification procedure to the Lagrangian ℒ( , , , ˙ , ˙ , ˙ ) defined in (3). The complexification process only applies to and ˙ = by considering ℒ as a function of ( , , * , ˙ , , * ):

ℒ( , ˙ , , * ) = 2 ˙ 2 + 2 ( + ) ( * + - ) .
Then the usual equations (1) read

( ∂ℒ ∂ ˙ ) = ∂ℒ ∂ -, 2 ( ∂ℒ ∂ * ) = - since ∂ℒ ∂ * = 0 and ∂ℒ ∂ ˙ * = ∂ℒ ∂ * .
More generally, the magnetic Lagrangian ℒ is a real value function of , and * that is 2 periodic versus . Thus any Lagrangian ℒ PM representing a 3-phases permanent magnet machine admits the following form

ℒ PM = 2 ˙ 2 + ℒ ( , , * ) (4)
Consequently, any model (with saliency, saturation, spaceharmonics, ...) of permanent magnet machines admits the following structure: Many other formulations of ℒ are possible and depend on particular modeling issues. Usually, the dominant part of ℒ will be of the form

( ˙ ) = ∂ℒ ∂ -, ( 2 ∂ℒ ∂ * ) = - (
¯ 2
+ 2 ( ¯ , positive constants) to which is added corrections terms that are "small" scalar functions of ( , , * ).

The magnetic energy

does not coincides with ℒ . It is given by the Hamiltonian, , defined via a Legendre transform on ℒ . Following the complex formulation used in quantum electro-dynamics (see [Cohen-Tannoudji et al., 1989, page 88, equation (A.30)]) we have:

( , , * ) = ∂ℒ ∂ ( , , * ) + * ∂ℒ ∂ * ( , , * ) -ℒ ( , , * ) . (6)
Notice that, when ℒ = 2 + 2 with and constant, we get = 2 ( | | 2 -2 ) and we recover the usual magnetic energy 2 | | 2 up to the constant magnetizing energy 2 2 . In next two sub-sections, we introduce some modifications to this standard Lagrangian ℒ to take into account saliency and saturation effects and derive the corresponding magnetic energy .

Saliency models

Adding to ℒ the correction -2 ℜ ( 2 -2 ) with | | < (ℜ means real part) provides a simple way to represent saliency phenomena while the dominant part of the magnetic Lagrangian (and thus of the dynamics) remains attached to 2 + 2 . With magnetic Lagrangian of the form

ℒ = 2 ( + ) ( * + - ) - 4
( ( * ) 2 + ( - ) 2 ) (7)
where = ( + )/2 and = ( -)/2 (inductances > 0 and > 0), equations ( 5) become (

= ¯ ) ⎧  ⎨  ⎩ ( ˙ ) = ℑ ( ( * + ¯ - - - 2
)

) -

( + ¯ - * 2 ) = - (8
) and we recover the usual model with saliency effect. In this case the magnetic energy

= ∂ℒ ∂ + * ∂ℒ ∂ * -ℒ is given by: = 2 ( | | 2 -2 ) - 4 ( ( * ) 2 + ( - ) 2 )

Saturation and saliency models

We can also take into account magnetic saturation effects, i.e., the fact that inductances depend on the currents. Let us assume that only the mean inductance in (7) depends on the modulus of + and that remains constant:

= (| + |) = ( √ ( + )( * + - )
)

.

The derivative of versus the modulus of + is denoted by ′ . The magnetic Lagrangian now reads

ℒ = ( + ) 2 + 2 - 4 ( ( * ) 2 + ( - ) 2 ) . ( 9
)
The dynamics is given by ( 5) with such ℒ . Since

∂ ∂ = ℑ ( - ) | + | ′ , ∂ ∂ * = + 2 | + | ′
we get the following model structure with both saliency and magnetic saturation effects:

⎧ ⎨ ⎩ ( ˙ ) = ℑ (( Λ ( * + - ) - - 2
) )

- ( Λ ( + ) - * 2 ) = - (10)
where

Λ = + | + | 2 ′ .
It is interesting to compute the magnetic energy from general formula (6):

= + + ′ 2 | | 2 + + ′ 2 ℜ ( - ) - 2 2 - 4 ( ( * ) 2 + ( - ) 2 ) . (11)
Such magnetic energy formulae are not straightforward but there are a direct consequence of such variational formulation of the dynamics and its setting with complex electrical variables.

We will assume now and in the sequel that admits the following parametric form:

( ) = 2 0 √ 1 + ( sat ) 2 -1 ( sat ) 2 (12)
with two positive parameters 0 > 0 and sat > 0. It yields for Λ to the following simple expression:

Λ( ) = 0 √ 1 + ( / sat ) 2 (13)
Thus with ( 13), the model ( 10) describes simultaneously saliency and saturation with 4 physically meaning positive parameters , 0 , and sat . We will see that such expressions are well adapted for saturation modelling.

A FIRST EXPERIMENTAL VALIDATION

Description of the test

Assume that the rotor is blocked via a mechanical brake at position = 0. Consider the following voltage inputs In the sequel denotes the primitive of , = , that admits a zero mean, ∫ 2 0 ( ) = 0. Then the electrical dynamics obey to the following differential implicit equation:

(Λ ( + ) - * ) = + (Ω ) - ( 15
)
where Λ is given by ( 13). Set = / its asymptotically and hyperbolically stable solution when = 0. For Ω large enough, the solution of ( 15) converges towards a small periodic orbits around : ( ) = + (Ω ) where the complex-value function → ( ) is 2periodic with zero mean.

(Ω ) corresponds to the socalled current ripples resulting from the high frequency injection (Ω ). The above statement can be rigorously proved by averaging theorem (see, e.g., [Guckenheimer and Holmes, 1983, theorem 4.1.1, page 168]) since the unperturbed system is asymptotically stable (for | | small enough, it is in fact a strict contraction in the sense of [START_REF] Lohmiler | On metric analysis and observers for nonlinear systems[END_REF] for the Euclidian metric on the stator flux in ℂ).

Standard asymptotics for Ω tending to +∞ show that the current ripples satisfy to the following complex equation:

( Λ + | + | Λ ′ 2 ) + ( ( + ) 2 Λ ′ 2 | + | - ) * = Ω (Ω ) + ( | | 2 Ω 2 )
where Λ and its -derivative Λ ′ are evaluated at = | + |. Assume real, then = / is real too and the above formulae simplify a little

( Λ + | + | Λ ′ 2 ) + ( | + | Λ ′ 2 - ) * = Ω (Ω ) + ( | | 2 Ω 2 ) . (16)
For real, is real and given by:

(Λ + | + | Λ ′ -) = Ω (Ω ) + ( | | 2 Ω 2
) .

Using (13), we have We should observe experimentally that, for the same high frequency voltage excitation, the amplitudes of the current ripples depend on the current offset .

⎛ ⎜ ⎝ 0 ( 1 + ( + ) 2 2 sat ) 3 2 - ⎞ ⎟ ⎠ ≈ Ω (Ω ). (17)

Simulation vs experimental results for a 1.2 kW machine

We take a permanent magnet synchronous motor of 1.2 kW from the manufacturer Bernecker+Rainer, Industrie-Elektonik Ges. M.b.h, Eggelsberg, Austria (reference: 8YS-H0004R0.029-0). For this motor, we have = 6, the nominal current = 2.4 A and = 6.7 Ohms. The magnetizing current is around 2.6 , the saturation current sat is about 5 , the main inductance 0 is equal to 92.6mH and the saliency almost vanishes. Thus, for this motor and according to (17), the amplitude of the current-ripples should be an increasing function of when + > 0.

The parameters of the high frequency voltage injection are Ω/2 = 500 , | | = 100 and ( ) = sign(sin( )). We use five real values for the constant voltage injections leading to five levels for ∈ {2 , , 0, -, -2 }. For each value of , we estimate the amplitude of the currentripples via a simple PLL-filter. The experimental results are reported on figure 1.

We observe, as predicted via the theory, that this amplitude decreases when decreases. The simulation results are shown in figure 2. This dependance cannot be explained via the standard model ( 8) with linear inductances and results from non-negligible magnetic saturation effects. Indeed, the simulation results in figure 3 We chose here a particular context with = 0 in order to get the simplest computation and experimental test of such modelling. The above developments remain also valid when is no more fixed to 0 and when saliency (level and position) induced by saturation depends on the load torque.

CONCLUSION

The saturation models (10) for permanent-magnet machine with given by ( 12) are based on variational prin-ciples and Lagrangian formulation of the dynamics. Experimental data provide a first validation of such modelling procedures that preserve the physical insight while maintaining a synthetic view without describing all the technological and material details.

More complete validations could be done: at non zero rotor velocity, the proposed computations of the current ripples are still possible since they are obtained via usual perturbations techniques. More complex models can be developed with depending also on | + |. Such models can also be used for control purposes: adaptation of usual control schemes to take into account saturation effects are under study.

  • the pulsation Ω is large (typically around 1kHz).

Fig. 1 .

 1 Fig. 1. Experimental data. The blue curve corresponds to the amplitudes of the current-ripples (obtained via a simple PLL-filter) ; the red curve to the current measures ( -axis aligned with the rotor = 0); we observe an increasing dependance of the ripples versus the current offset .

  , obtained with linear inductances, show constant ripples independently of the total magnetizing current | + |. In the case of the standard model, we define the value of the inductance by = 0 √ 1 + ( / sat ) 2 equals to 82.2mH.

Fig. 2 .

 2 Fig.2. Simulation data with saturation model. The first curve (up) corresponds to the amplitudes of the current-ripples ; the red curve to the current measures ( -axis aligned with the rotor = 0); as predicted by theory, we observe in simulation an increasing dependance of the ripples versus the current offset .

Fig. 3 .

 3 Fig.3. Simulation data with standard model without saturation. The first curve (up) corresponds to the amplitudes of the current-ripples ; the red curve to the current measures ( -axis aligned with the rotor = 0); as predicted by theory, we observe in simulation that ripples stay constant independently of the current offset .