MPH, MSc, PhD Azmeraw T Amare

MD, PhD Klaus Oliver Schubert

PhD Liping Hou

MBBS Scott R Clark

Sergi Papiol

BSc Micah Cearns

Urs Heilbronner

MD Franziska Degenhardt

Fasil

PhD ; Sergi Papiol

PhD ; Urs Heilbronner

PhD ; Fasil Tekola-Ayele

PhD Yi-Hsiang Hsu

MSc Tatyana Shekhtman

MD Mazda Adli

PhD Nirmala Akula

MD Kazufumi Akiyama

MD Raffaella Ardau

PhD Bárbara Arias

MD Jean-Michel Aubry

MD, PhD Lena Backlund

; Abesh

MD Kumar Bhattacharjee

MD, PhD Frank Bellivier

MD, PhD Antonio Benabarre

MD Susanne Bengesser

PhD Joanna M Biernacka

MD Armin Birner

MD Clara Brichant-Petitjean

MD Pablo Cervantes

MD, PhD ; Hsi-Chung Chen

MD Caterina Chillotti

PhD Sven Cichon

PhD Cristiana Cruceanu

PhD Piotr M Czerski

MSc Nina Dalkner

MD Alexandre Dayer

MD Maria Del Zompo

MD ; J Raymond Depaulo

MD, PhD Bruno Étain

PhD Stephane Jamain

MD Peter Falkai

MD Andreas J Forstner

MD Louise Frisen

MD Mark A Frye

PhD Janice M Fullerton

MD Sébastien Gard

BN Julie S Garnham

MD Fernando S Goes

PhD Maria Grigoroiu- Serbanescu

MD, PhD Paul Grof

MD, PhD Ryota Hashimoto

MD Joanna Hauser

Dipl.Biol Stefan Herms

PhD ; Per Hoffmann

PhD Andrea Hofmann

PhD Esther Jiménez

MD, PhD Jean-Pierre Kahn

PhD Layla Kassem

PhD Po-Hsiu Kuo

MD, PhD Tadafumi Kato

MD John Kelsoe

MD Sarah Kittel-Schneider

MD Sebastian Kliwicki

MSc Barbara König

MD Ichiro Kusumi

MD Gonzalo Laje

MD Mikael Landén

PhD Catharina Lavebratt

MD, PhD Marion Leboyer

BSc Susan G Leckband

MD Alfonso Tortorella

MD, PhD Mirko Manchia

MD Lina Martinsson

MD, PhD Michael J Mccarthy

MD Susan Mcelroy

PhD Francesc Colom

PhD Marina Mitjans

MD Francis M Mondimore

MD Palmiero Monteleone

PhD Caroline M Nievergelt

MD Markus M Nöthen

MD Tomas Novák

MB Claire O'donovan

MD Norio Ozaki

MD, PhD Urban Ösby

MD Andrea Pfennig

MD, MPH James B Potash

MD Andreas Reif

; Major

MD Eva Reininghaus

MD Guy A Rouleau

MD Janusz K Rybakowski

MD Martin Schalling

PhD, DSc Peter R Schofield

RN Barbara W Schweizer

MD Giovanni Severino

PhD Paul D Shilling

MD Katzutaka Shimoda

MD Christian Simhandl

RN Claire M Slaney

PhD Alessio Squassina

MD Thomas Stamm

MD Pavla Stopkova

MD Mario Maj

MD Gustavo Turecki

MD, PhD ; Eduard Vieta

PhD Julia Veeh

PhD Stephanie H Witt

MCP Adam Wright

PhD Peter P Zandi

MD Philip B Mitchell

MD,PhD Michael Bauer

MD Martin Alda

MD Marcella Rietschel

MD Francis J Mcmahon

MD Thomas G Schulze

MD, PhD Bernhard T Baune
email: bernhard.baune@ukmuenster.de

Association of polygenic score for major depression with response to lithium in patients with bipolar disorder

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ABSTRACT

Lithium is a first-line medication for bipolar disorder (BD), but only one in three patients respond optimally to the drug. Since evidence shows a strong clinical and genetic overlap between depression and bipolar disorder, we investigated whether a polygenic susceptibility to major depression is associated with response to lithium treatment in patients with BD.

Weighted polygenic scores (PGSs) were computed for major depression (MD) at different GWAS p-value thresholds using genetic data obtained from 2,586 bipolar patients who received lithium treatment and took part in the Consortium on Lithium Genetics (ConLi + Gen) study. Summary statistics from genome-wide association studies in MD (135,458 cases and 344,901 controls) from the Psychiatric Genomics Consortium (PGC) were used for PGS weighting. Response to lithium treatment was defined by continuous scores and categorical outcome (responders versus non-responders) using measurements on the Alda scale. Associations between PGSs of major depression and lithium treatment response were assessed using a linear and binary logistic regression modelling for the continuous and categorical outcomes, respectively. The analysis was performed for the entire cohort, and for European and Asian sub-samples.

The PGSs for MD were significantly associated with lithium treatment response in multiethnic, European or Asian populations, at various p-value thresholds. Bipolar patients with a low polygenic load for major depression were more likely to respond well to lithium, compared to patients with high polygenic load, [lowest vs highest PGS quartiles: multiethnic sample: OR =1.54 (95%CI: 1.18-2.01) and European sample: OR =1.75 (95%CI: 1.30-2.36)]. While our analysis in the Asian sample found equivalent effect size in the same direction: OR =1.71 (95%CI: 0.61-4.90), this was not statistically significant. Using PGS decile comparison, we found a similar trend of association between a high genetic loading for MD and lower response to lithium. Our findings underscore the genetic contribution to lithium response in BD and support the emerging concept of a lithium-responsive biotype in BD.

Keywords: Lithium treatment, depression, bipolar disorder, polygenic score, pharmacogenomics

INTRODUCTION

Bipolar disorder (BD) is a chronic and severe psychiatric illness characterized by episodic, abnormal manic and depressive mood states. An estimated 48.8 million people are affected by BD globally [START_REF] Ferrari | The prevalence and burden of bipolar disorder: findings from the Global Burden of Disease Study 2013[END_REF] . The disorder accounts for 9.9 million years of life lived with disability worldwide, and substantially increases all-cause mortality and risk of suicide [START_REF] Ferrari | The prevalence and burden of bipolar disorder: findings from the Global Burden of Disease Study 2013[END_REF][START_REF] Chesney | Risks of all-cause and suicide mortality in mental disorders: a meta-review[END_REF] . Amongst available treatment options, lithium is regarded as a gold standard by several clinical guidelines [START_REF] Schulze | The International Consortium on Lithium Genetics (ConLiGen): An Initiative by the NIMH and IGSLI to Study the Genetic Basis of Response to Lithium Treatment[END_REF][START_REF] Malhi | Royal Australian and New Zealand College of Psychiatrists clinical practice guidelines for mood disorders[END_REF] . Lithium uniquely protects against both manic and depressive illness phases, has demonstrated protective effects against suicide [START_REF] Hou | Genetic variants associated with response to lithium treatment in bipolar disorder: a genome-wide association study[END_REF][START_REF] Grof | Is response to prophylactic lithium a familial trait?[END_REF][START_REF] Garnham | Prophylactic treatment response in bipolar disorder: results of a naturalistic observation study[END_REF] , and is particularly effective in preventing rehospitalization [START_REF] Lahteenvuo | Real-World Effectiveness of Pharmacologic Treatments for the Prevention of Rehospitalization in a Finnish Nationwide Cohort of Patients With Bipolar Disorder[END_REF] . However, not all patients with BD fully benefit from lithium and only about 30% show full response to the drug [START_REF] Hou | Genetic variants associated with response to lithium treatment in bipolar disorder: a genome-wide association study[END_REF][START_REF] Grof | Is response to prophylactic lithium a familial trait?[END_REF][START_REF] Garnham | Prophylactic treatment response in bipolar disorder: results of a naturalistic observation study[END_REF] . In current psychiatric practice, no biological or clinical markers exist that could reliably predict responsiveness to lithium [START_REF] Grande | Bipolar disorder[END_REF] , and prescribing cannot be targeted to patients who benefit most while avoiding side effects and sub-optimal treatment for poor responders 10 11 12, 13 .

In order to develop objective response markers and to move forward towards personalized prescribing of lithium for BD patients, a better understanding of the biological mechanisms underlying lithium response is urgently required. Recent genome-wide association studies (GWAS) carried out by our International Consortium on Lithium Genetics (ConLi + Gen) [START_REF] Hou | Genetic variants associated with response to lithium treatment in bipolar disorder: a genome-wide association study[END_REF] and others [START_REF] Chen | Variant GADL1 and response to lithium therapy in bipolar I disorder[END_REF][START_REF] Song | Genomewide association study identifies SESTD1 as a novel risk gene for lithium-responsive bipolar disorder[END_REF] have indicated that genetic variation could be an important mediator of response to long-term lithium treatment response in BD patients. Additionally, we have recently demonstrated that high genetic loading for schizophrenia (SCZ) risk variants in people with BD decreases the likelihood of favourable response to lithium [START_REF] Amare | Association of Polygenic Score for Schizophrenia and HLA Antigen and Inflammation Genes With Response to Lithium in Bipolar Affective Disorder: A Genome-Wide Association Study[END_REF] , suggesting that polygenic score (PGS) analysis of mental and physical traits could yield important information on the genetic architecture of BD phenotypes 17 18 19 . BD and MD show 47% genetic overlap [START_REF] Mitchell | Comparison of depressive episodes in bipolar disorder and in major depressive disorder within bipolar disorder pedigrees[END_REF][START_REF] Genomics | Genetic relationship between five psychiatric disorders estimated from genome-wide SNPs[END_REF][START_REF] Group | of the Psychiatric Genomics C. Identification of risk loci with shared effects on five major psychiatric disorders: a genome-wide analysis[END_REF] , and shared risk genes and biological pathways have been described [START_REF] Genomics | Genetic relationship between five psychiatric disorders estimated from genome-wide SNPs[END_REF][START_REF] Amare | The genetic overlap between mood disorders and cardiometabolic diseases: a systematic review of genome wide and candidate gene studies[END_REF][START_REF] Amare | Pharmacogenomics in the treatment of mood disorders: Strategies and Opportunities for personalized psychiatry[END_REF] . Lithium can be effective as an augmentation strategy in MD patients who have experienced an insufficient response to first-line antidepressants [START_REF] Abou-Saleh | Lithium in the episode and suicide prophylaxis and in augmenting strategies in patients with unipolar depression[END_REF][START_REF] Zhou | Comparative efficacy, acceptability, and tolerability of augmentation agents in treatment-resistant depression: systematic review and network meta-analysis[END_REF] and is protective against further MD episodes after symptom remission has been achieved [START_REF] Tiihonen | Pharmacological treatments and risk of readmission to hospital for unipolar depression in Finland: a nationwide cohort study[END_REF] .

Moreover, a large observational study based on the Finnish registry showed that lithium is the most effective agent preventing rehospitalization in MD [START_REF] Tiihonen | Pharmacological treatments and risk of readmission to hospital for unipolar depression in Finland: a nationwide cohort study[END_REF] .

On the other hand, in BD, lithium is more effective in preventing manic than depressive episodes [START_REF] Popovic | Polarity index of pharmacological agents used for maintenance treatment of bipolar disorder[END_REF][START_REF] Vieta | Bipolar disorders[END_REF] , leading to the notion that better lithium responders might be more likely to experience manic predominant polarity, as opposed to depressive predominant polarity [START_REF] Colom | Clinical and therapeutic implications of predominant polarity in bipolar disorder[END_REF] . In support of this view, one study found that excellent lithium responders were characterized by a manic but not depressive polarity of the index episode [START_REF] Kessing | Predictors of excellent response to lithium: results from a nationwide register-based study[END_REF] . Another study described an episodic illness pattern of 'mania-depression-interval' as a predictor for a good response, whereas a 'depression-mania-interval' predicted poorer outcomes [START_REF] Kleindienst | Which clinical factors predict response to prophylactic lithium? A systematic review for bipolar disorders[END_REF] . Inter-episode residual mood symptoms, as opposed to full remission [START_REF] Grof | Is response to prophylactic lithium a familial trait?[END_REF][START_REF] Garnham | Prophylactic treatment response in bipolar disorder: results of a naturalistic observation study[END_REF][START_REF] Pfennig | Influence of atypical features on the quality of prophylactic effectiveness of long-term lithium treatment in bipolar disorders[END_REF] , a rapid cycling pattern [START_REF] Kleindienst | Which clinical factors predict response to prophylactic lithium? A systematic review for bipolar disorders[END_REF][START_REF] Pfennig | Influence of atypical features on the quality of prophylactic effectiveness of long-term lithium treatment in bipolar disorders[END_REF] , and a history of mixed episodes [START_REF] Fountoulakis | Treatment of mixed bipolar states[END_REF][START_REF] Sportiche | Clinical factors associated with lithium response in bipolar disorders[END_REF] have also been described as predictors of poor response.

On the background of these complex interactions between BD, MD, and lithium treatment, we asked whether BD patients with a high genetic susceptibility for major depression, expressed by their PGS, would respond better or worse to lithium than BD patients with a low genetic loading [START_REF] Wray | Genome-wide association analyses identify 44 risk variants and refine the genetic architecture of major depression[END_REF] .

METHODS AND MATERIALS

Discovery GWAS summary dataset

The polygenic score for this study was computed using individual genetic data from the International Consortium on Lithium Genetics ( ConLi + Gen) [START_REF] Hou | Genetic variants associated with response to lithium treatment in bipolar disorder: a genome-wide association study[END_REF] , and GWAS summary statistics for MD from PGC [START_REF] Wray | Genome-wide association analyses identify 44 risk variants and refine the genetic architecture of major depression[END_REF] .

The summary GWAS for MD was produced from a meta-analysis of 9.6 million SNPs (Psychiatric Genomics Consortium-PGC; http://www.med.unc.edu/pgc/), obtained from 7 cohorts (deCODE, Generation Scotland, GERA, iPSYCH, UK Biobank, PGC29 and 23andMe) containing 135,458 MD cases and 344,901 healthy controls [START_REF] Wray | Genome-wide association analyses identify 44 risk variants and refine the genetic architecture of major depression[END_REF] .

Target Study Sample

For the PGS analysis, clinical data on lithium treatment response and genetic information were obtained from the International Consortium on Lithium Genetics (ConLi + Gen; www.ConLiGen.org) for n=2,586 patients, including 23 patients in the replication sample [START_REF] Schulze | The International Consortium on Lithium Genetics (ConLiGen): An Initiative by the NIMH and IGSLI to Study the Genetic Basis of Response to Lithium Treatment[END_REF][START_REF] Hou | Genetic variants associated with response to lithium treatment in bipolar disorder: a genome-wide association study[END_REF][START_REF] Amare | Association of Polygenic Score for Schizophrenia and HLA Antigen and Inflammation Genes With Response to Lithium in Bipolar Affective Disorder: A Genome-Wide Association Study[END_REF] . A series of quality control procedures were implemented on the genotype data before and after imputation as described below.

Target outcome

Lithium treatment response was assessed using the validated "Retrospective Criteria of Long-Term Treatment Response in Research Subjects with Bipolar Disorder" scale, also known as the Alda scale [START_REF] Garnham | Prophylactic treatment response in bipolar disorder: results of a naturalistic observation study[END_REF][START_REF] Duffy | A consecutive series of treated affected offspring of parents with bipolar disorder: is response associated with the clinical profile?[END_REF][START_REF] Manchia | Assessment of Response to Lithium Maintenance Treatment in Bipolar Disorder: A Consortium on Lithium Genetics (ConLiGen) Report[END_REF] . This scale quantifies symptom improvement over the course of treatment (A score, range 0-10), which is then weighted against five criteria (B score) that assess the quality of evidence for the response score [START_REF] Hou | Genetic variants associated with response to lithium treatment in bipolar disorder: a genome-wide association study[END_REF] , to arrive at a total Alda score. For dichotomized assessment of treatment response, patients with a total score of 7 or higher were categorized as "good responders", and the remainder were categorized as poor responders [START_REF] Hou | Genetic variants associated with response to lithium treatment in bipolar disorder: a genome-wide association study[END_REF][START_REF] Manchia | Assessment of Response to Lithium Maintenance Treatment in Bipolar Disorder: A Consortium on Lithium Genetics (ConLiGen) Report[END_REF] . For continuous assessment of treatment response, Alda A scores were used [START_REF] Scott | An examination of the quality and performance of the Alda scale for classifying lithium response phenotypes[END_REF] .

In addition to the Alda scale scores, information on covariates such as age and gender was collected, as described in detail elsewhere [START_REF] Hou | Genetic variants associated with response to lithium treatment in bipolar disorder: a genome-wide association study[END_REF] .

Genotyping and quality control

The genome-wide genotypes, as well as clinical and demographic data, were collected by 22 participating sites. Quality control (QC) procedures were implemented on the genotype data using PLINK, version 1.09 prior to imputation [START_REF] Purcell | PLINK: a tool set for whole-genome association and population-based linkage analyses[END_REF] . Samples with low genotype rates <95%, sex inconsistencies (based on X-chromosome heterozygosity), and one of a pair of genetically related individuals were excluded. SNPs were excluded based on the following criteria: a poor genotyping rate (<95%), strand ambiguity (A/T and C/G SNPs), a low minor allele frequency (MAF<1%), or those deviated from genotype frequency expectations under the Hardy-Weinberg Equilibrium (p<10 -6 ).

Imputation

The genotype data passing QC were imputed on the Michigan server [START_REF] Das | Next-generation genotype imputation service and methods[END_REF] (https://imputationserver.sph.umich.edu) separately for each genotype platform using reference data from the 1000 Genomes Project Phase 3 (Version 5). The European reference panel was used for all the samples except for those from Japan and Taiwan, for which an East Asian reference population data was used. After excluding low-frequency SNPs (MAF<10%); low-quality variants (imputation INFO < 0.9); and indels, the imputed dosages were converted to best guess genotypes. The subsequent polygenic analyses were performed using these best guess genotypes.

STATISTICAL ANALYSES

Polygenic score (PGS) association analysis

PGSs were calculated using the approach previously described by the International Schizophrenia Consortium [START_REF] Purcell | Common polygenic variation contributes to risk of schizophrenia and bipolar disorder[END_REF] . Prior to the PGS computation, independent SNPs were identified through a clumping procedure. Quality-controlled SNPs were clumped for linkage disequilibrium based on GWAS association p-value informed clumping at r 2 = 0.1 within a 250-kilobase window to create a SNP-set in linkage equilibrium using PLINK software, version 1.09 run on Linux (plink

--clump-p1 1 --clump-p2 1 --clump-r2 0.1 --clump-kb 250).
PGSs of MD were calculated for each patient in the ConLi + Gen sample at 10 p-value thresholds (<1x10 -4 , <1x10 -3 , <0.01, <0.05, <0.1, <0.2, <0.3, <0.4, <0.5, <1). For a patient, a PGS was calculated at each p-value threshold (P T ) as the sum of allelic counts (from 0 to 2) for the reference alleles across independent SNPs on a genome-wide scale weighted by their effect sizes estimated as beta or log10 (odds ratio), obtained from previously published GWASs of MD [START_REF] Wray | Genome-wide association analyses identify 44 risk variants and refine the genetic architecture of major depression[END_REF] .

Once the PGSs were constructed, a binary logistic regression model was applied for the binary outcome (lithium response versus non-response) and a linear regression modelling was implemented for the continuous outcome (Alda score on subscale A) to evaluate the association of the PGSs for MD with lithium treatment response at each P T . Using the PGS at the most optimal thresholds, we divided the study samples into quartiles and deciles, ranging from the lowest polygenic load (1 st quartile or 1 st decile) to the highest polygenic load (4 th quartile or 10 th decile). Then, we compared BP patients in the lower polygenic load quartiles (1 st to 3 rd quartiles or 1 st to 9 th decile) with patients in the highest polygenic load quartile (4 th quartile or 10 th decile), to quantify the effect of MD polygenic load on lithium treatment response. The analysis was performed for the European sample (N=2366), Asian sample(N=220) and all the sample combined (N=2586). Associations were considered significant at p < 0.05 after adjusting for covariates.

The PGS association analyses were adjusted for the covariates age, gender, genotyping platform, a polygenic score for schizophrenia [START_REF] Amare | Association of Polygenic Score for Schizophrenia and HLA Antigen and Inflammation Genes With Response to Lithium in Bipolar Affective Disorder: A Genome-Wide Association Study[END_REF] , a polygenic score for bipolar disorder [START_REF] Stahl | Genomewide association study identifies 30 loci associated with bipolar disorder[END_REF]

Sensitivity analysis

To evaluate the robustness of our findings, we ran sensitivity analyses using GWAS summary data from bone traits [lumbar spine bone mineral density, femoral neck mineral density and forearm bone mineral density] [START_REF] Zheng | Whole-genome sequencing identifies EN1 as a determinant of bone density and fracture[END_REF] that have previously shown non-significant genetic correlations with psychiatric disorders [START_REF] Amare | Bivariate genomewide association analyses of the broad depression phenotype combined with major depressive disorder, bipolar disorder or schizophrenia reveal eight novel genetic loci for depression[END_REF] . Once we compute polygenic scores for lumbar spine bone mineral density, femoral neck mineral density and forearm bone mineral density, we evaluated its association with lithium treatment response, both continuous and categorical outcomes, in the combined sample (N=2586). Each analysis was adjusted for covariates age, gender, genotyping platform, polygenic score for schizophrenia [START_REF] Amare | Association of Polygenic Score for Schizophrenia and HLA Antigen and Inflammation Genes With Response to Lithium in Bipolar Affective Disorder: A Genome-Wide Association Study[END_REF] , polygenic score for bipolar disorder [START_REF] Stahl | Genomewide association study identifies 30 loci associated with bipolar disorder[END_REF] and 7 PCs.

RESULTS

Sample characteristics and lithium treatment response rate

After QC, 2,586 patients (3,193 before QC) remained for analysis. While n=2,366 were of European ancestry, the remaining (n=220) were of Asian ancestry. In all, 704 patients (27.2%) responded optimally to lithium treatment (total Alda score ≥7). Detailed sample and demographics details have been described previously [START_REF] Amare | Association of Polygenic Score for Schizophrenia and HLA Antigen and Inflammation Genes With Response to Lithium in Bipolar Affective Disorder: A Genome-Wide Association Study[END_REF] . Analysis of the correlation between the PGSs for MD and the self-reported number of depressive episodes available for a subset of the ConLi + Gen sample (N=1140) showed a statistically significant positive correlation, with estimates ranging from 0.08 to 0.12, suggesting that the PGS for MD may be an approximation to a more severe depressive phenotype in BD (Supplementary Figure 1). The polygenic score for MD is inversely associated with lithium treatment response in BD Statistically significant associations were found at various p-value thresholds between the PGSs for MD and lithium treatment response. In the combined multi-ethnic sample, the strongest association were found at P T <5x10 -2 ; p<0.001, R 2 = 0.8% with the continuous outcome (Alda A score) and p<0.001, R 2 = 0.7% with the categorical outcome (total Alda score ≥7) (Figure 1A).

In European ancestry patients, the PGS at most of the tested p-value thresholds showed significant associations of MD PGS with lithium response across continuous and dichotomized outcomes. Strongest associations were found at P T <5x10 -2 ; p<0.001, R 2 = 0.7% with the continuous outcome and p<0.001, R 2 = 0.9% with the categorical outcome (Figure 1B). However, in the Asian sub-sample, the association of the PGS for MD and lithium treatment response was less robust and marginal associations were found only with the continuous outcome at P T <1x10 -2 (p=0.034, R 2 = 0.85%) and P T <5x10 -2 (p=0.042, R 2 = 0.75%) (Figure 1C). Using PRSice2 software, we found consistent results of association between the PGSs for depression and lithium treatment response [START_REF] Choi | PRSice-2: Polygenic Risk Score software for biobank-scale data[END_REF] (Supplementary Figure 2A-C). After adjusting for multiple testing using the Bonferroni method [START_REF] Bland | Multiple significance tests: the Bonferroni method[END_REF] , associations remained statistically significant in the muti-ethnic and European sample, but not in the Asian sample (Supplementary Table 1). Beta coefficients for all associations were negative, indicating that high genetic loadings for MD are associated with poorer response to lithium in BD.

[Insert figure 1 A-C about here]

To further evaluate the impact of MD PGS on lithium treatment response, we divided the study population into quartiles and deciles based on their polygenic loading for MD. As shown in Figure 2 and Table 1, BD patients who carry a lower polygenic load (1 st quartile or 1 st decile) for MD have higher odds of favourable lithium treatment response, compared to patients carrying a high polygenic load (4 th quartile or 10 th decile). In the combined sample, the odds ratio (OR) of favourable response for patients in the 1 st quartile compared with those in the 4 th quartile was 1.54 (95%CI:1.18 -2.01) and the OR of patients in 1 st decile compared to the 10 th decile was 1.49 (95%CI: 0.97, 2.31). Stratified analysis by ethnicity found a stronger association in the European sample than the Asian sample (Table 1, Figure 2 & Supplementary Figure 3). Title Table 1: Odds ratios of favourable lithium treatment response in patients with BDcomparing the response status of patients in the low PGS decile for MD with patients with the highest polygenic load (10 th decile).

[Insert table 1 & figure 2 about here]

Legend Table 1: Reference (10 th decile) are the PGSs categories with the highest polygenic load for MD at the most significant threshold.

¥ adjusted for the covariates age -gender -genotyping platform -polygenic score for schizophrenia [START_REF] Amare | Association of Polygenic Score for Schizophrenia and HLA Antigen and Inflammation Genes With Response to Lithium in Bipolar Affective Disorder: A Genome-Wide Association Study[END_REF] -polygenic score for bipolar disorder [START_REF] Stahl | Genomewide association study identifies 30 loci associated with bipolar disorder[END_REF]

Sensitivity analysis

To ensure the robustness of our findings, we performed a sensitivity analysis and found no significant association between the polygenic scores for lumbar spine bone mineral density, femoral neck mineral density or forearm bone mineral density and lithium treatment response in bipolar patients, p<0.05 for all polygenic scores computed at different p-value thresholds (Supplementary Figure 4A-C).

DISCUSSION

Our study represents the first direct molecular evidence of an association between a genetic predisposition for major depression and poorer response to lithium treatment in patients with BD.

Using PGS analyses of genetic variants related to MD, we found that BD patients with low genetic loading for these variants were about 1.5 times more likely to have favourable long-term outcomes following lithium treatment compared to BD patients with high MD genetic loading.

Higher MD PGSs were associated with a higher number of reported life-time depressive episodes. Analyses following stratification of our sample into European and Asian ancestries indicated that these associations were particularly robust in the European subsample. Adjustment for the potential effects of psychiatric traits that show genetic overlap with MD (SCZ, BD), and sensitivity analyses with medical traits that are unrelated to psychiatric disorders [START_REF] Zheng | Whole-genome sequencing identifies EN1 as a determinant of bone density and fracture[END_REF] underscored the overall robustness of our findings.

Our findings could form part of a genetic explanation for the previously described clinical observations in relation to mania, depression and lithium response in BD [START_REF] Grof | Is response to prophylactic lithium a familial trait?[END_REF][START_REF] Garnham | Prophylactic treatment response in bipolar disorder: results of a naturalistic observation study[END_REF][START_REF] Popovic | Polarity index of pharmacological agents used for maintenance treatment of bipolar disorder[END_REF][START_REF] Vieta | Bipolar disorders[END_REF][START_REF] Colom | Clinical and therapeutic implications of predominant polarity in bipolar disorder[END_REF][START_REF] Kessing | Predictors of excellent response to lithium: results from a nationwide register-based study[END_REF][START_REF] Kleindienst | Which clinical factors predict response to prophylactic lithium? A systematic review for bipolar disorders[END_REF][START_REF] Pfennig | Influence of atypical features on the quality of prophylactic effectiveness of long-term lithium treatment in bipolar disorders[END_REF][START_REF] Fountoulakis | Treatment of mixed bipolar states[END_REF][START_REF] Sportiche | Clinical factors associated with lithium response in bipolar disorders[END_REF] and supports the notion that better lithium responsiveness could be associated with a 'core' bipolar phenotype in the Kraepelinian form of manic depression [START_REF] Sportiche | Clinical factors associated with lithium response in bipolar disorders[END_REF][START_REF] Malhi | The science and practice of lithium therapy[END_REF] , characterized by a predominant maniadepression-interval (MDI) sequence pattern [START_REF] Grof | Lithium response and the sequence of episode polarities: preliminary report on a Hamilton sample[END_REF][START_REF] Hui | A systematic review and meta-analysis of clinical predictors of lithium response in bipolar disorder[END_REF] . That such a phenotype is complex and difficult to clinically identify is exemplified by the lack of meta-analytic evidence for a more straightforward association between lithium response and mania over depression dominance in BD [START_REF] Hui | A systematic review and meta-analysis of clinical predictors of lithium response in bipolar disorder[END_REF] . Similarly, previous family studies found no association of a family psychiatric history of MD and poorer lithium response in BD [START_REF] Grof | Lithium response and genetics of affective disorders[END_REF] .

Together with the previously reported inverse association of lithium response and schizophrenia PGS [START_REF] Amare | Association of Polygenic Score for Schizophrenia and HLA Antigen and Inflammation Genes With Response to Lithium in Bipolar Affective Disorder: A Genome-Wide Association Study[END_REF] , in the same cohort, our finding suggests that the presence of psychiatric co-morbid genetic traits in BD diminishes the likelihood of optimal treatment response to lithium. Given the substantial overlap between schizophrenia-and MD risk alleles [START_REF] Stahl | Genomewide association study identifies 30 loci associated with bipolar disorder[END_REF] , the possibility that these effects are driven by similar molecular mechanisms warrants further clarification in future studies.

In addition to its effects in BD, lithium's effectiveness as an adjunct antidepressant treatment for people with treatment-resistant MD is well established [START_REF] Bschor | Lithium in the Treatment of Major Depressive Disorder[END_REF][START_REF] Alevizos | Low dosage lithium augmentation in venlafaxine resistant depression: an open-label study[END_REF][START_REF] Bauer | Lithium augmentation in treatment-resistant depression: Metaanalysis of placebo-controlled studies[END_REF][START_REF] Bauer | Role of lithium augmentation in the management of major depressive disorder[END_REF][START_REF] Bauer | Doubleblind, placebo-controlled trial of the use of lithium to augment antidepressant medication in continuation treatment of unipolar major depression[END_REF][START_REF] Bauer | Lithium Augmentation Therapy in Refractory Depression: Clinical Evidence and Neurobiological Mechanisms[END_REF][START_REF] Bschor | Efficacy and mechanisms of action of Lithium augmentation in refractory major depression[END_REF] , and lithium is a first-line treatment for BD type 2 that shows a substantial genetic overlap with MD [START_REF] Kan | Genetic overlap between type 2 diabetes and depression in Swedish and Danish twin registries[END_REF] . Therefore, our finding raises the intriguing possibility that lithium possesses specific antidepressant mechanisms of action that are different from the mechanisms conferring long-term treatment response in BD.

Our finding of a more robust effect of the MD PGS association with lithium response in European compared to Asian patients is interesting but needs to be interpreted with caution.

First, our Asian sub-sample was small (n=220) and may not have been powered sufficiently to detect more consistent effects. Second, the polygenic basis of MD in East Asian and European populations is only partially shared with reported trans-ancestry genetic correlation of 0.33-0.41 [START_REF] Bigdeli | Genetic effects influencing risk for major depressive disorder in China and Europe[END_REF] . The projection of MD risk alleles obtained from the global PGC study onto the Asian ConLi + Gen cohort for PGS analysis may, therefore, be less precise and underestimate the true MD PGS effect. It is notable that ethnic differences with regards to lithium response have not been studied extensively and are not supported by a smaller previous study [START_REF] Shan | Long term use of lithium and factors associated with treatment response among patients with bipolar disorder[END_REF] .

The main limitation of our study is that PGSs for MD explain only a small proportion of the variance in lithium treatment response (<1%), and on their own have no utility as clinical tests.

However, since we detected significant effects in our relatively small sample, it is likely that in the future increased sample sizes will further improve the predictive power of PGSs [START_REF] Wray | Research review: Polygenic methods and their application to psychiatric traits[END_REF] . Further, the current version of the Alda scale assesses only overall lithium efficacy but not effects specific to predominant illness polarity or episode sequence pattern. Availability and incorporation of such information would have refined our results. While our findings, in isolation, are not yet ripe for clinical applications, they could serve as a component of multimodal prediction models incorporating clinical and other biological data. The development of such models and the demonstration of their potential clinical utility in prospective study designs are beyond the scope of the current investigation but need to be attempted to translate our research findings into actionable clinical applications.

In conclusion, we demonstrated that high genetic loadings for MD are predictive of unfavourable long-term response to lithium in patients with BD. Our study underscores the potential of PGS analysis to contribute to predictive models for medication response in psychiatry. The results of our study support clinical observations that have pointed to better lithium responsiveness in a BD subtype characterized by lower psychiatric co-morbidity and more dominant mania-related clinical features. Table 1: Odds ratios of favourable lithium treatment response in patients with BD -comparing the response status of patients in the low PGS quartile or decile for MD with patients with the highest polygenic load (4 th quartile or 10 th decile).

Tables

Legend: reference (4 th quartile and 10 th decile) are the PGSs categories with the highest polygenic load for MD at the most significant threshold.

¥ adjusted for the covariates age -gender -genotyping platform -polygenic score for schizophrenia [START_REF] Amare | Association of Polygenic Score for Schizophrenia and HLA Antigen and Inflammation Genes With Response to Lithium in Bipolar Affective Disorder: A Genome-Wide Association Study[END_REF] -polygenic score for bipolar disorder [START_REF] Stahl | Genomewide association study identifies 30 loci associated with bipolar disorder[END_REF] Legend Figure 1: The y-axis refers to the percentage of variance in treatment response to lithium accounted for by the PGSs for major depression at particular P-value thresholds. On the x-axis, are the GWAS P-value thresholds used to select single-nucleotide polymorphisms for the PGSs. On the top of each bar are the p-values for the association between the PGSs for major depression and lithium treatment response. Beta coefficients (not shown) were negative for all associations, indicating an inverse effect of MD PGS on lithium response. Legend: reference (4 th quartile and 10 th decile) are the PGSs categories with the highest polygenic load for MD at the most significant threshold.

¥ adjusted for the covariates age -gender -genotyping platform -polygenic score for schizophrenia [START_REF] Ferrari | The prevalence and burden of bipolar disorder: findings from the Global Burden of Disease Study 2013[END_REF] -polygenic score for bipolar disorder 2 -and 7 principal components (PCs) in the combined sample or 5 PCs in the European sample and 4 PCs in the Asian sample.

  , and 7 principal components (PCs) in the combined sample or 5 PCs in the European sample and 4 PCs in the Asian sample. The PCs were computed using a --pca command in PLINK and then the top PCs with an eigenvalue of >2.0 were extracted and used as covariates to correct for population stratification. The analyses were performed using R for Statistical Computing and PLINK, version 1.09 for Linux 40 . Prediction accuracy, the percentage of variance in lithium response accounted for by the PGS at each P T , was estimated as the variance explained by the full model including each PGS and covariates minus the variance explained by the model including only covariates.

  and 7 principal components (PCs) in the combined sample or 5 PCs in the European sample and 4 PCs in the Asian sample.

FiguresFigure 1 :

 1 Figures

Figure 2 :Legend Figure 2 :

 22 Figure 2: Odds ratios (ORs) for favourable treatment response to lithium for patients with BD.

1 .

 1 Amare AT, Schubert KO, Hou L, Clark SR, Papiol S, Heilbronner U et al. Association of Polygenic Score for Schizophrenia and HLA Antigen and Inflammation Genes With Response to Lithium in Bipolar Affective Disorder: A Genome-Wide Association Study. JAMA psychiatry 2018; 75(1): 65-74.

2 .

 2 Stahl EA, Breen G, Forstner AJ, McQuillin A, Ripke S, Trubetskoy V et al. Genome-wide association study identifies 30 loci associated with bipolar disorder. Nat Genet 2019; 51(5): 793-803.

Table 1 :

 1 Odds ratios of favourable lithium treatment response in patients with BD -comparing the response status of patients in the low PGS quartile or decile for MD with patients with the highest polygenic load (4 th quartile or 10 th decile).

	Categories	Multi-ethnic (N=2586)	European (N=2366)	Asian (N=220)
	Quartile	unadjusted OR	¥ Adjusted OR	unadjusted OR	¥ Adjusted OR	unadjusted OR	¥ Adjusted OR
		(95% CI)	(95% CI)	(95% CI)	(95% CI)	(95% CI)	(95% CI)
	1 st lowest score	1.50(1.17 -1.92)	1.54(1.18 -2.01)	1.86(1.41 -2.47)	1.75(1.30 -2.36)	1.43(0.55 -3.79)	1.71(0.61 -4.90)
	2 nd	1.45(1.13 -1.86)	1.46(1.12 -1.90)	1.81(1.37 -2.40)	1.77(1.31 -2.38)	1.54(0.60 -4.04)	1.74(0.64 -4.85)
	3 rd	1.16(0.90 -1.49)	1.12(0.85 -1.47)	1.25(0.94 -1.66)	1.21(0.89 -1.64)	0.63(0.21 -1.81)	0.55(0.17 -1.66)
	4 th highest	1 (reference)	1 (reference)	1 (reference)	1 (reference)	1 (reference)	1 (reference)
	score
	Decile
	1 st lowest score	1.50(1.00 -2.27)	1.49(0.97 -2.31)	1.92(1.23 -2.99)	1.74(1.08 -2.81)	1.77(0.40 -8.52)	2.12(0.41 -12.13)
	2 nd	2.06(1.39 -3.09)	2.09(1.38 -3.20)	2.1(1.35 -3.28)	1.98(1.24 -3.18)	1.25(0.27 -6.08)	1.50(0.28 -8.61)
	3 rd	1.76(1.18 -2.64)	1.68(1.10 -2.59)	1.90(1.22 -2.96)	1.65(1.03 -2.66)	1.62(0.37 -7.73)	2.26(0.46 -12.55)
	4 th	1.84(1.23 -2.75)	1.80(1.18 -2.77)	1.77(1.14 -2.77)	1.72(1.08 -2.78)	1.25(0.27 -6.08)	1.68(0.33 -9.44)
	5 th	1.49(1.00 -2.25)	1.43(0.93 -2.20)	1.92(1.23 -2.99)	1.80(1.13 -2.90)	1.25(0.27 -6.08)	1.40(0.28 -7.49)
	6 th	1.38(0.92 -2.10)	1.23(0.80 -1.92)	1.00(0.63 -1.59)	0.91(0.56 -1.50)	0.41(0.05 -2.43)	0.43(0.05 -2.87)
	7 th	1.39(0.92 -2.10)	1.39(0.90 -2.15)	1.65(1.06 -2.58)	1.65(1.03 -2.66)	0.93(0.18 -4.68)	0.95(0.17 -5.46)
	8 th	1.30(0.86 -1.97)	1.19(0.77 -1.85)	1.36(0.87 -2.14)	1.19(0.74 -1.94)	0.41(0.05 -2.43)	0.40(0.04 -2.74)
	9 th	1.25(0.82 -1.90)	1.17(0.75 -1.81)	0.92(0.57 -1.47)	0.91(0.56 -1.50)	1.25(0.27 -6.08)	0.61(0.12 -3.34)
	10 th highest	1 (reference)	1 (reference)	1 (reference)	1 (reference)	1 (reference)	1 (reference)
	score

DA1151/5-2 to UD; SFB-TRR58, Project C09 and Z02 to UD), and the Interdisciplinary Center for Clinical Research (IZKF) of the Medical Faculty of the University of Münster (grant Dan3/012/17 to UD) supported this work. The collection of the Romanian sample was supported by U.E.F.I.S.C.D.I., Romania, grant awarded to Maria Grigoroiu-Serbanescu. The collection of the Czech sample was supported by the project Nr. LO1611 with a financial support from the MEYS under the NPU I program and by the Czech Science Foundation, grant Nr. 17-07070S.

Funding

ATA received a Postgraduate Research Scholarship support from the University of Adelaide through the Adelaide Scholarship International (ASI) program. We thank 23andMe, Inc. staffs for giving us the permission to utilize GWAS summary data for depression which was included as part of the Psychiatric Genomics Consortium (PGC) study. The primary sources of funding were the Deutsche Forschungsgemeinschaft (DFG; grant no.RI 908/7-1; grant FOR2107, RI 908/11-1 to Marcella Rietschel, NO 246/10-1 to Markus M. Nöthen, WI3429/3-1 to Stephanie H. Witt) and the Intramural Research Program of the National Institute of Mental Health (ZIA-MH00284311; ClinicalTrials.gov identifier: NCT00001174). The genotyping was in part funded by the German Federal Ministry of Education and Research (BMBF) through the Integrated Network IntegraMent (Integrated Understanding of Causes and Mechanisms in Mental Disorders), under the auspices of the e:Med Programme (grants awarded to Thomas G. Schulze, Marcella Rietschel, and Markus M. Nöthen).

Conflict of interest

All authors declare that they have no competing interests.

Acknowledgment

The authors are grateful to all patients who participated in the study and we appreciate the contributions of clinicians, scientists, research assistants and study staffs who have helped with patient recruitment, data collection and sample preparation for the studies. We are also indebted to the members of the ConLi + Gen Scientific Advisory Board (http://www.conligen.org/) for critical input over the course of the project.

The analysis of this study was carried out using the high-performance computational capabilities of the University of Adelaide, Phoenix supercomputer https://www.adelaide.edu.au/phoenix/.


Qishuai Liu
email: qishuai.liu@huskers.unl.edu

Fayrouz Isfoula
email: fayrouz.isfoula@univ-poitiers.fr

Emmanuel Bernuau
email: emmanuel.bernuau@agroparistech.fr

Emmanuel Moulay
email: emmanuel.moulay@univ-poitiers.fr

Patrick Coirault
email: patrick.coirault@univ-poitiers.fr

Qing Hui
email: qing.hui@unl.edu

Practical consensus tracking of multi-agent systems with linear controllers

come

Practical consensus tracking of multi-agent systems with linear controllers

Fayrouz Isfoula, Emmanuel Bernuau, Emmanuel Moulay, Patrick Coirault,

I. INTRODUCTION

Since a decade, the consensus problem of multi-agent systems (MAS) has attracted great attention in many fields, such as biology, physics, robotics, and control engineering [START_REF] Ren | Distributed consensus in multi-vehicle cooperative control[END_REF]- [START_REF] Ren | Distributed coordination of multi-agent networks: emergent problems, models, and issues[END_REF]. This is mainly due to its wide applications in distributed formation control. Consensus means the agreement of a group of agents on their common states via local interaction. A variety of topics related to consensus problems have been studied such as consensus with switching topology [START_REF] Cheng | Sampled-data based average consensus of second-order integral multi-agent systems: Switching topologies and communication noises[END_REF]- [START_REF] Munz | Consensus in multiagent systems with coupling delays and switching topology[END_REF] or finite-time consensus [START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF]- [START_REF] Wang | Finite-time consensus for multi-agent networks with second-order agent dynamics[END_REF]. Existing consensus algorithms can be categorized into two classes: consensus without a leader called leaderless consensus or simply consensus and consensus with a leader called consensus tracking where the leader is a special agent whose motion is independent from other agents [START_REF] Ren | Distributed consensus in multi-vehicle cooperative control[END_REF]- [START_REF] Ren | Distributed coordination of multi-agent networks: emergent problems, models, and issues[END_REF]. The leader is followed by all of other agents which communicate with each other and receive instructions from the leader. The agents following the leader are called the followers.

Since the seminal work of Jadbabaie [START_REF] Jadbabaie | Coordination of groups of mobile autonomous agents using nearest neighbor rules[END_REF], several control protocols have been proposed to solve the consensus tracking problem [START_REF] Cao | Distributed coordinated tracking with reduced interaction via a variable structure approach[END_REF]- [START_REF] Zhang | Leader-following consensus for linear and lipschitz nonlinear multiagent systems with quantized communication[END_REF] or the finite time consensus tracking problem [START_REF] Guan | Finite-time consensus for leader-following second-order multi-agent networks[END_REF]- [START_REF] Zuo | Fixed-time consensus tracking for multi-agent systems with high-order integrator dynamics[END_REF] with a given graph topology and constant or varying leader's velocity. In all the previous mention work, the proposed controllers are discontinuous or nonlinear. In our article, it is supposed that we have an undirected connected graph between the followers, a varying leader's velocity and continuous controllers.

All studies done until now consider a non-perceptive leader being focused only on its motion and whose trajectory is independent of the states of the followers [START_REF] Hong | Distributed observers design for leader-following control of multi-agent networks[END_REF], [START_REF] Li | Distributed tracking control for linear multiagent systems with a leader of bounded unknown input[END_REF], [START_REF] Ni | Leader-following consensus of multi-agent systems under fixed and switching topologies[END_REF], [START_REF] Guan | Finite-time consensus for leader-following second-order multi-agent networks[END_REF], [START_REF] Zhao | Distributed finite-time tracking control for multi-agent systems: an observer-based approach[END_REF], [START_REF] Zuo | Fixed-time consensus tracking for multiagent systems with high-order integrator dynamics[END_REF], [START_REF] Cao | Leader-follower consensus of linear multiagent systems with unknown external disturbances[END_REF]. Firstly, we use in this article a nonperceptive leader which can be real or virtual and which is supposed to track a given preassigned trajectory. Its control input is bounded to obtain the practical consensus tracking of a second-order MAS controlled by a linear controller adapted from the classical linear controller proposed in [START_REF] Ren | Distributed consensus in multi-vehicle cooperative control[END_REF] for the consensus. The proof is based on input-to-state stability notions. The practical consensus means that the states' convergence is not exact but in a ball and it has been studied for instance in [START_REF] Ding | Network-based practical consensus of heterogeneous nonlinear multiagent systems[END_REF], [START_REF] Dong | Practical consensus for high-order linear time-invariant swarm systems with interaction uncertainties, time-varying delays and external disturbances[END_REF]. Then, in the second part, we use a perceptive leader that tracks a given preassigned trajectory and receives information from its followers and changes its motion according to the evolution of the fleet. Hence, the control input of the perceptive leader contains two components: the first one is its own control objective ensuring the tracking of a given preassigned trajectory and the second one is composed by the pinned nodes' states. A compromise between the control objective of the perceptive leader and the connectivity maintenance with the followers must be found. A study of the speed of convergence, precision and connectivity maintenance of both strategies with a non perceptive and a perceptive leader is provided in simulation and on a fleet of mobile robots.

The article is organized as follows. After some notations and definitions provided in Section II, the problem statement is given in Section III. Then, the main results of the paper are developed in Section IV for a non perceptive and a perceptive leader. Simulation results are presented in Section V. An experimental platform of a fleet of four mobile robots is used in Section VI to show the effectiveness of the proposed methods. Finally, a conclusion is reached in Section VII.

II. NOTATIONS AND DEFINITIONS

Let us introduce the following notations:

• R + = {x ∈ R : x ≥ 0}, where R is the set of real numbers. • We denote 1 n = (1, . . . , 1) T ∈ R n .

• I n denotes the unit matrix of size n.

• |•| denotes the absolute value in R and • the Euclidean norm on R n .

• If A is an m × n matrix and B a p × q matrix, then the Kronecker product A ⊗ B is the mp × nq block matrix

A ⊗ B =    a 11 B • • • a 1n B . . . . . . . . . a m1 B • • • a mn B   .
• If P is an n × n real symmetric matrix then:

λ min (P ) is the smallest eigenvalue of the matrix P ; λ max (P ) is the greatest eigenvalue of the matrix P .

• A continuous function α : R + → R + belongs to the class K if α(0) = 0 and the function is strictly increasing. A function α : R + → R + belongs to the class K ∞ if α ∈ K and it is unbounded.

A. Graph theory

Let us recall some basic definitions about graph theory given for instance in [START_REF] Ren | Distributed consensus in multi-vehicle cooperative control[END_REF]Appendix B].

Definition 1: An undirected graph G = (V, E) consists of a set of vertices V = {1, . . . , N } and a set of edges E ⊂ V × V such that (i, j) ∈ E iff (j, i) ∈ E. An edge (i, j) ∈ E in an undirected graph G represents that vertices i and j obtain information from each other.

Remark 2: An undirected path is a sequence of edges in an undirected graph of the form (i 1 , i 2 ), (i 2 , i 3 ), . . . Remark 3: An undirected graph is connected if there is an undirected path between every pair of distinct vertices.

Definition 4: The adjacency matrix of an undirected graph

(V, E) is defined by A = [a ij ] ∈ R N ×N where a ij = a ji = 1 if (i, j) ∈ E and a ij = 0 otherwise.
Definition 5: The Laplacian matrix of an undirected graph associated with adjacency matrix A is given as

L = [ ij ] ∈ R N ×N where ii = N j=1,j =i a ij and ij = -a ij if i = j.
It is clear that the matrices L and A are symmetric.

Lemma 6 ( [START_REF] Olfati-Saber | Consensus and cooperation in networked multi-agent systems[END_REF]): All of the nonzero eigenvalues of the Laplacian matrix L are real and positive for an undirected graph. Zero is a simple eigenvalue of L and the associated eigenvector is 1 N if and only if the undirected graph is connected.

B. Input-to-state stability

Consider the following system

ẋ = f (x, u) (1)
where f : R n × R m → R n is continuously differentiable and satisfies f (0, 0) = 0. Let us denote L ∞ loc the set of locally essentially bounded functions u : R → R m .

Let us recall the definition of input-to-state stability (ISS) and the definition of ISS Lyapunov function.

Definition 7 ([34]):

The system ( 1) is (globally) input-tostate stable (ISS) if there exist a KL function β : R + ×R + → R and a class K function γ such that, for each input u ∈ L ∞ loc and each x 0 ∈ R n , the following inequality holds

|x(t)| ≤ β( x 0 , t) + γ(ess sup s∈[0,t] u(s) ), ∀t ≥ 0. (2)
III. PROBLEM STATEMENT Consider a MAS of N agents labeled 1, 2, . . . , N that communicate with each other. We consider a graph G where the vertices are the agents and two agents are linked by an edge when the communication between them is possible. Through the whole of the paper, the communication topology between the agents satisfies the following assumption.

Assumption 8: The graph G is undirected and connected. The dynamics of each agent is given by

qi = p i i = 1, . . . , N ṗi = u i (3)
where q i ∈ R m is the position, p i ∈ R m is the velocity and u i is the control input of the agent i. The agents are called followers and track the trajectory of a special agent called the leader labeled by 0 which has its own control objective. In general, only a small fraction of the followers can communicate with the leader. Definition 9: The matrix D of a graph (V, E) with a leader denoted 0 is defined by D = Diag{a 10 , . . . , a N 0 }, where a i0 = 1 if the agent i communicate with the leader and a i0 = 0 otherwise. If a i0 = 1 then the node i is called a pinned node. We suppose that we have the following assumption Assumption 10: The matrix D has at least one non-zero entry. The dynamics of the leader is given by

q0 = p 0 ṗ0 = u 0 (4)
where q 0 ∈ R m is the position of the leader, p 0 ∈ R m is its velocity and u 0 is the time varying leader's acceleration.

q i (t) -q ref (t) ≤ ε, lim sup t→+∞ p i (t) -p ref (t) ≤ ε,
Remark 12: If ε = 0, we recover the consensus tracking problem.

The aim of this paper is to design controls u i , i ∈ {0, . . . , N } such that the system (3)-( 4) achieves practical consensus tracking.

IV. MAIN RESULTS

We consider the following control protocol of the MAS (3) proposed in [START_REF] Ren | Distributed consensus in multi-vehicle cooperative control[END_REF] [START_REF] Gao | Sampled-data based consensus of continuous-time multi-agent systems with time-varying topology[END_REF] for i ∈ {1, . . . , N }, where k 1 > 0 and k 2 > 0 are positive constants, a ij are the components of the adjacency matrix A and a i0 are the diagonal components of the matrix D.

u i = -k 1 N j=0 a ij (q i -q j ) -k 2 N j=0 a ij (p i -p j ),
In the sequel, we propose two control laws for the leader. The first one is based only on the leader's state and the reference trajectory (and therefore not on the followers' state), which justifies the terminology of non-perceptive leader used below. On the contrary, the second one takes into consideration the followers' state, hence the name of perceptive leader.

A. The non-perceptive leader case

In this section, the leader's acceleration is given by

u 0 = u ref -k 1 (q 0 -q ref ) -k 2 (p 0 -p ref ) (6)
where k 1 > 0 and k 2 > 0 are the same constants than in [START_REF] Gao | Sampled-data based consensus of continuous-time multi-agent systems with time-varying topology[END_REF].

Remark that, if q 0 (0) = q ref (0) and p 0 (0) = p ref (0), and in absence of external disturbances, we have q 0 (t) = q ref (t) and p 0 (t) = p ref (t) for all t ≥ 0. We have the following result.

Theorem 13: Consider a MAS with dynamics (3) under the control protocol ( 5) and the non-perceptive leader (4) with control [START_REF] Olfati-Saber | Consensus problems in networks of agents with switching topology and time-delays[END_REF]. Suppose that Assumption 8 and Assumption 10 hold. If there exists u max > 0 such that sup t≥0 |u ref (t)| ≤ u max , then the MAS solves the practical consensus tracking problem.

Proof:

We define the vectors e q and e p ∈ R n , n = m(N + 1) by e q = (q 0 -q ref ) T , (q 1 -q ref ) T , . . . , (q N -q ref ) T T and e p = (p 0 -p ref ) T , (p 1 -p ref ) T , . . . , (p N -p ref ) T T . We also define the matrix M ∈ R n×n by blocks:

M = -I m 0 m×(N m) a ⊗ I m -S ,
where a = (a 10 , . . . , a N 0 ) T and S = (L f + D) ⊗ I m . We obtain ėq = e p ėp = k 1 M e q + k 2 M e p -bu ref [START_REF] Xiao | Asynchronous consensus in continuous-time multiagent systems with switching topology and time-varying delays[END_REF] with b = (0 T m , 1 T mN ) T ∈ R n and L f the Laplacian matrix of the graph of the followers. If we denote e = (e T q , e T p )

T ∈ R 2n , B = -(0 T n , b T ) T and A = 0 n I n k 1 M k 2 M
, then the system ( 7) is equivalent to

ė = Ae + Bu ref . (8)
Let us show that all the real parts of the eigenvalues of A are strictly negative. Denoting χ A the characteristic polynomial of the matrix A, we find

χ A (λ) = det(λ 2 I n -k 2 λM -k 1 M )
and then

χ A (λ) = det λ 2 I m + k 2 λI m + k 1 I m 0 m×(mN ) -k 2 λa ⊗ I m -k 1 a ⊗ I m λ 2 I mN + k 2 λS + k 1 S .
We know that I m and S commute. Then, expanding this determinant by minors we get

χ A (λ) = (λ 2 + k 2 λ + k 1 ) m det λ 2 I mN + (k 1 + k 2 λ)S .
We see that the eigenvalues of A are the roots of the polynomial λ 2 + k 2 λ + k 1 , which are of negative real part provided that k 1 > 0 and k 2 > 0, and the roots of π(λ) = det

λ 2 I mN + (1 + λ)S . Note that π(-k 1 /k 2 ) = 1 thus -k 1 /k 2 is not a root of π.
Therefore we can write that

π(λ) = (k 1 + k 2 λ) mN det S + λ 2 k 1 + k 2 λ I n = (k 1 + k 2 λ) mN χ S λ 2 k 1 + k 2 λ .
Under Assumption 8 and from Lemma 6, the matrix L f is symmetric positive-definite and D is diagonal with positive entries. Hence S is a symmetric, positive-definite matrix and thus the eigenvalues of S are positive. Let µ > 0 be an eigenvalue of S. By computing the solutions of the equation µ = -λ 2 k1+k2λ , it follows that the real part of λ is always negative and we conclude that A is Hurwitz. It follows that there exist P and Q, two symmetric, positive-definite matrices, such that P A + A T P = -Q. with c = λ min (P -1/2 QP -1/2 ). Fix ε ∈ (0, 1). As long as

√ e T P e ≥ 2umax √ B T P B c(1-ε)
, we have V ≤ -εcV (e) and thus V (e) ≤ exp(-εct)V (e(0)). Therefore we have either

V (e) ≤ exp(-εct)V (e(0)) or V (e) ≤ 2umax √ B T P B c(1-ε) 2 , that is V (e) ≤ max    exp(-εct)V (e(0)); 2u max √ B T P B c(1 -ε) 2    .
Using λ min (P ) e 2 ≤ V (e) ≤ λ max (P ) e 2 we finally get e ≤ max λ max (P )

λ min (P ) exp(- εct 2 ) e(0) ; γ u max with γ = 2 √ B T P B c(1-ε) √ λmin(P )
. Therefore, the system error ( 8

i.e. the MAS solves the practical consensus tracking problem.

B. The perceptive leader case

Consider the following assumption on the communication topology between the leader and its followers.

Assumption 14: The paths between the leader and its followers are undirected. Due to Assumption 14, the MAS (3) can be controlled by a perceptive leader with the following control

u 0 = -k 1 (q 0 -q ref ) -k 2 (p 0 -p ref ) -k 1 N i=1 a i0 (q 0 -q i ) -k 2 N i=1 a i0 (p 0 -p i ). (10)
The first part of the controller ensures the tracking after the predefined trajectory and the second part ensures the connectivity between the followers and the leader.

Theorem 15: Consider a MAS with dynamics (3) under the control protocol ( 5) and the perceptive leader (4) with control [START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF]. Suppose that Assumption 8, Assumption 10 and Assumption 14 hold. If there exists u max > 0 such that sup t≥0 |u ref (t)| ≤ u max , then the MAS solves the practical consensus tracking problem.

Proof: Consider the graph ( V , Ē) obtained by adding the leader to the followers' graph. Given Assumption 14, the graph ( V , Ē) verifies Assumption 8. We will now add a "virtual leader" to this MAS. This virtual leader is only linked to the real leader. Hence the D matrix associated with this virtual leader has exactly one non-zero entry at the first line, first column; this yields that Assumption 10 holds. Now, assume that the virtual leader follows the dynamics and control rule defined by ( 4) and ( 6) but with initial condition q ref (0) and p ref (0). As we already pointed out, this means that the trajectory of the virtual leader will stick to the reference trajectory. Therefore, we see that the control law of the leader (10) corresponds the control law of a follower (5) that is linked with the leader. This means that we can apply Theorem 13 and we conclude that the MAS solves the practical consensus tracking problem.

Remark 16: In the proof of Theorem 15, the dynamics of the virtual leader is only known by the leader. Using this virtual leader is just a technique to prove the theorem and does not need additional online simulations. Moreover, the eventual practical consensus can be improved by considering the following more general control law of the leader

u 0 =u ref -1 (q 0 -q ref ) -2 (p 0 -p ref ) -k 1 N i=1 a i0 (q 0 -q i ) -k 2 N i=1 a i0 (p 0 -p i ). (11)
The proof of this fact is similar to the proof of Theorem 13. The gains 1 and 2 > 0 allow the leader to change the balance between following the reference trajectory and waiting for the followers.

V. SIMULATIONS

In this section, we demonstrate in simulations the effectiveness of the proposed linear controllers for the practical consensus of MAS with both perceptive and non-perceptive leaders. We consider a second-order MAS with N = 10 agents. All agents are moving in the plane (m = 2) with the dynamics given by (3). The leader is denoted by 0 and the graph between the agents is undirected. The network topology is given by Fig. 1. It can be noted that the undirected graph G is connected and the leader is in the neighborhood of agents 1, 2 and 6. 0

F 1 F 6 F 2 F 3 F 7 F 5 F 4 F 9 F 8 F 10
Fig. 1: Topology of the MAS of 10 followers with a leader denoted 0.

The initial positions of the followers (q i (0) are randomly chosen from the box [-10, 10]

2 while the initial velocities are p i (0) = (0, 0). The initial position of the leader q 0 (0) is randomly chosen from the square box [-10, 10] 2 and its initial velocity is p 0 (0) = (0, 0). The simulations are computed with a sampling rate of 0.01s. We define the barycenter G of the N follower's position q i (t) by q G (t) = 1 N N i=1 q i (t). Fig. 2 presents the trajectory of the barycenter G and the reference trajectory q ref (t) of the virtual leader for a non-perceptive leader. The reference trajectory q ref (t) is chosen as q refx (t) = Q x cos(ω x t) + q 0,refx and q refy (t) = Q y sin(ω y t) + q 0,refy with ω x = 0.005(2π)rd/s, ω y = 0.03(2π)rd/s, Q x = 5 and Q y = 10. It can be seen from Fig. 2 and Fig. 3 that the practical consensus is reached, i.e. the followers remain in a closed neighborhood of the virtual leader.

Fig. 4 (resp. Fig. 5) presents the barycenter position q Gx (t), q Gy (t) (resp. barycenter speed) and the leader position q 0x (t), q 0y (t) for a perceptive leader with q ref = (5, 5) and p ref = (0, 0). At t = 0, the perceptive leader and the virtual leader are at the same position. Due to the perceptive part in the control of the leader, the leader leaves its initial position and tends to reach the agents, while these ones tend to reach the leader. Once the leader is in the neighborhood of the agents, the followers and the leader ultimately converge in the vicinity of q ref . In this section, we show the effectiveness of the practical consensus of a MAS with both perceptive and non-perceptive leaders via an experimental platform of a fleet of four Pioneer 3-DX mobile robots that use odometer to record their own positions. Their initial positions are given by Fig. 6. The robots can communicate with each other through ethernet with TCP/IP protocols. The control algorithm is written in C++ language and executed at 0.01s sampling rate. A PC is used to generate the desired trajectory and acts as a virtual leader wheel mobile robot. We approximate the modelling of the robots with a double integrator dynamics (3). Assumption 8 is maintained for this experiment. The coordinate of the initial positions of the robots are chosen as q 1x = -0.915m, q 1y = 0.915m; q 2x = 2.205m, q 2y = 0m; q 3x = 1.22m, q 3y = 1.22m; q 4x = -0.62m, q 4y = 2.44m and all the velocities are initiated by zero. Moreover, we set q ref = p ref = (0, 0). In practical applications, robots have limited speed so gains must be chosen small enough so that the speed limit does not exceed. We take k 1 , k 2 , l 1 , l 2 = 0.3. The evolution of the x-position of the fleet with controller (5) and the non-perceptive leader (4) is given by Fig. 7. The evolution of the x-position of the fleet with controller (5) and the perceptive leader [START_REF] Cao | An overview of recent progress in the study of distributed multi-agent coordination[END_REF] is given by Fig. 8.

The practical consensus tracking is said to be achieved when the robots are 1m away from the leader and the deviation between the leader and the followers is the angle As in simulations, we observe that the use of a non perceptive and perceptive leader can ensure the practical consensus with interesting results in terms of speed of convergence, precision and connectivity maintenance.

VII. CONCLUSION

In this article, the practical consensus tracking of secondorder multi-agent systems under a fixed topology and a varying leader's velocity has been studied. Two cases of practical consensus tracking were investigated, with a non-perceptive leader and with a perceptive leader. It was proved that the practical consensus tracking can be achieved in simulation and on a fleet of mobile robots for both approaches. The case of switching topology has been anticipated as future work.

  Consider now the candidate Lyapunov function V (e) = e T P e. The time derivative of V is given by V = ėT P e + e T P ė = (Ae + Bu ref ) T P e + e T P (Ae + Bu ref ) = e T A T P e + e T P Ae + (B T P e + e T P B)u ref = -e T Qe + 2e T P Bu ref . By common manipulations, we get V ≤ √ e T P e -c √ e T P e + 2u max √ B T P B ,

  ) is ISS with respect to the input u ref . It leads to lim sup t→+∞ e(t) ≤ γ u max ,

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

 234 Fig. 2: Barycenter position (blue line) and reference trajectory (red line) for a non-perceptive leader

Fig. 5 :Fig. 6 :

 56 Fig. 5: x-barycenter speed (blue line) and y-barycenter speed (red line) for a perceptive leader

Fig. 7 :

 7 Fig. 7: The evolution of the x-position of the fleet with a non-perceptive leader

Fig. 8 :

 8 Fig. 8: The evolution of the x-position of the fleet with a perceptive leader

  • A continuous function β : R + × R + → R + belongs to the class KL if β(•, t) ∈ K ∞ for each fixed t ∈ R + and if for each fixed s ∈ R + the function t → β(s, t) is decreasing to 0.

  The leader tries to track a preassigned trajectory (q ref , p ref ) which dynamics is given by qref= p ref , ṗref = u ref .Let us stress that this reference trajectory is only known by the leader. The state (q ref , p ref ) is also defined as the state of a virtual leader.Let us now define the practical consensus tracking.

	Definition 11 (Practical consensus tracking): The MAS
	(3) is said to solve the practical consensus tracking problem
	if there exists ε ≥ 0 such that for all i ∈ {0, . . . , N }
	lim sup
	t→+∞


Nima Yeganefar

Nader Yeganefar

Olivier Bachelier

Emmanuel Moulay

Exponential stability for 2D systems: the linear case

Keywords: 2D systems, Lyapunov theorem, linear discrete systems, exponential stability

This short paper deals with a 2D discrete linear Roesser model. The results introduced here are a follow-up of a paper we proposed recently and where we explained and motivated the reasons we need to adopt a new definition of exponential stability for 2D systems. However this previous result left aside a crucial point that we would like to asses here: is our new definition of exponential stability coherent with the existing stability criterion in the linear case? We hereby show that, in the linear case, 1. our new definition of exponential stability is equivalent to asymptotic stability and 2. the characteristic polynomial-based stability criterion is a sufficient and necessary condition for the exponential stability we have introduced.

I. INTRODUCTION

A. Background

Multidimensional systems were first introduced in the mid 1970s. Working on digital filters, several authors proposed to extend the framework which was mainly based on transfer functions to the multidimensional case: instead of having one independent variable, they looked at polynomial functions of higher dimensions. The notion of stability was also discussed in the BIBO (bounded input bounded output) sense [START_REF] Shanks | Stability and synthesis of twodimensional recursive filters[END_REF]. This allowed to extend a well-known stability criterion to the nD case and it was based on the characteristic polynomial equation of the studied system.

This famous criterion simply states that the transfer function is BIBO stable for systems devoid of poles outside the stability region, which seems natural as it is similar to the 1D case. However the criterion is not as simple as in the 1D case, the number of zeros are usually infinite and further techniques need to be used in order to have a tractable condition (see for instance [START_REF] Ghamgui | Robust stability of hybrid Roesser models against parametric uncertainty : A general approach[END_REF]). It was quoted as a necessary and sufficient condition, led to several stability tests [START_REF] Huang | Stability of two-dimensional recursive filters[END_REF], [START_REF] Anderson | Stability test for two-dimensional recursive filters[END_REF], and broadened to include exponential stability in [START_REF] Pandolfi | Exponential stability of 2-D systems[END_REF]. However Goodman later showed in [START_REF] Goodman | Some stability properties of two dimensional linear shiftinvariant digital filters[END_REF] that the condition is necessary only if the transfer function does not present a particular type of point called nonessential singularity of the second kind.

State-space models for multidimensional systems were given by different authors around the same time (see [START_REF] Roesser | Discrete state-space model for linear image processing[END_REF] and [START_REF] Fornasini | State-space realization theory of twodimensional filters[END_REF], [START_REF]Doubly-indexed dynamical systems: State-space models and structural properties[END_REF]). Fornasini-Marchesini also introduced a definition of asymptotic stability which is still in use today in the field. Introducing state space models allowed a large number of researchers to engage in the field of multidimensional systems as several techniques based on LMIs (linear matrix inequalities) and Lyapunov techniques were extended from the 1D case to the nD case ( [START_REF] Galkowski | LMI approach to statefeedback stabilization of multidimensional systems[END_REF], [START_REF] Kaczorek | LMI approach to stability of 2D positive systems[END_REF]).

Exponential stability, on the other hand, has been seldom studied directly in the 2D systems literature. In the linear case, one of the only works dealing with this problem is given by Pandolfi in [START_REF] Pandolfi | Exponential stability of 2-D systems[END_REF]. Pandolfi proved that asymptotic stability and exponential stability are equivalent, similarly to what happens in the 1D case. This equivalence could explain why this particular type of stability has not been very much investigated in the 2D field. Later in [START_REF] Dymkov | Exponential stability of discrete linear repetitive processes[END_REF], the authors extended this definition to repetitive systems but we are not discussing this special type of multidimensional systems in this paper; this work could however impact the results given in [START_REF] Dymkov | Exponential stability of discrete linear repetitive processes[END_REF]. In the continuous case, we also have proposed a definition in [START_REF] Ghamgui | Exponential stability conditions for 2D continuous state-delayed systems[END_REF]. However, in a recently published paper [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF] we have carefully explained the shortcomings of these definitions and strongly motivated the introduction of a new one (for the discrete case only).

B. Goal of the paper

As pointed out above, in [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF], we have shown that the existing definition of exponential stability [START_REF] Pandolfi | Exponential stability of 2-D systems[END_REF] used in the literature of 2D systems is not correct. We have therefore introduced and motivated the use of a new definition that allowed us to investigate the exponential stability of nonlinear 2D systems and to propose the first converse Lyapunov theorem for 2D systems under certain assumptions. By doing so, we have left unanswered several questions, especially in the linear case. This paper therefore tries to fill the gap introduced by our work.

The questions we would like to answer in this short paper are the following:

1) Do we still have equivalence between our newly defined exponential stability and asymptotic stability in the linear case? 2) There is a well-known characteristic polynomial-based stability criterion from which a lot stability results are derived. Is this stability criterion still valid (ie. necessary and sufficient) if one uses our newly proposed definition? The answers to both of these questions are positive and the proofs of these answers are the main results of this article.

Let us now recall the results we are talking about.

C. Previous important results

In order to have a self-contained article, we think it is important to quickly recall a few of the important results we published in the previously mentioned work [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF]. Moreover, to make it easier to follow for the reader, we are going to use a Fornasini-Marchesini type model [START_REF] Fornasini | State-space realization theory of twodimensional filters[END_REF], [START_REF]Doubly-indexed dynamical systems: State-space models and structural properties[END_REF] contrary to [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF] where the model is a nonlinear version of the well-known Roesser model [START_REF] Roesser | Discrete state-space model for linear image processing[END_REF]. Both models are equivalent and therefore working with one or the other is the same. That being said, our system is described as follows:

x(i + 1, j + 1) = Ax(i, j + 1) + Bx(i + 1, j) (1)
where x is a real vector of dimension n and matrices A, B are of appropriate dimensions. i and j are positive integers so that the initial conditions are taken in the first quadrant (ϕ(i) = x(i, 0), and ψ(j) = x(0, j) for i, j ∈ N), and we study the behavior of trajectories in N 2 as i + j → +∞. Note that the approach adopted by Fornasini-Marchesini ( [START_REF] Fornasini | State-space realization theory of twodimensional filters[END_REF], [START_REF]Doubly-indexed dynamical systems: State-space models and structural properties[END_REF]) for 2D systems is different: they consider initial conditions given on the subset {(i, j) ∈ Z 2 , i + j = 0}, and they study the behavior of trajectories as i + j → +∞. This choice of initial conditions is not without consequence on the stability definitions as we have pointed out in [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF], for further remarks on the importance of the initial conditions for 2D systems one can refer to eg. [START_REF] Valcher | Characteristic cones and stability properties of twodimensional autonomous behaviors[END_REF], [START_REF] Napp | Time-relevant stability of 2D systems[END_REF]. For such a system, observe that the definition of asymptotic stability we adopted in [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF] is more restrictive than the one usually found in the literature, as we define the concept of stability and attractivity (as in the 1D case).

Definition 1.1 ( [START_REF] Liu | Stability analysis of state-space realizations for two-dimensional filters with overflow nonlinearities[END_REF], [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF]): The point x e = 0 is said to be asymptotically stable (in the sense of Lyapunov) if:

1) x e = 0 is stable (for all > 0, there exists δ( ) > 0 such that if ϕ(i) < δ, ψ(j) < δ then x(i, j) < for all i, j > 0), 2) lim i+j→∞ x(i, j) = 0 when lim i→∞ ϕ(i) = 0 and lim j→∞ ψ(j) = 0. This property will be called attractivity as in the 1D case. Exponential stability is defined as follows:

Definition 1.2 ( [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF]): The equilibrium point x e = 0 of system (1) is said to be exponentially stable if there are constants q ∈ (0, 1) and M > 0 such that for any initial conditions ϕ and ψ, and for all (i, j) ∈ N 2 , we have:

x(i, j) ≤ M i r=0 ϕ(r) q r+1 + j s=0 ψ(s) q s+1 q i+j (2)
Remark 1: This definition was mainly supported by two observations in [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF]. First, notice that each term x(i, j) is determined by the knowledge of ϕ(r) and ψ(s), for r = 0, . . . i, and s = 0, . . . , j, so that it is natural to express the bound (2) in terms of these quantities (see for instance (5) which gives the full expression of the solutions). Second, analyze the following special example of an upper triangular matrix with a single eigenvalue 0 < q < 1:

x(i + 1, j + 1) = q 1 0 0 x(i, j + 1) + 0 0 0 q x(i + 1, j) (3)
As expected, because q ∈ (0, 1), 0 is an exponentially stable equilibrium point as defined in Definition 1.2. However, the trivial solution of system (3) is not exponentially stable using the definition introduced in [START_REF] Pandolfi | Exponential stability of 2-D systems[END_REF] (for further details, please see [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF]).

II. MAIN RESULTS

Proposition 2.1: Exponential stability implies asymptotic stability (Definition 1.1).

Proof: Let us first show that exponential stability implies stability. This comes from the fact that for a given > 0, there exists a δ = (1-q) 4M > 0 such that if for all i, j ≥ 0, ϕ(i) ≤ δ and ψ(j) ≤ δ, then

x(i, j) ≤ M j k=0 ψ(k) q k+1 + i k=0 ϕ(k) q k+1 q i+j ≤ M δ j k=0 1 q k+1 + i k=0 1 q k+1 q i+j ≤ M δ q j-1 + q i-1 -2q i+j 1 -q ≤
Attractivity follows by similar arguments. Note that this proof is independent of the linearity of the system. Theorem 2.2: The linear system (1) is exponentially stable if and only if

det (I -z 1 A -z 2 B) = 0 (4) in the closed set (z 1 , z 2 ) ∈ C 2 , |z 1 | ≤ 1, |z 2 | ≤ 1 Proof:
The proof is very similar to the one given in [START_REF] Pandolfi | Exponential stability of 2-D systems[END_REF] (and inspired by [START_REF]Doubly-indexed dynamical systems: State-space models and structural properties[END_REF]) only the final steps are different. The idea of the proof can be easily explained following this path:

(ES) ⇒ (AS) ⇔ (4) ⇒ (ES)
Indeed from Proposition 2.1, exponential stability (ES) implies asymptotic stability (AS). From the work of Fornasini-Marchesini [START_REF]Doubly-indexed dynamical systems: State-space models and structural properties[END_REF], we know that asymptotic stability is equivalent to the characteristic polynomial condition (4). We still need to prove that if (4) is verified then the system is exponentially stable. Note that this will also prove that exponential and asymptotic stability are equivalent in the linear case.

To do so, we need to explicitly find the solutions of the linear system [START_REF] Shanks | Stability and synthesis of twodimensional recursive filters[END_REF]. Following Pandolfi's work in [START_REF] Pandolfi | Exponential stability of 2-D systems[END_REF], we know that the solutions of system (1) can be explicitly computed using a (z, ζ)-transform of x(i, j):

x(i, j) = i r=0 E(i -r, j)Φ(r, 0) + j s=0 E(i, j -s)Ψ(0, s) (5)
where E(i, j), Φ(i, j) and Ψ(i, j) are defined by their (z, ζ)transform.

E(z, ζ) = I - A z - B ζ -1 Φ(z, ζ) = I - A z ϕ(z) Ψ(z, ζ) = I - B ζ ψ(ζ).
Let us also give the expression of Φ (Ψ is analogous, see [START_REF] Pandolfi | Exponential stability of 2-D systems[END_REF] for details). Φ(i, j) = Φ 1 (i, j) -AΦ 2 (i, j) with

Φ 1 (i, j) = ϕ(i), i ≥ 0, j = 0 0, otherwise, Φ 2 (i, j) = ϕ(i -1), i ≥ 1, j = 0 0, otherwise.
From inequality [START_REF] Pandolfi | Exponential stability of 2-D systems[END_REF] and the definitions of Φ and Ψ, we can derive the following bound:

x(i, j) ≤ i r=0 E(i -r, j) ϕ(r)

+ A i r=0 E(i -r, j) ϕ(r -1) + j s=0 E(i, j -s) ψ(s) + B j s=0 E(i, j -s) ψ(s -1) . (6)
As shown in [START_REF]Doubly-indexed dynamical systems: State-space models and structural properties[END_REF], [START_REF] Pandolfi | Exponential stability of 2-D systems[END_REF], if condition ( 4) is verified then there exists a number α ∈ (0, 1) such that the series

∞ r=0 ∞ s=0 z -r ζ -s E(r, s) is absolutely convergent for |z| > α, |ζ| > α. Choosing α < q < 1, then ∞ r=0 ∞ s=0 q -r q -s E(r, s) < M.
As we are dealing only with positive terms, the following inequality holds:

q -(i+j) E(i, j) ≤ ∞ r=0 ∞ s=0 q -r q -s E(r, s) , so that E(i, j) ≤ M q i+j .
Therefore we can bound the different terms in (6) as follows:

i r=0 E(i -r, j) ϕ(r) ≤ M i r=0 q i+j-r ϕ(r) i r=0 E(i -r, j) ϕ(r -1) = i-1 r=0 E(i -r -1, j) ϕ(r) ≤ M i-1 r=0 q i+j-r-1 ϕ(r) = M 1 q i-1 r=0 q i+j-r ϕ(r)
and similarly for the other two terms in [START_REF] Goodman | Some stability properties of two dimensional linear shiftinvariant digital filters[END_REF]. Playing with the indices and the constants, we can easily show that there exists a constant M such that

x(i, j) ≤ M i r=0 ϕ(r) q r+1 + j s=0 ψ(s) q s+1 q i+j
Thus, (4) implies ES which concludes the proof. Remark 2: The attentive reader could notice that Pandolfi in [START_REF] Pandolfi | Exponential stability of 2-D systems[END_REF] showed that his definition of ES is equivalent to the criterion (4). We have just proved that using our definition of ES, we also have equivalence with (4). However we pointed out earlier that both definitions are not equivalent (see Remark 1). The problem lies in the fact that Pandolfi wrongly assumed that his definition of ES implies AS: this is not true. But as indirectly pointed out several times in this paper, he did all the hard work and his contribution to our paper is invaluable.

III. CONCLUSION

In this short paper, we wanted to complete the results introduced in a recently published work on exponential stability [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF]. Whereas [START_REF] Yeganefar | Lyapunov theory for 2D nonlinear roesser models: application to asymptotic and exponential stability[END_REF] was focused on nonlinear systems, this paper deals with linear systems and deals with several questions left unanswered. Indeed, by introducing a new definition of exponential stability, it was important to show that this definition is in coherence with the huge literature in the linear case. This is done by showing, first, that exponential stability and asymptotic stability are equivalent in the linear case, second, that a necessary and sufficient condition for exponential stability is given by the well-known characteristic polynomial condition. Whether this work impacts the definition of exponential stability given for repetitive systems remains to be investigated.

  Nima Yeganefar and Olivier Bachelier are with the University of Poitiers, LIAS, Bâtiment B25, 2 rue Pierre Brousse, B.P. 633, 86022 Poitiers Cedex, France, surname.name@univ-poitiers.fr Emmanuel Moulay is with the University of Poitiers, Xlim (UMR-CNRS 6172), 11 Bvd Marie et Pierre Curie -BP 30179, 86962 Futuroscope Chasseneuil Cedex, France, emmanuel.moulay@univ-poitiers.fr Nader Yeganefar is with University of Provence, CMI (UMR 6632), Technopole Chateau-Gombert, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13, France nader.yeganefar@cmi.univ-mrs.fr