Allison Agus

Julien Planchais

Harry Sokol
email: harry.sokol@aphp.fr

Affiliations

Harry Pr

Sokol

Gut microbiota regulation of tryptophan metabolism in health and disease

Keywords: tryptophan metabolism, gut microbiota, AhR, serotonin, kynurenine, IBD, Crohn's disease, ulcerative colitis, autism

come

Introduction

The intestine is a complex ecosystem harboring a dense and diverse microbial community called the gut microbiota that co-evolved with the host to develop a mutualistic relationship.

It is increasingly obvious that loss of the fragile equilibrium within this complex ecosystem, termed dysbiosis, is implicated in numerous human diseases. The intestinal microbiota has an important impact on several key physiological host functions, including metabolic and nutritional homeostasis, immune system maturation and stimulation and even brain activity. These effects are mediated by direct cell-to-cell interactions and by metabolites that are either produced by the microbes or derived from the transformation of environmental or host molecules. The gut microbiota is considered a virtual endocrine organ, producing molecules that are able to interact with the host physiology and trigger responses at the local and distant levels [START_REF] Zhang | Microbial metabolism of dietary components to bioactive metabolites: opportunities for new therapeutic interventions[END_REF]. Any perturbation in host-microbiota crosstalk can be an initiating or reinforcing factor in disease pathogenesis. A large array of metabolites drives the crosstalk between the host and its microbiome. The three currently most studied categories of metabolites involved in host-microbiota interactions are : (i) short-chain fatty acids (SCFAs), produced by bacteria from the fermentation of fibers; (ii) bile acids produced in the liver and transformed by the gut microbiota before re-impacting the host; and (iii) tryptophan (Trp) metabolites, which are the topic of this review [START_REF] Blacher | Microbiome-Modulated Metabolites at the Interface of Host Immunity[END_REF].

In this article, we review the most recent insights regarding the role of the gut microbiota in Trp metabolism, with a focus on the consequences on both physiology and diseases. In addition to the direct transformation of Trp into bioactive molecules by the gut microbiota, we discuss the way in which the gut microbiota controls host Trp metabolism in the gut.

Tryptophan: origin and production

Trp is an essential aromatic amino acid composed of a β carbon connected to the 3-position of an indole group. Of the 20 common canonical amino acids, Trp is the largest by molecular weight. Although Trp is the least abundant amino acid in protein and cells, it is a biosynthetic precursor of a large number of microbial [START_REF] Alkhalaf | Biosynthetic Manipulation of Tryptophan in Bacteria: Pathways and Mechanisms[END_REF] and host metabolites.

The ability of specific bacteria to produce Trp has been known for a century and has been significantly exploited in industry. Although some members of the bacterial microbiota, such as Escherichia coli, are able to produce Trp, no data support the significant contribution of bacteria-derived Trp in human physiology. Since Trp is not produced by animal cells, humans rely on exogenous, mostly dietary, intake. Common natural food sources of Trp include oats, bananas, dried prunes, milk, tuna fish, cheese, bread, poultry, peanuts and chocolate. The World Health Organization set the recommended Trp intake to 4 mg/kg/day, and to date, no adverse effects of excess Trp in the diet have been reported.

Intestinal tryptophan metabolism: pathways and physiology

Trp metabolism follows three major pathways in the gastrointestinal tract (Figure 1): (i) the direct transformation of Trp into several molecules, including ligands of the aryl hydrocarbon receptor (AhR), by the gut microbiota [START_REF] Zelante | Tryptophan catabolites from microbiota engage aryl hydrocarbon receptor and balance mucosal reactivity via interleukin-22[END_REF]; (ii) the kynurenine pathway (KP) in both immune and epithelial cells via indoleamine 2,3-dioxygenase 1 (IDO1) [START_REF] Clarke | The microbiome-gut-brain axis during early life regulates the hippocampal serotonergic system in a sex-dependent manner[END_REF]; and (iii) the serotonin (5-HT) production pathway in enterochromaffin cells via Trp hydroxylase 1 (TpH1) [START_REF] Yano | Indigenous bacteria from the gut microbiota regulate host serotonin biosynthesis[END_REF].

Direct metabolism of Trp by microorganisms

Trp metabolism in the gut includes the direct transformation of Trp by intestinal microorganisms into several molecules, such as indole and its derivatives. Many indole derivatives, such as indole-3-aldehyde (IAld), indole-3-acid-acetic (IAA), indole-3-propionic acid (IPA), indole-3acetaldehyde (IAAld), and indoleacrylic acid (IA), are ligands for AhR [START_REF] Alexeev | Microbiota-Derived Indole Metabolites Promote Human and Murine Intestinal Homeostasis through Regulation of Interleukin-10 Receptor[END_REF][START_REF] Hubbard | Indole and Tryptophan Metabolism: Endogenous and Dietary Routes to Ah Receptor Activation[END_REF]. AhR signaling is considered a key component of the immune response at barrier sites and is thus crucial for intestinal homeostasis by acting on epithelial renewal, barrier integrity and many immune cell types, such as intraepithelial lymphocytes (IELs), Th17 cells, innate lymphoid cells (ILCs), macrophages, dendritic cells (DCs) and neutrophils [START_REF] Lamas | Aryl hydrocarbon receptor and intestinal immunity[END_REF]. AhR is directly activated by dietary molecules and xenobiotics. Additionally, many AhR ligands are processed and inactivated by cytochrome p450 family proteins, such as Cyp1A1, which is a direct AhR transcriptional target constituting a feedback loop for AhR signaling [START_REF] Schiering | Feedback Control of AHR Signaling Regulates Intestinal Immunity[END_REF].

However, the role of microbial metabolism is preponderant in intestinal AhR activity. Indeed, the intestinal contents of germ-free or dysbiotic mice are deficient for AhR agonists [START_REF] Lamas | CARD9 impacts colitis by altering gut microbiota metabolism of tryptophan into aryl hydrocarbon receptor ligands[END_REF]. Only a few commensal species able to produce AhR ligands, such as Peptostreptococcus russellii [START_REF] Wlodarska | Indoleacrylic Acid Produced by Commensal Peptostreptococcus Species Suppresses Inflammation[END_REF] and Lactobacillus ssp [START_REF] Lamas | CARD9 impacts colitis by altering gut microbiota metabolism of tryptophan into aryl hydrocarbon receptor ligands[END_REF][START_REF] Zelante | Tryptophan catabolites from microbiota engage aryl hydrocarbon receptor and balance mucosal reactivity via interleukin-22[END_REF], have been characterized, and many likely remain to be discovered. Trp metabolizing pathways have been identified in some members of the human gut microbiota such as Clostridium sporogenes which is able to decarboxylate Trp leading to the production of the neurotransmitter tryptamine [START_REF] Williams | Discovery and characterization of gut microbiota decarboxylases that can produce the neurotransmitter tryptamine[END_REF]. In addition, oxidative and reductive pathways have also been described in this species and lead to the production of indoleacetic acid (IAA) and indolepropionic acid (IPA), two Trp metabolites known to affect intestinal permeability and host immunity [START_REF] Dodd | A gut bacterial pathway metabolizes aromatic amino acids into nine circulating metabolites[END_REF][START_REF] Galligan | Beneficial actions of microbiota-derived tryptophan metabolites. Neurogastroenterol[END_REF][START_REF] Lamas | CARD9 impacts colitis by altering gut microbiota metabolism of tryptophan into aryl hydrocarbon receptor ligands[END_REF] (Figure 2).

Trp and indole active transporters have been identified in E. coli. Tryptophanase, which converts Trp into indole, is expressed in E. coli and Lactobacilli, but the precise microbial enzymatic pathways involved in further indole processing as well as their existence and activity in other commensal species, have yet to be described [START_REF] Hubbard | Indole and Tryptophan Metabolism: Endogenous and Dietary Routes to Ah Receptor Activation[END_REF]. Indole is also an interspecies signaling molecule that is able to control aspects of bacterial physiology such as antibiotic resistance, sporulation, and biofilm formation. In non-indole-producing bacteria, indole and its derivatives notably inhibit quorum sensing and modulate virulence factors [START_REF] Lee | Roles of Indole as an Interspecies and Interkingdom Signaling Molecule[END_REF]. However, the importance of these complex phenomena in the gut ecosystem has not yet specifically been addressed.

Kynurenine pathway

Trp metabolism through the KP in the gut is mediated by the rate-limiting enzyme IDO1 and leads to the production of kynurenine (Kyn) and downstream products such as quinolinic acid (QA), niacin, nicotinamide adenine dinucleotide (NAD) and kynurenic acid (Kna) [START_REF] Cervenka | Kynurenines: Tryptophan's metabolites in exercise, inflammation, and mental health[END_REF][START_REF] Kennedy | Kynurenine pathway metabolism and the microbiota-gut-brain axis[END_REF]. The key role of the gut microbiota in stimulating IDO1 activity has been clearly demonstrated, notably in germ-free and antibiotic-treated mice. KP end products are implicated in the regulation of a number of host biological processes involving neurotransmission, inflammation and the immune response. Moreover, some metabolites appear to exert specific effects in the gut. This is the case for Kna, whose concentration increases gradually along the gastrointestinal tract. Kna exhibits mucosal protective and immunoregulatory effects, probably through its G protein-coupled receptor, GPR35, which is mostly expressed in epithelial and immune cells [START_REF] Gao | Impact of the Gut Microbiota on Intestinal Immunity Mediated by Tryptophan Metabolism[END_REF]. Two other enzymes, Trp 2,3-dioxygenase (TDO) and IDO2, metabolize Trp to Kyn, but these enzymes are not expressed in the gut and are not discussed here. In addition, several intestinal bacteria encode enzymes homologous to those of the eukaryotic KP and are thus also able to produce Kyn and downstream metabolites such as 3-hydroxyanthranilic acid (Vujkovic-Cvijin et al., 2013) which has neurotoxic effects [START_REF] O'farrell | Stress-related regulation of the kynurenine pathway: Relevance to neuropsychiatric and degenerative disorders[END_REF].

Serotonin pathway

The neurotransmitter 5-HT is produced in the brain through the Trp hydroxylase 2 enzyme (TpH2), where it plays an important role. However, more than 90% of the body's 5-HT is produced in the gut and particularly in enterochromaffin cells (ECs), a specialized subtype of intestinal epithelial cell. This process occurs through Trp hydroxylase 1 enzyme (TpH1) that produces 5-hydroxytryptophan (5-HTP), which is further metabolized into 5-HT. Under physiological conditions, peripheral 5-HT does not cross the blood-brain barrier.

Peripheral 5-HT triggers numerous functions in the gastrointestinal tract and is implicated in a wide number of human physiological functions by activating specific 5-HT receptors [START_REF] Mawe | Serotonin Signaling in the Gastrointestinal Tract: Functions, dysfunctions, and therapeutic targets[END_REF]. Specifically, 5-HT is an important gastrointestinal signaling molecule that conveys signals from the gut to intrinsic or extrinsic neurons and influences intestinal peristalsis and motility, secretion, vasodilatation and the absorption of nutrients. In addition, the serotonin-selective reuptake transporter (SERT, encoded by SLC6A4 gene), expressed in the apical and basolateral membrane of intestinal epithelial cells, acts as a sponge to remove 5-HT from the interstitial space after production by ECs. This pivotal molecule involved in the local regulation of 5-HT availability is also responsible for 5-HT reuptake in the brain.

The gut microbiota is a major actor in intestinal 5-HT production [START_REF] Yano | Indigenous bacteria from the gut microbiota regulate host serotonin biosynthesis[END_REF]. Its role has been demonstrated in germ-free mice that exhibit impaired 5-HT production in the colon (but not in the small intestine) and low 5-HT concentrations in the blood. The mechanisms by which the gut microbiota modulates 5-HT production are not fully understood, but the role of SCFAs in the stimulation of TpH1 expression has been suggested [START_REF] Reigstad | Gut microbes promote colonic serotonin production through an effect of short-chain fatty acids on enterochromaffin cells[END_REF] Moreover some secondary bile acids, such as deoxycholate produced by microbial biotransformation of cholate can also stimulate 5-HT biosynthesis [START_REF] Yano | Indigenous bacteria from the gut microbiota regulate host serotonin biosynthesis[END_REF].

Perturbations in tryptophan metabolism

In the last few decades, the role of the gut microbiota has been suggested in many diseases associated with Western lifestyles, such as inflammatory bowel diseases (IBD, regrouping Crohn's disease and ulcerative colitis), irritable bowel syndrome (IBS), metabolic syndrome and associated complications (diabetes, obesity, nonalcoholic fatty liver disease (NAFLD), insulin resistance, and atherosclerosis), and neuropsychiatric traits (notably anxiety, depression and autism). Many of these diseases are also impacted by end products of Trp metabolism (Figure 3), suggesting that the effects of the microbiota are at least partially mediated by impaired Trp metabolism.

IBD and intestinal immunity

The role of an altered gut microbiota is clearly demonstrated in IBD, and several seminal publications have highlighted alterations in gut Trp metabolism with potential links to intestinal microorganisms. We recently uncovered the decreased production of AhR ligands by the microbiota of IBD patients, which was influenced by genetic factors [START_REF] Lamas | CARD9 impacts colitis by altering gut microbiota metabolism of tryptophan into aryl hydrocarbon receptor ligands[END_REF], and AhR expression in intestinal tissues was decreased compared with that in healthy subjects [START_REF] Monteleone | Aryl hydrocarbon receptor-induced signals up-regulate IL-22 production and inhibit inflammation in the gastrointestinal tract[END_REF]. Moreover, IPA is decreased in the serum of patients with ulcerative colitis (UC) [START_REF] Alexeev | Microbiota-Derived Indole Metabolites Promote Human and Murine Intestinal Homeostasis through Regulation of Interleukin-10 Receptor[END_REF]. Overactivation of IDO1 is observed both locally in the gut [START_REF] Lamas | CARD9 impacts colitis by altering gut microbiota metabolism of tryptophan into aryl hydrocarbon receptor ligands[END_REF] and systemically, likely reflecting immune system activation. This hypothesis is supported by higher IDO1 activity in active compared to nonactive IBD patients and by the negative correlation between serum levels of Trp and C-reactive protein, a commonly used biomarker increased in response to inflammation [START_REF] Nikolaus | Increased Tryptophan Metabolism Is Associated With Activity of Inflammatory Bowel Diseases[END_REF]. The state of 5-HT pathway activation in IBD is controversial. Increased expression of the rate-limiting enzyme TpH1 and elevated intestinal 5-HT levels have been reported in Crohn's disease, while most studies have found the opposite in UC [START_REF] Manocha | Serotonin and GI Disorders: An Update on Clinical and Experimental Studies[END_REF]. These alterations in Trp metabolism have a potential role in IBD pathogenesis.

Murine studies have also shown that AhR deficiency increases the severity of mice to experimental colitis that is driven either by T-cell transfer or chemically via administration of Dextran sulfate sodium (DSS). In these models, the AhR deficiency drives colitis partially through the altered production of IL-22, a cytokine with a well-known effect on intestinal homeostasis [START_REF] Qiu | The Aryl Hydrocarbon Receptor Regulates Gut Immunity through Modulation of Innate Lymphoid Cells[END_REF][START_REF] Zelante | Tryptophan catabolites from microbiota engage aryl hydrocarbon receptor and balance mucosal reactivity via interleukin-22[END_REF]. Our group demonstrated that the dysbiotic gut microbiota of mice deficient in caspase recruitment domain 9 (Card9), an IBD susceptibility gene, fails to catalyze Trp into AhR ligands, leading to reduced IL-22 release and ultimately to higher susceptibility of Card9 -/-mice to DSS-induced colitis [START_REF] Lamas | CARD9 impacts colitis by altering gut microbiota metabolism of tryptophan into aryl hydrocarbon receptor ligands[END_REF]. Some functional relevance is also seen in humans, as pharmacological activation of AhR reduces the production of the pro-inflammatory cytokine IFNɣ and increases the production of IL-22 in lamina propria mononuclear cells from IBD patients [START_REF] Monteleone | Aryl hydrocarbon receptor-induced signals up-regulate IL-22 production and inhibit inflammation in the gastrointestinal tract[END_REF]. Additionally, as observed in humans with UC, IPA and indole are decreased in serum of mice with DSS-induced colitis with additional evidence indicating that oral administration of IPA has protective properties in this model system [START_REF] Alexeev | Microbiota-Derived Indole Metabolites Promote Human and Murine Intestinal Homeostasis through Regulation of Interleukin-10 Receptor[END_REF].

Alterations in the KP may also be mechanistically involved in IBD pathogenesis. IDO1 -/-mice are more susceptible to colitis, demonstrating that IDO1 is a negative regulator of intestinal inflammation. The pathological damages associated with IDO1 deficiency are partially due to the activation of proinflammatory cytokines and a decreased number of CD4+ Foxp3+ regulatory T cells in the colon. However, the precise mechanisms and metabolites involved remain unknown [START_REF] Takamatsu | IDO1 Plays an Immunosuppressive Role in 2,4,6-Trinitrobenzene Sulfate-Induced Colitis in Mice[END_REF]. Kyn is known to be an AhR agonist, but the concentration required to elicit reporter AhR activity in a hepatoma cell line casts doubt on its relevance as an AhR activator under physiological conditions [START_REF] Hubbard | Indole and Tryptophan Metabolism: Endogenous and Dietary Routes to Ah Receptor Activation[END_REF]. Alterations to downstream metabolic pathways leading to deficiency in anti-inflammatory metabolites, such as Kna, might be involved, but this remains to be demonstrated. In the context of IBD, abnormal signals from a dysbiotic microbiota may be a driver of a skewed KP.

The severity of chemically induced colitis is attenuated in TpH1 -/-mice and in mice treated with the 5-HT synthesis inhibitor parachlorophenylalanine, suggesting that 5-HT worsens intestinal inflammation [START_REF] Ghia | Serotonin has a key role in pathogenesis of experimental colitis[END_REF]. Moreover, deletion of SERT, leading to increased 5-HT availability, induces the exacerbation of experimental colitis [START_REF] Spohn | Non-conventional features of peripheral serotonin signaling[END_REF]. These proinflammatory effects might be partly driven by the activation of 5-HT7 receptors on DCs. However, new clues suggest that 5-HT also exerts anti-inflammatory effects by acting on the 5-HT4 receptor, with positive consequences on intestinal epithelial cell barrier functions [START_REF] Spohn | Non-conventional features of peripheral serotonin signaling[END_REF].

Taken together, these data suggest that the alterations in Trp metabolism observed in IBD might have an active role in disease pathogenesis. The involvement of the microbiota is obvious in terms of the impaired ability of these microorganisms to produce AhR agonists but might also account for the exacerbated local activation of IDO and TpH1 that occurs under the direct influence of the microbiota under physiological conditions.

Irritable bowel syndrome

Although the etiology of IBS is largely unknown, the role of a dysbiotic gut microbiota in its pathogenesis is suspected, at least for some subgroups of patients [START_REF] Fan | Close association between intestinal microbiota and irritable bowel syndrome[END_REF][START_REF] Hyland | Microbiota-host interactions in irritable bowel syndrome: Epithelial barrier, immune regulation and brain-gut interactions[END_REF]. Additionally, there might be connections with impaired Trp metabolism. Kyn is increased in the serum of IBS patients [START_REF] Clarke | A Distinct Profile of Tryptophan Metabolism along the Kynurenine Pathway Downstream of Toll-Like Receptor Activation in Irritable Bowel Syndrome[END_REF], and peripheral IDO1 activity is positively correlated with IBS severity [START_REF] Fitzgerald | Tryptophan catabolism in females with irritable bowel syndrome: relationship to interferon-gamma, severity of symptoms and psychiatric co-morbidity[END_REF]. Alterations in gut motility, one of the key features in IBS, are linked to 5-HT dysmetabolism. TpH1 and SERT expression levels were found to be decreased in rectal biopsies from IBS patients compared to those from healthy controls [START_REF] Kerckhoffs | SERT and TPH-1 mRNA expression are reduced in irritable bowel syndrome patients regardless of visceral sensitivity state in large intestine[END_REF]. In addition, 5-HT colon contents are decreased and increased in constipation-and diarrhea-predominant IBS, respectively [START_REF] Manocha | Serotonin and GI Disorders: An Update on Clinical and Experimental Studies[END_REF]. The pleiotropic effects of 5-HT are related to the diversity of its receptors that are able to trigger specific functions in specific organs. 5-HT3 and 5 -HT4 subtypes, which are the most expressed in gastrointestinal tract, link 5-HT to visceral nociception and motility disorders. The role of 5-HT has already exploited as a therapeutic target with the use of 5-HT3 receptor antagonists and 5-HT4 receptor agonists that showed some efficacy in diarrhea-and constipation-predominant IBS, respectively [START_REF] Binienda | Efficacy and safety of serotonin receptor ligands in the treatment of irritable bowel syndrome: a review[END_REF].

However, perturbed central serotonin action -which is also potentially modulated by gut microbiota (Figure 1)-may participate in IBS pathogenesis as well [START_REF] Stasi | Serotonin receptors and their role in the pathophysiology and therapy of irritable bowel syndrome[END_REF]. The effects of the gut microbiota on 5-HT production and gut motility have been demonstrated in mice [START_REF] Yano | Indigenous bacteria from the gut microbiota regulate host serotonin biosynthesis[END_REF] and suggest that IBS pathogenesis is partly related to the dysfunctional control of 5-HT production by the microbiota.

Metabolic syndrome and obesity

In human patients with metabolic syndrome, overactivation of IDO1 has been reported with increased serum Kyn levels and a correlation between the Kyn/Trp ratio and obesity, metabolic syndrome, BMI and blood TG [START_REF] Mallmann | Dysregulation of Tryptophan Catabolism in Metabolic Syndrome[END_REF]. Increased gene expression of IDO1, and of downstream enzymes of the KP such as kynureninase (KYNU), kynurenine aminotransferase (KAT) and kynurenine 3-monooxygenase (KMO) (Figure 2) has been observed in the adipose tissues of obese patients, suggesting the local activation of IDO1 [START_REF] Favennec | The kynurenine pathway is activated in human obesity and shifted toward kynurenine monooxygenase activation[END_REF]. However, circulating 5-HT levels are decreased in metabolic syndrome and are negatively correlated with BMI and body fat [START_REF] Hodge | Obesity, Whole Blood Serotonin and Sex Differences in Healthy Volunteers[END_REF]. Several indole derivatives produced via Trp transformation by the microbiota may have a role in metabolic syndrome pathogenesis. Indole itself has been shown to stimulate enteroendocrine L-cells to produce glucagon-like peptide-1 (GLP-1), an incretin stimulating the secretion of insulin by pancreatic betacells. This mechanism involves the rapid inhibition of voltage-gated K + channels stimulating GLP-1 secretion but is controlled by longer-term effects on ATP synthesis inhibition, reducing GLP-1 secretion [START_REF] Chimerel | Bacterial metabolite indole modulates incretin secretion from intestinal enteroendocrine L cells[END_REF]. Indole is also absorbed and metabolized to indoxyl sulfate in the liver. During kidney failure, this metabolite accumulates, and its proinflammatory and oxidant effects are implicated in the pathogenesis of atherosclerosis, arteriosclerosis, congestive heart failure and other cardiovascular complications that are particularly over-represented in patients with chronic kidney failure [START_REF] Hung | Indoxyl Sulfate: A Novel Cardiovascular Risk Factor in Chronic Kidney Disease[END_REF]. The role of indoxyl sulfate in subjects with normal kidney function remains to be determined. The KP has also been implicated in atherosclerosis. In a mouse model, IDO1 deficiency reduced the development of atherosclerosis lesions through the dysregulation of IL-10 production, a phenotype reversed by the administration of Kna. In humans, high Kna levels are correlated with an unstable plaque phenotype [START_REF] Metghalchi | Indoleamine 2,3-Dioxygenase Fine-Tunes Immune Homeostasis in Atherosclerosis and Colitis through Repression of Interleukin-10 Production[END_REF]. The low-grade chronic inflammation that has been described in these pathologies may contribute to IDO1 activation. Overactivation of the KP is also likely to participate in the onset of insulin resistance in low-grade inflammatory situations such as obesity, depression, hepatitis C virus (HCV) infection, and cardiovascular diseases. Human and experimental data suggest that xanthurenic acid and other products of the KP have deleterious effects on insulin production and release and consequences on target tissues [START_REF] Oxenkrug | Insulin Resistance and Dysregulation of Tryptophan-Kynurenine and Kynurenine-Nicotinamide Adenine Dinucleotide Metabolic Pathways[END_REF].

Serotonin produced in the brain induces satiety, but gut-derived 5-HT produced under the direct influence of the microbiota does not cross the blood-brain barrier. However, Trp and the direct 5-HT precursor 5-HTP do cross the blood-brain barrier and thus indirectly modulate central 5-HT production and function. Moreover, peripheral 5-HT impacts host metabolism independent of any central effect: gut-derived 5-HT is able to induce hypophagia and satiety [START_REF] Voigt | Serotonin controlling feeding and satiety[END_REF], its level increases during fasting and stimulates lipolysis in adipose tissue and gluconeogenesis in hepatocytes, favoring impaired glycemic control [START_REF] Sumara | Gut-derived serotonin is a multifunctional determinant to fasting adaptation[END_REF]. Subsequently, mice with genetic or chemical ablation of TpH1 given a high-fat diet are protected against obesity, insulin resistance and Non-alcoholic fatty liver disease (NAFLD) by a mechanism involving greater energy expenditure by thermogenic brown adipose tissue [START_REF] Crane | Inhibiting peripheral serotonin synthesis reduces obesity and metabolic dysfunction by promoting brown adipose tissue thermogenesis[END_REF]. However, these results might not be applicable to adult humans in whom the abundance of brown adipose tissue is low and decreases with age. In addition, obesity in humans has been associated with decreased peripheral 5-HT, suggesting a complex role in pathogenesis [START_REF] Hodge | Obesity, Whole Blood Serotonin and Sex Differences in Healthy Volunteers[END_REF]. The role of AhR has been investigated in metabolic syndrome using mouse models, but no clear conclusion has been reached. This might be related to the multiple effects of AhR, which is expressed in various cell types (enterocytes, hepatocytes, and immune cells) involved in metabolic syndrome pathogenesis.

Infectious diseases

The production of Trp-derived AhR agonists, such as IAld, by the gut microbiota plays an important role in the protection against mucosal candidiasis, a fungal infection caused by Candida species. The underlying mechanisms involve the production of IL-22, a key cytokine in colonization resistance against fungi [START_REF] Zelante | Tryptophan catabolites from microbiota engage aryl hydrocarbon receptor and balance mucosal reactivity via interleukin-22[END_REF]. Similarly, the degradation of AhR ligands in response to Cyp1A1 dysregulation as well as the deletion of AhR lead to increased susceptibility to Citrobacter rodentium infection [START_REF] Qiu | The Aryl Hydrocarbon Receptor Regulates Gut Immunity through Modulation of Innate Lymphoid Cells[END_REF][START_REF] Schiering | Feedback Control of AHR Signaling Regulates Intestinal Immunity[END_REF]. Susceptibility to these two pathogens is reversed by the restoration of intestinal AhR activity, highlighting the importance of balanced Trp metabolism in intestinal homeostasis and the response to infection. Independent of the production of AhR ligands, local Trp metabolism might be an adaptive component of the interactions between the host and microorganisms. Trp auxotrophy by certain pathogens, such as the intracellular bacteria Chlamydia and parasite Leishmania, is a flaw that CD4+ T cells exploit to limit infection by overactivating IDO1, hijacking Trp to the KP and thus starving bacteria. Some bacteria, such as Mycobacterium tuberculosis, escape CD4-mediated defense by synthesizing their own Trp under stress conditions [START_REF] Zhang | Tryptophan biosynthesis protects mycobacteria from CD4 T cell-mediated killing[END_REF]. Such a mechanism has not yet been described in the gut but could be relevant for both pathogen-and microbiota-host interactions.

Neuropsychiatric disorders

The gut microbiota influences the brain and may be involved in neuropsychiatric disorders, partly by modulating circulating Trp availability. Although the blood-brain barrier is highly selective, Trp and Kyn cross it and have notable effects on the metabolism of neurotransmitters.

Kna and QA, produced in the brain from Trp or directly from Kyn, impact brain chemistry differently by acting on glutamate receptors such as the N-methyl-D-aspartate receptor (NMDAR) important in memory function. Indeed, Kna and QA act respectively by decreasing or increasing extracellular levels of glutamate [START_REF] Schwarcz | Kynurenines in the mammalian brain: when physiology meets pathology[END_REF], which has been shown to be involved in anxiety and stress-related disorders. In this context, IDO1 activation in the periphery, notably in the case of inflammation, may remotely impact these cerebral processes, although the mechanisms involved remain unclear [START_REF] Schwarcz | Kynurenines in the mammalian brain: when physiology meets pathology[END_REF]. Recently, the pathogenesis of autism spectrum disorder (ASD), one of the most serious neurodevelopmental conditions worldwide, has been suggested to involve an altered gut microbiota. Moreover, patients with ASD display altered Trp metabolism characterized by reduced plasma and urine levels of Trp [START_REF] Kałużna-Czaplińska | How important is tryptophan in human health?[END_REF], high IDO1 activity (assessed by Kyn/Trp ratio) [START_REF] Lim | Altered kynurenine pathway metabolism in autism: Implication for immune-induced glutamatergic activity[END_REF] and high 5-HT blood levels [START_REF] Muller | The serotonin system in autism spectrum disorder: from biomarker to animal models[END_REF]).

An experimental survey in mice mimicking autism syndrome reported a 50% reduction in small and large intestine mucosal 5-HT with some correlations among 5-HT levels, intestinal transit time and the abundance of certain bacterial taxa such as Blautia [START_REF] Golubeva | Microbiota-related Changes in Bile Acid & Tryptophan Metabolism are Associated with Gastrointestinal Dysfunction in a Mouse Model of Autism[END_REF]. These results are in line with the frequent constipation observed in patients with ASD. Thus, if low intestinal production of 5-HT is confirmed in humans with ASD, the origin of the increased 5-HT blood levels observed in these patients must be identified outside of the gut. In addition to 5-HT, other Trp metabolism pathways might participate. Total neuroprotective metabolites such as picolinic acid (PA), a metabolite of the KP, are decreased in ASD patients despite an excess of QA production [START_REF] Lim | Altered kynurenine pathway metabolism in autism: Implication for immune-induced glutamatergic activity[END_REF]. Moreover, AhR signaling might also be involved, as polymorphisms in the gene encoding AhR nuclear translocator (ARNT), which binds and facilitates AhR functions are associated with ASD severity [START_REF] Fujisawa | Association of Aryl Hydrocarbon Receptor-Related Gene Variants with the Severity of Autism Spectrum Disorders[END_REF].

Emerging evidence implicates the microbiome-gut-brain axis in depression as well. Decreased availability of 5-HT in the brain is a key feature in the pathogenesis of depression. In the case of IDO1 pathway overactivation, such as that occurring in chronic inflammatory diseases or in patients treated with interferon for hepatitis C, Trp is massively diverted to the production of Kyn, causing a deficiency in brain Trp and in 5-HT production, subsequently leading to depression. A similar hypothesis exists regarding depression in obesity, a condition also characterized by chronic inflammation and IDO1 activation [START_REF] Chaves Filho | IDO chronic immune activation and tryptophan metabolic pathway: A potential pathophysiological link between depression and obesity[END_REF]. A comparable effect is observed in the case of Trp deficiency due to low dietary intake, supporting the role of Trp pool depletion in depression.

Tryptophan metabolism: from a disrupted equilibrium to clinical opportunities

Since Trp metabolism is affected in pathological situations, the use of Trp and its metabolites as biomarkers to support diagnosis and prognosis and to orientate therapeutic choices is attractive. For example, plasma levels of Trp and Kna predict adverse cardiovascular outcomes in patients hospitalized for acute myocardial infarction [START_REF] Metghalchi | Indoleamine 2,3-Dioxygenase Fine-Tunes Immune Homeostasis in Atherosclerosis and Colitis through Repression of Interleukin-10 Production[END_REF]. This approach can likely be expanded and applied to the other diseases described above.

In addition to being biomarkers, the biological effects of Trp metabolites and their alterations in disease suggest that they may be therapeutic targets. This is directly achieved by the use of Trp metabolites or by targeting their receptors or by indirectly manipulating the gut microbiota. For example, the administration of Lactobacillus, which naturally produces AhR agonists, improves colitis severity in mice with genetically induced dysbiosis suggesting potential therapeutic applications in IBD [START_REF] Lamas | CARD9 impacts colitis by altering gut microbiota metabolism of tryptophan into aryl hydrocarbon receptor ligands[END_REF]. Similarly, L. reuteri through the production of the AhR agonist indole-3-lactic acid, is able to reprogram intraepithelial CD4+ T cells into immunoregulatory CD4 + CD8αα + T cells (Cervantes- [START_REF] Cervantes-Barragan | Lactobacillus reuteri induces gut intraepithelial CD4+CD8αα+ T cells[END_REF].

In situations where IDO is over-activated, decreased gastro-intestinal Trp availability may contribute to lower production of AhR agonists by the gut microbiota. This is the case in settings of intestinal inflammation, such as DSS-induced colitis in mice in which dietary Trp supplementation alleviates colitis severity through restoration of AhR ligand production by the gut microbiota [START_REF] Islam | Dietary tryptophan alleviates dextran sodium sulfate-induced colitis through aryl hydrocarbon receptor in mice[END_REF]. Similar protective effects have been reported in a porcine colitis model [START_REF] Kim | l-Tryptophan exhibits therapeutic function in a porcine model of dextran sodium sulfate (DSS)-induced colitis[END_REF]. This type of approach can be expanded to other inflammatory conditions such as multiple sclerosis, in which decreased levels of circulating AhR agonists have been observed. In an encephalomyelitis mouse model, supplementation with Trp or Trp-derived AhR agonists has been shown to potentiate IFN suppressive effects in an AhR-dependent manner and to limit CNS inflammation [START_REF] Rothhammer | Type I interferons and microbial metabolites of tryptophan modulate astrocyte activity and CNS inflammation via the aryl hydrocarbon receptor[END_REF]. The pathological role of indoxyl sulfate in kidney diseases might also be a feasible target, as suggested by a recent study showing that manipulation of the gut microbiota with diet and genetically modified bacteria controls indoxyl sulfate levels [START_REF] Devlin | Modulation of a circulating uremic solute via rational genetic manipulation of the gut microbiota[END_REF]. Manipulation of the IDO1 and 5-HT pathways in the gut using microbiota-based approaches is also attractive, but better knowledge of the microorganisms and mechanisms involved is needed for further development.

To date, a large part of Trp-metabolizing microorganisms, as well as the associated biochemical pathways, remains to be characterized. Factors complicating this task include the diversity of Trp-derived bioactive molecules, the variety of microorganisms involved in their transformation, and the partial overlap of microbial and host pathways. These impediments can be addressed with high sensitivity and/or high throughput methods that are constantly under development. Combining different methods, particularly metabolomics, with metagenomics and/or metatranscriptomics seems a particularly promising strategy to identify microbes and microbial genes involved in the modulation of Trp metabolism. Once identified, one can imagine using natural microbes, genetically engineered bacteria or, more directly, microbial products to modulate Trp metabolism in a therapeutic context. This type of intervention might be used either alone or more likely in combination to target several pathways and with concomitant modulation of Trp intake.

Concluding remarks

Trp metabolism has a central role in physiology and physiopathology. The major pathways described above, specifically the 5-HT, Kyn, and AhR pathways, are differentially impacted in diseases but remain tightly interconnected. Moreover, intestinal Trp metabolism in these pathways is directly or indirectly controlled by the microbiota (Figure 3). Trp metabolism in the gut is therefore an actionable actor from a therapeutic perspective, using either molecules targeting a specific pathway or exploiting microorganisms manipulating Trp metabolism as probiotics. However, the complexity of microbiota-host interactions and the sophistication of the diseases and models studied demands further investigation to refine targets and therapeutic interventions.

Fig. 1

 1 Fig. 1 Integrated Trp metabolism under the control of the gut microbiota in host physiology. Dietary Trp can be directly converted by gut microbiota into AhR ligands that are able to tune local and distant host functions, including immune homeostasis and barrier physiology. Gut microbiota also influence the kynurenine-producing IDO pathway, which plays a critical role in inflammatory mechanisms, immune responses, and neurobiologic functions. Peripheral production of serotonin by enterochromaffin cells is also under the influence of the gut microbiota. Gut-produced serotonin have many local effects, such as stimulating gut motility and, even if it does not cross blood-brain barrier, gut microbiota indirectly affects central serotoninergic pathways by modulating Trp and tryptamine availability.

Fig. 2

 2 Fig. 2 Pathways of Trp metabolism through the 5-HT, Kyn and Indole/AhR pathways. Overview of Trp metabolic fate through eukaryotic and bacterial pathways to major products and derivatives.

Fig. 3

 3 Fig. 3 Perturbations to Trp metabolism in diseases. The three major pathways of Trp metabolism, differentially impacted in diseases, remain tightly interconnected. Diagrams indicate repartitioning of Trp fluxes in diseases based on the available clinical data. Weights of arrows indicate strength of pathway activation. The restoration of disrupted equilibrium using molecules or probiotics represents a promising therapeutic strategy. AhR, Aryl hydrocarbon receptor; IDO1, indoleamine 2,3-dioxygenase 1; TpH1, Tryptophan hydroxylase 1; 5-HT, 5-hydroxytryptamine; CD, Crohn's disease; UC, Ulcerative colitis; IBS-C, irritable bowel syndrome with

Acknowledgements

Harry Sokol received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (ERC-2016-StG-71577) and from JPI HDHL Intestinal Microbiomics (ANR-15-HDIM-0001-1).

Declaration of interest:

Patents related to this work (HS) : EP 15306303.7 ; EP16306300.1 HS is a co-founder of Nextbiotix.


Baptiste Chomette
email: baptiste.chomette@upmc.fr

Adrien Mamou-Mani

Modal control based on direct modal parameters estimation

Modal active control is based on a state model that requires the identification of modal parameters. This identification can typically be done through a rational fraction polynomial algorithm applied in the frequency domain. This method generates numerical problems when estimating high-order models, particularly when moving from the basis of orthogonal polynomials for the modal basis. This algorithm must therefore be applied independently on multiple frequency ranges with a low order for each range. In this case, the controller design cannot be automated and requires a lot of human intervention especially to build the state space model. To address this issue, this paper presents the application of the direct modal parameters estimation algorithm for active modal control design. The identification algorithm is presented in a simplified version with only positive frequencies. Unlike other classical identification methods in the frequency domain, the direct modal parameters estimation algorithm provides a solution with a great numerical stability and allows estimating models with a higher order. Using this method, the design of the controller can be largely automated and requires a minimal of human intervention. After a theoretical presentation, the proposed method is experimentally validated by controlling the vibration modes of a suspended plate.

Introduction

In current industrial structures, severe stress conditions like high-level vibrations cause robustness problems and affect the lifetime of structures. One way to reduce vibrations is to apply active control to increase the natural damping of systems and consequently its lifetime. A review of different control techniques such as collocated control, active and passive strategies, state space approach and modal control can be found in [START_REF] Preumont | Vibration Control of Active Structures: An Introduction, Third Edition[END_REF]. There are many active control applications on different industrial structures like aircraft [START_REF] Prakash | Active vibration control of a full scale aircraft wing using a reconfigurable controller[END_REF], manipulator [START_REF] Zhang | Active Vibration Control and Coupled Vibration Analysis of a Parallel Manipulator with Multiple Flexible Links[END_REF], multi-axis systems [START_REF] Hanieh | Multi-axis vibration isolation using different active techniques of frequency reduction[END_REF] or on shell structures [START_REF] Yue | Experiments on vibration control of a piezoelectric laminated paraboloidal shell[END_REF]. The application of active modal control is less common cause of the complexity of design and of its lack of robustness. Nevertheless, there are a few applications in the industrial domain. For example, in aerospace domain to reduce damage induced by vibrations on electronic systems [START_REF] Chomette | Semi-adaptive modal control of on-board electronic boards using an identification method[END_REF][START_REF] Chomette | Damage reduction of on-board structures using piezoelectric components and active modal control-Application to a printed circuit board[END_REF], in the field of sound transmission [START_REF] Kaizuka | Active control of sound transmission using structural modal filters[END_REF] or in musical domain where active modal control has become a way to modify the sound of instruments [START_REF] Benacchio | Modal proportional and derivative state active control applied to a simplified string instrument[END_REF][START_REF] Benacchio | Active control and sound synthesis-two different ways to investigate the influence of the modal parameters of a guitar on its sound[END_REF]. Recently, a new approach to the synthesis of active modal control is introduced in [START_REF] Braghin | A new approach to the synthesis of modal control laws in active structural vibration control[END_REF]. This approach permits to easily evaluate spillover effects on non-modeling modes. Modal active control is based on a state model that requires the identification of modal parameters, frequency, damping, actuators and sensors properties (modal amplitudes). There are a lot of identification algorithms in the frequency domain which can be used to achieve it. The traditional rational fraction polynomial algorithm (RFP) [START_REF] Formenti | Parameter estimation from frequency response measurements using rational fraction polynomials (twenty years of progress)[END_REF] based on the use of orthogonal polynomials [START_REF] Forsythe | Generation And Use Of Orthogonal Polynomials For Data-Fitting With A Digital Computer[END_REF][START_REF] Van Der | Multiple Input Orthogonal Polynomial Parameter-Estimation[END_REF] is a first solution. A control procedure based on the RFP algorithm is detailed in [START_REF] Chesne | Experimental identification of smart material coupling effects in composite structures[END_REF]. This procedure is based on the identification of the product of actuator and sensor properties. Indeed, it's important to note that just this product has to be identified to establish the controller. There is no need to identify independently the actuator and sensor matrices. For example, in the case of piezoelectric sensors and actuators, the measure of the electromechanical properties of the transducers is very difficult without a dedicated system [START_REF] Porfiri | Identification of electromechanical modal parameters of linear piezoelectric structures[END_REF][START_REF] Maurini | On the identification of modal couplings and inherent capacitances of piezoelectric structures -art[END_REF]. In spite of its application, the RFP algorithm is not suitable in the case of high modal density or high model order. In this case, the identification of modal parameters is in general not optimal due to the ill-conditioned problem to estimate the power polynomial matrices from the orthogonal polynomial matrices. One approach that is used to avoid this problem consists in cutting the spectrum in several frequency ranges that need only a few modes in the identification procedure. But this solution does no permit to automate the control procedure. An alternative approach consists in building an identification algorithm based on well conditioned matrices for higher order model. Recently, the dual fitting method [START_REF] Hh | A dual fitting algorithm for modal parameters identification[END_REF] or the Direct Modal Parameters Estimation (DMPE) [START_REF] El-Kafafy | Direct calculation of modal parameters from matrix orthogonal polynomials[END_REF] propose a stable numerical solution to estimate high order models. The DMPE algorithm is introduced using complex orthogonal polynomials with both negative and positive frequencies for the generation of orthogonal polynomials as explained in [START_REF] Forsythe | Generation And Use Of Orthogonal Polynomials For Data-Fitting With A Digital Computer[END_REF]. The principle of the method is based on companion matrices introduced in ( [START_REF] Levine | Modal parameter estimation for stable filters[END_REF]). Moreover, in control procedures, the interpretation of identification results has to be automated. One way to select physical modal parameters is to use stabilization charts as explained in [START_REF] Mohanty | Automated Interpretation of Stability Plots for Analysis of a Non-Stationary Structure[END_REF] and [START_REF] Lau | Automatic modal analysis: Reality or myth?[END_REF]. In this paper, we propose a new control procedure based on the DMPE algorithm and using automated stabilization charts interpretation to design the state model of the controller. The formulation of the DMPE algorithm is simplified using real polynomials with only positive frequencies. The identification results will be then included in a control procedure. This paper is organized as follows. The identification and the control design are theoretically presented in the first section. The efficiency of the proposed method is then experimentally validated and compared to the classical procedure based on the RFP algorithm in the second section.

Identification and control design

The aim of the identification step is to determine the state model of the structure which can be described by ẋ(t) = Ax(t) + Bu(t) + Gw(t)

y(t) = Cx(t) , (1)
where A is the dynamical system matrix, B and C are the actuator and sensor matrices respectively and G is the disturbance matrix. x and y are the state and the output vectors respectively. u is the control and w the disturbance. In the case of one actuator and one disturbance point, u and w are scalar quantities. The state vector can be written for n modes

x = q q , ( 2
)
where q is the modal displacement vector. The state matrices can be written for n modes

A = 0 n×n Id n×n -diag ω 2 k n×n -diag (2ξ k ω k ) n×n 2n×2n , B = 0 n×1 Π a n×1 2n×1 , C = Π s 1×n 0 1×n 1×2n , G = 0 n×1 Π w n×1 2n×1 , (3)
where ω k and ξ k are the modal frequency and damping respectively. Π a , Π w and Π s are the actuator, sensor and disturbance vectors respectively. This model is then used to design the observer and the controller. The crucial point of the identification is the numerical stability of the procedure, that's why we propose to use the DMPE algorithm to identify the state model of the structure.

Simplified DMPE algorithm

The measured transfer function between one sensor output y and one actuator input u can be expressed in a rational form using orthogonal polynomials and for n modes

H(s i ) = y u = 2n-1 k=0 β k φ i,k (s i ) 2n k=0 α k θ i,k (s i ) , (4)
where s i = jω i is the Laplace variable evaluated at frequency ω i . φ i,k and θ i,k are the numerator and denominator orthogonal polynomials matrix associated to the coefficients β k and α k respectively for frequency i and order k. These coefficients are supposed to be estimated from a first step. More details are given in [START_REF] Forsythe | Generation And Use Of Orthogonal Polynomials For Data-Fitting With A Digital Computer[END_REF][START_REF] Van Der | Multiple Input Orthogonal Polynomial Parameter-Estimation[END_REF][START_REF] Formenti | Parameter estimation from frequency response measurements using rational fraction polynomials (twenty years of progress)[END_REF]. We remind that the zeros of the denominators are the poles of the system associated to the modal parameters, frequency and damping. The generation of complex Forsythe polynomials for negative and positive frequencies using a recurrence relation is detailed in [START_REF] Forsythe | Generation And Use Of Orthogonal Polynomials For Data-Fitting With A Digital Computer[END_REF]. The simplification of this method using real polynomials and just positive frequencies is explained in [START_REF] Formenti | Parameter estimation from frequency response measurements using rational fraction polynomials (twenty years of progress)[END_REF]. In the case of Forsythe polynomials, the real orthogonal polynomials S i,k of the numerator can be calculated for a complex Frequency Response Function (FRF) of length L using the recurrence relation

S i,k = ω i R i,k-1 -V k-1 R i,k-2 , (5)
where

V k-1 = 2 L i=1 ω i R i,k-1 R i,k-2 q i . (6)
The normalized real Forsythe polynomial is obtained from

R i,k = S i,k D k , (7)
with

D k = 2 L i=1 S 2 i,k q i 1/2 . ( 8
)
The complex polynomials P i,k are obtained from the real Forsythe polynomials using

P i,k = (j) k R i,k . (9)
Equation ( 5) can be written using ( 7)

ω i R i,k-1 = D k R i,k + V k-1 R i,k-2 . (10)
Multiply the above through by j k and with equation ( 9), the recurrence relation can be written

(jω i ) P i,k-1 = D k P i,k -V k-1 P i,k-2 (11)
and finally

P i,k = s P i,k-1 D k + V k-1 P i,k-2 D k . ( 12
)
This relation is available for the denominator using P i,k = θ i,k . The 2n poles of the system can be found by seeking the roots of the equation

2n k=0 α k θ i,k (s i ) = 0. ( 13
)
Using the polynomial fraction-matrix model and the fact that α 2n = 1, the roots of the denominator of the transfer function ( 4) can be calculated with

2n k=0 α k θ i,k (s i ) = 0 ⇔ θ i,2n = - 2n-1 k=0 α k θ i,k (s i ). ( 14
)
Equation ( 14) can be written in an equivalent matrix form

θ i,2n θ i,2n-1 • • • θ i,1 1 × 2n = θ i,2n-1 θ i,2n-2 • • • θ i,0 1 × 2n D den (15)
with

D den =         -α 2n-1 1 0 . . 0 -α 2n-2 0 1 . . 0 . . . . . . . . . . . . -α 1 0 . . 1 0 -α 0 0 . . 0 0         2n × 2n . ( 16
)
Using recurrence relation ( 12) applied for the denominator with

P i,k = θ i,k θ i,2n θ i,2n-1 • • • θ i,1 1 × 2n = θ i,2n-1 θ i,2n-2 • • • θ i,0 1 × 2n (s Z den + P den ) (17)
with

Z den =       D -1 2n 0 . . 0 0 D -1 2n-1 0 . 0 . . . . . . . . . . . . . . D -1 1       2n × 2n , P den =       0 . . . 0 V 2n-1 D -1 2n 0 . . 0 . . . . . . . . . . 0 . . V 2 D -1 2 0       2n × 2n . (18)
Equation ( 17) can be written in the form of an eigenvalue problem

D den = s Z den + P den ⇔ (D den -P den ) Z -1 den -s Id = 0. (19)
In practice, the estimation is achieved using scaled frequency to avoid numerical problems. The simplest scaling procedure is to scale all frequency to the unit interval by dividing all frequency by the highest value ω max . The eigenvalues λ k and its conjugate λ k * of the matrix G den = (D den -P den ) Z -1 den can be written using the scaled frequency w k and modal damping

ξ k λ k , λ k * = -ξ k ω k ± jω k 1 -ξ 2 k . (20)
The modal frequency ω k and modal damping ξ k can be finally obtained using

ω k = |λ k |ω max , ξ k = - Re(λ k )ω max ω k . ( 21
)

Selection of the stable poles and modal amplitude coefficients estimation

The stable poles are then automatically extracted using a stabilization chart. We remind that a stabilization chart is based on several runs of the pole identification process by using models of increasing order. Physical poles always appears at a nearly frequency whereas mathematical poles tend to scatter around the frequency range. The typical stabilization criteria are chosen as equal to 1% for the frequency and 5% for the damping between two runs at order n and n + 1. The identified transfer function can then be written in terms of stable modal parameters corresponding to physical poles

H DMP E (s) = n k=0 R k s 2 + 2sξ k ω k + ω 2 k , (22)
where R k denotes the complex modal amplitudes which can be calculated using the linear square problem evaluated at each frequency ω i . The optimal order is automatically determined using the error criterion ε between the measured (f rf m ) and the estimated (f rf e ) FRF obtained from the stable poles

ε = L i=1 ( f rf e -f rf m ) 2 L i=1 f rf m 2 . ( 23
)

From modal parameters to state space model

The transfer function between the control u applied on the actuator and the output y measured on one sensor can be written in a matrix format and without disturbance

H(s) = y u = C (sId -A) -1 B, (24)
where s denotes the Laplace variable. This transfer function can also be written using the modal parameters summing up each vibration mode

H(s) = n k=0 Π a k Π s k s 2 + 2sξ k ω k + ω 2 k , (25)
where the numerator Π a k Π s k is a real quantity. The modal frequency and damping of equation ( 25) are directly obtained using the denominator of equation [START_REF] Lau | Automatic modal analysis: Reality or myth?[END_REF] and can be used to define the dynamical system matrix A. The product value Π a k Π s k can be identify using the complex numerator R k of equation ( 22) which can be written using its modulus and phase

R k = |R k | cos(ϕ k ) + j sin(ϕ k ) , (26)
where ϕ k denotes the phase angle. In the case of a lightly damped system with a diagonal damping matrix, the phase angle can be supposed equal to ± π and the complex numerator becomes a real quantity

R k = |R k |cos(± π). (27)
The sign of the numerator is determined by the sign of the cosinus and consequently by the sign of the real part of the numerator ℜ (R k ). The numerator can then be written

R k = |R k |sign ℜ (R k ) . (28)
Finally, the product of the actuator and sensor modal coordinates can be identified using

Π a k Π s k = |R k |sign ℜ (R k ) . ( 29
)
The actuator and sensors matrices are finally obtained with

B = 0 n×1 |R k |sign ℜ (R k ) n×1 2n×1 , C = 1 1×n 0 1×n 1×2n . (30)
If the assumption about damping is not completely verified, the imaginary part of the complex numerator is reflected in a phase shift on the real modal displacement. This phase shift may have an impact on the control performances.

Controller and observer design

The modal control is classically design using a pole placement algorithm [START_REF] Valasek | Efficient Egenvalue Assignement for General Linear MIMO System[END_REF] for both the control gain K and the observer gain L used in the Luenberger observer [START_REF] David | Observing the state of a linear system[END_REF]. The control u is obtained from the estimated state vector x with u(t) = -Kx(t).

The dynamic of the controller without disturbance and including the observer is defined by

ẋ(t) = A m x(t) + B m u(t) + L y(t) -ŷ(t) ŷ(t) = C m x(t) , ( 32
)
where ŷ is the estimated output vector. The subscription m denotes the matrix used in the observer and build up using the DMPE identification.

Application

The proposed method is applied to a suspended 250 x 380 x 3.5 mm rectangular plate composed of Duralumin and presented in Figure 1(a). The structure is instrumented with one exciter HIAX19C01-8 and one collocated piezoelectric PZT sensor located behind the plate. This structure was chosen for its rich non harmonic sound properties close to a bell plate sound. The acquisition and control loop are carried out using a COALA system, presented in Figure 1(b), including a signal pre-amplifier, a 20W power amplifier and a real time control system based on the real time frame work Xenomai on a BeagleBone Black [START_REF] Piéchaud | A Lightweight C++ Framework for Real Time Active Control[END_REF].

RFP versus DMPE identification

In this section, we compare the identification procedures using the RFP and the DMPE algorithm. The plate is excited with a swept sine applied on the actuator. The output is measured on the piezoelectric sensor. The complex FRF between the actuator and the sensor is used to applied the two identification algorithms.

RFP identification

A first identification is realized using the RFP algorithm on the single frequency range 100-1735 Hz. The stabilization chart presented in Figure 2 is achieved for an order varying from n = 10 to n = 50. To make it clearer, only the stable poles in frequency and damping marked by '•' are represented on the chart. The optimal order is 24 and the identified modal parameters (frequency f k , modal damping ξ k and amplitudes R k ) are detailed in Table 1. Only three modes are identified but the algorithm do not keep the numerical stability when the order increase more than 24.

Table 1: Modal parameters identified using the RFP algorithm on the frequency range 100-1735 Hz for n=24. As explained in the introduction, a solution to identify more modes consists in cutting the spectrum in several frequency ranges that need only a few modes in the identification procedure. Results corresponding to this second identification are detailed in Table 2. Six different frequency ranges are needed to identify modes 1, 2, 3, 4, 5, 6, 7 and 8: 100-350 Hz for mode 1, 350-500 Hz for mode 2, 300-700 Hz for mode 3, 600-1000 Hz for modes 4 and 5, 1200-1745 Hz for modes 6 and 8 and finally 1370-1490 Hz for mode 7. In this case, the identification procedure can not be automated and requires a lot of adjustments.

Mode f k [Hz] ξ k [%] R k 1 148

DMPE identification

A third identification is realized using the DMPE algorithm on the single frequency range 100-1735 Hz. The stabilization chart is presented in Figure 2 ('+' denotes the stable poles). The optimal order is 47 and 8 modes are identified. The modal parameters are detailed in Table 3. We see that the numerical stability of the DMPE algorithm permits to easily increase the order of the identification on the single frequency range to include more modes than using the RFP algorithm. These results are very similar to those from the RFP algorithm applied on several frequency ranges. In this case, the identification procedure can be automated and requires a minimum number of adjustable parameters: the frequency and the order range to build the stabilization chart.

Table 3: Modal parameters identified using the DMPE algorithm on the frequency range 100-1735 Hz for n=47.

Mode f k [Hz] ξ k [%] R k 1 148.7 0.13 -2.

Controller and observer design

The controller and the observer focus on the vibration modes with a maximum amplitude located in the center of the plate: modes 1, 2, 4, 5 and 8 in Table 3. The other modes correspond to corner modes and are not taken into account in this example. The numerical FRF obtained from the reduced state space model with real modal amplitudes based on equation ( 30) is presented in Figure 3. ).

Poles 1, 2, 4, 5 and 8 whose frequencies are equal to 149, 361, 690, 839 and 1468 Hz respectively are damped with a coefficient 10 applied on the real part. The control gain K 1×2n = [0 1×n K 2 ] and the observer gain L 2n×1 = [L 1 L 2 ] t are conventionally calculated using a pole placement algorithm. The actuator matrix B m , the control and observer gains are detailed in Table 4. The poles of the uncontrolled system corresponding to the eigenvalues of A m and the poles of the controlled system associated to the eigenvalues of (A m -B m K) are shown on a pole chart in Figure 4.

Table 4: Actuator matrix, control and observer gains.

Mode Bm K2 L1 L2 1 -2.5×10 4 3.4×10 -3 1.4×10 2 -4.9×10 3 2 -5.7×10 4 -2.3×10 -3 2.0×10 2 3.1×10 3 4 -1.4×10 5 1.7×10 -2 2.1×10 2 -1.0×10 5 5 -2.5×10 5 -3.4×10 -2 4.0×10 2 3.5×10 5 8
-4.3×10 5 -3.6×10 -2 4.9×10 2 6.5×10 5

Application with a collocated swept sine excitation

In this first step, a swept sine is applied to the actuator. The uncontrolled and controlled FRF of the system are presented in Figure 5. A zoom on modes 4 and 5 for the numerical and for the experimental FRF is presented in Figure 6 and 7 respectively. Frequency f k c and damping ξ k c of the controlled system are obtained using the DMPE algorithm applied on the controlled FRF. Values are detailed in Table 5.

The frequency shift induced by the controller is very low and remains under 1.1% for the controlled modes.

The increase of damping induced by the controller should be equal to 10. The controller performances are closed to this value for modes 1, 2, 4 and 5 but significantly worse for mode 8 with a coefficient of increase only equal to 4.6. This can be explained by the value of its modal amplitudes. In the case of mode 8, the imaginary part of the modal amplitude is higher than the real part and the phase shift of the modal displacement is not properly taking into account. This error must have a significant impact on the control performances.

Application with a non-collocated percussive excitation

In a second experimental case, the controller is tested with a percussive excitation, applied with a mallet reproducing the gesture of a percussionist as indicated in Figure 1. The aim of this experiment is to validate the control system in real life context, for any unknown and non repeatable excitation, as in a musical context. The Experimental uncontrolled and controlled sensor signal y (t) using the DMPE modal model and a mallet excitation are shown in Figure 8. The effect of the controller is clearly visible on the time signal and shows the amount of damping induce by the controller. We can note that the amplitude of the second percussive excitation is slightly larger than the first one, which explains that the amplitude after control is also slightly larger. The spectrum of the sensor signal in dB for the uncontrolled and controlled system is shown in Figure 9. These results show a large decrease of the amplitude of controlled modes 1, 2, 4, 5 and 8. A subjective analysis of the authors' perceived sounds showed that the metal plate sounds like a wooden plate.

Summary

This paper deals with the use of the DMPE algorithm to design active modal control. The presented procedure is particularly efficient and requires a minimum of adjustments. Moreover, the DMPE algorithm is presented here in a simplified version with only positive frequencies and real polynomials. The procedure is experimentally validated on a suspended rectangular plate firstly with a collocated swept sine excitation and secondly with a non-collocated percussive excitation. For the identification procedure, in the case of the RFP algorithm, it is necessary to cut the spectrum in several frequency ranges with only a few modes. This procedure makes the controller design more complicated. In the case of the DMPE algorithm, a single frequency range is sufficient to identify all the modes and the modal model identification can be largely automated. It also shows that the controller performances are closely linked to the modal amplitude values. In the case of modal amplitudes with a large real part, the controller performances are close to those numerically obtained. In the case of modal amplitudes with a large imaginary part, the real modal model doesn't take into account the phase shift and this error has a negative impact on the controller performances. Future works will deal with the control of the phase shift to ensure the performances of the controller even in the case of modal amplitude with higher imaginary part.

Figure 1 :

 1 Figure 1: Studied plate (a) and control system COALA (b).

5×10 4 - 4 Figure 2 :

 442 Figure 2: Transfer function (a) and phase angle (b) of the system ( ); stabilization chart obtained from the RFP (•) and from the DMPE algorithm (+); synthesized FRF obtained from the RFP (---) and from the DMPE ( ) modal parameters.

Figure 3 :

 3 Figure 3: Transfer function (a) and phase angle (b) of the system ( ) and numerical FRF (a) and phase angle (b) obtained from the reduced state space model with real modal amplitudes ().

Figure 4 :

 4 Figure 4: Poles chart of the uncontrolled system (•) and of the controller (×).

Figure 5 :

 5 Figure 5: Experimental uncontrolled ( ) and controlled ( ) FRF (a) and phase angle (b) using the DMPE modal model and a swept sine excitation.

Figure 6 :

 6 Figure 6: Numerical uncontrolled ( ) and controlled ( ) FRF (a) and phase angle (b) using the DMPE modal model and a swept sine excitation, zoom on modes 4 and 5.

Figure 7 :

 7 Figure 7: Experimental uncontrolled ( ) and controlled ( ) FRF (a) and phase angle (b) using the DMPE modal model and a swept sine excitation, zoom on modes 4 and 5.

Figure 8 :

 8 Figure 8: Experimental uncontrolled () and controlled sensor signal ( ) using the DMPE modal model and a mallet excitation.

Figure 9 :

 9 Figure 9: Experimental uncontrolled () and controlled spectrum in dB of the sensor signal ( ) using the DMPE modal model and a mallet excitation.

Table 2 :

 2 Modal parameters identified using the RFP algorithm on several frequency ranges.

	Mode f k [Hz] ξ k [%]	R k	Frequency range Optimal order
	1	148.7	0.15	-2.7×10 4 -3.9×10 3 j	100-350 Hz	12
	2	360.8	0.08	-5.8×10 4 -5.8×10 3 j	350-500 Hz	12
	3	477.4	0.15	-1.8×10 3 -2.6×10 3 j	300-700 Hz	18
	4	690.3	0.04	-1.2×10 5 + 6.3×10 4 j	600-1000 Hz	11
	5	839.4	0.07	-2.0×10 5 + 1.5×10 5 j	600-1000 Hz	11
	6	1327.2	0.05	-1.3×10 4 + 3.5×10 4 j	1200-1745 Hz	12
	7	1382.0	0.03	-8.6×10 3 + 1.4×10 4 j	1370-1490 Hz	06
	8	1467.9	0.05	-1.3×10 5 + 4.1×10 5 j	1200-1745 Hz	12

Table 5 :

 5 Frequency and damping of the controlled system and coefficient of the damping increase.

	Mode f k c [Hz] ξ k c [%]	ξ k c ξ k
	1	147.2	1.11	8.5
	2	360.1	0.85	12.1
	4	691.5	0.47	11.7
	5	842.8	0.66	9.4
	8	1473.5	0.23	4.6
	0	5	10	15	20
			t [s]