Camille Vatier

Sophie Christin-Maitre

Corinne Vigouroux
email: corinne.vigouroux@inserm.fr

Role of insulin resistance on fertility -focus on polycystic ovary syndrome Rôle de l'insulino-résistance sur la fertilité -focus sur le syndrome des ovaires polykystiques

Keywords: insulin resistance, polycystic ovaries, infertility

Several lines of evidence show that gonadal functions and insulin sensitivity display multifaceted relationships, which extend far beyond the well-known association between polycystic ovary syndrome (PCOS), obesity, and metabolic syndrome.

In this brief review, we will summarize the main findings showing the pathophysiological role of insulin resistance in impairing reproductive functions. Extreme phenotypes of severe insulin resistance, due to primary defects in insulin receptor or to lipodystrophy syndromes, provide unique opportunities for the modeling of interactions between insulin signaling and ovarian endocrine functions. In addition, recent studies further suggest that common forms of dysfunctional adiposity, as well as altered production of adipokines, could represent important pathophysiological links between metabolic syndrome and infertility.

PCOS, insulin resistance and ovarian dysfunction

PCOS, which affects 5 to 15 % of reproductive aged women worldwide, is the most common disorder leading to female infertility [START_REF] Azziz | Polycystic ovary syndrome[END_REF][START_REF] Rosenfield | The Pathogenesis of Polycystic Ovary Syndrome (PCOS): The Hypothesis of PCOS as Functional Ovarian Hyperandrogenism Revisited[END_REF]. Although the diagnosis of PCOS refers to different phenotypes of ovarian dysfunction, ovarian hyperandrogenism is the common denominator in most patients [START_REF] Rosenfield | The Pathogenesis of Polycystic Ovary Syndrome (PCOS): The Hypothesis of PCOS as Functional Ovarian Hyperandrogenism Revisited[END_REF]. Insulin resistance, frequently associated with different clusters of comorbidities referred to as the metabolic syndrome, affects about one to two-thirds of women with PCOS. Obesity is an important risk factor for both metabolic syndrome and PCOS. However, insulin resistance is significantly more frequent in body-mass index (BMI)matched women with PCOS than without PCOS, showing that obesity is not the primary driver of insulin resistance associated with the disease, and that intrinsic defects in insulin responsiveness could contribute to PCOS [START_REF] Dunaif | Profound peripheral insulin resistance, independent of obesity, in polycystic ovary syndrome[END_REF]. In addition, the severity of insulin resistance, adjusted for BMI, is correlated with that of menstrual cycle abnormalities in women with PCOS, further stressing the pathophysiological link between insulin resistance and ovarian dysfunction [START_REF] Ezeh | Association of severity of menstrual dysfunction with hyperinsulinemia and dysglycemia in polycystic ovary syndrome[END_REF].

Insulin increases the production of androgens by the ovary, mainly in synergy with the luteinizing hormone (LH). Its effects are basically mediated through the insulin receptor, which is expressed in granulosa and theca cells [START_REF] Poretsky | The insulin-related ovarian regulatory system in health and disease[END_REF]. In accordance, the theca cell-selective knockout of insulin receptor has been shown to prevent hyperandrogenic anovulation induced by high fat diet-related insulin resistance in mice [START_REF] Wu | Obesity-induced infertility and hyperandrogenism are corrected by deletion of the insulin receptor in the ovarian theca cell[END_REF]. Insulin-growth-factor 1 (IGF1) signaling, which also promotes ovarian steroidogenesis, can be stimulated by insulin, although insulin binds the IGF1 receptor with far less affinity than its own receptor.

Nevertheless, IGF1-R mediated effects are probably important, at least in the context of severe hyperinsulinemia, as discussed below.

It seems paradoxical to observe at the same time, on the one hand, exaggerated effects of insulin on the ovary, and, on the other hand, a state of systemic insulin resistance, characterized by impaired effects of insulin on glucose metabolism, as routinely assessed by absolute or relative hyperinsulinemia with regards to glycemia. The main hypotheses to reconcile those two concomitant situations were supported by in vitro studies showing that PCOS-associated insulin resistance could be tissue-specific, and could affect metabolic, but not mitogenic or steroidogenic post-receptor signaling pathways [START_REF] Wu | Obesity-induced infertility and hyperandrogenism are corrected by deletion of the insulin receptor in the ovarian theca cell[END_REF][START_REF] Diamanti-Kandarakis | Insulin resistance and the polycystic ovary syndrome revisited: an update on mechanisms and implications[END_REF]. Selective insulin resistance, responsible for compensatory hyperinsulinemia, could thus contribute to excessive ovarian androgen production, as thecal cells are very sensitive to insulin. This has been demonstrated by in vitro culture of thecal cells from patients with polycystic ovarian syndrome [START_REF] Diamanti-Kandarakis | Insulin resistance and the polycystic ovary syndrome revisited: an update on mechanisms and implications[END_REF]. Selective insulin resistance could also explain the preserved inhibitory effects of insulin on the production of IGF-and sex-hormone-binding proteins (IGFBP1 and SHBG) by the liver, which raises the levels of bioavailable fractions of IGF-1 and testosterone, respectively, further enhancing hyperandrogenism [START_REF] Diamanti-Kandarakis | Insulin resistance and the polycystic ovary syndrome revisited: an update on mechanisms and implications[END_REF].

Ovarian dysfunctions in patients with extreme phenotypes of insulin resistance

Women with pathogenic variants in INSR, encoding the insulin receptor gene, autoantibodies directed against the insulin receptor, or with genetic or acquired forms of lipodystrophy syndromes, are typically described with severe insulin resistance and ovarian hyperandrogenism, in the absence of obesity [START_REF] Kahn | The syndromes of insulin resistance and acanthosis nigricans. Insulin-receptor disorders in man[END_REF][START_REF] Taylor | Insulin resistance associated with androgen excess in women with autoantibodies to the insulin receptor[END_REF][START_REF] Sollier | Lipodystrophic syndromes: From diagnosis to treatment[END_REF][START_REF] Musso | The long-term effect of recombinant methionyl human leptin therapy on hyperandrogenism and menstrual function in female and pituitary function in male and female hypoleptinemic lipodystrophic patients[END_REF][START_REF] Young | Type A insulin resistance syndrome revealing a novel lamin A mutation[END_REF][START_REF] Vambergue | Follow-up study of two sisters with type A syndrome of severe insulin resistance gives a new insight into PCOS pathogenesis in relation to puberty and pregnancy outcome: a case report[END_REF][START_REF] Vantyghem | Fertility and obstetrical complications in women with LMNA-related familial partial lipodystrophy[END_REF][START_REF] Joy | Prevalence of reproductive abnormalities among women with familial partial lipodystrophy[END_REF][START_REF] Keller | Lipodystrophy: an unusual diagnosis in a case of oligomenorrhea and hirsutism[END_REF][START_REF] Lungu | Insulin resistance is a sufficient basis for hyperandrogenism in lipodystrophic women with polycystic ovarian syndrome[END_REF][START_REF] Brown | Type B Insulin Resistance Masquerading as Ovarian Hyperthecosis[END_REF][START_REF] Huang-Doran | Ovarian Hyperandrogenism and Response to Gonadotropin-releasing Hormone Analogues in Primary Severe Insulin Resistance[END_REF]. However, a few studies have reported the natural history of ovarian dysfunctions and consequences on fertility associated with these rare and severe insulin resistance syndromes. Vantyghem et al. have evaluated that PCOS and infertility could affect, respectively, 54% and 28% of women with LMNA-linked familial partial lipodystrophy [START_REF] Vantyghem | Fertility and obstetrical complications in women with LMNA-related familial partial lipodystrophy[END_REF], pointing to ovarian dysfunction as a major burden of the disease in women. A recent report involved a large cohort of patients with severe insulin resistance, either due to lipodystrophy (n=185) or insulin signaling defects (n=65), or considered as idiopathic (n=29) [START_REF] Huang-Doran | Ovarian Hyperandrogenism and Response to Gonadotropin-releasing Hormone Analogues in Primary Severe Insulin Resistance[END_REF]. Serum total testosterone, which was significantly positively associated with fasting insulin, was above the normal range in 34% of the whole cohort, but was highly variable, suggesting that ovarian response to hyperinsulinemia strongly vary among individuals. However, whatever the cause of insulin resistance, the ovarian phenotype, including ovarian histopathology in 9 patients, was close to that observed in PCOS [START_REF] Huang-Doran | Ovarian Hyperandrogenism and Response to Gonadotropin-releasing Hormone Analogues in Primary Severe Insulin Resistance[END_REF]. Five patients, with different causes of extreme insulin resistance, developed ovarian tumors associated with virilization, which could be favored by long-term severe hyperinsulinemia. Importantly, testosterone was not elevated in pre-pubertal patients, showing that the pulsatile release of gonadotrophins plays a permissive role in insulin resistance-related hyperandrogenism [START_REF] Huang-Doran | Ovarian Hyperandrogenism and Response to Gonadotropin-releasing Hormone Analogues in Primary Severe Insulin Resistance[END_REF], as previously reported [START_REF] Vambergue | Follow-up study of two sisters with type A syndrome of severe insulin resistance gives a new insight into PCOS pathogenesis in relation to puberty and pregnancy outcome: a case report[END_REF]. The fact that the ovarian phenotype of patients with primary insulin receptor dysfunctions versus postreceptor insulin resistance states was highly similar, supports the hypothesis that hyperinsulinemia drives ovarian hyperandrogenism via its action on IGF1 receptors. Indeed, INSR loss-of-function mutations, which affect all tissues, and are predicted to disrupt all insulin signaling pathways, may not induce tissue-specific or selective insulin resistance. In accordance, serum concentrations of SHBG, IGFBP-1 and adiponectin are preserved, or even increased, in patients with insulin receptor defects, resulting from disruption of insulinmediated effects on the liver and adipose tissue. In contrast, these markers are decreased in patients with post-receptor insulin resistance, characterized by the preservation of some, but not all, insulin signaling pathways [START_REF] Huang-Doran | Ovarian Hyperandrogenism and Response to Gonadotropin-releasing Hormone Analogues in Primary Severe Insulin Resistance[END_REF][START_REF] Semple | Plasma adiponectin as a marker of insulin receptor dysfunction: clinical utility in severe insulin resistance[END_REF].

Dysfunctional adiposity as a driver of insulin resistance and PCOS

Several studies have suggested that, rather than obesity or insulin receptor dysfunctions per se, adipose dysfunction could contribute to the pathophysiology of PCOS in the general population. Adipose tissue, in addition to its well-known effects on whole-body energy regulation, is an important endocrine organ whose altered distribution and/or dysfunction could lead to inflammation, oxidative stress, and/or lipotoxicity, all being associated with insulin resistance. Adipokines produced by adipose tissue also display important regulatory roles on the reproductive axis [START_REF] Nikanfar | Role of adipokines in the ovarian function: Oogenesis and steroidogenesis[END_REF]. Defects in adipocyte differentiation and/or functions are the main determinants of lipodystrophy syndromes, associated with insulin resistance and PCOS features [START_REF] Sollier | Lipodystrophic syndromes: From diagnosis to treatment[END_REF][START_REF] Musso | The long-term effect of recombinant methionyl human leptin therapy on hyperandrogenism and menstrual function in female and pituitary function in male and female hypoleptinemic lipodystrophic patients[END_REF][START_REF] Vantyghem | Fertility and obstetrical complications in women with LMNA-related familial partial lipodystrophy[END_REF][START_REF] Joy | Prevalence of reproductive abnormalities among women with familial partial lipodystrophy[END_REF][START_REF] Keller | Lipodystrophy: an unusual diagnosis in a case of oligomenorrhea and hirsutism[END_REF][START_REF] Lungu | Insulin resistance is a sufficient basis for hyperandrogenism in lipodystrophic women with polycystic ovarian syndrome[END_REF][START_REF] Huang-Doran | Ovarian Hyperandrogenism and Response to Gonadotropin-releasing Hormone Analogues in Primary Severe Insulin Resistance[END_REF][START_REF] Semple | Plasma adiponectin as a marker of insulin receptor dysfunction: clinical utility in severe insulin resistance[END_REF][START_REF] Nikanfar | Role of adipokines in the ovarian function: Oogenesis and steroidogenesis[END_REF][START_REF] Brown | The Diagnosis and Management of Lipodystrophy Syndromes: A Multi-Society Practice Guideline[END_REF][START_REF] Musso | The long-term effect of recombinant methionyl human leptin therapy on hyperandrogenism and menstrual function in female and pituitary function in male and female hypoleptinemic lipodystrophic patients[END_REF][START_REF] Zammouri | Molecular and Cellular Bases of Lipodystrophy Syndromes[END_REF]. Leptin deficiency due to lipoatrophy contributes to insulin resistance, as shown by the metabolic benefit of substitutive therapy with the leptin analog metreleptin in generalized lipoatrophy [START_REF] Brown | The Diagnosis and Management of Lipodystrophy Syndromes: A Multi-Society Practice Guideline[END_REF]. Interestingly, metreleptin also improves ovarian function in women with lipodystrophy, by increasing insulin sensitivity and restoring LH pulsatility [START_REF] Musso | The long-term effect of recombinant methionyl human leptin therapy on hyperandrogenism and menstrual function in female and pituitary function in male and female hypoleptinemic lipodystrophic patients[END_REF]. Importantly, leptin deficiency by itself is not sufficient to inhibit pubertal development in patients with congenital lipodystrophy, and metreleptin therapy does not modify the onset of puberty in these patients. Therefore, the insulin-sensitizing effect of metreleptin is probably the major driver of the decrease in testosterone concentrations and the restoration of normal menses in women with lipodystrophy [START_REF] Musso | The long-term effect of recombinant methionyl human leptin therapy on hyperandrogenism and menstrual function in female and pituitary function in male and female hypoleptinemic lipodystrophic patients[END_REF]. Besides, Dumesic et al.

have recently reported that preadipocytes originating from subcutaneous adipose tissue of women with common forms of PCOS display an abnormal adipogenic behavior as compared to controls [START_REF] Dumesic | Accelerated subcutaneous abdominal stem cell adipogenesis predicts insulin sensitivity in normal-weight women with polycystic ovary syndrome[END_REF]. Expression of 11β-hydroxysteroid dehydrogenase type 1 (11β-HSD1), which convert inactive cortisone to active cortisol, is increased in subcutaneous adipose tissue of women with PCOS as compared to age-and BMI-matched women [START_REF] Li | The expression of 11β-HSDs, GR, and H6PDH in subcutaneous adipose tissue from polycystic ovary syndrome subjects[END_REF], as well as in granulosa cells [START_REF] Li | Addressing the role of 11β-hydroxysteroid dehydrogenase type 1 in the development of polycystic ovary syndrome and the putative therapeutic effects of its selective inhibition in a preclinical model[END_REF]. Furthermore, in rats, 11β-HSD1 ovarian overexpression induces PCOS, while 11β-HSD1 knockdown or pharmacological inhibition reverses PCOS, increases fertility and improves insulin sensitivity [START_REF] Li | Addressing the role of 11β-hydroxysteroid dehydrogenase type 1 in the development of polycystic ovary syndrome and the putative therapeutic effects of its selective inhibition in a preclinical model[END_REF]. Adipose tissue from women with PCOS also differentially expresses several genes involved in inflammation and lipid metabolism, as well as in adipocytokines production and responsiveness [START_REF] Chazenbalk | Abnormal expression of genes involved in inflammation, lipid metabolism, and Wnt signaling in the adipose tissue of polycystic ovary syndrome[END_REF][START_REF] Chazenbalk | Regulation of adiponectin secretion by adipocytes in the polycystic ovary syndrome: role of tumor necrosis factor-alpha[END_REF][START_REF] Juan | Increased regulated on activation, normal T-cell expressed and secreted levels and cysteine-cysteine chemokine receptor 5 upregulation in omental adipose tissue and peripheral blood mononuclear cells are associated with testosterone level and insulin resistance in polycystic ovary syndrome[END_REF]. However, studies fail to draw a integrated picture of the pathophysiological mechanisms linking metabolic and secretory functions of adipose tissue with development of PCOS.

Genetic and epigenetic contributors could lead to both insulin resistance and PCOS

As observed in severe insulin resistance syndromes, with or without lipodystrophy, and as confirmed by mendelian randomization studies, insulin resistance is probably an important determinant of PCOS [START_REF] Zhu | Causes and Consequences of Polycystic Ovary Syndrome: Insights From Mendelian Randomization[END_REF]. In accordance, all life-style interventions and treatments increasing insulin sensitivity globally decrease hyperandrogenism and improve ovulation in women with PCOS [START_REF] Azziz | Polycystic ovary syndrome[END_REF][START_REF] Rosenfield | The Pathogenesis of Polycystic Ovary Syndrome (PCOS): The Hypothesis of PCOS as Functional Ovarian Hyperandrogenism Revisited[END_REF]. However, congenital virilization leads to PCOS metabolic and reproductive features [START_REF] Rosenfield | The Pathogenesis of Polycystic Ovary Syndrome (PCOS): The Hypothesis of PCOS as Functional Ovarian Hyperandrogenism Revisited[END_REF] and a bidirectional relationship between insulin resistance and hyperandrogenism is not ruled out in common PCOS [START_REF] Azziz | Polycystic ovary syndrome[END_REF][START_REF] Rosenfield | The Pathogenesis of Polycystic Ovary Syndrome (PCOS): The Hypothesis of PCOS as Functional Ovarian Hyperandrogenism Revisited[END_REF][START_REF] Zhu | Causes and Consequences of Polycystic Ovary Syndrome: Insights From Mendelian Randomization[END_REF]. A unifying hypothesis could be that genetic and epigenetic factors could lead to both adipose tissue dysfunction, insulin resistance, and PCOS. Familial studies strongly support that PCOS is inherited as a polygenic disease in which multiple risk loci with small effects contribute to disease, a pattern similar to the genetics of type 2 diabetes [START_REF] Rosenfield | The Pathogenesis of Polycystic Ovary Syndrome (PCOS): The Hypothesis of PCOS as Functional Ovarian Hyperandrogenism Revisited[END_REF]. In line, a polygenic model of type 2 diabetes in rats was recently shown to spontaneously recapitulate not only the metabolic features, but also the reproductive characteristics of PCOS, which are developmentally programmed [START_REF] Bourgneuf | The Goto-Kakizaki rat is a spontaneous prototypical rodent model of polycystic ovary syndrome[END_REF].

Several susceptibility loci which could modulate metabolism and reproductive functions have been identified from genome wide association studies (GWAS) in humans. However, although two-third of the susceptibility to PCOS could be due to genetic factors, only a minority of the genetic loci involved has been identified [START_REF] Azziz | Polycystic ovary syndrome[END_REF]. GWAS performed in PCOS patients from China, the United States and in United Kingdom have identified a susceptibility locus near the insulin receptor INSR gene [START_REF] Welt | Genetics of Polycystic Ovary Syndrome: What is New?[END_REF]. However, the menstruation number per year, ovarian follicular number, ovarian volume, and insulin sensitivity index were not associated with any single nucleotide polymorphisms in 1,810 women with PCOS [START_REF] Hong | Relationship between the characteristic traits of polycystic ovary syndrome and susceptibility genes[END_REF].

Recent studies have highlighted the major role of epigenetic modifications, including in utero epigenetic programming of postnatal development, in the pathophysiology of PCOS [START_REF] Pugeat | Epigenetics suggests new perspectives for the treatment of polycystic ovary syndrome: Transgenerational effect of fetal exposure to Antimüllerian Hormone[END_REF][START_REF] Mimouni | Polycystic ovary syndrome is transmitted via a transgenerational epigenetic process[END_REF]. Upon Anti-Mullerian Hormone prenatal exposure (AMH), mice develop reproductive and metabolic PCOS features later in life, which are transmitted in unexposed offspring through several generations, along with hypomethylation of several genes involved in ovulation defects, hyperandrogenism and metabolic syndrome. Furthermore, pharmacological treatment with methyl donors rescued the PCOS phenotype in mice, and similar epigenetic alterations were observed in women with PCOS [START_REF] Mimouni | Polycystic ovary syndrome is transmitted via a transgenerational epigenetic process[END_REF]. Last, PCOS could be considered as an evolutionary adaptation to both genetic factors that conferred a survival advantage for ancestral population, and epigenetic modulators resulting from our current environmental conditions [START_REF] Parker | Polycystic Ovary Syndrome: An Evolutionary Adaptation to Lifestyle and the Environment[END_REF].

The relationships between insulin resistance and PCOS are tight and complex. From a clinical point of view, it is important to keep in mind that improvement of insulin sensitivity and prevention of diabetes and cardiovascular diseases should be integrated into the medical care of patients with PCOS. Conversely, since hirsutism and oligomenorrhea can reveal severe insulin resistance syndromes [START_REF] Young | Type A insulin resistance syndrome revealing a novel lamin A mutation[END_REF][START_REF] Keller | Lipodystrophy: an unusual diagnosis in a case of oligomenorrhea and hirsutism[END_REF], a careful clinical examination should allow to detect lipodystrophy and/or signs of primary insulin receptor defects, and to set up appropriate diagnosis and care protocols [START_REF] Sollier | Lipodystrophic syndromes: From diagnosis to treatment[END_REF][START_REF] Musso | The long-term effect of recombinant methionyl human leptin therapy on hyperandrogenism and menstrual function in female and pituitary function in male and female hypoleptinemic lipodystrophic patients[END_REF][START_REF] Young | Type A insulin resistance syndrome revealing a novel lamin A mutation[END_REF][START_REF] Vambergue | Follow-up study of two sisters with type A syndrome of severe insulin resistance gives a new insight into PCOS pathogenesis in relation to puberty and pregnancy outcome: a case report[END_REF][START_REF] Vantyghem | Fertility and obstetrical complications in women with LMNA-related familial partial lipodystrophy[END_REF][START_REF] Joy | Prevalence of reproductive abnormalities among women with familial partial lipodystrophy[END_REF][START_REF] Keller | Lipodystrophy: an unusual diagnosis in a case of oligomenorrhea and hirsutism[END_REF][START_REF] Lungu | Insulin resistance is a sufficient basis for hyperandrogenism in lipodystrophic women with polycystic ovarian syndrome[END_REF][START_REF] Brown | Type B Insulin Resistance Masquerading as Ovarian Hyperthecosis[END_REF][START_REF] Huang-Doran | Ovarian Hyperandrogenism and Response to Gonadotropin-releasing Hormone Analogues in Primary Severe Insulin Resistance[END_REF][START_REF] Semple | Plasma adiponectin as a marker of insulin receptor dysfunction: clinical utility in severe insulin resistance[END_REF]. Indeed, severe insulin resistance syndromes, and in particular lipodystrophy, are probably not as rare as previously evaluated [START_REF] Gonzaga-Jauregui | Clinical and Molecular Prevalence of Lipodystrophy in an Unascertained Large Clinical Care Cohort[END_REF].

Funding and Acknowledgments

The French Reference Centers of Rare Diseases of Insulin Secretion and Insulin Sensitivity (PRISIS) and of Rare Endocrine Diseases of Growth and Development (CRMERCD) at Assistance Publique-Hôpitaux de Paris are supported by the French Ministry of Solidarity and Health. The research group "Lipodystrophy, metabolic adaptations, and aging" directed by Prof. Bruno Fève at Saint-Antoine Research Center (Paris, France) is funded by the Institut National de la Santé et de la Recherche Médicale, Sorbonne Université, and the Fondation pour la Recherche Médicale.


Pierre Barraud

Frédéric Dardel

Carine Tisné
email: carine.tisne@univ-paris5.fr

Optimizing HSQC experiment for the observation of exchange broadened signals in RNA-protein complexes

Keywords: NMR, Chemical exchange, CPMG, HIV, Reverse transcription, Nucleocapsid

Sites for interaction in protein-RNA complexes are often regions of conformational exchange. Although the study of exchange processes could bring valuable information about the recognition mode between protein and RNA, chemical exchange can be detrimental to the NMR spectra quality, resulting in broad, very weak or even unobservable signals.

In the present report, we used CPMG-like experiments to improve HSQC spectra of an RNA-protein complex in fast exchange on the chemical shift time scale. The use of such improvement will allow us to handle the resolution of the three-dimensional structure of the complex by NMR.

Introduction

In the absence of radiofrequency radiation, the contribution of chemical exchange to relaxation is dependent on the rate of the exchange process, the chemical shift differences between involved sites and the population of each site. Near coalescence, the rate of exchange is of the order of the chemical shift difference, and the contribution of chemical exchange to relaxation is therefore maximal. Regions of conformational exchange often correspond to some of the functionally most important sites in proteins, for example, sites for interactions with RNA. Although the study of exchange processes could bring valuable information about the recognition process between protein and RNA, chemical exchange is often detrimental to theNMR spectra quality resulting in broad, very weak or even unobservable signals. Several methods have been developed to characterize chemical exchange processes (for reviews, see Refs. [1,[START_REF] Palmer | Nuclear Magnetic Resonance of Biological Macromolecules[END_REF]3]) and to improve NMR signal intensities in the presence of such exchange, i.e. to preserve spin coherence [4,5,6]. In the present report, we used previously described CPMG-like experiment [5,6] to improve HSQC spectrum of an RNA-protein complex. In this kind of experiment, the intensity of the signals involved in chemical exchange process is increased by refocusing spins during coherence transfer period. The HSQC experiment modified with a CPMG pulse train during INEPT period allowed us to improve the overall signal-to-noise ratio and to tear some signals off the noise. These NMR experiments were carried out on a complex between the HIV-1 nucleocapsid protein (NC) and an RNA hairpin that mimicks the D-arm of tRNA Lys 3 . The HIV-1 nucleocapsid protein (NC) is a short, basic, nucleic acid binding protein with two zinc finger domains. NC is a versatile protein which is involved in multiple functions during the virus replication cycle, many of these functions relying on the nucleic acid chaperone activity of NC (for recent reviews see Refs. [7,8,9]). In previous studies [10], we have highlighted a strong and specific interaction between the NC protein and the Darm of tRNA Lys 3 . This interaction could be the key factor driving the unfolding process of tRNA Lys 3 , thereby allowing complete annealing of tRNA Lys 3 to the primer binding site (PBS), the first event in the HIV-1 reverse transcription. However, the NMR study of the complex between the NC protein and the D-arm of tRNA Lys 3 is rendered tricky by chemical exchange process in the microsecond to millisecond time scale, leading to unobservable or broadened signals.

Materials and methods

NMR sample

The recombinant NC 55 protein was overexpressed from a recombinant plasmid (kind gift of M.F. Summers) and purified as previously described [11,12]. The RNA oligonucleotide mimicking the tRNA Lys 3 D-arm (24 nucleotides length) was purchased from Dharmacon Research with 2'-o-bis (acetoxyethoxy)methyl (ACE) protection. The sample was deprotected by following the manufacturer's recommendations and dialyzed several times against deionised water. The pH was adjusted between 6 and 6.5 before freeze-drying. The 15 N-labelled NC 55 and the unlabelled RNA D-arm were prepared at about 2 mM and 10 mM, respectively, in an NMR buffer (25 mM Na-d 3 -acetate pH 6.5, 25 mM NaCl, 0.1 mM BME, 0.1 mM ZnCl 2 , 10 % 2 H 2 O). The titration experiment is achieved by adding increased amount of RNA D-arm in a 0.5 mM sample of 15 N-labelled NC 55 . After RNA addition, the sample is immediately vortexed and kept at 30˚C during 30 min before NMR data acquisition.

NMR experiments

NMR experiments were carried out on a Bruker Avance DRX spectrometer operating at 600 MHz and equipped with a 5-mm triple-resonance gradient probe with z-axis gradients. All sequences used sine-bell shaped pulsed field gradients (PFG) along the z-axis. Acquisitions of data were performed at 303 K. All experiments were recorded with 512 points in the 1 H dimension with a spectral width of 4.6 ppm and 256 points in the 15 N dimension with a spectral width of 30.0 ppm. NMR data were processed with NMRPipe/ NMRDraw software [13]. A pure squared sine bell and a pure squared cosine bell were applied along t 2 and t 1 , respectively. The data were zero-filled to 1024 points along t 1 and to 4096 along t 2 prior to Fourier transform. Spectra were analyzed with the software SPARKY (T.D. Goddard and D.G. Kneller, SPARKY 3, University of California, San Francisco). The SE-HSQC experiment is a standard sensitivityenhanced 2D 1 H/ 15 N correlation via double INEPT transfer [14]. Frequency discrimination with retention of absorption mode lineshape is achieved by the echo anti-echo method [15]. Sensitivity enhancement is optimized for amide cross peaks by setting both reverse transfer periods to (1/2J). Heteronuclear decoupling during acquisition was achieved by GARP sequence. The SE-CPMG-HSQC experiment (Fig. 1) is adapted from the SE-HSQC experiment in which the INEPT and reverse-INEPT periods are modified by the inclusion of a phase-modulated pulse train. This experiment is based on previously described HSQC experiments for the observation of exchange broadened signals [6]. The sequence was modified to optimize sensitivity enhancement for amide cross peaks by setting both reverse transfer periods to (1/2J). See caption of Fig. 1 for more details on NMR pulse sequence.

Results and discussion

Biological context

The HIV-1 virus uses a cellular tRNA, the human tRNA Lys 3 as a primer to initiate the reverse transcription of the viral genome. In the virus, the 3' end of the primer tRNA is strongly bound by 18 Watson-Crick base pairs to a complementary sequence of the viral RNA called the primer binding site (PBS). The first event in viral replication is thus the annealing of tRNA Lys it was shown that NC has an inherent conformational flexibility that could be crucial to allow its binding to different nucleic acid sequences and structures along the HIV-1 replication cycle [12,18]. In the currently available structures [11,20,19] of NC with nucleic acids, most of the specific contacts between the protein and the nucleic acids occur via the zinc knuckles, notably residues F 16 and W 37 , interacting with the guanine-rich loop of RNA hairpin. We have shown by NMR of 15 N-labelled tRNA Lys 3 that NC interacts preferentially with the acceptor and D stems and with the nucleotides involved in tertiary interactions between the D and T loops [21]. We also performed the reciprocal experiment, i.e. the NMR footprint of tRNA Lys 3 binding on 15 N-labelled NC (data not shown). The signals are so broadened that roughly only the C-teminal amide group and the amino groups of side chains remain observable. Indeed, the multiplicity of NC binding sites on tRNA Lys 3 rendered analyses using the full-length tRNA intractable, even when stringency was raised by increasing salt concentration (data not shown). Reducing the target RNA length could decrease the number of non-specific sites and still have a biological interest. Indeed, studying separate hairpins cannot obviously mimic the entire tRNA, in particular at the level of the TΨC/D loops interaction. However, in the annealing process, there is a step where the TΨC/D loops interaction are broken and where the NC protein plays a crucial role [22]. Therefore, we have previously demonstrated that NC makes a specific interaction with the D hairpin mimicking the D-arm of tRNA Lys 3 (dissociation constant K d = 28 nM at 100 mM NaCl). This interaction could be a key factor driving the unfolding process, thereby allowing complete annealing of tRNA Lys 3 to the PBS [10]. Fig. 2a shows the HSQC experiment of 15 Nlabelled NC with the tRNA Lys 3 D hairpin. The NMR signals are quite broad for the size of the complex (14.2 kDa) and may indicate that the NC in the context of the complex is involved in a chemical exchange process in the microsecond to millisecond time scale. This chemical exchange is attributable to the equilibrium between the free and bound forms of the protein.

Analysis of the exchange process and NMR optimization of HSQC spectra

In the presence of such chemical exchange, multidimensional NMR experiments suffer from dephas-ing of spin coherence during periods of evolution and coherence transfer by scalar coupling [START_REF] Palmer | Nuclear Magnetic Resonance of Biological Macromolecules[END_REF]3]. The result of chemical exchange during periods of evolution is a broadening of the resonances, while it leads to loss of magnetization during periods of coherence transfer and therewith of signal intensity. Therefore, resonances are broad, very weak or even unobservable (Fig. 2a). The loss of magnetization during the period of coherence transfer can be minimized by using a modified INEPT block, i.e. the CPMG-INEPT block first described by Mueller and co-workers [5]. The intensity of the signals involved in the chemical exchange process is increased by refocusing them during the coherence transfer period. We then compared CPMG-HSQC [6] and SE-CPMGHSQC described in Fig. 1, since the gain of intensity from the sensitivity-enhanced HSQC could be counterbalanced by the increase of the pulse sequence length. As a result, the SE-CPMG-HSQC spectrum shows more intense signals than those from the CPMG-HSQC (data not shown). The comparison of the performance of the SE-HSQC and SE-CPMG-HSQC experiment is given in Fig. 2. The CPMG-derived sequence shows improved sensitivity for almost all resonances, resulting in an average enhancement of about 40 %. This result confirms that the exchange process lies in the microsecond to millisecond time scale. Moreover, some unobservable signals in the classic SE-HSQC experiment grow up in the SECPMG-HSQC and become observable (see signals 1, 2, 3 in Fig. 2). This could be of significant importance, as these signals, which show considerably higher intensity, belong to residues directly involved in RNA binding. Fig. 3 shows the titration of 15 N-labelled NC by increased amounts of D hairpin until reaching the equivalence. Two kinds of behaviour are observed:

(1) a progressive variation of chemical shifts as the amount of RNA increases (see for instance the NMR signals of C 28 and C 49 on Fig. 3); (2) a disappearance at the substoichiometric level of NC signals and their reappearance close to the 1:1 ratio of NC:RNA (Fig. 3). These results are consistent with an exchange relaxation constant (R ex ) behaviour in the fast exchange regime. Indeed, for a simple model of two-site exchange process in the fast exchange limit, the exchange relaxation constant is given by

R ex = p A p B ∆ω 2 k ex
where k ex is the exchange rate constant, ∆ω is the chemical shift difference between the two sites, and p A and p B the relative population of each site. For a complex with a dissociation constant more than 3 orders of magnitude smaller than the species concentrations, we can assume that the concentration of the complex in bound form is rather equal to the partner added in substoichiometric molarity. Thus, during the titration, for a fraction p of added RNA in the protein (p ∈ [0, 1]), the fraction of the bound protein is p and the fraction of the free one is (1 -p) and so R ex becomes

R ex = p(1 -p)∆ω 2
k ex Even if these equations apply only to the fast exchange limit (k ex > ∆ω), they provide useful qualitative information for the interpretation of the relaxation behaviour of a complex in a fast exchange regime. For instance, the R ex maximum during the titration occurs for p = 0.5, and depends quadratically on the chemical shift difference between the free and the bound forms. Therefore, the NMR signals that experience the largest variation of chemical shifts between the free and bound forms of NC are also those that will experience the largest broadening during the titration due to the exchange regime. We can reasonably think that these signals which present the largest differences in their chemical shifts between the free and bound forms of NC are those involved in the interaction with the RNA partner. Actually, they belong to amino acids surrounding the major aromatic residue of each zinc knuckle, i.e. F 16 and W 37 , and to residues with hydrophobic side chain, i.e. I 24 , A 25 and M 46 (Fig. 3). These residues have been identified to largely participate in the recognition by the NC of RNA encapsidation signals SL2 and SL3 [19,11]. Indeed, in these structures, each zinc knuckle forms a hydrophobic cleft by means of the hydrophobic side chains listed above and binds to an exposed guanine nucleobase of the RNA loop. Thus the NMR footprint of the RNA Darm on the NC (Fig. 3) suggests that the recognition mode between the NC and the tRNA Lys 3 D-arm is similar to that of the encapsidation signals.

Conclusion

In this paper, the use of modified HSQC experiment allowed us to significantly improve the signal to noise ratio in the spectra of the complex between the NC and the tRNA Lys Unless indicated, the pulses are applied along the x-axis. The delay τ was set to 140 ms such that the total length of the spin-echo pulse train equalled 5.55 ms, i.e. 1/2J with J = 90 Hz for N-H 1 J coupling. 15 N pulses were achieved using a field strength γB1 = 47 kHz. The phase modulation scheme for φ 16 is x, y, x, y, y, x, y, x, -x, -y, -x, -y, -y, -x, -y, -x.

The following phase cycling was used: φ 2 = y; φ 3 = x, -x; φ 7 = 2(x), 2(-x); φ 4 = 2(x), 2(-x); φ 5 = 2(y), 2(-y); φrec = x, -x, -x, x. Heteronuclear decoupling during acquisition was achieved by GARP. For each t 1 increment, axial peaks were shifted to the sides of the spectrum by inversion of φ 3 in concert with the receiver phase, similar as in States-TPPI [23]. The sequence uses sine-bell shaped pulsed field gradients (PFG) along the z-axis. To yield absorption mode spectra P-and N-type coherence selection is achieved by inversion of G 2 in concert with inversion of φ 5 . The duration of all the gradients was 1 ms, followed by a 400 ms recovery delay. Gradient relative strengths were G 1 = 50 %; G 2 = 80 %; G 3 = 8.1 %.

3 D 3 DFigure 1 .

 331 Figure1. Pulse scheme of the15 N-1 H SE-CPMG-HSQC experiment. Narrow (filled) and wide (open) rectangles represent 90 and 180 pulses, respectively. Unless indicated, the pulses are applied along the x-axis. The delay τ was set to 140 ms such that the total length of the spin-echo pulse train equalled 5.55 ms, i.e. 1/2J with J = 90 Hz for N-H 1 J coupling.15 N pulses were achieved using a field strength γB1 = 47 kHz. The phase modulation scheme for φ 16 is x, y,x, y, y, x, y, x, -x, -y, -x, -y, -y, -x, -y, -x. The following phase cycling was used: φ 2 = y; φ 3 = x, -x; φ 7 = 2(x), 2(-x); φ 4 = 2(x), 2(-x); φ 5 = 2(y), 2(-y); φrec = x, -x, -x, x. Heteronuclear decoupling during acquisition was achieved by GARP. For each t 1 increment, axial peaks were shifted to the sides of the spectrum by inversion of φ 3 in concert with the receiver phase, similar as in States-TPPI[23]. The sequence uses sine-bell shaped pulsed field gradients (PFG) along the z-axis. To yield absorption mode spectra P-and N-type coherence selection is achieved by inversion of G 2 in concert with inversion of φ 5 . The duration of all the gradients was 1 ms, followed by a 400 ms recovery delay. Gradient relative strengths were G 1 = 50 %; G 2 = 80 %; G 3 = 8.1 %.

Figure 2 . 3 D 3 D

 233 Figure 2. Contour plot of the 2D HSQC-type spectra of the complex between uniformly 15 N-labelled NC and (a) unlabelled tRNA Lys 3 D-arm (SE-HSQC experiment), (b) unlabelled tRNA Lys 3 D-arm (SE-CPMGHSQC experiment), (c) 1D SE-HSQC trace of signal 1 and (d) 1D SE-CPMG-HSQC trace of signal 1.

2 Figure 3 .

 23 Figure 3. Titration experiment (SE-CPMG-HSQC) of 15 N-labelled NC protein by unlabelled tRNA 15 D hairpin. In red: 1/0 protein/RNA ratio; in orange: 1/0.3; in yellow: 1/0.6, in green: 1/0.9, in blue: 1/1. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) The circled signals disappear during the titration. The corresponding residues are highlighted on the NC sequence. The major aromatic residue of each zinc knuckle is written in open letter.

to the PBS, this step being promoted by the viral NC. Most of the known functions of the HIV-1 NC involve interactions with nucleic acids (for recent reviews, see Refs.[7,8,9]). The structure of the free NC consists in two zinc knuckle domains, connected by a flexible linker attached to flexible N-and C-terminal tails[12,16,17]. By NMR,

Acknowledgments

P. Barraud is supported by a studentship from Ministère de la Recherche. The French AIDS National Agency (ANRS) and Ensemble contre le sida (Sidaction) supported this work.


Didier Theilliol Mickaël Rodrigues
email: mickael.rodrigues@cran.uhp-nancy.fr

Dominique Sauter

FAULT TOLERANT CONTROL DESIGN OF NONLINEAR SYSTEMS USING LMI GAIN SYNTHESIS

Keywords: Fault Tolerant Control, Nonlinear, Multi-model, LMI, Stability

In this paper, an active Fault Tolerant Control (FTC) strategy is developed to nonlinear systems described by multiple linear models to prevent the system deterioration by the synthesis of adapted controllers. By considering that Fault Detection, Isolation (FDI) and estimation is realized, the synthesis of an appropriate combination of predesigned gains is performed. The main contribution concerns the design of state feedback gains through LMI both in fault-free and faulty cases in order to preserve the system performances. For each separate actuator, a robust pole placement is designed by pole clustering. The effectiveness and performances of the method are illustrated in simulation by considering a nonlinear system: a Three-Tank system.

INTRODUCTION

To overcome the limitations of conventional feedback control, new controllers have been developed with accommodation capabilities or tolerance to faults. The objective of Fault Tolerant Control system (FTC) is to maintain current performances closed to desirable performances and preserve stability conditions in the presence of component and/or instrument faults ; in some circumstances reduced performance could be accepted as a trade-off. Accommodation capability of a control system depends on many factors such as the severity of the failure, the robustness of the nominal system, and the actuators redundancy. FTC can be motivated by different goals depending on the application under consideration, for instance, safety in flight control or reliability or quality improvements in industrial processes. Various approaches for FTC have been suggested in the literature [START_REF] Patton | Fault-tolerant control: the 1997 situation[END_REF], [START_REF] Zhang | Bibliographical review on reconfigurable Fault-Tolerant Control systems[END_REF] but often deal with linear systems. For nonlinear systems, the design of Fault Tolerant controller is far more complicated. Nonlinear systems based on multiple linear models, represents an attractive solution to deal with the control of nonlinear systems [START_REF] Leith | Survey of gainscheduling analysis and design[END_REF], [START_REF] Banerjee | Hinf control of nonlinear processes using multiple linear models[END_REF] or FDI methods as in the chapter nine of [START_REF] Chen | Robust model-based fault diagnosis for dynamic systems[END_REF] where nonlinear dynamic systems are described by a number of locally linearized models based on the idea of Tagaki-Sugeno fuzzy models or as interpolated multiple linear models (Murray-Smith and [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF]. Various recent FDI/FTC studies, based on a multiple model method have been developed in order to detect, isolate and estimated an accurate state of a system in presence of faults/failures around an operating point [START_REF] Maybeck | Multiple model adaptive algorithms for detecting and compensating sensor and actuator/surface failures in aircraft flight control systems[END_REF], [START_REF] Zhang | Integrated active Fault-Tolerant Control using IMM approach[END_REF]. Compared to multi-model based reconfigurable control method presented by [START_REF] Aström | Control of complex systems[END_REF], this paper not consider some redundant hardware which is very useful when failures are supposed to occur on the system. In this paper, an active fault tolerant strategy is developed so as to avoid actuator fault effect on nonlinear system where faults are assumed to be incipient, abrupt but not generate a total actuator failure. Under the assumption that the fault is detected, isolated and estimated, the developed method preserves the system performances through an appropriate gain synthesis in faulty case. It is a big difference with robust control which does not deal with a FDI module and does not take into account fault estimation for reconfiguration. Compared to recent work applied to similar nonlinear system (Theilliol et al., 2003a), where a multi-model representation is considered, the proposed FTC strategy is not based on an additional control law but on the redesign of appropriate gain in faulty case allowing stability and performances of the system. This appropriate design is inspired from a previous work made by [START_REF] Kanev | Reconfigurable robust fault-tolerant control and state estimation[END_REF] where a FTC strategy is devoted for each actuator. However, this work considers only linear case with a single operating point. Our paper contributes to improve it on real systems with multi-operating points and a relaxed LMI region for stability. The paper is organized as follows. In section II, an active state space representation is derived from an additive one by underlying the links between them. A global state space representation of nonlinear system is given through a multi-model approach. In section III, we introduce a pole placement by LMI region and then a gain synthesis for each actuator generate an active global state feedback synthesis. A simulation example is given in section IV to illustrate the proposed method. Finally, concluding remarks are given in the last section.

PROBLEM STATEMENT

It is assumed that dynamic behaviour of the system operating at different operating points can be approximated by a set of N linear time invariant models. Consider the following state space representation of a nonlinear system around j-th operating point with actuator faults:

x j k+1 = A j x j k + B j u j k + F j f k + ∆x j y j k = C j x j k + ∆y j (1)
where x j ∈ R n represents the state vector, u j ∈ R p is the input vector, y j ∈ R m is the output vector and f k ∈ R p represents the actuator fault vector. A j ∈ R n×n , B j ∈ R n×p , C j ∈ R m×n and the fault distribution matrix F j ∈ R n×f are invariant matrices defined around the j th operating point (OP j ). We consider the terms ∆x j , ∆y j : ∆x j = -A j x j e -B j u j e + x j e ∆y j = -C j x j e + y j e

(2)

where (x j e , y j e , u j e ) are a family of operating points of the nonlinear plant. As considered in [START_REF] Johansen | Off-equilibrium linearisation and design of gain-scheduled control with application to vehicle speed control[END_REF], [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF] and (Theilliol et al., 2003b), a possible model that would be able to catch the full range of operation is made from N weighting local models OP j by interpolation functions ρ j k . These activation functions ρ

j k ∀j ∈ [1, 2, .., N ] lie in a convex set Ω = {ρ j k ∈ R N , ρ k = [ρ 1 k ....ρ N k ] T , ρ j k ≥ 0 ∀j and N j=1 ρ j k = 1}
and these functions are generated via works of (Adam-Medina et al., 2003) and (Theilliol et al., 2003b), which permit to generate insensitive residual to faults and some uncertainties. So, activation functions are robust against faults and errors modeling and the dynamic system is well represented. The plant dynamics are formulated as a blended multiple representation as in (Theilliol et al., 2003a):

x k = N j=1 ρ j k x j k y k = N j=1 ρ j k y j k (3)
The representation considers additive fault representation but there exists multiplicative representation for specific actuator fault as in [START_REF] Kanev | Reconfigurable robust fault-tolerant control and state estimation[END_REF]. So, consider a local multiplicative actuator fault representation as:

x j k+1 = A j x j k + B j (I -γ a )u j k + ∆x j y j k = C j x j k + ∆y j (4) with γ a diag[γ a 1 , γ a 2 , ..., γ a p ], γ a 1 ∈ R, such that γ a i = 1
represents a total lost, a failure of i-th actuator and γ a i = 0 implies that i-th actuator operates normally. The relation between state space representations (1) and ( 4) is equivalent to

F j f k = -B j γ a u j k (5)
In closed-loop, the fault occurrence could be detected as described in [START_REF] Noura | Fault-tolerant control in dynamic systems: Application to a winding machine[END_REF] for linear case and (Theilliol et al., 2003a) for multilinear systems and Fault Tolerant Control could be performed via an additional control law as in [START_REF] Theilliol | Fault diagnosis and accommodation of three-tank system bsaed on analytical redundancy[END_REF] which permits to avoid fault on a system based on a state space representation as (1). In these papers, the goal was to synthesize a new control law U F T C with a nominal one u nom and additional one u ad . The term u ad was performed in order to vanish fault on the system. The global control law is obtained by interpolating gains of each local controller [START_REF] Leith | Survey of gainscheduling analysis and design[END_REF] and is defined as:

U F T C = N j=1 ρ j k (u j nom + u j ad ) (6)
with u j nom = -K j nom x j k . The gains were only performed for nominal cases and do not take into account fault occurrence. Based on a multiplicative fault representation defined in (4), the new control law u j F T C must vanish all faults on the system as:

u j F T C = [I -γ a ] + u j nom (7) Note that u j F T C = [I -γ a ] + u j nom = -[I - γ a ] + K j nom x j k = -K j F T C x j k .
So, without considering total fault, this specific control law in the state space representation (4) leads to:

B j (I -γ a )u j k = B j (I -γ a )(I -γ a ) + u j nom = B j u j nom (8)
As previously defined in (6), the model probability is viewed as a scheduled variable in the synthesis of the controller and the global control law is defined as:

U F T C = N j=1 ρ j k u j F T C (9)
with u j F T C the output of each local controller defined around each operating point. In order to synthesize state feedback for FTC ensuring both active control in multi-model philosophy and quadratic stability, use of LMI provide a well toolbox for these purposes. Total failures are not consider in this paper but only partial ones. We attract attention that the system has to be observable in all around each operating points and we will consider now only nonlinear plant defined around Equilibrium Points (EP) i.e. ∆x j = 0.

FAULT TOLERANT CONTROL ON MULTIPLE OPERATING POINTS

Pole clustering

In the synthesis of control system, some desired performances should be considered in addition to stability. In fact, classical stability conditions do not deal with transient responses of the closedloop system. In contrast, a satisfactory transient response can be guaranteed by confining its poles in a prescribed region. For many real problems, exact pole assignment may not be necessary: it suffices to locate the closed-loop poles in a prescribed subregion in the complex plane. We will discuss about pole clustering by introducing the following LMI-based representation of stability regions.

Definition 1. LMI stability region [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF]. A subset D of the complex plane is called an LMI region if there exist a symmetric

matrix α = [α kl ] ∈ R n×n and a matrix β = [β kl ] ∈ R n×n such that D = {z ∈ C : f D (z) < 0} (10)
where the characteristic function

f D (z) is given by f D (z) = [α kl + β kl z + β lk z] 1≤k,l≤n (11)
(f D is valued in the space of n × n Hermitian matrices). 2 Moreover, LMI regions are convex and symmetric with respect to the real axis since for any z ∈ Df D (z) = f D (z) < 0. Then, a matrix A has all its eigenvalue in D, if and only if there exists a symmetric matrix P such that [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF]:

M D (A, P ) = α ⊗ P + β ⊗ (AP ) + β T ⊗ (AP ) T = [α kl P + β kl AP + β lk P A T ] 1≤k,l≤m (12)
with M = [µ kl ] 1≤k,l≤n means that M is an n × n matrix (respectively, bloc matrix) with generic entry (respectively bloc) µ kl . Note that M D (A, P ) in ( 12) and f D (z) in ( 11) are related by the substitution (P, AP, P A T ) ↔ (1, z, z). It is easily seen that LMI regions are convex and symmetric with respect to real axis. Specifically, the circular LMI region D is considered:

D = {x + jy ∈ C : (x + q) 2 + y 2 < r 2 } (13)
centered at (-q, 0) with radius r > 0, where the characteristic function f D (z) is given by:

f D (z) = -r z + q z + q -r (14)
Therefore, this circular region puts a lower bound on both the exponential decay rate and the damping ratio of the closed-loop response, and thus is very common in practical control design. It is obvious that well chosen LMI region is needed for ensuring stability and good results: the parameters q, r have to be defined by the engineer.

Control law synthesis in fault-free case

Let consider the state space representation (4) of nonlinear system defined around the Equilibrium Points EP j , ∀j = 1, 2, .., N

x j k+1 = A j x j k + B i j (I -γ a )u j k y j k = C j x j k ( 15
)
with i = 1, 2, .., p the actuators for each EP j . Consider the matrix representing total faults in all actuators but the i-th:

B i j = [0, .., 0, b i j , 0, ..., 0] (16)
and

B j = [b 1 j , b 2 j , ..., b p j , ] with b i j ∈ R n×1 .
It is assumed that each column of B j is full column rank whatever the EP j . The pairs (A j , b i j ), ∀i = 1, ..., p are assumed to be controllable for all ∀j = 1, ..., N . Let D, a LMI region defining a disk with a center (-q, 0), and a radius r with (q + r) < 1 for defining pole assignment in the unit circle. Assume that for each B i j , there exist matrices X i = X T i > 0 and Y i , ∀j = 1, ..., N, ∀i = 1, 2, ..., p such as:

-rX i qX i + (A j X i -B i j Y i ) T qX i + A j X i -B i j Y i -rX i < 0 (17)
It can be noticed that if q = 0 and r = 1, the previous equation ( 17) is equivalent to solve a classical quadratic stability problem.

Based on the assumptions that for each OP j each pairs (A j , b i j ) are controllable, it is possible to find a Lyapunov matrices X i > 0 and state-feedback K i with Y i = K i X i and finally form a global statefeedback gain K nom .

Theorem 1. Consider the system (15) in fault-free case (γ a = 0) defined for all EP j , j = 1, 2, .., N : it is possible to develop a mixing of pre-designed state-feedback gains matrices K i = Y i X -1 i for each actuator i with i = 1, 2, .., p such that ( 17) holds for all j = 1, ..., N . The state feedback control for each operating point is given by:

u j k = u j nom = -( p i=1 G i Y i )( p i=1 X i ) -1 x j k = -Y X -1 x j k = -K nom x j k (18) with p i=1 G i Y i = Y , X = p i=1 X i and G i = B i+
j B i j is matrix that has zeros everywhere except in entry (i, i) where it has a one. The general control law for all EP j could be defined as:

U nom = N j=1 ρ j k u j nom (19)
2 Proof: Summation of ( 17) for i = 1, 2, .., p gives for one equilibrium point

j p i=1 -rX i qX i + (A j X i -B i j Y i ) T qX i + (A j X i -B i j Y i ) -rX i < 0 (20)
related to the quadratic D-stability in a prescribed LMI region as in [START_REF] Chilali | H∞ design with pole placement constraints: an LMI approach[END_REF].

Next, denote X = p i=1 X i (with X = X T > 0) to obtain      -rX qX + (A j X - p i=1 B i j Y i ) T qX + (A j X - p i=1 B i j Y i ) -rX      < 0 (21)
Now, denote the l-th row of the matrix Y i as Y l i , i = 1, ..., p et l = 1, .., p, i.e.

Y l i = G l Y i (22) Therefore, p i=1 B i j Y i = p i=1 [0, .., 0, b i j , 0, ..., 0]Y i i = B j p i=1 Y i i (23) leading to p i=1 B i j Y i = B j ( p i=1 G i Y i ) (24) Thus, taking Y = p i=1 G i Y i , equation (24) be- comes p i=1 B i j Y i = B j Y (25)
which, substituted into LMI (21), finally makes

-rX qX + (A j X -B j Y ) T qX + (A j X -B j Y ) -rX < 0 (26)
for EP j , j = 1, 2, .., N . By multiplying each LMI (26) by ρ j k and summing all of them, we obtain

      -rX qX + N j=1 ρ j k (A j X -B j Y ) T qX + N j=1 ρ j k (A j X -B j Y ) -rX       < 0 (27)
it is equivalent to (17). However, other system such piecewise linear system could use the same approach with a multiple gain synthesis as in [START_REF] Ozkan | Control of a solution copolymerization reactor using multimodel predictive control[END_REF].

-rX qX + (A(ρ)X -B(ρ)Y ) T qX + (A(ρ)X -B(ρ)Y ) -rX < 0 (28) with A(ρ) = n j=1 ρ j k A j and B(ρ) = n j=1 ρ j k B j . Hence quadratic D-

Active Fault Tolerant Control design

As indicated in equation ( 8) and based on the previous synthesis control law, the FTC method can be developed in this section where only actuator faults are considered under assumptions that fault occurrence and fault magnitude γ a are known.

Theorem 2. Consider the system (15) in faulty case (γ a = 0) coupled with regulators with gains K i = Y i X -1 i for all equilibrium point j = 1, ..., N and for each actuator i with i = 1, 2, .., p. Let introduce the set of indexes of all actuators that are not completely lost, i.e. Θ {i : i ∈ (1, 2, .., p), γ a i = 1}. The control action is

u j k = u j F T C = -(I -γ a ) + i∈Θ G i Y i i∈Θ X i -1 x j k (29) where G i = B i+ j B i j
, applied to the faulty system allows to constrain pole placement in prescribed LMI region.

2 Proof: Applying the new control law (29) to the faulty system (15), leads to the following equation

B j (I -γ a )u j k = B j Γ a ( i∈Θ G i Y i )( i∈Θ X i ) -1 x j k (30) with Γ a = I p-h 0 0 O h (31)
Γ a is a diagonal matrix which contains only entries zero (representing total faults) and one (no fault). But here h = 0, which is the number of actuators completely lost, due to the fact that only the set Θ is considered. Since B j Γ a = i∈Θ B j i models only the actuators that are not completely lost, then performing the summations in the proof of Theorem (1) over the elements of Θ shows that (

p i=1 G i Y i )( p i=1 X i ) -1
is the state-feedback gain matrix for the faulty system

(A j , i∈Θ B j i , C j ). ⋄
The control law in equation ( 29) implies that

u j k = u j F T C = -K F T C x j k (32) with K F T C = (I -γ a ) + i∈Θ G i Y i ( i∈Θ X i ) -1 (33)
The global control law U F T C of the system is realized as:

U F T C = N j=1 ρ j k u j F T C (34)
4. APPLICATION

Process description

The approach presented in this paper has been applied to the well known three tanks benchmark as in (Theilliol et al., 2003a). As all the three liquid levels are measured by level sensors, the output vector Y is [l 1 l 2 l 3 ] T . The control input vector is U = [q 1 q 2 ] T . The goal is to control the system around three equilibrium points with ∆x j = 0. Thus, 3 linear models have been identified around each of these equilibrium points and the operating conditions are given in Table (1). The linearized system is described by a discrete Table 1. state space representation with a sampling period T s = 1s. For each OP, each control matrix pair (A j , b i j ) is controllable. Controllers have been designed for levels l 1 and l 2 to track reference input vector Y r ∈ R 2 . Nominal controllers have been designed through Theorem (1), leading to two state feedback gain matrices K 1 , K 2 (due to 2 actuators) for all the three OP in order to achieve satisfying tracking performances. The simulation of actuator faults on the system does not affect the controllability and observability of the system.

Results and comments

Simulations have been performed such as the 3 operating conditions described in Table (1 Consequently, the dynamic behaviour of the other levels is also affected by this fault and control system tries to cancel the static error created by the corrupted input. Consequently, the real output is different from the reference input and the control law is different from its nominal value.

Since an actuator fault acts on the system as a perturbation, and in spite of the presence of an integral controller, the system outputs can not reach again their nominal values. In the same way, the actuator Fault Tolerant Control method's ability to compensate faults is illustrated in the presence of the same fault. Once the fault is isolated and simultaneously estimated, a new control law (34) is computed in order to reduce the fault effect on the system. Indeed, since the effect of an actuator fault is quite similar to the effect of a perturbation, the system outputs reach again their nominal values, as illustrated in figure (4).

A time delay between fault occurrence and fault compensation equals to 10 samples is considered in our simulation. Computation of the tracking error norm in fault-free case, in faulty case without and with FTC underlines the performances of this approach as seen in Table (2). With FTC, the tracking error norm for output l 1 is a bit larger than with the fault-free case but it still widely smaller than the one without FTC. The actuator Fault Tolerant Control is able to maintain performances as close as possible to nominal ones and to ensure the closed-loop stability despite the presence of instruments malfunction.

CONCLUSION

The method developed in this paper emphasises the importance of the active Fault Tolerant Control of nonlinear systems based on multi-model representation. This method is suitable for actuator faults on the whole operating range of the system. A robust controller is designed for each separate actuator through an LMI pole placement in fault-free case and faulty case. It allows the system to continue operating safely, to avoid stopping it immediately and to ensure stability. The synthesis of this active state feedback control takes into account the information provided by FDI scheme. The performances and the effectiveness of this active Fault Tolerant Control based on multiple model approach have been illustrated in this simulation example. Futures works will deal with total failures, restructuration and will improve the limits of the strategy.

Fig. 2 .

 2 Fig.1. Weighting functions with respect to set-point changes occur at time instant 500s and after at time instant 2500s. In the simulation, gaussian noises (N (0, 1e -4 2 )) are added to each output signal. The reference inputs correspond to step changes for l 1 and l 2 . The

Fig. 3 .

 3 Fig. 3. System outputs with pump degradation and a nominal controller

Fig. 4 .

 4 Fig. 4. System outputs with pump degradation and a FTC controller

Table 2 .

 2

	Error	Fault-free case	Actuator fault
	norm		without FTC	with FTC
	l1	1.1063	3.3365	1.1176