What men should know about the impact of physical activity on their health.

What men should know about the impact of physical activity on their health.

Introduction

Length of life and quality of health depend on a complex mix of factors of individual variable weighting. Genotype, lifestyle variables, country of residence and luck all have a role to play [START_REF] Yates | Exceptional longevity in men: modifiable factors associated with survival and function to age 90 years[END_REF][START_REF] Murray | Mortality by cause for eight regions of the world: Global Burden of Disease Study[END_REF]. Some of these an individual has little control over. Others such as the lifestyle variablesobesity, diet, physical activity levels and smoking have at least an element of personal choice and control [START_REF] Yates | Exceptional longevity in men: modifiable factors associated with survival and function to age 90 years[END_REF][START_REF] Murray | Mortality by cause for eight regions of the world: Global Burden of Disease Study[END_REF][START_REF] Lee | Combined effects of cardiorespiratory fitness, not smoking, and normal waist girth on morbidity and mortality in men[END_REF][START_REF] Lee | Does physical activity ameliorate the health hazards of obesity?[END_REF]. This perspective piece aims to discuss one of these variables -physical activity -in relation to the health of men. Although the focus of this piece is on men the messages on physical activity are relevant to both sexes across all ages.

Even though for thousands of years it has been suggested that physical activity is beneficial to health, serious scientific research into the topic has only occurred over the last 60 years [START_REF] Blair | Healthy Hearts -and the universal benefits of being physically active: physical activity and health[END_REF]. During this period strong evidence has accumulated to verify that physical activity is indeed good for health. Regular physical activity reduces the risks of many diseases, improves longevity and improves function and quality of life [START_REF] Blair | Healthy Hearts -and the universal benefits of being physically active: physical activity and health[END_REF][START_REF] Warburton | Health benefits of physical activity: the evidence[END_REF]. Indeed, Church and Blair [START_REF] Church | When will we treat physical activity as a legitimate medical therapy….even though it does not come in a pill?[END_REF] claim that the beneficial effects are so great that if it could be taken in pill form it would radically alter the world as we know it. It is with certainty that all health care professionals can recommend some form of physical activity to their patients.

Why should I exercise?

Living a sedentary lifestyle is a risk factor for many chronic diseases and increases an individual's chance of premature mortality [START_REF] Holtermann | The interplay between physical activity at work and during leisure time--risk of ischemic heart disease and all-cause mortality in middle-aged Caucasian men[END_REF][START_REF] Marcus | Physical activity intervention studies: what we know and what we need to know: a scientific statement from the American Heart Association Council on Nutrition, Physical Activity, and Metabolism (Subcommittee on Physical Activity); Council on Cardiovascular Disease in the Young; and the Interdisciplinary Working Group on Quality of Care and Outcomes Research[END_REF][START_REF] Ogilvie | Interventions to promote walking: systematic review[END_REF]. The link between increased physical activity and reduced risk of cardiovascular disease (CVD) is well established [START_REF] Blair | Healthy Hearts -and the universal benefits of being physically active: physical activity and health[END_REF][START_REF] Warburton | Health benefits of physical activity: the evidence[END_REF]. Regular moderate to intense physical activity is associated with decreased risk of coronary heart disease, ischemic and haemorrhagic stroke [START_REF] Yung | Exercise, vascular wall and cardiovascular diseases: an update part 2[END_REF][START_REF] Vogel | Health benefits of physical activity in older patients: a review[END_REF]. The benefits of regular physical activity also extend to secondary prevention in patients with established CVD [START_REF] Warburton | Health benefits of physical activity: the evidence[END_REF]. In addition a growing body of evidence suggests that increasing physical activity can also reduce risk of certain types of cancers, osteoporosis, type 2 diabetes, depression, obesity and hypertension [START_REF] Blair | Healthy Hearts -and the universal benefits of being physically active: physical activity and health[END_REF][START_REF] Marcus | Physical activity intervention studies: what we know and what we need to know: a scientific statement from the American Heart Association Council on Nutrition, Physical Activity, and Metabolism (Subcommittee on Physical Activity); Council on Cardiovascular Disease in the Young; and the Interdisciplinary Working Group on Quality of Care and Outcomes Research[END_REF][START_REF] Teychenne | Physical activity and likelihood of depression in adults: a review[END_REF]. Rhodes et al [START_REF] Rhodes | Characteristic of physical activity guidelines and their effect on adherence[END_REF] note that regular physical exercise is presently considered to be beneficial in the primary and secondary prevention of about 25 conditions. Regular exercise and physical activity can also reduce or slow the deterioration associated with age related changes in many of the body's systems [START_REF] Chodzko-Zajko | American College of Sports Medicine position stand. Exercise and physical activity for older adults[END_REF].

Evidence of the beneficial effects of physical activity in the primary prevention and management of cancer is growing [START_REF] Warburton | Health benefits of physical activity: the evidence[END_REF][START_REF] Newton | Exercise in prevention and management of cancer[END_REF]. Cancer mortality has been found to be inversely related to physical activity levels [START_REF] Mctiernan | Mechanisms linking physical activity with cancer[END_REF][START_REF] Laukkanen | Intensity of leisuretime physical activity and Cancer mortality in men[END_REF]. In addition to decreased mortality Orsini et al [START_REF] Orsini | Association of physical activity with cancer incidence, mortality, and survival: a population-based study of men[END_REF] found that walking or cycling for at least an hour a day was associated with a 16% reduction in overall cancer incidence. For specific cancer sites the evidence is mixed. For colon cancer in both sexes there is a strong relationship between increased physical activity and decreased incidence [START_REF] Wolin | Physical activity and colon cancer prevention: a meta-analysis[END_REF][START_REF] Moradi | Occupational physical activity and risk for cancer of the colon and rectum in Sweden among men and women by anatomic subsite[END_REF]. Other cancers in men have yet to demonstrate such a clear relationship. Prostate cancer incidence seems to be less for those who are active at work, defined as not sitting for most of the day [START_REF] Orsini | A prospective study of lifetime physical activity and prostate cancer incidence and mortality[END_REF][START_REF] Johnsen | Physical activity and risk of prostate cancer in the European Prospective Investigation into Cancer and Nutrition (EPIC) cohort[END_REF]. The link with leisure time physical activity is less clear. The European Prospective Investigation into Cancer and Nutrition did not find a link between prostate cancer and leisure time physical activity [START_REF] Johnsen | Physical activity and risk of prostate cancer in the European Prospective Investigation into Cancer and Nutrition (EPIC) cohort[END_REF]. Orsini et al [START_REF] Orsini | A prospective study of lifetime physical activity and prostate cancer incidence and mortality[END_REF] however did find that walking or cycling for more than 30 mins a day was associated with decreased incidence. Also of note is that post diagnosis, physical activity can aid recovery and improve outcomes [START_REF] Newton | Exercise in prevention and management of cancer[END_REF].

As men age, the risk of erectile dysfunction increases. There are many potential factors that can contribute to the problem. Amongst these a number of modifiable lifestyle variables have been identified. Physical inactivity and obesity are associated with increased risk of erectile dysfunction [START_REF] Orsini | A prospective study of lifetime physical activity and prostate cancer incidence and mortality[END_REF]. Conversely being physically active is associated with decreased risk.

How much physical activity should I do?

The American College of Sports Medicine and the American Heart Association have put out physical activity recommendations for healthy adults aged between 18-65 [START_REF] Haskell | Physical activity and public health: updated recommendation for adults from the American College of Sports Medicine and the American Heart Association[END_REF]. The core recommendations are that individuals should aim for a minimum of 30 minutes on 5 days each week of moderate intensity aerobic exercise (brisk walking is an example). This can be substituted by vigorous intensity exercise for a minimum of 20 minutes 3 days each week (jogging is an example). A combination of moderate and vigorous exercise can be used to meet the guidelines. In addition strength training 2 days a week using the major muscle groups with a load that allows 8-12 repetitions to volitional fatigue is also recommended. For healthy older adults the recommendations are essentially the same but the addition of balance and flexibility training should also be considered. Frequently in the research literature when advising/assessing level of physical activity, 150 minutes/week of moderate intensity physical activity is used as a level to aim for. For both clinical and research purposes this is a useful guide as it is clear and easily remembered.

Although these guidelines give a benchmark there is a strong dose response rate and doing more is associated with increased benefits [START_REF] Haskell | Physical activity and public health: updated recommendation for adults from the American College of Sports Medicine and the American Heart Association[END_REF]. This is highlighted by the Cancer Research Fund/American Institute for Cancer Research who recommend at least 60 minutes a day of physical activity. This is based on research findings that indicate that the benefits of decreased risk of cancer are more evident at higher levels of physical activity [START_REF] Orsini | Association of physical activity with cancer incidence, mortality, and survival: a population-based study of men[END_REF][START_REF] Vogel | Health benefits of physical activity in older patients: a review[END_REF].

It must be noted however that physical activity that does not reach the level of the guidelines is still beneficial [START_REF] Rhodes | Characteristic of physical activity guidelines and their effect on adherence[END_REF][START_REF] Chodzko-Zajko | American College of Sports Medicine position stand. Exercise and physical activity for older adults[END_REF]. Bouts of appropriately intense activity of 10 minutes or more can also count towards daily quota [START_REF] Haskell | Physical activity and public health: updated recommendation for adults from the American College of Sports Medicine and the American Heart Association[END_REF]. Although episodic accumulation has been shown to improve fitness to the same degree as a continuous session it has not been firmly established whether equivalent health benefits are also gained [START_REF] Murphy | Accumulated versus continuous exercise for health benefit: a review of empirical studies[END_REF].

What should I do?

Walking, cycling, running, aerobic sports, land/water based aerobics classes, dancing, swimming, stair climbing, gym, gardening/yard work and other similar activities done at an appropriate intensity all count towards physical activity. Which is most appropriate depends on an individual's lifestyle, what is enjoyable to the individual and presence of other pathologies. An example of a pathology that might influence the type of advised physical activity could be OA. For a patient with severe knee OA it may be better to recommend cycling/pool based exercise rather than jogging or high impact running sports [START_REF] Stehling | Patellar Cartilage: T2 Values and Morphological Abnormalities at 3.0-T MR Imaging in Realtion to Physical Activity in Asymptomatic Subjects from the Osteoarthritis Initiative[END_REF]. Other factors such as osteopenia should also be taken into account. Weight bearing impact exercise such as running based sports, high impact aerobics or walking with a weighted back pack are better for maintenance of bone density than swimming or cycling [START_REF] Skelton | Physical activity and Osteoporosis[END_REF][START_REF] Rector | Participation in road cycling vs running is associated with lower bone mineral density in men[END_REF]. If sarcopenia is a concern the addition of a strengthening program may be appropriate. Exercise prescription may also vary with age, an example being the addition of balance training exercises in older adults [START_REF] Church | When will we treat physical activity as a legitimate medical therapy….even though it does not come in a pill?[END_REF][START_REF] Vogel | Health benefits of physical activity in older patients: a review[END_REF]. Recommending increased physical activity is the first step, but for motivated individuals tailored programs can be designed to best match individual needs [START_REF] Church | When will we treat physical activity as a legitimate medical therapy….even though it does not come in a pill?[END_REF]. The current evidence base suggests that, even in those with health problems there should be some performance of exercise/physical activity [START_REF] Newton | Exercise in prevention and management of cancer[END_REF][START_REF] Kujala | Evidence on the effects of exercise therapy in the treatment of chronic disease[END_REF]. Is it too late?/ Am I too old to start?

The scientific evidence suggests; those who are middle aged or elderly and physically active should continue to be so. Those who are not should increase their physical activity levels [START_REF] Vogel | Health benefits of physical activity in older patients: a review[END_REF]. Stressman et al [START_REF] Stessman | Physical activity, function, and longevity among the very old[END_REF] found that continuing and indeed commencing physical activity once very old was associated with better survival and function. King et al [START_REF] King | Turning back the clock: adopting a healthy lifestyle in middle age[END_REF] found those in middle age who adopted a healthy lifestyle (5 fruit and vegetables, exercise, healthy weight and no smoking) had statistically significant reduction in CVD and all cause mortality. Similarly, Yates et al [START_REF] Yates | Exceptional longevity in men: modifiable factors associated with survival and function to age 90 years[END_REF] found that smoking abstinence, weight management, blood pressure control and regular exercise were associated with extended lifespan, good health and improved function during older age. Byberg et al [START_REF] Byberg | Total mortality after changes in leisure time physical activity in 50 year old men: 35 year follow-up of population based cohort[END_REF] found that a cohort of 50 year old men who increased their physical activity levels achieved a similar reduction in mortality to that of men who had always been physically active. It took 10 years however for the groups to reach parity.

In addition to the benefits previously mentioned, there is growing evidence that physical activity helps to maintain cognitive performance and decreases the risk of dementia in the elderly [START_REF] Vogel | Health benefits of physical activity in older patients: a review[END_REF].

What is the point I've never been able to lose weight or give up smoking?

Ideally, to gain maximal health benefits, men should exercise, not smoke, eat a healthy diet and have a BMI of less than 25 [START_REF] Lee | Combined effects of cardiorespiratory fitness, not smoking, and normal waist girth on morbidity and mortality in men[END_REF]. The more of these healthy traits an individual has the less likely they are to develop a range of chronic disorders. It is obviously desirable for an individual to give up smoking and maintain a healthy weight range but if they can not they will still gain health benefits from increasing their physical activity [START_REF] Lee | Combined effects of cardiorespiratory fitness, not smoking, and normal waist girth on morbidity and mortality in men[END_REF][START_REF] Hu | Joint effects of physical activity, body mass index, waist circumference, and waist-to-hip ratio on the risk of heart failure[END_REF][START_REF] Sui | The Influence of Cardiorespiratory Fitness on Lung Cancer Mortality[END_REF][START_REF] Fulton | Physical activity: an investment that pays multiple health dividends: comment on "combined effects of cardiorespiratory fitness, not smoking, and normal waist girth on morbidity and mortality in men," "physical activity and survival in male colorectal cancer survival[END_REF].

It is important to explain the health benefits of losing weight or giving up smoking, but if a patient can not, they should still be encouraged to be more physically active.

I'm too busy/ I don't have time Many people have very busy lives but given the health benefits, time should be found for regular physical activity. Ideally time should be set aside each day for exercise. If this is not possible then increased physical activity can be incorporated into other activities. More data is emerging on the impact of active commuting (walking/cycling). Current evidence from epidemiological studies indicate that active commuting is associated with decreased cardiovascular risk factors [START_REF] Von | Commuting physical activity is favourably associated with biological risk factors for cardiovascular disease[END_REF][START_REF] Gordon-Larsen | Active commuting and cardiovascular disease risk: the CARDIA study[END_REF]. Hamer and Chida [START_REF] Hamer | Active commuting and cardiovascular risk: A meta-analytic review[END_REF] in their meta-analytical review found that active commuting decreased the risk of cardiovascular disease but the effect was more robust amongst women than men. Given that blood pressure, BMI, insulin and triglyceride levels are all positively influenced, active commuting should be encouraged as a strategy for increasing physical activity [START_REF] Gordon-Larsen | Active commuting and cardiovascular disease risk: the CARDIA study[END_REF].

Multiple small bursts of physical activity can also be useful. Parking further from destination and walking some of the way, taking stairs instead of the lift and doing more energetic housework/gardening are all ways that individuals can increase their physical activity levels in bursts without having to find lots of extra time. Although studies show that from a fitness perspective accumulated short bouts of physical activity have a similar effect as the same amount of time of continuous activity, the research has not been conducted to confirm that the health benefits are the same [START_REF] Murphy | Accumulated versus continuous exercise for health benefit: a review of empirical studies[END_REF].

Summary

Apart from not smoking, being physically active is the most powerful lifestyle choice an individual can make for improved health outcomes [START_REF] Church | When will we treat physical activity as a legitimate medical therapy….even though it does not come in a pill?[END_REF]. Booth and Leye [40] note that long term exercise and physical activity result in positive adaptations to almost every cell type, tissue and system in the body. The same authors make the point that this could also be viewed as, physical inactivity results in widespread pathophysiological changes [40]. Perhaps our bodies have evolved to function optimally on a certain level of physical activity that many people just do not achieve in our modern day society? The results are witnessed by a broad range of health care professionals, across many different specialties and in many health care settings. Men and women of all ages who do not meet the minimal physical activity guidelines should be encouraged to be more physically active.

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11 of 13 International Journal of Clinical Practice International Journal of Clinical Practice

  40. Booth FW, Laye MJ. Lack of adequate appreciation of physical exercise's complexities can pre-empt appropriate design and interpretation in scientific discovery. J Physiol. 2009 Dec;587(23):1-13.

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Santos Martins

M Rodrigues

A Proportional Integral Feedback for Open Channels Control trough LMI Design

This paper deals with the controller design for systems described by Partial Differential nonlinear Equation (PDE) of Saint-Venant. The proposed approach is based on Multi-Models concept which takes into account Linear Time Invariant models defined around a set of operating points. This method allows to describe the dynamic of this nonlinear system over a wide operating range. By the means of an Internal Model Boundary Control (IMBC), the design of a Proportional Integral (PI) feedback is performed through Linear Matrix Inequality (LMI). The method is applied to simulation and also compared to previous experimentations on a micro-channel, illustrating the new theoretical results developed in the paper.

INTRODUCTION

Regulation of open channels represents an economic and environment interest and many research are done in this area. Indeed, water losses in open channels, due to inefficient management and control, may be large. In order to deliver water, it is important to ensure that the water level and the flow rate in the open channel remain at certain values [START_REF] Mareels | Systems engineering for irrigation systems: Successes and challenges[END_REF]. The difficulty of this control system is that only the gates positions are able to meet performance specifications so a specific design with boundary control laws satisfying the control objectives is required. This problem has been previously considered in the literature using a wide variety of technics see [START_REF] Malaterre | Classification of canal control algorithms[END_REF][START_REF] Weyer | Decentralised pi controller of an open water channel[END_REF]. Some of them take into account the uncertainties and apply robust control approach (see [START_REF] Litrico | Robust continuous-time and discrete-time flow control of a dam-river system: (i) modelling[END_REF][START_REF] Litrico | Robust continuous-time and discrete-time flow control of a dam-river system: (ii) controller design[END_REF] e.g.). Studying directly the nonlinear dynamics is also possible as in [START_REF] Zaccarian | Anti-windup for marginally stable plants and its application to open water channel control systems[END_REF][START_REF] Litrico | Experimental validation of a methodology to control irrigation canals based on saint-venant equations[END_REF][START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF]. Recent approaches consider the distributed feature of the system. Using the Riemann coordinates approach on the Saint-Venant equations, stability results are given in [START_REF] Greenberg | The effect of boundary damping for the quasilinear wave equations[END_REF] for a system of two conservation laws, and for system of larger dimension. Lyapunov technics have been used in [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF][START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF].

In practice, process industries as chemical, water treatment processes are characterized by complex processes which often operate in multiple operating regimes. It is often difficult to obtain nonlinear models that accurately describe plants in all regimes. Also, considerable effort is required for development of nonlinear models. An attractive alternative to nonlinear technique is to use a multi-linear model strategy. Multi-linear models methods are based on the partitioning of the operating range of a system into separate regions and applying local linear models to each region [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF]. The Multi-Models structure is well adapted for nonlinear systems because this structure allows to determine a set of linear models defined around some predefined operating points. Each local model (submodel) is defined as a LTI dynamic system defined for a specific operating point. The Multi-Models philosophy is based on weighting functions which ensure the transi-tion between the different locals models. These functions represent the degree of validity of each local models and it depends on the system inputs and outputs which vary with time. The multi-model approach has been often used in recent years for modelling and control of nonlinear systems [START_REF] Theilliol | A multiple model based approach for Fault Tolerant Control in nonlinear systems[END_REF][START_REF] Porfirio | Multi-model predictive control of an industrial c3/c4 splitter[END_REF][START_REF] Athans | Issues on robust adaptive feedback control[END_REF] and for fault diagnosis [START_REF] Bhagwat | Multilinear model-based fault detection during process transitions[END_REF][START_REF] Gatzke | Use of multiple models and qualitative knowledge for on-line moving horizon disturbance estimation and fault diagnosis[END_REF][START_REF] Rodrigues | A fault detection and isolation scheme for industrial systems based on multiple operating models[END_REF]. In the Multi-Models concept, some authors speak about gain scheduling strategy which is well detailed in [START_REF] Leith | Survey of gain-scheduling analysis and design[END_REF] or for interpolated controllers or switching controllers [START_REF] Narendra | Adaptation and learning using multiple models, switching and tuning[END_REF][START_REF] Aberkane | Output feedback control of a class of stochastic hybrid systems[END_REF].

The use of Multi-Models representation for stability study of systems described by nonlinear PDE is not present in the literature. More generally, common approaches are based on a finite dimensional approximation of the nonlinear PDE and adaptive control. The stability for such systems is still an open problem. In this paper, an analysis of the stability of the nonlinear PDE of Saint-Venant is proposed by the use of the Multi-Models and IMBC [START_REF] Toure | Semigroup formalism and internal model control for a heat exchanger[END_REF] structures. A previous work from the authors has been published with the use of an Integral control. The goal of this paper is to extend the theory with the use of a PI Control. The stability in Multi-Models framework is often performed by Linear Matrix Inequality (LMI) due to the effectiveness for calculating a gain solution for multiple models [START_REF] Lopez-Toribio | A mutiple-model approach to fault-tolerant control using takagi-sugeno fuzzy modelling: real application to an induction motor drive system[END_REF][START_REF] Rodrigues | Fault tolerant control design for polytopic lpv systems[END_REF].

The paper is organized as follows: firstly, the Saint-Venant equations are presented as well as the control problem. The Internal Model Boundary Control is explained and the physical constraints are given. Secondly, the linearized systems are developed around equilibrium sets which depend on the space variable. Their insertion into the LMI formalism are also described into this second part. The third part of the paper is dedicated to the design of the feedback gain by LMI which ensures the stability of the system: a Proportional Integral (PI) controller is implemented using a LMI approach and the local stability of each systems. The last section is dedicated to the simulations and comparison with previous experimentations. Comparisons between initial experimental results using a PI-controller (done some years ago) and simulations with the new PI-controller using the LMI gain calculated through this paper, are realized.

PROBLEM STATEMENT ABOUT CHANNEL REGULATION

Let us consider the following class of water channels represented on the figure (1), i.e. a reach of an open channel delimited by underflow and/or overflow gates where:

• Q(x, t) is the water flow rate,

• Z(x, t) is the height of water channel,

• L is the length of the reach taken between the upstream x = 0 and the downstream x = L, • U 0 (t), U L (t) are the opening of the gates at upstream and downstream. The regulation problem concerns the stabilization of the water flow rate and/or the height of the water around an equilibrium for a reach denoted by (z e (x), q e (x)). A linear model with variable coefficients can be deduced from the nonlinear PDE, in order to describe the variation of the water level and flow on an open channel. Let recall these models of [START_REF] De Saint-Venant | Théorie du mouvement non permanent des eaux avec applications aux crues des rivières et à l'introduction des marées dans leur lit[END_REF].

A model of a reach

We suppose that the channel has a sufficient length L such that we can consider that the lateral movement is uniform. Nonlinear PDE of de Saint-Venant which describe the channel are the following [START_REF] Georges | Automatique pour la Gestion des Ressources en Eau[END_REF][START_REF] Malaterre | Le contrôle automatique des canaux d'irrigation : Etat de l'art et perspectives[END_REF]:

∂ t Z = -∂ x Q b , (1)
∂ t Q = -∂ x ( Q 2 bZ + 1 2 gbZ 2 ) + gbZ(I -J), (2)
Z 0 (x) = Z(x, 0), Q 0 (x) = Q(x, 0), ( 3
)
where I is the slope, b is the channel width, g is the gravity constant. J is the friction slope from the formula of Manning-Strickler and R is the hydraulic radius. J and R are defined such that:

J = n 2 Q 2 (bZ) 2 R 4/3 , R = bZ b + 2Z . ( 4
)
The different limits conditions bring us to consider 2 control cases (mono or multi variable control). Here, the control of two underflow gates is considered.

Multi-variable control :

The equation of the upstream condition of the reach

(x = x up ) is given by Q(x up , t) = U up (t)Ψ 1 (Z(x up , t)), with Ψ 1 (Z) = K 1 2g(z up -Z).
The other control at downstream of the reach, i.e. in x = x do (Fig. 1) is given by: Q(x do , t) = U do (t)Ψ 3 (Z(x do , t)), where Ψ 3 (Z) = K 2 2g(Zz do ) and U do (t) is the downstream control of the reach, z do is the water height on downstream of the gate (cf. figure 1). Remark 1. Upstream and downstream depend of the considered reach, it is the same thing for abscissa and gates.

A regulation model

An equilibrium point of the system verifies the following equations:

∂ x q e = 0 ∂ x z e = gbz e I + J e + 4 3 J e 1 1+2ze/b gbz e -q 2 e /bz 2 e , (5)
Remark 2. The fluvial case is considered and it follows that:

z e > 3 q 2 e /(gb 2 ). ( 6
)
Let denote that q e is constant but that z e depends of variable space. The linearized model around an equilibrium point (z e (t) q e (t)) t is, with ξ(x, t) = (z(x, t) q(x, t)) t , ∀t > 0, x ∈ Ω =]0, L[ the linearized state variables:

∂ t ξ(x, t) = A 1 (x)∂ x ξ(x, t) + A 2 (x)ξ(x, t) (7)
ξ(x, 0) = ξ 0 (x) q(x up ) = u up,e ∂ z Ψ 1 (z e (x up ))z(x up ) + u up Ψ 1 (z e (x up )), q(x do ) = u do,e ∂ z Ψ 3 (z e (x do ))z(x do ) + u do Ψ 3 (z e (x do )) ∀t > 0, where u up,e , u do,e are the openings gates for the upstream and downstream at the equilibrium and u up (t), u do (t) are the variations of these openings gates to be controlled. The matrices A 1 (x), A 2 (x) are given by:

A 1 = 0 -a 1 -a 2 -a 3 , A 2 = 0 0 a 4 -a 5 , (8)
with

a 1 (x) = 1/b, a 2 (x) = gbz e (x) - q 2 e bz 2 e (x) , a 3 (x) = 2qe bze(x) , a 5 (x) = 2gbJe(x)ze(x) qe , a 4 (x) = gb(I + J e (x) + 4 3 Je(x) 1+2ze(x)/b ).
The control problem is to find the variations of u up (t) at extremity x = x up and u do (t) at the extremity x = x do of the reach such that downstream water level, z(x do , t) = z(L, t) (measured variables), track a reference signal r(t). The reference signal r(t) is chosen for all cases or constant or non-persistent (a stable step answer of a non-oscillatory system). In this paper, the control scheme based on the Internal Model Boundary Control (IMBC) [START_REF] Santos | Contrôle frontière par modèle interne de systèmes hyperboliques : Application à la régulation de canaux d'irrigation[END_REF][START_REF] Santos | Multivariable boundary control approach by internal model, applied to irrigations canals regulation[END_REF] is adopted as illustrated on figure [START_REF] Athans | Issues on robust adaptive feedback control[END_REF]. This control strategy integrates the process model in real time and allows to regulate the water height in all the points of the channel by taking into account the error between the model and the system.

• M f is the linear filtering model of finite dimension.

• M r is the pursuit model which allows to set a dynamic in regards of the fixed instruction r(t).

Fig. 2. IMBC structure: Internal Model Boundary Control

In order to control the water level over a wide operating range, we consider a set of models established around judicious operating points: each model is an approximation of the process in a small interval of the operating range and it should be activated to synthesize a control on this interval. The idea is to construct a set of predefined models in order to control the system over all the operating range.

A Multi-Models representation

The Multi-Models structure [START_REF] Rodrigues | A fault detection and isolation scheme for industrial systems based on multiple operating models[END_REF] allows to control the system over a wide operating range because it takes into account the different sub-models which can be activated under different operating regimes [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF][START_REF] Rodrigues | A fault detection and isolation scheme for industrial systems based on multiple operating models[END_REF]. The representation of Saint-Venant's PDE around N operating points by the Multi-Models approach is defined by the following equations:

∂ t ξ(x, t) = N i=1 µ i (ζ(t))A i (x)ξ(x, t), ∀t > 0, x ∈ Ω (9) A i (x) = A 1,i (x)∂ x + A 2,i (x) (10)
ξ 0 (x) = ξ(x, 0) F b ξ(t) = B b u(t) on Γ = ∂Ω, ∀t > 0 (11)
• A i (x) is the operator which corresponds to the i th equilibrium state. • ζ(t) is a function depending of some decision variables directly linked with the mesurables states variables and eventually to the input.

• µ i (ζ(t))
is the weighting functions which determines the sub-model for the control law synthesis depending of the output height of the process z L .

The equation (9) describes the system dynamic in open loop. In this representation, the state vector ξ(x, t) is not explicitly linked with the boundary control. In order to design an output feedback and to study the stability in closed loop, an operator D of distribution of the boundary control is introduced, it is a bounded operator such that Du ∈ Ker(A), [START_REF] Santos | Contrôle frontière par modèle interne de systèmes hyperboliques : Application à la régulation de canaux d'irrigation[END_REF]:

ξ(x, t) = ϕ(x, t) + Du(t). ( 12
)
This operator is naturally null in the domain of A(x) as it is active only on the boundary of the domain. This change of variables allows to get a Kalman representation of the system:

∂ t ϕ(x, t) = N i=1 µ i (ζ(t)) A i (x)ϕ(x, t) -D u (13) ϕ(x, 0) = ϕ 0 (x) = ξ 0 (x) -Du(0). (14)
A Multi-Model approach can be developed and made possible the study of the stability by the second theory of Lyapunov.

In the following paragraph, we will focus on the synthesis of a control law by LMI technics. An output feedback is considered under a hypothesis of an integral control, so as to do a synthesis of a gain by LM I which ensures the stability of the system.

STABILITY STUDY BY LM I

In this part, the closed loop structure (Fig. 2) is studied under a proportional integral feedback. The pursuit model (M r ) and filtering model (M f ) are not considered.

Closed-loop structure for a Proportional-Integral feedback

For a control with an output feedback, K i and K p are defined as the gains, u (t) = K p ε(t) + K i ε(t)dt , it follows that [START_REF] Santos | Contrôle frontière par modèle interne de systèmes hyperboliques : Application à la régulation de canaux d'irrigation[END_REF]: [START_REF] Litrico | Robust continuous-time and discrete-time flow control of a dam-river system: (i) modelling[END_REF] with y(x, t) = Cξ(x, t) and equation ( 12), one deduce:

ε(t) = r(t) -y(t) (15)
u(t) = K p [r(t) -y(t)] + K i [r(τ ) -y(τ )]dτ
y(x, t) = Cϕ(x, t) + CDu(t) (17)
To clarify equations, the index i of the i-th equilibrium is omitted in the following (e.g. A replace A i ). By replacing y(x, t) into the control equation:

u = K p [r -Cϕ -CDu] + K i [r -Cϕ -CDu] dτ ⇒ u = K p [ ṙ -C φ -CD u] + K i [r -Cϕ -CDu] = K p [ ṙ -CAϕ] + K i [r -Cϕ -CDu]
using the change of variables [START_REF] Georges | Automatique pour la Gestion des Ressources en Eau[END_REF] and the fact that Im(D) ⊂ Ker(A) (See [START_REF] Santos | Contrôle frontière par modèle interne de systèmes hyperboliques : Application à la régulation de canaux d'irrigation[END_REF]). Then u is introduced into the equation ( 13) and, the closed-loop expression is then

∂ t ϕ(x, t) = N i=1 µ i (ζ(t)) DK i (CDu(t) -r(t)) -DK p ṙ(t) + (A i (x) + DK p CA i + DK i C) ϕ(x, t) (18)
Let define:

K in = DK i K pr = DK p (19)
The equation ( 18) can be written as

∂ t ϕ = N i=1 µ i (ζ) A i + K pr CA i + K in C ϕ + K in (CDu -r) -K pr ṙ . ( 20
)
The conditions which ensure the stability are ensured by using a quadratic Lyapunov function [START_REF] Rodrigues | Fault tolerant control design for polytopic lpv systems[END_REF][START_REF] Chadli | Output stabilisation in multiple model approach[END_REF], in order to guarantee the convergence of the water height to the reference r(t), over the widest operating range.

Stability study with a quadratic Lyapunov function

Let us consider:

V (ϕ(x, t), t) = ϕ T (x, t)P ϕ(x, t).

(21) The Multi-Models representation of the linearized PDE of Saint-Venant defined by equation ( 20) is asymptotically stable if there exists a matrix P > 0 such that 1 : V (ϕ, t) = φT P ϕ + ϕ T P φ < 0, [START_REF] Murray-Smith | Multiple Model Approaches to Modelling and Control[END_REF] Then, it follows this inequality

N i=1 µ i (ζ) A i + K pr CA i + K in C ϕ (23) + K in (CDu -r) -K pr ṙ T P ϕ +ϕ T P N i=1 µ i (ζ) A i + K pr CA i + K int C ϕ (24) + K in (CDu -r) -K pr ṙ < 0 ( 25
)
The development of this inequality leads us to consider an inequality for each i such that:

ϕ T A i + K pr CA i + K in C T P ϕ +ϕ T P A i + K pr CA i + K in C ϕ +[ K in (CDu -r)] T P ϕ + ϕ T P [ K in (CDu -r)]
-( K pr ṙ) T P ϕϕ T P ( K pr ṙ) < 0 (26) Remark 3. In a first approach, the particular case K prop = K int is taken and so one get K pr = K in = K. Then equation ( 26) become:

ϕ T A i + K CA i + KC T P ϕ + [ K(CDu -r)] T P ϕ +ϕ T P A i + K CA i + KC ϕ + ϕ T P [ K(CDu -r)] -( K ṙ) T P ϕ -ϕ T P ( K ṙ) < 0 ( 27
)
In the inequality [START_REF] Theilliol | A multiple model based approach for Fault Tolerant Control in nonlinear systems[END_REF], which defines the stability condition of the system, the control parameter u appears in this inequality and it is a difficulty for the design of the gain K. Let us consider the following equality deduced from [START_REF] Litrico | Robust continuous-time and discrete-time flow control of a dam-river system: (ii) controller design[END_REF]:

CDu(t) -r(t) = Cξ(x, t) -r(t) -Cϕ(x, t) (28)
Proposition 1. If there exists a matrix P positive definite, a matrix W and a scalar α such that the following statements hold true:

a) ϕ T P K (CDu(t) -r(t)) ≤ αϕ T P KCϕ, (29) b) A T i P + P A i + W C i + C T i W T < 0, with K = P -1 W and with C i = CA i + (1 + α)C c)
r is constant by piecewise, then the system (13) with a proportional integral control input ( 16) is stable. 1 We suppose that ∂tψ = ψ whatever the function ψ.

Proof 1. Only a sketch of the proof is given here. For the integral case, one can found it in [START_REF] Diagne | Une approche Multi-modèles des équations de Saint-Venant : une analyse de la stabilité par techniques LMI[END_REF].

Let consider the quadratic Lyapunov function V (ϕ(x, t), t) = ϕ T (x, t)P ϕ(x, t)

then one can wrote V (t) < 0 such that ( 27) can be upper bounded. Indeed, the use of inequality [START_REF] Weyer | Decentralised pi controller of an open water channel[END_REF] implies that (the same is done for the transposed expression)

ϕ T P A i + K CA i + KC ϕ + ϕ T P [ K(CDu -r)] -ϕ T P ( K ṙ) < ϕ T P A i + K CA i + (1 + α) KC ϕ -ϕ T P ( K ṙ) < ϕ T P A i + KC i = ϕ T [P A i + W C i ] ϕ < 0 (30)
with C i = CA i + (1 + α)C, W = P K and using the condition a) firstly, the third c) and the second condition b) of the proposition.

The gain K has been implemented into the model of simulations so as to verify the stability of the system. The results have been obtained for a single reach with two underflow gates. The aim is to compare the simulations and experimental curves obtained with this method and the ones obtained experimentally by Dos Santos Martins in this works [START_REF] Santos | Contrôle frontière par modèle interne de systèmes hyperboliques : Application à la régulation de canaux d'irrigation[END_REF][START_REF] Santos | Multivariable boundary control approach by internal model, applied to irrigations canals regulation[END_REF].

SIMULATIONS RESULTS

Firstly, let describe the benchmark used for the simulations and the experimentations, described in the second and third subsections respectively.

Configuration and data of the channel

For this study, the following set of parameters from the practical Valence's channel (Fig. 3) is considered where the data are defined such that: In this single reach with two gates, the regulation of the water height Z L at x = L, is done by controlling the openings U 0 (t) and U L (t) of the gates at upstream and downstream respectively: it is a multi-variable control (cf Fig.

( 1)).

The equilibria profiles have been chosen such that the calculated control law from the local models can be efficient over all the operating range of the water height [START_REF] Santos | Contrôle frontière par modèle interne de systèmes hyperboliques : Application à la régulation de canaux d'irrigation[END_REF]. Let notice that it has been experimentally verified that a local model is valid around ±20% of an equilibrium profile. In order to assign references which are included between 0.6dm and 2dm, the operating points at x = 0 are the following:

Simulations

These results are obtained from an IMB Control and a Multi-Models approach with a LM I gain previously calculated. The figure [START_REF] Chadli | Output stabilisation in multiple model approach[END_REF] shows that the output converges to the reference even if this one strongly varies (variations> 100%). The reference tracks a slow dynamic and one can see that the convergence of the output is good. The curves that describe the upstream and downstream gates openings of the reach are given by the figure [START_REF] Coron | A strict lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]. The convergence of the output to the reference is ensured even when the reference is decreasing or increasing.

The following simulations compare an integral and a PI controllers which gains have been calculated by the LMI approach, Fig ( 6)- [START_REF] Diagne | Une approche Multi-modèles des équations de Saint-Venant : une analyse de la stabilité par techniques LMI[END_REF]. The PI performs better than the integral control as expected.

Next simulations are a comparison between simulations using the theoretical gain obtains through LMI approach (I and PI control) with the first tests realized some years ago by [START_REF] Santos | Multivariable boundary control approach by internal model, applied to irrigations canals regulation[END_REF], using an experimental Multi-Models gain, without any theoretical study. The figure [START_REF] Santos | Contrôle frontière par modèle interne de systèmes hyperboliques : Application à la régulation de canaux d'irrigation[END_REF] represents the dynamic evolution of the simulated system and the experimental data obtained with the quite same references. This study is based on the previous works of Rodrigues [START_REF] Rodrigues | A fault detection and isolation scheme for industrial systems based on multiple operating models[END_REF] and from works of Dos Santos [START_REF] Santos | Contrôle frontière par modèle interne de systèmes hyperboliques : Application à la régulation de canaux d'irrigation[END_REF][START_REF] Santos | Multivariable boundary control approach by internal model, applied to irrigations canals regulation[END_REF].

Experimentations have been realized into the Valence channel (Fig. 3) with a Multi-Models approach and a gain calculated via the LMI approach. For these experimentations, the wide range of the accessible water level is attempt for an integral controller, but authors did not get time to implement their PI controller. For the integral case, relevant experimentations have been included into an article, which is actually under revision.

CONCLUSION

The first results trough the use of multi-models approach dedicated to irrigation channels control, under an IMBC structure, have been realized some years ago [START_REF] Santos | Contrôle frontière par modèle interne de systèmes hyperboliques : Application à la régulation de canaux d'irrigation[END_REF]. These good experimental results, but without stabilizable theoretical approach, were obtained. In this paper, the authors have formalized and extended their LMI approach of this regulation problem by the synthesis of a PI Controller. Simulations have shown the improvements realized towards the initial multi-models approach trough this new PI feedback controller designed by LMI.

Fig. 1 .

 1 Fig. 1. Channel scheme: two underflow gates

Fig. 3 .

 3 Fig. 3. Valence channel scheme

Fig. 4 .

 4 Fig. 4. Variations of the reference along the valued domain

Fig. 6 .Fig. 8 .Fig. 9 .

 689 Fig. 5. Gates opening

Table 1 .

 1 Initial set points for the simulations

		Simulations
	z e1 (x = 0)	z e2 (x = 0)	z e3 (x = 0)
	0.625dm	0.9375dm	1.40625dm
	In this application, the weighting function µ i (ζ(t)) is equal
	to 1 if the output's height is included into the validity
	domain of the model and 0 in the other case for each
	operating state. The parameter ζ(t) exclusively depends
	on the output which is the only one variable of decision in
	this precise case.