D AUROUX AND S BONNABEL
13 pages
English
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D AUROUX AND S BONNABEL

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13 pages
English

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Niveau: Secondaire
D. AUROUX AND S. BONNABEL 1 Symmetry-based observers for some water-tank problems Didier Auroux and Silvere Bonnabel Abstract—In this paper we consider a tank containing fluid and we want to estimate the horizontal currents when the fluid surface height is measured. The fluid motion is described by shallow water equations in two horizontal dimensions. We build a simple non-linear observer which takes advantage of the symmetries of fluid dynamics laws. As a result its structure is based on convolutions with smooth isotropic kernels, and the observer is remarkably robust to noise. We prove the convergence of the observer around a steady-state. In numerical applications local exponential convergence is expected. The observer is also applied to the problem of predicting the ocean circulation. Realistic simulations illustrate the relevance of the approach compared with some standard oceanography techniques. Index Terms—Observer, symmetries, wave equation, shallow water model, estimation, Lie group, oceanography, data assimi- lation. I. INTRODUCTION The following study is derived from a data assimilation problem in oceanography. The problem considered in this paper consists in estimating the state of a fluid in a water tank where the surface height is measured everywhere. In this paper we propose a symmetry-based non-linear infinite dimensional observer and we prove the convergence when the fluid motion is described by linearized wave equations under shallow water approximations.

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D. AUROUX AND S. BONNABEL
Symmetry-based observers for some water-tank problems Didier Auroux and Silvere Bonnabel
1
Abstract —In this paper we consider a tank containing uid and output estimation error (correction term)”. For this reason, they we want to estimate the horizontal currents when the uid surface do not take into account the symmetries of the model. There height is measured. The uid motion is described by shallow water equations in two horizontal dimensions. We build a simple ehnasginbeeeerninrgeceprnotbwleomrksownhoenbstehrveermdoedseilgnisandnistyemdmimeterinessiofnoarl non-linear observer which takes advantage of the symmetries of uid dynamics laws. As a result its structure is based on and when there is a Lie group acting on the state space [3], convolutions with smooth isotropic kernels, and the observer is [2], [32], [10], [11]. Symmetries provide a helpful guide to remarkably robust to noise. We prove the convergence of the si observer around a steady-state. In numerical applications local de gn non-linear correction terms. Indeed the only difference exponential convergence is expected. The observer is also applied between the observer and model equations comes from the to the problem of predicting the ocean circulation. Realistic correction term. Linear systems are invariant by scaling, and simulations illustrate the relevance of the approach compared so is the correction term in general (Luenberger observer, with some standard oceanography techniques. Kalman lter). But when the system is non-linear, there is Index Terms —Observer, symmetries, wave equation, shallow no reason why the correction term should have a linear water model, estimation, Lie group, oceanography, data assimi-form (extended Kalman lter). When this term is bound to lation. preserve symmetries, it has a non-linear structure based on the specic nonlinearities of the system, and the observer is I. I NTRODUCTION calledinvariant”,orsymmetry-preserving”.Theresulitsthat Thefollowingstudyisderivedfromadataassimilationtchoeoredsitniamtaetsi,oansnddothneotesdtiempeatnedsosnhaarrebitcroarmymcohnoicpehsysoifcaulniptrsopor -problem in oceanography. The problem considered in this paper consists in estimating the state of a uid in a water tank erties with the true physical variables (in the examples given where the surface height is measured everywhere. In this paper in [10], estimated chemical concentrations are automatically we propose a symmetry-based non-linear innite dimensional p S o O si ( ti 3 v ) e),.Ienstismoamteedcarsoetsa,titohnemerartorircessysateutmomevateincalplryesbeenltosnvgertyo observer and we prove the convergence when the uid motion isdescribedbylinearizedwaveequationsundershallowwaternicLeoporkoinpegrtaite[s3(4a],ut[o1n2o],m[o1u0s],etrhreordeesqiugantimonetihnod[1o1f],sy[2m8m])e.try-approximations. preserving observers could be summed up this way: the non-Over the last years much attention has been devoted to the motionplanningandfeedbackstabilizationofauidunderltihneeagraifnosrmareofttuhneedobassesrigvneirnigstghievepnolbeystohfetshyememrreotrriessy,staenmd shallow water approximations, problem raised by [18], [37]. A related problem is the control of ows described by Saint- around a trajectory or a steady-state. This is always possible as Venantequationsinchannels[15],[14],[13],[33],[17].FewertaoroLuunednabnerygestreoadbsye-rstvaetres,[i1n0v]a.riantobserverscanbeidentied efforts have been put on the theory of observers for this kind of innite dimensional systems. Nevertheless a natural extension This paper is an extension to the innite-dimensional case ofthistheoreticalobserverproblemconsistsinoceanographicsoyfsttehemsredceesnctriibdeedasbyonordoibnsaerryvedriffdeerseigntnialanedqusaytimonmset(rOieDsEfso)r. applications, as we will see later, and extended Kalman lters-type observers are frequently used to tackle these related The Saint-Venant equations considered in this paper are indeed problems [23], [42], [20]. A different approach for observer invariant by rotation and translation ( SE (2) -invariance). In the design for ows in channels is to approximate the motion case of systems described by PDEs, the design of observers by non-linear ordinary differential equations at critical points based on the symmetries of the physical system is new to the alongthechannels[33],[9].Moregenerally,pasteffortsintheautThhoers'krsntotwhleeodrgeet.icalcontributionofthispaperistoderive theory of observers for systems described by partial differential equations (PDEs) include innite dimensional Luenberger a SE (2) -invariant observer for the problem. The correction observersforlinearsystems[29],[40].SomeotherproblemstTerhemyscdoorrneostpodnedpetnodaocnoannvyoluntoino-ntripvrioadlucchtooicfethoefocoutoprudtineartreosr. have also drawn attention recently [43], [16], [22], [21]. and a smooth isotropic kernel, a feature which ensures re-Kalman-type lters, or Luenberger observers, are usually in thestandardformcopyofthesystemplusinjectionofthefmeaartukraeb,lethreoboubsstenrevsesrtioswclhoitseeninoiistes.Waitvhourrestpoec[t4t1o]twhihserlaetttehre LaboratoireJ.A.Dieudonn´e,Universite´deNiceSophiaAntipolis,Parc authors derive a non-linear observer to estimate the velocity Valrose, 06108 Nice cedex 2, France ( auroux@unice.fr ). and pressure in an innite channel. Their observer consists CAOR, Mines-ParisTech (Ecole des Mines de Paris), 60 Bd St-Michel, 75272 Paris Cedex 06, France in a copy of the system and a correction term corresponding ( silvere.bonnabel@mines-paristech.fr ). to a one-dimensional convolution product of the output error