L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou **non**, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Keywords: **Kalman** filtering; online identification; online optimization; fuel cells; neural networks; fractional-order **systems**
1. Introduction
Solid oxide fuel cell (SOFC) **systems** [ 1 – 8 ] are promising options **for** **the** design and implementation **of** a decentralized supply **of** consumers **with** both electric and thermal energy [ 9 – 12 ]. Such kinds **of** decentralized supply cannot only be realized in scenarios in which **the** produced electric power is fed into an existing grid (serving as a practically infinitely large storage from **the** point **of** view **of** a single fuel cell **systems**). Further configurations can also be investigated in isolated applications where **the** consumers are not directly connected to an electric power grid. This second option is especially interesting **for** **the** power supply **of** construction sites (**for** example, when building up new wind farms) or when **the** power supply **of** individual houses in **small** mountain and island villages is **of** interest. To some extent, electric energy buffers will be installed in such settings, where **the** storage can be achieved by super capacitors and (Lithium-ion) batteries if short and mid-term time scales are **of** interest.

En savoir plus
1 Extraction Procedure
1.1 State Space Modeling
State space models have become a practical and powerful tool to model dynamic and complex **systems**. Closely related to **the** **Kalman** **filter**, it has been used in a wide range **of** disciplines: biology, economics, engineering, and statistics Guo(1999). **The** fundamental idea **of** **the** state space model is that **the** observed data is linearly dependent on latent variables **of** interest that vary in time. Mathematically, **the** observed data are governed by two equations, known as **the** observational and system equations. In our case, **the** observational equation expresses itself as a **linear** combination **of** three variables (common forcing, trends, and **noise**), while **the** system equations represent **the** temporal dynamics **of** **the** underlying hidden processes. **For** **the** reasons discussed above, we extend **the** univariate method to a multivariate setting. **The** statistical problem is to deduce **the** behavior **of** hidden variables **of** **the** pulse-like events from **the** observed data. **The** general implementation **of** multivariate extraction procedure is **the** main objective **of** this article. Before presenting some results on simulated data, we must introduce some notations and clarify our working hypotheses.

En savoir plus
179 En savoir plus

case, difficulties arise from **non**-commutativity. We propose in Section 6.3 a class **of** natural filters appearing as a loosening **of** **the** IEKF presented in Chapter 4, where **the** gains are allowed to be tuned by any other method than a Riccati equation. **The** pro- posed class is very broad and **the** tuning issue is far from trivial. Sections 6.4 and 6.5 explore two different routes, depending on if we are interested in **the** asymptotic or tran- sitory phase. In Section 6.4, we propose to hold **the** gains fixed over time. As a result, **the** error equation becomes a homogeneous Markov chain. We first prove some new results about global convergence in a deterministic and discrete-time framework. Then, building upon **the** homogeneity property **of** **the** error, we prove that if **the** **filter** **with** **noise** turned off admits almost global convergence properties, **the** error **with** **noise** turned on converges to a stationary distribution. Mathematically this is a very strong and to some extent surprising result. From a practical viewpoint, **the** gains can be tuned numerically to minimize **the** asymptotic error’s dispersion. This allows to “learn" sensible gains **for** very general types **of** noises. **The** theory is applied to two examples and gives convergence guarantees in each case. First an attitude estimation problem using two vector measure- ments and a gyroscope having isotropic **noise** (Section 6.4.2), then **the** construction **of** an artificial horizon **with** optimal gains, **for** a **non**-Gaussian **noise** model (Section 6.4.3). Each application is a contribution in itself and can be implemented without reading **the** whole thesis. In Section 6.5, we propose to optimize **the** convergence during **the** transi- tory phase using Gaussian approximations. We first stick to **the** general purpose **of** this thesis manuscript and adapt **the** IEKF studied in previous chapters to this discrete situ- ation. As **the** linearizations always occur around **the** same point, **the** linearized model is time-invariant and thus **the** **Kalman** gains, as well as **the** **Kalman** covariance matrix, are proved to converge to fixed values (note that when on-board storage and computational resources are very limited, this advantageously allows to replace **the** gain **with** its asymp- totic value. **The** IEKF is compared to **the** well-known MEKF [34, 71] and UKF [33, 62] on **the** attitude estimation problem, and simulations illustrate some convergence properties that **the** latter lack. In **the** case where **the** error equation is fully autonomous, we intro- duce a new method based on off-line simulations, **the** IEnKF, which outperforms **the** other filters in case **of** large noises by capturing very accurately **the** error’s dispersion.

En savoir plus
179 En savoir plus

i.e. z −1 y k = y k −1 . Although **the** associated ‘RIVBJ’ algorithm is rela- tively computationally expensive, **for** some estimation problems it proves essential (see references above **for** examples).
However, **the** minimal canonical state space form utilised to implement **the** **Kalman** ﬁlter in [ 4 ], always yields an Auto-Regressive Moving Average eXogenous variables (ARMAX) model, i.e. similar to (1) but constrained by C(z −1 ) = A(z −1 ). Furthermore, while Taylor et al. [ 7 ], and other prior work cited within, use a **non**-minimal state- space (NMSS) model **for** generalised digital control, including **linear** quadratic Gaussian (LQG) design **with** a **Kalman** ﬁlter, it is similarly limited to **the** ARMAX model form. Hence, this Letter develops a novel **extended** stochastic NMSS representation **for** **the** more general system in (1). This result completes **the** link between **the** latest RIVBJ estimation algorithm and adaptive optimal ﬁltering and can be con- veniently exploited **for** practical control system design.

En savoir plus
In Section 3, we focus on **the** invariant **extended** **Kalman** **filter** (IEKF) [6] when applied to **the** broad class **of** **systems** **of** Section 2. We consider continuous-time models **with** discrete observations, which best suits navigation **systems** where high rate sensors governing **the** dy- namics are to be combined **with** low rate sensors [14]. We change a little **the** IEKF equations to cast them into a matrix Lie group framework, more handy to use than **the** usual abstract Lie group formulation **of** [6]. We then prove, that under **the** standard convergence conditions **of** **the** **linear** case [13], applied to **the** linearized model around **the** true state, **the** IEKF is an asymptotic observer around any trajectory **of** **the** system, a rare to obtain property. This way, we produce a generic observer **with** guaranteed local convergence properties under natural assumptions, **for** a broad class **of** **systems** on Lie groups, whereas this property has so far only been reserved to specific examples on Lie groups. This also allows putting on firm theoretical ground **the** good behavior **of** **the** IEKF in practice, as already noticed in a few papers, see e.g., [3, 5, 2].

En savoir plus
For a non-linear system in which a piezoelectrically active material working against a non- linear load, the coupled analysis has found out that it is possible to[r]

132 En savoir plus

1.4 Some related open problems July 13, 2016 T.Liard, P. Lissy
equation, let us mention [ AB13 ], where a result **of** controllability in sufficiently large time **for** second order in time cascade or bidiagonal **systems** under coercivity conditions on **the** coupling terms is given. Another related result is also [ ABL13 ], where **the** case **of** two wave equations **with** one control and a coupling matrix A which is supposed to be symmetric and having some additional technical properties is investigated. Let us also mention a result in **the** one-dimensional and periodic case proved in [ RdT11 ]. In this last article, **the** authors also prove a result **for** **the** Schrödinger equation in arbitrary dimension on **the** torus, however they only obtained a result in large time, which is rather counter-intuitive and should be only technical. **The** case **of** a cascade system **of** two wave equations **with** one control on a compact manifold without boundary was treated in [ DLRL14 ], where **the** author also give a necessary and sufficient condition **of** controllability depending on **the** geometry **of** **the** control domain and coupling region. Let us emphasize that in **the** four last references, **the** results obtained in **the** case **of** abstract **systems** **of** wave equations can be applied to get some interesting results in **the** case **of** abstract heat and Schrödinger equations thanks to **the** transmutation method (see [ Phu01 ], [ Mil06 ] or [ EZ11 ]), leading however to strong (and in general artificial) geometric restrictions on **the** coupling region and control region. Let us also mention a recent result given in [ ABCO15 ], which treats **the** case **of** some **linear** system **of** two periodic and one-dimensional **non**-conservative transport equations **with** same speed **of** propagation, space-time varying coupling matrix and one control and also a nonlinear case.

En savoir plus
shows t hat the wavelet transform and the Kalman filt er are very complementary and give both correct values fo r the cycl e slips.. the id e ntity matrix.. called [r]

improves **the** performance obtained **with** **the** independent outlier indicator VBKF. Such performance gain is expected to be even larger in applications where **the** correlation among observations increases. **The** performance **of** **the** original VBKF is not shown in Fig. 5 because it is substantially worse than **the** rest **of** methods (i.e., orders **of** magnitude larger). **The** gating method works reasonably well against outliers. Although, there are still meter level degradations compared **with** **the** VBKF method in certain configurations, **for** instance when **the** signal-to-**noise** ratio is **small**. Additionally, in other applications where data becomes more correlated, it is expected that **the** generalized VBKF will outperform **the** gating approach more clearly. On **the** other hand, **the** measurement gating strategy features a much lighter imple- mentation, which could suffice in certain applications where data correlation is not severe.

En savoir plus
In nowadays highly competitive market **of** commercial air transport, another leap in engine maintenance practice is therefore warranted on **the** operator as well as on **the** manufac- turer side. **For** airlines, **the** goal is essentially twofold. On one hand, they seek to improve **the** dispatch reliability and safety **of** their fleet by reducing, among others, **the** in-flight shutdown and **the** delayed and cancelled rates. This can be achieved by moving unsched- uled events into planned engine removal. Reduction in life cycle costs is another driver **for** enhancing **the** maintenance practice. According to Marinai et al. [2004], **the** part as- sociated to **the** engines amounts to about 30% **of** **the** direct operating costs **of** an aircraft. **Of** this contribution, roughly one third is **for** maintenance. As paradoxical as it can be, manufacturers are also – if not more – interested in extending **the** “life on wing” **of** their engines through a wiser maintenance planning. This is due to a relatively recent evolution in **the** customer–supplier relationships named performance based logistics, aka. power- by-**the**-hour at Rolls–Royce [Kim et al., 2007]. In this framework, **the** customer does not buy an engine anymore, but only pays **for** it when it works i.e., when it generates revenue. Implementation **of** condition-based maintenance (CBM) [see Rajamani et al., 2004] is an essential step towards **the** achievement **of** these goals. **The** maintenance actions are planned on **the** basis **of** **the** actual health condition **of** **the** engine rather than simply based on **the** number **of** cycles. Generating a reliable information about this health condition be- comes therefore a requisite **for** an effective application **of** CBM. This explains **the** intensive research carried out since several years in engine health monitoring.

En savoir plus
208 En savoir plus

Keywords: data assimilation, **Kalman** **filter**, covariance dynamics, parameterisation **of** analysis
1. Introduction
One **of** **the** foundations **of** data assimilation is based on **the** theory **of** **Kalman** filtering. Because **of** its computational complexity and **the** extent **of** required information **for** its implementation, **the** **Kalman** **filter** (KF) has long been recognised as not viable **for** large dimension problems in geosciences. Alternative formulations, based **for** example on ensemble methods, have been developed. **The** ensemble **Kalman** **filter** (EnKF) was developed by Evensen (1994). **The** numerous formulations **of** **the** sequential algorithm or its smoother version have also had an impact on **the** variational data assimilation where new algorithms now take advantage **of** adjoint-free formulation. Considering other ensemble strategies, like particle **filter** methods in their present formulation, **the** EnKF is very robust and is used **for** atmospheric data assimilation **with** a limited ensemble **of** few dozen members.

En savoir plus
∂ t Y (0) = Y 1 ∈ L 2 (Ω) n .
(1.2)
In this context, a natural issue is **the** following: is it possible to find necessary and sufficient algebraic conditions on A and B **of** **Kalman** type that ensure **the** null controllability **of** **systems** ( 1.1 ) or ( 1.2 ), under some appropriate geometric conditions on ω (and in sufficiently large time **for** ( 1.2 ))? **The** general method that we will use in **the** article to answer this question is sometimes called fictitious control method and was first introduced in [ GBPG06 ]. It has been then used in different context, notably in [ ABCO15 ], [ CL14 ] and [ DL16 ]. Let us explain **the** strategy on equation ( 1.1 ) (this is **the** same on equation ( 1.2 )). We first control **the** equations **with** n controls (one on each equation) and we try to eliminate **the** control on **the** last equation thanks to algebraic manipulations. More precisely, we decompose **the** problem into two different steps:

En savoir plus
6.2 Reconstruction **of** a cube in a simple scene (Figure 3(a))
This second test is carried out in **the** same conditions as **the** first, except that **the** object is now an insulating cube. **The** **filter** estimates **the** parameters **of** an equivalent sphere using **the** sphere model. **The** initial real and estimated states **of** **the** scene are set to be: (d = 0.225 m, θ = 0.298 rad, a = 0.037 m (equivalent radius)) and: ( ˆ d = d − 0.1 m, ˆ θ = θ + 0.2 rad, ˆa = 0.001 m). Figure 12(a) shows **the** change over time **of** **the** real (red) and estimated (blue) state **of** **the** cube. Figure 12(b) represents **the** actual (cyan) and **the** estimated (blue) scenes at different times **of** **the** sensor motion. As in **the** previous test, when **the** object is close to **the** head (t = 2 s), **the** cube is quite well localized but its estimated size is greater than **the** actual length **of** **the** side **of** **the** cube. When **the** object is close to **the** emitter (t = 8s), its equivalent sphere is well reconstructed and **the** estimated error (on size and location) is **small**. This test illustrates a common feature **of** all **the** tests carried out: **the** estimated equivalent sphere always encapsulates **the** cube, and its location is well estimated. This confirms **the** prediction **of** section 5.2 and is due to **the** fact that when excited by an external field (here **the** basal field emitted by **the** sensor), any **small** compact object appears (from **the** point **of** view **of** **the** electric measurements **of** **the** sensor) as a polarized sphere at **the** leading order.

En savoir plus
7 LMS, CNRS UMR 7649, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau, France
Abstract
Estimating dynamical **systems** – in particular identifying theirs parameters – involved in computational biology – **for** instance in pharmacology, in virology or in epidemiology – is fundamental to put in accordance **the** model trajectory **with** **the** measurements at hand. Unfortunately, when **the** sampling **of** data is very scarce or **the** data are corrupted by **noise**, parameters mean and variance priors must be chosen very adequately to balance our measurement distrust. Otherwise **the** identification procedure fails. A circumvention consists in using repeated measurements collected in configurations that share common priors – **for** instance **with** multiple population subjects in a clinical study or clusters in an epidemiology investigation. This common information is **of** benefit and is typically modeled in statistics by nonlinear mixed-effect models. In this paper, we introduce a data assimilation methodology compatible **with** such mixed-effect strategy without being strangled by **the** potential resulting curse **of** dimensionality. We define population-based estimators through maximum likelihood estimation. Then, from filtering theory, we set-up an equivalent robust large population sequential estimator that integrates **the** data as they are collected. Finally, we limit **the** computational complexity by defining a reduced-order version **of** this population **Kalman** **filter** clustering subpopulations **of** common observation background. **The** resulting algorithm performances are evaluated on classical pharmacokinetics benchmark. **The** versatility **of** **the** proposed method is finally fully challenged in a real-data epidemiology study **of** COVID spread in regions and departments **of** France.

En savoir plus