Haut PDF Efficiency of the extended Kalman filter for non linear systems with small noise

Efficiency of the extended Kalman filter for non linear systems with small noise

Efficiency of the extended Kalman filter for non linear systems with small noise

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Kalman Filter-Based Real-Time Implementable Optimization of the Fuel Efficiency of Solid Oxide Fuel Cells

Kalman Filter-Based Real-Time Implementable Optimization of the Fuel Efficiency of Solid Oxide Fuel Cells

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.
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Three examples of the stability properties of the invariant extended Kalman filter

Three examples of the stability properties of the invariant extended Kalman filter

the absence of noise is a great indication that the filter should not diverge for sufficiently small errors and noises. 3. FIRST EXAMPLE: ATTITUDE ESTIMATION In this section we consider the problem of attitude esti- mation with gyrometers and observation of two vectors with known directions in the fixed frame. This problem has received a lot of attention over the past decades, and the literature is too broad to be covered here (see e.g. Mahony et al. (2008); Izadi and Sanyal (2014); Sanyal and Nordkvist (2012); Zamani et al. (2014); Lee et al. (2007)). We insist that this example is very simple, and that we included it essentially for tutorial reasons. However, note that, contrarily to the non-linear attitude estimation lit- erature just mentioned, the following filter accomodates discrete-time observations, which can be relevant in prac- tice (for instance a star tracking system in a sattelite could use infrequent snapshots to save energy), and time varying model uncertainties (for instance the quasi-static assumption below for a UAV does not hold in dynamic flight phases, so that the noise variance on the gravity measurement should be temporarily increased when ma- noeuvers are made). Note also that we already published the equations of the IEKF for this problem, but we never proved its local stability properties.
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Extracting Common pulse like signals from multivariate time series with a non linear Kalman Filter

Extracting Common pulse like signals from multivariate time series with a non linear Kalman Filter

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.
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Non-linear state error based extended Kalman filters with applications to navigation

Non-linear state error based extended Kalman filters with applications to navigation

The Kalman filter as an observer An interesting property of the Kalman filter is to be also a convergent deterministic observer. Indeed, if the data are noiseless (i.e. V = 0, w = 0), and the equations are linear, for any choice of positive definite covari- ance matrices for the trusted noises (here they should be referred to as tuning matrices Q, R since we assume there is no noise) the estimate ˆ X converges to the true state X under natural assumptions [38]. In the non-linear case, the EKF does not possess any convergence guarantee, and its efficiency is aleatory. Indeed, its main flaw lies in its very nature: the Kalman gain is computed assuming the estimation error is sufficiently small to be propagated analytically through a first-order linearization of the dynamics about the estimated trajectory. When the estimate is actually far from the true state variable, the linearization is not valid, and results in an ill-adapted gain that may amplify the error. In turn, such positive feedback loop may lead to divergence of the filter. This is the reason why most of the papers dealing with the stability of the EKF (see [23, 24, 92, 102]) rely on the highly non-trivial assumption that the eigenvalues of the Kalman covariance matrix P(t) computed about the estimated trajectory are lower and upper bounded by strictly positive scalars. Under this assumption, the error as long as it is sufficiently small can not be amplified in a way that makes the filter diverge. To the author’s knowledge, only a few papers deal with the stability of the EKF without invoking this assumption [69]. It is then replaced by second-order properties whose verification can prove difficult in practi- cal situations. This is also due to the fact that the filter can diverge indeed in a number of applications. Moreover, the lack of guarantees goes beyond the general theory: there are few engineering examples where the EKF is proved to be (locally) stable.
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Convergence of discrete-time Kalman filter estimate to continuous-time estimate for systems with unbounded observation

Convergence of discrete-time Kalman filter estimate to continuous-time estimate for systems with unbounded observation

The minimum variance state estimate for linear systems with Gaussian noise processes is given by the continuous-time Kalman filter. However, for obvious reasons, in a practical implementation the continuous-time system is often first discretized, and then the discrete-time Kalman filter is used on the discretized system. The objective of this article is to expand the recent results presented in [1] by the author on the convergence of the state estimate given by the discrete-time Kalman filter on the sampled system to the continuous-time estimate. There convergence results were shown for finite-dimensional systems and infinite-dimensional systems with bounded observation operators. The expansion in this paper covers systems with un- bounded observation operators and systems whose dynamics are governed by an analytic semigroup. In particular, we shall show convergence rate estimates for the variance of the discrepancy between the discrete- and continuous-time estimates.
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Non-linear state error based extended Kalman filters with applications to navigation

Non-linear state error based extended Kalman filters with applications to navigation

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.
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Robust linearly constrained extended Kalman filter for mismatched nonlinear systems

Robust linearly constrained extended Kalman filter for mismatched nonlinear systems

For nonlinear systems, the EKF is one of the most popular estimation techniques and therefore has been largely investigated. 1,3,24 The sensitivity of the EKF to the filter initialization and its inevitable divergence if the noise matrices have not been chosen appropriately, together with the analysis of the related stability and robustness of the filter, has been widely reported in the open literature. 25,26 The EKF performance also depends on the accurate knowledge of the observation parametric model (linear or nonlinear system functions) and is particularly sensitive to different types of mismatches between the assumed signal model and the real signal. 27,28 Thus, in order to provide additional robustness to modeling errors, state constraints may be introduced, yielding the so-called constrained EKF (CEKF). 29,30 Recently, the CEKF has attracted a lot of attention and the use of state constraints has increased in practical engineering applications such as robotics, 31 navigation, 32 as well as target tracking. 33 The use of linear equality constraints in nonlinear systems has been briefly explored in [ 23, section IV] where the Unscented KF is applied to the gain-constrained linear KF, which in turn is a particular instance of the LCKF. 21,22 But the analysis in Reference 23 only applies the linear constrained gain computation with the unscented innovation covariance approximation, and no further discussion is provided. Moreover, this approach is not intended to be a robust filtering solution under model mismatch. As previously stated, 21 recently proposed a different approach where linear constraints (LCs) are used to mitigate process and/or measurement model matrices mismatch in linear systems. The use of constraints for robust filtering under model mismatch in nonlinear systems has not been yet explored, thus being an important missing point.
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Non-minimal state-space polynomial form of the Kalman filter for a general noise model

Non-minimal state-space polynomial form of the Kalman filter for a general noise model

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.
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The Invariant Extended Kalman Filter as a Stable Observer

The Invariant Extended Kalman Filter as a Stable Observer

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].
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Actuation efficiency and work flow in piezoelectrically driven linear and non-linear systems

Actuation efficiency and work flow in piezoelectrically driven linear and non-linear systems

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]

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A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups

A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups

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.
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Application of the Wavelet Transform for GPS Cycle Slip Correction and Comparison with Kalman Filter

Application of the Wavelet Transform for GPS Cycle Slip Correction and Comparison with Kalman Filter

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]

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Robust Variational-based Kalman Filter for Outlier Rejection with Correlated Measurements

Robust Variational-based Kalman Filter for Outlier Rejection with Correlated Measurements

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.
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Variations on the Kalman filter for enhanced performance monitoring of gas turbine engines

Variations on the Kalman filter for enhanced performance monitoring of gas turbine engines

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.
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Parametric Kalman filter for chemical transport models

Parametric Kalman filter for chemical transport models

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.
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A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups

A Kalman rank condition for the indirect controllability of coupled systems of linear operator groups

∂ 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:
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A self-mixing laser sensor design with an extended kalman filter for optimal online structure analysis and damping evaluation

A self-mixing laser sensor design with an extended kalman filter for optimal online structure analysis and damping evaluation

The lager the number of measured points, the more accurate the AVL value will be (i.e., closed to the theoretical value). However, to measure the performance of a plate, it has been proved experimentally that 10 points of measurement results in an AVL value with sufficient precision. As an example, Fig. 10 represents the respective responses of a plate without damping and with a layer of EAR for the first seven modes. For each frequency, ten measurements have been performed at different points of the plate, the laser sensor being translated manually between each FRF acquisition. By averaging the results obtained from a given sample of ten measurements, the eigen frequencies of the coated plate can be determined with a better accuracy.
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Environment reconstruction and navigation with electric sense based on kalman filter

Environment reconstruction and navigation with electric sense based on kalman filter

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.
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Estimation for dynamical systems using a population-based Kalman filter - Applications to pharmacokinetics models

Estimation for dynamical systems using a population-based Kalman filter - Applications to pharmacokinetics models

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.
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