Elsevier

Aerospace Science and Technology

Volume 55, August 2016, Pages 366-376
Aerospace Science and Technology

Terminal height estimation using a Fading Gaussian Deterministic filter

https://doi.org/10.1016/j.ast.2016.06.013Get rights and content

Abstract

In a recent work by the authors the concept of Fading Gaussian Deterministic filter was investigated. The algorithm is based on a set of equations derived from the minimization of a cost function where earlier data are progressively de-weighted by a fading factor. In such a way, the estimation was proved to be less prone to problem unknowns. A tuning procedure was proposed that allows the resulting globally best estimator to evaluate the covariance of an effective measurement noise and the true estimation error, without any a-priori assumption. In the present paper, a general formulation is derived where the observed system is influenced by a control input. Also, a proof is derived for the proposed tuning criterion, which is shown to provide, under certain assumptions, the fading factor that best dampens the modeling errors with respect to measurement noise. The validity of the proposed approach is investigated by means of both numerical simulations and an experimental campaign, where height estimation is performed by fusing information from MEMS accelerometers and a barometric altimeter.

Introduction

This paper presents an estimation filter that features online tuning capabilities and its application to a data-fusion problem for height estimation. The algorithm is based on a recursive two-step set of equations known as Fading Gaussian Deterministic (FGD) filter, recently investigated by the authors in Ref. [1]. In such a framework, the filter was shown to be “optimal” because the gain matrix is computed through the formal minimization of a cost function, with no other assumption. In particular, earlier data are progressively de-weighted by a fading (or “forgetting”) factor, in order to make the estimation less prone to unknown external disturbances and/or model uncertainties and/or non-modeled dynamics, regardless of their nature (deterministic or stochastic). The introduction of the fading factor and the elimination of the so-called process noise covariance matrix Q make the difference with respect to a classical Kalman-like estimation process [2]. The approach was first developed by Norman Morrison in Ref. [3] and preliminarily elaborated in Ref. [4]. A similar concept was also investigated within the digital processing community in Refs. [5], [6], [7]. More recently, the Morrison filter was applied, with both non-recursive and recursive formulations, to a nonlinear tracking problem, where Levenberg–Marquardt methods were incorporated for improved convergence [8].

The main contribution provided in Ref. [1] consisted in a criterion for filter tuning, in order to minimize the estimation error, by selecting the best estimator in an ensemble of filters where only the fading factor value is varied. Once the filter is tuned, an effective measurement noise covariance R is directly estimated from the data (not as an a-priori assumption) and, as a by-product, the true covariance of the estimation error P is evaluated.

In the present paper, the recursive formulation introduced in Ref. [1] is extended to the general case where the system is influenced by a control input. In addition to the goodness-of-fit interpretation given in Ref. [1], a formal justification is also derived for the proposed tuning criterion, which is shown to provide the fading factor that best dampens the modeling errors with respect to the measurement noise. The effectiveness of the proposed approach and its ability to identify a globally optimum filter (within its own class) is investigated by means of both numerical simulations and an experimental campaign, where it is stressed that the proposed proof to the tuning technique allows quantifying the degree of knowledge of system dynamics. In particular, the algorithm is implemented in a test case relative to height estimation by fusing information from MEMS accelerometers and a barometric altimeter. A comparison is provided between the FGD algorithm and a simple method based on the so-called complementary filtering [9]. A simulation scenario is finally described where, in order to outline an evolving dynamic environment, the modeling accuracy is artificially corrupted, with the consequent deterioration of the goodness-of-fit and the necessity to increase the filter's fading effect.

Section snippets

Basic specifications of the FGD filter

Suppose that, at time k, there is some true state vector xk with dimension m which propagates approximately according to xk+1=Φkxk+Γkuk, where Φk is the state transition matrix, uk is a known input vector with dimension l, and Γk is the control-input model. The observation zk with dimension n approximately tracks this process as in zk=Hkxk, where Hk is the observation matrix.

Given the time series of data z0,z1,z2,,zk, the aim is to compute estimates in a time series to the state vector, namely

Global optimization of the FGD filter

FGD filter specifications given in Section 2 allow its global optimization. In particular, the design of the optimal value of β (tuning), the calibration of the weighting matrix R (scaling), and the estimation of the true error covariance Pk will be performed. The complete procedure closely follows the work described by the authors in Ref. [1], thus only the main steps will be recalled in what follows.

First, define a scalar normalized residualρk=(zkHkx˜k)TR1(zkHkx˜k) which, by taking into

Tuning and scaling

It is assumed that the actual measurement noise covariance R can be sufficiently approximated as constant while taking measurements and that the assigned R matches R through a scaling factor θ, namelyR=θR Note that the multiplicative factor θ applied to the assigned R matrix has no effect on the estimation, since the scaling of the R matrix implies a scaling of the entire cost function in Eq. (2). On the other hand, the computed filter covariance Ek results to be scaled by the same factor in

Height estimation

In this Section, filter validation is addressed by implementing the proposed algorithm within a test case relative to altitude estimation by fusing the information represented by the local vertical acceleration and the barometric height.

When a pressure sensor is used as an altimeter that measures changes in height, Δhbaro=hhref, with respect to a reference height href, the pressure p as measured in Pascal is approximately converted to change in height in meters by [12]:Δhbaro=η(λ)ln(ppref)

Conclusions

In this paper, an estimation technique based on a set of recursive equations was analyzed and applied to a linear dynamical system subject to a known input. Following previous works on the subject, the algorithm is named Fading Gaussian Deterministic filter, and is based on a recursive two-step formulation whose equations derive from the minimization of a cost function where earlier data are progressively de-weighted by a fading factor. The introduction of the fading factor and the consequent

Conflict of interest statement

There is no conflict of interest.

Acknowledgements

This work was funded in part by the European Space Agency (ESA) through European Student Earth Orbiter contract 4000107255/12/NL/SFe. P.T. is grateful to M.S. Hodgart from University of Surrey for his careful introduction and guidance to estimation problems back in 1999, and for his constructive comments to this work. The invaluable inspiring contribution of N. Morrison, from the University of Cape Town, through his pioneering book [3] is also greatly acknowledged.

References (13)

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