A square root formulation of the Kalman- Schmidt filter

Bellantoni, Juan F.; Dodge, K.

doi:10.2514/6.1967-90

Cited by 19 publications

(20 citation statements)

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“…6,8,18,32,33 One solution to this problem is found in the form of various square-root KF implementations. [34][35][36][37][38] The importance of square-root filtering and other least-square estimations is investigated and confirmed in the excellent theoretical studies of Verhaegen and Van Dooren. 39,40 Nowadays, it is almost compulsory, for each new state and/or parameter estimator, to derive additionally its numerically robust square-root form.…”

Section: Introductionmentioning

confidence: 76%

“…More formally, we have applied the manipulation of Särkkä 30(appendix) for converting the full covariance matrix equation (31) into its square-root form (34). Then, numerical integrations of Equations (33)- (35) with initial values (32) by the MATLAB code ode45 with the aforementioned accuracy conditions complete the time update steps of both square-root MATLAB-based EKFs in each sampling interval [t k − 1 , t k ]. After that, the measurement update steps of these filters will be applied in the form of the square-root formulas (26), (27) if the conventional matrix inversion is implemented or in the form of formulas (28), (29) if the Moore-Penrose matrix pseudoinversion (19), (20) is utilized.…”

Section: Matlab-based Ekfmentioning

confidence: 99%

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Numerical robustness of extended Kalman filtering based state estimation in ill‐conditioned continuous‐discrete nonlinear stochastic chemical systems

Kulikov

Kulikova

2018

Intl J Robust & Nonlinear

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This paper presents a case study investigation of numerical robustness of extended Kalman filters used for estimation of stochastic chemical systems with ill-conditioned measurements. Here, we consider both a batch reactor model and that of a continuously stirred tank reactor. Our purpose is to explore performance of extended Kalman filtering-based state estimators when the measurement model becomes increasingly ill conditioned. In this way, we determine numerically robust methods, which are suitable for accurate estimation of stochastic chemical systems in the presence of round-off and other disturbances. We examine both conventional filters and their square-root forms. All these algorithms are implemented by means of the conventional matrix inversion used in their measurement update steps and the Moore-Penrose pseudoinversion as well. Furthermore, the square-root filters under investigation are obtained in two ways, namely, by solving square-root moment differential equations and by square rooting the filter itself. We show that only the square-root filters grounded in the second approach (with use of stable orthogonal decompositions) are numerically robust and provide the excellent estimation accuracy within all our ill-conditioned stochastic chemical system scenarios considered in this paper. In addition, the convectional filters and the square-root variants based on solving moment equations are rather sensitive to round-off and may be useful and accurate if the chemical system at hand is rather well conditioned. KEYWORDS batch and continuously stirred tank reactors, extended Kalman filtering methods, ill-conditioned continuous-discrete nonlinear stochastic system, numerical robustness, round-off error Int J Robust Nonlinear Control. 2019;29:1377-1395.wileyonlinelibrary.com/journal/rnc

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Section: Introductionmentioning

confidence: 76%

Section: Matlab-based Ekfmentioning

confidence: 99%

Numerical robustness of extended Kalman filtering based state estimation in ill‐conditioned continuous‐discrete nonlinear stochastic chemical systems

Kulikov

Kulikova

2018

Intl J Robust & Nonlinear

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“…Soon after the introduction of these algorithms, it was shown that their implementation in practice, especially on short-word-length computers, may lead to the computation of negative-definite covariance matrices, 14 ' 16 which, in turn, may cause filter divergence. Squareroot (SR) filtering algorithms were developed to overcome these difficulties.…”

Section: (T -S);mentioning

confidence: 99%

“…Now, because of the orthogonality of the covariance eigenvectors and the relation (14), the "true" matrix (i.e., the matrix that would have been computed using an infinite word length)…”

Section: Orthogonalization Of the Eigenvector Matrixmentioning

confidence: 99%

Square-root state estimation for second-order large space structuresmodels

Oshman

Inman

Laub

1989

Journal of Guidance, Control, and Dynamics

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Two square-root filtering algorithms are developed for large space structures that are modeled by secondorder, continuous-time, finite, dynamic models. The first filter, which assumes a continuous-time measurement system, is a single-stage continuous algorithm that is based on the V-Lambda square-root method for the solution of a generalized Riccati equation. The second measurement system considered is of a discrete-time type, for which the resulting estimator is a hybrid continuous/discrete one. Both estimators are based on the spectral decomposition of the estimation error covariance matrix. Thus, they continuously provide the user with the covariance spectral factors. This distinct feature of the V-Lambda algorithms is valuable in ill-conditioned cases, in which an insight into the estimation process is needed to reveal singularities and to identify state subsets that become nearly dependent. Moreover, using the orthogonality property of the covariance eigenvectors, an orthogonalization step is added to the algorithms to enhance their accuracy in cases where simple, unsophisticated software is to be used. Two different methods for performing the orthogonalization are suggested. A typical filtering example is used to demonstrate the square-root nature of the new filters.

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“…For the state estimation of nonlinear systems, the extended Kalman filter (EKF) [21,22] has been widely used. The EKF is an IIR-type filter because it is based on the KF with an IIR structure.…”

Section: Introductionmentioning

confidence: 99%

Weighted average extended FIR filter bank to manage the horizon size in nonlinear FIR filtering

Pak¹,

Yoo²,

Lim³

et al. 2014

Int. J. Control Autom. Syst.

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In this paper, we propose a novel approach to manage the horizon size in nonlinear finite impulse response (FIR) filtering. The proposed approach is to perform state estimation through a bank of FIR filters called a weighted average extended FIR filter bank (WAEFFB). In the WAEFFB, the state estimate is obtained by weighting the average of multiple estimates from a bank of extended FIR filters that uses different horizon sizes. The horizon sizes used for the WAEFFB are adjusted constantly by maximizing the likelihood function. We show through simulations that the WAEFFB yields better results than the conventional approach that uses a constant (i.e., fixed) horizon size.

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A square root formulation of the Kalman- Schmidt filter

Cited by 19 publications

References 0 publications

Numerical robustness of extended Kalman filtering based state estimation in ill‐conditioned continuous‐discrete nonlinear stochastic chemical systems

Numerical robustness of extended Kalman filtering based state estimation in ill‐conditioned continuous‐discrete nonlinear stochastic chemical systems

Square-root state estimation for second-order large space structuresmodels

Weighted average extended FIR filter bank to manage the horizon size in nonlinear FIR filtering

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