Cubature Kalman Filtering for Continuous-Discrete Systems: Theory and Simulations
In this paper, we extend the cubature Kalman filter (CKF) to deal with nonlinear state-space models of the continuous-discrete kind. To be consistent with the literature, the resulting nonlinear filter is referred to as the continuous-discrete cubature Kalman filter (CD-CKF). We use the Itô-Taylor...
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| Published in: | IEEE transactions on signal processing Vol. 58; no. 10; pp. 4977 - 4993 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York, NY
IEEE
01.10.2010
Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Subjects: | |
| ISSN: | 1053-587X, 1941-0476 |
| Online Access: | Get full text |
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| Abstract | In this paper, we extend the cubature Kalman filter (CKF) to deal with nonlinear state-space models of the continuous-discrete kind. To be consistent with the literature, the resulting nonlinear filter is referred to as the continuous-discrete cubature Kalman filter (CD-CKF). We use the Itô-Taylor expansion of order 1.5 to transform the process equation, modeled in the form of stochastic ordinary differential equations, into a set of stochastic difference equations. Building on this transformation and assuming that all conditional densities are Gaussian-distributed, the solution to the Bayesian filter reduces to the problem of how to compute Gaussian-weighted integrals. To numerically compute the integrals, we use the third-degree cubature rule. For a reliable implementation of the CD-CKF in a finite word-length machine, it is structurally modified to propagate the square-roots of the covariance matrices. The reliability and accuracy of the square-root version of the CD-CKF are tested in a case study that involves the use of a radar problem of practical significance; the problem considered herein is challenging in the context of radar in two respects- high dimensionality of the state and increasing degree of nonlinearity. The results, presented herein, indicate that the CD-CKF markedly outperforms existing continuous-discrete filters. |
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| AbstractList | In this paper, we extend the cubature Kalman filter (CKF) to deal with nonlinear state-space models of the continuous-discrete kind. To be consistent with the literature, the resulting nonlinear filter is referred to as the continuous-discrete cubature Kalman filter (CD-CKF). We use the Ito-Taylor expansion of order 1.5 to transform the process equation, modeled in the form of stochastic ordinary differential equations, into a set of stochastic difference equations. Building on this transformation and assuming that all conditional densities are Gaussian-distributed, the solution to the Bayesian filter reduces to the problem of how to compute Gaussian-weighted integrals. To numerically compute the integrals, we use the third-degree cubature rule. For a reliable implementation of the CD-CKF in a finite word-length machine, it is structurally modified to propagate the square-roots of the covariance matrices. The reliability and accuracy of the square-root version of the CD-CKF are tested in a case study that involves the use of a radar problem of practical significance; the problem considered herein is challenging in the context of radar in two respects- high dimensionality of the state and increasing degree of nonlinearity. The results, presented herein, indicate that the CD-CKF markedly outperforms existing continuous-discrete filters. In this paper, we extend the cubature Kalman filter (CKF) to deal with nonlinear state-space models of the continuous-discrete kind. To be consistent with the literature, the resulting nonlinear filter is referred to as the continuous-discrete cubature Kalman filter (CD-CKF). We use the Itô-Taylor expansion of order 1.5 to transform the process equation, modeled in the form of stochastic ordinary differential equations, into a set of stochastic difference equations. Building on this transformation and assuming that all conditional densities are Gaussian-distributed, the solution to the Bayesian filter reduces to the problem of how to compute Gaussian-weighted integrals. To numerically compute the integrals, we use the third-degree cubature rule. For a reliable implementation of the CD-CKF in a finite word-length machine, it is structurally modified to propagate the square-roots of the covariance matrices. The reliability and accuracy of the square-root version of the CD-CKF are tested in a case study that involves the use of a radar problem of practical significance; the problem considered herein is challenging in the context of radar in two respects- high dimensionality of the state and increasing degree of nonlinearity. The results, presented herein, indicate that the CD-CKF markedly outperforms existing continuous-discrete filters. In this paper, we extend the cubature Kalman filter (CKF) to deal with nonlinear state-space models of the continuous-discrete kind. To be consistent with the literature, the resulting nonlinear filter is referred to as the continuous-discrete cubature Kalman filter (CD-CKF). We use the Itô-Taylor expansion of order 1.5 to transform the process equation, modeled in the form of stochastic ordinary differential equations, into a set of stochastic difference equations. Building on this transformation and assuming that all conditional densities are Gaussian-distributed, the solution to the Bayesian filter reduces to the problem of how to compute Gaussian-weighted integrals. To numerically compute the integrals, we use the third-degree cubature rule. For a reliable implementation of the CD-CKF in a finite word-length machine, it is structurally modified to propagate the square-roots of the covariance matrices. The reliability and accuracy of the square-root version of the CD-CKF are tested in a case study that involves the use of a radar problem of practical significance; the problem considered herein is challenging in the context of radar in two respects- high dimensionality of the state and increasing degree of nonlinearity. The results, presented herein, indicate that the CD-CKF markedly outperforms existing continuous-discrete filters. |
| Author | Arasaratnam, Ienkaran Haykin, Simon Hurd, Thomas R |
| Author_xml | – sequence: 1 givenname: Ienkaran surname: Arasaratnam fullname: Arasaratnam, Ienkaran email: haran@ieee.org organization: Center for Mechatron. & Hybrid Technol., McMaster Univ., Hamilton, ON, Canada – sequence: 2 givenname: Simon surname: Haykin fullname: Haykin, Simon email: haykin@mcmaster.ca organization: Cognitive Syst. Lab., McMaster Univ., Hamilton, ON, Canada – sequence: 3 givenname: Thomas R surname: Hurd fullname: Hurd, Thomas R email: hurdt@mcmaster.ca organization: Dept. of Math. & Stat., McMaster Univ., Hamilton, ON, Canada |
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| Snippet | In this paper, we extend the cubature Kalman filter (CKF) to deal with nonlinear state-space models of the continuous-discrete kind. To be consistent with the... |
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| SubjectTerms | Applied sciences Bayesian filters Cadmium Computer science; control theory; systems Control theory. Systems Cuba cubature Kalman filter (CKF) Detection, estimation, filtering, equalization, prediction Difference equations Differential equations Exact sciences and technology Filtering theory Gaussian processes Information, signal and communications theory Integral equations Integrals Itô-Taylor expansion Kalman filters Mathematical analysis Mathematical models Miscellaneous nonlinear filtering Nonlinear filters Nonlinearity Radar Signal and communications theory Signal processing Signal, noise square-root filtering Stochastic processes Stochasticity System theory Telecommunications and information theory Transforms |
| Title | Cubature Kalman Filtering for Continuous-Discrete Systems: Theory and Simulations |
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