## Closed-Form Prediction of Nonlinear Dynamic Systems by Means of Gaussian Mixture Approximation of the Transition Density

Venue: | in IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems |

Citations: | 15 - 13 self |

### BibTeX

@INPROCEEDINGS{Huber_closed-formprediction,

author = {Marco Huber and Dietrich Brunn and Uwe D. Hanebeck},

title = {Closed-Form Prediction of Nonlinear Dynamic Systems by Means of Gaussian Mixture Approximation of the Transition Density},

booktitle = {in IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems},

year = {},

pages = {98--103}

}

### OpenURL

### Abstract

Abstract — Recursive prediction of the state of a nonlinear stochastic dynamic system cannot be efficiently performed in general, since the complexity of the probability density function characterizing the system state increases with every prediction step. Thus, representing the density in an exact closed-form manner is too complex or even impossible. So, an appropriate approximation of the density is required. Instead of directly approximating the predicted density, we propose the approximation of the transition density by means of Gaussian mixtures. We treat the approximation task as an optimization problem that is solved offline via progressive processing to bypass initialization problems and to achieve high quality approximations. Once having calculated the transition density approximation offline, prediction can be performed efficiently resulting in a closed-form density representation with constant complexity. I.

### Citations

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Citation Context ...tional effort. Only for some special cases full analytical solutions are available. For linear systems with Gaussian random variables the Kalman filter provides exact solutions in an efficient manner =-=[7]-=-. Versatile approximative techniques exist for the case of nonlinear systems: To overcome the problem of representing the whole predicted density, particle filters use samples instead [2]. They are ea... |

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Citation Context ...cient manner [7]. Versatile approximative techniques exist for the case of nonlinear systems: To overcome the problem of representing the whole predicted density, particle filters use samples instead =-=[2]-=-. They are easy to implement and to parallelise, but it is still a hard task to obtain adequate samples at every prediction step. Another possibility arises from the usage of generic parameterized den... |

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Citation Context ...ses linearization to apply the Kalman filter equations on nonlinear systems [10], while the unscented Kalman filter offers in addition higher order accuracy by using a deterministic sampling approach =-=[6]-=-. The resulting single Gaussian density of both estimation Marco Huber, Dietrich Brunn, and Uwe D. Hanebeck are with the Intelligent Sensor-Actuator-Systems Laboratory, Institute of Computer Science a... |

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Citation Context ...ty, Gaussian mixtures [8] are a much better approach for parameterized density functions. The bandwidth of estimators using Gaussian mixtures is wide. It ranges from the efficient Gaussian sum filter =-=[1]-=- that allows only an individual updating of the mixture components up to computationally more expensive but precise methods [5]. In this paper, we introduce a new closed-form prediction approach for n... |

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Citation Context ... so-called progression step the distance measure G(η, γ) between the parameterized transition density ˜ f T (xk+1, γ) and its approximation f T (xk+1, η) is minimized by employing of the BFGS formula =-=[3]-=-, a well known optimization method. The approximation f T (xk+1, η), or more precisely the parameter vector η, follows gradually ˜ f T (xk+1, γ) until thedesired true transition density ˜ f T (xk+1) ... |

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Citation Context ...ocus in this paper on the system equation (1). Given an estimate f x 0 (x0) for x0 at k = 0, this equation is used in a Bayesian setting for a recursive system state propagation in time. According to =-=[11]-=- this so-called prediction step of the Bayesian estimator results in a density f x ∫ k+1(xk+1) = f T (xk+1)f x k (xk)dxk (2) for xk+1, where f T (xk+1) is the transition density R f T (xk+1) = f(xk+1|... |

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Citation Context ...y. {mhuber|brunn}@ira.uka.de, uwe.hanebeck@ieee.org methods is typically not a sufficient representation for the true complex density. Due to their universal approximation property, Gaussian mixtures =-=[8]-=- are a much better approach for parameterized density functions. The bandwidth of estimators using Gaussian mixtures is wide. It ranges from the efficient Gaussian sum filter [1] that allows only an i... |

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Citation Context ...ssian mixtures is wide. It ranges from the efficient Gaussian sum filter [1] that allows only an individual updating of the mixture components up to computationally more expensive but precise methods =-=[5]-=-. In this paper, we introduce a new closed-form prediction approach for nonlinear systems by means of a Gaussian mixture approximation of the transition density. The transition density is used to prop... |

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Citation Context ...)dxkdxk+1 R Ω } {{ } =:I + 1 ∫ ∫ ( T f (xk+1, η) 2 )2 dxkdxk+1 , (12) R R where merely the integral I cannot be solved analytically. Numerical integration methods like the adaptive Simpson quadrature =-=[4]-=- have to be applied. The necessary condition for the existence of a minimum of G(η, γ) for a given γ is ∂G(η, γ) = 0 . (13) ∂η gradient ∂G(η,γ) ∂η Since (13) allows no closed-form solution we use the ... |

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Citation Context ...essing Instead of attempting to directly approximate the transition density, we pursue a progressive approach for finding η min as shown in Figure 2. This type of processing has been proposed in [5], =-=[9]-=-. In doing so, a parameterized transition density ˜f T (xk+1, γ) with the progression parameter γ ∈ [0, 1] is introduced. This progression parameter ensures a continuous transformation of the solution... |