Conference Paper

Efficient Nonlinear Measurement Updating based on Gaussian Mixture Approximation of Conditional Densities

Univ. Karlsruhe, Karlsruhe
DOI: 10.1109/ACC.2007.4282269 Conference: American Control Conference, 2007. ACC '07
Source: IEEE Xplore

ABSTRACT Filtering or measurement updating for nonlinear stochastic dynamic systems requires approximate calculations, since an exact solution is impossible to obtain in general. We propose a Gaussian mixture approximation of the conditional density, which allows performing measurement updating in closed form. The conditional density is a probabilistic representation of the nonlinear system and depends on the random variable of the measurement given the system state. Unlike the likelihood, the conditional density is independent of actual measurements, which permits determining its approximation off-line. By treating the approximation task as an optimization problem, we use progressive processing to achieve high quality results. Once having calculated the conditional density, the likelihood can be determined on-line, which, in turn, offers an efficient approximate filter step. As result, a Gaussian mixture representation of the posterior density is obtained. The exponential growth of Gaussian mixture components resulting from repeated filtering is avoided implicitly by the prediction step using the proposed techniques.

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    • "Alternatively, Expectation Maximization (EM) algorithm can be used to simultaneously predict and reduce the Gaussian Mixture [26]–[29], e.g. by running the EM algorithm on synthetically generated data [27], [29]. Among the reduction schemes mentioned above, the methods that rely on removing clusters are the least computationally complex. "
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    • "The key idea of this paper is to enable a closed-form calculation of the prediction and measurement step by means of wavelet expansions of the conditional densities. Compared to similar approaches using axis-aligned Gaussian mixtures as in [6] or [11], wavelets have some significant advantages . Besides their good approximation quality, no complex optimization problem has to be solved in order to obtain the decomposition of a given density. "
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    • "This special type of a squared integral measure is defined over the corresponding cumulative distribution functions of both densities and allows quantifying the distance especially in case of the used Dirac delta functions. In contrast to our purely Gaussian mixture conditional density approximation approach in [12], this optimization problem can be solved analytically and is not restricted to time-invariant systems. The hybrid structure of the approximate conditional density allows an analytical and efficient evaluation of the prediction step as well as the measurement update step. "
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