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...ct, the bounds are universally valid and thus applicable in further information-theoretic tasks like capacity calculation of communication channels [45], parameter estimation [39], image registration =-=[182]-=-, and many others. Furthermore, providing a tight lower and upper bound of the entropy value allows deciding whether a direct approximation is meaningful or not, i.e., some kind of confidence interval...

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...2.6. Related Work 23 2.6.2 Myopic or Greedy Approaches A collection of different objective functions for myopic sensor management is described in detail and compared for a target tracking scenario in =-=[43, 199]-=-. In accordance with [57], it is shown that objective functions based on the expected posterior density function are merely a measure of the predicted density and thus of limited use. More appropriate...

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...d, where the state estimates are represented by means of Gaussian mixture densities. In contrast to existing estimators employing statistical linearization like the well-known unscented Kalman filter =-=[91, 183]-=-, estimation quality of the novel Gaussian estimator can be adapted by adjusting the number of regression points used. Chapter 6 In nonlinear state estimation, it is generally inevitable to incorporat...

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...the desired number of components. The first is Williams’ reduction algorithm [193], which can be considered as global reduction approach. The second is a local reduction algorithm proposed by M. West =-=[188]-=-, while the method of A. Runnalls [153] represents a compromise between local and global approaches. For a more detailed introduction to these algorithms and their classification into local and global...

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...the regressions points of the set ...

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...g, that in optimum experimental design theory similar scalar objective functions are defined based on the Fisher information matrix (or its inverse) instead of the ones based on the covariance matrix =-=[11, 178]-=-. Minimizing such objective functions corresponds to minimizing a lower bound of the covariance matrix.20 Chapter 2. Considered Problem 2.5.2 Information Theoretic Objective Functions Shannon’s infor...

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... Especially, the concept of the virtual measurements and pruning via the probabilistic branch-and-bound algorithm can be found here. The upper and lower bounds on the differential entropy are part of =-=[213]-=-. The combination of all these techniques to the information theoretic sensor manager and further improvements on the entropy bounds are the main contributions of this chapter. 4.1 Closed-loop Control...

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... always tight [20], the resulting performance can divergence arbitrarily from the closed-loop performance. For target tracking applications with linear system dynamics and additive measurement noise, =-=[40, 190]-=- present open-loop control approaches, where the nonlinear sensor models are linearized multiple time steps ahead, based on the current system estimate. This facilitates the use of the Kalman filter w...

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...mit their measurement values simultaneously. Furthermore,4 Chapter 1. Introduction often only those measurements are informative that are provided by sensors in close vicinity of the observed system =-=[189]-=-. To increase the operational lifetime of the sensor network, the measurement rate should be as low as possible, which on the other hand leads to a decrease in information gain and consequently in est...

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...97]. For any nonlinear transformation, the propagated regression points capture the posterior mean and covariance matrix accurately up to the second order of the corresponding Taylor-series expansion =-=[180]-=-. For further improving the accuracy, the set of regression points used for the Gaussian estimator proposed in this chapter is not restricted to a fixed size. Increasing the number of regression point...

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...r divergence (2.30) can also be used, especially as it is the ideal deviation measure for mixture7.1. Gaussian Mixture Reduction via Homotopy Continuation 113 reduction in a maximum likelihood sense =-=[153, 193]-=-. Due to the fact that it is impossible to evaluate this measure in closed form for Gaussian mixtures, numerical integration schemes have to be employed, which leads to increased computational costs. ...

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...trix. As the PCRLB depends on the future system state, the PCRLB is calculated approximately in a predictive manner based on particle filtering [41, 80], a priori and constant estimation of the state =-=[143, 179]-=-, or adaptive discretization [79]. As the PCRLB is merely a lower bound, which is above all not always tight [20], the resulting performance can divergence arbitrarily from the closed-loop performance...

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...er’s number. Mutual Information R ...

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...etermination of the covering ellipsoid that contains the ellipsoids of all sensor information matrices. This problem is similar to the so-called covariance union or Löwner ellipsoid problem (see e.g. =-=[23, 198]-=-). Thanks to the fact that all sensor information ellipsoids have the same center, the covariance union problem can be significantly simplified. At first, it is sufficient to consider the determinatio...

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...ty. The complete Gaussian estimator and an extension to Gaussian mixtures is presented in Section 5.4 and Section 5.5, respectively. The Gaussian estimator introduced in this section was published in =-=[217]-=- and its applicability on the real-world localization problem of tracking a human in a telepresence environment was demonstrated in experiments in [209]. Extensions to these publications are the grid ...

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... ˆ...

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...en applied to the problems of sensor subset selection [103], sensor scheduling [104], or sensor placement selection [105]. Extensions to sensor management problems for dynamic systems can be found in =-=[191, 192]-=-. 2.6.3 Non-myopic Sensor Management For discrete state spaces, solutions to the closed-loop control sensor management problem based on dynamic programming can be found for example in [33, 112]. Espec...

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...the regressions points of the set ...

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...mponents of the Bayesian estimator, while all other components contain the approximations that are necessary for performing estimations by means of the HDF. The HDF for scalar states was published in =-=[215, 216]-=-, while the extension to the multivariate case represents unpublished material. The proposed estimator was applied to stochastic model predictive control problems [226, 230] and parameter identificati...

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...of the proposed reduction method, closed-form solutions for all necessary calculations are derived. The fundamentals of the proposed progressive Gaussian mixture reduction algorithm were published in =-=[219]-=- for univariate Gaussian mixtures. This chapter extends the results of this paper to the multivariate case. 1 For an introduction to homotopy continuation see for example [3].110 Chapter 7. Progressi...

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...e the performance of the estimators for stronger and more heavily tailored noise. For the given simulation setup the following four estimators are considered: 2 For tracking with glint noise see e.g. =-=[195]-=-.6.3. Extension to Multivariate States 105 HDF A HDF exploiting that merely the position [...

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... Cramér-von Mises distance [24] ...

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...ensor management could be to employ submodular objective functions [104, 192], which provide a tight bound on the estimation performance in case of greedy/myopic sensor management. As demonstrated in =-=[220]-=-, myopic sensor management in turn simplifies a distributed implementation as only local knowledge is necessary. Constrained Sensor Management The objective functions employed in this thesis, namely c...

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...nt policy from the measurement values. The separation principle still holds for the additional consideration of running measurement costs, which are associated with the requested level of information =-=[194]-=-. Works toward pure sensor management strategies neglecting the control of the dynamic system and employing more adequate objective functions can be found in [37, 88, 119, 138, 148]. In [119], mutual ...

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...mponents of the Bayesian estimator, while all other components contain the approximations that are necessary for performing estimations by means of the HDF. The HDF for scalar states was published in =-=[215, 216]-=-, while the extension to the multivariate case represents unpublished material. The proposed estimator was applied to stochastic model predictive control problems [226, 230] and parameter identificati...

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... depending on the concrete application and required accuracy. The novel open-loop model predictive sensor management approach named quasi-linear sensor management proposed in this section is based on =-=[218, 221]-=-. Extensions to these publications are in particular the linearization procedure as well as the derivation of the bounding sensor for pruning.26 Chapter 3. Quasi-linear Sensor Management 3.1 Linear G...

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...extension to the multivariate case represents unpublished material. The proposed estimator was applied to stochastic model predictive control problems [226, 230] and parameter identification problems =-=[222]-=-. 6.1 Conditional Density Approximation In this section, only scalar states are considered for brevity and clarity. The extension to multivariate states is content of Section 6.3. Furthermore, it is a...

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...ar states was published in [215, 216], while the extension to the multivariate case represents unpublished material. The proposed estimator was applied to stochastic model predictive control problems =-=[226, 230]-=- and parameter identification problems [222]. 6.1 Conditional Density Approximation In this section, only scalar states are considered for brevity and clarity. The extension to multivariate states is ...

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... steps have to be performed consecutively without a measurement update in-between or when the measurement noise is strong. For the Gaussian sum filter, a simultaneous weight calculation is derived in =-=[177]-=-, which can be adapted for the proposed Gaussian mixture estimator in a straightforward manner. 5.5.2 Measurement Update Step Analogously to the prediction step, state and noise Gaussian mixtures are ...

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... a computationally tractable determination of the configuration sequence, their application is not restrict to the proposed sensor management approach. The foundation of this chapter was published in =-=[230]-=- in the context of closed-loop model predictive control of stochastic nonlinear systems. Especially, the concept of the virtual measurements and pruning via the probabilistic branch-and-bound algorith...

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...nd thus of limited use. More appropriate are objective functions that evaluate the expected information gain as for example mutual information or the expected entropy do. This finding is exploited in =-=[52, 186]-=-. For a reduced computational burden, entropy differences defined over the measurement space are used in [186] for approximating mutual information instead of considering the joint state and measureme...

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...en applied to the problems of sensor subset selection [103], sensor scheduling [104], or sensor placement selection [105]. Extensions to sensor management problems for dynamic systems can be found in =-=[191, 192]-=-. 2.6.3 Non-myopic Sensor Management For discrete state spaces, solutions to the closed-loop control sensor management problem based on dynamic programming can be found for example in [33, 112]. Espec...

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...nding weights that exactly captures the mean ˆ...

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...tor introduced in this section was published in [217] and its applicability on the real-world localization problem of tracking a human in a telepresence environment was demonstrated in experiments in =-=[209]-=-. Extensions to these publications are the grid approach for the extension to multivariate densities and the extension to Gaussian mixtures. 5.1 Problem Formulation In order to derive a Gaussian estim...

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...rithm is used, which ensures even in this particular case that the optimal solutions is always found. In the following, a brief review of PBAB is given. A detailed description of PBAB can be found in =-=[187]-=- and [230]. Since PBAB operates on negative objective function values for maximization, the step objectives are converted according to ¯...

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...(...

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...see next section), where the noise dimensions can be considered as less relevant. Furthermore, the techniques proposed in Section 6.3.2 for decomposing the state vector can also be employed here (see =-=[208]-=-). By this means, only parts of the state vector needs to be represented by a set of regression points, which facilitates the application of the grid approach even for many high-dimensional problems....

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... depending on the concrete application and required accuracy. The novel open-loop model predictive sensor management approach named quasi-linear sensor management proposed in this section is based on =-=[218, 221]-=-. Extensions to these publications are in particular the linearization procedure as well as the derivation of the bounding sensor for pruning.26 Chapter 3. Quasi-linear Sensor Management 3.1 Linear G...