Nonlinear state and parameter estimation of spatially distributed systems

In this thesis two probabilistic model-based estimators are introduced that allow the reconstruction and identification of space-time continuous physical systems. The Sliced Gaussian Mixture Filter (SGMF) exploits linear substructures in mixed linear/nonlinear systems, and thus is well-suited for identifying various model parameters. The Covariance Bounds Filter (CBF) allows the efficient estimation of widely distributed systems in a decentralized fashion

Nonlinear Multidimensional Bayesian Estimation with Fourier Densities

Efficiently implementing nonlinear Bayesian estimators is still an unsolved problem, especially for the multidimensional case. A trade-off between estimation quality and demand on computational resources has to be found. Using multidimensional Fourier series as representation for probability density functions, so called Fourier densities, is proposed. To ensure non-negativity, the approximation is performed indirectly via Psi-densities, of which the absolute square represent the Fourier density. It is shown that PSI-densities can be determined using the efficient fast Fourier transform algorithm and their coefficients have an ordering with respect to the Hellinger metric. Furthermore, the multidimensional Bayesian estimator based on Fourier Densities is derived in closed form. That allows an efficient realization of the Bayesian estimator where the demands on computational resources are adjustable

Simultaneous State and Parameter Estimation of Distributed-Parameter Physical Systems based on Sliced Gaussian Mixture Filter

This paper presents a method for the simultaneous state and parameter estimation of finite-dimensional models of distributed systems monitored by a sensor network. In the first step, the distributed system is spatially and temporally decomposed leading to a linear finite-dimensional model in state space form. The main challenge is that the simultaneous state and parameter estimation of such systems leads to a high-dimensional nonlinear problem. Thanks to the linear substructure contained in the resulting finite-dimensional model, the development of an overall more efficient estimation process is possible. Therefore, in the second step, we propose the application of a novel density representation - sliced Gaussian mixture density - in order to decompose the estimation problem into a (conditionally) linear and a nonlinear problem. The systematic approximation procedure minimizing a certain distance measure allows the derivation of (close to) optimal and deterministic results. The proposed estimation process provides novel prospects in sensor network applications. The performance is demonstrated by means of simulation results

Distributed Greedy Sensor Scheduling for Model-based Reconstruction of Space-Time Continuous Physical Phenomena

A novel distributed sensor scheduling method for large-scale sensor networks observing space-time continuous physical phenomena is introduced. In a first step, the model of the distributed phenomenon is spatially and temporally decomposed leading to a linear probabilistic finite-dimensional model. Based on this representation, the information gain of sensor measurements is evaluated by means of the so-called covariance reduction function. For this reward function, it is shown that the performance of the greedy sensor scheduling is at least half that of the optimal scheduling considering long-term effects. This finding is the key for distributed sensor scheduling, where a central processing unit or fusion center is unnecessary, and thus, scaling as well as reliability is ensured. Hence, greedy scheduling in combination with a proposed hierarchical communication scheme requires only local sensor information and communication

Modellbasierte Vermessung verteilter Phänomene und Generierung optimaler Messsequenzen

Dieser Beitrag befasst sich mit modellbasierten Methoden zur Vermessung verteilter physikalischer Phänomene. Diese Methoden zeichnen sich durch eine systematische Behandlung von Unsicherheiten aus, so dass neben der Rekonstruktion der vollständigen Wahrscheinlichkeitsdichte der relevanten Grö\ss{}en aus einer geringen Anzahl von zeit-, orts- und wertdiskreten Messungen auch die Generierung optimaler Messsequenzen möglich ist. Es wird dargestellt, wie eine Beschreibung für ein verteilt"=parametrisches System in Form einer partiellen Differentialgleichung, welche einen unendlich-dimensionalen Zustandsraum beschreibt, in eine konzentriert-parametrische Form konvertiert wird. Diese kann als Grundlage für den Entwurf klassischer Schätzer, wie z. B. des Kalman-Filters, dienen. Ferner wird eine Methode zur Sensoreinsatzplanung vorgestellt, mit der eine optimale Sequenz von Messparametern bestimmt werden kann, um mit einem minimalen Messaufwand die Unsicherheit auf ein gewünschtes Ma\ss{} zu reduzieren. Die Anwendung dieser Methoden wird an zwei Beispielen, einer Temperaturverteilung und der Verformung einer Führungsschiene, demonstriert. Zusätzlich werden die Herausforderungen bei der Behandlung nichtlinearer Systeme und die Probleme bei der dezentralen Verarbeitung, wie sie typischerweise beim Einsatz von Sensornetzwerken auftreten, diskutiert

Decentralized State Estimation of Distributed Phenomena based on Covariance Bounds

This paper addresses the problem of decentralized state estimation of distributed physical phenomena observed by a sensor network. The centralized approaches are not scalable for large sensor networks, because all information has to be transmitted to a powerful central processing node requiring an extensive amount of communication bandwidth and a lot of processing power. Thus, for a decentralized reconstruction of distributed phenomena, we propose a novel methodology consisting of three steps: (a) conversion of the distributed phenomenon into a lumped-parameter system description, (b) decomposition of the resulting system in order to map the description to the actual sensor network, and (c) decomposition of the density representation leading to a decentralized estimation approach. The main problem of a decentralized approach is that due to the propagation of local information through the network, unknown correlations are caused. This fact needs to be considered during the reconstruction process in order to get correct and consistent estimation results. For that reason, we employ a robust estimator (based on Covariance Bounds) for the local reconstruction update on each sensor node. By this means, the individual sensor nodes are able to estimate the local state of the distributed phenomenon using local estimates obtained and communicated by adjacent nodes only. The information about their correlations is not stored in the sensor network

Parameter Identification and Reconstruction for Distributed Phenomena Based on Hybrid Density Filter

This paper addresses the problem of model-based reconstruction and parameter identification of distributed phenomena characterized by partial differential equations. The novelty of the proposed method is the systematic approach and the integrated treatment of uncertainties, which naturally occur in the physical system and arise from noisy measurements. The main challenge of accurate reconstruction is that model parameters, i.e., diffusion coefficients, of the physical model are not known in advance and usually need to be identified. Generally, the problem of parameter identification leads to a nonlinear estimation problem. Hence, a novel efficient recursive procedure is employed. Unlike other estimators, the so-called Hybrid Density Filter not only assures accurate estimation results for nonlinear systems, but also offers an efficient processing. By this means it is possible to reconstruct and identify distributed phenomena monitored by autonomous wireless sensor networks. The performance of the proposed estimation method is demonstrated by means of simulations

Informationsfusion für verteilte Systeme

Dieser Beitrag befasst sich mit modellbasierten Methoden zur Vermessung verteilter physikalischer Phänomene. Diese Methoden zeichnen sich durch eine systematische Behandlung stochastischer Unsicherheiten aus, so dass neben der Rekonstruktion der vollständigen Wahrscheinlichkeitsdichte der relevanten Grössen aus einer geringen Anzahl von zeit-, orts- und wertdiskreten Messungen auch die Generierung optimaler Messsequenzen möglich ist. Es wird dargestellt, wie eine Beschreibung für ein verteilt-parametrisches System in Form einer partiellen Differentialgleichung, welche einen unendlich-dimensionalen Zustandsraum beschreibt, in eine konzentriert-parametrische Form konvertiert wird. Diese kann als Grundlage für den Entwurf klassischer Schätzer, wie z.B. des Kalman Filters, dienen. Ferner wird eine Methode zur Sensoreinsatzplanung vorgestellt, mit der eine optimale Sequenz von Messparametern bestimmt werden kann, um mit einem minimalen Messaufwand die Unsicherheit auf ein gewünschtes Maß zu reduzieren. Die Anwendung dieser Methoden wird an zwei Beispielen, einer Temperaturverteilung und der Verformung einer Führungsschiene, demonstriert. Zusätzlich werden die Herausforderungen bei der Behandlung nichtlinearer Systeme und die Probleme bei der dezentralen Verarbeitung, wie sie typischerweise beim Einsatz von Sensornetzwerken auftreten, diskutiert

Including Expert Knowledge in Finite Element Models by Means of Fuzzy Based Parameter Estimation

Abstract: In this paper we present a novel approach for modelling spatial distributed biochemical and environmental processes like the growth of plants and the related biochemical reactions. One of the main challenges for modelling of spatial distributed phenomena is the estimation of the model parameters. The physical phenomena like flow and mass transport can be described by PDEs of fluid dynamics, but for effects like growth rates often no analytic models are available. However, in many cases experts have knowledge about the system behaviour that can be formulated by a set of if-then-rules. As this kind of knowledge can easily be handled by so-called Fuzzy models we propose the coupling of FEM models with such Fuzzy models. By this means one or more parameters of the classical PDEs are estimated by Fuzzy Models. Besides the natural inclusion of expert knowledge a second benefit of this approach consists in the fact that Fuzzy Models can describe even very nonlinear phenomena. The proposed approach is applied for modelling the growth of algae of Orbetello lake in Italy