Abstract
This article addresses the model-based reconstruction and prediction of distributed phenomena characterized by partial differential equations, which are monitored by sensor networks. The novelty of the proposed reconstruction method is the systematic approach and the integrated treatment of uncertainties, which occur in the physical model and arise naturally from noisy measurements. By this means it is possible not only to reconstruct the entire phenomenon, even at non-measurement points, but also to reconstruct the complete density function of the state characterizing the distributed phenomenon. In the first step, the partial differential equation, i.e., distributed-parameter system, is spatially and temporally decomposed leading to a finite-dimensional state space form. In the next step, the state of the resulting lumped-parameter system, which provides an approximation of the solution of the underlying partial differential equation, is dynamically estimated under consideration of uncertainties. By using the estimation results, several additional tasks can be achieved by the sensor network, e.g. optimal sensor placement, optimal scheduling, model improvement, and system identification. The performance of the proposed model-based reconstruction method is demonstrated by means of simulations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Anderson, B. D. O. and Moore, J. B. (1979). Optimal Filtering. Prentice-Hall.
Baker, A. J. (1983). Finite Element Computational Fluid Mechanics. Taylor and Francis.
Brunn, D. and Hanebeck, U. D. (2005). A model-based framework for optimal measurements in machine tool calibration. In IEEE International Conference on Robotics and Automation (ICRA 2005), Barcelona, Spanien.
Culler, D. E. and Mulder, H. (2004). Smart sensors to network the world. In Scientific American, volume 6.
Fagherazzi, S., Furbish, D. J., Rasetarinera, P., and Hussaini, M. Y. (2004). Application of the discontinuous spectral Galerkin method to groundwater flow. In Advances in Water Resources, volume 27, pp. 129–140.
Faulds, A. L. and King, B. B. (2000). Sensor location in feedback control of partial differential equation systems. In Proceedings of the 2000 IEEE International Conference on Control Applications (CCA 2000), pp. 536–541, Anchorage, AK.
Fournier, A., Bunge, H.-P., Hollerbach, R., and Vilotte, J.-P. (2004). Application of the spectral-element method to the axisymmetric Navier-Stokes equation. In Geophysical Journal International, number 156, pp. 682–700.
Hanebeck, U. D. and Briechle, K. (2001). New results for stochastic prediction and filtering with unknown correlations. In Proceedings of the IEEE Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2001), pp. 147–152, Baden-Baden, Germany.
Hanebeck, U. D., Briechle, K., and Rauh, A. (2003). Progressive Bayes: a new framework for nonlinear state estimation. In Proceedings of SPIE, AeroSense Symposium, pp. 256–267, Orlando, Florida.
Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. In Transactions of the ASME Journal of Basic Engineering, issue 82, pp. 34–45.
Karniadakis, G. E. and Sherwin, S. (2005). Spectral/HP Element Methods for Computational Fluid Dynamics. Oxford University Press.
Komatitsch, D., Ritsema, J., and Tromp, J. (2002). The spectral-element method, Beowulf computing, and global seismology. In Science, volume 298, pp. 1737–1742.
Komatitsch, D., Vilotte, J.-P., Vai, R., Castillo-Covarrubias, J. M., and Sanchez-Sesma, F. J. (1999). The spectral element method for elastic wave equations – application to 2-D and 3-D seismic problems. In International Journal for Numerical Methods in Engineering, volume 45, pp. 1139–1164.
Kumar, T., Zhao, F., and Shepherd, D. (2002). Collaborative signal and information processing in microsensor networks. In IEEE Signal Processing Magazine, volume 19, pp. 13–14.
Levin, J. G., Iskandarani, M., and Haidvogel, D. B. (2000). A nonconforming spectral element ocean model. In International Journal for Numerical Methods in Fluids, volume 34, pp. 495–525.
Ortmaier, T., Groeger, M., Boehm, D. H., Falk, V., and Hirzinger, G. (2005). Motion estimation in beating heart surgery. In IEEE Transactions on Biomedical Engineering, volume 52, issue 10, pp.1729–1740.
Papoulis, A. (1991). Probability, Random Variables and Stochastic Processes. McGraw-Hill.
Rao, B., Durrant-Whyte, H., and Sheen, J. (1993). A fully decentralized multisensor system for tracking and surveillance. In International Journal of Robotics Research, volume 12, pp. 20–44.
Roberts, K. and Hanebeck, U. D. (2005). Prediction and reconstruction of distributed dynamic phenomena characterized by linear partial differential equations. In The 8th International Conference on Information Fusion (Fusion 2005), Philadelphia, USA.
Sawo, F., Brunn, D., and Hanebeck, U. D. (2006a). Parameterized joint densities with gaussian mixture marginals and their potential use in nonlinear robust estimation. In IEEE International Conference on Control Applications (CCA 2006), Munich, Germany.
Sawo, F., Roberts, K., and Hanebeck, U. D. (2006b). Bayesian estimation of distributed phenomena using discretized representations of partial differential equations. In Proceedings of the 3rd International Conference on Informatics in Control, Automation and Robotics (ICINCO 2006), pp. 16–23, Setubal, Portugal.
Thuraisingham, B. (2004). Secure sensor information management and mining. In IEEE Signal Processing Magazine, volume 5, pp. 14–19.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Sawo, F., Roberts, K., Hanebeck, U.D. (2008). Model-based Reconstruction of Distributed Phenomena using Discretized Representations of Partial Differential Equations. In: Cetto, J.A., Ferrier, JL., Costa dias Pereira, J., Filipe, J. (eds) Informatics in Control Automation and Robotics. Lecture Notes Electrical Engineering, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79142-3_24
Download citation
DOI: https://doi.org/10.1007/978-3-540-79142-3_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79141-6
Online ISBN: 978-3-540-79142-3
eBook Packages: EngineeringEngineering (R0)