Sensor network design for the estimation of spatially distributed processes

Dariusz Uciński ; Maciej Patan

International Journal of Applied Mathematics and Computer Science, Tome 20 (2010), p. 459-481 / Harvested from The Polish Digital Mathematics Library

Résumé

In a typical moving contaminating source identification problem, after some type of biological or chemical contamination has occurred, there is a developing cloud of dangerous or toxic material. In order to detect and localize the contamination source, a sensor network can be used. Up to now, however, approaches aiming at guaranteeing a dense region coverage or satisfactory network connectivity have dominated this line of research and abstracted away from the mathematical description of the physical processes underlying the observed phenomena. The present work aims at bridging this gap and meeting the needs created in the context of the source identification problem. We assume that the paths of the moving sources are unknown, but they are sufficiently smooth to be approximated by combinations of given basis functions. This parametrization makes it possible to reduce the source detection and estimation problem to that of parameter identification. In order to estimate the source and medium parameters, the maximum-likelihood estimator is used. Based on a scalar measure of performance defined on the Fisher information matrix related to the unknown parameters, which is commonly used in optimum experimental design theory, the problem is formulated as an optimal control one. From a practical point of view, it is desirable to have the computations dynamic data driven, i.e., the current measurements from the mobile sensors must serve as a basis for the update of parameter estimates and these, in turn, can be used to correct the sensor movements. In the proposed research, an attempt will also be made at applying a nonlinear model-predictive-control-like approach to attack this issue.

Détail

Publié le : 2010-01-01

Zbl 1211.93041

EUDML-ID : urn:eudml:doc:208000

Bibliographie

[000] Akçelik, V., Biros, G., Ghattas, O., Long, K. R. and van Bloemen Waanders, B. (2003). A variational finite element method for source inversion for convective-diffusive transport, Finite Elements in Analysis and Design 39: 683-705.

[001] Atkinson, A.C., Donev, A.N. and Tobias, R.D. (2007). Optimum Experimental Designs, with SAS, Oxford University Press, Oxford. | Zbl 1183.62129

[002] Banks, H.T. (1992). Computational issues in parameter estimation and feedback control problems for partial differential equation systems, Physica D 60: 226-238. | Zbl 0783.93036

[003] Banks, H.T., Smith, R.C. and Wang, Y. (1996). Smart Material Structures: Modeling, Estimation and Control, Research in Applied Mathematics, Masson, Paris. | Zbl 0882.93001

[004] Biegler, L.T., Ghattas, O., Heinkenschloss, M., Keyes, D. and van Bloemen Waanders, B. (Eds.) (2007). Real-Time PDEConstrained Optimization, Society for Industrial and Applied Mathematics, Philadelphia, PA. | Zbl 1117.49004

[005] Boggs, P.T., Long, K.R., Margolis, S.B. and Howard, P.A. (2006). Rapid source inversion for chemical/biological attacks. Part 1: The steady-state case, SIAM Journal on Optimization 17(2): 430-458. | Zbl 1121.49037

[006] Butkovskiy, A.G. and Pustyl'nikov, A.M. (1987). Mobile Control of Distributed Parameter Systems, John Wiley & Sons, New York, NY.

[007] Cassandras, C. G. and Li, W. (2005). Sensor networks and cooperative control, European Journal of Control 11(4-5): 436-463. | Zbl 1293.93069

[008] Chavent, G. (1991). On the theory and practice of non-linear least-squares, Advances in Water Resources 14(2): 55-63.

[009] Chong, C.-Y. and Kumar, S.P. (2003). Sensor networks: Evolution, opportunities, and challenges, Proceedings of the IEEE 91(8): 1247-1256.

[010] Christofides, P.D. (2001). Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes, Systems & Control: Foundations & Applications, Birkhäuser, Boston, MA. | Zbl 1018.93001

[011] Daescu, D.N. and Navon, I.M. (2004). Adaptive observations in the context of 4D-Var data assimilation, Meteorology and Atmospheric Physics 85: 205-226.

[012] Demetriou, M.A. (2006a). Detection and containment policy of moving source in 2D diffusion processes using sensor/actuator network, Proceedings of the European Control Conference 2007, Kos, Greece, (on CD-ROM).

[013] Demetriou, M.A. (2006b). Power management of sensor networks for detection of a moving source in 2-D spatial domains, Proceedings of the 2006 American Control Conference, Minneapolis, MN, (on CD-ROM).

[014] Demetriou, M.A. (2007). Process estimation and moving source detection in 2-D diffusion processes by scheduling of sensor networks, Proceedings of the 2007 American Control Conference, New York, NY, USA, (on CD-ROM).

[015] Demetriou, M.A. (2009). Natural consensus filters for second order infinite dimensional systems, Systems & Control Letters 58(12): 826-833. | Zbl 1191.93017

[016] Demetriou, M.A. and Hussein, I.I. (2009). Estimation of spatially distributed processes using mobile spatially distributed sensor network, SIAM Journal on Control and Optimization 48(1): 266-291. | Zbl 1182.93111

[017] Fedorov, V.V. and Hackl, P. (1997). Model-Oriented Design of Experiments, Lecture Notes in Statistics, Springer-Verlag, New York, NY. | Zbl 0878.62052

[018] Ford, I., Titterington, D.M. and Kitsos, C.P. (1989). Recent advances in nonlinear experimental design, Technometrics 31(1): 49-60. | Zbl 0668.62048

[019] Gevers, M. (2005). Identification for control: From the early achievements to the revival of experiment design, European Journal of Control 11(4-5): 335-352. | Zbl 1293.93206

[020] Gnot, S., Rafajłowicz, E. and Urbańska-Motyka, A. (2001). Statistical inference in a linear model for spatially located sensors and random input, Annals of the Institute of Statistical Mathematics 53(2): 370-379. | Zbl 1027.62038

[021] Goodwin, G.C. and Payne, R.L. (1977). Dynamic System Identification. Experiment Design and Data Analysis, Mathematics in Science and Engineering, Academic Press, New York, NY. | Zbl 0578.93060

[022] Gruver, W.A. and Sachs, E. (1980). Algorithmic Methods in Optimal Control, Pitman Publishing Limited, London. | Zbl 0456.49001

[023] Hirsch, M.J., Pardalos, P.M., Murphey, R. and Grundel, D. (Eds.) (2008). Advances in Cooperative Control and Optimization. Proceedings of the 7th International Conference on Cooperative Control and Optimization, Springer-Verlag, Berlin. | Zbl 1121.93006

[024] Hjalmarsson, H. (2005). From experiment design to closed-loop control, Automatica 41(3): 393-438. | Zbl 1079.93016

[025] Hussein, I. I. and Demetriou, M. A. (2007). Estimation of distributed processes using mobile spatially distributed sensors, Proceedings of the 2007 American Control Conference, New York, NY, USA, (on CD-ROM). | Zbl 1182.93111

[026] Isakov, V. (1998). Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, Springer-Verlag, New York, NY. | Zbl 0908.35134

[027] Jacobson, M.Z. (1999). Fundamentals of Atmospheric Modeling, Cambridge University Press, Cambridge. | Zbl 0916.76001

[028] Jain, N. and Agrawal, D.P. (2005). Current trends in wireless sensor network design, International Journal of Distributed Sensor Networks 1: 101-122.

[029] Jennings, L.S., Fisher, M.E., Teo, K.L. and Goh, C.J. (2002). MISER 3: Optimal Control Software, Version 2.0. Theory and User Manual, Department of Mathematics, University of Western Australia, Nedlands, http://www.cado.uwa.edu.au/miser/.

[030] Jeremić, A. and Nehorai, A. (1998). Design of chemical sensor arrays for monitoring disposal sites on the ocean floor, IEEE Transactions on Oceanic Engineering 23(4): 334-343.

[031] Jeremić, A. and Nehorai, A. (2000). Landmine detection and localization using chemical sensor array processing, IEEE Transactions on Signal Processing 48(5): 1295-1305.

[032] Kubrusly, C.S. and Malebranche, H. (1985). Sensors and controllers location in distributed systems-A survey, Automatica 21(2): 117-128. | Zbl 0555.93035

[033] Kusiak, S. and Weatherwax, J. (2008). Identification and characterization of a mobile source in a general parabolic differential equation with constant coefficients, SIAM Journal on Applied Mathematics 68(3): 784-805. | Zbl 1142.49016

[034] Lefèvre, F. and Niliot, C.L. (2002). The BEM for point heat source estimation: Application to multiple static sources and moving sources, International Journal of Thermal Sciences 41: 536-546.

[035] Lehmann, E.L. and Romano, J.P. (2005). Testing Statistical Hypotheses, 3rd Edn., Springer-Verlag. | Zbl 1076.62018

[036] Ljung, L. (1999). System Identification: Theory for the User, 2nd Edn., Prentice Hall, Upper Saddle River, NJ. | Zbl 0615.93004

[037] Maksimov, V.I. (2000). Problems of Dynamic Input Reconstruction of Infinite-Dimensional Systems, Russian Academy of Sciences Press, Ekaterinburg, (in Russian).

[038] Martínez, S. and Bullo, F. (2006). Optimal sensor placement and motion coordination for target tracking, Automatica 42(4): 661-668. | Zbl 1110.93050

[039] Navon, I.M. (1997). Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography, Dynamics of Atmospheres and Oceans 27: 55-79.

[040] Nehorai, A., Porat, B. and Paldi, E. (1995). Detection and localization of vapor-emitting sources, IEEE Transactions on Signal Processing 43(1): 243-253.

[041] Ögren, P., Fiorelli, E. and Leonard, N.E. (2004). Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment, IEEE Transactions on Automatic Control 49(8): 1292-1302.

[042] Patan, M. and Patan, K. (2005). Optimal observation strategies for model-based fault detection in distributed systems, International Journal of Control 78(18): 1497-1510. | Zbl 1122.93018

[043] Patan, M. and Uciński, D. (2005). Optimal activation strategy of discrete scanning sensors for fault detection in distributedparameter systems, Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, (on CD-ROM).

[044] Patan, M. and Uciński, D. (2008). Configuring a sensor network for fault detection in distributed parameter systems, International Journal of Applied Mathematics and Computer Science 18(4): 513-524, DOI: 10.2478/v10006-008-00454. | Zbl 1155.93426

[045] Polak, E. (1997). Optimization. Algorithms and Consistent Approximations, Applied Mathematical Sciences, Springer-Verlag, New York, NY. | Zbl 0899.90148

[046] Porat, B. and Nehorai, A. (1996). Localizing vapor-emitting sources by moving sensors, IEEE Transactions on Signal Processing 44(4): 1018-1021.

[047] Pytlak, R. (1999). Numerical Methods for Optimal Control Problems with State Constraints, Springer-Verlag, Berlin. | Zbl 0928.49002

[048] Quereshi, Z.H., Ng, T.S. and Goodwin, G.C. (1980). Optimum experimental design for identification of distributed parameter systems, International Journal of Control 31(1): 21-29. | Zbl 0431.93017

[049] Rafajłowicz, E. (1981). Design of experiments for eigenvalue identification in distributed-parameter systems, International Journal of Control 34(6): 1079-1094. | Zbl 0476.93071

[050] Rafajłowicz, E. (1983). Optimal experiment design for identification of linear distributed-parameter systems: Frequency domain approach, IEEE Transactions on Automatic Control 28(7): 806-808. | Zbl 0521.93066

[051] Rafajłowicz, E. (1986). Optimum choice of moving sensor trajectories for distributed parameter system identification, International Journal of Control 43(5): 1441-1451. | Zbl 0581.93066

[052] Sastry, S. and Iyengar, S.S. (2005). Real-time sensor-actuator networks, International Journal of Distributed Sensor Networks 1: 17-34.

[053] Schwartz, A. L., Polak, E. and Chen, Y. (1997). A Matlab Toolbox for Solving Optimal Control Problems. Version 1.0 for Windows, http://www.schwartz-home.com/ãdam/RIOTS/.

[054] Sinopoli, B., Sharp, C., Schenato, L., Schaffert, S. and Sastry, S.S. (2003). Distributed control applications within sensor networks, Proceedings of the IEEE 91(8): 1235-1246.

[055] Sivergina, I.F. and Polis, M.P. (2002). Comments on “Modelbased solution techniques for the source localization problem”, IEEE Transactions on Control Systems Technology 10(4): 633-633.

[056] Sivergina, I.F., Polis, M.P. and Kolmanovsky, I. (2003). Source identification for parabolic equations, Mathematics of Control, Signals, and Systems 16: 141-157. | Zbl 1029.93012

[057] Song, Z., Chen, Y., Sastry, C. R. and Tas, N. C. (2009). Optimal Observation for Cyber-physical Systems: A FisherInformation-Matrix-Based Approach, Springer-Verlag, London. | Zbl 1219.93002

[058] Sun, N.-Z. (1994). Inverse Problems in Groundwater Modeling, Theory and Applications of Transport in Porous Media, Kluwer Academic Publishers, Dordrecht.

[059] Uciński, D. (1999). Measurement Optimization for Parameter Estimation in Distributed Systems, Technical University Press, Zielona Góra.

[060] Uciński, D. (2000a). Optimal selection of measurement locations for parameter estimation in distributed processes, International Journal of Applied Mathematics and Computer Science 10(2): 357-379. | Zbl 0965.93041

[061] Uciński, D. (2000b). Optimal sensor location for parameter estimation of distributed processes, International Journal of Control 73(13): 1235-1248. | Zbl 1004.93015

[062] Uciński, D. (2005). Optimal Measurement Methods for Distributed-Parameter System Identification, CRC Press, Boca Raton, FL. | Zbl 1155.93003

[063] Uciński, D. and Atkinson, A. C. (2004). Experimental design for time-dependent models with correlated observations, Studies in Nonlinear Dynamics & Econometrics 8(2), Article No. 13. | Zbl 1082.62514

[064] Uciński, D. and Bogacka, B. (2005). T-optimum designs for discrimination between two multivariate dynamic models, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67: 3-18. | Zbl 1060.62084

[065] Uciński, D. and Chen, Y. (2005). Time-optimal path planning of moving sensors for parameter estimation of distributed systems, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, Seville, Spain, (on CD-ROM).

[066] Uciński, D. and Chen, Y. (2006). Sensor motion planning in distributed parameter systems using Turing's measure of conditioning, Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA, (on CDROM).

[067] Uciński, D. and Demetriou, M.A. (2008). Resource-constrained sensor routing for optimal observation of distributed parameter systems, Proceedings of the 18th International Symposium on Mathematical Theory of Networks and Systems, Blacksburg, VA, (on CD-ROM).

[068] Uciński, D. and Korbicz, J. (2001). Optimal sensor allocation for parameter estimation in distributed systems, Journal of Inverse and Ill-Posed Problems 9(3): 301-317. | Zbl 0994.35123

[069] Uciński, D. and Patan, M. (2007). D-optimal design of a monitoring network for parameter estimation of distributed systems, Journal of Global Optimization 39: 291-322. | Zbl 1180.90173

[070] Uspenskii, A.B. and Fedorov, V.V. (1975). Computational Aspects of the Least-Squares Method in the Analysis and Design of Regression Experiments, Moscow University Press, Moscow, (in Russian).

[071] van de Wal, M. and de Jager, B. (2001). A review of methods for input/output selection, Automatica 37(4): 487-510. | Zbl 0995.93002

[072] Vogel, C.R. (2002). Computational Methods for Inverse Problems, Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA. | Zbl 1008.65103

[073] von Stryk, O. (1999). User's Guide for DIRCOL, a Direct Collocation Method for the Numerical Solution of Optimal Control Problems. Version 2.1, Simulation, Systems Optimization and Robotics Group, Technical University of Darmstadt. http://www.sim.informatik.//tu-darmstadt.de/index/leftnav.html.en.

[074] Walter, É. and Pronzato, L. (1990). Qualitative and quantitative experiment design for phenomenological models-A survey, Automatica 26(2): 195-213. | Zbl 0703.62072

[075] Walter, É. and Pronzato, L. (1997). Identification of Parametric Models from Experimental Data, Communications and Control Engineering, Springer-Verlag, Berlin. | Zbl 0864.93014

[076] Zhao, F. and Guibas, L.J. (2004). Wireless Sensor Networks: An Information Processing Approach, Morgan Kaufmann Publishers, Amsterdam.

[077] Zhao, T. and Nehorai, A. (2006). Detecting and estimating biochemical dispersion of a moving source in a semiinfinite medium, IEEE Transactions on Signal Processing 54(6): 2213-2225.