An Optimal Scanning Sensor Activation Policy for Parameter Estimation of Distributed Systems

  • Dariusz Uciński
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 4)


A technique is proposed to solve an optimal node activation problem in sensor networks whose measurements are supposed to be used to estimate unknown parameters of the underlying process model in the form of a partial differential equation. Given a partition of the observation horizon into a finite number of consecutive intervals, the problem is set up to select nodes which will be active over each interval while the others will remain dormant such that the log-determinant of the resulting Fisher information matrix associated with the estimated parameters is maximized. The search for the optimal solution is performed using the branch-and-bound method in which an extremely simple and efficient technique is employed to produce an upper bound to the maximum objective function. Its idea consists in solving a relaxed problem through the application of a simplicial decomposition algorithm in which the restricted master problem is solved using a multiplicative algorithm for D-optimal design. The performance evaluation of the technique is additionally presented by means of simulations.


Sensor Network Fisher Information Matrix Distribute Parameter System Restricted Master Problem Sensor Configuration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported by the Polish Ministry of Science and Higher Education under grant No. N N519 2971 33.


  1. 1.
    Amouroux, M., Babary, J.P.: Sensor and control location problems. In: M.G. Singh (ed.) Systems & Control Encyclopedia. Theory, Technology, Applications, vol. 6, pp. 4238–4245. Pergamon Press, Oxford (1988)Google Scholar
  2. 2.
    Armstrong, M.: Basic Linear Geostatistics. Springer-Verlag, Berlin (1998)zbMATHCrossRefGoogle Scholar
  3. 3.
    Atkinson, A.C., Donev, A.N., Tobias, R.D.: Optimum Experimental Designs, with SAS. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  4. 4.
    Banks, H.T., Kunisch, K.: Estimation Techniques for Distributed Parameter Systems. Systems & Control: Foundations & Applications. Birkhäuser, Boston (1989)Google Scholar
  5. 5.
    Banks, H.T., Smith, R.C., Wang, Y.: Smart Material Structures: Modeling, Estimation and Control. Research in Applied Mathematics. Masson, Paris (1996)zbMATHGoogle Scholar
  6. 6.
    Bensoussan, A., Da Prato, G., Delfour, M.C., Mitter, S.K.: Representation and Control of Infinite Dimensional Systems, 2nd edn. Birkhäuser, Boston (2007)zbMATHGoogle Scholar
  7. 7.
    Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Optimization and Computation Series. Athena Scientific, Belmont, MA (1999)zbMATHGoogle Scholar
  8. 8.
    Boer, E.P.J., Hendrix, E.M.T., Rasch, D.A.M.K.: Optimization of monitoring networks for estimation of the semivariance function. In: A.C. Atkinson, P. Hackl, W. Müller (eds.) mODa 6, Proc. 6th Int. Workshop on Model-Oriented Data Analysis, Puchberg/Schneeberg, Austria, 2001, pp. 21–28. Physica-Verlag, Heidelberg (2001)Google Scholar
  9. 9.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  10. 10.
    Cassandras, C.G., Li, W.: Sensor networks and cooperative control. European Journal of Control 11(4–5), 436–463 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chong, C.Y., Kumar, S.P.: Sensor networks: Evolution, opportunities, and challenges. Proceedings of the IEEE 91(8), 1247–1256 (2003)CrossRefGoogle Scholar
  12. 12.
    Christofides, P.D.: Nonlinear and Robust Control of PDE Systems: Methods and Applications to Transport-Reaction Processes. Systems & Control: Foundations & Applications. Birkhäuser, Boston (2001)Google Scholar
  13. 13.
    Cook, D., Fedorov, V.: Constrained optimization of experimental design. Statistics 26, 129–178 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Cressie, N.A.C.: Statistics for Spatial Data, revised edn. John Wiley & Sons, New York (1993)Google Scholar
  15. 15.
    Culler, D., Estrin, D., Srivastava, M.: Overview of sensor networks. IEEE Computer 37(8), 41–49 (2004)CrossRefGoogle Scholar
  16. 16.
    Curtain, R.F., Zwart, H.: An Introduction to Infinite-Dimensional Linear Systems Theory. Texts in Applied Mathematics. Springer-Verlag, New York (1995)zbMATHCrossRefGoogle Scholar
  17. 17.
    Daescu, D.N., Navon, I.M.: Adaptive observations in the context of 4D-Var data assimilation. Meteorology and Atmospheric Physics 85, 205–226 (2004)CrossRefGoogle Scholar
  18. 18.
    Demetriou, M.A.: Detection and containment policy of moving source in 2D diffusion processes using sensor/actuator network. In: Proceedings of the European Control Conference 2007, Kos, Greece, July 2–5 (2006). Published on CD-ROMGoogle Scholar
  19. 19.
    Demetriou, M.A.: Power management of sensor networks for detection of a moving source in 2-D spatial domains. In: Proceedings of the 2006 American Control Conference, Minneapolis, MN, June 14–16 (2006). Published on CD-ROMGoogle Scholar
  20. 20.
    Demetriou, M.A.: Process estimation and moving source detection in 2-D diffusion processes by scheduling of sensor networks. In: Proceedings of the 2007 American Control Conference, New York City, USA, July 11–13 (2007). Published on CD-ROMGoogle Scholar
  21. 21.
    Demetriou, M.A.: Natural consensus filters for second order infinite dimensional systems. Systems & Control Letters 58(12), 826–833 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Demetriou, M.A., Hussein, I.I.: Estimation of spatially distributed processes using mobile spatially distributed sensor network. SIAM Journal on Control and Optimization 48(1), 266–291 (2009). DOI 10. 1137/060677884MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Fedorov, V.V.: Optimal design with bounded density: Optimization algorithms of the exchange type. Journal of Statistical Planning and Inference 22, 1–13 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Fedorov, V.V., Hackl, P.: Model-Oriented Design of Experiments. Lecture Notes in Statistics. Springer-Verlag, New York (1997)zbMATHCrossRefGoogle Scholar
  25. 25.
    Floudas, C.A.: Mixed integer nonlinear programming, MINLP. In: C.A. Floudas, P.M. Pardalos (eds.) Encyclopedia of Optimization, vol. 3, pp. 401–414. Kluwer Academic Publishers, Dordrecht, The Netherlands (2001)CrossRefGoogle Scholar
  26. 26.
    Gerdts, M.: Solving mixed-integer optimal control problems by branch&bound: A case study from automobile test-driving with gear shift. Journal of Optimization Theory and Applications 26, 1–18 (2005)MathSciNetGoogle Scholar
  27. 27.
    Gevers, M.: Identification for control: From the early achievements to the revival of experiment design. European Journal of Control 11(4–5), 335–352 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Goodwin, G.C., Payne, R.L.: Dynamic System Identification. Experiment Design and Data Analysis. Mathematics in Science and Engineering. Academic Press, New York (1977)zbMATHGoogle Scholar
  29. 29.
    Hearn, D.W., Lawphongpanich, S., Ventura, J.A.: Finiteness in restricted simplicial decomposition. Operations Research Letters 4(3), 125–130 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Hearn, D.W., Lawphongpanich, S., Ventura, J.A.: Restricted simplicial decomposition: Computation and extensions. Mathematical Programming Study 31, 99–118 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Hjalmarsson, H.: From experiment design to closed-loop control. Automatica 41, 393–438 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    von Hohenbalken, B.: Simplicial decomposition in nonlinear programming algorithms. Mathematical Programming 13, 49–68 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge, UK (1986)Google Scholar
  34. 34.
    Hussein, I.I., Demetriou, M.A.: Estimation of distributed processes using mobile spatially distributed sensors. In: Proceedings of the 2007 American Control Conference, New York, USA, July 11–13 (2007). Published on CD-ROMGoogle Scholar
  35. 35.
    Jain, N., Agrawal, D.P.: Current trends in wireless sensor network design. International Journal of Distributed Sensor Networks 1, 101–122 (2005)CrossRefGoogle Scholar
  36. 36.
    Jeremić, A., Nehorai, A.: Design of chemical sensor arrays for monitoring disposal sites on the ocean floor. IEEE Transactions on Oceanic Engineering 23(4), 334–343 (1998)CrossRefGoogle Scholar
  37. 37.
    Jeremić, A., Nehorai, A.: Landmine detection and localization using chemical sensor array processing. IEEE Transactions on Signal Processing 48(5), 1295–1305 (2000)CrossRefGoogle Scholar
  38. 38.
    Kammer, D.C.: Sensor placement for on-orbit modal identification and correlation of large space structures. In: Proc. American Control Conf., San Diego, California, 23–25 May 1990, vol. 3, pp. 2984–2990 (1990)Google Scholar
  39. 39.
    Kammer, D.C.: Effects of noise on sensor placement for on-orbit modal identification of large space structures. Transactions of the ASME 114, 436–443 (1992)Google Scholar
  40. 40.
    Kincaid, R.K., Padula, S.L.: D-optimal designs for sensor and actuator locations. Computers & Operations Research 29, 701–713 (2002)zbMATHCrossRefGoogle Scholar
  41. 41.
    Kubrusly, C.S., Malebranche, H.: Sensors and controllers location in distributed systems — A survey. Automatica 21(2), 117–128 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Lam, R.L.H., Welch, W.J., Young, S.S.: Uniform coverage designs for molecule selection. Technometrics 44(2), 99–109 (2002)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Lange, K.: Numerical Analysis for Statisticians. Springer-Verlag, New York (1999)zbMATHGoogle Scholar
  44. 44.
    Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Encyclopedia of Mathematics and Its Applications, vol. I and II. Cambridge University Press, Cambridge (2000)Google Scholar
  45. 45.
    Le, N.D., Zidek, J.V.: Statistical Analysis of Environmental Space-Time Processes. Springer-Verlag, New York (2006)zbMATHGoogle Scholar
  46. 46.
    Liu, C.Q., Ding, Y., Chen, Y.: Optimal coordinate sensor placements for estimating mean and variance components of variation sources. IEE Transactions 37, 877–889 (2005)CrossRefGoogle Scholar
  47. 47.
    Ljung, L.: System Identification: Theory for the User, 2nd edn. Prentice Hall, Upper Saddle River, NJ (1999)Google Scholar
  48. 48.
    Meyer, R.K., Nachtsheim, C.J.: The coordinate-exchange algorithm for constructing exact optimal experimental designs. Technometrics 37(1), 60–69 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Müller, W.G.: Collecting Spatial Data. Optimum Design of Experiments for Random Fields, 2nd revised edn. Contributions to Statistics. Physica-Verlag, Heidelberg (2001)Google Scholar
  50. 50.
    Munack, A.: Optimal sensor allocation for identification of unknown parameters in a bubble-column loop bioreactor. In: A.V. Balakrishnan, M. Thoma (eds.) Analysis and Optimization of Systems, Part 2, Lecture Notes in Control and Information Sciences, volume 63, pp. 415–433. Springer-Verlag, Berlin (1984)Google Scholar
  51. 51.
    Navon, I.M.: Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography. Dynamics of Atmospheres and Oceans 27, 55–79 (1997)CrossRefGoogle Scholar
  52. 52.
    Nehorai, A., Porat, B., Paldi, E.: Detection and localization of vapor-emitting sources. IEEE Transactions on Signal Processing 43(1), 243–253 (1995)CrossRefGoogle Scholar
  53. 53.
    Nychka, D., Piegorsch, W.W., Cox, L.H. (eds.): Case Studies in Environmental Statistics. Lecture Notes in Statistics, volume 132. Springer-Verlag, New York (1998)Google Scholar
  54. 54.
    Nychka, D., Saltzman, N.: Design of air-quality monitoring networks. In: D. Nychka, W.W. Piegorsch, L.H. Cox (eds.) Case Studies in Environmental Statistics, Lecture Notes in Statistics, volume 132, pp. 51–76. Springer-Verlag, New York (1998)CrossRefGoogle Scholar
  55. 55.
    Ögren, P., Fiorelli, E., Leonard, N.E.: Cooperative control of mobile sensor networks: Adaptive gradient climbing in a distributed environment. IEEE Transactions on Automatic Control 49(8), 1292–1302 (2004)CrossRefGoogle Scholar
  56. 56.
    Omatu, S., Seinfeld, J.H.: Distributed Parameter Systems: Theory and Applications. Oxford Mathematical Monographs. Oxford University Press, New York (1989)zbMATHGoogle Scholar
  57. 57.
    Patan, M.: Optimal activation policies for continous scanning observations in parameter estimation of distributed systems. International Journal of Systems Science 37(11), 763–775 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Patan, M., Patan, K.: Optimal observation strategies for model-based fault detection in distributed systems. International Journal of Control 78(18), 1497–1510 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Patriksson, M.: Simplicial decomposition algorithms. In: C.A. Floudas, P.M. Pardalos (eds.) Encyclopedia of Optimization, vol. 5, pp. 205–212. Kluwer Academic Publishers, Dordrecht, The Netherlands (2001)Google Scholar
  60. 60.
    Pázman, A.: Foundations of Optimum Experimental Design. Mathematics and Its Applications. D. Reidel Publishing Company, Dordrecht, The Netherlands (1986)zbMATHGoogle Scholar
  61. 61.
    Point, N., Vande Wouwer, A., Remy, M.: Practical issues in distributed parameter estimation: Gradient computation and optimal experiment design. Control Engineering Practice 4(11), 1553–1562 (1996)CrossRefGoogle Scholar
  62. 62.
    Porat, B., Nehorai, A.: Localizing vapor-emitting sources by moving sensors. IEEE Transactions on Signal Processing 44(4), 1018–1021 (1996)CrossRefGoogle Scholar
  63. 63.
    Pronzato, L.: Removing non-optimal support points in D-optimum design algorithms. Statistics & Probability Letters 63, 223–228 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Pukelsheim, F.: Optimal Design of Experiments. Probability and Mathematical Statistics. John Wiley & Sons, New York (1993)zbMATHGoogle Scholar
  65. 65.
    Quereshi, Z.H., Ng, T.S., Goodwin, G.C.: Optimum experimental design for identification of distributed parameter systems. International Journal of Control 31(1), 21–29 (1980)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Rafajłowicz, E.: Design of experiments for eigenvalue identification in distributed-parameter systems. International Journal of Control 34(6), 1079–1094 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Rafajłowicz, E.: Optimal experiment design for identification of linear distributed-parameter systems: Frequency domain approach. IEEE Transactions on Automatic Control 28(7), 806–808 (1983)zbMATHCrossRefGoogle Scholar
  68. 68.
    Rafajłowicz, E.: Optimum choice of moving sensor trajectories for distributed parameter system identification. International Journal of Control 43(5), 1441–1451 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Reinefeld, A.: Heuristic search. In: C.A. Floudas, P.M. Pardalos (eds.) Encyclopedia of Optimization, vol. 2, pp. 409–411. Kluwer Academic Publishers, Dordrecht, The Netherlands (2001)Google Scholar
  70. 70.
    Russell, S.J., Norvig, P.: Artificial Intelligence: A Modern Approach, 2nd edn. Pearson Education International, Upper Saddle River, NJ (2003)Google Scholar
  71. 71.
    Silvey, S.D., Titterington, D.M., Torsney, B.: An algorithm for optimal designs on a finite design space. Communications in Statistics — Theory and Methods 14, 1379–1389 (1978)Google Scholar
  72. 72.
    Sinopoli, B., Sharp, C., Schenato, L., Schaffert, S., Sastry, S.S.: Distributed control applications within sensor networks. Proceedings of the IEEE 91(8), 1235–1246 (2003)CrossRefGoogle Scholar
  73. 73.
    Song, Z., Chen, Y., Sastry, C.R., Tas, N.C.: Optimal Observation for Cyber-physical Systems: A Fisher-information-matrix-based Approach. Springer-Verlag, London (2009)zbMATHCrossRefGoogle Scholar
  74. 74.
    Sun, N.Z.: Inverse Problems in Groundwater Modeling. Theory and Applications of Transport in Porous Media. Kluwer Academic Publishers, Dordrecht, The Netherlands (1994)Google Scholar
  75. 75.
    Titterington, D.M.: Aspects of optimal design in dynamic systems. Technometrics 22(3), 287–299 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Torsney, B.: Computing optimising distributions with applications in design, estimation and image processing. In: Y. Dodge, V.V. Fedorov, H.P. Wynn (eds.) Optimal Design and Analysis of Experiments, pp. 316–370. Elsevier, Amsterdam (1988)Google Scholar
  77. 77.
    Torsney, B., Mandal, S.: Construction of constrained optimal designs. In: A. Atkinson, B. Bogacka, A. Zhigljavsky (eds.) Optimum Design 2000,  chap. 14, pp. 141–152. Kluwer Academic Publishers, Dordrecht, The Netherlands (2001)CrossRefGoogle Scholar
  78. 78.
    Torsney, B., Mandal, S.: Multiplicative algorithms for constructing optimizing distributions: Further developments. In: A. Di Bucchianico, H. Läuter, H.P. Wynn (eds.) mODa 7, Proc. 7th Int. Workshop on Model-Oriented Data Analysis, Heeze, The Netherlands, 2004, pp. 163–171. Physica-Verlag, Heidelberg (2004)Google Scholar
  79. 79.
    Uciński, D.: Measurement Optimization for Parameter Estimation in Distributed Systems. Technical University Press, Zielona Góra (1999)Google Scholar
  80. 80.
    Uciński, D.: Optimal sensor location for parameter estimation of distributed processes. International Journal of Control 73(13), 1235–1248 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Uciński, D.: Optimal Measurement Methods for Distributed-Parameter System Identification. CRC Press, Boca Raton, FL (2005)zbMATHGoogle Scholar
  82. 82.
    Uciński, D.: D-optimum sensor activity scheduling for distributed parameter systems. In: Preprints of the 15th IFAC Symposium on System Identification, Saint-Malo, France, July 6–8, (2009). Published on CD-ROMGoogle Scholar
  83. 83.
    Uciński, D., Atkinson, A.C.: Experimental design for time-dependent models with correlated observations. Studies in Nonlinear Dynamics & Econometrics 8(2) (2004). Article No. 13Google Scholar
  84. 84.
    Uciński, D., Bogacka, B.: T-optimum designs for discrimination between two multivariate dynamic models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67, 3–18 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Uciński, D., Bogacka, B.: A constrained optimum experimental design problem for model discrimination with a continuously varying factor. Journal of Statistical Planning and Inference 137(12), 4048–4065 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Uciński, D., Chen, Y.: Time-optimal path planning of moving sensors for parameter estimation of distributed systems. In: Proc. 44th IEEE Conference on Decision and Control, and the European Control Conference 2005, Seville, Spain (2005). Published on CD-ROMGoogle Scholar
  87. 87.
    Uciński, D., Chen, Y.: Sensor motion planning in distributed parameter systems using Turing’s measure of conditioning. In: Proc. 45th IEEE Conference on Decision and Control, San Diego, CA (2006). Published on CD-ROMGoogle Scholar
  88. 88.
    Uciński, D., Demetriou, M.A.: An approach to the optimal scanning measurement problem using optimum experimental design. In: Proc. American Control Conference, Boston, MA (2004). Published on CD-ROMGoogle Scholar
  89. 89.
    Uciński, D., Demetriou, M.A.: Resource-constrained sensor routing for optimal observation of distributed parameter systems. In: Proc. 18th International Symposium on Mathematical Theory of Networks and Systems, Blacksburg, VA, July 28–August 1 (2008). Published on CD-ROMGoogle Scholar
  90. 90.
    Uciński, D., Korbicz, J.: Optimal sensor allocation for parameter estimation in distributed systems. Journal of Inverse and Ill-Posed Problems 9(3), 301–317 (2001)MathSciNetzbMATHGoogle Scholar
  91. 91.
    Uciński, D., Patan, M.: Optimal location of discrete scanning sensors for parameter estimation of distributed systems. In: Proc. 15th IFAC World Congress, Barcelona, Spain, 22–26 July 2002 (2002). Published on CD-ROMGoogle Scholar
  92. 92.
    Uciński, D., Patan, M.: D-optimal design of a monitoring network for parameter estimation of distributed systems. Journal of Global Optimization 39, 291–322 (2007)zbMATHCrossRefGoogle Scholar
  93. 93.
    Uspenskii, A.B., Fedorov, V.V.: Computational Aspects of the Least-Squares Method in the Analysis and Design of Regression Experiments. Moscow University Press, Moscow (1975). (In Russian)Google Scholar
  94. 94.
    Vande Wouwer, A., Point, N., Porteman, S., Remy, M.: On a practical criterion for optimal sensor configuration — Application to a fixed-bed reactor. In: Proc. 14th IFAC World Congress, Beijing, China, 5–9 July, 1999, vol. I: Modeling, Identification, Signal Processing II, Adaptive Control, pp. 37–42 (1999)Google Scholar
  95. 95.
    Ventura, J.A., Hearn, D.W.: Restricted simplicial decomposition for convex constrained problems. Mathematical Programming 59, 71–85 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Vogel, C.R.: Computational Methods for Inverse Problems. Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (2002)zbMATHCrossRefGoogle Scholar
  97. 97.
    van de Wal, M., de Jager, B.: A review of methods for input/output selection. Automatica 37, 487–510 (2001)zbMATHCrossRefGoogle Scholar
  98. 98.
    Walter, É., Pronzato, L.: Identification of Parametric Models from Experimental Data. Communications and Control Engineering. Springer-Verlag, Berlin (1997)zbMATHGoogle Scholar
  99. 99.
    Zhao, F., Guibas, L.J.: Wireless Sensor Networks: An Information Processing Approach. Morgan Kaufmann Publishers, Amsterdam (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute of Control and Computation EngineeringUniversity of Zielona GóraZielona GóraPoland

Personalised recommendations