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Optimal location of sensors for linear stochastic distributed parameter systems

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Distributed Parameter Systems: Modelling and Identification

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 1))

Abstract

In this paper we propose a method which gives the optimal location of N sensors for linear, time-invariant distributed-parameter systems, subject to noise disturbances in the dynamics and in the observation.

The state of the system, described by a set of partial differential equations of parabolic type, is disturbed by a noise assumed white in time but not in space. The boundary conditions are homogeneous and therefore deterministic.

The observations, realized in a fixed number of measurement points, is assumed continuous and subject to an additive noise white in time and space.

The location of sensors is determined in such a way that we obtain the best estimation of a number of variables necessary to the control of the system. We propose the solution of this problem in two steps:

  • -Assuming that the sensor locations are fixed, the Kalman-Bucy optimal filter is used to construct the best estimation of the state

  • -The optimality criterion for the location of sensors is chosen to be the spatial integral of the trace of the steady state estimation error covariance matrix, augmented by a term characterising the error covariance matrix, for the variables necessary to the control law.

Different examples are studied and numerical results, obtained via a modified gradient algorithm, are given. The effects of system parameters changes and noise statistics are discussed.

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References

  1. S.E. AIDAROUS, “Optimal Allocation Strategies in Stochastic Distributed Systems“ Thesis presented at Université Catholique de Louvain Belgium (1976)

    Google Scholar 

  2. M. ATHANS “On the Determination of Optimal Costly Measurement Strategies for Linear Stochastic Systems”, Automatica, Vol. 8, pp 397–412 (1972)

    Article  Google Scholar 

  3. A. BENSOUSSAN “Optimization of Sensors Location in a Distributed Filtering Problem”, Proc. Int. Symp. on Stability of Stochastic Dynamic Systems, Coventry, England (1971)

    Google Scholar 

  4. A. BENSOUSSAN “On the Separation Principle for Distributed Parameter Systems”, Proc. IFAC Symp. on Control of Distributed Parameter Systems, Banff, Canada (1971)

    Google Scholar 

  5. S.P. BINGULAC “An alternate Approach to Expanding PA+A'P=− Q“ IEEE Trans. on A.C. Vol. 15, pp 135–137 (1970)

    Article  Google Scholar 

  6. W.H. CHEN and I.H. SEINFELD “Optimal Location of process Measurements”, Int. J. Control, Vol. 21, pp 1003–1014 (1975)

    Google Scholar 

  7. P.L. FALB “Infinite Dimension Filtering: The Kalman Bucy Filter in Hilbert Space”, Inf. and Control, Vol. 11, pp 102–137 (1967)

    Article  Google Scholar 

  8. R.E. GOODSON and R.E. KLEIN “A Definition and Some Results for Distributed Systems Observability“ IEEE Trans. on A.C. Vol. 15, no 2, pp 165–174 (1970)

    Article  Google Scholar 

  9. R.E. KALMAN and R.S. BUCY New Results in Linear Filtering and Prediction Theory“ J. Basic Eng. Trans; ASME, Vol. 83, pp 95–108 (1961)

    Google Scholar 

  10. D.L. KLEINMAN “On a Iterative Technique for Riccati Equations Computations“ IEEE Trans. on A.V. Vol. 13 pp 114–115 (1968)

    Google Scholar 

  11. V. KUCERA “A contribution to Matrix Quadratic Equations“ IEEE Trans. on A.C. Vol. 17, pp. 344–347 (1972)

    Article  Google Scholar 

  12. Y. SAKAWA “Optimal Filtering in Linear Distributed Parameter Systems“ Int. J. Control, Vol. 18, pp 117–127 (1972)

    Google Scholar 

  13. F.E. THAU “On Optimum Filtering for a Class of Linear Distributed Parameter Systems”, ASME Trans. J. Basic. Engng., Vol 91, pp 173–178 (1969)

    Google Scholar 

  14. S.G. TZAFESTAS and J.M. NIGHTINGALE “Optimal Filtering Smoothing and Prediction in Linear Distributed Parameter Systems”, Proc. IEE, Vol. 115, No 8 pp 1207–1212 (1968)

    Google Scholar 

  15. S.G. TZAFESTAS and J.M. NIGHTINGALE “Optimal Control of a Class of Linear Stochastic Distributed Parameter Systems“. Proc. IEE, Vol. 115, No 8, pp 1213–1220 (1968)

    Google Scholar 

  16. S.G. TZAFESTAS “On the Distributed Parameter Least-Squares State Estimation Theory“. Int. J. Systems Sci. Vol 4, no 6, pp 833–858 (1973)

    Google Scholar 

  17. T.K. YU and J.H. SEINFELD “Observability and Optimal Measurement Location in Linear Distributed Parameter Systems”, Int. J. Control, Vol. 18 pp. 787–789 (1973)

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire d'Automatique et d'Analyse des Systèmes du Centre National de la Recherche Scientifique, 7, avenue du Colonel Roche, 31400, Toulouse, France

    M. Amouroux, J. P. Babary & C. Malandrakis

Authors
  1. M. Amouroux
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  2. J. P. Babary
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  3. C. Malandrakis
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Editor information

Antonio Ruberti

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© 1978 Springer-Verlag

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Amouroux, M., Babary, J.P., Malandrakis, C. (1978). Optimal location of sensors for linear stochastic distributed parameter systems. In: Ruberti, A. (eds) Distributed Parameter Systems: Modelling and Identification. Lecture Notes in Control and Information Sciences, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0003733

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  • DOI: https://doi.org/10.1007/BFb0003733

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08405-1

  • Online ISBN: 978-3-540-37195-3

  • eBook Packages: Springer Book Archive

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