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Observability with bearing-only observations and smoothness of the attainable set


The attainable set of a linear control system can have both smooth and nonsmooth boundary. This smoothness property is known to be used to classify such systems. One approach, suggested by A. I. Ovseevich in the case of a smooth control set, is based on connecting smoothness of the attainable set with spherical observability of the dual system. This paper generalizes these results to the case of nonsmooth control sets. The corresponding property of the spherical observability notion can be treated as observability with several bearing-only observations.

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  1. 1.

    V. J. Aidala and S. C. Nardone, “Biased estimation properties of the pseudolinear tracking filter,” IEEE Trans. Aerospace Electron. Systems, 18, No. 4, 432–441 (1982).

    Article  Google Scholar 

  2. 2.

    A. A. Alexandrov, V. G. Boltyansky, S. S. Lemak, N. A. Parusnikov, and V. M. Tihomirov, Optimization of Dynamics of Controlled Systems [in Russian], Izd. Mosk. Univ., Moscow (2000).

    Google Scholar 

  3. 3.

    V. I. Arnold, Mathematical Methods of Classical Mechanics, Nauka, Moscow (1978).

    MATH  Google Scholar 

  4. 4.

    M. Athans and P. Falb, Optimal Control, McGraw-Hill, New York (1966).

    MATH  Google Scholar 

  5. 5.

    Yu. V. Bolotin, “A TLS approach to angle-only trajectory estimation,” in: IEEE Workshop on Real Time Computing, Prague (1994).

  6. 6.

    Yu. V. Bolotin, “Generalized least squares in bearing-only estimation,” Avtomat. Telemekh., No. 2 (1997).

  7. 7.

    Yu. V. Bolotin and S. N. Morgunova, “Observability with bearing-obly observations,” Vestn. Mosk. Univ. Ser. 1 Mat. Mekh., No. 3 (2000).

  8. 8.

    A. M. Formalsky, Controllability and Stability with Limited Resources [in Russian], Nauka, Moscow (1974).

    Google Scholar 

  9. 9.

    E. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley and Sons (1970).

  10. 10.

    S. C. Nardone and V. J. Aidala, “Observability criteria for bearings-only target motion analysis,” IEEE Trans. Aerospace Electron. Systems, 17, No. 2, 162–166 (1981).

    Article  MathSciNet  Google Scholar 

  11. 11.

    A. I. Ovseevich, “Properties of boundaries of attainable sets and the problem of observability,” in: Proc. 5th All-Russian Workshop “Mathematical Problems of Navigation” [in Russian], MSU, Moscow (1994).

    Google Scholar 

  12. 12.

    A. N. Payne, “Observability problem for bearing-only tracking,” Internat. J. Control, 49, No. 3, 761–768 (1989).

    MATH  MathSciNet  Google Scholar 

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 8, pp. 119–130, 2005.

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Bolotin, Y.V., Morgunova, S.N. Observability with bearing-only observations and smoothness of the attainable set. J Math Sci 147, 6631–6638 (2007).

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  • Projective Measurement
  • Dual System
  • Linear Control System
  • Outer Normal
  • Spherical Measurement