Reducibility of Linear Time-Varying Systems of Special Form with Control and Measurements

  • V. I. KalenovaEmail author
  • V. M. MorozovEmail author


This paper considers some classes of linear time-varying systems with control and output vectors that can be reduced to time-invarint systems using a constructive transformation with state-space extension. As an illustrative example, the calibration problem of a gimballess inertial navigation system on a single degree-of-freedom turn table is solved.



  1. 1.
    A. M. Lyapunov, General Problem of the Stability of Motion (Gostekhizdat, Moscow, 1950; CRC, Boca Raton, FL, 1992).Google Scholar
  2. 2.
    V. M. Morozov and V. I. Kalenova, Estimation and Control in Nonstationary Linear Systems (Mosk. Gos. Univ., Moscow, 1988) [in Russian].zbMATHGoogle Scholar
  3. 3.
    V. I. Kalenova and V. M. Morozov, Linear Nonstationary Systems and their Applications to Problems of Mechanics (Fizmatlit, Moscow, 2010) [in Russian].Google Scholar
  4. 4.
    R. Bellman, Introduction to Matrix Analysis (SIAM, Philadelphia, 1997; Nauka, Moscow, 1969).Google Scholar
  5. 5.
    V. I. Kalenova and V. M. Morozov, “The reducibility of linear second-order time-varying systems with control and observation,” J. Appl. Math. Mech. 76, 413 (2012).MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. I. Kalenova and V. M. Morozov, “On the control of linear time-varying systems of a special form,” J. Comput. Syst. Sci. Int. 52, 333 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    M. L. J. Hautus, “Controllability and observability conditions of linear autonomous systems,” Proc. Koninkl. Nederl. Akad. Wetensch., Ser. A 72, 443–448 (1969).Google Scholar
  8. 8.
    N. N. Krasovskii, Theory of Control of Motion. Linear Systems (Nauka, Moscow, 1968) [in Russian].Google Scholar
  9. 9.
    A. Chang, “An algebraic characterization of controllability,” IEEE Trans. Autom. Control. 10, 112–113 (1965).CrossRefGoogle Scholar
  10. 10.
    M.-Y. Wu, “Transformation of linear time-varying systems into a linear time-invariant system,” Int. J. Control 27, 589–602 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    R. E. Kalman, P. Falb, and M. Arbib, Topics in Mathematical System Theory (McGraw-Hill, New York, 1969; Mir, Moscow, 1971).Google Scholar
  12. 12.
    A. A. Golovan and N. A. Parusnikov, Mathematical Foundations of Navigation Systems. Part 2. Application of Optimal Estimation Methods to Navigation Problems (MAKS Press, Moscow, 2012) [in Russian].Google Scholar
  13. 13.
    N. B. Vavilova, N. A. Parusnikov, and I. Yu. Sazonov, “Calibration of gimballess inertial navigation systems using coarse single-stage stands,” in Modern Problems of Mathematics and Mechanics, Vol. 1: Applied Studies (Mosk. Gos. Univ., Moscow, 2009), pp. 212–223 [in Russian].Google Scholar
  14. 14.
    I. Yu. Sazonov, “Identification of parameters of instrumental errors of a gimballess inertial navigation system using coarse single-stage stands,” Cand. Sci. (Phys.-Math.) Dissertation (Lomonosov Moscow State Univ., Moscow, 2012).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Institute of Mechanics, Moscow State UniversityMoscowRussia

Personalised recommendations