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Stabilization of Steady Motions for Systems with Redundant Coordinates

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Moscow University Mechanics Bulletin Aims and scope

Abstract

The vector-matrix Shulgin’s equations are used to stabilize the steady motions of mechanical systems with nonlinear geometric constraints in the case of incomplete information on the state. The momenta are introduced only for the cyclic coordinates that are not used to control. Three variants of the measurement vector are used to prove a theorem on the stabilization of control with the help of a part of the cyclic coordinates described by Lagrange variables. The control coefficients and the estimation system coefficients are specified by solving the corresponding Krasovskii linear-quadratic problem for a linear controlled subsystem without the critical variables corresponding to the redundant coordinates and to the introduced momenta. The stability of the complete closed nonlinear system is proved by reducing to a special Lyapunov case and by the application of the Malkin stability theorem in the case of time-varying perturbations.

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References

  1. M. F. Shul’gin, On Some Differential Equations of Analytical Dynamics and Their Integration (Sredneazitsk. Gos. Univ., Tashkent, 1958) [in Russian].

    Google Scholar 

  2. V. V. Rumyantsev, Stability of Steady Motions of Satellites (Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1967) [in Russian].

    Google Scholar 

  3. A. I. Lur’e, Analytical Mechanics (Fizmatgiz, Moscow, 1961; Springer, Berlin, 2002).

    Book  Google Scholar 

  4. E. J. Routh, Dynamics of a System of Rigid Bodies (Dover, New York, 1960; Nauka, Moscow, 1983).

    MATH  Google Scholar 

  5. A. Ya. Krasinskii and E. M. Krasinskaya, “A Stabilization Method for Steady Motions with Zero Roots in the Closed System,” Avtom. Telemekh., No. 8, 85–100 (2016) [Autom. Remote Control 77 (8), 1386–1398 (2016)].

    MATH  Google Scholar 

  6. A. Ya. Krasinskii, E. M. Krasinskaya, and K. B. Obnosov, “On the Development of Shul’gin’s Scientific Methods in the Application to the Stability and Stabilization of Mechanotropic Systems with Redundant Coordinates,” in Collection of Scientific-Methodological Papers in Theoretical Mechanics (Mosk. Gos. Univ., Moscow, 2012), Issue 28, pp. 169–184.

    Google Scholar 

  7. A. Ya. Krasinskii and E. M. Krasinskaya, “A Method to Study a Class of Stabilization Problems in the Case of Incomplete Information on the State,” in Proc. Int. Conf. on Dynamics of Systems and Control Processes, Ekaterinburg, Russia, September 15–20, 2014 (Institute of Mathematics and Mechanics, Ekaterinburg, 2014), pp. 228–235.

    Google Scholar 

  8. A. Ya. Krasinskiy and A. N. Ilyina, “The Mathematical Modeling of the Dynamics of Systems with Redundant Coordinates in the Neighborhood of Steady Motions,” Vestn. Yuzhn. Ural. Gos. Univ. Ser. Mat. Model. Programm. 10 (2), 38–50 (2017).

    Google Scholar 

  9. A. M. Lyapunov, Collected Works (Izd. Akad. Nauk SSSR, Moscow, 1956), Vol. 2 [in Russian].

    MATH  Google Scholar 

  10. I. G. Malkin, Stability Theory of Motion (Nauka, Moscow, 1968) [in Russian].

    Google Scholar 

  11. M. A. Aizerman and F. R. Gantmacher, “Stabilität der Gleichewichtslage in Einem Nicht Holonomen System // Z. Angew. Math. Mech. 37 (1–2), 74–75 (1957).

    Article  MATH  Google Scholar 

  12. R. E. Kalman, P. L. Falb, and M. A. Arbib, Topics in Mathematical System Theory (McGraw-Hill, New York, 1969; Mir, Moscow, 1971).

    MATH  Google Scholar 

  13. N. N. Krasovskii, “Problems of Stabilization of Controlled Motion,” in Stability Theory of Motion (Nauka, Moscow, 1966), pp. 475–515.

    Google Scholar 

  14. A. Ya. Krasinskii and E. M. Krasinskaya, “A Method to Study the Stability and Stabilization of Steady Motions for Mechanical Systems with Redundant Coordinates,” in Proc. Int. Conf. on Control Problems, Moscow, Russia, June 16–19, 2014 (Inst. Probl. Upravl., Moscow, 2014), pp. 1766–1778.

    Google Scholar 

  15. A. Ya. Krasinskii and E. M. Krasinskaya, “Linearization of Geometric Constraint Equations in the Problems of Stability and Stabilization of Equilibria,” in Collection of Scientific-Methodological Papers in Theoretical Mechanics (Mosk. Gos. Univ., Moscow, 2015), Issue 29, pp. 54–65.

    Google Scholar 

  16. A. Ya. Krasinskii, A. N. Il’ina, and E. M. Krasinskaya, “Modeling of the Ball and Beam System Dynamics as a Nonlinear Mechatronic System with Geometric Constraint,” Vestn. Udmurt Gos. Univ. 27 (3), 414–430 (2017).

    MATH  Google Scholar 

  17. A. Ya. Krasinskii and E. M. Krasinskaya, “Modeling of Dynamics of Manipulators with Geometrical Constraints as a Systems with Redundant Coordinates,” Int. Rob. Automat. J. 3 (2017). doi 10.15406/iratj.2017.03.00056

    Google Scholar 

  18. A. S. Klokov and V. A. Samsonov, “Stabilizability of Trivial Steady Motions of Gyroscopically Coupled Systems with Pseudo-Cyclic Coordinates,” Prikl. Mat. Mekh. 49 (2), 199–202 (1985) [J. Appl. Math. Mech. 49 (2), 150–153 (1985)].

    MATH  Google Scholar 

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Authors and Affiliations

  1. Moscow State University of Food Production, Institute of Economics and Management, Volokolamskoe Shosse 11, Moscow, 125080, Russia

    A. Ya. Krasinskii

  2. Moscow Aviation Institute, Faculty of Information Technologies and Applied Mathematics, Volokolamskoe Shosse 4, Moscow, 125993, Russia

    A. N. Il’ina

  3. Bauman Moscow State Technical University, Faculty of Fundamental Sciences, ul. Baumanskaya 5/1, Moscow, 105005, Russia

    E. M. Krasinskaya

Authors
  1. A. Ya. Krasinskii
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  2. A. N. Il’ina
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  3. E. M. Krasinskaya
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Correspondence to A. Ya. Krasinskii, A. N. Il’ina or E. M. Krasinskaya.

Additional information

Russian Text © A.Ya. Krasinskii, A.N. Il’ina, E.M. Krasinskaya, 2019, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2019, Vol. 74, No. 1, pp. 46–51.

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Krasinskii, A.Y., Il’ina, A.N. & Krasinskaya, E.M. Stabilization of Steady Motions for Systems with Redundant Coordinates. Moscow Univ. Mech. Bull. 74, 14–19 (2019). https://doi.org/10.3103/S0027133019010035

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  • DOI: https://doi.org/10.3103/S0027133019010035

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