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.
Similar content being viewed by others
References
M. F. Shul’gin, On Some Differential Equations of Analytical Dynamics and Their Integration (Sredneazitsk. Gos. Univ., Tashkent, 1958) [in Russian].
V. V. Rumyantsev, Stability of Steady Motions of Satellites (Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1967) [in Russian].
A. I. Lur’e, Analytical Mechanics (Fizmatgiz, Moscow, 1961; Springer, Berlin, 2002).
E. J. Routh, Dynamics of a System of Rigid Bodies (Dover, New York, 1960; Nauka, Moscow, 1983).
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)].
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.
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.
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).
A. M. Lyapunov, Collected Works (Izd. Akad. Nauk SSSR, Moscow, 1956), Vol. 2 [in Russian].
I. G. Malkin, Stability Theory of Motion (Nauka, Moscow, 1968) [in Russian].
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).
R. E. Kalman, P. L. Falb, and M. A. Arbib, Topics in Mathematical System Theory (McGraw-Hill, New York, 1969; Mir, Moscow, 1971).
N. N. Krasovskii, “Problems of Stabilization of Controlled Motion,” in Stability Theory of Motion (Nauka, Moscow, 1966), pp. 475–515.
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.
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.
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).
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
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)].
Author information
Authors and Affiliations
Corresponding authors
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.
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S0027133019010035