Abstract
Based on previous results, we consider stabilization problems for both non-asymptotic stability and asymptotic stability with respect to all variables for equilibrium positions and stationary motions of mechanical systems with redundant coordinates. The linear stabilizing control is defined by the solution of a linear–quadratic stabilization problem for an allocated linear controllable subsystem of as small dimension as possible. We find sufficient conditions under which a complete nonlinear system closed by this control is ensured asymptotic stability despite the presence of at least as many zero roots of the characteristic equation as the number of geometric relations. We prove a theorem on the stabilization of the control equilibrium applied only with respect to redundant coordinates and constructed from the estimate of the phase state vector obtained by a measurement of as small dimension as possible.
Similar content being viewed by others
References
Krasovskii, N.N., Stabilization Problems for Controlled Motions, in Teoriya ustoichivosti dvizheniya (Theory of Stability of Motion), Malkin, I.G., Moscow: Nauka, 1966, pp. 475–514.
Lyapunov, A.M., Sobranie sochinenii (Collected Works), Moscow: Akad. Nauk SSSR, 1956, vol. 2.
Malkin, I.G., Teoriya ustoichivosti dvizheniya (Theory of Stability of Motion), Moscow: Nauka, 1966.
Kamenkov, G.V., Ustoichivost’ i kolebaniya nelineinykh sistem. Izbr. tr. (Stability and Oscillations of Nonlinear Systems. Collected Works), Moscow: Nauka, 1972, vol. 2.
Karapetyan, A.V. and Rumyantsev, V.V., Stability of Conservative and Dissipative Systems, in Itogi Nauki i Tekhniki. Obshchaya Mekhanika (Results in Science and Technology. General Mechanics), Moscow: VINITI, 1983, vol. 6, pp. 3–128.
Krasinskaya, E.M. and Krasinskii, A.Ya., On Stability and Stabilization of Non-Isolated Steady Modes of Motion for Mechanical Systems. Holonomic Systems, in Prikladnaya Matematika i Mekhanika: Sb. Nauchn. Tr. (Applied Mathematics and Mechanics), Ul’yanovsk: UlGTU, 2011, pp. 301–322.
Krasinskii, A.Ya., On One Method for Studying Stability and Stabilization for Non-Isolated Steady Motions of Mechanical Systems, Proc. VIII Int. Seminar “Stability and Oscillations of Nonlinear Control Systems,” Moscow: Inst. Probl. Upravlen., 2004, http://wwwipuru/ semin/arhiv/stab04, pp. 97–103.
Krasinskaya, E.M. and Krasinskii, A.Ya., On One Method for Studying Stability and Stabilization for Steady Motions of Mechanical Systems with Redundant Coordinates, Proc. XII Russian Seminar on Control Problems VSPU-2014, Moscow, June 1–19, 2014, pp. 1766–1778.
Krasinskaya, E.M. and Krasinskii, A.Ya., On Stability and Stabilization of the Equilibrium of Mechanical Systems with Redundant Coordinates, Nauka Obrazovan., Bauman MSTU, Electr. J., 2013, no. 3, DOI: 10.7463/0313.0541146.
Shul’gin, M.F., On Some Differential Equations of Analytical Dynamics and Their Integration, Proc. SAGU, Tashkent, 1958, no. 144.
Rumyantsev, V.V., Ob ustoichivosti statsionarnykh dvizhenii sputnikov (On the Stability of Stationary Motions of Satellites), Moscow: VTs ANSSSR, 1967.
Krasinskii, A.Ya., Stabilization of Steady Motions of Systems with Cyclic Coordinates, Prikl. Mat. Mekh., 1992, vol. 56, no. 6, pp. 939–950.
Aiserman, M.A. and Gantmacher, F.R., Stabilitaet der Gleichgewichtslage in einem nichtholonomen System, ZAMM, 1957, vol. 37, no. 1/2, pp. 74–75.
Kalman, R.E., Falb, P.L., and Arbib, M.A., Topics in Mathematical System Theory, New York: McGraw-Hill, 1969. Translated under the title Ocherki po matematicheskoi teorii sistem, Moscow: Mir, 1971.
Gabasov, R. and Kirillova, F.M., Kachestvennaya teoriya optimal’nykh protsessov (Qualitative Theory of Optimal Processes), Moscow: Nauka, 1971.
Krasinskii, A.Ya. and Krasinskaya, E.M., Modeling the Dynamics of a GBB 1005 BALL AND BEAM Testbed as a Controllable Mechanical System with a Redundant Coordinate, Nauka Obrazovanie, Bauman MSTU, Electronic Journal, 2014, no. 1, DOI: 10.7463/0114.0646446.
Kuntsevich, V.M. and Lychak, M.M., Sintez sistem avtomaticheskogo upravleniya s pomoshch’yu funktsii Lyapunova (Synthesis of Automated Control Systems and Lyapunov Functions), Moscow: Nauka, 1977.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.Ya. Krasinskii, E.M. Krasinskaya, 2016, published in Avtomatika i Telemekhanika, 2016, No. 8, pp. 85–100.
Rights and permissions
About this article
Cite this article
Krasinskii, A.Y., Krasinskaya, E.M. A stabilization method for steady motions with zero roots in the closed system. Autom Remote Control 77, 1386–1398 (2016). https://doi.org/10.1134/S0005117916080051
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117916080051