Mechanics of Solids (about journal) Mechanics of Solids
A Journal of Russian Academy of Sciences
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936

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B. N. Sokolov, "A single-channel bang-bang controller for guaranteed stabilization of translational motions of a rigid body with internal degrees of freedom," Mech. Solids. 38 (1), 7-12 (2003)
Year 2003 Volume 38 Number 1 Pages 7-12
Title A single-channel bang-bang controller for guaranteed stabilization of translational motions of a rigid body with internal degrees of freedom
Author(s) B. N. Sokolov (Moscow)
Abstract The problem of stabilization of a rigid body with internal particles (point masses) about a prescribed fixed position is considered. The particles are connected to one another and to the carrier body by means of linear spring-and-dashpot elements. The body moves translationally along a straight line under the action of a constant disturbance force and a bang-bang control force, both these forces being directed along the line of motion. It is assumed that the controller does not have information about the kinematic state of the internal particles and that there is a fixed delay in the control channel and, hence, arbitrarily frequent switchings of the control are not allowed. To solve this problem, we utilize the game approach [1-3]. In accordance with this approach, the influence of the internal system of particles on the motion of the carrier body is treated as the action of an external disturbance force which is unknown in advance but is bounded in absolute value. Estimates are obtained for the magnitude of this disturbance, as well as for the amplitudes of vibrations of the particles appearing under the action of inertial forces as a result of a switching of the bang-bang control. A guaranteed attainable estimate is obtained for the accuracy of stabilization of the prescribed position of the body, depending on the mechanical parameters of the system and the magnitude of the control force. As an example, the controlled motion of a two-mass vibrating system is considered. This paper is closely related to those of [4-6] and continues investigations of guaranteed optimal linear controllers with delay in the control channel [7-9]. The dynamics of a rigid body with elastic and dissipative members has been investigated in [10] with the assumption that the period of free vibrations and the time of their damping are small in comparison with the characteristic time of motion of the entire system.
1.  R. Issacs, Differential Games [Russian translation], Mir, Moscow, 1967.
2.  A. I. Subbotin and A. G. Chentsov, Optimization of Guarantee in Control Problems [in Russian], Nauka, Moscow, 1981.
3.  N. N. Krasovskii, Control of a Dynamical System: Guaranteed Result Minimization Problem [in Russian], Nauka, Moscow, 1985.
4.  K. B. Alekseev and G. G. Bebenin, Control of a Flying Spacecraft [in Russian], Mashinostroenie, Moscow, 1964.
5.  E. V. Gaushus and N. D. Smol`yaninov, "Investigation of a bang-bang stabilization system for a flying vehicle," Izv. AN SSSR. MTT. No. 2, pp. 5-13, 1970.
6.  E. V. Gaushus, Investigation of Dynamical Systems by the Point Transformation Method [in Russian], Nauka, Moscow, 1976.
7.  N. V. Banichuk, I. I. Karpov, D. M. Klimov, et al., Mechanics of Large-scale Space Structures [in Russian], Faktorial, Moscow, 1997.
8.  V. F. Ivanova and B. N. Sokolov, "Maximum guaranteed accuracy of a bang-bang controller in a one-dimensional stabilization problem," Izv. AN. MTT. No. 2, pp. 26-36, 1999.
9.  B. N. Sokolov, "On the structure of a bang-bang controller possessing the maximum-efficiency behavior for a guaranteed accuracy," Izv. AN. MTT. No. 3, pp. 17-33, 2002.
10.  F. L. Chernousko, "On the motion of a rigid body with elastic and dissipative elements," PMM [Applied Mathematics and Mechanics], Vol. 42, No. 1, pp. 34-42, 1978.
Received 24 April 2001
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