# The synthesis of asymptotically stable motions of a gyrostat variable structure

### Mathematics. Physics. Mechanics

### Аuthors

^{*},

^{**}

Samara National Research University named after Academician S.P. Korolev, 34, Moskovskoye shosse, Samara, 443086, Russia

*e-mail: bezglasnsp@rambler.ru

**e-mail: motya31087@list.ru

### Abstract

The necessity to realize of the set motions and orientation of satellites and spacecraft in orbit requires the development of a mathematical theory of control and new algorithms of control. Work is devoted to the active control of randomly set of the spacial motions of a satellite-gyrostat variable structure relative to its center of mass.

The main purpose of this work is analytical construction of active external program and stabilizing controls to ensure the stability randomly set of program motions of a satellite-gyrostat.

A satellite-gyrostat variable structure is simulated by a system of two co-axial solids with moments of inertia which depend on the time. The equations of motion of the system are displayed in the form of the Lagrange equations of the second kind. Program control is constructed as solution of the inverse problems of dynamics; stabilizing control is synthesized on the principle of feedback. The property of asymptotic stability of motions was proved on the basis of Lyapunov's direct method of classical stability theory by using the method of limit functions and limit systems, allowing to build and apply Lyapunov's function having constant signs derivatives.

In this work the equations of a spacial motion relative to the center of mass of a balanced gyrostat with variable structure are derived. Active control attached to the satellite-gyrostat to realize the manifold of its randomly set program motion in orbit is received. Stabilizing control which ensuring the property of asymptotic stability realized motion is synthesized.

Results which were received in this work are develop and generalize the corresponding results in area of a control of the gyrostat’s motion in two directions: firstly, Lyapunov’s function having constant signs derivatives allows to build control is much easier without the terms of the third order of smallness of the variation; secondly, this problem was solved for the gyrostat with variable moments of inertia in the first time. Results of work can be used for the design of systems of active control of the satellites-gyrostats in orbit.

The necessity to realize of the set motions and orientation of satellites and spacecraft in orbit requires the development of a mathematical theory of control and new algorithms of control. Work is devoted to the active control of randomly set of the spacial motions of a satellite-gyrostat variable structure relative to its center of mass.

The main purpose of this work is analytical construction of active external program and stabilizing controls to ensure the stability randomly set of program motions of a satellite-gyrostat.

A satellite-gyrostat variable structure is simulated by a system of two co-axial solids with moments of inertia which depend on the time. The equations of motion of the system are displayed in the form of the Lagrange equations of the second kind. Program control is constructed as solution of the inverse problems of dynamics; stabilizing control is synthesized on the principle of feedback. The property of asymptotic stability of motions was proved on the basis of Lyapunov's direct method of classical stability theory by using the method of limit functions and limit systems, allowing to build and apply Lyapunov's function having constant signs derivatives.

In this work the equations of a spacial motion relative to the center of mass of a balanced gyrostat with variable structure are derived. Active control attached to the satellite-gyrostat to realize the manifold of its randomly set program motion in orbit is received. Stabilizing control which ensuring the property of asymptotic stability realized motion is synthesized.

Results which were received in this work are develop and generalize the corresponding results in area of a control of the gyrostat’s motion in two directions: firstly, Lyapunov’s function having constant signs derivatives allows to build control is much easier without the terms of the third order of smallness of the variation; secondly, this problem was solved for the gyrostat with variable moments of inertia in the first time. Results of work can be used for the design of systems of active control of the satellites-gyrostats in orbit.

### Keywords:

gyrostat, program motion, functions with a constant sign, Lyapunov's function, asymptotical stability### References

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