Stabilization problem of steady motions

of S.A.Chaplygin non-holonomic systems

б.Ya.Krasinsky

Moscow State University of applied biotechnology

33, Talalikhin, Moscow, Russia

B.Atajanov

National University of Uzbekistan

Yunus-Abad, Tashkent-114, 100114, Uzbekistan

Chaplygin non-holonomic system with the cyclic coordinates which are under the influence of potential and non-potential generalized forces are considered in this paper. The motion is described by Routhian variables, and momentums are introduced not for all cyclic coordinates.

Earlier the stabilization problems of the unstable steady motions with respect to all phase variables were investigated by applying linear controlling forces on the part of cyclic momentums.

Here the stabilization problem up to non-asymptotic stability is investigated by application of linear control forces for all or only for some cyclic coordinates corresponding to Lagrangian or Hamiltonian variables. It is necessary to state that the stabilization problem is solved in complete statement: not only the coefficients of stabilizing control are defined, but also estimation system is constructed for concrete measured signal and all consideration will be carried out at a choice of the real executive device for realization of stabilizing control in concrete tasks. It is obvious that depending on a choice of concrete executive device research can lead to absolutely different problems of mechanics. Consideration is based on regular use of vector-matrix equations of disturbed movement in the form which, unlike their traditional use, allows not only to define an arrangement of roots of the characteristic equation, but also to analyze structure of nonlinear members in view of non-holonomicity members. The advantages and disadvantages of above-stated variables are shown in research of different types of problems.

Movement of considered system is described by Chaplygin's equations irrespective of the constraints equations; stability and stabilization research of steady motions in relation to positional coordinates, positional velocitiesљ and momentums (if those are entered) defines the character of unconditional stability, but relative to all variables including coordinates, whose velocities are dependent, defines conditional stability.

If in considered system there are cyclic coordinates then at the certain initial conditions system can make stationary movement. Thus cyclic coordinates vary linearly in time, and this implies instability of steady motions in relation to cyclic coordinates. Considered non-holonomic system with 2n-m degrees of freedom comes to non-holonomic system with 2n-m-k degrees of freedom with the generalized coordinates q₁,љ q₂,:,љ q_m,љ q_m+k+1,:, q_n.

For Chaplygin's systems traditionally the constraints equations are not added to the system of disturbed movement equations. In the elementary tasks (for example, in case of stability research of equilibrium positions) when to independent velocities and their coordinates there correspond roots of characteristic equation with negative real parts, stability will occur in relation to the coordinates, corresponding to dependent velocities. In general case of steady motions, in disturbed movement equations for the constraints corresponding to dependent velocities, linear members of arbitrary structure can appear. As a result there is a big variety of problem statements in the field of stability and stabilization. Therefore it is necessary to formulate considered task of stability and stabilization precisely and correctly (e.g. one of possible tasks of stabilization for a disk (one-wheeled robot) is keeping steady state, and another task - that it has also to move into required state).

Stability research and a problem of equilibrium state and stationary motions stabilization have their distinctive features and differences.

One can achieve stabilization of equilibrium position with help of non-potential generalized and control forces. And stationary movement is stabilized not only because of these forces, but also due to linear non-holonomicity members appearing from their series expansion. The problem of unstable steady motions stabilization up to non-asymptotic stability relative to all phase variables (or a part of them) is solved with help of control forces for all the cyclic coordinates (or for their part) corresponding to Lagrange or Hamilton variables. Stabilizing force is defined by solution of linearly square-law task of optimum stabilization for controlled subsystem. Thus controls are realized in the form of a feedback on estimation of a part of state vector, constructed of phase variables disturbances measurement. Control is formed at presence of measuring information only about disturbances of coordinates or velocities; information about disturbances of all positional coordinates and all velocities is necessary for a general case.

atajanov_b@rambler.ru