RESEARCH OF BEHAVIOUR OF NONLINEAR SYSTEMS IN VICINITIES OF SETS WITH VARIED BORDER

PILISHKIN, V.

Abstract

The most known approaches of solving the problem of control systems remains a problem of providing those or other restrictions, imposed on movement of dynamic system, are based on using of L.S.Pontryagin’s principle of maximum and a method of dynamic programming of R.Bellman. Others direct approaches are known to synthesis control on system's motion at restrictions. Methods of numerical synthesis, methods on the basis of use of functions of Lyapunov and methods of an inverse problem of dynamics can be related to them. However application of the listed approaches does not eliminate numerous difficulties at the solving of a considered problem. In such a way, developing of methods of synthesis of motion system’s control laws, proceeding from providing of the given restrictions,is urgent. Thus it is necessary to take into consideration such circumstances, as: the analysis and providing of resolvability of problem of synthesis at the expense of allowable change (deformation) of some restrictions; an opportunity of "non-rigid" (deformable) restrictions;non-sensitivity (robustness property) of movements to factors revolting it (not precisely set parameters of system, external influences). In this work new approach based on general representation of functional systems is offered.The nonlinear system on which variable conditions restrictions are imposed is generally considered.The purpose of control is maintenance of required three-dimensional motion in state space at presence of restrictions on control and action of poorly-controled indignations. The "rigid" (unchangeable) assignment of set of phase restrictions frequently leads to insolubility of problem of synthesis.The assignment itself is enough heuristic,to some extent arbitrary.Therefore generally a deformable set instead of "rigid" is used. It is carried out due to introduction of scalar non-negative function (a measure of proximity), determining remoteness, in that or other sense, of points of a trajectory of the system from initial set of phase restrictions. On a basis "measure of proximity" are formed the limited vicinities of set,beyond which limits a researched system should not exist. With the help of a “measure of proximity” the so-called connected phase plane is formed, in which movement of the system in relation to the pre-set set is completely characterized: remoteness from him and speed of change of remoteness. On the connected plane properties of movement of system in relation to initial set in sense of a chosen measure of proximity are researched. Sufficient conditions of this limitation are achieved by splitting border of set into "transparent" and "non-transparent" sites for trajectories of system, and also introductions of the "transparent" and "non-transparent" areas in the state space. The given conditions are based on limitation from above logarithmic derivative of the speed of change of remoteness of a trajectory in relation to initial set. It is shown, that if movement is limited, the corresponding "transparent" area adjoining to initial set, also is limited, and the logarithmic derivative is equal to negative infinity. Sufficient conditions represent the mixed system of algebraic equality and the inequalities accessible to the solution. On the connected phase plane the analogy with Hamiltonian behaviour of system is entered. Using the dissipativity of the system in relation to the some Hamiltonian, sufficient conditions of limitation of movement with consideration of action of poorly-controled indignations are defined.

Coresponding author e-mail: pilishkin[at]hotmail[dot]com

Session: Linear and Non-linear Control System Design