On a method for solving a local boundary problem for a nonlinear stationary system with perturbations in the class of piecewise constant controls

Abstract

Summary The wide class of nonlinear stationary systems of ordinary differential equations taking into account restrictions on control and external perturbation is considered. An algorithm for constructing a discrete control function that guarantees the transfer of the systems from the initial state to the origin and an arbitrary neighborhood of the origin is proposed. A constructive sufficient condition of the Kalman type, in which the specified translation is possible, is obtained. The problem of robot‐manipulator control is considered and its numerical simulation is carried out.

A Constructive Method of Solving Local Boundary Value Problems for Nonlinear Systems with Perturbations and Control Delays

“…We leave the long algebra outside the scope of this paper. The complete details of the transformations can be found in [50][51][52][53], where the same idea is applied to different control problems. However, for the sake of completeness, we briefly present some derivations from the mentioned papers here.…”

“…In [53], it is shown that the solution of the auxiliary system (24), closed with the auxiliary stabilizing control (30) in the domain (26), satisfies the estimate:…”

“…In [53], it is shown that ε g ∈ (0, ε g ) can be chosen so that any x 0 and g satisfying x 0 ≤ ε g , g ≤ ε g provides that z(τ) belongs to the domain ( 5), ( 26) and decays exponentially.…”

“…Suppose that the solution of the system (42) defined by formulas ( 44) and ( 45) belongs to the domain ( 5), (26). From ( 44) and ( 45) taking into account ( 28), ( 32), ( 34), ( 35), (37), and (43) analogous to the result in [53], we obtain the estimations…”

“…In the present paper, the method developed and applied in [50][51][52][53] to various systems is modified for systems with a delayed control.…”