Approximations of the Set of Trajectories and Integral Funnel of the Nonlinear Control Systems with Limited Control Resources

TL;DR

This paper develops methods to approximate the set of possible trajectories and the integral funnel of nonlinear control systems with limited resources, using piecewise-constant controls and Euler's broken lines.

Contribution

It introduces a novel approximation approach for trajectory sets and integral funnels of nonlinear systems with integral control constraints, using finite piecewise-constant controls.

Findings

Finite piecewise-constant controls approximate the trajectory set.

Euler's broken lines effectively approximate individual trajectories.

The integral funnel can be approximated by a finite set of points.

Abstract

In this paper approximations of the set of trajectories and integral funnel of the control system described by nonlinear ordinary differential equation with integral constraint on the control functions are considered. The set of admissible control functions is replaced by a set, consisting of a finite number of piecewise-constant control functions. It is shown that the set of trajectories generated by a finite number of piecewise-constant control functions is an internal approximation of the set of trajectories. Further, each trajectory generated by a piecewise-constant control function is substituted by appropriate Euler's broken line and it is proved that the set consisting of a finite number of Euler's broken lines is an approximation of the set of trajectories of given control system. An approximation of the system's integral funnel by a set consisting of a finite number of points is obtained.