Approximations of the set of trajectories, attainable sets and integral funnel of the control system with mixed constraints on the control functions

Abstract

In this paper, the set of trajectories, attainable sets and integral funnel of a control system described by an ordinary differential equation are studied. The system is nonlinear with respect to the phase state vector and affine with respect to the control vector. It is assumed that the admissible control functions satisfy mixed constraints, including both integral and geometric constraints. Step by step, the set of admissible control functions is replaced by a set consisting of a finite number of piecewise-constant control functions that generate a finite number of trajectories. First, an error evaluation between the set of trajectories and the set consisting of a finite number of trajectories is presented. Then, the trajectories generated by the piecewise-constant control functions are changed with Euler’s broken lines, and an error estimation between the set of trajectories of the system and the set consisting of a finite number of Euler’s broken lines is obtained. Similar estimations for attainable sets of the system are also provided. By applying these results, we derive an approximation with error evaluation for the integral funnel of the system. It is shown that by appropriately defining discretization parameters, the Hausdorff distance between the set of trajectories, the attainable sets, the integral funnel and their approximations can be made sufficiently small. The impact of upper bounds of the geometric and integral constraints on the presented approximations is discussed.