Construction of Attainable Sets and Integral Funnels of Nonlinear Controlled Systems in the Matlab Environment

Attainable sets play an important role in investigating the behavior of controlled systems. The pixel method [1,2] is a universal numerical method for the construction of attainable sets of nonlinear controlled system. In the first part of the article, we describe the pixel method that requires partition of the phase space into n-dimensional cubes and time discretization using Euler and Runge–Kutta methods of second order approximation. This method enables us to construct attainability and controllability sets as well as integral funnels for nonlinear controlled systems described by a system of differential equations with controlled parameters, both with and without phase constraints. For attainable sets constructed by the pixel method, we have obtained bounds on the Hausdorff distance between the exact attainable set and its discrete approximation. These bounds lead to a relationship between the partition parameters in time and phase coordinates. In the second part of the article, we present a description of the program PixelSet developed in the Matlab environment for the construction of attainable sets and integral funnels of two- and three-dimensional nonlinear controlled systems by the pixel method. We present examples of the construction of attainable sets of some two- and three-dimensional controlled systems. © 2014, Springer Science+Business Media New York.