Algorithm of Quasiuniform Filling of Reachable Set for Nonlinear Control System

Abstract

In this paper, we propose an algorithm of obtaining points that uniformly fill the volume of the reachable set, and even for a small number of elements results in a cloud quasiuniform approximation of the set. To solve the task of finding each additional point is to solve the optimization problem. Minimized function describes the uniformity and depends on the Euclidean distance between the elements of the approximation. It is designed to be equal or close to zero, if the distance exceeds the desired threshold value. Thus, a lower bound for the optimal value of the functional is pre-defined, so we save computing time for the random component of global optimization algorithms. "The tunnel ideology" underlies this algorithm. Besides local descent mechanisms it assumes that there are also transition mechanisms from a local extremum with the current record functional value to lower value extrema attraction domains. As a globalizing mechanism we use a nonlocal search in random directions repeated several times at each iteration of the algorithm. To improve the reliability of the proposed method of algorithm construction a recurrent random multistart is also included. The article includes the results of computational experiments on test examples and its comparison with calculations obtained by the method based on the Pontryagin maximum principle \cite{Gornov}. The designed method of reachable set approximation is applicable for two-dimensional systems and multidimensional ones as well. The experiments showed the efficiency of the approach and results comparison confirmed the accuracy the obtained approximations.

Keywords

• reachable set
• approximation algorithm
• optimal control problem