The explicit form of the switching surface in admissible synthesis problem

Abstract

In this article we consider the problem related to positional synthesis and controllability function method and more precisely to admissible maximum principle. Unlike the more common approach the admissible maximum principle method gives discontinuous solutions to the positional synthesis problem. Let us consider the canonical system of linear equations $\dot{x}_i=x_{i+1}, i=\overline{1,n-1}, \dot{x}_n=u$ with constraints $|u| \le d$. The problem for an arbitrary linear system $\dot{x} = A x + b u$ can be simplified to this problem for the canonical system. A controllability function $\Theta(x)$ is given as a unique positive solution of some equation $\Phi(x,\Theta) = 0$. The control is chosen to minimize derivative of the function $\Theta(x)$ and can be written as $u(x) = -d \text{ sign}(s(x,\Theta(x)))$. The set of points $s(x,\Theta(x)) = 0$ is called the switching surface, and it determines the points where control changes its sign. Normally it \mbox{contains} the variable $\Theta$ which is given implicitly as the solution of equation $\Phi(x, \Theta) = 0$. Our aim in this paper is to find a representation of the switching surface that does not depend on the function $\Theta(x)$. We call this representation the explicit form. In our case the expressions $\Phi(x, \Theta)$ and $s(x, \Theta)$ are both polynomials with respect to $\Theta$, so this problem is related to the problem of finding conditions when two polynomials have a common positive root. Earlier the solution for the 2-dimensional case was known. But during the exploration it was found out that for systems of higher dimensions there exist certain difficulties. In this article the switching surface for the three dimensional case is presented and researched. It is shown that this switching surface is a sliding surface (according to Filippov's definition). Also the other ways of constructing the switching surface using the interpolation and approximation are proposed and used for finding the trajectories of concrete points.