Stepwise synthesis of constrained controls for single input nonlinear systems of special form

Abstract

For control systems of the form \({dx/dt = a(x) + B(x)\beta(x, u)}\) with one-dimensional control, where a(x) is an n-dimensional vector function, B(x) is an \({(n \times m)}\)-matrix, and \({\beta(x, u)}\) is an m-dimensional vector function, the method of constructing of stepwise synthesis control is proposed. At first, under certain conditions we reduce such system to a system consisting of m subsystems; in each subsystem all equations are linear except of the last one. Further we propose the method for construction of controls which transfer an arbitrary initial point to the equilibrium point in a certain finite time. Each such control is constructed as a concatenation of a finite number of positional controls (we call it a stepwise synthesis control). On each step of our method we choose a new synthesis control. In this connection, nonlinearity of a system with respect to a control is essentially used. The obtained results are illustrated by examples. In particular, the problem of the complete stoppage of a two-link pendulum with the help of non-linear forces is solved. Finally, we introduce the class of nonlinear systems which is called the class of staircase systems that provides the applicability of our method.