Controllability of the linear switched dynamical systems of the special type

Abstract

Switched systems is a special case of hybrid dynamical systems with discrete and continuous dynamics. They are widely applied when a real system cannot be described by one single model. In theoretical works on switched systems, switching signals and times can be random or given by some law. Stability depends both on vector fields and on the switching law.
In the present paper, a different formulation of the problem is considered, that is the case, when switching signal is under our control. Namely, a switched system is called controllable if for any two points there exists a switching signal that allows to get from the first point to the second one. In the paper the controllability of linear switched systems of a special type is studied. More specifically, we consider a switch, that is carried out between two 2x2 matrices with purely imaginary eigenvalues of both matrices. In the first section we discuss the physical meaning of switched systems of this type. Namely, the problem of oscillation of a spring pendulum with a switchable stiffness coefficient is considered with the series and parallel connection of an additional spring to the system with one given spring. We prove that such a system is controllable, and propose the method of finding the controlling switching signal. In the second section we present the main result of the work. We formulate an algorithm that allows finding a set of switching signals for getting from any given initial point to any given end point. We present an example of such controlling switching signals, simulated in MATLAB. In the last section we propose a generalization of the obtained result and formulate the theorem that states the controllability of the special type switched system with a block-diagonal matrix of high dimension. The method presented in the paper can be generalized to study of controllability of linear switched systems of more general form.