Global Analysis of Third - Order Relayfeedback Systems

Relays are common in automatic control systems. It is well-known that a linear dynamical system under relay feedback can give complex oscillations. In this paper it is proved that several of these phenomena can actually be captured by third-order systems. It is shown that there exist systems giving arbitrarily fast relay switches similar to sliding modes. A novel method for analyzing linear dynamical systems under relay feedback is also introduced. Trajectories for a class of third order systems are shown to converge in a certain sense. 1. INTRODUCTION Analysis of relay feedback systems is a classical topic in control theory. The early work was motivated by relays in electromechanical systems and simple models for dry friction. An important property of a linear dynamical system under relay feedback is its tendency to oscillate. Design of simple relay controllers in aerospace applications is described in Fl ugge-Lotz (1953). This work gave inspiration to the development of the self-oscillating adaptive controller in the 1960's. Recently new interest of relay feedback appeared due to the idea of using relays for tuning of simple controllers, see strrm and HHgglund (1984). By simply replacing the controller by a relay, measure the amplitude and frequency of the possible oscillation, and out of these derive the controller parameters, a robust control design method is given. Even if this method is widely spread and accepted in practice, it has not been theoretically investigated in greater detail. The idea of putting the plant under relay feedback is also used in Smith and Doyle (1993) for estimating perturbation bounds in