Synthesis of the relay-linear control with the hysteresis region for the second order object example

The purpose of this work is a relay-linear control synthesis with using of a hysteresis region, that contains a switching surface and is build with piecewise linear approximation of the switching surface. An application of the hysteresis region together with linear control region allows to avoid sliding mode. A second order object is discussed for better visual presentation of many geometrical concepts that are used in relay-linear control theory.