The relationship between periodic solutions and stability in a class of third-order relay control systems

The global asymptotic stability of third-order relay control systems with real, nonpositive eigenvalues is related to conditions necessary for the existence of periodic solutions of such systems. By application of Popov´s theorem it is shown that conditions which guarantee that a system will have no symmetric periodic solutions are also sufficient to insure the absolute stability of the origin. This result allows a relay system to be designed by choosing the switching function subject to the constraint that the switching plane avoid a certain easily defined region of state space.