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.