In-the-large stability of relay and saturating control systems with linear controllers

It is shown that when the plant of a relay control system consists of a pure integrators, with n ≥ 3, and the control signal proceding the relay is a linear combination of state coordinates, the system is unstable in-the-large for all values of the controller coefficients. The same result holds if the relay is replaced by a. saturator. The method used is to prove that (except for a degenerate case) state trajectories exist which go to infinity. This method enables more general systems to be treated than those discussed in a previous paper, which demonstrated the existence of a periodic solution. Plants not consisting of pure integrators are also considered. The question of when it is worthwhile to incorporate non-linearities in the controller is discussed.