On the Stabilizability of Multiple Integrators by Means of Bounded Feedback Controls

It is known that a linear system x = Ax + Bu can be stabilized by means of a smooth bounded control if and only if it has no eigenvalues with positive real part, and all the uncontrollable modes have negative real part. Here we investigate, for single-input systems, the question whether such systems can be stabilized by means of a feedback u = oe(h(x)), where h is linear and oe(s) is a saturation function such as sign(s) min(jsj; 1). A stabilizing feedback of this particular form exists if A has no multiple eigenvalues, and also in some other special cases such as the double integrator. We show that for the multiple integrator of order n, with n 3, no saturation of a linear feedback can be globally stabilizing. 1. Introduction The problem of globally stabilizing a linear system x = Ax+Bu by means of a bounded smooth feedback has been considered in [1], where a proof was outlined of the fact that such a system admits a globally stabilizing bounded feedback u = k(x) if and only i..