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 a negative real part. The authors investigate, for single-input systems, the question of whether such systems can be stabilized by means of a feedback u=σ(h(x)), where h is linear and σ(s) is a saturation function such as sign(s) min(|s|,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. It is shown that for the multiple integrator of order n, with n ⩾3, no saturation of a linear feedback can be globally stabilizing