Stability of single saturated-input linear systems with one zero eigenvalue

In this work we address the stability properties of single saturated input linear systems with one zero open-loop eigenvalue. These systems arise naturally in hyperbolic linear systems with a proportional-integral feedback, or with input rate saturation. We prove that, if the eigenvalues of the open-loop hyperbolic subsystem have negative real part, all the trajectories of the closed-loop system are bounded. Similarly, if such eigenvalues have positive real part, it is proved that the asymptotic stability region of the closed-loop system is contained in a cylinder homeomorphic to R×D_n-1, where D_n-1 is the open (n-1)-dimensional disk