Stabilisation of the complex double integrator by means of a saturated linear feedback

Abstract

Consider the saturated complex double integrator, i.e., the linear control system $\dot{x}=Ax+B\sigma (u)$, where $x\in {\mathbb{R}}^4$, $u\in \mathbb{R}$, $B\in {\mathbb{R}}^4$, the $4\times 4$ matrix $A$ is not diagolizable and admits a non zero purely imaginary eigenvalue of multiplicity two, the pair $(A,B)$ is controllable and $\sigma :\mathbb{R}\to \mathbb{R}$ is a saturation function. We prove that there exists a linear feedback $u=K^{T}x$ such that the resulting closed loop system given by $\dot{x}=Ax+B\sigma (K^{T}x)$ is globally asymptotically stable with respect to the origin.