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

TL;DR

This paper demonstrates that a linear feedback can stabilize a saturated complex double integrator system, which has non-diagonalizable dynamics and purely imaginary eigenvalues, ensuring global asymptotic stability.

Contribution

It proves the existence of a linear feedback law that guarantees global stability for a complex double integrator with saturation, non-diagonalizable matrix, and purely imaginary eigenvalues.

Findings

Existence of a linear feedback stabilizing the system

Global asymptotic stability achieved with saturation

Applicable to systems with non-diagonalizable matrices and imaginary eigenvalues

Abstract

Consider the saturated complex double integrator, i.e., the linear control system $\dot x=Ax+B\sigma(u)$, where $x\in\R^4$, $u\in\R$, $B\in\R^4$, 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 $\sigma:\R\to\R$ is a saturation function. We prove that there exists a linear feedback $u=K^Tx$ such that the resulting closed loop system given by $\dot x=Ax+B\sigma(K^Tx)$ is globally asymptotically stable with respect to the origin.