A H $_{\infty}$ PI $_{\varepsilon}$ D-Based Observer Designed by LMI for Some Special System

This paper considers the problems of observing system states of some special system and proposes a novel
$H\infty $
proportional–integral–derivative (
$PI^{\varepsilon }D$
)-based observer designed by linear matrix inequality (LMI). First, an augmented system is designed to ensure the asymptotic stability of the system with the
$H\infty $
-based
$PI^{\varepsilon }D$
observer. Then, the proportional, integral, and derivative gains of
$PI^{\varepsilon }D$
observer are designed based on
$H\infty $
synthesis. Finally, the
$H\infty $
synthesis problem is translated into LMI, and through solving the LMI, the proposed observer can ensure the performance of
$H\infty $
. The observation of vehicle longitudinal tire force is followed as an example, whose closed-loop system with traditional PID observer could not guarantee asymptotic stability. Both simulation and experimental results illustrate the great performance of the proposed observer.