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.