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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.

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A H&lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$_{\infty}$ &lt;/tex-math&gt; &lt;/inline-formula&gt; PI&lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$_{\varepsilon}$ &lt;/tex-math&gt; &lt;/inline-formula&gt;D-Based Observer Designed by LMI for Some Special System