A H<inline-formula> <tex-math notation="LaTeX">$_{\infty}$ </tex-math></inline-formula> PI<inline-formula> <tex-math notation="LaTeX">$_{\varepsilon}$ </tex-math></inline-formula>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 <inline-formula> <tex-math notation="LaTeX">$H\infty $ </tex-math></inline-formula> proportional&#x2013;integral&#x2013;derivative (<inline-formula> <tex-math notation="LaTeX">$PI^{\varepsilon }D$ </tex-math></inline-formula>)-based observer designed by linear matrix inequality (LMI). First, an augmented system is designed to ensure the asymptotic stability of the system with the <inline-formula> <tex-math notation="LaTeX">$H\infty $ </tex-math></inline-formula>-based <inline-formula> <tex-math notation="LaTeX">$PI^{\varepsilon }D$ </tex-math></inline-formula> observer. Then, the proportional, integral, and derivative gains of <inline-formula> <tex-math notation="LaTeX">$PI^{\varepsilon }D$ </tex-math></inline-formula> observer are designed based on <inline-formula> <tex-math notation="LaTeX">$H\infty $ </tex-math></inline-formula> synthesis. Finally, the <inline-formula> <tex-math notation="LaTeX">$H\infty $ </tex-math></inline-formula> synthesis problem is translated into LMI, and through solving the LMI, the proposed observer can ensure the performance of <inline-formula> <tex-math notation="LaTeX">$H\infty $ </tex-math></inline-formula>. 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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