This study aims to explore the problem of $H_{\mathrm{\infty }}$ fuzzy control with an adjustable convergence rate for switched nonlinear systems with time-varying delays under the persistent dwell time (PDT) switching. Compared to the widely studied dwell time (DT) switching or average dwell time (ADT) switching in existing literature, PDT switching provides a more comprehensive consideration of the switching frequency and has a broader range of applicability. Subsequently, by combining the interval stability definition, T-S fuzzy model, PDT technique, and Lyapunov-Krasovskii (L-K) functional, a new $H_{\mathrm{\infty }}$ fuzzy control criterion for adjusting the convergence rate of switched nonlinear systems with time-varying delays is proposed. This criterion enables the development of a novel method for constructing $H_{\mathrm{\infty }}$ fuzzy controllers, which can regulate the system's convergence rate and achieve the specified $H_{\mathrm{\infty }}$ performance. Combining the above methods, an algorithm is introduced to precisely control the convergence rate of the target system. Finally, the effectiveness of this method is validated through a control example of a single-link robot arm.