This paper presents a novel adaptive control strategy for nonlinear cascade systems interconnected through rate-dependent hysteresis, which frequently arises in smart material actuators such as piezoelectric devices and shape memory alloys. Unlike traditional methods that require explicit inversion of hysteresis models or full knowledge of the plant dynamics, the proposed approach integrates a filtered integral feedback structure with an adaptive neural operator that approximates hysteretic behavior in real time. This design avoids signal differentiation and hysteresis inversion while ensuring robustness to model uncertainty. Theoretical analysis establish input-to-state stability (ISS) of the closed-loop system with respect to the hysteresis approximation error. Under a standard persistence of excitation (PE) condition, we further prove uniform global convergence of the tracking error using Barbalat’s Lemma. Simulation results demonstrate that the proposed controller outperforms conventional PI and PID controllers in terms of tracking accuracy, convergence speed, and robustness to parameter variations. This work extends inversion-free control frameworks to nonlinear systems with dynamic hysteresis, offering a practical and scalable framework for advanced smart actuator control.