The underactuated rotary double inverted pendulum (RDIP) system is widely utilized in applications such as exoskeletons, humanoid robots, and space vehicles. Designing an effective controller for such RDIP system poses significant challenges in system modeling, control, and optimization. This study presents a linearization approach for the RDIP system and proposes a simulated annealing optimized state feedback controller with integrator (SAOSFCI) to stabilize the RDIP in the upright position while enabling the rotary arm to track a time-varying trajectory. The performance of the SAOSFCI is evaluated against a linear quadratic regulator (LQR) across two RDIP models. Simulation results demonstrate that the SAOSFCI achieves superior stabilization with reduced deviations in pendulum and rotary arm angles. Additionally, hardware validation is conducted on one of the RDIP model. The results obtained through simulation and hardware validation are presented using several performance indices. These indices corroborate to the effectiveness of the SAOSFCI in achieving effective trajectory tracking capabilities in real-world scenarios.