# Soft Fuzzy Reinforcement Neural Network Proportional-Derivative Controller
## 4 Results and Discussion
### Testing Performance Evaluation  
To evaluate the effectiveness of the proposed method, a series of controlled experiments were conducted, comparing its performance with Soft Actor-Critic (SAC) and traditional Proportional-Derivative (PD) controllers. The experiments were designed to assess response speed, accuracy, stability, noise robustness, and disturbance rejection capability. The initial evaluation was conducted under fixed initial conditions (\(\theta=-135^{\circ}\), \(\dot{\theta}=2\) rad/s) to ensure a controlled comparison. The control parameters of the traditional PD algorithm were determined using the Ziegler-Nichols tuning method (\(K_{p}=5.708\), \(K_{d}=9.510\)). As shown in Figure 7a, the proposed SFPD algorithm exhibited superior speed, accuracy, and stability compared to SAC and traditional PD.  
Figure 7: Testing results on the pendulum: (**a**) without Gaussian noise; (**b**) with Gaussian noise; (**c**) with Gaussian noise and external disturbance.  
To analyze the impact of sensor noise on control performance, a Gaussian white noise measurement disturbance with \(\mu\sim\mathcal{N}(0,0.1^{2})\) was introduced. Figure 7b demonstrates that SFPD exhibited strong resilience to noise, owing to its Markov-based control strategy, which ensured decisions were made based on the current state, thereby mitigating the accumulation of errors over time. To further assess disturbance rejection capability, an external torque perturbation was applied [43] between steps 100 and 102. As shown in Figure 7c, SFPD demonstrated the fastest recovery with minimal oscillations, highlighting its exceptional robustness. SAC maintained stability but exhibited a slower response, while traditional PD showed significant overshoot and oscillations, indicating its limited disturbance rejection capability.  
A detailed quantitative analysis of the control performance is presented in Table 4. In noise-free conditions, SFPD achieved a 12.15% reduction in Integral of Squared Error (ISE) and a 40.19% reduction in Integral of Absolute Error (IAE) compared to SAC. Additionally, SFPD exhibited a 59.16% improvement in ISE and a 74.20% improvement in IAE over traditional PD, highlighting its superior error minimization. Even under measurement noise interference, SFPD remained highly effective, achieving a 7.54% reduction in ISE over SAC. Although SAC outperformed SFPD in IAE under these conditions (SAC: 1231, SFPD: 1368), SFPD still significantly outperformed traditional PD, achieving improvements of 53.69% in ISE and 51.70% in IAE, further demonstrating its robustness in noisy environments.  
Overall, the experimental results demonstrated that SFPD outperformed SAC and traditional PD across various control scenarios. The proposed method achieved faster response, reduced steady-state errors, superior robustness to measurement noise, and enhanced disturbance rejection, making it a more effective solution for nonlinear control applications.  
Figure 8: Random initial positions test on the pendulum: (**a**) without Gaussian noise; (**b**) with Gaussian noise.  
Noise introduced noticeable fluctuations in angular error across all controllers as shown in Figure 8b. However, SFPD continued to maintain superior convergence performance with minimal oscillations compared to SAC and traditional PD. SAC showed improved noise tolerance compared to traditional PD, but its fluctuations remained larger than those of SFPD. Traditional PD suffered from larger amplitude oscillations and slower stabilization under noisy conditions, indicating its vulnerability to measurement noise.