In soft robotics, actuators using both positive and negative pressures are notable for their high payload-to-weight ratios and wide operating ranges, but they require separate power sources. A single-pump system generating dual pressures presents a promising solution, though addressing pressure fluctuations due to coupled dynamics remains a challenge. In this work, we propose a reinforcement learning (RL)-based controller capable of tracking both pressures over a wide range. To facilitate RL training, we built a simulator that models not only airflow dynamics but also the pump's kinematics and the electromagnetic behavior of pneumatic components. Our controller employs Model-Predicted Observation (MPObs) to predict future input effects and mitigate nonlinearities, and uses a Conditioning for Action Policy Smoothness (CAPS)-based action smoothing to reduce abrupt input changes. Experimental results show that the proposed RL controller achieves root-mean-square errors (RMSEs) of 0.6935 kPa (positive) and 0.2646 kPa (negative), outperforming the Disturbance Observer (DOB)-based approach. Ablation studies confirm the synergistic effect of MPObs and CAPS, underscoring their importance in control. Furthermore, robustness tests with external loads from 0 to 20 kg demonstrate a maximum RMSE of 0.7906 kPa (positive) and 0.1186 kPa (negative), indicating strong robustness. This study verifies that our proposed RL-based controller overcomes the nonlinear challenges of pneumatic power sources and highlights its potential for future stand-alone systems in field applications.