Softsign-Based Nonlinear Control of Steam Condenser via Gbest-Guided Atom and Pattern Search Approach

This paper introduces a novel cascaded softsign function-based PID (CSoft-PID) controller designed for precise pressure regulation in highly nonlinear shell-and-tube steam condenser systems. For the first time in the literature, the classical PID control structure is enhanced through a cascaded nonlinear transformation using the softsign function, which dynamically adjusts the controller input according to the magnitude of the error. This architecture allows for high sensitivity near the setpoint while gracefully limiting excessive control efforts during larger deviations, thereby improving stability and transient performance. To optimally tune the six parameters of the proposed controller, a new hybrid optimization algorithm, termed hGASO-PS, is proposed. This method synergistically integrates an adaptive gbest-guided atom search optimization (ASO) strategy with the precision of the pattern search (PS) technique, ensuring both effective global exploration and fine-tuned local exploitation. The controller parameters are optimized by minimizing the integral of time-weighted absolute error (ITAE), subject to a step change in the condenser pressure setpoint. Extensive simulations and statistical evaluations demonstrate the superiority of the proposed approach. The hGASO-PS-based CSoft-PID controller achieved the lowest ITAE value of 2.1608, with an average of 2.2746 across 30 runs. It also demonstrated the fastest settling time (12.51 s) and the lowest overshoot (1.98%) among all tested controllers. Comparisons with recent PI, FOPID, and cascaded PI-PDN controllers confirm the consistent outperformance of the proposed method in both transient response and control precision, making it a promising candidate for industrial condenser applications.