Development of a PID controller for passenger inflow control in congested metro stations using elliptic bivariate relationship

Metro systems move large numbers of commuters, but limited capacity can leave passengers stranded during peak periods, increasing platform density and safety risks. This study proposes a passenger inflow control strategy using a PID controller driven by two parameters, ϕ and |λ|, derived from complex eigenvalue analysis of the elliptic bivariate relationship between pedestrian accumulation and time spent in the paid area. Here, ϕ identifies whether congestion is worsening or easing, while |λ| represents its severity. The controller estimates the net inflow and supports the allocation of entry and exit turnstiles in AFC systems. The strategy was implemented and evaluated in VISWALK using a calibrated real-world metro station. Compared with no control, it reduced average time spent by 10.2% and stranded passengers by 18.7%, and it also outperformed a threshold-based benchmark in average operational performance. Robustness tests under step disturbances and gain perturbations confirmed stable performance under non-recurrent demand conditions.