Abstract: Proportional-integral-derivative (PID) control maintains its dominant position in oscillation suppression due to its operational simplicity, excellent adaptability, and strong robustness. As power system oscillation problems grow increasingly complex, enhancing PID-based additional damping control carries significant engineering importance. This paper presents an innovative sub-synchronous damping control (SSDC) strategy based on generalized PID principles. The methodology first incorporates additional historical trajectory information to expand SSDC dimensionality, deriving the general expression of the generalized PID control algorithm. Subsequently, the introduction of step factors yields a more adaptable generalized PID relaxation algorithm. To optimize computational efficiency, historical data within specified time windows undergoes sparsification, endowing the generalized PID-based SSDC with both computational efficiency and superior oscillation suppression capabilities. Theoretical analysis and simulation results demonstrate the effectiveness and superiority of this generalized PID-based sub-synchronous oscillation damping method. Compared with conventional PID-based SSDC, the proposed control strategy exhibits stronger suppression performance and demonstrates promising engineering application potential.