Application of fractional order proportional integral derivative controller to functional electrical stimulation induced knee swinging in paraplegia

Functional Electrical Stimulation (FES) has emerged as an effective rehabilitation technique for restoring motor function in individuals with paraplegia. However, conventional Proportional-Integral-Derivative (PID) controllers used in FES systems often fail to compensate for the nonlinear and time-varying nature of muscle dynamics, resulting in performance degradation and early onset of fatigue. This study presents the design and implementation of a hybrid Fractional Order Proportional-Integral-Derivative (FOPID+PID) controller for FES-induced knee swinging, aimed at enhancing precision, stability, and fatigue resistance. A comprehensive biomechanical model incorporating a two-link leg structure, Hill-type muscle dynamics, and the Ding fatigue model was developed and simulated in MATLAB/Simulink. Performance analysis under non-fatigue and fatigue conditions demonstrated that the FOPID+PID controller achieved superior results compared to traditional FOPID and PID controllers. Without fatigue, the hybrid controller achieved a rise time of 2.28 ms, settling time of 1.11 ms, and an error of 16.55, outperforming both FOPID (Tr = 2.31 ms, Ts = 1.19 ms, error = 17.22) and PID (Tr = 38.57 ms, Ts = 5.43 ms, error = 15.12). Under fatigue, the hybrid controller maintained a rise time of 2.23 ms, settling time of 1.08 ms, and reduced error of 9.55, achieving near-constant muscle force (FCE = 5922 N; FMAX = 5980–6106 N). The FOPID+PID’s adaptive compensation for nonlinear muscle behavior and fatigue yielded smoother joint trajectories and improved rehabilitation efficiency. The results indicate that the hybrid controller offers a clinically viable approach for enhancing the stability, endurance, and natural movement of FES-assisted knee rehabilitation in paraplegic patients.