Lyapunov-Based Nonlinear Model Predictive Control of Input-Delayed Functional Electrical Stimulation: Investigative Simulations and Experiments

Existing closed-loop controllers for functional electrical stimulation are prone to exceeding subject-specific stimulation limits, thereby limiting performance and also accelerating stimulation-induced muscle fatigue. In view of these challenges, this paper develops a Lyapunov-based model predictive control method to control knee flexion and extension during input-delayed stimulation. The method incorporates a contractive constraint under an electromechanical delay (EMD) compensation control law that achieves system stability despite an unknown constant input delay, bounded control constraints, and imperfectly estimated model parameters. A Lyapunov stability analysis proves that the Lyapunov constraint renders the closed-loop error ultimately bounded, and gain conditions are provided to guarantee recursive feasibility. LMPC's performance is explored in simulation and experiments and compared against an analytical proportional derivative-dynamic surface controller (PD-DSC) and a proportional-derivative-delay compensation (PD-DC) controller. In simulation, LMPC improved tracking root-mean-square error by 75.57% and 71.71%, on average, compared to PD-DSC and PD-DC, respectively. We observed that incorporating a slackening term often improved LMPC's tracking performance, although strict enforcement of the Lyapunov constraint was superior when there was greater EMD estimation error. Additionally, unlike PD-DSC and PD-DC, LMPC was not destabilized when EMD was overestimated or underestimated, nor did it violate input constraints. In knee extension experiments, LMPC respected input constraints, which PD-DSC did not. The LMPC was also validated in overground walking experiments to test its ability to produce both knee flexion and extension in participants with and without spinal cord injury.