Learning as a Controlled Dynamical System: EKF–ODE Identification and Effort-Constrained Practice Scheduling from LMS Logs

Learning in higher education is shaped not only by the amount of practice students complete but also by the temporal allocation of practice opportunities. This study presents and retrospectively evaluates a continuous-time framework that connects learning analytics with model-based practice scheduling. Using the Open University Learning Analytics Dataset (OULAD), latent mastery and a latent fatigue/effort state are modeled as a controlled ordinary differential equation (ODE), with daily submission count used as the practice-intensity input. Model parameters are identified by maximum likelihood using a continuous–discrete extended Kalman filter (EKF), and practice schedules are subsequently derived through effort-constrained optimal control using direct collocation. To prevent learner-level information leakage, model identification was restricted to training partition and predictive evaluation was performed on 240 disjoint held-out learners. Prediction of each held-out learner’s final observed assessment from the pre-update state yielded AUC = 0.714 and RMSE = 0.197, while event-level one-step-ahead prediction yielded AUC = 0.637 and RMSE = 0.238 across the 2096 test events. Under the 28-day control horizon and matched cumulative effort, the collocation-derived schedule produced terminal mastery pT=0.94228, compared with pT=0.93990 for the constant baseline, corresponding to a small model-based difference of 0.00238, or approximately 0.24 percentage points. Sensitivity analyses across 21-, 28-, and 35-day horizons and alternative objective weights confirmed that the numerical policy contrast remains modest, although its magnitude depends on the control specification. The results demonstrate the computational feasibility of integrating continuous-time state estimation and constrained policy optimization in learning analytics, while prospective validation remains necessary to establish educational and causal effectiveness.