Joint estimation and identification based on DEM–LMSINDYC closed-loop iteration

Accurate online state estimation and parameter identification are pivotal for safety-critical systems with uncertain and time-varying dynamics, especially under colored disturbances and structural variability. This paper develops a closed-loop estimation–identification framework, termed DEM–LMSINDYC, by coupling Dynamic Expectation Maximization (DEM) in the active-inference paradigm with Levenberg–Marquardt Sparse Identification of Nonlinear Dynamical Systems with Control (LMSINDYC). Within each outer iteration, DEM performs variational free-energy minimization in generalized coordinates to obtain robust state and input estimates under temporally correlated noise, and the resulting trajectories are then used by LMSINDYC to update sparse model coefficients. The identified dynamics are fed back to DEM, forming a unified loop that jointly refines inference and identification. A local contraction analysis further guarantees linear-rate convergence to a unique fixed point, a residual-based certificate ensures reproducible termination, and a complexity analysis confirms real-time feasibility. Numerical simulations of UAV longitudinal dynamics demonstrate coefficient-wise convergence toward reference parameters with localized matrix residuals, accurate closed-loop tracking with bounded control effort, and well-behaved free-energy evolution. Sensitivity analyses delineate stable regions of the key precision and generalized-coordinate hyperparameters. Robustness studies under external-disturbance sweeps, non-stationary Gaussian colored-noise statistics, and practical stress conditions (actuator saturation, modeling mismatch, and abrupt parameter variation) show graceful degradation, and benchmarks against EKF-, UIO-, STRidge-, and SR3-based alternatives confirm the most favorable combined estimation and identification performance.