Koopman Observer with 42% Better Accuracy Certifies Nonlinear State Estimation

Nonlinear state estimation remains a critical challenge in control systems, especially when models are approximate and measurements noisy. Traditional observers like the Extended Kalman Filter (EKF) lack formal convergence guarantees under model mismatch. Syed Pouladi's new work on arXiv tackles this by combining Koopman operator theory — which lifts nonlinear dynamics into a linear (but infinite-dimensional) space — with generalized Persidskii systems. The key insight: the error dynamics of a Koopman latent-space observer exactly fit the Persidskii structure, which admits diagonal Lyapunov functions and incremental sector conditions. This allows the observer correction gain to be computed via a linear matrix inequality (LMI) that certifies input-to-state stability (ISS) with respect to lifting residuals and external disturbances. Under nominal conditions, exponential convergence is proven; under bounded perturbations, ultimate boundedness holds.

Numerical experiments on the Van der Pol oscillator and a nonlinear robotic arm with friction uncertainty demonstrate clear superiority. The proposed observer achieves up to 42% reduction in steady-state root-mean-square error (RMSE) compared to both the EKF and a standard linear Koopman observer. This is a significant step toward making Koopman-based methods practically deployable in safety-critical systems like robotics, autonomous vehicles, and industrial control, where stability guarantees are non-negotiable.