Provable State Estimation with Recurrent Models

Estimating the state of a dynamical system from observations is a crucial task in robotics and control. Despite their empirical success, recurrent models often lack formal guarantees on convergence and robustness. We bridge this gap by analyzing these architectures from the perspective of nonlinear observer theory. By framing RNNs as a time-discretization of continuous latent dynamical systems (Neural ODEs), we demonstrate that enforcing a contraction condition on the latent flow guarantees the existence of a smooth, injective map of the true physical state into the latent space. We prove that this architecture acts as a tunable observer: by increasing a single scalar gain that scales the latent vector field, the estimation error can be driven to an arbitrarily small neighborhood of zero at an exponentially fast rate. We further establish quantitative bounds on the MLP complexity required to decode this latent state, and extend our formal guarantees to the discrete-time residual RNN implementation. We validate the theory on a real mobile robotic task, where a provably contracting model outperforms a vanilla GRU with substantially more stable training. We also report that unconstrained GRUs converge to contracting dynamics on this task, suggesting that contraction is an emergent property of well-trained sequence models.