KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics

Abstract

Real-time state estimation of dynamical systems is a fundamental task in signal processing and control. For systems that are well-represented by a fully known linear Gaussian state space (SS) model, the celebrated Kalman filter (KF) is a low complexity optimal solution. However, both linearity of the underlying SS model and accurate knowledge of it are often not encountered in practice. Here, we present KalmanNet, a real-time state estimator that learns from data to carry out Kalman filtering under non-linear dynamics with partial information. By incorporating the structural SS model with a dedicated recurrent neural network module in the flow of the KF, we retain data efficiency and interpretability of the classic algorithm while implicitly learning complex dynamics from data. We numerically demonstrate that KalmanNet overcomes nonlinearities and model mismatch, outperforming classic filtering methods operating with both mismatched and accurate domain knowledge.

A Closed-Loop Output Error Approach for Physics-Informed Trajectory Inference Using Online Data

“…Proof: Consider the Lyapunov function (15). The timederivative of V along the system trajectories (22) and the update rule ( 14) is…”

“…One main drawback of this technique is that requires that the physics to be normalized to avoid instability which is difficult to achieve for multiagent systems and if the upper bound of the dynamic model is unknown. Other approaches use recurrent neural networks [15]- [17] to add memory and take into account previous states. However, in general this kind of neural networks are not able to capture the physics of the system and the trajectory inference is not accurate.…”

Closed-Loop Output Error Approaches for Drone's Physics Informed Trajectory Inference

“…Proof: First, consider the first element of the right-hand side of (15). Using partial fractions gives…”

“…informed model (PIM) [11], [12], also known as model-based, that allows to estimate unknown states with noise attenuation capabilities [13], [14]. A wrong model may cause large estimation error and instability in the estimation process [15]. Other techniques are based on Gaussian processes that capture all the prior information [16], [17] that we have from the real system in the form of non-parametric function approximators, which allow to infer the drone's trajectories in future time steps [18].…”