Extended Kalman Filter
The extended Kalman filter applies the Kalman update to nonlinear systems by linearizing the dynamics around the current estimate.
Kalman for nonlinear systems
Most physical systems, including plasmas and thermal-hydraulic loops, are nonlinear. The extended Kalman filter keeps the Kalman structure of predict and update but replaces the linear system matrices with the Jacobians, the local first derivatives, of the nonlinear model evaluated at the current estimate. In effect it linearizes the world afresh at every step.
How it works
The state is advanced through the full nonlinear model, but the covariance is propagated using the Jacobian of that model. Likewise the measurement is predicted through the full nonlinear observation function, while the gain uses the Jacobian of that function. Everything else matches the classical filter.
Strengths
- Reuses the mature and cheap Kalman machinery
- Works well when nonlinearity is mild over one time step
- Provides a covariance estimate at low cost
Weaknesses
- Requires Jacobians, which may be hard or expensive to derive
- Linearization error grows with strong nonlinearity and can diverge
- Assumes the posterior stays roughly Gaussian, which fails near instabilities
When to reach for something else
If deriving Jacobians is impractical or the dynamics turn sharply, the unscented filter avoids linearization by propagating chosen sample points, and the ensemble filter handles very high-dimensional states. For strongly non-Gaussian, multi-modal situations, such as competing plasma scenarios, a particle filter may be required.
In a Kronos twin the extended filter is a reasonable default for moderately nonlinear subsystems, for example tracking coupled temperatures and flows in a cooling loop of the breeder Hyperion, where the physics is smooth and Jacobians are available from the thermal model. Its covariance output feeds directly into remaining-life and control decisions.