Extended and Unscented Kalman Filters

Fusion dynamics are nonlinear; the EKF linearizes the model while the UKF propagates sample points, giving the twin nonlinear state estimation.

Handling nonlinearity

The plant models - Grad-Shafranov evolution, ambipolar dynamics - are nonlinear, so the plain Kalman filter does not apply directly. The extended Kalman filter (EKF) linearizes the model about the current estimate each step; the unscented Kalman filter (UKF) instead propagates a small set of deterministically chosen sample points through the true nonlinear model. Both keep a Gaussian belief but differ in accuracy and cost.

EKF: linearize with Jacobians each step
  F_k = df/dx |_x^ ,  H_k = dh/dx |_x^
  then apply Kalman predict/update with F_k, H_k

UKF: unscented transform (sigma points)
  choose 2n+1 sigma points X_i from (x^, P)
  propagate: Y_i = f(X_i)
  x^- = sum w_i Y_i ,  P^- = sum w_i (Y_i-x^-)(Y_i-x^-)' + Q

EKF versus UKF trade

The EKF is cheap but its first-order linearization degrades when dynamics are strongly nonlinear, and it needs Jacobians (available by autodiff from the PINN/twin models). The UKF captures mean and covariance to higher order without Jacobians, at the cost of multiple model evaluations. The stack picks per subsystem: EKF where near-linear and fast, UKF where nonlinearity bites, such as near stability boundaries.

# UKF predict via sigma points (schematic)
X = sigma_points(x, P, alpha, kappa)    # 2n+1 points
Y = [f(Xi, u) for Xi in X]              # nonlinear propagation
x_pred = sum(wm[i]*Y[i] for i in range(len(Y)))
P_pred = sum(wc[i]*outer(Y[i]-x_pred, Y[i]-x_pred)
             for i in range(len(Y))) + Q

Where each is used

For the breeder, near-equilibrium tracking of shape and current is well served by the EKF using twin Jacobians; near disruption-relevant boundaries, where the response curves sharply, the UKF's better covariance estimate matters. For the burner, strong nonlinearity in the ambipolar response favors the UKF, though its estimates inherit the 166-830x regime uncertainty and are reported as such.

• EKF: Jacobian-based, cheap, accurate near-linear regimes.
• UKF: sigma-point, Jacobian-free, better for strong nonlinearity.
• Selection per subsystem by nonlinearity and latency budget.
• Both still report covariance; both feed innovation monitoring.

When the state dimension is very large - a full field twin - even the UKF's covariance is unwieldy, and the stack turns to the ensemble Kalman filter covered next.