Unscented Kalman Filter
The unscented Kalman filter captures nonlinearity by propagating a small deterministic set of sample points instead of linearizing.
Sampling instead of linearizing
The extended Kalman filter approximates a nonlinear function by its slope; the unscented filter approximates the probability distribution instead. It chooses a small set of carefully placed points, called sigma points, that capture the mean and covariance of the current estimate, pushes each through the full nonlinear model, and reconstructs the mean and covariance of the result from where the points land.
The unscented transform
For a state of dimension n the filter uses about 2n+1 sigma points, spread around the mean along the covariance directions. Because the points go through the real model, the transform captures curvature that a single Jacobian misses, and it is accurate to a higher order for many distributions. No derivatives are needed.
Advantages over the extended filter
- No Jacobians, so it works with black-box or non-differentiable models
- Higher accuracy for moderately strong nonlinearity
- Comparable computational cost for modest state dimension
Limits
The number of sigma points grows with the state dimension, so for very large fields the unscented filter becomes expensive and the ensemble approach is preferred. It still assumes the posterior is well described by a mean and covariance, so it struggles with strongly multi-modal distributions where a particle filter is appropriate.
Use in a fusion twin
The unscented filter suits subsystems that are nonlinear but not enormous, and where the model is available only as a solver rather than as clean equations. An example in a Kronos twin is estimating coupled magnet current and temperature margin in the burner's high-field plug region, where the model is a solver and derivatives are awkward, yet the state dimension is small enough for sigma points to remain cheap. Its uncertainty output feeds the same decision and control layers as any other filter.