Symmetry-preserving filter

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Symmetry-preserving filters, also known as invariant observers, are estimation methods that use the natural symmetries of a nonlinear model to estimate its state. The key idea is that many physical systems behave the same under certain transformations, such as rotations, shifts, or scaling. A filter designed to respect these invariances tends to work well over a larger range of conditions than standard filters like the Extended Kalman Filter (EKF) or the Unscented Kalman Filter (UKF).

Mathematically, you describe the system’s symmetries with a mathematical group and show how the state, input, and output transform under those symmetries. If applying these transformations leaves the system equations unchanged, the system is called invariant. An invariant filter is built in a constructive way that uses these transformations to determine how to correct estimates, how to weight the state error, and how to incorporate information that stays consistent with the symmetries.

A special feature of these filters is the way they measure error. Instead of the usual difference between the estimated state and the true state, they use an invariant error that respects the system’s symmetries. This invariant error leads to an error dynamics that are autonomous, meaning the rate of change of the error depends only on the error itself and the fixed invariant information, not on the current estimate. This often results in a very large basin of attraction for convergence and makes tuning the filter easier.

There are standard ways to choose the filter’s gain, taking the invariant structure into account. In practice, invariant observers have been applied across many areas to reliably estimate the state of complex systems.