View source for Symmetry-preserving filter

{{Technical|date=January 2022}}
In [[mathematics]], '''Symmetry-preserving observers''',<ref name="Bonnabel_tac" /><ref name="Bonnabel_cdc09">S. Bonnabel, Ph. Martin and E. Salaün, "Invariant Extended Kalman Filter: theory and application to a velocity-aided attitude estimation problem", 48th IEEE Conference on Decision and Control,  pp. 1297-1304, 2009.</ref> also known as '''invariant filters''', are estimation techniques whose structure and design take advantage of the natural symmetries (or invariances) of the considered [[nonlinear model]]. As such, the main benefit is an expected much larger domain of convergence than standard filtering methods, e.g. [[Kalman filter#Extended Kalman filter|Extended Kalman Filter]] (EKF) or [[Kalman filter#Unscented Kalman filter|Unscented Kalman Filter]] (UKF).

== Motivation ==

Most physical systems possess natural symmetries (or invariance), i.e. there exist [[Transformation (geometry)|transformations]] (e.g. rotations, translations, scalings) that leave the system unchanged. From mathematical and engineering viewpoints, it makes sense that a filter well-designed for the system being considered should preserve the same invariance properties.

== Definition ==

Consider <math>G</math> a Lie group, and 
(local) transformation groups <math>\varphi_g, \psi_g, \rho_g</math>, where <math>g \in G</math>.

The nonlinear system
:<math>
\begin{align}
 \dot x&=f(x,u)\\
 y &=h(x,u)
\end{align}
</math>
is said to be ''invariant'' if it is left unchanged by the action of <math>\varphi_g, \psi_g, \rho_g</math>, i.e.

:<math>
 \begin{align}
\dot X&=f(X,U)\\
 Y &=h(X,U) ,
\end{align}
</math>

where <math> (X,U,Y)=(\varphi_g(x),\psi_g(u),\rho_g(y))</math>.

<br />
The system <math>\dot{\hat x}=F(\hat{x},u,y)</math> is then an '''invariant filter''' if

* <math> F(x,u,h(x,u)) = f(x,u)</math>, i.e. that it can be written <math>\dot{\hat x}=f(\hat x,u)+C</math>, where the correction term <math>C</math> is equal to <math>0</math> when <math>\hat y=y</math>
* <math>\dot{\hat X}=F(\hat X,U,Y)</math>, i.e. it is left unchanged by the [[transformation group]].

== General equation and main result ==

It has been proved <ref name="Bonnabel_tac">S. Bonnabel, Ph. Martin, and P. Rouchon, “Symmetry-preserving observers,”
''IEEE Transactions on Automatic and Control'', vol. 53, no. 11, pp. 2514–2526, 2008.</ref> that every [[invariant observer]] reads

:<math>\dot{\hat x}=f(\hat x,u)
 +W(\hat x)L\Bigl(I(\hat x,u),E(\hat x,u,y)\Bigr)E(\hat x,u,y) ,</math>

where

* <math>E(\hat x,u,y)</math> is an [[invariant output error]], which is different from the usual output error <math>\hat y-y</math>
*<math>W(\hat x)=\bigl(w_1(\hat x),..,w_n(\hat x)\bigr)</math> is an invariant frame
*<math>I(\hat x,u)</math> is an invariant vector
*<math>L(I,E)</math> is a freely chosen gain matrix.

Given the system and the associated transformation group being considered, there exists a constructive method to determine <math> E(\hat x,u,y),W(\hat x),I(\hat x,u)</math>, based on the moving frame method.

To analyze the error convergence, an invariant state error <math> \eta(\hat x,x)</math> is defined, which is different from the standard output error <math> \hat x-x </math>, since the standard output error usually does not preserve the symmetries of the system. One of the main benefits of symmetry-preserving filters is that the error system is "''autonomous''", but for the free known invariant vector <math>I(\hat x,u)</math>, i.e. <math>\dot\eta=\Upsilon\bigl(\eta,I(\hat x,u)\bigr)</math>. This important property allows the estimator to have a very large domain of convergence, and to be easy to tune.<ref name="Martin_ifac">Ph. Martin and E. Salaün, "An invariant observer for Earth-velocity-aided attitude heading reference
systems", 17th IFAC World Congress,  pp. 9857-9864, 2008.</ref><ref name="Martin_cep">Ph. Martin and E. Salaün, "Design and implementation of a low-cost observer-based Attitude and Heading Reference System", ''Control Engineering Practice'', 2010.</ref>

To choose the gain matrix <math>L(I,E)</math>, there are two possibilities:

* a ''deterministic approach'', that leads to the construction of truly nonlinear symmetry-preserving filters (similar to Luenberger-like observers)
* a ''stochastic approach'', that leads to [[Invariant extended Kalman filter|Invariant Extended Kalman Filters]] (similar to Kalman-like observers).

== Applications ==

There has been numerous applications that use such invariant observers to estimate the state of the considered system. The application areas include

* [[Attitude and Heading Reference Systems|attitude and heading reference systems]] with <ref name="Martin_ifac"/> or without <ref name="Martin_cep"/> position/velocity sensor (e.g. GPS)
* ground vehicle localization systems
* chemical reactors<ref name="Bonnabel_tac" />
* [[oceanography]]

== References ==

<references/>

[[Category:Nonlinear filters]]
[[Category:Signal estimation]]