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In mathematics, Symmetry-preserving observers,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. 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 In mathematics, nonlinear modelling is empirical or semi-empirical modelling which takes at least some nonlinearities into account. Nonlinear modelling in practice therefore means modelling of phenomena in which independent variables affecting the ...
. As such, the main benefit is an expected much larger domain of convergence than standard filtering methods, e.g.
Extended Kalman Filter In estimation theory, the extended Kalman filter (EKF) is the nonlinear version of the Kalman filter which linearizes about an estimate of the current mean and covariance. In the case of well defined transition models, the EKF has been considered t ...
(EKF) or
Unscented Kalman Filter For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estima ...
(UKF).


Motivation

Most physical systems possess natural symmetries (or invariance), i.e. there exist 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 G a Lie group, and (local) transformation groups \varphi_g, \psi_g, \rho_g, where g \in G. The nonlinear system : \begin \dot x&=f(x,u)\\ y &=h(x,u) \end is said to be ''invariant'' if it is left unchanged by the action of \varphi_g, \psi_g, \rho_g, i.e. : \begin \dot X&=f(X,U)\\ Y &=h(X,U) , \end where (X,U,Y)=(\varphi_g(x),\psi_g(u),\rho_g(y)).
The system \dot=F(\hat,u,y) is then an invariant filter if * F(x,u,h(x,u)) = f(x,u), i.e. that it can be witten \dot=f(\hat x,u)+C, where the correction term C is equal to 0 when \hat y=y * \dot=F(\hat X,U,Y), i.e. it is left unchanged by the
transformation group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is t ...
.


General equation and main result

It has been proved S. Bonnabel, Ph. Martin, and P. Rouchon, “Symmetry-preserving observers,” ''IEEE Transactions on Automatic and Control'', vol. 53, no. 11, pp. 2514–2526, 2008. that every
invariant observer Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iterat ...
reads :\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) , where * E(\hat x,u,y) is an
invariant output error Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iterat ...
, which is different from the usual output error \hat y-y *W(\hat x)=\bigl(w_1(\hat x),..,w_n(\hat x)\bigr) is an invariant frame *I(\hat x,u) is an invariant vector *L(I,E) is a freely chosen gain matrix. Given the system and the associated transformation group being considered, there exists a constructive method to determine E(\hat x,u,y),W(\hat x),I(\hat x,u), based on the moving frame method. To analyze the error convergence, an invariant state error \eta(\hat x,x) is defined, which is different from the standard output error \hat x-x , 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 I(\hat x,u), i.e. \dot\eta=\Upsilon\bigl(\eta,I(\hat x,u)\bigr). This important property allows the estimator to have a very large domain of convergence, and to be easy to tune.Ph. Martin and E. Salaün, "An invariant observer for Earth-velocity-aided attitude heading reference systems", 17th IFAC World Congress, pp. 9857-9864, 2008.Ph. Martin and E. Salaün, "Design and implementation of a low-cost observer-based Attitude and Heading Reference System", ''Control Engineering Practice'', 2010. To choose the gain matrix L(I,E), 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 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 An attitude and heading reference system (AHRS) consists of sensors on three axes that provide attitude information for aircraft, including roll, pitch, and yaw. These are sometimes referred to as MARG (Magnetic, Angular Rate, and Gravity) sen ...
with or without position/velocity sensor (e.g. GPS) * ground vehicle localization systems * chemical reactors * oceanography


References

Nonlinear filters Signal estimation