Symmetry-preserving filter

Last updated January 07, 2022

In mathematics, Symmetry-preserving observers,^[1]^[2] 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. Extended Kalman Filter (EKF) or Unscented Kalman Filter (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{aligned}{\dot {x}}&=f(x,u)\\y&=h(x,u)\end{aligned}}

is said to be invariant if it is left unchanged by the action of $\varphi _{g},\psi _{g},\rho _{g}$ , i.e.

{\begin{aligned}{\dot {X}}&=f(X,U)\\Y&=h(X,U),\end{aligned}}

where $(X,U,Y)=(\varphi _{g}(x),\psi _{g}(u),\rho _{g}(y))$ .

The system ${\dot {\hat {x}}}=F({\hat {x}},u,y)$ is then an invariant filter if

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

General equation and main result

It has been proved ^[1] that every invariant observer 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, 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.^[3]^[4]

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 with ^[3] or without ^[4] position/velocity sensor (e.g. GPS)
ground vehicle localization systems
chemical reactors^[1]
oceanography

Related Research Articles

In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".

In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions.

In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.

Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918, after a special case was proven by E. Cosserat and F. Cosserat in 1909. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.

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 estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.

The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other being fermions) would be considered massless, but measurements show that the W⁺, W⁻, and Z⁰ bosons actually have relatively large masses of around 80 GeV/c². The Higgs field resolves this conundrum. The simplest description of the mechanism adds a quantum field (the Higgs field) that permeates all Hilbert space to the Standard Model. Below some extremely high temperature, the field causes spontaneous symmetry breaking during interactions. The breaking of symmetry triggers the Higgs mechanism, causing the bosons it interacts with to have mass. In the Standard Model, the phrase "Higgs mechanism" refers specifically to the generation of masses for the W^±, and Z weak gauge bosons through electroweak symmetry breaking. The Large Hadron Collider at CERN announced results consistent with the Higgs particle on 14 March 2013, making it extremely likely that the field, or one like it, exists, and explaining how the Higgs mechanism takes place in nature.

In mathematics, specifically in operator theory, each linear operator $on a Euclidean vector space defines a Hermitian adjoint operator on that space according to the rule$

In quantum mechanics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates :

In control theory, a state observer or state estimator is a system that provides an estimate of the internal state of a given real system, from measurements of the input and output of the real system. It is typically computer-implemented, and provides the basis of many practical applications.

In physics, a symmetry of a physical system is a physical or mathematical feature of the system that is preserved or remains unchanged under some transformation.

In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on $, functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle.$

In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature. More precisely, they are representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields.

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 the de facto standard in the theory of nonlinear state estimation, navigation systems and GPS.

In geometry, a ball is a region in a space comprising all points within a fixed distance, called the radius, from a given point; that is, it is the region enclosed by a sphere or hypersphere. An $n$ -ball is a ball in an $n$ -dimensional Euclidean space. The volume of a $n$ -ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a $n$ -ball of radius $R$ is $where is the volume of the unit n -ball, the n -ball of radius 1 .$

In physics, a gauge theory is a type of field theory in which the Lagrangian does not change under local transformations according to certain smooth families of operations.

The invariant extended Kalman filter (IEKF) was first introduced as a version of the extended Kalman filter (EKF) for nonlinear systems possessing symmetries, then generalized and recast as an adaptation to Lie groups of the linear Kalman filtering theory. Instead of using a linear correction term based on a linear output error, the IEKF uses a geometrically adapted correction term based on an invariant output error; in the same way the gain matrix is not updated from a linear state error, but from an invariant state error. The main benefit is that the gain and covariance equations have reduced dependence on the estimated value of the state. In some cases they converge to constant values on a much bigger set of trajectories than is the case for the EKF, which results in a better convergence of the estimation.

Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems.

In quantum probability, the Belavkin equation, also known as Belavkin-Schrödinger equation, quantum filtering equation, stochastic master equation, is a quantum stochastic differential equation describing the dynamics of a quantum system undergoing observation in continuous time. It was derived and henceforth studied by Viacheslav Belavkin in 1988.

Group actions are central to Riemannian geometry and defining orbits. The orbits of computational anatomy consist of anatomical shapes and medical images; the anatomical shapes are submanifolds of differential geometry consisting of points, curves, surfaces and subvolumes,. This generalized the ideas of the more familiar orbits of linear algebra which are linear vector spaces. Medical images are scalar and tensor images from medical imaging. The group actions are used to define models of human shape which accommodate variation. These orbits are deformable templates as originally formulated more abstractly in pattern theory.

References

1 2 3 S. Bonnabel, Ph. Martin, and P. Rouchon, “Symmetry-preserving observers,” IEEE Transactions on Automatic and Control, vol. 53, no. 11, pp. 2514–2526, 2008.
↑ 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.
1 2 Ph. Martin and E. Salaün, "An invariant observer for Earth-velocity-aided attitude heading reference systems", 17th IFAC World Congress, pp. 9857-9864, 2008.
1 2 Ph. Martin and E. Salaün, "Design and implementation of a low-cost observer-based Attitude and Heading Reference System", Control Engineering Practice, 2010.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Bonnabel_tac-1] 1 2 3 S. Bonnabel, Ph. Martin, and P. Rouchon, “Symmetry-preserving observers,” IEEE Transactions on Automatic and Control, vol. 53, no. 11, pp. 2514–2526, 2008.

[Bonnabel_cdc09-2] 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.

[Martin_ifac-3] 1 2 Ph. Martin and E. Salaün, "An invariant observer for Earth-velocity-aided attitude heading reference systems", 17th IFAC World Congress, pp. 9857-9864, 2008.

[Martin_cep-4] 1 2 Ph. Martin and E. Salaün, "Design and implementation of a low-cost observer-based Attitude and Heading Reference System", Control Engineering Practice, 2010.

[1]

[2]

[3]

[4]