In mathematics, Symmetry-preserving observers,[1] 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).
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.
Consider
G
\varphig,\psig,\rhog
g\inG
The nonlinear system
\begin{align}
x&=f(x,u)\\ |
y&=h(x,u) \end{align}
\varphig,\psig,\rhog
\begin{align} X&=f(X,U)\\ |
Y&=h(X,U), \end{align}
where
(X,U,Y)=(\varphig(x),\psig(u),\rhog(y))
The system
\hatx |
=F(\hat{x},u,y)
F(x,u,h(x,u))=f(x,u)
\hatx |
=f(\hatx,u)+C
C
0
\haty=y
\hatX |
=F(\hatX,U,Y)
It has been proved [2] that every invariant observer reads
\hatx |
=f(\hatx,u) +W(\hatx)Ll(I(\hatx,u),E(\hatx,u,y)r)E(\hatx,u,y),
where
E(\hatx,u,y)
\haty-y
W(\hatx)=l(w1(\hatx),..,wn(\hatx)r)
I(\hatx,u)
L(I,E)
Given the system and the associated transformation group being considered, there exists a constructive method to determine
E(\hatx,u,y),W(\hatx),I(\hatx,u)
To analyze the error convergence, an invariant state error
η(\hatx,x)
\hatx-x
I(\hatx,u)
η=\Upsilonl(η,I(\hat |
x,u)r)
To choose the gain matrix
L(I,E)
There has been numerous applications that use such invariant observers to estimate the state of the considered system. The application areas include