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Spurgeon ( 2008 ). It is interesting to point out that an engineering approach for fast
estimation is to build an electronic analogue realization of a sliding mode observer
( L'Hernault et al. 2008 ).
1.4
Optimal Filtering
Optimal filtering is a class of observers that achieve optimal performance by
minimizing some metrics of the estimation error. Due to the optimality requirement,
the online computational load required for optimal filters is usually higher than that
needed for asymptotic observers.
1.4.1
Kalman Filters
Consider a system with random noise
x D f.x/ C G
w
(1.19)
y D h.x/ C D
v
where w and v are standard white Gaussian noises. Suppose the estimated state is
x.t/
x.t/
.If( 1.19 ) is nonlinear, we linearize it around
x D A.t/x C w
y D C.t/x C v
(1.20)
where
A.t/ D @f
@x .x.t//; C.t/ D @h
@x .x.t//
A Kalman filter based upon the linearization of a nonlinear system is called an
extended Kalman filter (EKF). It includes the estimates of the state variable,
x
,and
P.t/ 2 R n . More specifically, EKF is an
the estimation error covariance matrix,
observer with a dynamic gain
x D f.x/ C K.t/.y y/
P.t/ D A.t/P.t/ C P.t/A T .t/ C Q.t/ P.t/C T .t/R 1 C.t/P.t/
x.t 0 / D x 0 ;P.t 0 / D P 0
y D h.x/
Q.t/ D G.t/G T .t/
R.t/ D D.t/D T .t/
K.t/ D P.t/C T .t/R 1 .t/
 
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