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Note that the theorem guarantees the existence of a local change of coordinates
around an equilibrium. Therefore, the observers are limited in a local neighborhood
of an equilibrium point. Among a large number of publications on the observer
design by achieving linearized error dynamics, we would like to bring up (
Kazantzis
and Kravaris 1998
). In this work, the formulation of the observer design problem
is realized via a system of singular first-order linear partial differential equations
(PDE). The theory is applicable to a larger family of systems than that addressed
in
Krener and Isidori
(
1983
). In fact, after a nonlinear change of coordinates, the
resulting system is not required to have a linear output like in (
1.15
). Another
advantage of the work in
Kazantzis and Kravaris
(
1998
) is that the solution to
the PDEs is locally analytic and this enables the development of a series solution
method, that is programmable using symbolic software packages. In the presence
of noise, some types of output injection, such as a
y
2
term, may result in a biased
EŒ.y
C
n/
2
D
EŒy
2
C
EŒn
2
estimation because
is a random noise.
Other related work includes Zeitz's extended Luenberger observer based upon
a local linearization technique (
Zeitz 1987
). Nonlinear coordinate transformations
have also been employed to transform the nonlinear system to a suitable observer
canonical form, where the observer design problem may be solved (
Bestle and Zeitz
1983
;
Ding et al. 1990
;
Xia and Gao 1989
;
Zheng et al. 2007
).
,where
n
1.3.3
Observers Based on Lyapunov Functions
For systems that do not admit a linear error dynamics, nonlinear observers can
be derived so that its stability is guaranteed by a Lyapunov function. A widely
used approach is based on the high gain observer proved in
Gauthier et al.
(
1992
).
Once again, consider the nonlinear system (
1.13
). Using a single output case as an
example, consider the mapping,
z
D
z
.x/
W
R
n
!
R
n
,definedby
2
4
3
5
h.x/
L
f
h.x/
:
:
:
L
n
f
h.x/
z
.x/
D
(1.17)
R
n
. Under this
transformation, the original system is equivalent to the system in the form
We assume that
z
D
z
.x/
is a diffeomorphism on a region
2
4
3
5
z
z
3
:
:
z
n
.
z
D
(1.18)
z
/
y
D
z
1
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