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x
2
R
n
is the state variable,
y
2
R
p
is the output variable
is a linear system in which
A
2
R
n
n
and
C
2
R
n
p
are known constant or
whose value can be measured,
time varying matrices. Given
A
,
C
, and the past sensor information about
y.t/
,the
problem is to estimate
or a function of the state variable in the presence of noise
and uncertainties. A nonlinear system is defined similarly,
x
x
D
f.x/
y
D
h.x/
x.0/
D
x
0
(1.2)
An immediate question to be answered before observer design is whether
asystem(
1.1
)or(
1.2
) admits a convergent estimator. In other words, how to
determine that the past values of
y.t/
contain adequate information to achieve a
reliable estimate of
x.t/
. This leads to the concept of observability. Two initial
states
x
01
and
x
02
are said to be distinguishable if the outputs
y
1
.t/
and
y
2
.t/
of (
1.2
) satisfying the initial conditions
x
0
D
x
01
and
x
0
D
x
02
differ at some time
t
0
x
02
are distinguishable.
Observability can be easily verified for linear systems. The output of (
1.1
) and its
derivatives at time
. The system is said to be observable if every pair
x
01
,
t
D
0
are
y.0/
D
Cx
0
y.0/
D
CAx
0
y.0/
D
CA
2
x
0
:
:
:
y
.n
1/
.0/
D
CA
n
1
x
0
(1.3)
Obviously, (
1.1
) is observable if the mapping from
y.t/
is one-to-one. In fact, it can be proved that (
1.1
) is observable if and only if the
following observability matrix has full rank
x
0
to the derivatives of
2
3
C
CA
CA
2
:
:
:
CA
n
1
4
5
O
D
For nonlinear systems, the output and its derivatives are given by the iterated Lie
derivatives
y.0/
D
y.x
0
/
y.0/
D
L
f
.h/.x
0
/
D
@h
@x
.x
0
/f.x
0
/
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