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where the control sequence is known; we derive a linear system of equations,
where the unknown vector is
x
(0) when
k
varies from 0 to
n
(
n
is the dimension
of the state vector):
k−
1
HA
k
x
(0) =
y
(
k
)
HA
k−
1
−j
Bu
(
j
)
.
−
j
=0
The solution of that linear system is unique if and only if the rank of
the matrix [
H
;
...
;
HA
n
]is
n
. In that case, the system (
H
,
A
)issaidtobe
completely observable
.
That concept may be extended to nonlinear dynamical systems [Sontag
1990; Slotine et al. 1991] using differential geometry tools such as Lie brackets,
which are not in the scope of this topic.
4.4.1.2 Filtering State Noise and Reconstructing the State
Tra jectory
When the evolution is not deterministic, uncertainty at time
k
is modeled by
a random vector
v
(
k
). Therefore, the state equation takes the following form:
x
(
k
+1)=
f
[
x
(
k
)
,
u
(
k
)
,
v
(
k
+1)]
.
In the linear additive model, it has the particular form
x
(
k
+1)=
Ax
(
k
)+
Bu
(
k
)+
v
(
k
+1)
.
In the section “Regression modeling of controlled dynamical systems”, we
mentioned that, in that case, the model of the state evolution is a particu-
lar stochastic process, namely a Markov chain. Assume that the state is not
completely observed. Then we define the measurement process through the
following measurement equation:
y
(
k
)=
h
[
x
(
k
)]
,
which takes on the particular form for linear system
y
(
k
)=
Hx
(
k
)
.
In the following, we assume, in a first approach, that the model is linear.
Further, when the nonlinear extension will be considered, we will specify it
explicitly.
To reconstruct the state trajectory, it would be necessary to solve recur-
sively the following linear equation where
v
(
k
+ 1) is the unknown variable:
Hv
(
k
+1)=
y
(
k
+1)
−
HA
x
(
k
)
−
HBu
(
k
)
.
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