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sensor measurements. The basis for the algorithm synthesis is the method
of the exact pole placement [2].
The method of the exact pole placement for solving observation
problems.
Let’s look at a linear multidimensional dynamic system, set in
a state space by the equations of the type [4].
D
x =
A
x +
B
u; y =
C
x
,
(1)
Where
x
∈
R
n
is vector of state;
u
∈
R
r
is input vector;
y
∈
R
m
is
output vector;
R
is the set of real numbers;
D
is the symbol designating
either the differentiating operator
D
x(
t
) = ˙x(
t
)
, or the shift operator
D
x(
t
) = x(
t
+ 1)
.
Let a pair of matrices (
A, C
) be fully observed, i.e. Kalman condition
is fulfilled
rank
C
CA
...
CA
n
−
m
=
n,
then it is possible to build the observer to estimate the state vector of
x
object according to
u
input and
y
output vectors. If the observer forms the
estimation of the total
x
vector, then we speak about full rank observer, if
only some part of this vector is estimated, the observer is called reduced.
Full rank observer is defined by the equation:
D
ˆx = (
A
−
LC
) ˆx +
L
y +
B
u
here
ˆx
∈
R
n
is the observer condition representing the estimation sought
for;
L
is the matrix of the observer feedback.
To solve the observer synthesis problem (the
L
matrix definition) it is
possible to apply any method of modal control. Similar to the work [5],
let’s use the method used in works [2, 6]. Let’s introduce a multilevel
decomposition of the system (1) given by a pair of matrices (
A, C
):
a zero (starting) level
A
0
=
A
T
, B
0
=
C
T
;
(2)
k
th
level
k
= 1
, J
,
J
= ceil
n
m
−
1
A
k
=
B
⊥
k
−
1
A
k
−
1
B
⊥−
k
−
1
, B
k
=
B
⊥
k
−
1
A
k
−
1
B
k
−
1
.
here
B
⊥
i
is annihilator (zero divisor) of the matrix
B
i
,
B
⊥
i
B
i
= 0
;
B
⊥−
i
is 2-semireversible matrix for
B
i
, i.e. the matrix satisfying the regularity
conditions
B
⊥
i
B
⊥−
i
B
⊥
i
=
B
⊥
i
, B
⊥−
i
B
⊥
i
B
⊥−
i
=
B
⊥−
i
.
4
ISSN 0236-3933. HERALD of the BMSTU. Series “Instrument Engineering”. 2014. No. 5