2 / 15
x
= (
x
1
, . . . , x
n
)
T
∈
R
n
, u
= (
u
1
, . . . , u
m
)
T
∈
R
m
;
F
(
x
) = (
F
1
(
x
)
, . . . , F
n
(
x
))
T
, G
j
(
x
) = (
G
1
j
(
x
)
, . . . , G
nj
(
x
))
T
;
F
i
(
x
)
, G
ij
(
x
)
∈
C
∞
(
R
n
)
, i
= 1
, n, j
= 1
, m,
which is not linearizable by a feedback, it is required to find such continuous
controls
u
1
=
u
1
(
t
)
, . . . , u
m
=
u
m
(
t
)
,
t
∈
[0
, t
∗
]
that for given time
t
∗
can
transform system (1) from the initial state
x
(0) =
x
0
to the final state
x
(
t
∗
) =
x
∗
.
Transformation of the system to a quasicanonical form.
The
following theorem [11] sets the necessary and sufficient conditions under
which system (1) is transformed to a quasicanonical form
˙
z
i
1
=
z
i
2
;
. . . . . . . . . . . . . . . .
˙
z
i
r
i
−
1
=
z
i
r
i
;
˙
z
i
r
i
=
f
i
(
z
1
, . . . , z
m
, η
) +
m
X
j
=1
g
ij
(
z
1
, . . . , z
m
, η
)
u
j
, i
= 1
, m
;
˙
η
=
q
(
z
1
, . . . , z
m
, η
);
(2)
r
1
+
. . .
+
r
m
=
n
−
ρ, z
i
= (
z
i
1
, . . . , z
i
r
i
)
T
, η
= (
η
1
, . . . , η
ρ
)
T
;
q
(
z
1
, . . . , z
m
, η
) = (
q
1
(
z
1
, . . . , z
m
, η
)
, . . . , q
ρ
(
z
1
, . . . , z
m
, η
))
T
.
In the formulation of the theorem, vector fields are used
F
=
n
X
i
=1
F
i
(
x
)
∂
∂x
i
, G
j
=
n
X
i
=1
G
ji
(
x
)
∂
∂x
i
, j
= 1
, m,
which one-to-one correspond to system (1) in the range of states
R
n
and
the vector fields
ad
0
F
G
j
=
G
j
,
ad
k
F
G
j
= [
F,
ad
k
−
1
F
G
j
]
,
k
= 1
,
2
, . . .
, where
[
X, Y
]
is a commutator of the vector fields
X
and
Y
.
Theorem 1.
For the transformation of the affine system
(1)
on the set
Ω
⊆
R
n
to a quasicanonical form
(2)
it is necessary and sufficient to have
the following features
:
1)
functions
ϕ
i
(
x
)
∈
C
∞
(Ω)
,
i
= 1
, m
,
satisfying the system of the
first-order partial differential equations in the set
Ω
ad
k
F
G
j
ϕ
i
(
x
) = 0
, k
= 0
, r
i
−
2
, i, j
= 1
, m, x
∈
Ω;
2)
functions
ϕ
n
−
ρ
+
l
(
x
)
∈
C
∞
(Ω)
,
l
= 1
, ρ
that for all
x
∈
Ω
G
j
ϕ
n
−
ρ
+
l
(
x
) = 0
, j
= 1
, m, l
= 1
, ρ
ISSN 1812-3368. Herald of the BMSTU. Series “Natural Sciences”. 2014. No. 5
17