Баландин Д.В., Коган М.М. Использование LMI toolbox пакета Matlab в синтезе законов управления
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˙x = Ax + Bu ,
x ∈ R
n
x
u ∈ R
n
u
u = Θx ,
Θ
x = 0
(A, B)
A
˙x = A
c
x , A
c
= A + BΘ
V (x) = x
T
Xx X = X
T
> 0
˙
V = x
T
(A
T
c
X + XA
c
)x < 0 , ∀x 6= 0 .
A
T
X + XA + Θ
T
B
T
X + XBΘ < 0 ,
X
−1
Y = X
−1
Y A
T
+ AY + Y Θ
T
B
T
+ BΘY < 0 , Y > 0 .
(Y, Θ)
Z = ΘY
Y A
T
+ AY + Z
T
B
T
+ BZ < 0 , Y > 0
Y Z (Y, Z)
Θ = ZY
−1
(A, B)
Y Z
Θ = ZY
−1
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0
Ψ = Y A
T
+ AY P = Y Q = B
T
P 6= 0
Θ
W
T
B
T
(Y A
T
+ AY )W
B
T
< 0 , Y > 0 .
W
B
T
B
T
W
B
T
B
T
W
B
T
= 0
Y
Θ
Y
Y
Θ
Y
Y A
T
+ AY −µBB
T
< 0 , Y > 0
Y µ > 0
Θ = −(µ/2)B
T
Y
−1
.
¨ϕ − ϕ = u .
x = (x
1
, x
2
) x
1
= ϕ x
2
= ˙ϕ
˙x
1
= x
2
,
˙x
2
= x
1
+ u ,
A =
0 1
1 0
!
, B =
0
1
!
.
2y
12
y
11
+ y
22
+ z
1
? 2(y
12
+ z
2
)
!
< 0 ,
y
11
y
12
? y
22
!
> 0 ,
Z = (z
1
z
2
)
y
12
< 0 , 4y
12
(y
12
+ z
2
) − (y
11
+ y
22
+ z
1
)
2
> 0 ,
y
11
> 0 , y
11
y
22
− y
2
12
> 0 ,
Y Z
Θ
Y =
90, 9732 −30, 3244
∗ 90, 9732
, Z = (−181, 9464 − 15, 1622) ,
Θ = (−2, 3125 −0, 9375) .
u = −2, 3125ϕ − 0, 9375 ˙ϕ .
W
B
T
B
T
W
B
T
= 0 .
W
B
T
=
1
0
!
.
(1 0)
2y
12
y
11
+ y
22
? 2y
12
1
0
!
= 2y
12
< 0 .
Y > 0 y
12
< 0
Θ
Y =
2 −1
? 1
!
Θ = (−3, 4963 −3, 9925)
u = −3, 4963ϕ − 3, 9925 ˙ϕ
u = Θ
1
ϕ +
Θ
2
˙ϕ s
2
−θ
2
s −(1 + θ
1
)
Θ
1
< −1 Θ
2
< 0
¨ϕ
1
= 2ϕ
1
− ϕ
2
+ u ,
¨ϕ
2
= −2ϕ
1
+ 2ϕ
2
.
x = (x
1
, x
2
, x
3
, x
4
) x
1
= ϕ
1
x
2
= ϕ
2
x
3
= ˙ϕ
1
x
4
= ˙ϕ
2
˙x
1
= x
3
,
˙x
2
= x
4
,
˙x
3
= 2x
1
− x
2
+ u ,
˙x
4
= −2x
1
+ 2x
2
,
A =
0 0 1 0
0 0 0 1
2 −1 0 0
−2 2 0 0
, B =
0
0
1
0
.
Y =
136, 0315 95, 7016 −35, 9087 −4, 6735
95, 7016 97, 7976 3, 4852 −32, 1794
−35, 9087 3, 4852 270, 8664 34, 0661
−4, 6735 −32, 1794 34, 0661 55, 9804
,
Z = (−447, 2277 − 127, 6716 −7, 7261 − 101, 6202) ,
Θ = (−18, 0248 19, 9613 − 4, 0071 10, 5928) ,
Y
Θ = (−17, 7036 19, 5432 − 3, 8653 10, 2930) .
N
˙x = A
i
x + B
i
u, i = 1, . . . , N.
u = Θx
N
N
V
i
(x) = x
T
X
i
x X
i
= X
T
i
> 0
(A
i
+ B
i
Θ)
T
X
i
+ X
i
(A
i
+ B
i
Θ) < 0, i = 1, . . . , N .
N
X
i
= X, i = 1, . . . , N
X
−1
Y = X
−1
Z = ΘY
Y A
T
i
+ A
i
Y + Z
T
B
T
i
+ B
i
Z < 0, i = 1, . . . , N ,
Y = Y
T
> 0 Z Θ = ZY
−1
Ψ
i
+ Q
T
i
ΘP
i
+ P
T
i
Θ
T
Q
i
< 0, i = 1, . . . , N .
B
i
= B, i = 1, . . . , N
Ψ
i
+ Q
T
Θ + Θ
T
Q < 0, i = 1, . . . , N ,
Ψ
i
= A
T
i
X + XA
i
, Q = B
T
X
Θ W
T
Q
Ψ
i
W
Q
< 0, i =
1, . . . , N W
Q
Q W
Q
= X
−1
W
B
T
W
T
B
T
(Y A
T
i
+ A
i
Y )W
B
T
< 0 , i = 1, . . . , N
Y = Y
T
> 0 Θ
Y A
T
i
+A
i
Y +Y Θ
T
B
T
+
BΘY < 0, i = 1, . . . , N, Y
˙x = Ax + Bu ,
y = Cx ,
x ∈ R
n
x
u ∈ R
n
u
y ∈ R
n
y
k
˙x
r
= A
r
x
r
+ B
r
y ,
u = C
r
x
r
+ D
r
y ,
x
r
∈ R
k
k = 0
u = D
r
y
k 6= 0
˙x
c
= A
c
x
c
, A
c
=
A + BD
r
C BC
r
B
r
C A
r
,
x
c
= (x, x
r
)
V (x
c
) = x
T
c
Xx
c
X
T
= X > 0
˙
V < 0 .
A
T
c
X + XA
c
< 0 .
Θ =
A
r
B
r
C
r
D
r
,
A
c
= A
0
+ B
0
ΘC
0
,
A
0
=
A 0
n
x
×k
0
k×n
x
0
k×k
,
B
0
=
0
n
x
×k
B
I
k
0
k×n
u
, C
0
=
0
k×n
x
I
k
C 0
n
y
×k
,
Θ
k
k 6= 0
˙
¯x = A
0
¯x + B
0
¯u ,
¯y = C
0
¯x ,
A
0
B
0
C
0
¯u = Θ¯y
Θ k
A
T
0
X + XA
0
+ C
T
0
Θ
T
B
T
0
X + XB
0
ΘC
0
< 0
X
−1
Y = X
−1
Y A
T
0
+ A
0
Y + Y C
T
0
Θ
T
B
T
0
+ B
0
ΘC
0
Y < 0 , Y > 0 .
Z = ΘC
0
Y
Y A
T
0
+ A
0
Y + Z
T
B
T
0
+ B
0
Z < 0 , Y > 0
Y Z (Y, Z)
Θ
(n
u
+ k)(n
x
+ k) (n
u
+ k)(n
y
+ k)
(n
x
> n
y
)
(Y, Z)
Ψ + P
T
Θ
T
Q + Q
T
ΘP < 0
Ψ = A
T
0
X + XA
0
P = C
0
Q = B
T
0
X
Θ
W
T
B
T
0
X
(A
T
0
X + XA
0
)W
B
T
0
X
< 0 , W
T
C
0
(A
T
0
X + XA
0
)W
C
0
< 0 ,
W
B
T
0
X
W
C
0
B
T
0
X C
0
W
B
T
0
X
= X
−1
W
B
T
0
W
B
T
0
B
T
0
k ≤ n
x
(n
x
+ k) ×
(n
x
+ k) X = X
T
W
T
C
0
(A
T
0
X + XA
0
)W
C
0
< 0 , X > 0 ,
W
T
B
T
0
(X
−1
A
T
0
+ A
0
X
−1
)W
B
T
0
< 0 .
X Θ
Θ
Y = X
−1
X Y
W
T
C
0
(A
T
0
X + XA
0
)W
C
0
< 0 , X > 0 ,
W
T
B
T
0
(Y A
T
0
+ A
0
Y )W
B
T
0
< 0 , Y > 0 .
(n
x
+ k) ×(n
x
+ k)
X = X
T
Y XY = I
˙x
1
= x
2
,
˙x
2
= x
1
+ u ,
y = x
1
.
Θ
A
r
, B
r
, C
r
, D
r
Θ
A
0
=
0 1 0
1 0 0
0 0 0
, B
0
=
0 0
0 1
1 0
, C
0
=
0 0 1
1 0 0
,