Баландин Д.В., Коган М.М. Использование LMI toolbox пакета Matlab в синтезе законов управления
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X 3×3
W
B
T
0
=
1
0
0
, W
C
0
=
0
1
0
.
X
X =
0.7549 −0.0001 −0.6737
? 1.5655 1.0186
? ? 4.1051
, Y =
1.6050 −0.2044 0.3141
? 0.7878 −0.2290
? ? 0.3520
,
Θ =
−0.1819 0.1197
0.5487 −1.5601
.
˙x
r
= −0.1819x
r
+ 0.1197y ,
u = 0.5487x
r
− 1.5601y ,
A
c
=
0 1 0
−0.5601 0 0.5487
0.1197 0 −0.1819
λ
1,2
= −0.0582 ± 0.7410i , λ
3
= −0.0655 ,
˙x
1
= x
3
,
˙x
2
= x
4
,
˙x
3
= 2x
1
− x
2
+ u ,
˙x
4
= −2x
1
+ 2x
2
,
y
1
= x
1
,
y
2
= x
2
.
Θ
A
r
, B
r
= (B
(1)
r
B
(2)
r
), C
r
, D
r
= (D
(1)
r
D
(2)
r
)
Θ
A
0
=
0 0 1 0 0
0 0 0 1 0
2 −1 0 0 0
−2 2 0 0 0
0 0 0 0 0
, B
0
=
0 0
0 0
0 1
0 0
1 0
, C
0
=
0 0 0 0 1
1 0 0 0 0
0 1 0 0 0
,
X 5 × 5
W
B
T
0
=
1 0 0
0 1 0
0 0 0
0 0 1
0 0 0
, W
C
0
=
0 0
0 0
1 0
0 1
0 0
.
X
X =
3.4330 −5.3960 −0.0027 −0.7563 2.1220
? 13.1102 0.8556 −1.5000 −8.2465
? ? 0.4706 −0.9072 −1.0650
? ? ? 3.2732 3.0418
? ? ? ? 7.0358
,
Y =
3.0358 2.7483 −0.1135 −0.3296 2.4310
? 3.2966 −0.4087 −1.2210 3.5011
? ? 4.9399 1.4611 −0.3288
? ? ? 1.8501 −1.9104
? ? ? ? 4.2886
,
Θ =
−8.4770 −3.8599 11.3614
−22.7817 −21.4799 40.8038
.
˙x
r
= −8.4770x
r
− 3.8599y
1
+ 11.3614y
2
,
u = −22.7817x
r
− 21.4799y
1
+ 40.8038y
2
,
λ
1
= −7.0975, λ
2,3
= −0.2964 ± 2.6608i, λ
4
= −0.7098, λ
5
= −0.0770,
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
W
B
T
0
W
C
0
B
T
0
W
B
T
0
=
0 C
0
W
C
0
= 0
W
B
T
0
=
W
B
T
0
, W
C
0
=
W
C
0
.
A
0
=
A 0
n
x
×k
0
k×n
x
0
k×k
X Y
X =
X
11
X
12
X
T
12
X
22
, Y =
Y
11
Y
12
Y
T
12
Y
22
.
W
T
C
(A
T
X
11
+ X
11
A)W
C
< 0 ,
W
T
B
T
(Y
11
A
T
+ AY
11
)W
B
T
< 0 .
X > 0 Y > 0 X
11
> 0
Y
11
> 0
X
11
Y
11
X Y
XY = I
X
11
Y
11
+ X
12
Y
T
12
= I ,
X
11
Y
12
+ X
12
Y
22
= 0 ,
X
T
12
Y
11
+ X
22
Y
T
12
= 0 ,
X
T
12
Y
12
+ X
22
Y
T
22
= I .
I − X
11
Y
11
= X
12
Y
T
12
.
k
k
(I − X
11
Y
11
) ≤ k .
X
11
− Y
−1
11
= X
12
X
−1
22
X
T
12
.
X
11
− Y
−1
11
≥ 0 ,
X
11
I
I Y
11
≥ 0 .
X
11
> 0 Y
11
> 0
X > 0 Y > 0 X
11
Y
11
X
11
Y
11
X
X
11
− Y
−1
11
X
11
− Y
−1
11
= (U
1
U
2
)
Σ 0
0 0
U
T
1
U
T
2
,
U
1
∈ R
n×r
U
2
∈ R
n×(n−r)
Σ = (λ
1
, ···, λ
r
) > 0
S
Σ 0
r×(k−r)
0
(k−r)×r
I
k−r
S
T
, S = (U
1
U
2
)
I
r
0
r×(k−r)
0
(n−r)×r
0
(n−r)×(k−r)
X
12
= S , X
22
= (λ
−1
1
, ···, λ
−1
r
, 1, ···, 1) .
X
Θ
(n
x
× n
x
)
X
11
= X
T
11
> 0 Y
11
= Y
T
11
> 0
k = n
x
k = n
x
(n
x
×n
x
)
X
11
= X
T
11
> 0 Y
11
= Y
T
11
> 0
W
T
C
(A
T
X
11
+ X
11
A)W
C
< 0 ,
W
T
B
T
(Y
11
A
T
+ AY
11
)W
B
T
< 0 ,
X
11
I
I Y
11
≥ 0 .
X
12
= X
22
= V > 0 V = X
11
−Y
−1
11
X
11
Y
11
X
11
− Y
−1
11
> 0 Θ
X =
X
11
V
V V
, V = X
11
− Y
−1
11
.
˙x
1
= x
3
,
˙x
2
= x
4
,
˙x
3
= 2x
1
− x
2
+ u ,
˙x
4
= −2x
1
+ 2x
2
,
y = x
1
.
Θ
A
(i,j)
r
, i, j = 1, ···, 4 B
(i)
r
, i =
1, ···, 4 C
(j)
r
, j = 1, ···, 4 D
r
Θ A
0
B
0
C
0
X 8 × 8
W
B
T
=
1 0 0
0 1 0
0 0 0
0 0 1
, W
C
=
0 0 0
1 0 0
0 1 0
0 0 1
,
X
11
Y
11
4 × 4 X
12
X
22
X
11
=
1871.7 165.9 −329.8 −51.7
? 544.1 44.2 −340.5
? ? 1434.8 522.1
? ? ? 445.7
,
X
12
=
−0.8125 0.5733 0.0660 −0.0819
−0.1152 0.0103 −0.8299 0.5458
0.5323 0.7412 −0.2679 −0.3090
0.2076 0.3491 0.4849 0.7745
,
X
22
=
0.0005 0 0 0
0 0.0007 0 0
0 0 0.0013 0
0 0 0 0.0917
,
Θ = 10
4
−6.0558 6.3583 1.1637 −58.602 16132.0
−5.6231 5.9042 1.0805 −54.416 14979.0
1.0674 −1.1207 −0.2052 10.329 −2843.0
0.0180 −0.0189 −0.0035 0.175 −48.0
0.0054 −0.0056 −0.0010 0.052 −14.0
.
λ
1
= −1813.1 , λ
2
= −5.4 , λ
3,4
= −0.4 ± 0.9i ,
λ
5,6
= −0.3 ± 0.6i , λ
7
= −1.1, λ
8
= −1.2 ,
Θ
Θ = Ψ, Q, P,
0
Xmin
0
.
Θ =
2.2561 −4.4298 −6.7194 1091.4 691.67
3.2879 −3.3957 −7.2738 982.0 −686.88
−0.4228 0.3313 −0.1676 −185.2 360.80
−0.1479 0.1586 0.0361 −4.5 381.11
−0.0023 0.0029 0.0066 −0.9 −2.16
,
λ
1
= −1.5847 , λ
2
= −1.1216 , λ
3,4
= −0.7534 ± 2.3388i ,
λ
5,6
= −0.5314 ± 1.9128 , λ
7
= −0.2723, λ
8
= −0.2206 .
A
r
= 0 B
r
= 0 C
r
= 0 Θ = D
r
(n
x
×n
x
)
X = X
T
> 0 Y = Y
T
> 0
W
T
C
(A
T
X + XA)W
C
< 0 , X > 0 ,
W
T
B
T
(Y A
T
+ AY )W
B
T
< 0 , Y > 0 .
˙x = Ax + Bu ,
y = Cx ,
x ∈ R
n
x
u ∈ R
n
u
y ∈ R
n
y
˙x
r
= Ax
r
+ Bu + L(Cx
r
− y) ,
u = Kx
r
,
x
r
∈ R
n
x
K L
e = x−x
r
(x, e)
˙x
˙e
=
A + BK −BK
0 A + LC
x
e
.
A + BK A + LC
(A, B) (A
T
, C
T
)
Y
1
A
T
+ AY
1
+ Z
T
1
B
T
+ BZ
1
< 0 ,
Y
2
A + A
T
Y
2
+ Z
T
2
C + C
T
Z
2
< 0
Y
1
= Y
T
1
> 0 Z
1
Y
2
= Y
T
2
> 0 Z
2
K = Z
1
Y
−1
1
, L = Y
−1
2
Z
T
2
.
C n
y
< n
x
n
z
= n
x
− n
y
˙z = F z + T Bu + Qy ,
z ∈ R
n
z
y u
F T Q
T A − F T = QC .
e = z −T x
˙e = F e .
F z
T x y
C =
I
n
y
0
n
y
×n
z
.
A B
A =
A
11
A
12
A
21
A
22
, B =
B
1
B
2
,
A
11
(n
y
×n
y
) B
1
(n
y
×n
u
)
F T Q
F = A
22
+ LA
12
, T = (L I
n
z
), Q = A
21
+ LA
11
− (A
22
+ LA
12
)L ,
L F
(A
T
22
, A
T
12
)
˙x
r
= (A
22
+ LA
12
)x
r
+ (B
2
+ LB
1
)u + [A
21
+ LA
11
− (A
22
+ LA
12
)L]y ,
u = K
1
x
r
+ K
2
y ,
K
1
K
2
x
r
= T x + e
˙x = (A + BK)x − BK
1
e ,
˙e = F e ,
K = (K
2
+ K
1
L K
1
) .
K A + BK
L
K
1
K
2
Y
1
A
T
+ AY
1
+ Z
T
1
B
T
+ BZ
1
< 0 ,
Y
2
A
22
+ A
T
22
Y
2
+ Z
T
2
A
12
+ A
T
12
Z
2
< 0
Y
1
= Y
T
1
> 0 Z
1
Y
2
= Y
T
2
> 0 Z
2
K = (K
2
+ K
1
L K
1
) = Z
1
Y
−1
1
, L = Y
−1
2
Z
T
2
.
˙x
1
= x
3
,
˙x
2
= x
4
,
˙x
3
= 2x
1
− x
2
+ u ,
˙x
4
= −2x
1
+ 2x
2
,
y = x
1
.
C = (1 0 0 0) .
A =
0 | 0 1 0
− − − − −
0 | 0 0 1
2 | −1 0 0
−2 | 2 0 0
, B =
0
−
0
1
0
.
Y
1
=
102.195 71.981 −27.278 −3.674
? 73.319 2.119 −24.064
? ? 203.296 25.975
? ? ? 41.835
,
Z
1
=
−335.704 −96.619 −5.270 −75.512
,
Y
2
=
46.007 −6.289 −27.506
? 339.231 81.770
? ? 44.784
,
Z
2
=
−175.690 −61.945 6.288
,
K =
−18.180 20.162 −4.044 10.707
,
L
T
=
10.372 −2.786 11.598
.
˙x
r
=
0 10.372 1
−1 −2.786 0
2 11.598 0
x
r
+
0
1
0
u +
17.298
4.610
9.567
y ,
u = (20.162 −4.044 10.707)x
r
− 362.730y ,
λ
1
= −0.903 , λ
2,3
= −0.941 ± 2.405i ,
λ
4,5
= −1.457 ± 2.862i , λ
6,7
= −0.564 ± 0.510i ,