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Багиян А.А. Авиационная Автоматика. Методическое пособие по проведению лабораторных занятий в среде MATLAB & Simulink

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MATLAB & Simulink
•
• 2016
«
30
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06.07.2015 .)
(
).
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:
MATLAB & Simulink /
.
.–
:
,
2016. – 45 .
,
,
«
»
«
,
»,
,
«
»
«
».
©
©
©
2
o-
, 2016
., 2016
:
, 2016
1.
1.1.
1.2.
1.3.
2.
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
3.
4.
5.
............................................................................................................................... 4
.......................................................................................... 4
......................................................... 7
................................................... 13
..................................... 18
.................................................... 27
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................................................................................................ 43
................................................................... 44
3
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MATLAB/Simulink
.
1.
(
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:
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–
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4
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(
)
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–
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:
. 1.
,
,
:
5
. 2.
,
(
. 3).
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. 3.
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(
,
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6
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,
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1.1.
.
,
7
U t
Y t :
Y t
(1)
A U t .
:
,
,
.
,
,
.
.
,
.
,
RLC
,
. 4
.
. 4. RLC
:
i t
d 2u t
dt 2
u t
R
C
1 du t
RC dt
du t
dt
1
u t dt.
L
1 di t
.
C dt
1
u t
LC
.
,
.
8
,
.
X s
x t e st dt ,
L x t
0
t (
x t
s
X s
,
)
j ,
–
X s
x t
, L –
.
:
L
,
L x t
x1 t
L x1 t
L x2 t ,
,
sL x t
L
x2 t
sX s ,
1
L x t
s
x t dt
1
X s .
s
.
Y s
y t
U s
:
u t
Y s
.
U s
W s
(2)
y t
u t ,v t
an y
n
an 1 y
n 1
... a0 y
n
bmu
m
m
ai y i
i 0
bm 1u
bj u
m 1
l
j
j 0
... b0u cl v
ck v k , m, l
l
cl 1v
n.
ai y
i 0
i
m
L
bju
j 0
j
,
l
L
ck v
k 0
9
k
(3)
(4)
:
n
... c0v
k 0
(4)
L
l 1
,
n
i 0
Y s
L y t ,U s
m
ai s iY s
l
b j s jU s
j 0
ck s kV s ,
(5)
k 0
L u t ,V s
L v t .
(5)
Wu s , Wv s
:
u t ,v t
m
Y s
Wu s
U s
bj s j
j 0
n
ai s i
bm s m bm 1s m 1 ... b0
,
an s n an 1s n 1 ... a0
i 0
l
Y s
V s
Wv s
ck s k
k 0
n
ai s i
cl s l cl 1 s l 1 ... c0
.
an s n an 1s n 1 ... a0
i 0
,
.
s,
,
–
,
.
.
,
,
.
.
:
:
. 5.
V s
W1 s U s , Y s
W s
W2 s V s
Y s
U s
:
10
Y s
W1 s W2 s .
W1 s W2 s U s
. 6.
V s
Y s
W1 s U s , Q s
W1 s
W2 s U s , Y s
V s
Q s
Y s
U s
W1 s
W2 s .
W s
W2 s U s
:
. 7.
V s
Y s
U s
Q s , Y s
W1 s U s
W2 s Y s
W1 s V s , Q s
Y s
U s
W s
W2 s Y s
W1 s
.
1 W1 s W2 s
.
.
,
,
,
.
,
,
.
,
.
11
xi , i 1, n
X
X
x1
AX
BU ,
x2 ... xn
T
,
(6)
n n, B –
A –
u1 u2 ... um
n m, U
T
–
.
(6)
.
Y
y1
y2 ... yr
T
:
Y
C
CX
DU ,
(7)
r n
D –
(6)
r m
.
(7)
:
. 8.
(6),
,
«
»,
.
(6)
(7),
sX s
Y s
CX s
AX s
BU s , sI
DU s , Y s
C sI
A X s
A
1
BU s , X s
B D U s
W s
sI
C sI
X,
X s , U s ,Y s –
Y,
I
A
1
BU s ,
A
1
B D,
U
n n,
–
12
W s
C sI
A
1
B D –
,
.
1.2.
,
,
.
.
.
,
,
.
.
,
,
.
:
,
0, t 0,
, t 0.
t
t dt 1.
–
,
,
:
1 t
1, t 0,
0, t 0.
t
1 t
d .
:
u t
at , t 0, a
0, t 0.
const,
at 2 , t
const,
:
u t
0, t
0, a
0.
:
u t, s
13
e st ,
s
j ,
j2
1,
:
u t, s
j
e
t
e t cos t
je t sin t.
,
:
ej
u t,
t
cos t
(
j sin t.
) g t
h t
. 9).
.
h t
.
.
.
,
–
.
. 9.
:
–
,
,
–
,
:
an y
n
an 1 y
n 1
... a0 y
bmu
m
bm 1u
m 1
... b0u.
:
Y s
Y s
L y t ,U s
W s U s ,
L u t ,
W s
bm s m bm 1 s m 1 ... b0
.
an s n an 1 s n 1 ... a0
14
(8)
,
Y s
L g t
1,
U s
y
g t
t .
u
(8)
W s
g t e st dt ,
L g t
(9)
0
.
,
Y s
L h t
U s
y
1 s,
L 1 t
u 1 t .
h t
(8)
L h t
1
,
s
W s
W s
sL h t .
,
(9)
dh t
,
dt
g t
(10)
.
(9)
(10)
,
.
,
.
an y
n
an 1 y
n 1
... a0 y
bmu
m
bm 1u
m 1
... b0u,
bm s m bm 1 s m 1 ... b0
.
an s n an 1 s n 1 ... a0
W s
W j
W j
W j
W j
bm j
m
an j
n
P
bm
an
jQ
–
1
1
j
m 1
... b0
j
n 1
... a0
A
, P
, Q
Im W j
15
–
ej
.
,
Re W j
–
,
A
P2
W j
arctg Q
Q2
–
,
–
P
.
W j
OC (
. 10),
,
A
,
.
,
,
W j
0
,
.
. 10.
W j
–
,
.
L
L
.
20 lg A
(
-1
(11)
).
.
,
10
,
.
lg
.
,
,
16
,
,
,
,
.
.
,
s
(9)
W j
j t
g t e
j :
(12)
dt.
0
t
W j
g t e
0,
g t
0
j t
(12)
(13)
dt.
,
,
,
:
1
2
g t
t
e j td .
W j
0
g
t
1
2
0
W j
1
g t
,
t
cos t
W j
1
P
P
Q
P
(15)
d .
(15),
t
e
2
j t
d .
jQ
j sin t
g t
Q
ej
W j
j t
e
(14)
ej
(14)
cos t d
i
1
Q
Q
P
,
(16)
cos t d .
P
(16)
,
,
:
g t
2
P
0
17
cos t d .
(17)
(10)
(17)
0
2 P
h t
t:
(18)
sin t d .
0
.
1.3.
,
.
,
. 11.
,
,
,
.
,
.
,
,
,
».
.11,
,
. 11,
,
.
»,
«
.
,
,
«
.
. 11.
–
«
»,
:
–
«
», –
, –
,
(
. 12.
. 12).
–
18
,
–
,
,
u t
an y
n
an 1 y
n 1
m
... a0 y
bmu
an y n
an 1 y n
bm 1u
1
m 1
... a0 y
... b0u cl v
fu t
l
cl 1v
l 1
... c0v,
fv t ,
bm s m bm 1 s m 1 ... b0
.
an s n an 1 s n 1 ... a0
W s
Q
v t
an
n
an
n 1
... a0
1
(19)
0.
.
,
,
.
,
Q
,
s
(
).
.
,
g t
t
lim g
t
c–
d
c,
t0
.
.
,
,
.
.
,
19
,
,
,
.
,
i
, i 1, n ,
(19)
:
an
(20)
0, i 1, n,
i
an
3
an
5
an
an
2
an
4
0
an
1
an
3
1
n
0
0
0
0
a0
0
0
0
0
a1
0
0
0
0
a2
a0
:
an 1 , an 2 ,
, a0 ,
,
–
,
n
,
,
0.
n
:
an 1 ,
1
an
2
an
1
an
an
3
an
2
,
3
1
an
3
an
5
4
3
an
an
2
an
0
an
1
an
…
.
,
.
,
.
(
,
. 13)
,
G j
l 2
1, j 0
,
l –
.
. 14.
20
. 13.
. 14.
G j
,
,
1, j 0 .
.
.
:
,
,
,
,
,
,
,
,
.
.
,
,
.
21
,
.
,
.
–
h t
(
1 t
. 15).
. 15.
,
,
,
:
t (
)–
,
5%):
h t
h
t
t .
(21)
%
hmax1 h
h
22
100%.
(22)
20-30%
.
t
1 t
(23)
h .
h t
2
,
T
T–
(24)
.
t –
.
hmax1 h
hmax 2
.
h
(25)
n
h t
t
–
.
.
(19),
.
. 16
:
min Re
i
i
min
Re
i
max Re
i
i
i
.
(26)
,
,
.
i
,
,
i
(
. 16
),
:
max tg
i
max
i
23
i
Im
Re
tg
i
i
max,
.
(27)
(28)
. 16.
,
,
.
,
.
,
.
(
. 17, 18):
Amax A 0 –
M
:
1,1 M
M,
.
,
1,5 .
–
,
,
.
–
0
A
0
0
,
2
A 0 .
2
,
.
,
A 0 :
A
A 0 .
,
24
t
1 2
2
.
(29)
. 17.
L ,
. 18).
. 18.
:
–
,
–
L
.
25
,
(
.
18, ).
,
:
L
20lg
1
, U
U
;
U
(30)
.
. 18, .
:
e t
y t
y t
(31)
y t ,
–
, y t
,
.
–
,
.
:
(
):
t
e2
I
d .
(32)
d .
(33)
0
(
):
t
I
e
0
(
):
t
I
e
(34)
d .
0
(
):
t
e2
I
(35)
d .
0
t .
,
(
).
,
26
.
,
,
t
I
f e
,u
,y
(36)
, d ,
0
f–
,
,
.
2.
,
,
,
bm s m bm 1s m 1 ... b0
an s n an 1 s n 1 ... a0
W s
1
1
1
k , s, , Ts 1,
, T 2 s 2 2 Ts 1, 2 2
,
s
Ts 1
T s 2 Ts 1
k–
,T–
,
(
(37)
–
).
,
,
,
.
,
an y
n
an 1 y
n 1
... a0 y
bmu
m
bm 1u
m 1
... b0u,
.
(37)
,
.
1.
W s
k.
(38)
W s
s.
(39)
W s
1
.
s
(40)
2.
3.
4.
27
W s
5.
(
Ts 1.
(41)
1
.
Ts 1
(42)
)
W s
6.
W s
T 2s2
W s
2 2
2 Ts 1.
(43)
1
.
2 Ts 1
(44)
7.
0,
(44)
,
T s
1 –
,
0
1
–
.
,
.
,
:
W s
e
2.1.
. 19, 20.
h t
. 19.
28
s
.
(45)
L
,
20 log k
,
. 20.
(
)
(
),
,
.
2.2.
. 21, 22.
h t
0
0
t, c
. 21.
(
. 23, ),
. 23, ).
29
L
,
,
. 22.
. 23.
2.3.
. 24, 25.
h t
. 24.
30
L
,
,
. 25.
. 26.
. 26.
:
–
,
–
2.4.
. 27, 28.
h t
. 27.
31
L
,
,
. 28.
2.5.
. 29, 30.
h t
. 29.
L
,
-20
-90 -1
10
100
. 30.
32
,
101
102
. 31.
. 31.
– RC
,
:
–
, –
2.6.
. 32, 33.
h t
. 32.
L
,
40
0
90
0
10 -1
100
. 33.
33
,
101
10 2
2.7.
. 34, 35.
h t
1
0
t, c
. 34.
L
,
,
. 35.
RLC
,
(
m
k
(
. 36.
34
. 36, ).
. 36, ),
2.8.
.
,
,
,
.
,
,
,
.
,
.
:
W j
e
j
cos
h t
1 t
j sin
, L
,
20 lg1 0,
.
3.
.
.
,
bm s m bm 1 s m 1 ... b0
.
an s n an 1 s n 1 ... a0
W s
:
1.
.
2.
h t
3.
.
g t
A
.
4.
L
,
.
5.
Simulink.
.
35
6.
W j
,
0,
.
7.
.
8.
.
.
) Control System Toolbox
MATLAB,
Simulink.
LTI (Linear Time Invariant
Models)-
.
Control System Toolbox
,
.
,
c
LTI-
:
tf([bm, …, b1, b0], [an, …, a1, a0]),
B = [bm, …, b0], A=[an, …, a0] –
.
,
1.
1.
step(<LTI>)
impulse(<LTI>)
freqs(num,den,w)
bode(<LTI>)
pole(<LTI>)
zero(<LTI>)
nyquist(<LTI>)
(
MATLAB
)
,
Simulink,
Simulink Control Design, Continuous, Sources, Sinks,
Simulink
Simulink Library Browser,
View.
,
:
1.
.
2.
MATLAB
3.
tf-
Simulink.
,
.
4.
.
5.
MATLAB (
–
–
,
,
.
36
.1)
6.
pole
zero,
.
7.
Simulink
.
8.
.
9.
.
10.
.
11.
.
,
,
,
,
,
,
.
,
:
–
(
–
,
. 2).
2.
W s
1
2
3
4
W s
W s
W s
1
2
3
4
b3 s 3 b2 s 2 b1s b0
a3 s 3 a2 s 2 a1s a0
a4 s 4
a5 s 5 a4 s 4
a4 s 4
1
2
3
4
b2 s 2 b1s b0
a3 s 3 a2 s 2 a1s a0
1
2
3
4
b1s b0
a3 s 3 a2 s 2 a1s a0
1
2
3
4
b3 s 3 b2 s 2 b1s
a3 s 3 a2 s 2 a1s a0
37
b3
0
3
1
0
b2
8
1
9
0
b1
4
0
7
6
b3
7
11
5
6
b2
5
4
5
2
b1
0
3
4
2
b0
11
1
9
0
b2
8
3
9
1
b1
2
0
4
13
b0
1
5
7
3
a5
1
0
5
1
b1
4
8
1
0
b0
1
4
2
7
a4
1
2
7
4
a4
5
0
1
2
a4
1
7
5
4
a3
1
7
8
0
a3
2
0
9
1
a3
3
4
2
5
a3
2
6
0
5
a2
7
5
11
7
a2
7
6
3
4
a2
7
9
3
4
a2
5
9
2
8
a1
9
6
0
5
a1
9
8
11
1
a1
4
1
6
8
a1
3
5
6
7
a0
1
1
9
4
a0
1
7
8
3
a0
1
5
4
7
a0
0
4
1
3
4.
.
W s
4s 3 3s 2 5s 7
.
3s 4 4 s3 5s 2 2 s 1
.
MATLAB.
1.
LTI-
W
B
A:
>> B = [4 3 5 7]; A = [3 4 5 2 1]; W = tf(B,A)
W =
4 s^3 + 3 s^2 + 5 s + 7
------------------------------3 s^4 + 4 s^3 + 5 s^2 + 2 s + 1
Continuous-time transfer function.
2.
Y s
U s
W s
4 s 3 3s 2 5 s 7
,
3s 4 4s 3 5s 2 2 s 1
Y s 3s 4 4 s3 5s 2 2 s 1
3 y (IV) t
4 y (III) t
5y t
2y t
U s 4 s 3 3s 2 5 s 7 ,
y t
4u (III) t
3.
3u t
step(W) (
5u t
7u t .
. 37):
>> step(W); grid
4.
impulse(W) (
. 38):
>> impulse(W); grid
5.
(
.
39)
(
.
40)
:
>> w = 0.1:0.1:30; h = freqs(B,A,w);
>> Amp = abs(h); phi = angle(h);
>>plot(w,Amp);
grid;
title('AFR');
xlabel('\omega,
rad/sec');
ylabel('Amplitude');
>>plot(w,phi*180/pi);
grid;
title('FFR');
ylabel('\phi, deg.');
38
xlabel('\omega,
rad/sec');
Step Response
12
10
8
6
4
2
0
0
5
10
15
20
25
30
35
40
45
35
40
45
Time (seconds)
. 37.
Impulse Response
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
0
5
10
15
20
25
30
Time (seconds)
. 38.
AFR
14
12
Amplitude
10
8
6
4
2
0
0
5
10
15
20
25
30
, rad/sec
. 39.
6.
(
. 41)
,
>> bode(W); grid;
39
bode(W):
FFR
200
150
100
50
0
-50
-100
-150
-200
0
5
10
15
20
25
30
, rad/sec
Phase (deg)
Magnitude (dB)
. 40.
. 41.
7.
pole(W),
zero(W):
>> pole(W)
ans =
-0.5000 + 0.8660i
-0.5000 - 0.8660i
-0.1667 + 0.5528i
-0.1667 - 0.5528i
>> zero(W)
ans =
0.1674 + 1.2590i
0.1674 - 1.2590i
-1.0848 + 0.0000i
40
8.
:
W s
7 0.9218s 1 0.6199s 2 0.2075s 1
s
2
1
1
.
2
s 1 3s s 1
,
(
. 42).
. 42.
9.
Simulink
. 43)
(
Linear Analysis Tool,
Analysis
. 45.)
Control Design
. 43.
Linear Analysis.
Simulink
10.
(
. 44)
nyquist(W):
Nyquist Diagram
15
0 dB
10
5
2 dB
-2 dB
-4 dB
-6 dB
4 dB 6 dB
0
-5
-10
-15
-8
-6
-4
-2
0
2
Real Axis
. 44.
41
4
6
8
. 45.
,
Simulink
11.
.
:
4 2 0 0
3 5 1 0
4
0 4 2 0
0 3 5 1
,
4 2 0
3 5 1,
3
0 4 2
4 2
2
3 5
,
1
MATLAB:
>> del4 = [4,2,0,0;3,5,1,0;0,4,2,0;0,3,5,1]; del4 = det(del4)
del4 =
12
>> del3 = [4,2,0;3,5,1;0,4,2]; del3 = det(del3)
del3 =
12
>> del2 = [4,2;3,5]; del2 = det(del2)
del2 =
42
4.
14
,
4
12,
12,
3
14,
2
1
4 0,
,
,
(
.
. 45).
.
5.
1.
.
2.
.
3.
.
4.
.
5.
.
6.
.
7.
.
8.
.
9.
.
10.
.
11.
.
12.
.
13.
.
14.
.
15.
,
.
16.
.
17.
.
18.
.
19.
.
20.
.
43
1.
.,
.:
.
«
. –
», 2003. – 752 .
2.
.
.:
.
. –
, 1978. – 736 .
3.
.,
.–
.
.:
4.
.
.
, 1991. – 332 .
.
.
.1.
. –
.:
, 2003. – 288 .
5.
:
5-
.
.1:
,
.
.
.
.
. –
, 2004. – 656 .
6.
.
.
.,
.
2.–
.
.1.
.:
, 1986. – 367 .
7. Dorf R.C., Bishop R.H. Modern Control Systems. – Prentice Hall, 2010. – 1104 pp.
44
.:
MATLAB & Simulink
10.03.2016.
29,7 × 42/2 (A4).
.
.
.- . . 3,5.
.
. . 4,3.
Times New Roman.
2
.
.
45
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