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- Chương 23: LÖnh LTIFR
a) C«ng dông:
§¸p øng tÇn sè cña hÖ tuyÕn tÝnh bÊt biÕn.
b) Có ph¸p:
ltifr(a,b,s)
c) Gi¶i thÝch:
LÖnh ltifr dïng ®Ó më réng ®¸p øng tÇn sè cña hÖ kh«ng gian
tr¹ng th¸i tuyÕn tÝnh bÊt biÕn.
G = Ltifr(a,b,s) t×m ®¸p øng tÇn sè cña hÖ thèng víi mét ngâ
vµo duy nhÊt :
G(s) = (sI – A)-1B
Vector s chØ ra sè phøc mµ t¹i ®ã ®¸p øng tÇn sè ®-îc x¸c
®Þnh. §èi víi ®¸p øng gi¶n ®å Bode hÖ liªn tôc, s n»m trªn trôc ¶o.
§èi víi ®¸p øng gi¶n ®å Bode hÖ gi¸n ®o¹n, s nhËn c¸c gi¸ trÞ
quanh vßng trßn ®¬n vÞ.
ltifr t¹o ra ®¸p øng tÇn sè d-íi d¹ng ma trËn phøc G víi sè
cét b»ng sè tr¹ng th¸i hay sè hµng cña ma trËn A vµ cã sè hµng lµ
length(s).
- C¸C BµI TËP VÒ §¸P øNG TÇN Sè
Bµi 1: hµm margin (bµi tËp nµy trÝch tõ trang 11-138 s¸ch
‘Control System Toollbox’
» hd=tf([0.04798 0.0464],[1 -1.81
0.9048],0.1)
Transfer function:
0.04798 z + 0.0464
---------------------
z^2 - 1.81 z + 0.9048
Sampling time: 0.1 ; Thêi gian lÊy mÉu: 0,1
» [Gm,Pm,Wcg,Wcp]=margin(hd);
» [Gm,Pm,Wcg,Wcp]
ans =
2.0517 13.5712 5.4374 4.3544
» margin(hd)
KÕt qu¶:
- B ode Diagram s
Gm = 6.2424 dB (at 5.4374 rad/s ec), P m = 13.571 deg. (at 4.3544 rad/s ec )
20
0
-20
P has e (deg); M agnitude (dB )
-40
-60
-80
0
-100
-200
-300
10 1
Frequency (rad/sec )
Bµi 2: lÖnh modred (bµi tËp nµy trÝch tõ trang 11-142 s¸ch
‘Control System Toollbox’
s3 11s 2 36 s 26
h( s )
s 4 14,6 s3 74,96 s 2 153,7 s 99,65
» h=tf([1 11 36 26],[1 14.6 74.96 153.7
99.65])
Transfer function:
s^3 + 11 s^2 + 36 s + 26
--------------------------------------------
s^4 + 14.6 s^3 + 74.96 s^2 + 153.7 s + 99.65
» [hb,g]=balreal(h)
- a =
x1 x2
x3 x4
x1 -3.6014 -0.82121
-0.61634 -0.058315
x2 0.82121 -0.59297
-1.0273 -0.090334
x3 -0.61634 1.0273
-5.9138 -1.1272
x4 0.058315 -0.090334
1.1272 -4.4918
b =
u1
x1 1.002
x2 -0.10641
x3 0.086124
x4 -0.0081117
c =
x1 x2
x3 x4
y1 1.002 0.10641
0.086124 0.0081117
d =
u1
y1 0
Continuous-time model.
g =
- 0.1394
0.0095
0.0006
0.0000
» g'
ans =
0.1394 0.0095 0.0006 0.0000
» hmdc=modred(hb,2:4,'mdc')
a =
x1
x1 -4.6552
b =
u1
x1 1.1392
c =
x1
y1 1.1392
d =
u1
y1 -0.017857
Continuous-time model.
» hdel=modred(hb,2:4,'del')
- a =
x1
x1 -3.6014
b =
u1
x1 1.002
c =
x1
y1 1.002
d =
u1
y1 0
Continuous-time model.
» bode(h,'-',hmdc,'x',hdel,'*')
KÕt qu¶:
- B ode Diagram s
From: U(1)
0
-20
P has e (deg); M agnitude (dB )
-40
-60
-80
0
-50
To: Y (1)
-100
-150
-200
10 -1 10 0 10 1 10 2 10 3
Frequency (rad/sec )
Bµi 3: (Trang 11-16 s¸ch ‘Control System Toollbox’)
Xem zero-pole-gain (zero-cùc-®é lîi) cña hÖ thèng sau:
» sys=zpk([-10 -20.01],[-5 -9.9 -20.1],1)
Zero/pole/gain:
(s+10) (s+20.01)
----------------------
(s+5) (s+9.9) (s+20.1)
»
» [sys,g]=balreal(sys)
a =
x1 x2
x3
x1 -4.9697 0.2399
-0.22617
- x2 -0.2399 -4.2756
9.4671
x3 -0.22617 -9.4671
-25.755
b =
u1
x1 1
x2 0.024121
x3 0.022758
c =
x1 x2
x3
y1 1 -0.024121
0.022758
d =
u1
y1 0
Continuous-time model.
g =
0.1006
0.0001
0.0000
» g'
ans =
- 0.1006 0.0001 0.0000
» sysr=modred(sys,[2 3],'del')
a =
x1
x1 -4.9697
b =
u1
x1 1
c =
x1
y1 1
d =
u1
y1 0
Continuous-time model.
» zpk(sysr)
Zero/pole/gain:
1.0001
--------
(s+4.97)
» bode(sys,'-',sysr,'x')
- B ode Diagram s
From: U(1)
-10
-20
P has e (deg); M agnitude (dB )
-30
-40
-50
0
-20
-40
To: Y (1)
-60
-80
-100
10 0 10 1 10 2
Frequency (rad/sec )
Bµi 4: TrÝch tõ trang 55 s¸ch ‘H-íng dÉn sö dông MATLAB’
t¸c gi¶ NguyÔn V¨n Gi¸p.
VÏ biÓu ®å nyquist cña hÖ thèng:
H(s) = (s+4)/(s2 + 3s – 8)
» num=[1 4];
» den=[1 3 -8];
» nyquist(num,den);
- Nyquist Diagram s
From: U(1)
0.3
0.2
0.1
Im aginary A x is
0
To: Y (1)
-0.1
-0.2
-0.3
-0.4
-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0
Real A x is
BaØi 5: TrÝch trang 11-147 s¸ch ‘Control System Toolbox’
VÏ ®¸p øng Nichols cña hÖ thèng cã hµm truyÒn:
4 s 4 48s 3 18s 2 250 s 600
H ( s)
s 4 30 s 3 282 s 2 525s 60
» H=tf([-4 48 -18 250 600],[1 30 282 525
60])
Transfer function:
-4 s^4 + 48 s^3 - 18 s^2 + 250 s + 600
--------------------------------------
s^4 + 30 s^3 + 282 s^2 + 525 s + 60
Nichols(H)
ngrid
- Nichols Charts
From: U(1)
20
15
10
Open-Loop Gain (dB )
5
To: Y (1)
0
-5
-10
-15
-600 -500 -400 -300 -200 -100 0
Open-Loop P hase (deg)
Bµi 6: Trang 131 s¸ch ‘øng dông MATLAB trong ®iÒu khiÓn tù
®éng’ t¸c gi¶ NguyÔn V¨n Gi¸p.
Trªn gi¶n ®å Nichols vÏ ®-êng cong logarit biªn ®é – pha cña
hµm truyÒn hÖ thèng
k
H(s) =
S3+52s2+100s
» k=438;
» num=k;
» den=[1 52 100 0];
» w=.1:.1:10;
» [mag,phase]=bode(num,den,w);
» ngrid,
KÕt qu¶:
- 40
0 dB
30 0.25 dB
0.5 dB
20 1 dB -1 dB
Open-Loop Gain (dB )
10 3 dB
-3 dB
6 dB
0 -6 dB
-10 -12 dB
-20 -20 dB
-30
-40 -40 dB
-350 -300 -250 -200 -150 -100 -50 0
Open-Loop P hase (deg)
nguon tai.lieu . vn
