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BI 1. PHN TCH H THNG IU KHIN TNG
(tip theo)
Sinh Vin Thc Hin: (Nhm 1Sng tht)
Nguyn nh Dng 09090421
Hunh Gia An - 09206461
Cao Trng Duy Dng - 09216381
3. Kho st h thng dng biu Nyquist
Hm truyn h:
()
( )( )
a/ Khi K= 10, thc hin v biu Nyquist trn MatLAB nh sau:
>> TS=10;MS=conv([1 0.2],[1 8 20]);
>> G=tf(TS,MS)
Transfer function:
10
--------------------------
s^3 + 8.2 s^2 + 21.6 s + 4
>> nyquist(G)
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b/Xc nh tn s ct pha, tn s ct bin, trpha, tr bin:
Do matlab cha h trxc nh trc tip tn s ct pha, tn s ct bin, trpha, tr
bin trn ca s biu Nyquist, nn ta sxc nh bng phng php th cng:
Trc tin ta s v biu Nyquist nh bc 1
Sau nhp lnh sau vng trn n v:
>> hold on
>> tron = linspace(0, 2*pi, 360);
x = cos(tron);
y = sin(tron);
plot(x,y,'m')
axis('equal')
grid on
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a l khong cch tim m pha bng 180on im -1
tr bin:
GM = 20*log10(a) [dB]
Kt qu ca cch tnh ny chmang tnh tng i v gn ging vi cch tnh biu bode, v mt sthng srt kh xc nh.
nhn c kt qu chnh xc ta thc hin lnh sau:
>> [Gm,Pm,wcp,wcg]=margin(G)
Gm = 17.3123 dB
Pm = 103.3350
wcp = 4.6476 rad/s
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wcg = 0.4546 rad/s
c/ Tiu chun n nh biu nyquist:
trng hp ny, dng lnh sau xc nh cc ca hm h:
>> [cuc,zero]=pzmap(G)
cuc =
-4.0000 + 2.0000i
-4.0000 - 2.0000i
-0.2000
zero =
Empty matrix: 0-by-1
Nh vy m=3(Nu nh th sbao im -1+0j s vng l 1.5 vng)
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Tht vy, t biu nyquist ta thy ng cong nyquist bao im -1+j0 ng 1.5 vngtheo chiu dng.
Vy h thng n nh theo tiu chun Nyquist v ging nh tiu chun Bode
d/ khi K=400:
Ta thc hin tng t:
Lnh thc hin:
>> TS=400;MS=conv([1 0.2],[1 8 20]);G=tf(TS,MS)
Transfer function:
400
--------------------------
s^3 + 8.2 s^2 + 21.6 s + 4
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>> nyquist(G)
>> grid on
>> [Gm,Pm,wcp,wcg]=margin(G)%Xc nh chnh xc cc tn s ct v d tr
Warning: The closed-loop system is unstable.
In warning at 26
In DynamicSystem.margin at 60
Gm = 0.4328
Pm = -23.4305
wcp = 4.6476
wcg = 6.7265
Nhn xt: ta thy sau lnh trn nhn c dng cnh bo sau: The closed-loop system isunstable [tm dch l h kn khng n nh] v theo tiu chun bode th l gii vic ny ln gin v Pm[ trpha] < 0.
Tht vy khi xt theo tiu chun Nyquist (lu l nn gii hn biu nyquist bng cchduoble click vo biu , ca s Properties hin ln hy gii hn X thuc [-3,2] v Ythuc [-2,2]) nh vy ta s dxc nh svng cng nh chiu bao im -1+j0 hn.
>> [cuc,zero]=pzmap(G)
cuc =
-4.0000 + 2.0000i
-4.0000 - 2.0000i
-0.2000
zero =
Empty matrix: 0-by-1
Suy ra m=3
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Nhn trn biu ta thy ng Nyquist bao im -1+j0 theo chiu m.
Vy h kn khng n nh
3.1 [Lm thm ] Dng tiu chun Nyquist kho st h thng:
()
( )( )
Nh bi 3.
a & b/ Khi k=10
>> TS=10;MS=conv([1 1],[1 1 4]);G=tf(TS,MS)
Transfer function:
10
---------------------
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s^3 + 2 s^2 + 5 s + 4
>> nyquist(G)
>> [Gm,Pm,wcp,wcg]=margin(G)
Warning: The closed-loop system is unstable.
> In warning at 26
In DynamicSystem.margin at 60
Gm = 0.6000
Pm = -23.8089
wcp = 2.2361
wcg = 2.5641
Nh vy ta cng nhn c cng bo l h thng kn khng n nhnh bi 3. L giitheo bode th ta thy do Pm = -23.8089 < 0 nn h kn skhng n nh.
c/ Xt theo tiu chun Nyquist
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Tm s cc:
>> [cuc,zero]=pzmap(G)
cuc =
-0.5000 + 1.9365i
-0.5000 - 1.9365i
-1.0000
zero =
Empty matrix: 0-by-1
Suy ra m=3(nu h thng kn n n th ng cong Nyquist se bao im -1+0j 1,5 vngtheo chiu dng.
Nh vi h thng ny th khng phi vy: ng cong Nyquist bao im -1+j0 theo chiudng.
d/ Lm li vi K=400
>> TS=400;MS=conv([1 1],[1 1 4]);G=tf(TS,MS)
Transfer function:
400
---------------------
s^3 + 2 s^2 + 5 s + 4
>> nyquist(G)
>> grid on
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Xc nh tn s cc v tr bin-pha:
>> [Gm,Pm,wcp,wcg]=margin(G)
Warning: The closed-loop system is unstable.
> In warning at 26
In DynamicSystem.margin at 60
Gm = 0.0150
Pm = -74.2451
wcp = 2.2361
wcg = 7.5037
Ta cng nhn c cnh bo: H kn khng n nh : Pm = -74.2451
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Xt c th theo tiu chun n nh Nyquist:
Ta cn phng to hn vi gii hn chiu X l [-2;2] v chiu Y l [-3;3].
>> [cuc,zero]=pzmap(G)
cuc =
-0.5000 + 1.9365i
-0.5000 - 1.9365i
-1.0000
zero =
Empty matrix: 0-by-1
Suy ra m=3(nu h thng kn n n th ng cong Nyquist se bao im -1+0j 1,5 vngtheo chiu dng.
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Nh vi h thng ny th khng phi vy: ng cong Nyquist bao im -1+j0 theo chiudng.
3.2 [Lm thm ] Dng tiu chun Nyquist kho st h thng:
()
( )
a/Khi K=10
Tin hnh nhp hm truyn h:
>> TS=10;MS=conv([2 1],conv([2 1],[2 1]));G=tf(TS,MS)
Transfer function:
10
------------------------
8 s^3 + 12 s^2 + 6 s + 1
>> nyquist(G)
>> grid on
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b/ Tm tn s ct v tr bin
>> [Gm,Pm,wcp,wcg]=margin(G)
Warning: The closed-loop system is unstable.
> In warning at 26
In DynamicSystem.margin at 60
Gm = 0.8001
Pm = -7.0310
wcp = 0.8661
wcg = 0.9541
Vi cnh bo sau khi thc hin lnh, ta bit rng h thng khng n nh Pm = -7.0310
c/ Xt theo tiu chun Nyquist
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Zoom theo gii hn chiu X l [-2;1.5] v chiu Y l [-2;2]
Tip tc thc hin lnh tm cc
>> [cuc,zero]=pzmap(G)
cuc =
-0.5000
-0.5000 + 0.0000i
-0.5000 - 0.0000i
zero =
Empty matrix: 0-by-1
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Theo m=3(nu h thng kn n n th ng cong Nyquist se bao im -1+0j 1,5vng theo chiu dng.
Nh vi h thng ny th khng phi vy: ng cong Nyquist bao im -1+j0 theo chiudng.
d/ Khi K=400
Ta nhp li TS=400
>> nyquist(G)
>> grid on
Tm tn s ct v tr bin:
>> [Gm,Pm,wcp,wcg]=margin(G)
Warning: The closed-loop system is unstable.
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> In warning at 26
In DynamicSystem.margin at 60
Gm = 0.0200
Pm = -66.5991
wcp = 0.8661
wcg = 3.6499
Nh vy h kn skhng n nh do c cnh bo v theo bode th do Pm < 0.
Xt theo tiu chun Nyquist:
Zoom theo gii hn chiu X l [-2;2] v Y l [-2;2]
Tm s cc:
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>> [cuc,zero]=pzmap(G)
cuc =
-0.5000
-0.5000 + 0.0000i
-0.5000 - 0.0000i
zero =
Empty matrix: 0-by-1
Theo m=3(nu h thng kn n n th ng cong Nyquist sbao im -1+0j 1,5vng theo chiu dng.
Nh vi h thng ny th khng phi vy: ng cong Nyquistbao im -1+j0 theo chiudng.
3.3 [Lm thm ] Dng tiu chun Nyquist kho st h thng:
() ( )
( )( )
a/Khi K=10
Tin hnh nhp hm truyn h:
>> TS=conv(10,[1 1]);MS=conv([1 0],conv([1 3],[1 8 20]));G=tf(TS,MS)
Transfer function:
10 s + 10
------------------------
s^3 + 11 s^2 + 44 s + 60
>> nyquist(G)
>> grid on
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b/ Tm tn s ct, tr bin:>> [Gm,Pm,wcp,wcg]=margin(G)
Gm = 32.2178
Pm = 92.4899
wcp = 5.8945
wcg = 0.1686
Nh vy khng c cnh bo no sau khi thc hin lnh v cng theo tiu chun bode thh kn lun n nh.
c/ Xt theo tiu chun Nyquist
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Zoom theo gii hn chiu X l [-0.1;0.1] v Y l [-0.1;0.1]
Tm s cc ca h thng:
>> [cuc,zero]=pzmap(G)
cuc =
0
-4.0000 + 2.0000i
-4.0000 - 2.0000i
-3.0000
zero =
-1
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Theo m=4 (nu h kn n nh th ng cong nyquist s bao theo chiu dng im -1 + 0j 2 vng).
Tht vy, t biu nyquist ta thy ng cong nyquist bao im -1+j0 ng 2 vngtheo chiu dng.
Vy h thng n nh theo tiu chun Nyquist v ging nh tiu chun Bode
d/ Khi K=400
Ta nhp li TS=400(s+1)
>> TS=conv(400,[1 1]);MS=conv([1 0],conv([1 3],[1 8 20]));G=tf(TS,MS)
Transfer function:
400 s + 400
----------------------------
s^4 + 11 s^3 + 44 s^2 + 60 s
>> nyquist(G)
>> grid on
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Xc nh tn s ct, tr bin
>> [Gm,Pm,wcp,wcg]=margin(G)
Warning: The closed-loop system is unstable.
> In warning at 26
In DynamicSystem.margin at 60
Gm = 0.8054
Pm = -7.1198
wcp = 5.8945
wcg = 6.4985
Xut hin cnh bo tc h thng btn nh v cnh bo cng ng theo tiu chun nnh Bode.
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Xt theo tiu chun Nyquist:
Tm s cc ca h thng:
>> [cuc,zero]=pzmap(G)
cuc =
0
-4.0000 + 2.0000i
-4.0000 - 2.0000i
-3.0000
zero =
-1
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Theo m=4(nu h thng kn n n th ng cong Nyquist sbao im -1+0j 2 vngtheo chiu dng.)
Nh vi h thng ny th khng phi vy: ng cong Nyquist bao im -1+j0 theo chiudng.
4. Kho st h thng dng phng php QNS:
Kho st h thng phn hi m n v c hm truyn vng h:
()
( )( )
a/ Tin hnh nhp hm truyn h(lu khng nhp K t s=1)
Th tc nh sau:
>> TS=1;MS=conv([1 3],[1 8 20]);G=tf(TS,MS)
Transfer function:
1
------------------------
s^3 + 11 s^2 + 44 s + 60
Sau vQNS bng hm rlocus()
+ Kgh chnh in giao gia hm truyn v trc o.
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Vy Kgh=426
b/Tm K h thng tn sdao ng t nhin wn = 4.
Ta tm trn QNS ni c gi trFrequency(rad/s) = 4[lu ch cn double click voQNS l thng s s hin ln]. T suy ra gi tr K.
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Vy K=51.4
c/ Tm K h thng c h s tt = 0.7.
Tng tta tm ni c Dampling=0.7
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Nh vy K=20
d/ Tm K h thng c vt l POT = 25%.
Tm ni c Overshoot(%)=25%. T suy ra K
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Vy K=77
e/ Tm K h thng c thi gian xc lp txl = 4s.
Ta bit thi gian xc lp(chn chn s tiu chun 2%)
Vy =1
Trn QHS ta tm im c tch trn bng 1: tng tnh cp s 5*0.2=1:
Nh trn hnh ta xc nh c im c v
Th li ta c
(Con s ny l chp nhn c)
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Tim ny ta c K=171
5. nh gi cht lng ca h thng:
Kho st h thng phn hi m n v c hm truyn vng h:
()
( )( )
a/Tcu trn ta tm c Kgh=426:
vp ng qu ta tm hm truyn kn ca h thng t cng thc sau:
() ()
()
Th tc vi Matlab nh sau:
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>> TS=426;MS=conv([1 3],[1 8 20]);G=tf(TS,MS)
Transfer function:
426
------------------------
s^3 + 11 s^2 + 44 s + 60
>> Gk=feedback(G,1)
Transfer function:
426
-------------------------
s^3 + 11 s^2 + 44 s + 486
>>step(Gk)
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p ng ng ra dao ng
b/Tcu 4d ta tm c K=77:
Thc hin tng t ta c:
>> TS=77;MS=conv([1 3],[1 8 20]);G=tf(TS,MS);Gk=feedback(G,1);step(Gk)
Lu tm vot l chnh xc nht th th tc nh sau:
T ca sp ng qu : Chut phi => Characteristics =>Peak Response, sau pchut vo nh vt l s hin ln.
Nh vy vt ll 20.9% so vi 25% cu 4
c/ Tm c K =171
TS=171;MS=conv([1 3],[1 8 20]);G=tf(TS,MS);Gk=feedback(G,1);step(Gk)
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T ca sp ng qu : Chut phi => Characteristics => Settling Time, sau pchut vo txl s hin ln.
Nh vy txl = 3.47s so vi txl =4s cu 4, iu ny chng t c sai strong lc xc nhtxl.
d/Th tc nh sau:
>> TS=77;MS=conv([1 3],[1 8 20]);G=tf(TS,MS);Gk=feedback(G,1);step(Gk)
>> hold on
>> TS=171;MS=conv([1 3],[1 8 20]);G=tf(TS,MS);Gk=feedback(G,1);step(Gk)
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