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Kho st ng dng MATLAB trong iu khin t ng

V GIN BODE, NyQuist, NicholsL THUYT: Gin Bode gm hai th: th logarith bin ca hm truyn v gc pha theo logarith tn s. (mt n v trc honh gi l mt decade). Bin : G(j ) = 20 log10 G(j ) dB Pha : = G(j ) (hay arg G(j )) Gin Bode ca cc khu c bn: * Khu khuch i: Hm truyn t G(s) = K Gin Bode L( ) = 20 lgM( ) = 20 lgK l 1 ng thng song song vi trc honh. * Khu qun tnh bc 1: Hm truyn t G(s) =K Ts +1

(2.22) (2.23)

Biu Bode L( ) = 20 lgM( ) = 20 lgK 20lg T 22 +1 c dc gim 20dB/decade * Khu vi phn bc 1: Hm truyn t G(s) = K(Ts + 1) Gin Bode L( ) = 20 lgM( ) = 20 lgK + 20lg T 22 +1 c dc tng 20dB/decade * Khu tch phn: Hm truyn t G(s) =K s

Gin Bode L( ) = 20 lgM( ) = 20 lgK 20lg * Khu bc 2:2 n Hm truyn t G(s) = 2 2 s + 2 s + n n

Gin Bode L( ) = -20lg

(1 t )

2 2 2

+ 4 2 2 t 2


BI TPBai 1: V gin Bode h thng hi tip n v ca hm truyn vng h sau: G(s) = num = 10; den = [0.1 1 0]; bode(num,den) Kt qu:Bode Di agram s

1 0 s 1+0.1 ) ( s

40 20 Phase (deg); Magnitude (dB) 0 -20 -40

-100 -120 -140 -160 10-1

10

0

10

1

10

2

Frequency (rad/sec)

H thng gm 1 khu khuch i bng 10, mt khu tch phn v mt khu qun tnh bc 1 Tn s gy: 10. | G(jw)| dB = 20dB 20log Ti tn s = 1rad/sec | G(jw)| dB = 20dB v dc 20dB/decade (do khu tch phn). dc 20dB/decade tip tc cho n khi gp tn s ct = 10rad/sec, ti tn s ny ta cng thm 20dB/decade (do khu qun tnh bc nht) v to ra dc -40dB/dec.

Bi 2:10 5 ( s +100 ) G(s) = ( s +1)( s +10 )( s +1000 )


num = 100000*[1 100]; den = [1 1011 11010 10000]; bode(num,den) Kt qu:Bode Diagram s

50

Phase (deg); Magnitude (dB)

0

-50

-50 -100 -150 10-1

10

0

10

1

10

2

10

3

10

4

Frequency (rad/sec)

H thng gm mt khu khuch i 105, mt khu vi phn bc nht v 3 khu qun tnh bc 1. Tn s gy: 1,10,100,1000. | G(jw)| dB| w = 0 = 60dB Ti tn s gy = 1rad/sec c li 60dB v dc 20dB/decade (v khu qun tnh bc 1). dc 20dB/decade c tip tc n khi gp tn s gy = 10rad/sec ti y ta cng thm -20dB/decade(v khu qun tnh bc 1), to ra dc 40dB/dec. dc - 20dB tn s = 100rad/dec (do khu vi phn bc 1). Ti tn s gy = 100rad/sec tng 20dB (v khu vi phn bc 1). To ra dc c dc -20dB. Ti tn s gy = 1000rad/sec gim 20dB (v khu qun tnh bc 1). To ra dc 40dB.

Bi 3: G(s) =10 s(1+ 0.1 )2 s

num = 10; den = [0.01 0.2 1 0 ]; bode(num,den)

Kho st ng dng MATLAB trong iu khin t ng Kt qu:Bode Diagram s

40 20 Phase (deg); Magnitude (dB) 0 -20 -40 -60 -100 -150 -200 -250 10-1

10

0

10

1

10

2

Frequency (rad/sec)

H thng gm mt khu khuch i 10, mt khu tch phn v 1 thnh phn cc kp. Tn s gy: 10. | G(jw)| dB = 20dB 20log Tn s gy nh nht = 0.1 rad/sec ti tn s ny c li 40dB v dc 20dB (do khu tch phn). dc ny tip tc cho ti tn s gy kp = 10. tn s ny s gim 40dB/decade, to ra dc 60dB/dec. Bi 4: G(s) =1 2 (s +1 ) 0 0 s(s +1 s +1 0) )( 0

num = 100*[1 10]; den = [1 101 100 0]; bode(num,den) Kt qu:


Bode Diagram s

50

Phase (deg); Magnitude (dB)

0

-50

-100 -120 -140 -160 10-2

10

-1

10

0

10

1

10

2

10

3

Frequency (rad/sec)

H thng gm mt khu khuch i 100, mt khu tch phn v 2 khu qun tnh bc 1, 1 khu vi phn. Tn s gy: 1,10,100 | G(jw)| dB| w = 0 = 20log10 20log Ta ch xt trc tn s gy nh nht 1decade. Ti tn s gy = 0.1rad/sec c li 40dB v dc 20dB/dec, dc 20dB/dec tip tc cho n khi gp tn s gy = 1rad/sec, ta cng thm 20dB/dec (v khu qun tnh bc 1) v to ra dc 40dB/dec. Ti tn s =10 s tng 20dB/dec (v khu vi phn) to ra dc 20dB/dec, dc 20db/dec c tip tc cho n khi gp tn s gy = 100rad/sec s gim 20dB/dec (v khu qun tnh bc 1) s to dc 40dB/decade. Bi 5: Bi ny trch t trang 11-21 sch Control System Toollbox V gin bode ca h thng hi tip SISO c hm sau: S2+01.s+7.5 H(s) = ----------------------S2+0.12s3+9s2 g=tf([1 0.1 7.5],[1 0.12 9 0 0]); bode(g)

Kho st ng dng MATLAB trong iu khin t ngB o d e D ia g ra m sFro m: U(1 ) 40

20

P h a s e (d e g ); M a g n it u d e (d B )

0

-20

-40 0

-50 To : Y (1 )

-100

-150

-200 1 0 -1

100

101

F re q u e n c y (ra d / s e c )

Bi 6: Trang 11-153 sch Control System Toolbox V gian o bode ca hm ri rc sau, vi thi gian ly mu l: 0,1. z3-2.841z2+2.875z-1.004 H(z) = ---------------------------------z3+2.417z2+2.003z-0.5488 H=tf([1 -2.841 2.875 -1.004],[1 -2.417 2.003 -0.5488],0.1); norm(H) ans = 1.2438 [ninf,fpeak]=norm(H,inf) ninf = 2.5488 fpeak = 3.0844


bode(H)B o d e D ia g ra m sFro m: U(1 ) 10

5

P h a s e (d e g ); M a g n it u d e (d B )

0

-5 100

0 -100

To : Y (1 )

-200 -300

-400 100 101

F re q u e n c y (ra d / s e c )

20*log(ninf) ans = 18.7127

Bi 7: Trch t trang 5-18 sch Control System Toolbox Bi ny cho ta xem cng dng ca lnh chia trc subplot h=tf([4 8.4 30.8 60],[1 4.12 17.4 30.8 60]); subplot(121) Kt qu:


h=tf([4 8.4 30.8 60],[1 4.12 17.4 30.8 60]); subplot(121) bode(h) Kt qu:


h=tf([4 8.4 30.8 60],[1 4.12 17.4 30.8 60]); subplot(222) bode(h) Kt qu:


h=tf([4 8.4 30.8 60],[1 4.12 17.4 30.8 60]); subplot(121) bode(h) subplot(222) bode(h) subplot(224) bode(h) Kt qu:


Biu NicholsL thuyt:

Kho st ng dng MATLAB trong iu khin t ng Cng dng: xc nh n nh v p ng tn s vng kn ca h thng hi tip ta s dng biu Nichols. S n nh c nh gi t ng cong v mi quan h ca li theo c tnh pha ca hm truyn vng h. ng thi p ng tn s vng kn ca h thng cng c xc nh bng cch s dng ng cong bin v di pha vng kn khng i ph ln ng cong bin pha vng h. C php: [mod,phase,puls]= nichols(A,B,C,D); [mod,phase,puls]= nichols(A,B,C,D,ui); [mod,phase]= nichols(A,B,C,D,ui,w); [mod,phase,puls]= nichols(num,den); [mod,phase]= nichols(num,den,w); Nhng cu trc trn cho ln l nhng gi tr t nhin, pha l v vect ca dim tn s l rad/s. S tn ti ca im tn s m p ng tn s c nh gi bng vect w, v ui l bin khai bo vi h thng nhiu ng vo. Ch : + khi s dng lnh nichols vi cu trc khng c bin ng ra th ta c biu nichols. + lnh nichols lun lun cho pha trong khong [-3600,00] Bi 8: cho h thng c hm truyn sau:G s) = 30 ( s 2 + 7s + 1 s (s + 1 )3

Cc bc thc hin: num=30*[1 7 1]; den=[poly([-1 -1 -1]) 0]; hold on, plot(-180,0,'*r'), hold on; nichols(num,den) Tr v biu nichols vi im ti hn critical point (-1800 ,0) c biu din nh hnh sau:

Kho st ng dng MATLAB trong iu khin t ng Hnh: Biu Nichols

DNG BI TP V BIU NYQUYST V KHO ST N NH DNG GIN BODE L THUYT: H thng n nh trng thi h, s n nh trng thi kn nu biu Nyquist khng bao im (-1+i0) trn mt phng phc. H thng khng n nh trng thi h, s n nh trng thi kn nu biu Nyquist bao im (-1+i0)p ln ngc chiu kim ng h (p l s cc GH nm phi mt phng phc). BI TP: T du nhc ca ca s MATLAB, ta nhp: num = [nhp cc h s ca t s theo chiu gim dn ca s m]. den = [nhp cc h s ca mu s theo chiu gim dn ca s m]. nyquist(num,den) Bi 9: GH(s) = num = 10; den = [-1 1]; nyquist(num,den) Kt qu:k (vi k =10, t =1) 1st


Nyquist Diagram s 5 4 3 2 Im aginary Ax is 1 0 -1

(A)

-2 -3 -4 -5 0 2 4 Real Ax is 6 8 10

Nhn xt: hm truyn vng h c 1 cc nm bn phi mt phng phc. Biu Nyquist khng bao im A (-1+j0). im 1 k hiu (+) nm trn trc thc m (Real Axis), im 0 nm trn trc o (Imaginary Axis). Kt lun: h khng n nh. * Dng lnh margin tm bin d tr v pha d tr. T du nhc ca ca s lnh MATLAB ta dng lnh margin: num = 10; den = [-1 1]; margin(num,den);


Bode Diagram s Gm = 0 dB, Pm = 0 (unstable cl osed loop) 20 15 Phase (deg); Magnitude (dB) 10 5 0 80 60 40 20 100

Frequency (rad/sec)

Kt lun: d tr bin (Gm = 0 dB). d tr pha (Pm = 0). Warning: Closed loop is unstable (h vng kn khng n nh). Bi 10: Cho hm ttuyn: GH(s) =k (k = 10, t = 1) s(1st)

num = 10; den = [-1 1 0]; nyquist(num,den)


Nyquist Diagrams 1000 800 600 400 Imaginary Ax is 200 0 -200 -400 -600 -800 -1000 0 2 4 Real Ax is 6 8 10

(A)

Nhn xt: hm truyn vng h c 1 cc nm bn phi mt phng phc v 1 cc nm ti gc ta . Biu Nyquist khng bao im A (-1+j0). im 1 k hiu (+) nm trn trc thc m (Real Axis) , im 0 nm trn trc o (Imaginary Axis). Kt lun: h khng n nh. * Dng lnh margin tm bin d tr v pha d tr. T du nhc ca ca s lnh MATLAB ta dng lnh margin: num = 10; den = [-1 1 0]; margin(num,den)


Bode Diagram s Gm = 0 dB, Pm = 0 (unstable closed loop) 60 40 Phase (deg); Magnitude (dB) 20 0 -20

-20 -40 -60 -80 10-1

10

0

Frequency (rad/sec)

Kt lun: d tr bin (Gm = 0 dB). d tr pha (Pm = 0). Warning: Closed loop is unstable (h vng kn khng n nh). Bi 11: Cho h thng sau k GH(s) = ( t1s +1)( t2s +1) (k =10, t1 = 1, t2 = 2) num = 10; den = [2 3 1]; nyquist(num,den)


Nyquist Diagram s

6

4

Im aginary Ax is

2

0

-2

(A)-4

-6 0 2 4 Real Ax is 6 8 10

Nhn xt: hm truyn vng h c 2 cc nm bn tri mt phng phc. Biu Nyquist khng bao im A (-1+j0). im 1 k hiu (+) nm trn trc thc m (Real Axis) , im 0 nm trn trc o (Imaginary Axis). Kt lun: h thng n nh. * Dng lnh margin tm bin d tr v pha d tr. T du nhc ca ca s MATLAB dng lnh margin. num = 10; den = [2 3 1]; margin(num,den)


Bode Diagram s Gm = Inf, Pm =38.94 deg. (at 2.095 rad/sec) 20 10 Phase (deg); Magnitude (dB) 0 -10 -20

-50 -100 -1500

10

Frequency (rad/sec)

Kt lun: h thng n nh. d tr bin (Gm = ). d tr pha (Pm = 38.94), ti tn s ct bin 2.095 rad/sec. Bi 12: Cho h thng c hm truyn sau: k GH(s) = s( t1s +1)( t2s +1 ) (k = 10 t1=1, t2 =2) num = 10; den = [2 3 1 0]; nyquist(num,den)


Nyquist Diagram s 1000 800 600 400 Imaginary Ax is 200 0 -200 -400 -600 -800 -1000 -30 -25 -20 -15 Real Ax is -10 -5

(A)

Nhn xt: hm truyn vng h c 2 cc nm bn tri mt phng phc v 1 cc zero. Biu Nyquist bao im A(-1+j0). im 1 k hiu (+) nm trn trc thc m (Real Axis) , im 0 nm trn trc o (Imaginary Axis). Kt lun: h khng n nh. * Dng lnh margin tm bin d tr v pha d tr. T du nhc ca ca s MATLAB ta dng lnh margin kim chng li h: num = 10; den = [2 3 1 0]; margin(num,den)


Bode Diagram s Gm = 0 dB, Pm = 0 (unstable closed loop) 60 40 Phase (deg); Magnitude (dB) 20 0 -20 -40

-100 -150 -200 -250 10-1

10

0

Frequency (rad/sec)

Kt lun: h thng khng n nh. d tr bin (Gm = 0 dB). d tr pha (Pm = 0) Bi 12: GH(s) =k s( t1s +1 t2s +1 t3s +1 )( )( )

( t1 =1, t2 = 2, t3 = 3, k = 10)

num = 10; den = [6 11 6 1 0]; nyquist(num,den)


Nyquist Diagram s 1000 800 600 400 Im aginary Ax is 200 0 -200 -400 -600 -800 -1000 -60 -50 -40 -30 Real Ax is -20 -10 0

(A)

Nhn xt: hm truyn vng h c 3 cc nm bn tri mt phng phc v 1 cc zero. Biu Nyquist bao im A (-1+i0). im 1 k hiu (+) nm trn trc thc m (Real Axis) , im 0 nm trn trc o (Imaginary Axis). Kt lun: h khng n nh. * Dng lnh margin tm bin d tr v pha d tr. T du nhc ca ca s MATLAB, dng lnh margin kim chng li h: num = 10; den = [6 11 6 1 0]; margin(num,den)


Bode Diagram s Gm = 0 dB, Pm = 0 (unstable closed loop) 50 Phase (deg); Magnitude (dB) 0 -50

-100

-200

-300-2 -1 0

10

10

10

Frequency (rad/sec)

Kt lun: h thng khng n nh. d tr bin (Gm = 0 dB). d tr pha (Pm = 0).


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