Waldemar Wodecki
Daniel Zając

III MM-DI, L14

Historia operacji w Matlabie

Kryterium Hurwitza:

s=tf('s')

G1=(1.1/(0.9*s+1));

G3=1/(0.02*s+1);

G2=2/(9.8*s+1);

p=[0.1]

G4=p*G1*G2;

Gz1=G4/(1+G3*G4)

Transfer function:

0.03881 s^3 + 1.987 s^2 + 2.358 s + 0.22

--------------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 134.5 s^2 + 23.77 s + 1.22

A2=[81.57 1.556; 134.5 191.4]

A2 =

81.5700 1.5560

134.5000 191.4000

det(A2)

ans =1.5403e+004

A3=[81.57 1.556 0; 134.5 191.4 81.57; 1.22 23.77 134.5]

A3 =

81.5700 1.5560 0

134.5000 191.4000 81.5700

1.2200 23.7700 134.5000

det(A3)

ans =1.9137e+006

A4=[81.57 1.556 0 0; 134.5 191.4 81.57 1.556; 1.22 23.77 134.5 191.4; 0 0 1.22 23.77]

A4 =

81.5700 1.5560 0 0

134.5000 191.4000 81.5700 1.5560

1.2200 23.7700 134.5000 191.4000

0 0 1.2200 23.7700

det(A4)

ans =4.1896e+007

Układ stabilny

p=[1]

G4=p*G1*G2;

Gz2=G4/(1+G3*G4)

Transfer function:

0.3881 s^3 + 19.87 s^2 + 23.58 s + 2.2

-----------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 152 s^2 + 44.96 s + 3.2

A22=[81.57 1.556; 152 191.4]

det(A22)

ans = 1.5376e+004

A23=[81.57 1.556 0; 152 191.4 81.57; 3.2 44.96 152]

det(A23)

ans = 2.0384e+006

A24=[81.57 1.556 0 0; 152 191.4 81.57 1.556; 3.2 44.96 152 191.4; 0 0 3.2 44.96]

det(A24)

ans = 8.2248e+007

Układ stabilny

p=[2]

G4=p*G1*G2;

Gz3=G4/(1+G3*G4)

Transfer function:

0.7762 s^3 + 39.75 s^2 + 47.17 s + 4.4

------------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 171.4 s^2 + 68.5 s + 5.4

A32=[81.57 1.556; 171.4 191.4]

det(A32)

ans = 1.5346e+004

A33=[81.57 1.556 0; 171.4 191.4 81.57; 5.4 68.5 171.4]

det(A33)

ans = 2.1752e+006

A34=[81.57 1.556 0 0; 171.4 191.4 81.57 1.556; 5.4 68.5 171.4 191.4; 0 0 5.4 68.5]

det(A34)

ans = 1.3319e+008

Układ stabilny

p=[10]

G4=p*G1*G2;

Gz4=G4/(1+G3*G4)

Transfer function:

3.881 s^3 + 198.7 s^2 + 235.8 s + 22

------------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 326.6 s^2 + 256.8 s + 23

A42=[81.57 1.556; 326.6 191.4]

det(A42)

ans = 1.5104e+004

A43=[81.57 1.556 0; 326.6 191.4 81.57; 23 256.8 326.6]

det(A43)

ans = 3.2273e+006

A44=[81.57 1.556 0 0; 326.6 191.4 81.57 1.556; 23 256.8 326.6 191.4; 0 0 23 256.8]

det(A44)

ans = 7.6303e+008

Układ stabilny

p=[50]

G4=p*G1*G2;

Gz5=G4/(1+G3*G4)

Transfer function:

19.4 s^3 + 993.7 s^2 + 1179 s + 110

-----------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 1103 s^2 + 1198 s + 111

A52=[ 81.57 1.556; 1103 191.4]

det(A52)

ans = 1.3896e+004

A53=[81.57 1.556 0; 1103 191.4 81.57; 111 1198 1103]

det(A53)

ans = 7.3705e+006

A54=[81.57 1.556 0 0; 1103 191.4 81.57 1.556; 111 1198 1103 191.4; 0 0 111 1198]

det(A54)

ans = 8.5515e+009

Układ stabilny

p=[100]

G4=p*G1*G2;

Gz6=G4/(1+G3*G4)

Transfer function:

38.81 s^3 + 1987 s^2 + 2358 s + 220

-----------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 2073 s^2 + 2375 s + 221

A62=[ 81.57 1.556; 2073 191.4]

det(A62)

ans = 1.2387e+004

A63=[81.57 1.556 0; 2073 191.4 81.57; 221 2375 2073]

det(A63)

ans = 9.9037e+006

A64=[81.57 1.556 0 0; 2073 191.4 81.57 1.556; 221 2375 2073 191.4; 0 0 221 2375]

det(A64)

ans = 2.3064e+010

Układ stabilny

p=[150]

G4=p*G1*G2;

Gz7=G4/(1+G3*G4)

Transfer function:

58.21 s^3 + 2981 s^2 + 3538 s + 330

-----------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 3043 s^2 + 3552 s + 331

A72=[ 81.57 1.556; 3043 191.4]

det(A72)

ans = 1.0878e+004

A73=[81.57 1.556 0; 3043 191.4 81.57; 331 3552 3043]

det(A73)

ans =9.5087e+006

A74=[81.57 1.556 0 0; 3043 191.4 81.57 1.556; 331 3552 3043 191.4; 0 0 331 3552]

det(A74)

ans = 3.3235e+010

Układ stabilny

p=[200]

G4=p*G1*G2;

Gz8=G4/(1+G3*G4)

Transfer function:

77.62 s^3 + 3975 s^2 + 4717 s + 440

-----------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 4013 s^2 + 4729 s + 441

A82=[ 81.57 1.556; 4013 191.4]

det(A82)

ans = 9.3683e+003

A83=[81.57 1.556 0; 4013 191.4 81.57; 441 4729 4013]

det(A83)

ans = 6.1857e+006

A84=[81.57 1.556 0 0; 4013 191.4 81.57 1.556; 441 4729 4013 191.4; 0 0 441 4729]

det(A84)

ans = 2.8725e+010

Układ stabilny

p=[230]

G4=p*G1*G2;

Gz9=G4/(1+G3*G4)

Transfer function:

89.26 s^3 + 4571 s^2 + 5424 s + 506

-----------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 4595 s^2 + 5436 s + 507

A92=[81.57 1.556; 4595 191.4]

det(A92)

ans = 8.4627e+003

A93=[81.57 1.556 0; 4595 191.4 81.57; 507 5436 4595]

det(A93)

ans = 2.7810e+006

A94=[81.57 1.556 0 0; 4595 191.4 81.57 1.556; 507 5436 4595 191.4; 0 0 507 5436]

det(A94)

ans = 1.4646e+010

Układ stabilny

p=[249]

G4=p*G1*G2;

Gz10=G4/(1+G3*G4)

Transfer function:

96.63 s^3 + 4949 s^2 + 5872 s + 547.8

-------------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 4964 s^2 + 5883 s + 548.8

A102=[81.57 1.556 ; 4964 191.4]

det(A102)

ans = 7.8885e+003

A103=[81.57 1.556 0; 4964 191.4 81.57; 548.8 5883 4964]

det(A103)

ans = 8.4728e+004

A104=[81.57 1.556 0 0; 4964 191.4 81.57 1.556; 548.8 5883 4964 191.4; 0 0 548.8 5883]

det(A104)

ans = 7.8897e+007

Układ stabilny

p=[250]

G4=p*G1*G2;

Gz11=G4/(1+G3*G4)

Transfer function:

97.02 s^3 + 4969 s^2 + 5896 s + 550

-----------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 4984 s^2 + 5906 s + 551

A112=[81.57 1.556 ; 4984 191.4]

det(A112)

ans = 7.8574e+003

A113=[81.57 1.556 0 ; 4984 191.4 81.57; 551 5906 4984]

det(A113)

ans = -6.5359e+004 < 0

A114=[81.57 1.556 0 0; 4984 191.4 81.57 1.556; 551 5906 4984 191.4; 0 0 551 5906]

det(A114)

ans = -8.0236e+008 < 0

Układ niestabilny

p=[300]

G4=p*G1*G2;

Gz12=G4/(1+G3*G4)

Transfer function:

116.4 s^3 + 5962 s^2 + 7075 s + 660

-----------------------------------------------------------

1.556 s^5 + 81.57 s^4 + 191.4 s^3 + 5954 s^2 + 7083 s + 661

A122=[81.57 1.556 ; 5954 191.4]

det(A122)

ans = 6.3481e+003

A123=[81.57 1.556 0 ; 5954 191.4 81.57 ; 661 7083 5954]

det(A123)

ans = -9.2476e+006 < 0

A124=[81.57 1.556 0 0; 5954 191.4 81.57 1.556; 661 7083 5954 191.4; 0 0 661 7083]

det(A124)

ans = -6.5711e+010 < 0

Układ niestabilny

Kryterium Hurwitza nie pozwala określić, ile pierwiastków równania charakterystycznego ma dodatnią część rzeczywistą, czyli „ile pierwiastków psuje stabilność”, jak również brak jest odpowiedzi na pytanie, jak daleko układ znajduje się od granicy stabilności.

Kryterium Nyquista:

>> nyquist(Gz1)

>> nyquist(Gz2)

>> nyquist(Gz3)

>> nyquist(Gz4)

>> nyquist(Gz5)

>> nyquist(Gz6)

>> nyquist(Gz7)

>> nyquist(Gz8)

>> nyquist(Gz8)

>> nyquist(Gz9)

>> nyquist(Gz10)

>> nyquist(Gz11)

>> nyquist(Gz11)

>> nyquist(Gz12)

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