C01_ASistemasICI

UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I

_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.3

SISTEMAS LINEALES INVARIANTES EN EL TIEMPO

1. Función de Transferencia Equivalente:

102

10)(

21ss

sG

>> num1=[0 0 10]

num1 = 0 0 10

>> den1=[1 2 10]

den1 = 1 2 10

5

5)(2

ssG

>> num2=[0 5] num2 = 0 5

>> den2=[1 5] den2 = 1 5

>> [num,den]=series(num1,den1,num2,den2) num = 0 0 0 50 den = 1 7 20 50

>> printsys(num,den)

num/den = 50 -------------------------------- s^3 + 7 s^2 + 20 s + 50

>> [num,den]=parallel(num1,den1,num2,den2) num = 0 5 20 100 den = 1 7 20 50

>> printsys(num,den) num/den = 5 s^2 + 20 s + 100

--------------------------------- s^3 + 7 s^2 + 20 s + 50

R(s) C(s)G2(s)

1

Transfer Fcn2

G1(s)

1

Transfer Fcn1

R(s)

C(s)

G2(s)

1

Transfer Fcn2

G1(s)

1

Transfer Fcn1



>> [num,den]=feedback(num1,den1,num2,den2) num = 0 0 10 50 den = 1 7 20 100

>> printsys(num,den) num/den = 10 s + 50

---------------------------------- s^3 + 7 s^2 + 20 s + 100

2. Función de Transferencia Espacio de Estado:

u

x

x

x

y

u

x

x

x

x

x

x

0001

120

25

0

5255

100

010

3

2

1

3

2

1

.

3

.

2

.

1

>> A=[0 1 0; 0 0 1; -5 -25 -5]

A = 0 1 0 0 0 1 -5 -25 -5

>> B=[0; 25; -120]

B = 0 25 -120

>> C=[1 0 0]

C = 1 0 0

>> D=[0]

D = 0

>> [num,den]=ss2tf(A,B,C,D)

num = 0 0.0000 25.0000 5.0000 den = 1.0000 5.0000 25.0000 5.0000

R(s) C(s)

G2(s)

1

Transfer Fcn2

G1(s)

1

Transfer Fcn1



5255

525)(

23 sss

ssG

>> [A,B,C,D]=tf2ss(num,den)

A = -5.0000 -25.0000 -5.0000 1.0000 0 0 0 1.0000 0 B = 1

0 0

C = 0.0000 25.0000 5.0000 D = 0

u

x

x

x

y

u

x

x

x

x

x

x

05250

0

0

1

010

001

5255

3

2

1

3

2

1

.

3

.

2

.

1

3. Respuesta en el tiempo:

254

252)(

2 ss

ssG

>> num=[0 2 25]

>> den=[1 4 25]

>> step(num,den)

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Am

plit

ude



>> step(num,den,10)

>> [y,x,t]=step(num,den,5);

>> size(y)

ans = 126 1

>> size(x) ans = 0 0

>> size(t) ans = 1 126

>> max(t) ans = 5

>> min(t) ans = 0

>> plot(t,y)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Am

plit

ude

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4



>> % Sistema 2 Input - 2 Output.

2

1

2

1

2

1

2

1

2

1

.

2

.

1

00

00

10

01

01

11

05.6

11

u

u

x

x

y

y

u

u

x

x

x

x

>> A=[-1 -1; 6.5 0] A = -1.0000 -1.0000 6.5000 0

>> B=[1 1; 1 0] B = 1 1

1 0

>> C=[1 0; 0 1] C = 1 0

0 1

>> D=[0 0; 0 0] D = 0 0

0 0

>> step(A,B,C,D)

Salidas para u1=step y u2=0 Salidas para u1=0 y u2= step.

-0.4

-0.2

0

0.2

0.4

From: In(1)

To: O

ut(

1)

0 2 4 6 8 10 120

0.5

1

1.5

2

To: O

ut(

2)

From: In(2)

0 2 4 6 8 10 12

Step Response

Time (sec)

Am

plit

ude



>> step(A,B,C,D,1) % Salidas para u1=step y u2=0.

>> step(A,B,C,D,2) % Salidas para u1=0 y u2= step.

>> % Sistema de Segundo Orden

22

2

.2)(

sssG

>> wn=5; >> damping_ratio=0.4;

>> [num0,den]=ord2(wn,damping_ratio)

num0 = 1 den = 1 4 25

254

1)(

2 sssG

-0.4

-0.2

0

0.2

0.4

To: O

ut(

1)

0 2 4 6 8 10 120

0.5

1

1.5

2T

o: O

ut(

2)

Step Response

Time (sec)

Am

plit

ude

-0.2

-0.1

0

0.1

0.2

0.3

To: O

ut(

1)

0 2 4 6 8 10 120

0.5

1

1.5

2

To: O

ut(

2)

Step Response

Time (sec)

Am

plit

ude



>> num=wn^2*num0

num = 25

254

25)(

2 sssG

>> printsys(num,den)

num/den = 25 ---------------------

s^2 + 4 s + 25

4. Respuesta al Impulso:

12,0

1)(

2 sssG

>> num=[0 0 1]; >> den=[1 0.2 1]; >> impulse(num,den)

>> num=[0 1 0]

)(

12,0

1)(

2

s

sssG

>> step(num,den)

0 10 20 30 40 50 60-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Impulse Response

Time (sec)

Am

plit

ude

0 10 20 30 40 50 60-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Step Response

Time (sec)

Am

plit

ude



5. Respuesta al Impulso y a la Rampa:

>> A=[0 1; -1 -1]; >> B=[0; 1]; >> C=[1 0]; >> D=[0]; >> step(A,B,C,D)

>> [num,den]=ss2tf(A,B,C,D) num = 0 0 1.0000 den = 1.0000 1.0000 1.0000

1

1)(

2 sssG

>> step(num,den)

>> impulse(num,den)

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Am

plit

ude

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

Step Response

Time (sec)

Am

plit

ude

0 2 4 6 8 10 12-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Impulse Response

Time (sec)

Am

plit

ude



>> % Respuesta al Impulso por I(t)=Du(t) >> num=[0 1 0]

num = 0 1 0

s

sssG

1

1)(

2

>> step(num,den)

>> % Respuesta a la Rampa Unitaria por R(t)=D-1u(t)

>> num=[0 0 0 1]

num = 0 0 0 1 >> den=[1 1 1 0]

den = 1 1 1 0

ssssG

1

1

1)(

2

>> step(num,den)

0 2 4 6 8 10 12-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Step Response

Time (sec)

Am

plit

ude

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

16

18

20

Step Response

Time (sec)

Am

plit

ude



6. Respuesta a Función Arbitraria:

>> num=[0 0 1] >> den=[1 1 1] >> t=0:.1:10; >> r=t;

>> y=lsim(num,den,r,t); >> plot(t,r,'-',t,y,'o')

>> num=[0 0 1] >> den=[1 1 1] >> t=0:.1:10; >> r=exp(-t);

>> y=lsim(num,den,r,t); >> plot(t,r,'-',t,y,'o')

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10-0.2

0

0.2

0.4

0.6

0.8

1

1.2



7. Fracciones Parciales y Respuesta al Escalón Unitario:

8096408

8072253)(

234

23

ssss

ssssG

>> num=[0 3 25 72 80] >> den=[1 8 40 96 80] >> step(num,den)

sssss

ssssGe

8096408

8072253)(

2345

23

>> nume=[0 0 3 25 72 80] nume = 0 0 3 25 72 80

>> dene=[1 8 40 96 80 0] dene = 1 8 40 96 80 0

)()(

)(.....

)2(

)2(

)1(

)1(

)(

)()( sk

nps

nr

ps

r

ps

r

sA

sBsGe

>> % Residuos, Polos y Termino Directo >> [r,p,k]=residue(nume,dene)

r = -0.2813 - 0.1719i -0.2813 + 0.1719i -0.4375 -0.3750 1.0000

p = -2.0000 + 4.0000i -2.0000 - 4.0000i -2.0000 -2.0000 0

k = []

0 0.5 1 1.5 2 2.5 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Step Response

Time (sec)

Am

plit

ude



8. Lugar Geométrico de las Raíces:

ssssG

23

1)(

23

>> num=[0 0 0 1] >> den=[1 3 2 0] >> rlocus(num,den)

>> num=[0 0 0 1] >> den=[1 3 2 0]

>> [R,K]=rlocus(num,den);

>> size(R) ans = 16 3

>> size(K) ans = 1 16

>> R R = 1.0e+002 *

0 -0.0100 -0.0200 -0.0005 -0.0090 -0.0205 -0.0013 -0.0078 -0.0209 -0.0041 -0.0044 -0.0215 -0.0042 + 0.0000i -0.0042 - 0.0000i -0.0215 -0.0042 + 0.0001i -0.0042 - 0.0001i -0.0215 -0.0041 + 0.0020i -0.0041 - 0.0020i -0.0218 -0.0034 + 0.0056i -0.0034 - 0.0056i -0.0232 -0.0023 + 0.0088i -0.0023 - 0.0088i -0.0254 -0.0007 + 0.0126i -0.0007 - 0.0126i -0.0286 0.0015 + 0.0172i 0.0015 - 0.0172i -0.0329 0.0044 + 0.0228i 0.0044 - 0.0228i -0.0388 0.0083 + 0.0300i 0.0083 - 0.0300i -0.0465 0.0133 + 0.0392i 0.0133 - 0.0392i -0.0567 0.5567 + 0.9816i 0.5567 - 0.9816i -1.1435 Inf Inf Inf

-6 -5 -4 -3 -2 -1 0 1 2-4

-3

-2

-1

0

1

2

3

4

Root Locus

Real Axis

Imagin

ary

Axis



>> K K = 1.0e+006 * 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 1.4561 Inf

9. Lugar Geométrico de las Raíces:

sssa 2)(

>> a=[1 1 0] a = 1 1 0

164)( 2 sssb

>> b=[1 4 16] b = 1 4 16

>> den=conv(a,b)

den = 1 5 20 16 0

sssssb 16205)( 234

>> roots(den)

ans = 0 -2.0000 + 3.4641i -2.0000 - 3.4641i -1.0000

>> num=[0 0 0 1 3] num = 0 0 0 1 3

3)( ssnum

>> roots(num) ans = -3

>> rlocus(num,den)

-10 -8 -6 -4 -2 0 2 4-10

-8

-6

-4

-2

0

2

4

6

8

10

Root Locus

Real Axis

Imagin

ary

Axis



10. Respuesta en Frecuencia:

254

25)(

2 sssG

>> num=[0 0 25] >> den=[1 4 25] >> bode(num,den)

sss

sssG

92.1

98.19)(

23

2

>> num=[0 9 1.8 9] >> den=[1 1.2 9 0] >> bode(num,den)

>> [mag,phase,w]=bode(num,den);

>> size(mag) ans = 53 1

>> size(phase) ans = 53 1

>> size(w) ans = 53 1

>> plot(w,mag) >> mag=20*log10(mag); >> w=logspace(-2,1,53);

-60

-40

-20

0

20

Magnitu

de (

dB

)

10-1

100

101

102

-180

-135

-90

-45

0

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

-30

-20

-10

0

10

20

Magnitu

de (

dB

)

10-1

100

101

102

-90

-45

0

45

90

Phase (

deg)

Bode Diagram

Frequency (rad/sec)

C01_ASistemasICI

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