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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.4
>> [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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.5
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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.6
>> 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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.7
>> % 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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.8
>> 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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.9
>> 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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.10
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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.11
>> % 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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.12
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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.13
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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.14
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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.15
>> 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
UNIVERSIDAD RICARDO PALMA INGENIERIA DE CONTROL I
_______________________________________________________________________________ DR.ING. FREEDY SOTELO V. PAG.16
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)