Trabalho 2

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3.1- 
a) 
Sem atraso 
 
Com atraso 
 
3.1- 
b) 
Sem atraso 
 
Com atraso 
 
3.1- 
c) 
Sem atraso 
 
Com atraso 
 
 
 
3.1- 
d) 
Sem atraso 
 
 
 
 
 
 
 
 
 
 
 
 
 
M
ag
ni
tu
de
 (d
B)
-150
-100
-50
0
10-2 10-1 100 101 102
Ph
as
e 
(d
eg
)
-270
-180
-90
0
Bode Diagram
Frequency (rad/s)
3.2- 
a) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
0 45 90 135 180 225 270 315 360
-20
-10
0
10
20
30
40
-20 dB
-12 dB
-6 dB
-3 dB
-1 dB
0 dB
0.25 dB
0.5 dB
1 dB
3 dB
6 dB
Nichols Chart
Open-Loop Phase (deg)
O
pe
n-
Lo
op
 G
ai
n 
(d
B)
3.2- 
b) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
-360 -315 -270 -225 -180 -135 -90 -45 0
-80
-60
-40
-20
0
20
40
-80 dB
-60 dB
-40 dB
-20 dB
-12 dB
-6 dB
-3 dB
-1 dB
0 dB
0.25 dB
0.5 dB
1 dB
3 dB
6 dB
Nichols Chart
Open-Loop Phase (deg)
O
pe
n-
Lo
op
 G
ai
n 
(d
B)
 
3.2- 
c) 
 
 
 
 
 
 
 
 
 
 
 
 
 
-180 -90 0 90 180 270 360
-100
-80
-60
-40
-20
0
20
40
-100 dB
-80 dB
-60 dB
-40 dB
-20 dB
-12 dB
-6 dB
-3 dB
-1 dB
0 dB
0.25 dB
0.5 dB
1 dB
3 dB
6 dB
Nichols Chart
Open-Loop Phase (deg)
O
pe
n-
Lo
op
 G
ai
n 
(d
B)
 
3.2- 
d) 
 
 
 
 
 
 
 
 
 
 
 
 
 
-360 -315 -270 -225 -180 -135 -90 -45 0 45 90 135 180
-120
-100
-80
-60
-40
-20
0
20
40
-120 dB
-100 dB
-80 dB
-60 dB
-40 dB
-20 dB
-12 dB
-6 dB
-3 dB
-1 dB
0 dB
0.25 dB
0.5 dB
1 dB
3 dB6 dB
Nichols Chart
Open-Loop Phase (deg)
O
pe
n-
Lo
op
 G
ai
n 
(d
B)
 
3.3- 
a) 
 
 
 
 
 
 
 
 
 
 
 
 
 
Im
ag
in
ar
y A
xi
s
3.3- 
b) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
-1 -0.5 0 0.5 1 1.5 2 2.5
-1.5
-1
-0.5
0
0.5
1
1.5
-4 dB
-6 dB
-2 dB
-10 dB
-20 dB
0 dB
20 dB
10 dB
4 dB
6 dB
2 dB
Nyquist Diagram
Real Axis
Im
ag
in
ar
y A
xi
s
3.3- 
c) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
-1 -0.8 -0.6 -0.4 -0.2 0 0.2
-5
-4
-3
-2
-1
0
1
2
3
4
5
-4 dB
-2 dB
-6 dB
-10 dB
0 dB
10 dB
4 dB6 dB
2 dB
Nyquist Diagram
Real Axis
Im
ag
in
ar
y A
xi
s
3.3- 
d) 
 
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
-10 dB
-20 dB
-6 dB-4 dB-2 dB0 dB2 dB4 dB6 dB10 dB
20 dB
Nyquist Diagram
Real Axis
Im
ag
in
ar
y A
xi
s
 
 
 
 
 
 
num = [1]; 
den = [1 2.5 1]; 
nyquist(num, den) 
grid 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.1- 
Nyquist 4.1 
 
 
 
 
 
 
 
 
 
num = [0.5 1]; 
den = [1 0 0]; 
nyquist(num, den) 
grid 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.2- 
Nyquist 4.2 
 
 
 
 
 
 
 
 
num = [1 10]; 
den = [1 6 10]; 
nyquist(num, den) 
grid 
title('Diagrama de Nyquist de G(s) = (10+s)/(s^2+6s+10)') 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
4.3- 
Nyquist 4.3 
 
 
 
 
 
 
 
 
 
 
 
for i = 1: 1001 
 
 t(i) = 0.8*(i-1)/1000; 
 
 x(i)=atand(2*t(i)*((1/(((4*(t(i)^4)+1)^0.5)-2*(t(i)^2)))^0.5)); 
 
 y(i)= t(i)/0.01; 
end 
 
plot(x,t) 
hold on 
plot(y,t) 
xlabel ('Margem de fase, graus'); 
ylabel ('Relação de amortecimento'); 
 
 
 
 
 
 
5.1- 
 
 
 
 
 
 
 
 
 
 
num=[1]; 
den=[0.2 1.2 1 0]; 
w=logspace(-1,1,400); 
axis([-210,0,-24,36]); 
nichols(num,den,w); 
ngrid 
 
 
 
 
 
 
 
 
 
 
5.2- 
 
 
 
 
 
 
 
 
K=16; 
num=K*[0.0743 -0.04460 0.8920]; d1=[1 1]; d2=[30 1]; 
d3=[1/9 1/3 1];d4=[0.0743 0.4460 0.8920]; 
den=conv(d1,conv(d2,conv(d3,d4))); 
% 
w=logspace(-2,1,400); 
[mag,fase,w]=bode(num,den,w); 
[Mg,Mf,Wcg,Wcp]=margin(mag,fase,w); 
% 
bode(num,den) 
 
 
M
ag
ni
tu
de
 (d
B)
-150
-100
-50
0
50
10-3 10-2 10-1 100 101 102
Ph
as
e 
(d
eg
)
-360
-180
0
180
360
Bode Diagram
Frequency (rad/s)
5.3- 
 
 
 
 
 
num=[0 0 31.5]; d1=[1 1]; d2=[30 1]; 
d3=[1/9 1/3 1]; 
den=conv(d1,conv(d2,d3)); 
% 
w=logspace(-2,1,400); 
[mag,fase,w]=bode(num,den,w); 
[Mg,Mf,Wcg,Wcp]=margin(mag,fase,w); 
% 
bode(num,den) 
 
 
M
ag
ni
tu
de
 (d
B)
-150
-100
-50
0
50
10-3 10-2 10-1 100 101 102
Ph
as
e 
(d
eg
)
-360
-270
-180
-90
0
Bode Diagram
Frequency (rad/s)