Servomec2 - Ativ Estrut 4

Prévia do material em texto

Nome: Jorge Leoncio A. Oliveira
Matrícula: 201202211461
Disciplina: Controle e Servomecanismos II
Professor: Felipe Caetano
1.1)
AV1
]2[3,0]1[6,0][3,0]2[9,0][ −+−+=−− nxnxnxnyny
)(3,0)(6,0)(3,0)(9,0)( 212 zXzzXzzXzYzzY −−− ++=−
)()3,06,03,0()()9,01( 212 zXzzzYz −−− ++=−
9,0²
3,06,0²3,0
9,01
3,06,03,0
)(
)()( 2
21
−
++
=
−
++
==
−
−−
z
zz
z
zz
zX
zY
zH
1 .
-0
Eq. a diferenças finitas
pela tabela
transf. Z
. z²
. z²
transf. Z inversa
-->num = poly([0.3 0.6 0.3], 'z' , 'c')
num =
 2 
 0.3 + 0.6z + 0.3z 
-->den=poly([-0.9 0 1], 'z' , 'c')
den =
 2 
 - 0.9 + z 
-->s1 = syslin('d' , num , den)
s1 =
 2 
 0.3 + 0.6z + 0.3z 
 ----------------- 
 2 
 - 0.9 + z 
-->x = [zeros(1,3), 1 ,zeros(1,10)];
-->y = flts(x , s1)
y =
 column 1 to 11
 0. 0. 0. 0.3 0.6 0.57 0.54 0.513 
0.486 0.4617 0.4374 
 column 12 to 14
 0.41553 0.39366 0.373977
Modelagem no SciLab:
→ -3 ≤ x ≤ 10
Atividade Estruturada 4
 
1.2)
n = 0 até n = 8:
n y[n] y[n - 2] x[n] x[n - 1] x[n - 2]
0 y[0] = 0,3 y[-2] = 0 x[0] = 1 x[-1] = 0 x[-2] = 0
1 y[1] = 0,6 y[-1] = 0 x[1] = 0 x[0] = 1 x[-1] = 0
2 y[2] = 0,57 y[0] = 0,3 x[2] = 0 x[1] = 0 x[0] = 1
3 y[3] = 0,54 y[1] = 0,6 x[3] = 0 x[2] = 0 x[1] = 0
4 y[4] = 0,513 y[2] = 0,57 x[4] = 0 x[3] = 0 x[2] = 0
5 y[5] = 0,486 y[3] = 0,54 x[5] = 0 x[4] = 0 x[3] = 0
6 y[6] = 0,4617 y[4] = 0,513 x[6] = 0 x[5] = 0 x[4] = 0
7 y[7] = 0,4374 y[5] = 0,486 x[7] = 0 x[6] = 0 x[5] = 0
8 y[8] = 0,41553 y[6] = 0,4617 x[8] = 0 x[7] = 0 x[6] = 0
cond. iniciais: y[-1] = 0; y[-2] = 0
{
{ {
{ { { {
{ { { {
{ { { {
{ { { {
{ { { {
{ { { {
{ { { {
41553,04617,09,0]6[3,0]7[6,0]8[3,0]6[9,0]28[3,0]18[6,0]8[3,0]28[9,0]8[
4374,0486,09,0]5[3,0]6[6,0]7[3,0]5[9,0]27[3,0]17[6,0]7[3,0]27[9,0]7[
4617,0513,09,0]4[3,0]5[6,0]6[3,0]4[9,0]26[3,0]16[6,0]6[3,0]26[9,0]6[
486,054,09,0]3[3,0]4[6,0]5[3,0]3[9,0]25[3,0]15[6,0]5[3,0]25[9,0]5[
513,057,09,0]2[3,0]3[6,0]4[3,0]2[9,0]24[3,0]14[6,0]4[3,0]24[9,0]4[
54,06,09,0]1[3,0]2[6,0]3[3,0]1[9,0]23[3,0]13[6,0]3[3,0]23[9,0]3[
57,03,0)3,09,0(]0[3,0]1[6,0]2[3,0]0[9,0]22[3,0]12[6,0]2[3,0]22[9,0]2[
6,0]1[3,0]0[6,0]1[3,0]1[9,0]21[3,0]11[6,0]1[3,0]21[9,0]1[
3,0]2[3,0]1[6,0]0[3,0]2[9,0]20[3,0]10[6,0]0[3,0]20[9,0]0[
0004617,0
000486,0
000513,0
00054,0
00057,0
0006,0
1003,0
0100
0010
=×=+++=−+−++−=
=×=+++=−+−++−=
=×=+++=−+−++−=
=×=+++=−+−++−=
=×=+++=−+−++−=
=×=+++=−+−++−=
=+×=+++=−+−++−=
=−+++−=−+−++−=
=−+−++−=−+−++−=
xxxyxxxyy
xxxyxxxyy
xxxyxxxyy
xxxyxxxyy
xxxyxxxyy
xxxyxxxyy
xxxyxxxyy
xxxyxxxyy
xxxyxxxyy
321321
321321321
Atividade Estruturada 4
 
2.1)
]1[5,0][]2[81,0]1[765,1][
]1[5,0][]2[81,0]1[)98,08,1(][]1[5,0][]2[81,0]1[
16
cos8,1][
98,0
−+=−+−−
−+=−+−×−⇒−+=−+−





−
nxnxnynyny
nxnxnynynynxnxnynyny
43421
pi
Transformada z:
81,0765,1²
5,0²
81,0765,11
5,01
)(
)()(
)()5,01()()81,0765,11()(5,0)()(81,0)(765,1)(
21
1
121121
+−
+
=
+−
+
==
+=+−⇒+=+−
−−
−
−−−−−−
zz
zz
zz
z
zX
zY
zH
zXzzYzzzXzzXzYzzYzzY
. z²
. z²
-->num = poly([0 , 0.5 , 1], 'z', 'c')
num =
 2 
 0.5z + z 
-->den = poly([0.81 , -1.765 , 1], 'z', 'c')
den =
 2 
 0.81 - 1.765z + z 
-->s1 = syslin('d', num, den)
s1 =
 2 
 0.5z + z 
 ---------------- 
 2 
 0.81 - 1.765z + z 
-->x = [zeros(1,10), 1 ,zeros(1,100)];
Modelagem no SciLab: -->y = flts(x , s1)
y =
 column 1 to 13
 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 2.265 3.187725 
 column 14 to 20
 3.7916846 4.1102661 4.1833551 4.0543063 3.7673329 3.3653545 2.888311 
 column 21 to 27
 2.3719318 1.8469278 1.3385627 0.8665517 0.4452279 0.0839204 - 0.2125151 
 column 28 to 34
 - 0.4430646 - 0.6098719 - 0.7175415 - 0.7724645 - 0.7821913 - 0.7548714 - 0.6987730 
 column 35 to 41
 - 0.6218885 - 0.5316271 - 0.4345922 - 0.3364372 - 0.2417920 - 0.1542488 - 0.0763976 
 column 42 to 48
 - 0.0099002 0.0444082 0.0863996 0.1165247 0.1356824 0.1450944 0.1461889 
 column 49 to 55
 0.1404970 0.1295641 0.1148781 0.0978129 0.0795886 0.0612453 0.0436313 
 column 56 to 62
 0.0274005 0.0130205 0.0007868 - 0.0091579 - 0.0168010 - 0.0222358 - 0.0256375 
 column 63 to 69
 - 0.0272391 - 0.0273107 - 0.0261397 - 0.0240149 - 0.0212131 - 0.0179891 - 0.0145681 
 column 70 to 76
 - 0.0111416 - 0.0078647 - 0.0048565 - 0.0022014 0.0000484 0.0018685 0.0032587
(...)
--> k = -10:100;
--> plot(k , y , '.r')
Atividade Estruturada 4
 
2.2) Resposta Analítica:
{
1]1[5,0]0[]2[81,0]1[765,1]10[5,0]0[]20[81,0]10[765,1]0[
0110
=−++−−−=−++−−−=
321321321
xxyyxxyyy
{ { {
265,25,0765,1]0[5,0]1[]1[81,0]0[765,1]11[5,0]1[]21[81,0]11[765,1]1[
1001
=+=++−−=−++−−−= xxyyxxyyy
321
{ { { {
187725,381,0)265,2765,1(]1[5,0]2[]0[81,0]1[765,1]12[5,0]2[]22[81,0]12[765,1]2[
001265,2
=−×=++−=−++−−−= xxyyxxyyy
{ { { {
3,7916846)265,281,0()187725,3765,1(]3[
]2[5,0]3[]1[81,0]2[765,1]13[5,0]3[]23[81,0]13[765,1]3[
00265,2187725,3
=×−×=
++−=−++−−−=
y
xxyyxxyyy
{ { { {
4,1102661)187725,381,0()3,7916846765,1(]4[
]3[5,0]4[]2[81,0]3[765,1]14[5,0]4[]24[81,0]14[765,1]4[
00187725,33,7916846
=×−×=
++−=−++−−−=
y
xxyyxxyyy
{ { { {
4,1833551)3,791684681,0()4,1102661765,1(]5[
]4[5,0]5[]3[81,0]4[765,1]15[5,0]5[]24[81,0]15[765,1]5[
003,79168464,1102661
=×−×=
++−=−++−−−=
y
xxyyxxyyy
{ { { {
4,0543063)4,110266181,0()4,1833551765,1(]6[
]5[5,0]6[]4[81,0]5[765,1]16[5,0]6[]26[81,0]16[765,1]6[
004,11026614,1833551
=×−×=
++−=−++−−−=
y
xxyyxxyyy
{ { { {
3,7673329)4,183355181,0()4,0543063765,1(]7[
]6[5,0]7[]5[81,0]6[765,1]17[5,0]7[]27[81,0]17[765,1]7[
004,18335514,0543063
=×−×=
++−=−++−−−=
y
xxyyxxyyy
{ { { {
3,3653545)4,054306381,0()3,7673329765,1(]8[
]7[5,0]8[]6[81,0]7[765,1]18[5,0]8[]28[81,0]18[765,1]8[
004,05430633,7673329
=×−×=
++−=−++−−−=
y
xxyyxxyyy
{ { { {
2,8883110)3,767332981,0()3,3653545765,1(]9[
]8[5,0]9[]7[81,0]8[765,1]19[5,0]9[]29[81,0]19[765,1]9[
003,76733293,3653545
=×−×=
++−=−++−−−=
y
xxyyxxyyy
{ { { {
2,3719318)3,365354581,0()2,8883110765,1(]10[
]9[5,0]10[]8[81,0]9[765,1]110[5,0]10[]210[81,0]110[765,1]10[
003,36535452,8883110
=×−×=
++−=−++−−−=
y
xxyyxxyyy
n y[n] y[n-1] y[n-2] x[n] x[n-1]
0 1 0 0 1 0
1 2,265 1 0 0 1
2 3,187725 2,265 1 0 0
3 3,7916846 3,187725 2,265 0 0
4 4,1102661 3,791685 3,187725 0 0
5 4,1833551 4,110266 3,791685 0 0
6 4,0543063 4,183355 4,110266 0 0
7 3,7673329 4,054306 4,183355 0 0
8 3,3653545 3,767333 4,054306 0 0
9 2,8883110 3,365355 3,767333 0 0
10 2,3719318 2,888311 3,365355 0 0
Atividade Estruturada 4