Ejercicio 1 (Tranformada inversa de Laplace)

Solapas principales

Enunciado del Ejercicio 1 del examen de la 1 semana de Febrero de Regulacion Automatica I

Solucion:

Vamos a descomponer la transformada de Laplace en fracciones simples:

Descomposicion de la transformada de Laplace en fracciones simples


 

Vamos a calcular a:

Descomposicion de la transformada de Laplace en fracciones simples parte 2


 

Vamos a calcular b y c:

Descomposicion de la transformada de Laplace en fracciones simples parte 3


Descomposicion de la transformada de Laplace en fracciones simples parte 4


Descomposicion de la transformada de Laplace en fracciones simples parte 5


 

Descomposicion de la transformada de Laplace en fracciones simples parte 6


 

Descomposicion de la transformada de Laplace en fracciones simples parte 7


 

valor de b


 

valor de c


 

Sustituyendo a,b y c nos queda:

Descomposicion de la transformada de Laplace en fracciones simples parte 8

 

Descomposicion de la transformada de Laplace en fracciones simples parte 10


 

Descomposicion de la transformada de Laplace en fracciones simples parte 9


 

Calculamos la transformada inversa:

Transformada inversa de Laplace de la descomposicion en fracciones simples


 

Vamos a comprobar los resultados con el scilab:

Programa en Scilab:

s=%s;
num=2*(s+1);
den=s*(s^2+s+2);
t=0:0.1:20;
ft=1-exp(-0.5*t).*cos(sqrt(7)*t/2)+3.*exp(-0.5*t).*sin(sqrt(7)*t/2)/sqrt(7);
g=syslin('c',num/den);
gf=tf2ss(g);
se=pfss(gf)
gt=csim('impulse',t,g);
clf;
subplot(2,1,1);
plot(t,gt); 
xgrid;
subplot(2,1,2);
plot(t,ft);
xgrid;
[ft;gt]
Resultados:
Comprobacion de los resultados con Scilab
->[ft;gt]
ans =
column 1 to 6
0. 0.1993503 0.3949429 0.5834191 0.7618747 0.9278726 0. 0.1993503
 0.3949429 0.5834191 0.7618747 0.9278726
 column 7 to 11
1.0794442 1.2150823 1.333726 1.4347382 1.5178775 1.0794442 1.2150823
 1.333726 1.4347382 1.5178775
column 12 to 16
1.5832649 1.6313473 1.662858 1.6787752 1.6802804 1.5832649 1.6313473
 1.662858 1.6787752 1.6802804 
column 17 to 21
1.6687165 1.6455468 1.6123162 1.5706133 1.5220369 1.6687165 1.6455468
 1.6123162 1.5706133 1.5220369
 column 22 to 26
1.4681639 1.4105222 1.3505656 1.2896535 1.2290344 1.4681639 1.4105222
 1.3505656 1.2896535 1.2290344
 column 27 to 31
1.169832 1.1130366 1.0594982 1.0099241 0.9648791 1.169832 1.1130366
 1.0594982 1.0099241 0.9648791
 column 32 to 36
0.9247879 0.8899411 0.8605016 0.8365142 0.8179158 0.9247879 0.8899411
 0.8605016 0.8365142 0.8179158
 column 37 to 41
0.8045471 0.7961644 0.7924524 0.7930372 0.7974989 0.8045471 0.7961644
 0.7924524 0.7930372 0.7974989
 column 42 to 46
0.8053836 0.8162159 0.8295094 0.8447773 0.8615417 0.8053836 0.8162159
 0.8295094 0.8447773 0.8615417
 column 47 to 51
0.8793415 0.8977401 0.9163308 0.9347422 0.9526414 0.8793415 0.8977401
 0.9163308 0.9347422 0.9526414
 column 52 to 56
0.9697372 0.9857811 1.0005684 1.0139377 1.0257699 0.9697372 0.9857811
 1.0005684 1.0139377 1.0257699
 column 57 to 61
1.0359865 1.0445471 1.0514466 1.056712 1.0603987 1.0359865 1.0445471
 1.0514466 1.056712 1.0603987
 column 62 to 66
1.062587 1.0633778 1.0628892 1.0612521 1.0586069 1.062587 1.0633778
 1.0628892 1.0612521 1.0586069
 column 67 to 71
1.0550999 1.0508797 1.0460944 1.0408885 1.0354012 1.0550999 1.0508797
 1.0460944 1.0408885 1.0354012
 column 72 to 76
1.0297634 1.0240966 1.0185112 1.0131056 1.0079654 1.0297634 1.0240966
 1.0185112 1.0131056 1.0079654
 column 77 to 81
1.003163 0.9987575 0.9947948 0.9913082 0.9883185 1.003163 0.9987575
 0.9947948 0.9913082 0.9883185
column 82 to 86
0.9858353 0.9838575 0.9823746 0.9813678 0.9808108 0.9858353 0.9838575
 0.9823746 0.9813678 0.9808108
column 87 to 91
0.9806714 0.9809125 0.9814934 0.9823706 0.9834994 0.9806714 0.9809125
 0.9814934 0.9823706 0.9834994
column 92 to 96
0.9848342 0.9863302 0.9879435 0.9896324 0.9913576 0.9848342 0.9863302
 0.9879435 0.9896324 0.9913576
column 97 to 101
0.9930828 0.9947753 0.9964060 0.9979498 0.9993856 0.9930828 0.9947753
 0.9964060 0.9979498 0.9993856
column 102 to 106
1.0006965 1.0018694 1.0028951 1.0037683 1.0044867 1.0006965 1.0018694
 1.0028951 1.0037683 1.0044867
column 107 to 111
1.0050515 1.0054666 1.0057384 1.0058752 1.0058874 1.0050515 1.0054666
 1.0057384 1.0058752 1.0058874
column 112 to 116
1.0057865 1.0055853 1.0052971 1.0049357 1.0045149 1.0057865 1.0055853
 1.0052971 1.0049357 1.0045149
column 117 to 121
1.0040484 1.0035493 1.0030303 1.0025031 1.0019785 1.0040484 1.0035493
 1.0030303 1.0025031 1.0019785
column 122 to 126
1.0014663 1.0009749 1.0005117 1.000083 0.9996934 1.0014663 1.0009749
 1.0005117 1.000083 0.9996934
column 127 to 131
0.9993467 0.9990455 0.9987910 0.9985837 0.9984231 0.9993467 0.9990455
 0.9987910 0.9985837 0.9984231
column 132 to 136
0.9983077 0.9982355 0.9982036 0.9982089 0.9982478 0.9983077 0.9982355
 0.9982036 0.9982089 0.9982478 
column 137 to 141
0.9983162 0.9984101 0.9985253 0.9986576 0.9988027 0.9983162 0.9984101
 0.9985253 0.9986576 0.9988027 
column 142 to 146
0.9989569 0.9991161 0.9992770 0.9994363 0.9995912 0.9989569 0.9991161
 0.9992770 0.9994363 0.9995912 
column 147 to 151
0.9997391 0.9998779 1.0000058 1.0001214 1.0002237 0.9997391 0.9998779
 1.0000058 1.0001214 1.0002237 
column 152 to 156
1.000312 1.000386 1.0004456 1.0004911 1.0005229 1.000312 1.000386
 1.0004456 1.0004911 1.0005229 
column 157 to 161
1.0005418 1.0005485 1.0005442 1.00053 1.000507 1.0005418 1.0005485
 1.0005442 1.00053 1.000507 
column 162 to 166
1.0004766 1.0004401 1.0003986 1.0003535 1.000306 1.0004766 1.0004401
 1.0003986 1.0003535 1.000306 
column 167 to 171
1.0002572 1.0002082 1.0001599 1.0001131 1.0000686 1.0002572 1.0002082
 1.0001599 1.0001131 1.0000686 
column 172 to 176
1.0000271 0.9999890 0.9999547 0.9999246 0.9998987 1.0000271 0.9999890
 0.9999547 0.9999246 0.9998987 
column 177 to 181
0.9998773 0.9998602 0.9998474 0.9998387 0.9998339 0.9998773 0.9998602
 0.9998474 0.9998387 0.9998339 
column 182 to 186
0.9998327 0.9998348 0.9998399 0.9998475 0.9998573 0.9998327 0.9998348
 0.9998399 0.9998475 0.9998573 
column 187 to 191
0.9998688 0.9998818 0.9998958 0.9999104 0.9999253 0.9998688 0.9998818
 0.9998958 0.9999104 0.9999253 column 192 to 196
0.9999402 0.9999549 0.9999690 0.9999823 0.9999948 0.9999402 0.9999549
 0.9999690 0.9999823 0.9999948 column 197 to 201
1.0000061 1.0000162 1.0000251 1.0000327 1.0000389 1.0000061 1.0000162
 1.0000251 1.0000327 1.0000389

 

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