modelado en matlab y simulink

8/17/2019 Modelado en Matlab y Simulink

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Modelado con Matlab y Simulink

Luis Sánchez



Funciones de Trasferencia en Matlab



Funciones de Trasferencia enMatlab



Ejemplos

Escribir en Matlab las siguientes funciones de

trasferencia:



Conversin de una funcin de trasferencia a

formato !anancia"polo"cero y viceversa



Ejemplo

#ada la funcin de trasferencia $%s&' e(presarla en formato

!anancia"polo"cero)



Cruise Control: Matlab Modeling

*sin! +e,ton-s .nd la,' the !overnin! e/uation for this system

becomes0 %1&

,here u is the force applied) For this e(ample' let-s assume that

m 2 1333 k!4 b 2 53 +)sec6m4 u 2 533 +

The model of the cruise control system is relatively simple) 7f it is assumed that rollin!

resistance and air dra! are proportional to the car-s speed' then the problem is reduced to

the simple mass and damper system sho,n below

bvuvm −=)



Transfer function model

Takin! the Laplace transform of the !overnin! differential e/uationand assumin! zero initial conditions' ,e find the transfer function

of the cruise control system to be0

bms sU

sV s P +

== 1&%&%&%

8e enter the transfer function model into M9TL9: usin! thefollo,in! commands0



Open-loop step response

The open-loop response of the system' ,ithout any feedback

control' to a step input force of 533 +e,tons is simulated inM9TL9: as follo,s0

>> Close all; clear all;

>> m = 1000; b = 50; u = 500;

>> N = [1]; D = [m b]

>> P_cruise = tf(N, D);

>> step(uP_cruise)

Alternativamente

>> s = tf(!s!);

>> P_cruise = 1"(ms#b);

>> step(uP_cruise)



Building the model in simulink

This system ,ill be modeled by summin! the forces actin! on the mass and inte!ratin! the

acceleration to !ive the velocity) ;pen Simulink and open a ne, model ,indo,) First' ,e ,ill

model the inte!ral of acceleration)

∫ = vdt dt

dv

<7nsert an 7nte!rator :lock %from the Continuous library& and dra, lines to and from its input

and output terminals)<Label the input line =vdot= and the output line =v= as sho,n belo,) To add such a label'

double click in the empty space just above the line)



Since the acceleration %dv6dt& is e/ual to the sum of the forces divided by mass' ,e

,ill divide the incomin! si!nal by the mass)<7nsert a $ain block %from the Math ;perations library& connected to the 7nte!rator

block input line and dra, a line leadin! to the input of the $ain block)

<Edit the $ain block by double"clickin! on it and chan!e its value to =16m=)<Chan!e the label of the $ain block to =inertia= by clickin! on the ,ord =$ain=

underneath the block)



<9ttach a Sum block %from the Math ;perations library& to the line leadin! to the inertia

$ain block)

<Chan!e the si!ns of the Sum block to =>"=)<7nsert a $ain block belo, the 7nertia block' select it by sin!le"clickin! on it' and select Flip

from the Format menu %or type Ctrl"F& to flip it left"to"ri!ht)

<Set the block-s value to =b= and rename this block to =dampin!=)<Tap a line %hold Ctrl ,hile dra,in!& off the 7nte!rator block-s output and connect it to the

input of the dampin! $ain block)

<#ra, a line from the dampin! $ain block output to the ne!ative input of the Sum :lock)



The second force acting on the mass is the control input, u. We will apply astep input.•Insert a tep bloc! "from the ources library# and connect it with a line tothe positi$e input of the um %loc!.• To $iew the output $elocity, insert a cope bloc! "from the in!s library#connected to the output of the Integrator.



To provide an appropriate step input of 533 at time e/ual zero' double"click the Step block

and set the Step Time to =3= and the Final ?alue to =u=)

@un the simulation %hit Ctrl"T or select Start from the

Simulation menu&) 8hen the simulation is finished'

double"click on the Scope and hit its autoscale button)

Aou should see the follo,in! output)



Modelado en Simulink de . blo/ues

acoplados

#eterminar las E#;s para los . blo/ues %5 Min&

The system that is bein!

analyzed is sho, in the

follo,in! dia!ram

7n the above' is to be taken as each of the follo,in!1) *nit impulse force)

.) *nit step force)

B) Sin%,t&

7t is re/uired to nd (1 y (. usin! Matlab-s Simulink soft,are for the analysis)



Modelado en Simulink de .blo/ue

acoplados



Modelado en Simulink



DC Motor Speed: Simulink Modeling

9 common actuator in control systems is the #C motor) 7t directly provides rotary motion

and' coupled ,ith ,heels or drums and cables' can provide translational motion) The

electric circuit of the armature and the free"body dia!ram of the rotor are sho,n in the

follo,in! fi!ure0

For this e(ample' ,e ,ill assume that the input of the system is the volta!e source %V &

applied to the motor-s armature' ,hile the output is the rotational speed of the shaft

d%theta&6dt) The rotor and shaft are assumed to be ri!id) 8e further assume a viscous

friction model' that is' the friction tor/ue is proportional to shaft an!ular velocity)



Darámetros fsicos para nuestro ejemplo son0

%& Moment of inertia of the rotor 3)31 k!)mG.

%b& Motor viscous friction constant 3)1 +)m)s

% K e& Electromotive force constant 3)31 ?6rad6sec

% K t& Motor tor/ue constant 3)31 +)m69mp

%@& Electric resistance 1 ;hm

%L& Electric inductance 3)5 H

i K T t =

)

θ e

K e =

Tor/ue %T &0

La fuerza electromotriz de

retroceso %e&0



unci!n de trasferencia

Las ecuaciones diferenciales /ue !obiernan el sistema son0

dt

d KeV eV Ri

dt

di L

i K T dt d b

dt d J

t

θ

θ θ

−+=−+=+

==+.

.

9pplyin! the Laplace transform' the above modelin! e/uations can be e(pressed in terms of the Laplace variable s)

&%&%&%&%

&%&%&%

s K sV s I R LS

s I K sb Js s

e

t

θ

θ

−=+

=+

The open&loop transfer function by eliminating I"s# between the twoabo$e e'uations, where the rotational speed is considered the outputand the armature $oltage is considered the input, is:.

et

t

K K b Js R Ls

K

sV

s

−++=

&%I&%&%

&%)

θ



Constru"endo el modelo con Simulink

This system ,ill be modeled by summin! the tor/ues actin! on the

rotor inertia and inte!ratin! the acceleration to !ive velocity) 9lso'

Jirchoff-s la,s ,ill be applied to the armature circuit) First' ,e ,ill

model the inte!rals of the rotational acceleration and of the rate of

chan!e of the armature current)

dt

d dt

dt

d θ θ =∫ .

.

idt

dt

di=∫

&%1

&%1.

.

.

.

dt

d KeV Ri

Ldt

dieV Ri

dt

di L

dt

d bi K

dt

d J

dt

d bT

dt

d J

t

θ

θ θ θ θ

−+−=⇒−+−=

−=⇒−=



To build the simulation model' open Simulink and open a

ne, model ,indo,)

modelado en matlab y simulink

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