contol system lab(matlab)[1]
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1.0 MATRICS,VECTORS AND SCALARS
In MATLAB a scalar is a variable with one row
and one column.
Scalars are the simple variables that we use and
manipulate in simple algebric
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EXAMPLE 1
>>a=[0:5]
a =
0 1 2 3 4 5
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Example 2
>>b=[10:20]
b =
10 11 12 13 14 15 16 17 18
19 20
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1.1 Creating scalar
To create a scalar simply introduce it on the left
hand side of an equal sign.
Example
>> x=1;
>> y=2;
>> Z=x+y
Z =
3
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1.2 Demostration on scalar addition,
subtraction, multiplication and division
Example1
>> u = 5;
>> v = 3;
>> w = u+v;
>> x = u-v;
>> y = u*v;
>> z = u/v;
>> w,x,y,z
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w =
8
X=
2
y =
15
z =
1.6667
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Example 3
>>x=3;
>> y=4;
>> z=x*y^2
z =
48
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1.3 Creating Vectors
In Matlab a vector is a matrix with either one row
or one column.
Example :
1. To create a row vector of length 5, filled with
5,
>>x=ones(1,5)
x =
1 1 1 1 1
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2. To create a column vector of length 5,filled
with zeros.
>>y=zeros(5,1)
y =
0
0
0 0
0
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Matrix
Matrix is a set of number arranged in a
rectangular grid of rows and columns.
Example:
>>A=[3 5]
A =
3 5
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The size of matrix is specified by the numbers of rowsand column.
If the matrix is same than it is called as square matrix.
3. Semicolon is used to separate the rows.
>>C=[1 2 3 ; 4 5 6 ; 7 8 9 ]
C =
1 2 3
4 5 6
7 8 9
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4. If we want to extract the third column of
matrix c, then we write
C(:,3)
ans =
3
6
9
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Simple Graphs
Example 1
To plot a simple graph
>>x=[1;2;3;4;5];
>> y=[0;.25;3;1.5;2];
>> plot(x,y)
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Example 2
>> x=0:5;
>> y=sinh(x);
>> plot(x,y)
>> xlabel('Time')
>> ylabel('Sinh')
>> title('Sinehyperbolic')
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Example 3
>> x=0:pi/100:2*pi;
>> y=sin(x);
>> plot(x,y)
>> xlabel('x=0:2\pi')
>> ylabel('Sin of x')
>> title('Plot of the sin function')
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Multiple Graphs
>> t=0:pi/100:2*pi;
>> y1=sin(t);
>> y2=sin(t+pi/2); >> plot(t,y1,t,y2)
>> grid on
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Exercise
For the graph above include the labels.
For X-axis, Y-axis and title.
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Solutions
>> t=0:pi/100:2*pi;
>> y1=sin(t);
>> y2=sin(t+pi/2);
>> plot(t,y1,t,y2)
>> grid on
>> xlabel('x=0:2\pi')
>> ylabel=('y1=sin(t)')
>> title('Multiple graphs')
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Root and convolution function
1. For the given polynomial, find the roots
a) P(s)=s2+6s+8
b) P(s)=s2-8s+5
c) P(s)=s2+2s-15
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Solution
a=
>> p=[1 6 8];
>> r=roots(p)
r =
-4
-2
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b=
>>p=[1 -8 15];
>> r=roots(p)
r =
5
3
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C =
>>p=[1 2 -15];
>>r =roots(p)
-5
3
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Pole and Zero
Find the zero and pole for the given
transfer function
T(s)= (s+2)
( s2+2s+1)
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Solution
>>sys=tf([1 2],[1 2 1]);
>> p=pole(sys);
>> z=zero(sys);
>> p,z
p =
-1
-1
z =
-2
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Pole - zero map
>>sys=tf([1 2],[1 2 1]);
>> p=pole(sys);
>> z=zero(sys); >> p,z
>> pzmap(sys)
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-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1Pole-Zero Map
Re al Ax is
Im
aginary
Axis
zero
pole
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Excersice
Show the pole and zero for the given transfer function in s- plane.
T(s)= 6s2+1
( s3 +3s2 + 3s +1)
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S- plane
>>sys=tf([6 0 1],[1 3 3 1]);
>> p=pole(sys);
>> z=zero(sys);
>> p,z
p =
-1.0000
-1.0000 + 0.0000i
-1.0000 - 0.0000i
z =
0 + 0.4082i
0 - 0.4082i
>> pzmap(sys)33
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-1 .4 -1 .2 -1 -0.8 -0 .6 -0 .4 -0 .2 0-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5Pole-Zero Map
Real Axis
Im
aginaryA
xis
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To construct transfer function
1. Given numerator =(s+5)
denominator =(s2 3s + 6)
>> num=[1 5];
>> den=[1 -3 6]; >> sys=tf(num,den)
Transfer function:
s + 5 -------------
s^2 - 3 s + 6
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Example 2
2. Given numerator =(-s+6)
denominator =(s2 + 4s + 4)
>>num=[-1 6];
>> den=[1 4 4]; >> sys=tf(num,den)
Transfer function:
- s + 6 -------------
s^2 + 4 s + 4
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Multiply the polynomials using conv
functions
1. Expand the following polynomials.
a) N(s) = (3s2+2s+1)(s+4)
b) Find the value of N(s) when s= -5
>>p=[3 2 1]; >> q=[1 4];
>> n=conv(p,q)
n =
3 14 9 4
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To evaluate the value of N(s) when s=-5
>>value=polyval(n,-5)
>>value =
-66
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Exersice
1. For the above polynomials, find the
value of N(s) when s = 2 and s = -7
2. Expand the following
N(s)= (-3s3+5s+6)(5s2-7s+1)
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Operator Addition (+)
Given G1(s) = 10 / s2+2s+5 and G2(s) = 1 / s+1
Using Matlab, perform the total of G1(s) + G2(s) .
>>num1=[10];den1=[1 2 5];
>> sys1=tf(num1,den1)
Transfer function:
10
------------- s^2 + 2 s + 5
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>>num2=[1];den2=[1 1]; >> sys2=tf(num2,den2)
Transfer function:
1
----- s + 1
>> sys=(sys1+sys2)
Transfer function:
s^2 + 12 s + 15 ---------------------
s^3 + 3 s^2 + 7 s + 5
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Exersice
Add the following transfer functions by
using Matlab. Given
G1(s) = s
2
+2s / 2s
2
- 4s+6
G2(s) = 2 / s+4
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Series Connection
Let the following transfer function be
G1(s) = 1 / 500s2
G2(s) = s+ 1 / s+2 When both are in cascade:
U(s) Y(s)
T(s) = Y(s) /U(s)
Sys1
( G1)
Sys2 (G2)
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Solution
>> numg1=[1];deng1=[500 0 0];
>> sys1=tf(numg1,deng1);
>> numg2=[1 1];deng2=[1 2];
>> sys2=tf(numg2,deng2);
>> sys=series(sys1,sys2)
Transfer function:
s + 1
------------------ 500 s^3 + 1000 s^2
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Parallel Connection
The following transfer functions is connected in parallel. Show the
transfer function.
numg1=[2 1];deng1=[1 2];
>> sys1=tf(numg1,deng1);
>> numg2=[1 3];deng2=[500 0 4]; >> sys2=tf(numg2,deng2);
>> sys=parallel(sys1,sys2)
Transfer function:
1000 s^3 + 501 s^2 + 13 s + 10
------------------------------
500 s^3 + 1000 s^2 + 4 s + 8
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Feedback
For the following block diagram, find the transfer
function. R(s) Y(s)
G(s)=1/ 500s2
H(s)=(s+1) /(s+2)
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Solution
>>numg=[1];deng=[500 0 0];
>> sysg=tf(numg,deng);
>> numh=[1 1];denh=[1 2];
>> sysh=tf(numh,denh);
>> sys=feedback(sysg,sysh)
Transfer function:
s + 2
--------------------------
500 s^3 + 1000 s^2 + s + 1
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Exercise
1. For the given bolck diagram , find the
transfer
R(s) y(s)
2/(s2 + 1) (S + 1)/ (s2 + 10)
((s2
-3) / (2s3
+ 2s +1)
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Solution
>> numg=[1];deng=[500 0 0];
>> sysg=tf(numg,deng);
>> numh=[1 1];denh=[1 2];
>> sysh=tf(numh,denh);
>> sys=feedback(sysg,sysh)
Transfer function:
s + 2
--------------------------
500 s^3 + 1000 s^2 + s + 1
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2. For the given block diagram, find the transfer
function.
R(s) Y(s)G1 =2 / (s
2 + 1) G2 =(s +1)/ (s2 +10)
H = ( s2 3) /
(2s3+2s+1)
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Solution
>> numg1=[2];deng1=[1 0 1];sys1=tf(numg1,deng1);
>> numg2=[1 1];deng2=[1 0 10];sys2=tf(numg2,deng2);
>> sysg=series(sys1,sys2);
>> numh1=[1 0 -3];denh1=[2 0 2 1];sysh=tf(numh1,denh1);
>> sys=feedback(sysg,sysh)
Transfer function:
4 s^4 + 4 s^3 + 4 s^2 + 6 s + 2
-------------------------------------------------
2 s^7 + 24 s^5 + s^4 + 44 s^3 + 13 s^2 + 14 s + 4
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Multiloop reduction
For the given multi loop feedback system, compute the closed
loop transfer function. Use series, parallel and feedback functions if
necessary.
G1 G2G3 G4
H2
H1
H3
R(s) Y(s)
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Five steps procedure is followed:
Step 1 = Input the system transfer function
into Matlab
Step 2 = Move H2
behind G4
Step 3 = Eliminate G3G4 H1loop.
Step 4 = Eliminate the loop containing
H2.
Step 5 = Eliminate the remaining loop and
calculate T(s)
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Solution
>>ng1=[1];dg1=[1 10];sysg1=tf(ng1,dg1);
>> ng2=[1];dg2=[1 1];sysg2=tf(ng2,dg2);
>> ng3=[1 0 1];dg3=[1 4 4];sysg3=tf(ng3,dg3);
>> ng4=[1 1];dg4=[1 6];sysg4=tf(ng4,dg4);
>> nh1=[1 1];dh1=[1 2];sysh1=tf(nh1,dh1);
>> nh2=[2];dh2=[1];sysh2=tf(nh2,dh2); >> nh3=[1];dh3=[1];sysh3=tf(nh3,dh3);
>> sys1=sysh2/sysg4;
>> sys2=series(sysg3,sysg4);
>> sys3=feedback(sys2,sysh1,+1);
>> sys4=series(sysg2,sysg3);
>> sys5=feedback(sys4,sys1);
>> sys6=series(sysg1,sys5);
>> sys=feedback(sys6,[1])
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Transfer function:
s^5 +4 s^4+ 6s^3 + 6s^2 + 5s + 2
-----------------------------------------------------------------------------------
12S^6 +20s^5 + 1066 s^4 + 2517s^3 +3128s^2 + 2196s + 712
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Root locus
The function covered are rlocus, rlocfind and residue.
rlocus and rlocfind :- used to obtain the root locus plots.
residue: - used for partial fraction expansion of rotational
functions.
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Closed loop transfer function
Consider the closed loop transfer function
T(s) = K(s+1)(s+3) / s(s+2)(s+3)+K(s+1)
Now use rlocus function to generate root locus plots.
The general form of characteristics is 1 + K G(S) = 0
Step 1: - Obtain the characeteristic equation in the formof 1 + K G(S) = 0 , where K is the parameter of interest.
Step 2:- Use the rlocus function to generate the plots.
[r,k]=rlocus(sys). r= complex and loot location
K= gain vector
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Generating a root locus plot
Let
>> p=[1 1]; q=[1 5 6 0];
>>sys=tf(p,q);
>>rlocus(sys)
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Root Locus
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-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5-8
-6
-4
-2
0
2
4
6
8Root Locus
Real Ax is
Im
agina
ry
Axis
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>>p=[1 1];q=[1 5 6 0];
>> sys=tf(p,q);
>> rlocus(sys);
>> rlocfind(sys)
Select a point in the graphics window
selected_point =
-2.4716 + 0.0248i
ans =
0.4195 This value will be vary. Depends on the selected value atgraph
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Find the step response and also find the
settling Time (Ts), Peak response(Ps)
Let the value of K is 20.5775
>>k=20.5775;num=k*[1 4 3];
>>den=[1 5 6+k k];
>> sys=tf(num,den); >> step(sys)
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S t ep Response
T ime (sec )
Amp
litude
0 0.5 1 1 .5 2 2 .5 30
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Sys tem : sysT ime (sec ) : 0 .423Amplitude: 4.49
Sys tem : sysTime (sec ) : 1.8Amplitude: 3.03
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Exersice
Compute step response for second order system when
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Bode Diagram
Bode The bode function is used togenerate a bode diagram.
Logspace The logspace function
generate a logarithmatically spaced vectorof frequencies utilized by the bodefunction.
The magnitude and phase characteristicsare placed in the workspace through thevariables mag and phase.
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Consider the transfer function given below:
Plot the bode diagram.
2
250
1
50
6.01)5.01(
1.015
ssss
s=
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Solution
% Bode plot
>> %
>> num=5*[0.1 1];
>> f1=[1 0];f2=[0.5 1]; f3=[1/2500 0.6/50 1]; >> den=conv(f1,conv(f2,f3));
>> %
>> sys=tf(num,den);
>> bode(sys)
>>
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Excersice
For the following transfer function sketch
the Bode plots, then verify with Matlab
a) G(s) =
)10)(1(
1
ss
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b) G(s) =)502(
)10(2
ss
s
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Gain Margin and Phase margin can be
determined from both Nyquist and Bode
diagram. The gain margin is a measure of how much the
system gain would have to be increased for the
GH(jw) locus to pass through the (-1,0) point,
thus resurlting in an unstable system.
Example
>> num=[0.5];den=[1 2 1 0.5];
>> sys=tf(num,den); >> margin (sys)
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-150
-100
-50
0
50
Magnitude(dB)
10- 2
10- 1
100
101
102
-270
-225
-180
-135
-90
-45
0
Phase(deg)
Bode DiagramG m = 9.55 dB (a t 1 rad/sec) , Pm = 49 deg (a t 0 .644 rad/sec)
F requency ( rad /sec )
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NYQUISTPLOT
This section covered Nyquist, Nichols,
margin ,pade and ngrid functions.
It is generally more difficult to manually
generate the Nyquist plot than Bode
diagram.
When Nyquist function generated,
automatically Nyquist plot is generated.
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Example
1. For the closed loop control system,
shown below, plot Nyquist plot.
5.02
5.0
23
ss
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Solution:
>>num=[0.5];den=[1 2 1 0.5];
>> sys=tf(num,den); >> nyquist (sys)
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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1.5
-1
-0.5
0
0.5
1
1.5Ny quist D iagram
Real Ax is
Imagina
ryAxis
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Example
The nyquist plot for the below system with
gain and phase margins.
5.02
5.0
23
sss
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% The Nyquist plot
% With a gain and phase margin calculation.
>> num=[0.5];den=[1 2 1 0.5];
>> sys=tf(num,den); >> [mag,phase,w]=bode(sys);
>> [Gm,Pm,Wcg,Wcp]=margin(mag,phase,w);
>> nyquist(sys);
>> title(['Gm=',num2str(Gm),'Pm=',num2str(Pm)])
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-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1.5
-1
-0.5
0
0.5
1
1.5Gm=3Pm=49.5753
Re al Ax is
Imaginary
Axis
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Using the Nyquist function, obtain the polar plot for the
following transfer function.
148
20)(
2
!
ss
sG
133
10
)( 23 ! ssssG
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Nichols Chart
Plot the nichols chart for the following
transfer function
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Solution
>> num=[1];den=[0.2 1.2 10];
>> sys=tf(num,den);
>> w=logspace(-1,1,400); >> nichols(sys,w);
>> ngrid
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-360 -315 -270 -225 -180 -135 -90 -45 0-40
-30
-20
-10
0
10
20
30
40
6 dB
3 dB
1 dB
0.5 dB
0.25 dB
0 dB
-1 dB
-3 dB
-6 dB
-12 dB
-20 dB
-40 dB
Nichols Chart
Open-Loop Phase (deg)
Open-Loop
Gain(dB)
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Exercise
Draw the Nichols chart for the given
transfer function.
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>>num=[1];den=[0.6 2.3 1];
>> sys=tf(num,den);
>> nichols(sys); >> ngrid
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-360 -315 -270 -225 -180 -135 -90 -45 0-100
-80
-60
-40
-20
0
20
40
6 dB3 dB
1 dB0.5 dB
0.25 dB
0 dB
-1 dB
-3 dB
-6 dB
-12 dB
-20 dB
-40 dB
-60 dB
-80 dB
-100 dB
Nichols Chart
Open-Loop Phase (deg)
Open-LoopG
ain(dB)
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Building a Simple Model
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Steps
1.Entersimulink in MATLAB commandWindow
2.Create a new model window by click
New
Library Simulink
Untitled
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3. To create the above model click the
follwing library.
- Sources Library
- Sinks Library
- Continuous Library
- Signal Routing library
4. Drag the respective blocks from the
browser and drop it in the model window.
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5. To view simulation output for 10
second,
- Open the scope block
- Click simulationparameters from the
simulation menu. Notice stop time for 10
second. Then click OK. Close the dialog
box.
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- Choose start from simulation and watch
the traces of the scope blocks input.
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6. To save choose save from the file
menu and enter the filename and location.
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Simulink
Running a Demo Model
1. Click the start button on the bottom left
corner of the MATLAB command window.
2. Select Demos from the menu.
3. Click the simulink entry in the Demos
panel.
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4. Select any one of features.
Simulink
- Features
- General
- Automotive- Aerospace.
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5. Click on demo link to start the demo.
6. Simulink start.
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END