contol system lab(matlab)[1]

8/3/2019 Contol System Lab(MATLAB)[1]

1/96

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

1


2/96

EXAMPLE 1

>>a=[0:5]

a =

0 1 2 3 4 5

2


3/96

Example 2

>>b=[10:20]

b =

10 11 12 13 14 15 16 17 18

19 20

3


4/96

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

4


5/96

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

5


6/96

w =

8

X=

2

y =

15

z =

1.6667

6


7/96

Example 3

>>x=3;

>> y=4;

>> z=x*y^2

z =

48

7


8/96

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

8


9/96

2. To create a column vector of length 5,filled

with zeros.

>>y=zeros(5,1)

y =

0

0

0 0

0

9


10/96

Matrix

Matrix is a set of number arranged in a

rectangular grid of rows and columns.

Example:

>>A=[3 5]

A =

3 5

10


11/96

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

11


12/96

4. If we want to extract the third column of

matrix c, then we write

C(:,3)

ans =

3

6

9

12


13/96

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)

13


14/96

14


15/96

Example 2

>> x=0:5;

>> y=sinh(x);

>> plot(x,y)

>> xlabel('Time')

>> ylabel('Sinh')

>> title('Sinehyperbolic')

15


16/96

16


17/96

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')

17


18/96

18


19/96

Multiple Graphs

>> t=0:pi/100:2*pi;

>> y1=sin(t);

>> y2=sin(t+pi/2); >> plot(t,y1,t,y2)

>> grid on

19


20/96

20


21/96

Exercise

For the graph above include the labels.

For X-axis, Y-axis and title.

21


22/96

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')

22


23/96

23


24/96

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

24


25/96

Solution

a=

>> p=[1 6 8];

>> r=roots(p)

r =

-4

-2

25


26/96

b=

>>p=[1 -8 15];

>> r=roots(p)

r =

5

3

26


27/96

C =

>>p=[1 2 -15];

>>r =roots(p)

-5

3

27


28/96

Pole and Zero

Find the zero and pole for the given

transfer function

T(s)= (s+2)

( s2+2s+1)

28


29/96

Solution

>>sys=tf([1 2],[1 2 1]);

>> p=pole(sys);

>> z=zero(sys);

>> p,z

p =

-1

-1

z =

-2

29


30/96

Pole - zero map

>>sys=tf([1 2],[1 2 1]);

>> p=pole(sys);

>> z=zero(sys); >> p,z

>> pzmap(sys)

30


31/96

-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

31


32/96

Excersice

Show the pole and zero for the given transfer function in s- plane.

T(s)= 6s2+1

( s3 +3s2 + 3s +1)

32


33/96

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


34/96

-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

34


35/96

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

35


36/96

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

36


37/96

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

37


38/96

To evaluate the value of N(s) when s=-5

>>value=polyval(n,-5)

>>value =

-66

38


39/96

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)

39


40/96

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

40


41/96

>>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

41


42/96

Exersice

Add the following transfer functions by

using Matlab. Given

G1(s) = s

2

+2s / 2s

2

- 4s+6

G2(s) = 2 / s+4

42


43/96

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)

43


44/96

Solution

>> numg1=[1];deng1=[500 0 0];

>> sys1=tf(numg1,deng1);

>> numg2=[1 1];deng2=[1 2];


>> sys=series(sys1,sys2)

Transfer function:

s + 1

------------------ 500 s^3 + 1000 s^2

44


45/96

Parallel Connection

The following transfer functions is connected in parallel. Show the

transfer function.

numg1=[2 1];deng1=[1 2];


>> 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

45


46/96

Feedback

For the following block diagram, find the transfer

function. R(s) Y(s)

G(s)=1/ 500s2

H(s)=(s+1) /(s+2)

46


47/96

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

47


48/96

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)

48


49/96

Solution

>> numg=[1];deng=[500 0 0];

>> sysg=tf(numg,deng);

>> numh=[1 1];denh=[1 2];

>> sysh=tf(numh,denh);


Transfer function:

s + 2

--------------------------

500 s^3 + 1000 s^2 + s + 1

49


50/96

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)

50


51/96

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);


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

51


52/96

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)

52


53/96

53


54/96

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)

54


55/96

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])

55


56/96

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

56


57/96

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.

57


58/96

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

58


59/96

Generating a root locus plot

Let

>> p=[1 1]; q=[1 5 6 0];

>>sys=tf(p,q);

>>rlocus(sys)

59

Root Locus


60/96

-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

60


61/96

>>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

61


62/96

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)

62

S t R


63/96

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

63


64/96

Exersice

Compute step response for second order system when

64


65/96

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.

65


66/96

Consider the transfer function given below:

Plot the bode diagram.

2

250

1

50

6.01)5.01(

1.015

ssss

s=

66


67/96

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)

>>

67


68/96

Excersice

For the following transfer function sketch

the Bode plots, then verify with Matlab

a) G(s) =

)10)(1(

1

ss

68


69/96

b) G(s) =)502(

)10(2

ss

s

69


70/96

70


71/96

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)

71


72/96

-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 )

72


73/96

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.

73


74/96

Example

1. For the closed loop control system,

shown below, plot Nyquist plot.

5.02

5.0

23

ss

74


75/96

Solution:

>>num=[0.5];den=[1 2 1 0.5];

>> sys=tf(num,den); >> nyquist (sys)

75


76/96

-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

76


77/96

Example

The nyquist plot for the below system with

gain and phase margins.

5.02

5.0

23

sss

77


78/96

% 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)])

78


79/96

-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

79


80/96

Using the Nyquist function, obtain the polar plot for the

following transfer function.

148

20)(

2

!

ss

sG

133

10

)( 23 ! ssssG

80


81/96

Nichols Chart

Plot the nichols chart for the following

transfer function

81


82/96

Solution

>> num=[1];den=[0.2 1.2 10];

>> sys=tf(num,den);

>> w=logspace(-1,1,400); >> nichols(sys,w);

>> ngrid

82


83/96

-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)

83


84/96

Exercise

Draw the Nichols chart for the given

transfer function.

84


85/96

>>num=[1];den=[0.6 2.3 1];

>> sys=tf(num,den);

>> nichols(sys); >> ngrid

85


86/96

-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)

86


87/96

Building a Simple Model

87


88/96

Steps

1.Entersimulink in MATLAB commandWindow

2.Create a new model window by click

New

Library Simulink

Untitled

88


89/96

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.

89


90/96

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.

90

Choose start from simulation and watch


91/96

- Choose start from simulation and watch

the traces of the scope blocks input.

91


92/96

6. To save choose save from the file

menu and enter the filename and location.

92


93/96

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.

93


94/96

4. Select any one of features.

Simulink

- Features

- General

- Automotive- Aerospace.

94


95/96

5. Click on demo link to start the demo.

6. Simulink start.

95


96/96

END

contol system lab(matlab)[1]

Documents