control sys demo 10 11

Good Evening

CONTROL SYSTEM TUTORIAL

with MATALAB

BY

RAVI KUMAR JATOTH

LECTURER

DEPT OF ECE

NIT, WARANGAL

16/03/2011 2Ravi Kumar Jatoth

INTRODUCTION

Computer software can be used to

aid computation and modeling of

system. A program that is often

used is

MATLAB

MATLAB MATLAB stands for MATrix LABoratory

MATLAB is a software package for high performance numerical computation and visualization.

There are many different toolboxes available which extend the basic functions of Matlab into different application areas.

we will make extensive use of the Control Systems Toolbox


Do Mech guys really need a computer for purposes other than CAD & CAM ? Can you recall these equation –

1∕√f = -2log(2.51/Re√f + 7/3.7d)

How about this one--

Q=(b*yn /n) {b*yn/(b+2yn)}2/3

They had cost many of us as much as 15 to 20

marks last sem in Fluid Mechanics course.

There are several equations for which only numerical

solutions exist.


MATLAB

Typical uses: Math and computation

Algorithm development

Data acquisition

Modeling, simulation, and prototyping

Data analysis, exploration, and visualization

Scientific and engineering graphics

Application development, including GUI building


MATLAB Powerful toolboxes

Extending the environment to solve particular classes of problems:

Symbolic Math

Control System

Neural Networks

Partial Differential Equations

Signal and Image Processing

...


Basics of MATLAB

MATLAB WINDOWS

MATLAB DESKTOP

1.Command window

2.current directory

3.workspace

4.command history

Figure window

Editor window 16/03/2011 8Ravi Kumar Jatoth

MATLAB Desktop

Command

Window

Launch Pad

History


MATLAB Desktop – cont’d

Command

Window

Workspace

Current

DIrectory


Matlab Help


Matlab Editor

Color keyed

text with auto

indents

tabbed sheets for other

files being edited

Access to

commands


Some Important Details Everything is a matrix.

Comma or space separates row elements.Semicolon separates rows.

Elements are referred to by indices in parentheses .

Assignment uses = (not :=)

Terminal semicolon (;) suppresses output.

Variable names are Case sensitive.

Two sets of arithmetic operations – vector, element-wise.

Previous inputs can’t be edited directly.


How to Use MATLAB? Command driven:

MATLAB processes single-line commands immediately and displays the result.

Script driven:

MATLAB is also capable of executing sequences of command that are stored in files.

These source files are called m-files, having a .m extension.


Matlab ProgramsMatlab is an extravagant calculator if all we can do is

execute commands typed into the Command Window… So how can we execute a “program?” Programs in Matlab are: Scripts, or Functions

Scripts: Matlab statements that are fed from a file into the Command Window and executed immediately

Functions: Program modules that are passed data (arguments) and return a result (i.e., sin(x))

These can be created in any text editor (but Matlab supplies a nice built-in editor)


Reserved Words…

Matlab has some special (reserved) words that you might use while writing programs

for

end

if

while

function

return

elsif

case

otherwise

switch

continue

else

try

catch

global

persistent

break


Matlab has Some Special VariablesSpecial Variable Description

ans default variable name for results

beep make sound

pi mathematical constant

eps smallest number that can be subtracted from 0 to

make a negative

inf infinity

i (or) j imaginary number

realmin, realmax smallest & largest positive real numbers

bitmax largest positive integer

nargin, nargout number of function in (or) out variables

varargin variable number of function in arg’s

varaout variable number of function out arg’s


Matrix Operators+ Unary Plus/Addition

- Unary Minus/Subtraction

* Multiplication

^ Power

‘ Conjugate transpose

/ or \ Matrix division

./ or .\ Array division


Matrix Operators


Relational and Logical Operators< Less than

<= Less than or equal to

> Greater than

>= Greater than or equal to

== Equal

~= Not equal

& AND

| OR

~ NOT

Note that = is used in an assignment statement, while == is used in a relation.


Built-in Functions

Trigonometric

functions sin, cos, tan, sin, acos, atan,

sinh, cosh, tanh, asinh,

acosh, atanh, csc, sec, cot,

acsc, … Exponential

functions exp, log, log10, sqrt

Complex

functions abs, angle, imag, real, conj

Rounding and

Remainder

functions

floor, ceil, round, mod, rem,

sign


Entering a matrix and assigning it to a variable name

Remember that row elements are separated by comma or space and rows by semicolon or enter


Matlab Basics Tutorial

Vectors

a = [1 2 3 4 5 6 9 8 7] Let's say you want to create a vector with elements between 0 and 20 evenly spaced in increments

of 2 (this method is frequently used to create a time vector):

t = 0:2:20

Functions

sin(pi/4)

Plotting

X=[0 1 2 3 4 5];

Y=[0 1 4 9 16 25];

plot(X,Y)


Cont..

Sine wave

t=0:0.25:7;

y = sin(t);

plot(t,y)

X=(0 1 2 3 4 5 6 7);

Y=exp(X);


Practice Entering Some Expressions

>>4+3*sqrt(2)

>>sin(pi/6)

>>i^2

>>j^2 (For electrical engineers)

>>x=log(10)

>>y=exp(x)


Matrices

Entering matrices into Matlab is the same as entering a vector, except each

row of elements is separated by a semicolon (;) or a return:

B = [1 2 3 4;5 6 7 8;9 10 11 12]

B = [ 1 2 3 4

5 6 7 8

9 10 11 12]

Matrices in Matlab can be manipulated in many ways. For one, you can find the transpose of a matrix using the apostrophe key:

C = B'

C = 1 5 9

2 6 10

3 7 11

4 8 12

D = B * C


Matrix Operations» C = [1 2 1 1;3 4 2 2]

C =

1 2 1 1

3 4 2 2

» C(1,:)

ans =

1 2 1 1

»

» C(:,2)

ans =

2

4

» C(:,2:end)

ans =

2 1 1

4 2 2

» 16/03/2011 27Ravi Kumar Jatoth

Matrix Functions» size(C)

ans =

2 4

» det(A)

ans =

-2

»

» A

A =

1 2

3 4

» inv(A)

ans =

-2.0000 1.0000

1.5000 -0.5000

» 16/03/2011 28Ravi Kumar Jatoth

Try the following.

X*B , B*X , pi*X , B^2 , X^2,X.^2 , Y=X’*X , Y^2, Y.^2,B.*B , B*B’ , B.*B’ ,sqrt(X) , exp(X) , cos(X)

1 2 3, 2 3

4 5 6B X


How to enter the following

matrix?


Computing Matrix Functions Characteristic Equation

– The roots of the characteristic equation of a square matrix A are the

same as the eigenvalues of A.

– The characteristic equation of A is computed with the function poly(A).

– Example

>> A = [0 1 0; 0 0 1; -6 -11 -6];

>> p = poly(A)

p =

1.0000 6.0000 11.0000 6.0000

This is the MATLAB representation of the characteristic equation

.06116 23 sss


Computing Matrix Functions

Eigenvalues and eigenvectors

– If A is an n n matrix, then the n numbers λ that satisfy Ax = λ are the eigenvalues of A.

– Eigenvalues of A are obtained with the function eig(A), which returns the eigenvalues in a column vector.

– Example:>> A = [0 1 0; -1 0 2; 3 0 5];

>> eig(A)

ans =

5.2130

-0.1065 + 1.4487i

-0.1065 - 1.4487i16/03/2011 32Ravi Kumar Jatoth

Polynomials

If the

to find the value of the above polynomial at s=2,

z = polyval([1 3 -15 -2 9],2)

Finding the roots would be as easy as entering

the following command;

roots([1 3 -15 -2 9])

92153 234 ssss


Evaluation of a Polynomial

a = x3 + 2x2 + 3x + 4

» polyval(a, 2)

ans =

26

»


Polynomial Multiplication

a = x3 + 2x2 + 3x + 4b = 4x2 + 9x + 16

» a = [1 2 3 4];

» b = [4 9 16];

» c = conv(a,b)

c =

4 17 46 75 84 64

»


Polynomial Curve Fitting

» x = [1 3 7 21];

» y = [2 9 20 55];

» polyfit(x,y,2)

ans =

-0.0251 3.1860 -0.8502

»


Using Matlab to Solve Simultaneous

Equations Set up the equation in matrix/vector form:

A = [3 4 5; 21 5 2; 1 -2 10]

u = [ x y z]’

b = [ 32 20 120]’

In other words, A u = b (this is linear algebra)

3 4 5

21 5 2

1 -2 10

x

y

z

*

32

20

120

=


The solution>> u = A\b

u =

1.4497 ( value of x)

-6.3249 ( value of y)

10.5901 ( value of z)

You can plug these values in the original

equation

test = A * u and see if you get b16/03/2011 38Ravi Kumar Jatoth

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Consider the rational fraction:

MATLAB solution to partial fraction

problem can be given by:

>> N=[10 40]; >> D=[1 4 3 0];

>> [A,p,K]=residue(N,D) A =

1.6667

-15.0000

13.3333

p =

-3

-1

0

K =

[ ] 39Ravi Kumar Jatoth

Programming in MATLABScript files

A Script file contains a set of MATLAB

command

Use them when you have a long sequence

of statements to solve a problem

Can run the program by typing its name in

the command window or from tools in the

editor window.


for loops

General form:for index=initial: increment: limit

statements

end

s=0

for i=1:3:11

s=s+i

end


Example MATLAB program to find the roots of

1)cos(2)( xxf

% program 1 performs four iterations of

% Newton’s Method

X=.7

for i=1:4

X=X – (2*cos(X)-1)/(-2*sin(X))

end

X =1.1111

X =1.0483

X =1.0472

X =1.0472

Result


Final Thoughts

Nobody could or would want to know

everything about MATLAB.

Start with a particular project or problem.

Use the Help files.

Be willing to try something.

Be prepared for frustration, but if you need

MATLAB it is worth the frustration.


Control System Tool Box

16/03/2011 Ravi Kumar Jatoth 45

Example: Train system

we will consider a toy train consisting of an engine

and a car. Assuming that the train only travels in one

direction

we want to apply control to the train so that it has a

smooth start-up and stop, along with a constant-speed

ride.


Modeling cont..

The two are held together by a spring, which has the

stiffness coefficient of k

u, represents the coefficient of rolling friction


Free body diagram and Newton's

law


When finding the transfer function, zero initial

conditions must be assumed. The transfer function

should look like the one shown below.


Since MATLAB can not manipulate symbolic variables, let's

assign numerical values to each of the variables.

Let

M1 = 1 kg

M2 = 0.5 kg

k = 1 N/sec

F= 1 N

u = 0.002 sec/m

g = 9.8 m/s^2


Create an new m-file and enter the following

commands.

M1=1;

M2=0.5;

k=1; F=1;

u=0.002;

g=9.8;

num=[M2 M2*u*g 1];

den=[M1*M2 2*M1*M2*u*g

M1*k+M1*M2*u*u*g*g+M2*k

M1*k*u*g+M2*k*u*g];


Transfer function

The transfer function can be of the

following form


The transfer function can be entered as

follows

G(s)=4(s+10)/(s+5)(s+15)


MATLAB code

num=4*[1 10];

den=conv([1 5],[1 15]);

printsys( num,den,’s’)


The function conv has been used to multiply

polynomials in the following MATLAB session for

the transfer function

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>> n1 = [5 1];

>> n2 = [15 1];

>> d1 = [1 0];

>> d2 = [3 1];

>> d3 = [10 1];

>> num = 100*conv(n1,n2);

>> den = conv(d1,conv(d2,d3));

>> GH = tf(num,den)

57Ravi Kumar Jatoth

Block Diagram Reduction


Series Blocks


Blocks are in Parellel


Unit Feed Back Systems


Building models

cloop : command provides a means of

generating a closed loop (unity feedback)

system description given an open loop

system .

Ex:


MATLAB code

ngo=[1 1];

dgo=conv([1 3],[1 5]);

[ngc2,dgc2]=cloop(ngo,dgo);

printsys(ngc2,dgc2,’s’)


feedback

feedback: for non unity feedback use feed

back command.

Ex:


Matlab code

ngo=[1 1];

dgo=conv([1 3],[1 5]);

nh=[1 6];

dh=[1 10];

[ngc2,dgc2]=feedback(ngo,dgo,nh,dh)

printsys(ngc2,dgc2,’s’)


Block Diagram Reduction


How to find out poles &zeros

G(s)=(5s^2+3s+4)/(s^3+2s^2+4s+7)


num=[5 3 4];

den=[1 2 4 7];

[z,p]=tf2zp(num,den)


Creating transfer function from

poles and zeroes

Zp2tf:

z=[0; 0; 0; -1];

p=[-1+i -1-i -2 -3.15+2.63*i

-3.15- 2.63*i];

k=2;

[num,den]=zp2tf(z,p,k);

printsys(num,den,'s')


Time Response Analysis


Time Response of a system to

different inputs

System Response

Step and impulse responses of LTI objects can be obtained by the commands

step and impulse . For example, to obtain the step response of the system

represented in LTI object G, enter

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num = [1 1];

den = [1 3 1];

G = tf(num,den)

>> step(G)

88Ravi Kumar Jatoth

Display response to step input

G(s)=5/(s^2+3s+12)

num=5;

den=[1 3 12];

step(num,den)


Step and impulse response data can be

collected into MATLAB variables by using

two left hand arguments. For example, the

following commands will collect step and

impulse response amplitudes in yt and time

samples in t.

[yt, t] = step(G)

[yt, t] = impulse(G)16/03/2011 91Ravi Kumar Jatoth

Changing the magnitude of the step

step(100*num,den)


Specifying the time scale

step (num,den,t)


Saving the response

If the time vector is used, the command is:

[y,x] = step(num,den,4); If the time vector is not used, the command is:


Impulse Response

num=5;

den=[1 3 12];

impulse(num,den)


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Response of LTI systems to arbitrary inputs can be obtained by the

command lsim. The command lsim(G,u,t) plots the time response

of the LTI model G to the input signal described by u and t. The

time vector t consists of regularly spaced time samples and u is a

matrix with as many columns as inputs and whose ith -row specifies

the input value at time t(i). Observe the following MATLAB session

to obtain the time response of LTI system G to sinusoidal input of

unity magnitude.

>> t=0:0.01:7;

>> u=sin(t);

>> lsim(G,u,t)

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The function gensig generates periodic signals for time response

simulation with lsim function. It can generate sine, square, and

periodic pulses. All generated signals have unit amplitude. Observe

the following MATLAB session to simulate G ( s ) for 20 seconds with

a sine wave of period 5 seconds.

>> [u,t]=gensig( 'sin' ,5,20); %Sine wave with period 5 sec and

duration 20 sec

>> lsim(G,u,t) %Simulate G(s) with u and t.

97Ravi Kumar Jatoth

16/03/2011

The following MATLAB script calculates

the step response of second-order system

with and various values of t=[0:0.1:12]; num=[1];

zeta1=0.1; den1=[1 2*zeta1 1];

zeta2=0.2; den2=[1 2*zeta2 1];

zeta3=0.4; den3=[1 2*zeta3 1];

zeta4=0.7; den4=[1 2*zeta4 1];

zeta5=1.0; den5=[1 2*zeta5 1];

zeta6=2.0; den6=[1 2*zeta6 1];

[y1,x]=step(num,den1,t); [y2,x]=step(num,den2,t);



plot(t,y1,t,y2,t,y3,t,y4,t,y5,t,y6)

xlabel( 't' ), ylabel( 'y(t)' )

grid

98Ravi Kumar Jatoth

Plotting the root locus of a transfer

function

Consider an open loop system which has a

transfer function of


num=[1 7];

den=conv(conv([1 0],[1 5]),conv([1 15],[1 20]));

rlocus(num,den)

axis([-22 3 -15 15])


http://www.engin.umich.edu/group/ctm/extras/plot.html

Choosing a value of K from the root locus

In our problem, we need an overshoot less than 5%

(which means a damping ratio Zeta of greater than

0.7) and a rise time of 1 second (which means a

natural frequency Wn greater than 1.8). Enter in the

Matlab command window:

zeta=0.7; Wn=1.8; sgrid(zeta, Wn)


Frequency Response Analysis and

Design

The frequency response is a representation of the system's response to sinusoidal inputs at varying frequencies. The output of a linear system to a sinusoidal input is a sinusoid of the same frequency but with a different magnitude and phase. The frequency response is defined as the magnitude and phase differences between the input and output sinusoids. In this tutorial, we will see how we can use the open-loop frequency response of a system to predict its behavior in closed-loop. I


http://www.engin.umich.edu/group/ctm/freq/freq.html

1.Bode plots

2.Gain and phase margin




code

50 /s^3 + 9 s^2 + 30 s + 40

bode(50,[1 9 30 40])


Gain and Phase Margin

Let's say that we have the following system:

where K is a variable (constant) gain and G(s) is the plant under

consideration. The gain margin is defined as the change in open

loop gain required to make the system unstable. Systems with

greater gain margins can withstand greater changes in system

parameters before becoming unstable in closed loop.


PHASE MARGIN

The phase margin is defined as the change in open loop phase shift required to make a closed loop system unstable.

The phase margin also measures the system's tolerance to time delay. If there is a time delay greater than 180/Wpc in the loop (where Wpc is the frequency where the phase shift is 180 deg), the system will become unstable in closed loop. The time delay can be thought of as an extra block in the forward path of the block diagram that adds phase to the system but has no effect the gain. That is, a time delay can be represented as a block with magnitude of 1 and phase w*time_delay (in radians/second).


SIMULINK

Simulink is started from the MATLAB

command prompt by entering the

following command:

simulink


Alternatively, you can hit the New

Simulink Model button at the top of the

MATLAB command window as shown

below:


When it starts, Simulink brings up two windows. The first

is the main Simulink window, which appears as:


Blocks

There are several general classes of blocks:

Sources: Used to generate various signals

Sinks: Used to output or display signals

Discrete: Linear, discrete-time system elements (transfer functions, state-space models, etc.)

Linear: Linear, continuous-time system elements and connections (summing junctions, gains, etc.)

Nonlinear: Nonlinear operators (arbitrary functions, saturation, delay, etc.)

Connections: Multiplex, Demultiplex, System Macros, etc.


Example


Modifying Blocks


to change the denominator to s^2+2s+1, enter the

following into the denominator field:

[1 2 1]


The "step" block can also be double-clicked, bringing up

the following dialog box.


The most complicated of these three blocks is the "Scope"

block. Double clicking on this brings up a blank

oscilloscope screen


Running Simulations


The simulation should run very quickly and the scope

window will appear as shown below


To fix this, hit the autoscale button (binoculars), which will

rescale the axes as shown below.


Thank you!


ALL THE BEST


control sys demo 10 11

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