Solving a System in Matlab

8/3/2019 Solving a System in Matlab

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Solving A system in Matlab

By Engr. Khurram Hashmi


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Introduction

This effort is dedicated to helping students solve

systems in Matlab (10). specially addressing the need

to utilize Matlab as a computational tool

Solving a System in Matlab 2


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Default Matlab view



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Writing an equation

Lets take any equation:

F(s)=3s3+4s2+11s+5

Usually this can be realized through a matrix of

coefficients

A=[3 4 11 5]which may be further manipulated.

To evaluate this at any value of s

polyval(A,3)

Where 3 is the value of s at which function is

evaluated



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Any shortcut to evaluating equations?

For equations of higher order solving manually to

obtain coefficients gives room to error

Alternatively, we can declare symbolic variables to be

treated separately. Matlab doesnt ask you to givevalues for them

syms

is a useful command in this regard



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Symbolic variables

syms s

F = 3*s^3 + 4*s^2 + 11 *s + 5

This appears in workspace

F=

3*s^3 + 4*s^2 + 11*s + 5

It works in previous versions of Matlab as well

You can further simplify this as well

simplify(F)

This one being already in simplified form

ans =3*s^3 + 4*s^2 + 11*s + 5



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Generating a transfer function

Usually coefficients of numerator and denominator are used as

>> num=[1 1];

>> den=[1 2 3 4];

>> sys=tf(num,den)

Transfer function:

s + 1

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

s^3 + 2 s^2 + 3 s + 4

however, there is another much direct approach..



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Generating a transfer function

>> s=tf(s)This prompts Matlab to treat all equations of the svariable as a transfer function. All followingequations of s will be automatically simplifiedand made transfer functions

>>N=s+1

Transfer function:s + 1

>> D=s^3 + 2*s^2 + 3*s + 4

Transfer function:

s^3 + 2 s^2 + 3 s + 4

Further we can play with this

>> G = N/D

Transfer function:s + 1

---------------------s^3 + 2 s^2 + 3 s + 4



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Workspace Variables

Have a look at your workspace variables.

This window is usually in top-right area

of your default Matlab layout. You can

go to each one of these and see their

contents by double-clicking them



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Variable Editor

Variable editor displays the contents of

your variables including transfer

functions.

Here Im showing the G transfer

function we just created

Ill double click the num and den to

show you



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Extracting variables


Numerator coefficients Denominator coefficients


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Seeing System Responses

We can view how the system plays to various standard inputs such as step

and impulse

through the command prompt

>> step(G)

Or

>> impulse(G)



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Step response

>> step(G)

This Gives the figure

shown left.

We can right click in the windowand enable viewing

characteristics as peak

response, settling time etc



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Impulse response

>> impulse(G)

This gives the impulse response

and can be manipulated

likewise



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LTI View

Alternatively we can use the Linear Time

Invarient View or LTIVIEW for short

>> ltiview

An ltiview window will pop up. Well have to select from the system(s)

present in the workspace.



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Well have to select one or more of these systems for the plot. Ill select G



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Same plot as before. But in LTI viewer.

Right click in the window and youll see a range of options to the type of plot you want to see.

you can plot the Bode Magnitude and Phase 17


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Bode Magnitude and Phase Plot

There is a range of options you can use.



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ROOT LOCUS

The Plot and its meaning



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Evans Root Locus

Evans Root Locus is one way of determining a

systems stability.

Root Locus is the Path poles and zeroes of a

system in closed loop take when multiplied

with a gain Factor (usually K ) that varies from

1 to +ve infinity

Systems behavior can be judged over this

range



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Root Locus

G(s)_

K

Input Output

>> rlocus(G)

This command plots the CLOSED LOOP poles of an OPEN LOOP transfer function .G.

Note: we do not have to solve the system in closed loop first. We just enter the OPEN

LOOP transfer function



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Poles and zeros on the right side of the imaginary axis means a stable system. Poles and zeros on

the left side of imaginary axis means an unstable system. Furthermore, closer the poles to the

imaginary axis, the less stable the system will be. 22


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You can also enable the grid through right click menu. Frequency values along imaginary

axis and zeta (damping ratio) along real axis 23


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Thank you..

Your feedback is welcome at:

[email protected]


Solving a System in Matlab

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