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
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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
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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:
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