# Laplace Transforms in Control Engineering

Hi Peter,
I'm back again. I have been working away at control theory and am happy with my understanding.
I used Matlab to convert my laplace transform to a z transform.
0.00375 X= --------------------- (s+0.15)*(s+0.25)
Which converts it to
0.000438*z + 0.000410 -------------------------- z^2-1.81*z+0.818
then the difference equation is got by dividing the num and dem them by the highest order z
0.000438 + 0.000410z^-1 ----------------------------- 1-1.81*z^-1+0.818*z^-2
therefore
x(n)-1.81*x(n-1)+0.818*x(n-2) = 0.000438*u(n)+0.0004104*u(n-1)
therefore
x(n) = 1.81*x(n-1)-0.818*x(n-2) +0.000438*u(n)+0.0004104*u(n-1)
Which is a little different to your result of
x(n) = 1.81*x(n-1)-.819*x(n-2)+.0004387*u(n-1)+.0004104*x(n-2)
Have I got lost somewhere again?
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Ann wrote:

No,I think you are on your way now. However, u(n) cannot affect x(n) as you have indicated in your equation. u(n) can only affect the following positions at x(n-1) and later. Think about it. You have neglected the ZOH ( zero order hold ). If you let us know how you found your answer we can help you with adding the ZOH. However, what you have should be good enough to start testing your fuzzy logic controller. You should add the ZOH to your simulator.
Peter Nachtwey
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Results of continuous system approximation in discret time domain depends on choosen sample time and method of approximation.
When you take other sampling time you get completely new Z-transform. The same about changing method.
One approach of aprroximation is to get exact time response. ( Same values as time response of continuous system -exact at sampling moments ofcourse). Next approach is to get exact frequency domain response.
You for almost sure are interested in time response maping. We can use step equivalent method. Remember that ideal impulsating device doesn't exist in real life. Read about Zero Order Holder (ZOH, thats what A/D devices do).
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Mikolaj

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Dnia Thu, 11 May 2006 09:19:33 +0200, Mikolaj
I meant D/A of course that turns numbers into energy with "sample and hold" method most often.
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Mikolaj

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napisa�:

There is usually a method given in any undergrad control textbook
From an earlier post there are 3 systems using zoh (with various sample times) - which just means that the o/p from your controller is a constant between outputted values - which i assume you want.
the final difference equation should include a z^-q delay in the controller output (as things don't happen instantaneously after the measurement is made) q being the integer number of sample time it takes for the control calculation and includes any delay (buffer code) that is necessary (consider that some control calcs may take longer than others - depending upon the error and the rate of change in error)
Would be surprised if your system is not close to
Transfer function: 1.85e-005 z + 1.826e-005 ------------------------ z^2 - 1.96 z + 0.9608
Sampling time: 0.1
Transfer function: 1.873e-007 z + 1.87e-007 ------------------------ z^2 - 1.996 z + 0.996
Sampling time: 0.01
Transfer function: 1.875e-009 z + 1.875e-009 ------------------------- z^2 - 2 z + 0.9996
Sampling time: 0.001
The above use the simple substitution for 's' and includes a zoh
There are other methods as i've said - an appreciation of these is important.
Kieran