Why we study continuous time systems if in the real life we control them using micro-controller which is discrete time .. ?
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3$\begingroup$ Engineers were designing control systems for real-world applications for centuries before micro-controllers were invented. For example there are plenty of planes still flying that use completely mechanical control systems like this: sandy-aircadets.org.uk/resources/senior/piston/… $\endgroup$– alephzeroAug 26, 2016 at 14:24
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$\begingroup$ Analog controllers are often used for control loops that have to be fast, faster than a digital controller can sample. $\endgroup$– Nick AlexeevAug 27, 2016 at 3:46
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A controller is built around a physical system. What is the open-loop behavior of the system?
You have to sample the physical (continuous-time!) system in order to provide feedback to your discrete controller. What sampling rate should you be using for that microcontroller? Are you sure?
Once you have the physical system modeled and the sampling mechanism designed, how do you develop the controller? Well, with Laplace transforms you could do PID control or state feedback control, but all of those use continuous time differentiators ($s$) or integrators ($1/s$).
So, how do you perform differentiation or integration in a discrete control system? Easy - the Z transform! But, which Z-transform? Do you use the Tustin Bilinear Z-transform? Well, that's great but it has some frequency warping. You could maybe avoid that by using a Z-transform that maps your poles correctly, or you could use a Z-transform that maps the impulse response, or you could do something more advanced.
Point being that, like all of engineering, there are trade-offs. Every method of converting a continuous time controller to a discrete time representation of that controller introduces unwanted aspects. It's up to you to understand the various drawbacks and apply the correct approach.
But, how can you understand a drawback unless you have a "perfect" system by which to compare all the others? This is a similar argument for studying ideal/frictionless systems in physics - you need to understand what the best possible outcome is in order to understand your limitations.
Nevermind the fact that you could also implement an analog controller.
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$\begingroup$ I'll ask another question sorry if it irrelevant to the main question, After i make the model for my system which one should i use z transform or laplace transform ..? because as you know it differ in z transform the system is stable if the poles are within the unit circle and in the laplace transform it's stable if real part of the poles is < 0 . $\endgroup$ Aug 26, 2016 at 15:48
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$\begingroup$ They're two sides of the same coin. Z-transform is the discrete version of the Laplace transform. I would say it's generally easiest to do all of your work in the continuous-time domain and later, when you've got a controller that simulates well, convert from continuous time to discrete time by applying the Z-transform anywhere you see an 's'. I'd say this is similar to algebra - do you substitute the numbers in the beginning or in the end? Either way you should end with the same answer, but by substituting from the start you lose both conceptual understanding and ease of later substitutions. $\endgroup$– ChuckAug 26, 2016 at 18:35
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1$\begingroup$ @JafarAbdi Why they are two sides of the same coin can be understood from the conversion between the two $z^{-1}=e^{-T\,s}$. Any $s$ in the right have plane will result into a $z$ which lies outside the unit disk. $\endgroup$ Aug 27, 2016 at 23:45