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CT, System Functions, and Frequency Response

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We are switching from discrete time (DT) modeling with difference equations to continuous time (CT) modeling with differential equations. Since our physical systems are innately CT and our controller (the microcontroller board) is innately DT, neither DT or CT is an obvious "best" choice. So, what possible reason could we have for switching from one to the other? That is, why change from modeling using DT approximations of the physics, to modeling using CT approximations of the controller?

We are introducing several new analysis tools, ones whose primarily goal is to help you develop insights about the systems you are modeling and controlling. And in our opinion (not a unanimous one, admittedly, even among the staff) the tools we will use (transfer functions, block diagrams and frequency responses) provide more insight in the continuous-time setting.

In this prelab, we are trying to introduce three ideas:

  1. The derivation and use of transfer (or system) functions.
  2. Modeling systems using block diagrams.
  3. Frequency response and sinusoidal-steady-state.

In the first of the problems below, we use a first order differential equation to introduce the CT natural frequency and compare to the DT case. In the second problem, we examine system functions more generally, the third problem is intended to give you practice with transfer functions and feedback, and the fourth problem (well just notes actually) describes frequency response.