A System Identification Example Problem
Al Dente's Oven

        The goal for you in this example problem is:

  Given the step response of a first order system
  Determine the parameters - DC gain and time constant of the system.
        Here's a problem to see how that goes.

        The great and famous Italian chef, Al Dente, has a quick heating oven which he uses to keep ingredients warm when that's necessary.  He has taken some temperature data for the oven from the time it was turned on until it seemed to be at a reasonably constant temperature.  Here's the data.  Is Al's oven a first order linear system?  If it is, then you should be able to get a mathematical description for the system, either a transfer function or the differential equation that relates input and output.  So, the first question is the following.


  What kind of evidence would lead us to accept the conclusion that it's a first order linear system?
  After you've considered the question, then continue.
        Here are some things to consider as evidence.  Click on each hot word for further information.
The steady state value.
The general shape of the curve.
The start-up portion of the curve.
        The items above are things that you can consider any time you have time-response data and you are trying to produce a model for the system.  However, you also need to consider how you go about obtaining that model, and the details of the model.  If you have reached this point, you should know how to produce a model from straight-forward data.  However, there is at least one detail of Al Dente's data that makes this problem a little more perverse.
How come the data doesn't start at zero?
If the data doesn't start at zero, can we still produce a block diagram model?

        Now, we can also note that the oven dynamics are described by our standard first order linear differential equation for this type of system.  Here's that differential equation again.

And, that is what we are going to accept - at least for now - for our model of Al Dente's oven.  Let's move on to using the model.


Using The Model

        Ok, you've gotten a model for this oven.  It's typical of the kinds of models you can get for many other systems.  What can you do with it?