A System Identification Example
Al Dente's Oven
Getting A Block Diagram

        We have noted the following aspects of the transient response of Al Dente's system.

    The response doesn't start at zero.  It starts at 32 degrees.

    The response of the system that is controlled by the input is the temperature rise above ambient.

        We're going to work with this idea to produce several models - in different forms - for Al's oven.  We'll start with a simple block diagram model.  There's an input to Al's oven when he turns it ON, and there's an output, the temperature.  Whenever you have a system to model, this is probably where you start.  Let's check that out.

        There is a problem with a simple linear model (shown below).

    A linear system, starting from zero initial conditions, has a response that starts at zero.  Al's oven starts out at room temperature.  The non-zero starting condition must be accounted for.

    In a linear system, if you double the input, you double the output.  If we could put twice as much heat into Al's oven,  we wouldn't double the output temperature.  As a matter of fact, if we didn't have any input at all, we would still probably have a non-zero temperature at the output since the oven is probably sitting in an environment where the ambient temperature is non-zero.

    We can conclude that the simple block diagram representation is not going to work for this system.  We need to modify it somehow.

        We need to account for the ambient temperature, but we also need to remember that any measurement we take will have the ambient temperature in it already.  Thinking in terms of the ambient temperature, remember the following.
    The input to the oven produces a rise in temperature above the ambient.

    Doubling the input would double the temperature rise above ambient.

        Using these observations, we can adjust the model to give a truer picture of what is going on.
    Consider the output of the oven to be the temperature rise above ambient temperature.
        Then, you can devise a strategy that will give you a model that better represents what actually happens.  Here is what we will do.
    Add in the ambient temperature to get the actual value of the oven temperature.

    And, remember that the oven dynamics are those of a first order linear system.

    Here is a first attempt at getting the ambient temperature into the model.


        Is this truly "a truer picture of what is going on"?

    The response doesn't start at zero.  We want the response to start from the ambient temperature - not from zero.

    However, this block diagram would indicate that a sudden change of ambient   temperature is reflected in a sudden change in the oven temperature.

    Do you think that could be true?  Could a sudden change in ambient temperature cause a sudden change in the internal oven temperature.

        Click on your answer:

 This may be a better model of what actually takes place.

    In this situation, if the ambient temperature changes suddenly, the output temperature will change eventually.
   For example, if enough heat is applied to raise the oven 20 degrees above ambient and the ambient is 30 degrees, the oven would reach a steady state of 50 degrees.  If the room becomes warmer - the ambient temperature goes up - to 40 degrees, eventually the oven would reach 60 degrees as long as the same amount of heat was input to the oven.
    While this cures one problem, another one crops up.  The ambient temperature is multiplied by the DC gain in this scheme, so we can't put just the ambient temperature in.  We need to divide it by the DC gain.  Here's that modification.

        This model accounts for ambient temperature better than the first model above.  Now, the next question is how to use the model.

        And what this really is is the end of this page.

        Now, click here to go back to Al Dente.