# Problem 2 -- What's the second element in the state of the system?

**Q: **
This problem requires a phase portrait of the model. I was wondering
what two variables would be plotted in the portrait? One is easy to
figure out, but I'm not really sure what the other one would be...

**A: **

You're right that a one-dimesional phase portrait would be rather boring (and in this case, incorrect).
In this problem, you are modeling the **closed-loop** behavior of
the controller built in simulink last week. Keep in mind that when modeling feedback control,
additional states can arise that do not appear in the original dynamics.

This model is going to take the same form as the finger&flame problem had last set -- it needs to return the derivative as a function of time and current state (but in in this case, state is a vector).

Let the state X = [x1;x2]; you need to calculate to calculate dX/dt as a function of X and t.
If you know x1, go ahead and try to write dx1/dt for the
closed-loop system as a function of time and x1. In doing this, I think you'll
figure out what the other state needs to be. It may also help to remember
the formal definition for state of a system, and to **re-read section 3.1**, where the cruise-control example is described in detail.

--Lindzey 19:33, 10 October 2006 (PDT)