Differential Equations for Block Diagram of Satellite Attitude Control System 
I am trying to understand the procedure to setup differential equations from a block diagram. The enclosed example is about the attitude control of a satellite. The ultimate goal is to find a state-space system representation of the model. Transfer functions are the intermediate step in this process; I understand how they are set up. I encouter problems as soon as differential equations must be determined. For example, $\dot{x}_{1}$ is stated to be $0.01K(\theta_{c}-\theta)$, which seems to only account for the lower block of the controller component. Also for $\dot{x}_{2}$ and $\dot{x}_{3}$, the $0.01$ disappears from the equations, which I don't understand. It would be very much appreciated if someone could tell me how I should approach these differential equations.
 A: I think I've understood the basic tricks that the book used, which are what are tripping you up.
The main difficulty I see you struggling with is the arbitraryness of the variable selection.  Could you write the state-space system differently?  Yes.  There are many ways you could write it.  The particular selection the book used is due to specific preferences dictated by consistency of the material they're teaching.
So let me explain how this answers your questions:

For example, x˙1 is stated to be 0.01K(θc−θ), which seems to only account for the lower block of the controller component. 

Yes.  But this is a true statement.  How?  Because x1 isn't what you think it is.  Here, I've tried to re-label the system with where I think they're selecting the variables.  I colored my marking in red.

You can see here that x1 is referring to a particular output.  This is why it's not the summation output.  It's just not that variable.  You could introduce a new variable which is the output of the summation.  They just don't have need for such a variable.
Then there are some more wonky elements of this.  For instance, since the last block is 2nd order, there are 2 independent variables introduced, one of which doesn't have a "location" on the diagram at all.  You would have to split the block into 2 blocks with a line in-between in order to label x3 on the diagram.
I hope that helps.
