2.010 Modeling Dynamics and Control III
SPRING 2001
Block Diagram Fundamentals
Going through the (simple!) algebra to derives Black's formula
for closing a negative feedback loop is a very worthwhile
exercise. 'Block diagram algebra' ain't called 'algebra' for
nothing: it's just a graphical way of representing algebraic
equations.
At left below is a generic negative feedback system.

'Closing the loop' results in the simplied system at the right. 


First, we will write a (standard) algebraic equation representing
the system at left;
In other words, the output, Y(s), is 'G' times the quantity
X(s) minus 'H' times that same output Y(s).
Now we place all terms in Y(s) on one side of the equation;
X(s) on the other.
Therefore, the transfer function from X(s) to Y(s):is:
This is an incredibly useful shortcut for the simplification of
feedback systems. If you feel comfortable with block diagram algebra,
you should have no problem rederiving this formula from scratch
(although it's quite likely you'll just end up memorizing it 
it's used so frequently in controls).
Last updated February 19, 2001.