(Control Theory : Classical Control : Black's formula)

Perhaps the simplest and yet most useful formula you will come across in feedback control theory. Black's formula is a quick method for deriving the equivalent transfer function of a feedback loop in a particular system, as defined by a block diagram.

For a given linear time-invariant (LTI) feedback system:

                       transfer function blocks
                          ________  ________
                  +       |      |  |      |
       input  -----O------| A(s) |--| B(s) |------------  output
                  -|      |      |  |      |      |
                   |      +------+  +------+      |
                   |           ________           | 
                   |           |      |           |
                   +-----------| C(s) |-----------+
                               |      |
                               +------+

(s is the complex frequency--see Laplace transform)


                                                        forward path
The equivalent closed-loop transfer function = G(s) = ----------------- 
                                                          1 - loop

Where "loop" is:
 * Positive for positive feedback (as shown in the formula)
 * Negative for negative feedback (resulting in a denominator of 1 + loop)

Usually, negative feedback is what you want since the name of the game is control, not "let's watch the system blow up". As the system runs, the output is continuously subtracted from the input, thus stabilizing the system, assuming you've designed the controller correctly, of course.

For example, using the block diagram above:

forward path = A(s)B(s)

        loop = -A(s)B(s)C(s) (negative feedback)


             A(s)B(s)               A(s)B(s)
G(s) = --------------------- = ------------------
        1 - (-A(s)B(s)C(s))     1 + A(s)B(s)C(s)

For more complicated systems with multiple and/or embedded feedback loops, simply repeat the procedure above as many times as needed.