Stabilization problem: Given plant , design controller so that the system has good stability.
The transfer function from to is . In order for the transfer function to be meaningful, we need . This is a well-posed system. Otherwise, it is said to be ill posed.
We can write all the transfer functions from one of the external signals to one of the internal signals as
where each element of the big matrix is the transfer function from the corresponding external signal to the corresponding internal signal.
We can note that there area really only four different transfer functions:
These are called the gang of four.
Among them, we further define the sensitivity function and complementary sensitivity function to be
such that
Note that we can also write the gang of four as