The transfer function
Due to the convolution theorem, the transfer function has the following relation with the input and output of a system:
The transfer function is defined only for linear time-invariant systems.
Example: For a system with transfer function
Find the zero-state response for input
of
Solution
We don’t need to convert transfer function back to the time domain. Instead, we can solve everything in the complex frequency domain and then convert back.
Our input is
, and . Then This formula does not match anything in the Laplace transform table directly, but we can use partial fraction decomposition: After some algebra we get
. Thus Now we can perform inverse Laplace transform according to the table:
For the above transfer function, write the differential equation relating the output
to the input assuming that the systems are controllable and observable
Solution
We know that
, and we can have . We can then performing inverse Laplace transform. As the is the zero-state response, each initial conditions will be zero. And thus we have