The transfer function is a function that models the system’s output for each possible input in the complex frequency domain. It is defined as the Laplace transform of the impulse response.

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

For the above transfer function, write the differential equation relating the output to the input assuming that the systems are controllable and observable