A transfer function relates inputs to outputs in the Laplace domain. In particular, the Laplace domain relationship between a manipulated input and output is called a process transfer function.
Consider an nth order differential equation,
Since we are assuming that the model is based on deviation variables, and that the system is initially at steady state, the initial conditions are
Taking the Laplace transform of each term,
Solving for Y(s), we find
The ratio of polynomials is called the ...