We derived the transfer function as the output divided by the input in the Laplace domain. Transfer functions will be used only for zero-state input, so initial conditions are always zero. You did Transfer function in E11 when you worked with Bode Plots / Frequency Domain (we'll go back to that later this semester).
A transfer function is typically given as a ratio of polynomials, where Y(s) represents the systems output and X(s) represents the system input. H(s) is the transfer function..
Though this looks daunting, it is less so if we take a specific case, e.g., M=2, N=3.
We also went over some of the reasons Transfer functions are useful are useful. To wit:
1) If the transfer function , H(s), is know then we can get the zero- state response easily for any input. X(s).
and we find y(t) by inverse Laplace Transform. Item 5, below, will use the transfer function to solve the zero-input problem, and hence enable a complete solution to be found.
2) The impulse response of the system is simply the inverse Laplace Transform of the Transfer Function and vice-versa. Therefore if we know either h(t) or H(s) the system is completely defined.
3) If the transfer function, H(s), is know we can easily find the system differential equation, and vice versa. If we consider the general case:
and if we take the inverse Laplace Transform:
Again, this is less daunting if we take a specific case (N=3, but right hand side is only second order (b0=0)).
4) If the transfer function , H(s), is know then we can easily get the systems characteristic equations by setting the denominator to zero.
or for the specific case N=3
5) We can use the Transfer Function to solve the zero-input problem (item 1, above used it to solve the zero-state problem, and hence enable a complete solution to be found.). First use the transfer function to get the zero-input differential equation:
Next, use the differentiation property of the Laplace transform:
Again if we use N=3, we get something more comprehensible
and we find y(t) from the inverse Laplace Transform.
6) The transfer function (or impulse response) is often easy to find experimentally:
Apply an impulse to the system, find h(t). From h(t) determine H(s).
Apply a step to the system. The derivative of the step response is the impulse response. From h(t) determine H(s).
Use frequency domain analysis to determine H(jω). From H(s) find h(t) if needed. We haven't discussed this in detail yet.