Transfer Function Review

The transfer function of the multiplicative feedback would be really interesting to figure out. Here's Murray's book (he taught the CDS class at Caltech). I was looking through chapter 8 - Transfer Functions to remember how to compute them.

All signals can be represented by a complex exponential - exp(st), s= \sigma + i\omega. Linear systems are represented by two equations, a diff eq and linear eq:

dx/dt = Ax + Bu
y = Cx + Du

where u is the input signal, and y is the output signal. The transfer function is simply the input-output function. For a linear system with where the output is some order derivative (or integral) of the input then the transfer function is just:

G(s) = b(s)/a(s)

Where a(s) and b(s) are polynomials. Poles and zeros are the roots of a(s) and b(s), respectively.

And then you can do "Block Diagram Algebra" for more complex systems.

And this can all be further formalized with laplace transforms, but that's basically what he was doing before.