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Oh, that PID example actually cleared up another confusion I had. Thanks for taking the time, really appreciate it.

Oh sorry, yes it is a made up exercise. It's not homework or anything, I would just like to understand maybe in a more general sense how difference equations relate to discrete transfer functions like differential equations relate to continuos time transfer functions.

Y is the output and U is the input. X is an intermediate variable I suppose. I did not make this up.

If you had a difference equation model of say x(k+1) = 0.2x(k)+u(k) and y(k+1)=-0.4y(k)+x(k)+u(k-1), how would you develop a transfer function in the discrete time domain. Likewise, working backwards, if you were given a transfer function in the continous domain of say (s+3)/(s^2+4s+16)...