How does one annotate initial conditions on a control system block diagram. Specifically, if $Y(z)$ corresponds to the system output, $X(z)$ to the system input, and $G(z)$ to the plant:
$$Y(z)=G(z)X(z)$$
If $y(0)\neq0$ is there a way to include the information $y(0)=y_0$ in the block diagram?
Generally no. Linear control theory frequently (though often tacitly), assumes the initial condition is zero. Since for a stable system the IC decays at the rate of the fastest timeconstant this simplifying assumption makes sense, since any action by the control system will have little effect on the IC transient.