  The PI control of first-order linear passive systems through a delayed
communication channel is revisited in light of the relative stability concept
called sigma-stability. Treating the delayed communication channel as a
transport PDE, the passivity of the overall control-loop is guaranteed,
resulting in a closed-loop system of neutral nature. Spectral methods are then
applied to the system to obtain a complete stability map. In particular, we
perform the D-subdivision method to declare the exact sigma-stability regions
in the space of PI parameters. This framework is then utilized to analytically
determine the maximum achievable exponential decay rate of the system while
achieving the PI tuning as an explicit function of the decay rate and the
system parameters.
