given a simple single-pole RC system

V = V0*(1 - exp(-t/RC))

where one is using a PID control system to set V to some desired setpoint,
is there a unique solution for the PID settings to achieve minimum response
time and minimum overshoot?

I would imagine this is somewhat trivial to solve for?

I have read of the Ziegler-Nichols empirical tuning methods, but havent come across an analytical solution for the simple case above.

What about a second order system of two cascaded RC low-pass circuits?

Also, I have read that the classical PID control system can be shown to be optimal for any second order system, is this true?

tia!

V = V0*(1 - exp(-t/RC))

where one is using a PID control system to set V to some desired setpoint,

I would imagine this is somewhat trivial to solve for?

I have read of the Ziegler-Nichols empirical tuning methods, but havent come across an analytical solution for the simple case above.

What about a second order system of two cascaded RC low-pass circuits?

Also, I have read that the classical PID control system can be shown to be optimal for any second order system, is this true?

tia!