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!

When you have a quadratic cost function, your system is linear and your
system's model is known exactly there's a fairly simple algorithm to
find the "optimal" controller (which turns out to be linear itself).
When you apply this algorithm to a 3rd-order system with an
uncontrollable but observable integrator, and you ask for a
reduced-order controller, the answer is a PID or can be reduced to one.
Since a 3rd-order system with an uncontrollable but observable
integrator is how you model a 2nd-order system with offset, that makes
the PID the optimal controller for such a system.
This is all really cool, except for the following minor bits:
a) No real system is linear.
b) No real system has unchangeable dynamics.
c) No real system has fully known dynamics.
d) No real system has a quadratic cost function.
If you have a linear system model and an idea of how close to linear
your system really is, how much your dynamics are going to change, at
what frequencies your model will break down and how close to quadratic
your cost function really is, then you can decide what will work.
If you have a system that matches the assumptions of the optimal
controller algorithm very closely then you can use it.
If you have a system that varies in the parameters and has a known
variance in the high-frequency departure from the model, and you can
either approximate the system nonlinearities or fold them into the
parameter variation then you can use the techniques of robust control
(either the old-line Bode plot methods or the new cool H-infinity and
H-2 methods) to get an "optimal in the face of variation" controller.
If you have a raspy, nasty system that can't be satisfactorily modeled,
and possibly can't even be fully tested economically, then your best bet
is to use intuition and experience to get a controller that works well
enough, then lock a panel down over the settings and go on to the next
emergency/project/whatever.

For a pure single time constant system such as you have given, the PID
is overkill - you can increase the gain without limit and not get
cycling. You have a maximum of 90 deg phase delay in your process and
nothing will make it even ring.
On the other hand, in the real world, you have a transmiter and final
control element in there somewhere which will add further elements to
the loop ...
Bruce.
perfb wrote:

I agree, a simple on off control is all that is needed for a first order lag
system. If you really want to use an analog output then all that is
required is a PI controller. WHY?
Peter Nachtwey

In general if you have a SISO system with N poles then you can
completely locate it's poles with a controller of order N - 1 -- so if
you don't care about offset you just need proportional control on that RC.
If you take your SISO system and tack on a controller with an integrator
to get rid of offset then you need N-1 additional poles in the
controller (assuming that you don't want any naked differentiators) --
so your RC system requires a PI and you have two parameters
(proportional and integral gain) to use to shove the two poles around.
If you had a 2-pole system to start then you'd use a PID and you'd have
a four-pole system (your original two plus the integrator plus the
differentiator pole) with four parameters (P, I and D gain plus the D
pole position) to use to shove the four poles around.
This all just falls out of the math for pole positioning.

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