To avoid interaction between the loops, the inner control loop should
respond AT LEAST 3 times faster than the outer loop. We cover cascade
controls, and tuning of cascade loops, in this recorded webinar:
http://www.expertune.com/pastWebinar.asp?name=HowToControlCascade2009Oct27&nameDesc=How%20To%20Control%20a%20Cascade%20System

Because it simply works out that way mathematically. At least in one
case.
I can prove that for a single pole system, such as a motor with a
velocity loop and a position loop, that the position loop must have a
lower bandwidth than the inner loop. This is not a general proof but
for a single pole system a few things should be obvious. First, the
closed loop transfer function of the inner loop will have two poles.
One for the system and on for the integrator that comes with the inner
loop integrator gain. The output loop will have 4 poles. There
will be two from the inner velocity loop. One for integrating
velocity to position and one from the outer loop integrator gain. It
should be obvious that the inner loop two poles system will be much
faster than the outer loop four pole systems unless the outer loop
poles are very fast relative to the inner loop poles. So how much
faster can the outer loop poles be made relative to the inner loop
poles? The answer is not fast enough. I tried an example where the
inner loop was tuned to be critically damped with the characteristic
equation being (s+lambda)^2 and then tried to chose outer loop poles
that would be faster. I chose a desired characteristic equation of (s
+mu)^2*(s+delta)*(s+gamma) and found the symbolic solutions for
them. I found there was a narrow range for mu relative to the inner
loop poles at -lambda. mu cannot be made to be greater than lambda
without moving delta and gamma to the right hand side and therefor
unstable. Also, mu had to be less than 1/3*lambda for Ki to be
positive and mu had to be less than 1/2*lambda for Kp to be positive.
By taking the derivative of either the formula for Ki or the formula
for Kp I found the relative value of mu compared to lambda that
provided the highest gains is about 0.21132*lambda. This is less than
1/3 lambda. As it turned out gamma=0.21132*lambda too but delta
turned out to be 1.366*lambda so the characteristic equation for the
outer loop turned out to be (s+0.21132*lambda)^3*(s+1.366*lambda)
which is much slower than the inner loop's characteristic equation of
(s+lambda)^2.
I don't consider this a proof but just one example but I can show
symbolically that the inner loop is going to be much faster than the
outer loop. I am pretty sure that more complicated systems will have
similar results for similar reasons. I don't agree with with the
statement on page 257 in the document JCH posted a link too. I can
prove the statement is wrong at least in this case.
Note, as a by product of this work it was found that inner loop AND
the outer loop can be tuned once I know the system gain and bandwidth
and by choosing inner closed loop pole locations at -lambda. The
outer loop Ki=0.01289*lambda^2 and outer loop Kp=0.192459*lambda
Peter Nachtwey

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