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:
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Thanks for your replies. The link to the book certainly helped me with more resources to better understand the operation of cascade architectures.
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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