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Let say we have a DC-DC converter operates with double loop control (inner and outer) , and with switching frequency of Fs= 20000 Hz. Please ,if you could answer the two following questions:

1- Usually, the inner current loop bandwidth is chosen based on the relation of Fs/10 , = 2000 Hz in this case Way?

2- in a two loop control of DC-DC converter such as : boost converter , always the bandwidth of inner current loop is faster or higher than the outer loop .why? Some paper mentioned that , The outer control loop bandwidth can be calculated as = inner loop bandwidth/10 = 200 Hz in this case Why?

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    \$\begingroup\$ In any control system the inner loops has to be faster than the outer. The rule of thumb is telling that it should be a factor of 10 at least. To work properly the inner loop has to stabilize quickly in response to any outer loop changes. \$\endgroup\$ – Eugene Sh. May 27 '19 at 15:57
  • \$\begingroup\$ @EugeneSh. You should make that an answer, it sums things up perfectly. \$\endgroup\$ – TimWescott May 27 '19 at 16:01
  • \$\begingroup\$ The reason that multi-loop control systems are designed with a fast inner loop and a slower outer loop is because doing it the other way around basically renders the inner loop pointless. If you can do the job with a fast outer loop and a slow inner loop, then in nearly all cases you can just leave the inner loop off. \$\endgroup\$ – TimWescott May 27 '19 at 16:03
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In any control system with an integrated function, such as a pulsed voltage thru an inductor and resulting current integrated to an output voltage makes is very sensitive to overshoot and settling time with say 25% step loads over the whole range of rated current. The result is a change in loop gain which makes the PID servo voltage with a single feedback voltage less than optimal.

When load current changes quickly you want to respond quickly with supply current so you have a 1st order or proportional feedback (P of PID). Similarly for regulating voltage with slower changing voltage one uses voltage feedback. Ideally, you always want 1st order feedback on disturbances (load current) and 1st order feedback on the output variable (voltage) giving a dual loop for better results with some integral and derivative gain to complete the optimization for overshoot and settling time.

The same is true for position servo's in HDD's, and all sorts of control systems. In some cases, there may be many variables and one feedback loop for each output variable only if the stability of each variable is important, such as acceleration, velocity and position. It is also true for PLL's with frequency being the integral of the phase comparator.

The block diagram will look something like below for a boost converter.

enter image description here

The time constants of each loop may be much faster than you suggested depending on the driving rate and energy storage time constants with the output Cap, C*ESR=Tau being one of them to handle very short term transients, such that the bandwidth of that servo must be faster than the cap storage bandwidth \$f_{-3dB}=0.35/(ESR*C)\$ in order to regulate step loads.

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