I am going through the documentation for Control System Toolbox 9.5 (R2013a) to learn more about how to use MATLAB with Bode diagram design. Specifically, I am looking at the section titled "Adjusting the Bandwidth" of the following page:
In this section, it says:
Because the design requirements include a 0.5-second rise time, try setting the gain so that the DC crossover frequency is about 3 rad/s. The rationale for setting the bandwidth to 3 rad/s is that, to a first-order approximation, this should correspond to about a 0.33-second time constant... For a crossover at 3 rad/s, the compensator gain should be about 38.
How are these numbers obtained? Specifically, I would like to know more details on the following:
1. How does a compensator gain of 38 correspond to a crossover frequency of 3 rad/s for the DC motor transfer function provided?
2. How does the desired rise time of 0.5 seconds lead to the conclusion that the crossover frequency should be 3 rad/s? What does this have to do with the time constant?
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