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I have found questions similar to mine in the forum but none were quite answered to my full satisfaction. I have two questions:

1) Why is controller design is traditionally done with the open-loop system? Is it simply that if the system responds in a stable and desirable manner in an open loop setting (e.g., step response), it will similarly do so with a changing input once the loop is closed? What if the system's feedback is not unity, but contains some transfer function?

2) Why is it that open-loop characteristics are always analyzed for a system? I understand that stability margins/crossover frequency are properties exclusive to the open loop, but I'm talking about more descriptive concepts. For example, a closed loop bode plot describes how the system will actually respond to a given forcing function, so why not report on the closed loop phase lag and system gain?

Thank you in advance.

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  • \$\begingroup\$ Why is it that open-loop characteristics are always analyzed for a system? As you said to analyze stability (e.g. gain and phase margin) . Once you know those things you have an idea about how the closed loop system will respond. I don't know why you have the idea that the closed loop system isn't analyzed and tested, I haven't seen many cases where it isn't. However you know a lot about how it should respond just from the open loop transfer function and whatever compensator you put in the loop. \$\endgroup\$
    – John D
    Apr 19, 2020 at 21:41
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    \$\begingroup\$ Regarding point 2: many SISO control engineering techniques (i.e. root locus and Bode plots) were developed in the first half of the 20th century. There was no MATLAB back then (in fact, no computers or even electronic calculators). So you needed a way to estimate whether a given compensator would produce the desired closed loop output characteristics before you went to the trouble of doing the laborious calculations involved in closing the loop. These techniques turned out to be so useful and give you insight into the performance of the system that they're still in use today. \$\endgroup\$
    – swineone
    Apr 19, 2020 at 21:50
  • \$\begingroup\$ Hi John, Swineone, Thanks very much for the helpful insight! \$\endgroup\$
    – Richard E
    Apr 26, 2020 at 1:42

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Open loop analysis (bode/Nyquist) is carried out out on the open loop because the stability margins can be measured, adjusted and set. If the stability margins (and encirclements of the -1 point) are adequate for the open loop then the system should be unconditionally stable when the loop is subsequently closed.

Both Bode and Nyquist are frequency response analysis techniques. The loop is opened, a signal is injected into the break and the phase & magnitude measured at the output of the loop over a range of input frequencies. When this is done it is the complete loop that is included in the measurements including the feedback transfer function. This would be called measuring the "loop" phase and magnitude as opposed to the "open loop" which usually refers to just the forward part of the loop.

Root-Locus analyses the closed loop transfer function by plotting the closed loop poles as some system parameter is varied from 0 to infinity. The parameter to be varied would normally be the open loop DC gain, K. Using this technique the value of K can be determined for a desired stability or damping factor.

Measuring closed loop gain and phase won't give you a measure of how close to instability the closed loop system is, where as open loop analysis allows the designer to set gain margin and phase margin for the desired amount of system stability.

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  • \$\begingroup\$ Hi James, That was very helpful, particularly the "broken loop" vs. "open loop" distinction when building the frequency response. Thank you! \$\endgroup\$
    – Richard E
    Apr 26, 2020 at 1:46

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