Closed loop vs. open loop system analysis and design

Question

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.

Answer 1

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.