# Why is closed loop frequency response hard to determine?

I'm learning about frequency domain control design and one of its advantages is said to be that: one can deduce the closed loop behaviour of a system by (experimental) data on frequency response on the open loop system through Bode/Nyquist plots. This is useful because I am told it is often easy to find frequency response of the open loop system, but not the closed loop system empirically. Why is this so? Can someone explain to me through a real-world example?

My background is in mathematics and I'm trying to learn control theory on my own, so the lack of practical experience makes some aspects of control design rather abstract and arbitrary to me.

Edit: Here is a quote from a book to clarify what I'm confused over: "In the early days of electronic communications, most instruments were judged in terms of their frequency response. It is therefore natural that when the feedback amplifier was introduced, techniques to determine stability in the presence of feedback were based on this response. Suppose the closed-loop transfer function of a system is known. We can determine the stability of a system by simply inspecting the denominator in factored form (because the factors give the system roots directly) to observe whether the real parts are positive or negative. However, the closed-loop transfer function is usually not known; in fact, the whole purpose behind understanding the root locus technique is to be able to find the factors of the denominator in the closed-loop transfer function, given only the open-loop transfer function. Another way to determine closed-loop stability is to evaluate the frequency response of the open-loop transfer function, then perform a test on that response. Note that this method also does not require factoring the denominator of the closed-loop transfer function"

• Why didn't you ask whoever made that statement? I do not see how measuring the frequency response of a system with feedback (closed loop) would be different from measusing the frequency response of a system without feedback (open loop). I can see that (practical) issues could exist if you would want to measure the open loop response of a system that is normally used in closed loop and where opening the loop prevents the system from working correctly. Nov 19, 2021 at 8:36
• The open loop is usually stable. Closing the loop introduces the possibility of instability, and meaningful measurements cannot then be made.
– Chu
Nov 19, 2021 at 9:57
• @Bimpelrekkie Because it's in a book. I quote it in an edit. If we know the open plant transfer function G, the closed loop transfer function for unity feedback is just G/(1+G), so I was wondering why the say the cltf is not 'known'. I suppose one way to interprete this is like what Neil_UK answered? Nov 24, 2021 at 3:14

This is useful because I am told it is often easy to find frequency response of the open loop system, but not the closed loop system empirically. Why is this so?

It depends what the closed loop servo system is. If it's something big, expensive or dangerous, you want to be 100% sure that it's not going to go unstable before you 'close the loop'. The classic example I had in my coursework was a servo-controlled aiming loop for a battleship main gun, but it could be a robot manipulator or a nuclear power plant. You can't measure the closed loop emperically because you have to actually close the loop and power it up. If it turns out to be unstable though, then it could damage itself or its mountings.

Even if it's not expensive or dangerous, if it's unstable, then you have nothing to measure. It will be stuck on an endstop, or slamming between the two endstops.

What you do with a system like that is measure the open loop response of each part of the chain, to small and controlled stimulii, then deduce whether it's going to be stable. Adjust the parts until you're sure it will be stable, then close the loop and power up.

• As stated by Neil_UK, it is possible to derive stability information (phase margin) from the loop gain Bode diagram. However, it is not possible to derive from it the "closed-loop frequency response" without knowing the feedback network.
– LvW
Nov 24, 2021 at 8:29

"One can deduce the closed loop behaviour of a system by (experimental) data on frequency response on the open loop system through Bode/Nyquist plots."

No this is not possible. Explanation: The classical closed-loop gain ( Acl) expression is:

Acl=Aol/(1+k*Aol) with k=feedback factor and Aol=open-loop gain.

That means:

When you have a Bode diagram only for the product loop gain=k*Aol you will have no information neither about Aol nor k.