Using single-loop analysis in multi-loop structure
Stability analyses for multi-loop systems are somewhat critical.
That does not mean that they are difficult - but the interpretation of the results is &quot;problematic&quot;.
Of course, there are systems where it is possible - for stability analyses - to break (open) two loops at the same time. Simple example: Opamp with positive as well as negative fedback. However, in most cases it will not be possible to open local loop(s) and an overall loop at the same time.
However, the question is: Does it make sense to open two (or even more) loops  at the same time? For which purpose ? What is the information we can gain by doing this?
(1) Case 1: The system is unstable. In this case, all of the existing loops reveal this instability. So - it is not necessary to open all the loops at the same time. In contrary - it is much better to open the loops one after another in order to see which loop has the worst stability margin. This is the best way to decide how the situation could be improved.
(2) Case 2: The system ist stable - but the closed-loop system analysis shows that the stability properties are poor (bad step response). For which purpose should we try to find a node for breaking all the loops at once? In this case, we will get  no information how to improve the step response. It is much better to open again the several loops one after the other - and to find the loop which is primary &quot;responsible&quot; for the bad closed-loop behaviour.
(2) In general: When a system with feedback has n feedback loops, we can define n loop gain functions - and all loops have different stability margins.
(3) Warning: When a system is given as a block diagram (control systems), some people think it would be wise to use the &quot;system block manipulation&quot; rules to create an equivalent single-loop system. But it should be kept in mind that this equivalence is valid for the closed-loop system only! If there are n loops, we can define n different ways for reducing the system to a single loop - with n different loop gain functions (and n different stability margins). Hence, this procedure does not help at all.
(4) Summary: A system with n loops has n stability margins and it is not possible (and not necessary) to find one single margin. It is best to identify that loop which has the smallest margin in order to find a way to improve this margin (if desired).
(5) Example:



simulate this circuit &ndash; Schematic created using CircuitLab
Three possible loop gain functions LG
(1) Break all 3 loops after F1: LG1=-(F1G1+F1F2G2)
(2) Break the lower loop: LG2=-F1F2G2/(1+F1G1)
(3) Break the upper loop: LG3=-F1G1/(1+F1F2G2)
(4) Closed-loop: H=F1F2/(1+F1F2G2+F1G1)