# Control system with nested loop

I have a question about a loop inside another one like this:

What is the constraint to ensure stability? Does the bandwidth need to be faster for the inner one?

• There's nothing specific to stop it being unstable if the inner loop had infinite bandwidth. The devil is in the detail and not in the generalizations. Commented Dec 13, 2023 at 15:02
• See here, for example: electronics.stackexchange.com/questions/518642/…
– LvW
Commented Dec 13, 2023 at 15:07

"What is the constraint to ensure stability? Does the bandwidth need to be faster for the inner one?"

No - the only constraint is that the poles of the closed-loop function must be within the left half of the s-plane. In many cases, this requirement is identical to a loop gain (loop open) which has a positive stability margin.

When there are more than one loop (as in your case) both loops must exhibit a positive stability margin. With respect to a "good" timely behaviour (step response) of the closed loop, in most cases the "inner" loop is made faster (larger loop bandwidth) than the outer loop - however, this is not a strict requirement.

• Supplement (edit): Perhaps the question arises "How to find the stability margin of the circuit"?

Answer: It is not possible to find the stability margin of the "circuit" because we can define three different loops with three different openings and three different stabiliy margins.

Three possible openings: Input B1, input B2 output H2. Each of these 3 loops will have its own stability margin. Without deeper investigation it is not posible to say which loop (or which margin) will mostly determine the behaviour of the complete circuit.

Another interesting property of the 3 loops: They have different loop gains and different stability margins - however, if one loop reaches the stability limit, this also applies to the other two loops: they reach this limit together.

• Thanks for your reply, well I thought that if the inner loop is lower bandwidth, it will impact the overall loop because of the phase shift and therefore the output of the inner stage can't follow the output of the first stage (A1) in case of reference is evolving. Yes you're right, the fianl question is how to find the stability of the system Commented Dec 18, 2023 at 13:54
• Hi, could you please explain more, why it's not a strict requirement to have the bandwith of the inner loop greater than the one of the outerloop ? I thought that if the innerloop bandwidth is greater than the outer loop bandwidth that could create instability if the system have to manage a signal with a frequency above the innerloop but below the outerloop Commented Jun 21 at 15:04
• Why should it be a "strict requirement" ? Only stability with a sufficient margin can be a "strict requirement". However, a better and more logical question is if it makes sense to require that the inner loop is the fastest one. The answer: Yes - it makes sense. Otherwise, the outer loop would react already before the inner loop - which determines the forward path of the whole sysytem - has settled. That means: The outer loop would try to stabilize the system before the heart of the system (the "plant" in terms of control theory) has stabilized internally.
– LvW
Commented Jun 21 at 18:27

Overall system stability is dependent upon location of poles only. You can simplify above block diagram and treat whole system as one and then proceed for stability test or you can analyse inner and outer block separately. when analysing outer block treat an inner block as variable. Here inner block poles and zeros act as open loop poles and zeros for outer block or overall system. so Root locus diagram of overall system starts from poles and zeros of inner block.

• While speaking of poles, I suppose you mean the closed-loop poles, right? That is obvious. However, simplification of the three-loop system does not help at all because you have exactly three different methods to develop a single-loop system - with THREE DIFFERENT LOOP GAINS. More than that, for loop gain analyses ,the input is grounded. Under this condition, you cannot distinguish anymore between inner and outer loops. Both loops meet in the most right summing junction - and are on equal footing.
– LvW
Commented Dec 18, 2023 at 15:05