Hi again,
To understand these in general the following might help...
Code:
FIG 1 +----A--->---o R*A
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R o---+-->--[Sum]---->---o R+H*X
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+--<----H-----o X
FIG2 +----A--->---o (R+H*X)*A
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R o---+-->--[Sum]--+-->---o R+H*X
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+--<----H-----o X
FIG 3 +----A--->---o R*A+H*X*A
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R o---+-->--[Sum]--+-->---o R+H*X
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+--<----H-----o X
+--<--(-H*X*A)
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FIG 4 +----A--->-------[Sum]---o R*A
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R o---+-->--[Sum]--+-->---o R+H*X
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+--<----H-----o X
+--<--(-H*A)--<--o X
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FIG 5 +----A--->-------[Sum]---o R*A
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R o---+-->--[Sum]--+-->---o R+H*X
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+--<----H-----o X
In the above Fig 1, we see the original diagram. We see there are two inputs and two outputs.
One output is R*A and the other output is R+H*X. The two inputs are R and X.
In Fig 2, we moved the pickoff point forward so now it is behind the summing junction. So the output of that section is no longer R*A but is (R+H*X)*A which factors into R*A+H*X*A (Fig 3). So what changed? The upper output changed from:
R*A
to:
R*A+H*X*A
So what do we have to do to R*A+H*X*A to get back the original R*A?
Simple, we have to subtract H*X*A.
In Fig 4, we see we've added another summing junction that sums R*A+H*X*A and just -H*X*A. We needed a negatve H*X*A so that it would subtract from R*A+H*X*A and leave us with R*A, the original output.
In Fig 5, we've just arranged it a little differently to show that the other input came from X, and that second X input can be tied to the first X input lower in the diagram. Note that the outputs are the same as they were when we started, the only difference being we moved a pickoff point and added another summing junction.
So the rule for moving a pickoff point forward past a summing junction (behind the summing junction) is that you must then subtract everything 'else' that came into the summing junction from the new path such that the path output is the same as it was before. In the example above we had all positives to start with, but there will sometimes be negatives like -H instead of H, and if we have a negative into a summing junction that just means subtraction so we would have had to add instead of subtract H*X*A like we did above. So it is also important to note the signs carefully.
This is still a good exercise because in real life often we do not have all the outputs that we would like to have because there is usually a limited number of variables being measured and/or we want to keep the measurements count down because we want as few sensors as possible. So we might end up having to tap off of a point that is not ideal so we'd have to know what we have to do to it in order to get the kind of measurement we want from that non ideal point in the system.
I assume you have a list of the other more simple rules. If you care to post them we can take a look.