How can I solve this block diagram problem?

Question

I practiced various examples of block algebra and I came across this example where I have to reduce the block diagram. The problem now is that I have there successively set branching points and sum points. How can I get rid of them?

Here is how I came to this block diagram:

Starting point and first step:

Second step:

Third step:

Final block diagram:

Don't know what to do now... I thought to merge (G1G2) with (G5/G2 +1) and (G3G4) but again there isn't so much improvement.

Answer 1

I'm not going to write in the answers, but if I redraw your stuff just a little:

One of the things that might make it easier is to duplicate some of your boxes. Consider doubling up the H2*G2*G3 section:

If I redraw THAT just a little then it should show you how to clear up the middle section:

I'll just call whatever the middle resolves to "A" so I can keep going without needing to do the work:

The last tricky business is that summing junction in the middle of that feedback line. What I'd like to is to push the summing junction to the left so it can be with the second summing junction (from the start). Then you can split each branch of the summation into its own feedback line. To do that, first get that H1 out of the way:

Now distribute the negative from the second summing junction and reorganize:

And then finally split the group of summing junctions up into their own lines, which makes the feedback branches clearer.

Now you just do like you'd do, resolve the feedback loops from inner to outer and you should have the reduction.