Block Diagram Reduction

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91divine

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Hello guys can you please solve this block diagram, i am stuck in this, and i have to solve this to get my project going on automated speed control of any rotating objects
 

We can call the output of each block: first sum, G1, second sum and G2 respectively a,b,c,d
And R is the input and out is the output, then:

1. a=R-H1d
2. b=aG1
3. c=b-Out
4. d=cG2
5. Out=d+aG3

from 2,3 and 4, --> d=(aG1-Out)G2 = aG1G2-OutG2
using now 1. --> d=(R-H1d)G1G2-OutG2 rearranging d=(RG1G2-OutG2)/(1*H1G1G2)

from 1. and 5. --> out=d+(R-H1d)G3 rearranging Out=d(1-H1G3)+RG3

substituting d with the value previously calculated from 1,2,3 and 1 --> Out=[(RG1G2-OutG2)/(1*H1G1G2)](1-H1G3)+RG3

you can now rearrange to find Out=f(R)

Please check all the calculations, there could be some mistake.
 

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