Block Diagram Reduction - Control Systems Engineering

[MechEngJordan]

Homework Statement

Determine the closed loop transfer functions of the systems shown below:

Homework Equations

Standard block diagram algebra.

The Attempt at a Solution

I've moved the take off point such that the diagram now looks like:

I am having difficulty determining G₁(s). The solution given is

G₁(s) = 4(s + 2) + 1 = 4s + 9

But it is not obvious to me where this is coming from. Taking G₁(s) as given, I am able to solve the rest of the question without issue.

Any help appreciated.

 

[FactChecker]

The question is a little confusing. The combination of the (s+2) and the gain 4 is clearly 4(s+2). Is that what you mean by G₁? If you want to consolidate that loop (second diagram) including the summation block, it would be 4(s+2)+1.

 

[LvW]

G₁(s) = 4(s + 2) + 1 = 4s + 9
But it is not obvious to me where this is coming from. Taking G₁(s) as given, I am able to solve the rest of the question without issue.
.

You have nothing to do than to write down the output of the summing block as a function of the input:
Vout=Vin + Vin*4*(s+2)=Vin(1+4(s+2)).
From this it follows Vout/Vin=4s+9