A control system is shown in the Figure. Which option represents the correct transfer function of the system?

The correct option/answer is:

$$\frac{1}{s^2+8s+17}$$

Step-by-Step Derivation and Analysis:

1. Identifying the Components from the Block Diagram:
By analyzing the provided control system diagram, we observe the following signals and blocks:
- Input signal: $R\left(s\right)$
- Output signal: $C\left(s\right)$
- First Summing Junction (on the left): Takes the input $R\left(s\right)$ with a positive sign (+) and the feedback signal with a negative sign (-).
- Forward path block: $G_1\left(s\right)=\frac{1}{s+4}$
- Feedback path block: $G_2\left(s\right)=\frac{1}{s+4}$
- Second Summing Junction (on the right): Combines the output of the upper path with a positive sign (+) and the output of the lower path feedback block with a negative sign (-).

2. Formulating the System Equations:
Let $X\left(s\right)$ be the output of the first summing junction.
The output $C\left(s\right)$ is the output of the upper forward path:

$$C\left(s\right)=X\left(s\right)\cdot \frac{1}{s+4}$$

This can be rewritten to express $X\left(s\right)$ in terms of $C\left(s\right)$:

$$X\left(s\right)=(s+4)C\left(s\right)$$

For a standard feedback system with a forward path $G\left(s\right)=\frac{1}{{(s+4)}^2}$ and a unity negative feedback loop $H\left(s\right)=1$, the closed-loop transfer function is defined by the relation:

$$T\left(s\right)=\frac{G\left(s\right)}{1+G\left(s\right)H\left(s\right)}$$

Substituting $G\left(s\right)=\frac{1}{{(s+4)}^2}$ and $H\left(s\right)=1$ into the equation:

$$T\left(s\right)=\frac{\frac{1}{{(s+4)}^2}}{1+\frac{1}{{(s+4)}^2}}$$

3. Simplifying the Transfer Function:
Multiply the numerator and denominator by ${(s+4)}^2$:

$$T\left(s\right)=\frac{1}{{(s+4)}^2+1}$$

Expand the quadratic term in the denominator:

$${(s+4)}^2=s^2+8s+16$$

Now, add 1 to the expanded form:

$${(s+4)}^2+1=s^2+8s+16+1=s^2+8s+17$$

Thus, the final closed-loop transfer function of the control system is:

$$T\left(s\right)=\frac{1}{s^2+8s+17}$$