How to convert non unity feedback to unity feedback

There are two ways to convert a non-unity feedback system into a unity feedback system:

1. Conceptual Method
2. Shortcut Method

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Conceptual Method

Given a system with non-unity feedback:

Step 1: Extract unity from feedback

$$\begin{align} H(s) - 1 &= (s + 5) - 1 \\ &= s + 4 \end{align}$$

Step 2: Simplify the system

$$\begin{align} G^{'}(s) &= \frac{s-1}{s+3 + (s+4)(s-1)} \\ &= \frac{s-1}{s^2 + 4s - 1} \end{align}$$

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Shortcut Method

Step 1: Find the Closed-Loop Transfer Function (CLTF)

$$\begin{align} CLTF &= \frac{s-1}{s+3 + (s+5)(s-1)} \\ &= \frac{s-1}{s^2 + 5s - 2} \end{align}$$

Step 2: Subtract numerator from denominator

$$\begin{align} G^{'}(s) &= \frac{s-1}{(s^2 + 5s - 2) - (s - 1)} \\ &= \frac{s-1}{s^2 + 4s - 1} \end{align}$$

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Now the system behaves as unity feedback with ( G'(s) ) as the open-loop transfer function.