Why does the denominator of TF change in MATLAB when multiplying by proportional gain?

Question

Trying to simulate a unity feedback closed loop system with gain of $K$

Let's say I want a proportional gain of $K = 5$. My plant's TF is $G(s) = \frac{10}{s^2+2s+1}$.

I thought that $KG(s) = \frac{50}{s^2+2s+1}$, but when I type it into matlab's feedback function $sys = feedback(k*G,1,-1)$, it returns: $\frac{50}{s^2+2s+49}$.

Am I missing something here? I'm just starting out in my design controls systems course.

Answer 1

Remember when you have feedback, the closed loop system can be calculated as

$$\frac{KG(s)}{1 + KG(s)} $$

But this function should give $\frac{50}{s^2 + 2s + 51}$