Analysis of Closed Loop Transfer Function

Question

I am trying to understand how to solve a homework problem I have. It states: Analytically show that closed-loop system is stable for all values of K.

By looking at the block diagram, I know that this is of course negative feedback with proportional control as the controller. Below, you can see my plant's transfer function and what I came up with for a closed loop transfer function.

\begin{array}{l} G(s) = \frac{1}{{{s^2} + 4s}}\\ H{(s)_{closedLoop}} = \frac{K}{{{s^2} + 4s + K}} = \frac{{N(s)}}{{D(s)}} \end{array}

I know that the poles here indicate whether the system is stable or not. Now I am not sure how I am supposed to show that every value of K satisfies stability because if I choose a vale such as K=-12, this will land a pole in the right half plane of the s-plane.

Did I mess up on getting the closed loop transfer function? Am I not understanding the question? Any hints on this one is appreciated!

• Andy aka: So, if you agree it's negative feedback, is it still negative feedback if k is negative?
• xxgiuzeppexx: I guess if the gain K was in the feedback look, it's gain should only be positive. But my gain is in the forward loop. Like in this link, so I was thinking I could take on any value of K: google.com/…:
• Andy aka: It's still in the loop whatever way you look at it.
• xxgiuzeppexx: Ah okay, I think I understand. By having a negative gain, when the error amount were to indicate an overshoot or undershoot, I basically screwed this information up by flipping this sign, essentially creating positive feedback, so if it undershot, it keeps undershooting or overshot, it keeps overshooting until it "rails". Is that right?
• Andy aka: That's positive feedback and disallowed how I read the question. Exactly the same as flipping the inputs of an opamp!

Answer 1

Basically, k is "within the loop" that has negative feedback applied. This means that if k was negative, you get positive feedback and this is out of bounds for the question. So, k has to be positive and you can only consider positive values in the question you have been set.