Control Systems Stability

So I asked this question previously in the control theory section and got no responses. Ill try it here. Basically I need to find the ranges of k and $\beta$ such that the steady state error will be less than 10% for unit step input.

I'm currently at the point where I know that the steady state error is given by:

$$9 < \beta k$$

Now I'm using the closed loop characteristic equation:

$$2s^4 +2s^3+ (2 + \beta)s^2 + (\beta - 10 + 2k)s + 1 +\beta k = 0$$

$$\begin{matrix} s^4 & 2 & (2+\beta) & (1+\beta k) \\ s^3 & 2 & \beta-10+2k \\ s^2 &12-2k & 1+\beta k \\ s^1 & \frac{(12-2k)(\beta-10+2k)-(2)(1+\beta k)}{12-2k} \\ s^0 & 1+\beta k \end{matrix}$$

So at this point I'm not sure how to find the ranges for k and $\beta$ and apply it to the steady state error? Can I say that:

$$12 - 2k > 0$$ $$- 2k > -12$$ $$2k < 12$$ $$k < 6$$

I'm just stuck as this point. Any thoughts? Thanks.

• FYI inline MathJax is [backslash][dollar sign] Math here! [backslash][dollarsign] Refresher here: meta.electronics.stackexchange.com/questions/5565/… Commented Mar 28, 2016 at 6:11
• Thanks for the tips! I didn't even know this sub existed and I'm very excited I found it. Commented Mar 28, 2016 at 6:14