Control Systems Stability

Question

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.

Answer 1

First, your Routh table is not correct. Third element in first column is wrong. When you get correct table, you need to find intersection of all inequalities in first column so that all elements are higher than zero including your condition for steady state error.