Finding K values for all poles of real parts are less than -2

Question

I have an open loop transfer function with k > 0 as below:

$$ L(s) = \dfrac{k}{s(s^2+8s+25)} $$

Now for the closed loop transfer function, I would like to find the k value which makes all the real parts of poles less than -2. Below is how I get the closed-loop transfer function:

$$ G(s) = \dfrac{L(s)}{1+L(s)} = \dfrac{k}{s^3+8s^2+25s+k} $$

Yet, I have no idea or method to locate the value k. I have used the Routh Criterion, and found that the boundary is $$200 > k > 0$$

However, this does not provide any useful information for me, is there any better way to solve this?

Thank you in advance.

Answer 1

Let's make the variable change \$u=s+2\$ in the polynomial \$P(s)=s^3+8s^2+25s+k\$. The result is the polynomial $$ Q(u)=P(u-2)=(u-2)^3+8(u-2)^2+25(u-2)+k=u^3 + 2u^2 + 5u + k - 26. $$ The Routh-Hurwitz criterion for this polynomial gives us the conditions \$k-26>0\$ and \$10>k-26\$, thus, \$26<k<36\$.

Since \$26<k<36\$ implies \${\rm Re}\; u<0\$, we also have \${\rm Re}\; (s+2)<0\$ and \${\rm Re}\; s<-2\$.