Q:

Determine the value of a constant for which the system is marginally stable

So I have the following transfer function
$$G(s)=\frac{s^2+0.1s}{10s^3 + 1.1s^2 + 0.01s + 2K}$$
Now I'm trying to determine the value of K so that I have a marginally stable system. I'm not supposed to use the Routh-Hurwitz method. I'm thinking hard but I seem to get to nowhere. I know that for the system to be marginally stable I will need a real pole in the left complex plane and two complex conjugate pure imaginary poles. But how can I determine the exact value of K that will provide me with those 3 specific poles?  

A:

I guess the simplest way of making sure the roots are located correctly is:
A polynomial with one real root and 2 imaginary roots generally looks as follows:
$$
A\cdot (s+a)\cdot (s^2 + b)
$$
This needs to match the denominator, so
$$\begin{align}
10\cdot (s + a)\cdot (s^2 + b) &= 10\cdot s^3 + (10a) \cdot s^2 + (10b)\cdot s + (10\cdot a\cdot b) \\
&= 10\cdot s^3 + 1.1 \cdot s^2 + 0.01\cdot s + 2K
\end{align}$$
From this it immediately follows that
$$\begin{align}
a &= \frac{1.1}{10} = 0.11\\
b &= \frac{0.01}{10} = 0.001\\
K &= \frac{10\cdot a\cdot b}{2} = \frac{0.11\cdot 0.001}{2} = 55\cdot 10^{-5}
\end{align}$$

