R-H range of k and oscillation frequency

Reading of the question: as printed, $\frac{k}{s(s^2+s+1)(s+2)+k}$ is the closed-loop transfer function of a unity feedback system whose open-loop TF is $G(s) = \frac{k}{s(s^2+s+1)(s+2)}$. So the characteristic equation is $s(s^2+s+1)(s+2) + k = 0$.

Characteristic equation

$$\begin{aligned} s(s^2+s+1)(s+2) + k &= 0 \\ s(s^3 + 3s^2 + 3s + 2) + k &= 0 \\ s^4 + 3s^3 + 3s^2 + 2s + k &= 0 \end{aligned}$$

Routh array

Range of k for stability

• $s^1$ row: $14 - 9k > 0 \Rightarrow k < \frac{14}{9} = 1.556$
• $s^0$ row: $k > 0$

$$0 < k < 1.556$$

Sustained oscillation

The system oscillates (is marginally stable) when the $s^1$ row is zero:

$$k_{mar} = \frac{14}{9} = 1.556$$

Auxiliary equation from the $s^2$ row:

$$\begin{aligned} \frac{7}{3}s^2 + \frac{14}{9} &= 0 \\ s^2 &= -\frac{14}{9}\times\frac{3}{7} = -\frac{2}{3} \\ \omega &= \sqrt{2/3} = 0.816\ \text{rad/s} \end{aligned}$$

Check: at $k = 14/9$ the roots are $\pm j0.816$ and $-1.5 \pm j0.289$.

Answer: stable for $0<k<1.556$; sustained oscillation at $k = 1.556$ with frequency $\omega = 0.816$ rad/s.

If $G(s)$ is taken literally as the open-loop TF with unity feedback, the characteristic equation becomes $s(s^2+s+1)(s+2) + 2k = 0$. Every $k$ above is then halved ($0<k<0.778$, $k_{mar}=0.778$), and the frequency stays the same, 0.816 rad/s.