Find K and a for oscillation at 2 rad/s

A system "just oscillates" when it is marginally stable. In the Routh array, the $s^1$ row then becomes zero, and the auxiliary equation from the $s^2$ row gives the frequency of oscillation.

Characteristic equation

$$\begin{aligned} 1 + \frac{K(s+1)}{s^3+as^2+2s+1} &= 0 \\ s^3 + as^2 + (2+K)s + (1+K) &= 0 \end{aligned}$$

Routh array

Condition for sustained oscillation

$s^1$ row $=0$:

$$a(2+K) = 1+K \qquad \text{...(1)}$$

Auxiliary equation from the $s^2$ row, with $s = j\omega$:

$$\begin{aligned} as^2 + (1+K) &= 0 \\ \omega^2 &= \frac{1+K}{a} \qquad \text{...(2)} \end{aligned}$$

From (1), $\frac{1+K}{a} = 2+K$, so (2) becomes $\omega^2 = 2+K$.

Substitute $\omega = 2$ rad/s

$$\begin{aligned} 4 &= 2+K \Rightarrow K = 2 \\ a &= \frac{1+K}{2+K} = \frac{3}{4} = 0.75 \end{aligned}$$

Check

With $K=2$, $a=0.75$: $s^3+0.75s^2+4s+3 = (s^2+4)(s+0.75)$. The roots are $\pm j2$ and $-0.75$, so the system oscillates at 2 rad/s.

Answer: $K = 2$, $a = 0.75$.