Error constants and K range for K/[s(s+1)(s²+2s+2)]

$G(s)$ has one pole at the origin, so the system is Type 1: the step error is zero and the ramp error is $1/K_v$.

Static error constants

$$\begin{aligned} K_p &= \lim_{s\to 0} G(s) = \infty \\ K_v &= \lim_{s\to 0} sG(s) = \frac{K}{(1)(2)} = \frac{K}{2} \\ K_a &= \lim_{s\to 0} s^2G(s) = 0 \end{aligned}$$

Steady state error for $r(t) = 10 + 50t$

$$e_{ss} = \frac{10}{1 + K_p} + \frac{50}{K_v} = 0 + \frac{50}{K/2} = \frac{100}{K}$$

Condition $e_{ss} < 0.5$

$$\frac{100}{K} < 0.5 \;\Rightarrow\; K > 200$$

Stability check (needed for the result to be valid)

Characteristic equation: $s^4 + 3s^3 + 4s^2 + 2s + K = 0$.

For stability: $2 - 0.9K > 0$ and $K > 0$, so $0 < K < 2.22$.

Answer: the error condition needs $K > 200$, but the system is stable only for $0 < K < 2.22$. The two ranges do not overlap, so no value of K meets $e_{ss} < 0.5$ with a stable system. A compensator (for example lag compensation) is needed to raise $K_v$ without causing instability.