Error constants of 5/[s(s+2)(s²+2s+8)], r = 2+5t+2t²

The steady state error of a unity feedback system follows from its static error coefficients. $G(s)$ has one pole at the origin, so the system is Type 1.

Stability check

$1 + G = 0$: $s^4 + 4s^3 + 12s^2 + 16s + 5 = 0$.

No sign change, so the system is stable.

Static error coefficients

$$\begin{aligned} K_p &= \lim_{s\to 0} G(s) = \infty \\ K_v &= \lim_{s\to 0} sG(s) = \frac{5}{(2)(8)} = 0.3125 \\ K_a &= \lim_{s\to 0} s^2G(s) = 0 \end{aligned}$$

Steady state error for $r(t) = 2 + 5t + 2t^2$

Write $2t^2 = 4\cdot\dfrac{t^2}{2}$.

$$\begin{aligned} e_{ss} &= \frac{2}{1 + K_p} + \frac{5}{K_v} + \frac{4}{K_a} \\ &= 0 + \frac{5}{0.3125} + \frac{4}{0} = 0 + 16 + \infty = \infty \end{aligned}$$

Answer: $K_p = \infty$, $K_v = 0.3125$, $K_a = 0$; $e_{ss} = \infty$. A Type 1 system cannot follow the parabolic part of the input.