Error constants of 1000/[s(s+10)(s+100)], r = 2+t

The steady state error of a unity feedback system follows from its static error constants. The system is Type 1 (one pole at the origin).

Stability check

$1 + G = 0$ gives $s^3 + 110s^2 + 1000s + 1000 = 0$. Routh: $110 \times 1000 > 1000$, so all first-column terms are positive and the system is stable.

Static error constants

$$\begin{aligned} K_p &= \lim_{s\to 0} G(s) = \lim_{s\to 0}\frac{1000}{s(s+10)(s+100)} = \infty \\ K_v &= \lim_{s\to 0} sG(s) = \frac{1000}{(10)(100)} = 1 \\ K_a &= \lim_{s\to 0} s^2G(s) = 0 \end{aligned}$$

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

By superposition:

$$\begin{aligned} e_{ss} &= \frac{2}{1 + K_p} + \frac{1}{K_v} \\ &= \frac{2}{1 + \infty} + \frac{1}{1} = 0 + 1 = 1 \end{aligned}$$

Answer: $K_p = \infty$, $K_v = 1\ \text{s}^{-1}$, $K_a = 0$; $e_{ss} = 1$.