Determine

(i) Error coefficient

(ii) Steady state error for input as $1+4t+\frac{t^2}{2}$.

G(S)=$\frac{10(S+1)}{S^2 (S+2)(S+10)}$

H(S)=1, unity feedback

Error coefficient,

Position error coefficient

$k_p$=$\lim_{s\to 0}$ G(S) H(S)=$\lim_{s\to 0}$ $\frac{10(S+1)}{S^2 (S+2)(S+10)}$ .1

=$\frac {10(1)}{(0)(2)(10)}$=10/0=∞

$k_p$=∞ Velocity error coefficient.

$k_v$=$\lim_{s\to 0}$ S G(S) H(S)=lim/(s→0) $\frac{S. (10(S+1)}{(S^2 (S+2)(S+10)}$ .1

=$\frac{10(1)}{(0(2)(10)}$=10/0=∞ $k_v$=∞

Acceleration error coefficient. $k_a$= $\lim_{s\to 0}$ $S^2$G(S) H(S)=$lim_{s\to 0}$ $\frac{(S^2)(10(S+1)}{S^2(S+2)(S+10)}$.1 =$\frac{10(1)}{(2)10}$ =$\frac{10}{20}$ $k_p$= 1/2

Steady state error is the combination of all the three types of inputs viz. step of magnitude $A_1$=1, Ramp of magnitude $A_2$=4 and parabolic of magnitude $A_3$=1.

Therefore, steady state error, $e_ss$=$e_ss$1+$e_ss$2+$e_ss$3

=$A_1$/(1+$k_p$ )+$A_2$/$k_v$ +$A_3$/$k_a$

=$1/(1+∞)+4/∞+1/0.5$

=$1/∞+4/∞+1/0.5$

=$0+0+1/0.5$

$e_ss$=2