A unity feedback system has

\(G(s) = \frac{{K(2s + 1)}}{{s(4s + 1){{(s + 1)}^2}}}\)

What is the value of K if the steady-state value of error is to be less than 0⋅1, when an input r(t) = 1 + 5t is applied?

Given,

 \(G(S)=\frac{K(2S+1)}{S(45+1)(5+1)^2}\)

Steady state error e_ss is less than 0.1

Given ramp input r(t) = 1 + 5t

Velocity constant is calculated for ramp input

\({K_V} = \mathop {Lt}\limits_{S \to 0} S.G(S) = \mathop {Lt}\limits_{S \to 0} \frac{{S.K(2S + 1)}}{{S(4S + 1){{(S + 1)}^2}}}\)

\({K_V} = \mathop {Lt}\limits_{S \to 0} \frac{{K(2S + 1)}}{{S(4S + 1){{(S + 1)}^2}}} = K\)

Slope of ramp input, \(V =\frac{dt}{dt}=5\)

Steady state error, 

\(e_{ss}=\frac{Slope}{K_V}=\frac{5}{K}.\)

e_ss < 0.1

\(\frac{5}{K}<0.1\)

\(\frac{5}{0.1}<K\)

K > 50 

If the steady gate error is less than 0.1 then value of K is