The unit step response y(t) of a unity feedback system with open loop transfer function \(G\left( s \right)H\left( s \right) = \frac{k}{{{{\left( {s + 1} \right)}^2}\left( {s + 2} \right)}}\)is shown in the figure. The value of K is _______ (up to 2 decimal places.)

\(G\left( s \right)H\left( s \right) = \frac{k}{{{{\left( {s + 1} \right)}^2}\left( {s + 2} \right)}}\)

From the given figure

steady state value of y(t) = 0.8

By using final value theorem,

steady state y(t) value of \(= \mathop {{\rm{lt}}}\limits_{s \to 0} s\;y\left( s \right)\)

\(\frac{T}{F} = \frac{{y\left( s \right)}}{{R\left( s \right)}} = \frac{{G\left( s \right)}}{{1 + G\left( s \right)H\left( s \right)}}\)

\(\frac{{Y\left( s \right)}}{{R\left( s \right)}} = \frac{k}{{\frac{{{{\left( {s + 1} \right)}^2}\left( {s + 2} \right)}}{{1 + \frac{k}{{{{\left( {s + 1} \right)}^2}\left( {s + 2} \right)}}}}}}\)

\(\Rightarrow \frac{{Y\left( s \right)}}{{R\left( s \right)}} = \frac{k}{{{{\left( {s + 1} \right)}^2}\left( {s + 2} \right) + k}}\)

As the input is unit step function,

R(s) = 1/s

\(\Rightarrow Y\left( s \right) = \frac{k}{{s\left( {{{\left( {s + 1} \right)}^2}\left( {s + 2} \right) + k} \right)}}\)

\(\mathop {{\rm{lt}}}\limits_{s \to 0} s\;Y\left( s \right) = 0.8\)

\(\Rightarrow \frac{k}{{\left( 1 \right)\left( 2 \right) + k}} = 0.8\)

⇒ k = 1.6 + 0.8k ⇒ k = 8