gate 2017 EE(Set-B) q41-solution

gate 2017 EE(Set-B) q41-solution

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Given $$A=\begin{bmatrix}
0 &1 \\
0 & -2
\end{bmatrix}$$
$$B=\begin{bmatrix}
0\\
1
\end{bmatrix}$$
$$C=\begin{bmatrix}
1 & 0
\end{bmatrix}$$\\
$$X(s)=(SI-A)^{-1}[X(0)+Bu(s)]=\frac{\begin{bmatrix}
s+2 & 1\\
0 & s
\end{bmatrix}}{s^2+2s}[\begin{bmatrix}
1\\
0
\end{bmatrix}+\begin{bmatrix}
0\\
1
\end{bmatrix}1/s]=\frac{\begin{bmatrix}
(s+2) &1 \\
0& s
\end{bmatrix}}{s(s+2)}[\frac{1}{s}]$$
$$x(s)=\frac{\begin{bmatrix}
(s+2)+\frac{1}{s}\\
1
\end{bmatrix}}{s(s+2)};y(s)=\frac{s(s+2)+1}{s^2(s+2)}$$
$$y(s)=\frac{1}{s}+\frac{1}{s^2(s+2)}=\frac{1}{s}-\frac{1}{4s}+\frac{1}{2s^2}+\frac{1}{4(s+2)}$$
$$y(1)=\frac{3}{4}4(t)+\frac{1}{2}t4(t)+\frac{1}{4}e^{-2t}4(t)=\frac{3}{4}+\frac{1}{2}+\frac{1}{4}e^{-2}$$
y(1)=1.2838