Question: The state-variable equations and the output equation for a dynamic system are given as dot{x}_{1}=x_{2} dot{x}_{2}=-x_{2}-x_{1}+u^{prime} quad y=frac{1}{2} x_{1}+x_{2} end{array} ight.] Find the transfer

The state-variable equations and the output equation for a dynamic system are given as

\dot{x}_{1}=x_{2} \\
\dot{x}_{2}=-x_{2}-x_{1}+u^{\prime} \quad y=\frac{1}{2} x_{1}+x_{2}
\end{array}\right.\]

Find the transfer function (or matrix) by determining the Laplace transforms of $x_{1}$ and $x_{2}$ in the state-variable equations and using them in the Laplace transform of the output equation.

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To find the transfer function lets take the Laplace transform of the statevariable equations beginalign mathcalLdotx1 mathcalLx2 sX1s x10 X2s endalign beginalign mathcalLdotx2 mathcalLx2 x1 ut sX2s x20 X2s X1s Us endalign Now lets express Us in terms of the Laplace transform of the input signal ut Us sUs u0 Given the output equation y frac12x1 x2 its Laplace transform is Ys frac12X1s X2s Substituting the Laplace transforms of the state ... View full answer

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