Chapter 3, Problem 14P

Find the transfer function G(s) = Y(s)/R(s) for each of the following systems represented in state space: [Section: 3.6]

1. $\begin{array}{l}\text{<mover accent="true"> <mo> x </mo> <mo> ˙ </mo> </mover>}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -3& -2& -5\end{array}\right]\text{x}+\left[\begin{array}{c}0\\ 0\\ 10\end{array}\right]r\\ y=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\text{x}\end{array}$
2. $\begin{array}{l}\text{<mover accent="true"> <mo> x </mo> <mo> ˙ </mo> </mover>}=\left[\begin{array}{ccc}2& -3& -8\\ 0& 5& 3\\ -3& -5& -4\end{array}\right]\text{x}+\left[\begin{array}{c}1\\ 4\\ 6\end{array}\right]r\\ y=\left[\begin{array}{ccc}1& 3& 6\end{array}\right]\text{x}\end{array}$
3. $\begin{array}{l}\text{<mover accent="true"> <mo> x </mo> <mo> ˙ </mo> </mover>}=\left[\begin{array}{ccc}3& -5& 2\\ 1& -8& 7\\ -3& -6& 2\end{array}\right]\text{x}+\left[\begin{array}{c}5\\ -3\\ 2\end{array}\right]r\\ y=\left[\begin{array}{ccc}1& -4& 3\end{array}\right]\text{x}\end{array}$