Chapter 3, Problem 26P

State-space representations are, in general, not unique. One system can be represented in several possible ways. For example, consider the following systems:

1. $\begin{array}{l}\stackrel{\dot{}}{x}=-5x+3u\\ y=7x\end{array}$
2. $\begin{array}{l}\left[\begin{array}{c}{\stackrel{\dot{}}{x}}_1\\ {\stackrel{\dot{}}{x}}_2\end{array}\right]=\left[\begin{array}{cc}-5& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}3\\ 1\end{array}\right]u\\ y=\left[\begin{array}{cc}7& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\end{array}$
3. $\begin{array}{l}\left[\begin{array}{c}{\stackrel{\dot{}}{x}}_1\\ {\stackrel{\dot{}}{x}}_2\end{array}\right]=\left[\begin{array}{cc}-5& 0\\ 0& -1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}3\\ 0\end{array}\right]u\\ y=\left[\begin{array}{cc}7& 3\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]\end{array}$

Show that these systems will result in the same transfer function. We will explore this phenomenon in more detail in Chapter 5.