State-space representations are, in general, not unique.

Chapter 3, Problem 26

(choose chapter or problem)

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

(a) \(\begin{aligned} \dot{x} & =-5 x+3 u \\ y & =7 x \end{aligned}\)

(b) \(\begin{aligned} {\left[\begin{array}{l} \dot{x}_1 \\ \dot{x}_2 \end{array}\right] } & =\left[\begin{array}{rr} -5 & 0 \\ 0 & -1 \end{array}\right]\left[\begin{array}{l} x_1 \\ x_2 \end{array}\right]+\left[\begin{array}{l} 3 \\ 1 \end{array}\right] u \\ y & =\left[\begin{array}{ll} 7 & 0 \end{array}\right]\left[\begin{array}{l} x_1 \\ x_2 \end{array}\right] \end{aligned}\)

(c) \(\begin{aligned} {\left[\begin{array}{l} \dot{x}_1 \\ \dot{x}_2 \end{array}\right] } & =\left[\begin{array}{rr} -5 & 0 \\ 0 & -1 \end{array}\right]\left[\begin{array}{l} x_1 \\ x_2 \end{array}\right]+\left[\begin{array}{l} 3 \\ 0 \end{array}\right] u \\ y & =\left[\begin{array}{ll} 7 & 3 \end{array}\right]\left[\begin{array}{l} x_1 \\ x_2 \end{array}\right] \end{aligned}\)

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