Similarity Transformations on State Equations
PROBLEM: Given the system represented in state space by Eqs. (5.73),
\dot{x} =\left[\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -2 & -5 & -7 \end{matrix} \right] x + \left[\begin{matrix} 0 \\ 0 \\ 1 \end{matrix} \right] uy=\left[\begin{matrix} 1 & 0 & 0\end{matrix} \right] x
transform the system to a new set of state variables, z, where the new state variables are related to the original state variables, x,
z_{1} =2x_{1}z_{2} =3x_{1} +2x_{2}
z_{3} =x_{1} +4x_{2} +5x_{3}