Coordinate transform

Sometimes we want to change the coordinate base of our states, either to give them physical meaning or to cast $A$ into a simpler form. This is rather easy linear algebra, lets say the new state $z=Tx \rightarrow x = T^{-1}z$. Then we can substitute that into the state space representation

$$ \dot x = Ax + Bu \rightarrow T^{-1}\dot z = AT^{-1}z + Bu \rightarrow \dot z = \underbrace{TAT^{-1}}_{\tilde A} z + \underbrace{TB}_{\tilde B}u = \tilde A z + \tilde B u.$$ and $$ y = Cx + Du \rightarrow y = \underbrace{CT^{-1}}_{\tilde C}z + Du = \tilde C z + Du.$$

Important: note that the input/output behaviour remains unchanged under state transformations. This is only a system-internal operation.

For systems with unique eigenvalues, the system is called diagonalisable, because taking the inverse transformation, $T^{-1}$, to be the horizontally stacked eigenvectors of $A$ results in a diagonal $\tilde A$. Sometimes stuff is named nice and descriptive.