Given the state space model $$ \dot x = Ax + Bu $$$$ y = Cx + Du $$

and the transformation matrix T, which maps the state vector x to another coordinate system $$ \bar{x} = Tx $$

the state-space model is transformed in a way that results in an equivalent state-space model which is based on the new state vector

$$ \dot{\bar{x}} = TAT^{-1}\bar{x} + TBu $$$$ y = CT^{-1}\bar{x} + Du $$

Please note: In the literature, T may be defined inversely:

$$ \bar{x} = T^{-1}x $$

References:

Control System Design, page 484 by Goodwin, Graebe, Salgado, 2000

https://de.mathworks.com/help/control/ref/ss2ss.html

Attention: T as defined by Matlab is the inverse of T as defined by Goodwin, Graebe, Salgado

Source Code: @ss/ss2ss