Hello, Richard! I believe you're implicitly assuming that $Im(A) \subset Im(B)$. Otherwise substituting $u$ like this wouldn't really work. Consider for instance A = [[0, -1], [1, 0]], B = [[0], [1]]; clearly the system is controllable, yet your argument doesn't work here.
![Degrees of Controllability and Gramians [Control Bootcamp]](/img/n.gif)
Amazing explanation!
Masterful
Thank you :)
Hello, Richard! I believe you're implicitly assuming that $Im(A) \subset Im(B)$. Otherwise substituting $u$ like this wouldn't really work. Consider for instance A = [[0, -1], [1, 0]], B = [[0], [1]]; clearly the system is controllable, yet your argument doesn't work here.