Final answer:
Controllability is preserved under state feedback; if (A, B) is controllable, so is (A+BK, B). The controllability matrix remains full rank even after the feedback transformation.
Step-by-step explanation:
To prove that the feedback system (A+BK, B) is controllable when the original system (A, B) is controllable, we rely on fundamental concepts of control theory. Controllability refers to the ability to move the state of a system from any initial state to any other final state within a finite time interval, using appropriate input functions.
In control theory, the controllability of the system does not change with the addition of state feedback. If (A, B) is controllable, then any state feedback matrix K will retain this property. The proof involves using the controllability matrix of system (A, B), which, by assumption, is full rank. The feedback transformation, which produces a new matrix A + BK, leads to a similar system dynamics in terms of controllability. Hence, the controllability matrix of (A+BK, B) will also be full rank, implying that the feedback system is controllable.