Question

Suppose (A, B) is controllable. Prove that the feedback system (A - BK, B) is controllable.

A) True
B) False
C) Controllability is not affected by feedback systems.
D) The question is not clear.

Answer 1

The statement is true. The feedback system (A - BK, B) is controllable.

Step-by-step explanation:

The statement is true. Let's prove it.

For a system to be controllable, the controllability matrix rank must be equal to the number of states in the system. The controllability matrix for the pair (A, B) is defined as C = [B, AB, A^2B, ..., A^(n-1)B], where n is the number of states in matrix A.

Now, let's consider the feedback system (A - BK, B). We can define the controllability matrix for this system as C_fb = [B, (A - BK)B, (A - BK)^2B, ..., (A - BK)^(n-1)B].

Since the controllability matrix for the feedback system includes the same terms as the controllability matrix for the original system, the rank of C_fb will also be equal to the number of states in the system. Therefore, the feedback system (A - BK, B) is controllable.