LTI Multi-input Control System. Proof that controllability holds given a state feedback.

Question

The question is : Prove that (A, B) is controllable if and only if (A + BK, B) is controllable for all K.

My proof thus far:

• Let $u=kx +v$
• Consider the Im(Qc) = Im(B) + (A+BK)*Im(B) + ... + (A+BK)^n-1 * Im(B)
• I know that it can be shown that for any vector 'v' contained in R^n that Im(B) + (A+BK)v = Im(B) + Av...however I am not sure how to show this

My idea is to show that Im(B) + (A+BK)v is contained in Im(B) + Av and vice versa. Any help would be greatly appreciated

Answer 1

Take $y,z \in \operatorname{Im} B$. We need to show that $y + (A+BK)z \in \operatorname{Im} B + A \operatorname{Im} B$, i.e. there exist $u,v \in \operatorname{Im} B$ such that $y+ (A+BK)z = u + Av$. Indeed, take $v=z$ and $u = y+BKz \in \operatorname{Im} B$.

For the other direction let $u,v \in \operatorname{Im} B$ arbitrary and show that there exist $y,z \in \operatorname{Im} B$ such that $u + Av = y + (A+BK)z$. Taking $z=v$ and $y=u-BKv \in \operatorname{Im} B$ we can conclude the result.

Answer 2

Alternatively, we can use the following property: $(A,B)$ is controllable if and only if $[\matrix{A-\lambda\mathbb{I} & B}]$ has full row rank for all $\lambda\in\mathbb{C}$.

Then we can write $$[\matrix{(A+BK)-\lambda\mathbb{I} & B}]=[\matrix{A-\lambda\mathbb{I} & B}]\left[\matrix{\mathbb{I} & 0\\K & \mathbb{I}}\right]$$ from which the desired property is obtained.