Lecture 9: Controllability and Observability - Illinois Institute of ...

Proof by Contradiction: Im(C(A, B)) ⊂ Im(W t )
Proof.
• This means that for all s ∈ [0, t],
d k
ds k BT e AT s x = B T ( A T ) k
e
A T s x = 0
• At s = 0, this implies B T ( A T ) k
x = 0 for all k.
• We conclude that
x T [ B AB · · · A n−1 B ] = 0
• Thus C(A, B) T x = 0, so x ∈ ker C(A, B) T . As before, this means
x ∈ Im(C(A, B)) ⊥ .
• We conclude that x ∉ Im(C(A, B)). This proves by contradiction that
Im(C(A, B)) ⊂ Im(W t ).
M. Peet Lecture 9: Controllability 6 / 15