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

Representation and Controllability
Question: Is the representation (A, B, C, D) of the system y = Gu,
unique
ẋ(t) = Ax(t) + Bu(t)
y(t) = Cx(t) + Du(t)
Question: Do there exist (Â, ˆB, Ĉ, ˆD) such that y and u also satisfy,
ẋ(t) = Âx(t) + ˆBu(t)
y(t) = Ĉx(t) + ˆDu(t)
Answer: Of Course! Recall the similarity transform: z(t) = T x(t) for any
invertible T . Then y and u also satisfy,
ż(t) = T ẋ(t) = T Ax(t) + T Bu(t)
= T AT −1 z(t) + T Bu(t)
y(t) = Cx(t) + Du(t)
= CT −1 x(t) + Du(t)
M. Peet Lecture 9: Controllability 8 / 15