[제어공학개론] Lec 09 - Canonical form

📢Precaution

본 게시글은 서울대학교 심형보 교수님의 23-2 제어공학개론 수업 내용을 바탕으로 작성되었습니다.

Canonical form

• 수많은 A, B, C, D로 describe 될 수 있는 system을 특정 표준형으로 설정해둔 것.
• 교과서, 참고자료에 따라 다르게 설정할 수 있음.

Controllability canonical form

\[\begin{aligned} A &= \begin{bmatrix}0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ -a_0 & -a_1 & -a_2 & \cdots & -a_{n-1} \end{bmatrix} \\ B &= [0, 0, 0, \cdots, 1]^T \\ C &= [b_0, b_1, b_2, \cdots, b_{n-1}] \\ D &= [D] \end{aligned}\]

it represents transfer function

\[T(s) = \frac{b_{n-1}s^{n-1}+\cdots + b_1s+b_0}{s^n+a_{n-1}s^{n-1}+\cdots+a_1s + a_0}+D\]

another book…’s controllability canonical form

• original canonical form

• also called as companion form

Observability canonical form

\[\begin{aligned} A &= \begin{bmatrix}0 & 0 & 0 & \cdots& 0 & -a_0 \\ 1 & 0 & 0 & \cdots & 0 & -a_1 \\ 0 & 1 & 0 & \cdots& 0 & -a_2 \\ \vdots & \vdots & \vdots &\vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots &1 & -a_{n-1} \end{bmatrix} \\ B &= [b_0, b_1, b_2, \cdots, b_{n-1}]^T \\ C &= [0, 0, 0, \cdots, 1] \\ D &= [D] \end{aligned}\]

another book…’s observability canonical form

• original canonical form

• companion canonical form