Control-Network-Systems/docs/Chapters/3-CANONICAL-FORMS.md

Canonical Forms

In order to see if we are in one of these canonical forms, just write the equations from the block diagram, and find the associated S(A, B, C, D).

Tip

In order to find a rough diagram, use Horner Factorization to find a_i values. Then put all the b_i to the right integrator by shifting them as many left places, starting from the rightmost, for the number of associated s

Control Canonical Form

It is in such forms when:

A = \begin{bmatrix}
- a_1 & -a_2 & -a_3 & \dots & -a_{n-1} &-a_n\\
1 & 0 & 0 & \dots  & 0 &  0\\
0 & 1 & 0 & \dots & 0 & 0\\
\dots & \dots & \dots & \dots  & \dots \\
0 & 0 & 0 & \dots & 1 & 0
\end{bmatrix}

B = \begin{bmatrix}
1 \\ 0 \\ \dots \\ \dots \\ 0
\end{bmatrix}

C = \begin{bmatrix}
b_1 & b_2 & \dots & b_n
\end{bmatrix}

D = \begin{bmatrix}
0
\end{bmatrix}

Modal Canonical Forms

Caution

This form is the most difficult to find, as this varies drastically in cases of double roots

A = \begin{bmatrix}
- a_1 & 0 & 0 & \dots & 0\\
0 & -a_2 & 0 & \dots & 0\\
0 & 0 & -a_3 & \dots & 0\\
\dots & \dots & \dots & \dots & \dots \\
0 & 0 & 0 & 0 & -a_n
\end{bmatrix}

B = \begin{bmatrix}
1 \\ 1 \\ \dots \\ \dots \\ 1
\end{bmatrix}

C = \begin{bmatrix}
b_1 & b_2 & \dots & b_n
\end{bmatrix}

D = \begin{bmatrix}
0
\end{bmatrix}

Observable Canonical Form