67 lines
1.6 KiB
Markdown
67 lines
1.6 KiB
Markdown
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# Canonical Forms
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In order to see if we are in one of these canonical forms, just write the
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equations from the block diagram, and find the associated $S(A, B, C, D)$.
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> [!TIP]
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> In order to find a rough diagram, use
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> [Horner Factorization](MODERN-CONTROL.md/#horner-factorization) to find
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> $a_i$ values. Then put all the $b_i$ to the right integrator by shifting them
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> as many left places, starting from the rightmost, for the number of
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> associated $s$
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## Control Canonical Form
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It is in such forms when:
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$$
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A = \begin{bmatrix}
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- a_1 & -a_2 & -a_3 & \dots & -a_{n-1} &-a_n\\
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1 & 0 & 0 & \dots & 0 & 0\\
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0 & 1 & 0 & \dots & 0 & 0\\
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\dots & \dots & \dots & \dots & \dots \\
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0 & 0 & 0 & \dots & 1 & 0
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\end{bmatrix}
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B = \begin{bmatrix}
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1 \\ 0 \\ \dots \\ \dots \\ 0
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\end{bmatrix}
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C = \begin{bmatrix}
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b_1 & b_2 & \dots & b_n
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\end{bmatrix}
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D = \begin{bmatrix}
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0
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\end{bmatrix}
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$$
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## Modal Canonical Forms
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> [!CAUTION]
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> This form is the most difficult to find, as this varies drastically in cases
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> of double roots
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>
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$$
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A = \begin{bmatrix}
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- a_1 & 0 & 0 & \dots & 0\\
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0 & -a_2 & 0 & \dots & 0\\
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0 & 0 & -a_3 & \dots & 0\\
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\dots & \dots & \dots & \dots & \dots \\
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0 & 0 & 0 & 0 & -a_n
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\end{bmatrix}
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B = \begin{bmatrix}
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1 \\ 1 \\ \dots \\ \dots \\ 1
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\end{bmatrix}
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C = \begin{bmatrix}
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b_1 & b_2 & \dots & b_n
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\end{bmatrix}
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D = \begin{bmatrix}
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0
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\end{bmatrix}
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$$
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## Observable Canonical Form
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<!--TODO: Correct here -->
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[^reference-input-pole-allocation]: [MIT | 06 January 2025 | pg. 2](https://ocw.mit.edu/courses/16-30-feedback-control-systems-fall-2010/c553561f63feaa6173e31994f45f0c60_MIT16_30F10_lec11.pdf)
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