49 lines
1.6 KiB
Markdown
49 lines
1.6 KiB
Markdown
# Kallman Decomposition
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## Some background
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- $X_r = R(K_c)$ : Reachable space is the range of the
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`controllable matrix`
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- $X_{no} = Ker(K_o)$ : Not observable Space is the kernel
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of the `observability matrix`
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- $X = X_r \bigoplus X_{nr}$ : Each possible state is sum of
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bases of `reachable` and `not-reachable states`
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## Full Decomposition
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> [!TIP]
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> Follow [Example 12](./Examples/EXAMPLE-12.md/#kallman-full-decomposition) To understand this part
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- $X_1 = X_r \cap X_{nr}$ : `Reachable` but `Not-Observable`
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space
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- $X_2 =$ Complement of $X_1$ to cover $X_r$ : Both `Reachable`
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and `Observable`
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- $X_3 =$ Complement of $X_1$ to cover $X_{no}$ : Both
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`Not-Reachable` and `Not-Observable`
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- $X_4 =$ Complement of all the others to cover $X$ : Bot
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`Not-Reachable` and `Observable`
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From here we have these blocks:
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$$
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\begin{align*}
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\hat{A} &= \begin{bmatrix}
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\hat{A}_{11} & \hat{A}_{12} & \hat{A}_{13} & \hat{A}_{14} \\
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0 & \hat{A}_{22} & 0 & \hat{A}_{24} \\
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0 & 0 & \hat{A}_{33} & \hat{A}_{34} \\
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0 & 0 & 0 & \hat{A}_{44}
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\end{bmatrix}\\
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\hat{B} &= \begin{bmatrix}
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\hat{B}_{1} \\ \hat{B}_{2} \\ 0 \\ 0
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\end{bmatrix}\\
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\hat{C} &= \begin{bmatrix}
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0 & \hat{C}_{2} & 0 & \hat{C}_{4}
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\end{bmatrix}
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\end{align*}
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$$
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Now, the eigenvalues of $\hat{A} = \cup_i^4 \hat{A}_{ii}$ and:
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- $eig(\hat{A}_{11})$: `Reachable` and `Not-Observable`
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- $eig(\hat{A}_{22})$: `Reachable` and `Not-Observable`
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- $eig(\hat{A}_{33})$: `Not-Reachable` and `Not-Observable`
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- $eig(\hat{A}_{44})$: `Not-Reachable` and `Observable`
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