2025-01-15 14:49:52 +01:00
|
|
|
# Kallman Decomposition
|
|
|
|
|
|
|
|
|
|
## Some background
|
|
|
|
|
- $X_r = R(K_c)$ : Reachable space is the range of the
|
|
|
|
|
`controllable matrix`
|
|
|
|
|
- $X_{no} = Ker(K_o)$ : Not observable Space is the kernel
|
|
|
|
|
of the `observability matrix`
|
|
|
|
|
- $X = X_r \bigoplus X_{nr}$ : Each possible state is sum of
|
|
|
|
|
bases of `reachable` and `not-reachable states`
|
|
|
|
|
|
|
|
|
|
## Full Decomposition
|
|
|
|
|
> [!TIP]
|
|
|
|
|
> Follow [Example 12](./Examples/EXAMPLE-12.md/#kallman-full-decomposition) To understand this part
|
|
|
|
|
- $X_1 = X_r \cap X_{nr}$ : `Reachable` but `Not-Observable`
|
|
|
|
|
space
|
|
|
|
|
- $X_2 =$ Complement of $X_1$ to cover $X_r$ : Both `Reachable`
|
|
|
|
|
and `Observable`
|
|
|
|
|
- $X_3 =$ Complement of $X_1$ to cover $X_{no}$ : Both
|
|
|
|
|
`Not-Reachable` and `Not-Observable`
|
|
|
|
|
- $X_4 =$ Complement of all the others to cover $X$ : Bot
|
|
|
|
|
`Not-Reachable` and `Observable`
|
|
|
|
|
|
|
|
|
|
From here we have these blocks:
|
|
|
|
|
$$
|
|
|
|
|
\begin{align*}
|
|
|
|
|
\hat{A} &= \begin{bmatrix}
|
|
|
|
|
\hat{A}_{11} & \hat{A}_{12} & \hat{A}_{13} & \hat{A}_{14} \\
|
|
|
|
|
0 & \hat{A}_{22} & 0 & \hat{A}_{24} \\
|
|
|
|
|
0 & 0 & \hat{A}_{33} & \hat{A}_{34} \\
|
|
|
|
|
0 & 0 & 0 & \hat{A}_{44}
|
|
|
|
|
\end{bmatrix}\\
|
|
|
|
|
|
|
|
|
|
\hat{B} &= \begin{bmatrix}
|
|
|
|
|
\hat{B}_{1} \\ \hat{B}_{2} \\ 0 \\ 0
|
|
|
|
|
\end{bmatrix}\\
|
|
|
|
|
|
|
|
|
|
\hat{C} &= \begin{bmatrix}
|
|
|
|
|
0 & \hat{C}_{2} & 0 & \hat{C}_{4}
|
|
|
|
|
\end{bmatrix}
|
|
|
|
|
\end{align*}
|
|
|
|
|
$$
|
|
|
|
|
|
|
|
|
|
Now, the eigenvalues of $\hat{A} = \cup_i^4 \hat{A}_{ii}$ and:
|
|
|
|
|
- $eig(\hat{A}_{11})$: `Reachable` and `Not-Observable`
|
|
|
|
|
- $eig(\hat{A}_{22})$: `Reachable` and `Not-Observable`
|
|
|
|
|
- $eig(\hat{A}_{33})$: `Not-Reachable` and `Not-Observable`
|
|
|
|
|
- $eig(\hat{A}_{44})$: `Not-Reachable` and `Observable`
|
|
|
|
|
|
