Control-Network-Systems/docs/Chapters/12-KALLMAN-DECOMPOSITION.md

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 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