# Kalman decomposition

In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant (LTI) control system to a form in which the system can be decomposed into a standard form which makes clear the observable and controllable components of the system. This decomposition results in the system being presented with a more illuminating structure, making it easier to draw conclusions on the system's reachable and observable subspaces.

## Definition

Consider the continuous-time LTI control system
    ${\dot {x}}(t)=Ax(t)+Bu(t)$,
    $\,y(t)=Cx(t)+Du(t)$,
or the discrete-time LTI control system
    $\,x(k+1)=Ax(k)+Bu(k)$,
    $\,y(k)=Cx(k)+Du(k)$.
The Kalman decomposition is defined as the realization of this system obtained by transforming the original matrices as follows:
    $\,{\hat {A}}=TA{T}^{-1}$,
    $\,{\hat {B}}=TB$,
    $\,{\hat {C}}=C{T}^{-1}$,
    $\,{\hat {D}}=D$,
where $\,T^{-1}$ is the coordinate transformation matrix defined as
    $\,T^{-1}={\begin{bmatrix}T_{r{\overline {o}}}&T_{ro}&T_{\overline {ro}}&T_{{\overline {r}}o}\end{bmatrix}}$,
and whose submatrices are
* $\,T_{r{\overline {o}}}$ : a matrix whose columns span the subspace of states which are both reachable and unobservable.
* $\,T_{ro}$ : chosen so that the columns of $\,{\begin{bmatrix}T_{r{\overline {o}}}&T_{ro}\end{bmatrix}}$ are a basis for the reachable subspace.
* $\,T_{\overline {ro}}$ : chosen so that the columns of $\,{\begin{bmatrix}T_{r{\overline {o}}}&T_{\overline {ro}}\end{bmatrix}}$ are a basis for the unobservable subspace.
* $\,T_{{\overline {r}}o}$ : chosen so that $\,{\begin{bmatrix}T_{r{\overline {o}}}&T_{ro}&T_{\overline {ro}}&T_{{\overline {r}}o}\end{bmatrix}}$ is invertible.
It can be observed that some of these matrices may have dimension zero. For example, if the system is both observable and controllable, then $\,T^{-1}=T_{ro}$, making the other matrices zero dimension.

## Consequences

By using results from controllability and observability, it can be shown that the transformed system $\,({\hat {A}},{\hat {B}},{\hat {C}},{\hat {D}})$ has matrices in the following form:
    $\,{\hat {A}}={\begin{bmatrix}A_{r{\overline {o}}}&A_{12}&A_{13}&A_{14}\\0&A_{ro}&0&A_{24}\\0&0&A_{\overline {ro}}&A_{34}\\0&0&0&A_{{\overline {r}}o}\end{bmatrix}}$
    $\,{\hat {B}}={\begin{bmatrix}B_{r{\overline {o}}}\\B_{ro}\\0\\0\end{bmatrix}}$
    $\,{\hat {C}}={\begin{bmatrix}0&C_{ro}&0&C_{{\overline {r}}o}\end{bmatrix}}$
    $\,{\hat {D}}=D$
This leads to the conclusion that
* The subsystem $\,(A_{ro},B_{ro},C_{ro},D)$ is both reachable and observable.
* The subsystem $\,\left({\begin{bmatrix}A_{r{\overline {o}}}&A_{12}\\0&A_{ro}\end{bmatrix}},{\begin{bmatrix}B_{r{\overline {o}}}\\B_{ro}\end{bmatrix}},{\begin{bmatrix}0&C_{ro}\end{bmatrix}},D\right)$ is reachable.
* The subsystem $\,\left({\begin{bmatrix}A_{ro}&A_{24}\\0&A_{{\overline {r}}o}\end{bmatrix}},{\begin{bmatrix}B_{ro}\\0\end{bmatrix}},{\begin{bmatrix}C_{ro}&C_{{\overline {r}}o}\end{bmatrix}},D\right)$ is observable.

## Variants

A Kalman decomposition also exists for linear dynamical quantum systems. Unlike classical dynamical systems, the coordinate transformation used in this variant requires to be in a specific class of transformations due to the physical laws of quantum mechanics.