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 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.
 * $$\, T_{r\overline{o}}$$ : 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.

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.