Kalman decomposition

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.

References

References

1. (February 2018). "The Kalman Decomposition for Linear Quantum Systems". IEEE Transactions on Automatic Control.