# Kalman decomposition

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In [control theory](/source/Control_theory), a **Kalman decomposition** provides a mathematical means to convert a representation of any [linear time-invariant](/source/LTI_system_theory) (LTI) [control system](/source/Control_system) to a form in which the system can be decomposed into a standard form which makes clear the [observable](/source/Observability) and [controllable](/source/Controllability) 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](/source/Reachability_problem) 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](/source/Realization_(systems)) 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](/source/Quantum_system). 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.[1]

## See also

- [Realization (systems)](/source/Realization_(systems))
- [Observability](/source/Observability)
- [Controllability](/source/Controllability)

## References

1. Zhang, Guofeng; Grivopoulos, Symeon; Petersen, Ian R.; Gough, John E. (February 2018). "The Kalman Decomposition for Linear Quantum Systems". *IEEE Transactions on Automatic Control*. **63** (2): 331–346. [arXiv:1606.05719](https://arxiv.org/abs/1606.05719). [Bibcode:2018ITAC...63..331Z](https://ui.adsabs.harvard.edu/abs/2018ITAC...63..331Z). [doi:10.1109/TAC.2017.2713343](https://doi.org/10.1109/TAC.2017.2713343). [hdl:10397/77565](https://hdl.handle.net/10397/77565). [ISSN 1558-2523](https://www.worldcat.org/issn/1558-2523). [S2CID 10544143](https://api.semanticscholar.org/CorpusID:10544143)

## External links

- [Lectures on Dynamic Systems and Control, Lecture 25](http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-241j-dynamic-systems-and-control-spring-2011/readings/MIT6_241JS11_chap25.pdf) - [Mohammed Dahleh](/source/Mohammed_Dahleh), [Munther Dahleh](/source/Munther_A._Dahleh), George Verghese — MIT OpenCourseWare

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Adapted from the Wikipedia article [Kalman decomposition](https://en.wikipedia.org/wiki/Kalman_decomposition) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Kalman_decomposition?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.