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.[1]

See also

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. Bibcode:2018ITAC...63..331Z. doi:10.1109/TAC.2017.2713343. hdl:10397/77565. ISSN 1558-2523. S2CID 10544143