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.
Consider the continuous-time LTI control system
,
,
or the discrete-time LTI control system
,
.
The Kalman decomposition is defined as the realization of this system obtained by transforming the original matrices as follows:
,
,
,
,
where
is the coordinate transformation matrix defined as
,
and whose submatrices are
: a matrix whose columns span the subspace of states which are both reachable and unobservable.
: chosen so that the columns of
are a basis for the reachable subspace.
: chosen so that the columns of
are a basis for the unobservable subspace.
: chosen so that
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
, making the other matrices zero dimension.
By using results from controllability and observability, it can be shown that the transformed system
has matrices in the following form:
![{\displaystyle \,{\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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/4e5c09fd04602e67b24dd2ffa0ab653dd4d95f60)
![{\displaystyle \,{\hat {B}}={\begin{bmatrix}B_{r{\overline {o}}}\\B_{ro}\\0\\0\end{bmatrix}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/6e1adbf4b22834e0d4d3472d75387f5927b9854d)
![{\displaystyle \,{\hat {C}}={\begin{bmatrix}0&C_{ro}&0&C_{{\overline {r}}o}\end{bmatrix}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/5e3eb208d870022cb397df0e97d5d6d02410c222)
![{\displaystyle \,{\hat {D}}=D}](//wikimedia.org/api/rest_v1/media/math/render/svg/05daf644820de03ca6c3099bf3e0b2cb7366ab65)
This leads to the conclusion that
- The subsystem
is both reachable and observable.
- The subsystem
is reachable.
- The subsystem
is observable.
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]