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

: $\dot(t) = Ax(t) + Bu(t)$,
: $\, y(t) = Cx(t) + Du(t)$,

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:

: $\, = TA^$,
: $\, = TB$,
: $\, = C^$,
: $\, = D$,

where $\, T^$ is the coordinate transformation matrix defined as

: $\, T^ = \begin T_ & T_ & T_ & T_\end$,

and whose submatrices are
* $\, T_$ : a matrix whose columns span the subspace of states which are both reachable and unobservable.
* $\, T_$ : chosen so that the columns of $\, \begin T_ & T_\end$ are a basis for the reachable subspace.
* $\, T_$ : chosen so that the columns of $\, \begin T_ & T_\end$ are a basis for the unobservable subspace.
* $\, T_$ : chosen so that $\,\begin T_ & T_ & T_ & T_\end$ 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 $\, T^ = T_$, making the other matrices zero dimension.

Consequences

By using results from controllability and observability, it can be shown that the transformed system $\, (\hat, \hat, \hat, \hat)$ has matrices in the following form:

: $\, \hat = \beginA_ & A_ & A_ & A_ \\ 0 & A_ & 0 & A_ \\ 0 & 0 & A_ & A_\\ 0 & 0 & 0 & A_\end$

: $\, \hat = \beginB_ \\ B_ \\ 0 \\ 0\end$

: $\, \hat = \begin0 & C_ & 0 & C_\end$

: $\, \hat = D$

This leads to the conclusion that
* The subsystem $\, (A_, B_, C_, D)$ is both reachable and observable.
* The subsystem $\, \left(\beginA_ & A_\\ 0 & A_\end,\beginB_ \\ B_\end,\begin0 & C_\end, D\right)$ is reachable.
* The subsystem $\, \left(\beginA_ & A_\\ 0 & A_\end,\beginB_ \\ 0 \end,\beginC_ & C_\end, D\right)$ 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.

See also

* Realization (systems)
*Observability
* Controllability

References

{{Reflist

External links

Lectures on Dynamic Systems and Control, Lecture 25
- Mohammed Dahleh, Munther Dahleh, George Verghese — MIT OpenCourseWare

Control theory