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

$\dot{x}(t)=Ax(t)+Bu(t)$,

$y(t)=Cx(t)+Du(t)$,

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

$T^{-1}=\left[\begin{array}{cccc}{T}_{r\bar{o}}& T_{ro}& T_{\overline{ro}}& T_{\bar{r}o}\end{array}\right]$,

and whose submatrices are

• $T_{r\bar{o}}$ : a matrix whose columns span the subspace of states which are both reachable and unobservable.
• $T_{ro}$ : chosen so that the columns of $\left[\begin{array}{cc}{T}_{r\bar{o}}& T_{ro}\end{array}\right]$ are a basis for the reachable subspace.
• $T_{\overline{ro}}$ : chosen so that the columns of $\left[\begin{array}{cc}{T}_{r\bar{o}}& T_{\overline{ro}}\end{array}\right]$ are a basis for the unobservable subspace.
• $T_{\bar{r}o}$ : chosen so that $\left[\begin{array}{cccc}{T}_{r\bar{o}}& T_{ro}& T_{\overline{ro}}& T_{\bar{r}o}\end{array}\right]$ 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:

$\hat{A}=\left[\begin{array}{cccc}{A}_{r\bar{o}}& A_{12}& A_{13}& A_{14}\\ 0& A_{ro}& 0& A_{24}\\ 0& 0& A_{\overline{ro}}& A_{34}\\ 0& 0& 0& A_{\bar{r}o}\end{array}\right]$

$\hat{B}=\left[\begin{array}{c}{B}_{r\bar{o}}\\ B_{ro}\\ 0\\ 0\end{array}\right]$

$\hat{C}=\left[\begin{array}{cccc}0& C_{ro}& 0& C_{\bar{r}o}\end{array}\right]$

$\hat{D}=D$

This leads to the conclusion that

• The subsystem $(A_{ro},B_{ro},C_{ro},D)$ is both reachable and observable.
• The subsystem $(\left[\begin{array}{cc}{A}_{r\bar{o}}& A_{12}\\ 0& A_{ro}\end{array}\right],\left[\begin{array}{c}{B}_{r\bar{o}}\\ B_{ro}\end{array}\right],\left[\begin{array}{cc}0& C_{ro}\end{array}\right],D)$ is reachable.
• The subsystem $(\left[\begin{array}{cc}{A}_{ro}& A_{24}\\ 0& A_{\bar{r}o}\end{array}\right],\left[\begin{array}{c}{B}_{ro}\\ 0\end{array}\right],\left[\begin{array}{cc}{C}_{ro}& C_{\bar{r}o}\end{array}\right],D)$ 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

External links

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