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.

Notation

The derivation is identical for both discrete-time as well as continuous time LTI systems. The description of a continuous time linear system is

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

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

where

\,x

is the "state vector",

\,y

is the "output vector",

\,u

is the "input (or control) vector",

\,A

is the "state matrix",

\,B

is the "input matrix",

\,C

is the "output matrix",

\,D

is the "feedthrough (or feedforward) matrix".

Similarly, a discrete-time linear control system can be described as

\,x(k+1)=Ax(k)+Bu(k)

\,y(k)=Cx(k)+Du(k)

with similar meanings for the variables. Thus, the system can be described using the tuple consisting of four matrices $\,(A,B,C,D)$ . Let the order of the system be $\,n$ .

Then, the Kalman decomposition is defined as a transformation of the tuple $\,(A,B,C,D)$ to $\,({\hat {A}},{\hat {B}},{\hat {C}},{\hat {D}})$ as follows:

\,{\hat {A}}={T^{-1}}AT

\,{\hat {B}}={T^{-1}}B

\,{\hat {C}}=CT

\,{\hat {D}}=D

$\,T$ is an $\,n\times n$ invertible matrix defined as

\,T={\begin{bmatrix}T_{r{\overline {o}}}&T_{ro}&T_{\overline {ro}}&T_{{\overline {r}}o}\end{bmatrix}}

where

$\,T_{r{\overline {o}}}$ is a matrix whose columns span the subspace of states which are both reachable and unobservable.
$\,T_{ro}$ is chosen so that the columns of $\,{\begin{bmatrix}T_{r{\overline {o}}}&T_{ro}\end{bmatrix}}$ are a basis for the reachable subspace.
$\,T_{\overline {ro}}$ is chosen so that the columns of $\,{\begin{bmatrix}T_{r{\overline {o}}}&T_{\overline {ro}}\end{bmatrix}}$ are a basis for the unobservable subspace.
$\,T_{{\overline {r}}o}$ is chosen so that $\,{\begin{bmatrix}T_{r{\overline {o}}}&T_{ro}&T_{\overline {ro}}&T_{{\overline {r}}o}\end{bmatrix}}$ is invertible.

By construction, the matrix $\,T$ 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 $\,T=T_{ro}$ , making the other matrices zero dimension.

Standard Form

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.

External links

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

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Kalman decomposition

Notation

Standard Form

See also

External links