Threefish

Template:Expert-subject In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant 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

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

where

• $T_{r\bar{o}}$ 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 $\left[\begin{array}{cc}{T}_{r\bar{o}}& T_{ro}\end{array}\right]$ are a basis for the reachable subspace.
• $T_{\overline{ro}}$ is 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.
• $T_{\bar{r}o}$ is 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.

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:

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

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

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

$\hat{D}=D$

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.

See also

• Observability
• Controllability

External links

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