Kalman decomposition

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

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

${\displaystyle \,y(t)=Cx(t)+Du(t)}$

where

${\displaystyle \,x}$ is the "state vector",

${\displaystyle \,y}$ is the "output vector",

${\displaystyle \,u}$ is the "input (or control) vector",

${\displaystyle \,A}$ is the "state matrix",

${\displaystyle \,B}$ is the "input matrix",

${\displaystyle \,C}$ is the "output matrix",

${\displaystyle \,D}$ is the "feedthrough (or feedforward) matrix".

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

${\displaystyle \,x(k+1)=Ax(k)+Bu(k)}$

${\displaystyle \,y(k)=Cx(k)+Du(k)}$

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

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

${\displaystyle \,{\hat {A}}=TA{T}^{-1}}$

${\displaystyle \,{\hat {B}}=TB}$

${\displaystyle \,{\hat {C}}=C{T}^{-1}}$

${\displaystyle \,{\hat {D}}=D}$

${\displaystyle \,T^{-1}}$ is an ${\displaystyle \,n\times n}$ invertible matrix defined as

${\displaystyle \,T^{-1}={\begin{bmatrix}T_{r{\overline {o}}}&T_{ro}&T_{\overline {ro}}&T_{{\overline {r}}o}\end{bmatrix}}}$

where

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

By construction, the matrix ${\displaystyle \,T^{-1}}$ 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 ${\displaystyle \,T^{-1}=T_{ro}}$, making the other matrices zero dimension.

By using results from controllability and observability, it can be shown that the transformed system ${\displaystyle \,({\hat {A}},{\hat {B}},{\hat {C}},{\hat {D}})}$ has matrices in the following form:

${\displaystyle \,{\hat {A}}={\begin{bmatrix}A_{r{\overline {o}}}&A_{12}&A_{13}&A_{14}\\0&A_{ro}&0&A_{24}\\0&0&A_{\overline {ro}}&A_{34}\\0&0&0&A_{{\overline {r}}o}\end{bmatrix}}}$

${\displaystyle \,{\hat {B}}={\begin{bmatrix}B_{r{\overline {o}}}\\B_{ro}\\0\\0\end{bmatrix}}}$

${\displaystyle \,{\hat {C}}={\begin{bmatrix}0&C_{ro}&0&C_{{\overline {r}}o}\end{bmatrix}}}$

${\displaystyle \,{\hat {D}}=D}$

This leads to the conclusion that

• The subsystem ${\displaystyle \,(A_{ro},B_{ro},C_{ro},D)}$ is both reachable and observable.
• The subsystem ${\displaystyle \,\left({\begin{bmatrix}A_{r{\overline {o}}}&A_{12}\\0&A_{ro}\end{bmatrix}},{\begin{bmatrix}B_{r{\overline {o}}}\\B_{ro}\end{bmatrix}},{\begin{bmatrix}0&C_{ro}\end{bmatrix}},D\right)}$ is reachable.
• The subsystem ${\displaystyle \,\left({\begin{bmatrix}A_{ro}&A_{24}\\0&A_{{\overline {r}}o}\end{bmatrix}},{\begin{bmatrix}B_{ro}\\0\end{bmatrix}},{\begin{bmatrix}C_{ro}&C_{{\overline {r}}o}\end{bmatrix}},D\right)}$ is observable.

• Observability
• Controllability

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