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.

Consider the continuous-time LTI control system

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

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

or the discrete-time LTI control system

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

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

The Kalman decomposition is defined as the realization of this system obtained by transforming the original matrices as follows:

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

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

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

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

where ${\displaystyle \,T^{-1}}$ is the coordinate transformation matrix defined as

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

and whose submatrices are

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

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.^[1]

• Realization (systems)
• Observability
• Controllability

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