Kalman decomposition explained

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

• 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 asT ^{-1} = \begin{bmatrix} T _{r\overline{o} } & T_ & T_ & T_\end,

and whose submatrices are

T _{r\overline{o} } : a matrix whose columns span the subspace of states which are both reachable and unobservable.T _{ro} : chosen so that the columns of \begin{bmatrix} T _{r\overline{o} } & T_\end are a basis for the reachable subspace.T _{\overline{ro} } : chosen so that the columns of \begin{bmatrix} T _{r\overline{o} } & T_\end are a basis for the unobservable subspace.T _{\overline{r} o} : chosen so that \begin{bmatrix} T _{r\overline{o} } & T_ & T_ & T_\end 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} = \begin{bmatrix}A _{r\overline{o} } & A_ & A_ & A_ \\0 & A_ & 0 & A_ \\0 & 0 & A_ & A_\\0 & 0 & 0 & A_\end\hat{B} = \begin{bmatrix}B _{r\overline{o} } \\ B_ \\ 0 \\ 0\end\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_\\ 0 & A_\end,\beginB_ \\ B_\end,\begin0 & C_\end, 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.

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.^[1]

See also

• Realization (systems)
• Observability
• Controllability

External links

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

Notes and References

1. Zhang. Guofeng. Grivopoulos. Symeon. Petersen. Ian R.. Gough. John E.. February 2018. The Kalman Decomposition for Linear Quantum Systems. IEEE Transactions on Automatic Control. 63. 2. 331–346. 10.1109/TAC.2017.2713343. 1558-2523. 10397/77565. 10544143 . free.