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.
Definition
Consider the continuous-time LTI control system
- [math]\displaystyle{ \dot{x}(t) = Ax(t) + Bu(t) }[/math],
- [math]\displaystyle{ \, y(t) = Cx(t) + Du(t) }[/math],
or the discrete-time LTI control system
- [math]\displaystyle{ \, x(k+1) = Ax(k) + Bu(k) }[/math],
- [math]\displaystyle{ \, y(k) = Cx(k) + Du(k) }[/math].
The Kalman decomposition is defined as the realization of this system obtained by transforming the original matrices as follows:
- [math]\displaystyle{ \, {\hat{A}} = TA{T}^{-1} }[/math],
- [math]\displaystyle{ \, {\hat{B}} = TB }[/math],
- [math]\displaystyle{ \, {\hat{C}} = C{T}^{-1} }[/math],
- [math]\displaystyle{ \, {\hat{D}} = D }[/math],
where [math]\displaystyle{ \, T^{-1} }[/math] is the coordinate transformation matrix defined as
- [math]\displaystyle{ \, T^{-1} = \begin{bmatrix} T_{r\overline{o}} & T_{ro} & T_{\overline{ro}} & T_{\overline{r}o}\end{bmatrix} }[/math],
and whose submatrices are
- [math]\displaystyle{ \, T_{r\overline{o}} }[/math] : a matrix whose columns span the subspace of states which are both reachable and unobservable.
- [math]\displaystyle{ \, T_{ro} }[/math] : chosen so that the columns of [math]\displaystyle{ \, \begin{bmatrix} T_{r\overline{o}} & T_{ro}\end{bmatrix} }[/math] are a basis for the reachable subspace.
- [math]\displaystyle{ \, T_{\overline{ro}} }[/math] : chosen so that the columns of [math]\displaystyle{ \, \begin{bmatrix} T_{r\overline{o}} & T_{\overline{ro}}\end{bmatrix} }[/math] are a basis for the unobservable subspace.
- [math]\displaystyle{ \, T_{\overline{r}o} }[/math] : chosen so that [math]\displaystyle{ \,\begin{bmatrix} T_{r\overline{o}} & T_{ro} & T_{\overline{ro}} & T_{\overline{r}o}\end{bmatrix} }[/math] 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 [math]\displaystyle{ \, T^{-1} = T_{ro} }[/math], making the other matrices zero dimension.
Consequences
By using results from controllability and observability, it can be shown that the transformed system [math]\displaystyle{ \, (\hat{A}, \hat{B}, \hat{C}, \hat{D}) }[/math] has matrices in the following form:
- [math]\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} }[/math]
- [math]\displaystyle{ \, \hat{B} = \begin{bmatrix}B_{r\overline{o}} \\ B_{ro} \\ 0 \\ 0\end{bmatrix} }[/math]
- [math]\displaystyle{ \, \hat{C} = \begin{bmatrix}0 & C_{ro} & 0 & C_{\overline{r}o}\end{bmatrix} }[/math]
- [math]\displaystyle{ \, \hat{D} = D }[/math]
This leads to the conclusion that
- The subsystem [math]\displaystyle{ \, (A_{ro}, B_{ro}, C_{ro}, D) }[/math] is both reachable and observable.
- The subsystem [math]\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) }[/math] is reachable.
- The subsystem [math]\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) }[/math] 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
References
- ↑ 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. doi:10.1109/TAC.2017.2713343. ISSN 1558-2523. https://ieeexplore.ieee.org/document/7942122.
External links
- Lectures on Dynamic Systems and Control, Lecture 25 - Mohammed Dahleh, Munther Dahleh, George Verghese — MIT OpenCourseWare
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