174.52.255.118: Explicit reference to Threefish being a symmetric-key cipher

2013-11-23T12:57:53Z

Explicit reference to Threefish being a symmetric-key cipher

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			In [[control theory]], a '''Kalman decomposition''' provides a mathematical means to convert a representation of any [[LTI system theory\|linear time-invariant]] [[control system]] to a form in which the system can be decomposed into a standard form which makes clear the [[Observability\|observable]] and [[Controllability\|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 [[Reachability problem\|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

			: <math>\dot{x}(t) = Ax(t) + Bu(t)</math>
			: <math>\, y(t) = Cx(t) + Du(t)</math>

			where
			: <math>\, x</math> is the "state vector",
			: <math>\, y</math> is the "output vector",
			: <math>\, u</math> is the "input (or control) vector",
			: <math>\, A</math> is the "state matrix",
			: <math>\, B</math> is the "input matrix",
			: <math>\, C</math> is the "output matrix",
			: <math>\, D</math> is the "feedthrough (or feedforward) matrix".

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

			: <math>\, x(k+1) = Ax(k) + Bu(k)</math>
			: <math>\, y(k) = Cx(k) + Du(k)</math>

			with similar meanings for the variables. Thus, the system can be described using the tuple consisting of four matrices <math>\, (A, B, C, D)</math>.
			Let the order of the system be <math>\, n</math>.

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

			: <math>\, {\hat{A}} = {T^{-1}}AT</math>
			: <math>\, {\hat{B}} = {T^{-1}}B</math>
			: <math>\, {\hat{C}} = CT</math>
			: <math>\, {\hat{D}} = D</math>

			<math>\, T</math> is an <math>\, n \times n</math> invertible matrix defined as

			: <math>\, T = \begin{bmatrix} T_{r\overline{o}} & T_{ro} & T_{\overline{ro}} & T_{\overline{r}o}\end{bmatrix}</math>

			where
			* <math>\, T_{r\overline{o}}</math> is a matrix whose columns span the subspace of states which are both reachable and unobservable.
			* <math>\, T_{ro}</math> is chosen so that the columns of <math>\, \begin{bmatrix} T_{r\overline{o}} & T_{ro}\end{bmatrix}</math> are a basis for the reachable subspace.
			* <math>\, T_{\overline{ro}}</math> is chosen so that the columns of <math>\, \begin{bmatrix} T_{r\overline{o}} & T_{\overline{ro}}\end{bmatrix}</math> are a basis for the unobservable subspace.
			* <math>\, T_{\overline{r}o}</math> is chosen so that <math>\,\begin{bmatrix} T_{r\overline{o}} & T_{ro} & T_{\overline{ro}} & T_{\overline{r}o}\end{bmatrix}</math> is invertible.
			By construction, the matrix <math>\, T</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>\, T = T_{ro}</math>, making the other matrices zero dimension.

			==Standard Form==
			By using results from controllability and observability, it can be shown that the transformed system <math>\, (\hat{A}, \hat{B}, \hat{C}, \hat{D})</math> has matrices in the following form:

			: <math>\, \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>\, \hat{B} = \begin{bmatrix}B_{r\overline{o}} \\ B_{ro} \\ 0 \\ 0\end{bmatrix}</math>

			: <math>\, \hat{C} = \begin{bmatrix}0 & C_{ro} & 0 & C_{\overline{r}o}\end{bmatrix}</math>

			: <math>\, \hat{D} = D</math>

			This leads to the conclusion that
			* The subsystem <math>\, (A_{ro}, B_{ro}, C_{ro}, D)</math> is both reachable and observable.
			* The subsystem <math>\, \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>\, \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.

			==See also==
			* [[Observability]]
			* [[Controllability]]

			==External links==
			*[http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-241j-dynamic-systems-and-control-spring-2011/readings/MIT6_241JS11_chap25.pdf Lectures on Dynamic Systems and Control, Lecture 25] - Mohammed Dahleh, Munther Dahleh, George Verghese — MIT OpenCourseWare

			[[Category:Control theory]]

en>Aednichols at 05:21, 30 July 2012

2012-07-30T05:21:36Z

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Threefish - Revision history

174.52.255.118: Explicit reference to Threefish being a symmetric-key cipher

en>Aednichols at 05:21, 30 July 2012