## FANDOM

97 Pages

State space analysis is an advanced method for the investigation of dynamic systems. It relies on the analysis of state variables that are usually vectors comprising a critical system variable, its derivative and optionally additional variables.

The state variable description of a system has the form

$\mathbf{\dot{x}}=\mathbf{Fx}+\mathbf{G}u$

y = Hx + ju

x: State of the system (column vector, n elements for an nth order system)

F: n * n system matrix

G: n * 1 input matrix (column matrix)

H: 1 * n output matrix (row matrix)

j: direct transmission term

y: system output

u: system input

## Example Edit

An ASIA element can be converted to state variable form with

${{dz} \over {dt}} = \alpha u(t) - \beta z(t)$.

The state variable form is then

$\left[ \begin{matrix} {\dot{z}} \\ {\ddot{z}} \\ \end{matrix} \right]=\left[ \begin{matrix} -\beta & 0 \\ 0 & 0 \\ \end{matrix} \right]\left[ \begin{matrix} z \\ {\dot{z}} \\ \end{matrix} \right]+\left[ \begin{matrix} \alpha \\ 0 \\ \end{matrix} \right]u(t)$

$y=\left[ \begin{matrix} 1 & 0 \\ \end{matrix} \right]\left[ \begin{matrix} z \\ {\dot{z}} \\ \end{matrix} \right]$

$x=\left[ \begin{matrix} z \\ {\dot{z}} \\ \end{matrix} \right];~\dot{x}=\left[ \begin{matrix} {\dot{z}} \\ {\ddot{z}} \\ \end{matrix} \right]$

$\mathbf{F}=\left[ \begin{matrix} -\beta & 0 \\ 0 & 0 \\ \end{matrix} \right];~\mathbf{G}=\left[ \begin{matrix} \alpha \\ 0 \\ \end{matrix} \right];~\mathbf{H}=\left[ \begin{matrix} 1 & 0 \\ \end{matrix} \right];~j=0$

## Reference Edit

1. Franklin GF, Powell JD, Emami-Naeini A. Feedback Control of Dynamic Systems. Delhi: Pearson Education, 2002. ISBN 8178086751
2. Dietrich, J. W. Signal Storage in Metabolic Pathways: The ASIA Element. kybernetiknet, Vol. 1, No. 3. (2000), pp. 1-9.