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Biomedical Cybernetics

State space analysis

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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

Asia 25

ASIA element

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.

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