State Space Models

Equations of motion for any physical system may be conveniently formulated in terms of its state $\underline{x}(t)$ :

$$\underline{{\dot x}}(t) = f_t[\underline{x}(t),\underline{u}(t)]$$

where

$$\begin{eqnarray*} \underline{x}(t) &=& \mbox{\emph{state} of the system at time $t$}\\ \underline{u}(t) &=& \mbox{vector of \emph{external inputs} (typically driving forces)}\\ f_t &=& \mbox{general function mapping the current state $\underline{x}(t)$\ and}\\ && \mbox{inputs $\underline{u}(t)$\ to the state time-derivative $\underline{{\dot x}}(t)$} \end{eqnarray*}$$

• The function $f_t$ may be time-varying, in general

• This potentially nonlinear time-varying model is extremely general (but causal)

• Even the human brain can be modeled in this form