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Numerical Integration

Recall the general state-space model in continuous time:

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

An approximate discrete-time numerical solution is

$\displaystyle \underline{x}(t_n+T_n) \eqsp \underline{x}(t_n) + T_n\,f_{t_n}[\underline{x}(t_n),\underline{u}(t_n)]
$

for $ n=0,1,2,\ldots\,.\quad$ (Forward Euler)

Let $ g_{t_n}[\underline{x}(t_n),\underline{u}(t_n)] \isdef \underline{x}(t_n) + T_n\,f_{t_n}[\underline{x}(t_n),\underline{u}(t_n)]$ :

\epsfig{file=eps/statemachineg.eps,width=5in}



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``Introduction to State Space Models'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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