In
statespace form (§
1.3.7) [
449],
^{8.7}a general class of
thorder
Ordinary Differential Equations (
ODE),
can be written as

(8.8) 
where
denotes time in seconds,
denotes a vector of
state variables at time
,
denotes the time derivative of
, and
is a vector (any
length) of the system input
signals, if any. Thus, Eq.
(
7.8) says
simply that the timederivative of the state vector is some function
depending on time
, the current state
, and the current
input signals
. The basic problem is to solve for the state
trajectory
given its initial condition
, the system
definition function
, and the input signals
for all
.
In the linear, timeinvariant (
LTI) case, Eq.
(
7.8) can be
expressed in the usual statespace form for LTI continuoustime
systems:

(8.9) 
In this case, standard methods for converting a
filter from continuous
to discrete time may be used, such as the
FDA (§
7.3.1) and
bilinear transform (§
7.3.2).
^{8.8}
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