First, a sampled-data system which can be described by a linear time-variant vector differential equation
X(
t)=
A(
t)
X(
t) in each step interval
T0 is considered. The step interval
T0 is defined as
T0=
T/
m, where
T is the sampling period and
m is a positive integer. Generally, this equation cannot be solved analytically. So, regarding
A(
t) as fixed in each step interval
T0 and introducing the time shift operator δ(·), we can solve the equation numerically. That is,
X(
t)=
An, iX(
t),
X(
nT+
iT0+)=
Bn, i(
T0)
X(
nT+
iT0-)+
Cn, i(
T0)
X(
nT-), for
nT+
iT0<
t≤
nT+(
i+1)
T0,
n=0, 1, 2, …,
i=0, 1, 2, …,
m-1, where
Bn, i(
T0) and
Cn, i(
T0) involve δ(·).
A sampled-data system with dead time can be analyzed similarly as a sampled-data system without dead time by introducing the time shift operator.
Next, a sampled-data system which can be described by a nonlinear time-variant vector differential equation in each step interval
T0 is considered.
In this case, by introducing the variable equivalent gain
ln, i of a nonlinear element, it can be solved similarly as in the linear case.
The error of the approximate solution is considered in chapter 4. It is proved that the error can be made arbitrarily small by choosing sufficiently small
T0.
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