Biomedical Engineering Reference
In-Depth Information
only on time. The solution of the system of Eq. (3.44) can be represented
by the convolution integral
t
x(t)
=
(t
−
ξ)
[
u(ξ)
+
v(ξ)
]
dξ.
(3.45)
0
To discretize Eq. (3.45), specify a time interval [0
,T
] for which the
motion of the system will be considered and divide this interval into
N
identical subintervals of size
h
=
T/N
. Denote the nodal points of the
discretization as
=
=
t
i
ih,
i
0
,
1
,...,N,
(3.46)
and the displacement
x(t
i
)
at the point
t
i
as
x
i
.
Approximate the control function
u(t)
and the disturbance
v(t)
by con-
stant values in each of the subintervals
t
i
−
1
≤
t<t
i
,thatis,
u(t)
=
u
i
for
(i
−
1
)h
≤
t<ih,
i
=
1
...,N,
(3.47)
v(t)
=
v
i
for
(i
−
1
)h
≤
t<ih,
i
=
1
...,N.
(3.48)
Substitute the piecewise constant functions of Eqs. (3.47) and (3.48) into
Eq. (3.45), set
t
kh
in the latter equation, and integrate the right-hand
side of Eq. (3.45) from 0 to
kh
to obtain
=
k
h
2
2
x
0
=
0
,
x
k
=
(u
i
+
v
i
)
[2
(k
−
i)
+
1]
,
k
=
1
...,N.
i
=
1
(3.49)
This relation is the discrete-time approximation of Eq. (3.45) and, hence,
of the system of Eq. (3.44). The variables
u
i
are to be determined, while
the variables
v
i
that approximate the prescribed function
v(t)
in the inter-
vals [
t
i
−
1
,t
i
)
are specified. The quantities
v
i
can be introduced in various
manners. For example,
v
i
may be defined as the mean between the nodal
values
v(t
i
−
1
)
and
v(t
i
)
:
2
v(t
i
−
1
)
v(t
i
)
,
1
=
+
=
v
i
i
1
...,N.
(3.50)
It is important that the
x
k
's of Eq. (3.49) are linear functions of the control
parameters
u
i
. This will enable the discrete-time analogues of Problems 3.1
and 3.2 to be reduced to a linear programming problem.
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