Biomedical Engineering Reference
In-Depth Information
However, analytical solutions are possible for only a few types of rather
simple
external
disturbances,
v(t)
.
Even
for
a
half-sine
disturbance,
which
is
frequently
used
to
model
an
automobile
crash
deceleration
pulse,
⎧
⎨
⎩
a
sin
πt
τ
≤
≤
if 0
t
τ,
v(t)
=
(3.43)
0
if
t > τ,
where
a
is the amplitude and
τ
is the duration of the pulse, a completely
analytical solution is impossible. Balandin, Bolotnik, and Pilkey (2001)
utilized a graphical-analytical technique to solve Problem 3.1 for such a
disturbance. This method was developed independently by Guretskii (1969)
and Sevin and Pilkey (1971). This technique can be useful, but it applies
effectively only to disturbances that have only one excursion beyond the
corridor
U
allowed for the value of the control variable
u
. Unlike
Eq. (3.16), this excursion does not necessarily start at the onset of the
pulse.
A numerical method that replaces the continuous-time model of Eq. (3.3)
by a discrete-time approximation and reduces an optimal control problem
to a constrained minimization of a function of a finite number of variables
is more universal. In this section, an approach for solving Problems 3.1 and
3.2 numerically by reducing them to linear programming will be discussed.
There are a number of highly reliable linear programming algorithms inte-
grated in most optimization software packages.
|
u
|≤
3.3.1 Discretization of the Equation of Motion and the
Performance Criteria
Discretization of Differential Equation
For the single-degree-of-
freedom model shown in Fig. 3.1, the motion of the object relative to
the base is governed by Eq. (3.3) subject to the initial conditions of
Eq. (3.4):
x
¨
=
u
+
v(t),
x(
0
)
=
0
,
x(
0
)
˙
=
0
,
(3.44)
where
x
is the displacement of the object relative to the base,
v(t)
is the
negative of the absolute acceleration of the base, and
u
is the absolute accel-
eration of the object. The function
v(t)
represents the impact deceleration
pulse of the base while the function
u
is regarded as a control variable.
For the limiting performance analysis, the function
u
is assumed to depend
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