INJURY CONTROL FOR WHEELCHAIR OCCUPANTS - Injury Biomechanics and Control: Optimal Protection from Impact - page 263

Performance index A Maximum magnitude of the absolute

acceleration of the object

Performance criterion
subjected to a constraint

R Difference between the maximum and

minimum values of the displacement of
the object relative to the base

Constraint D Maximum allowable value for the

criterion R

Optimal solution A(u 0 ) Optimal value of the performance

index A

u 0 Optimal control u

For the system of Eqs. (8.1) and (8.2) subject to the initial conditions

x( 0 ) = 0 , x( 0 ) ˙ = 0 (8.6)

find a piecewise continuous optimal control u u 0 (t) that minimizes the

peak magnitude of the absolute acceleration of the object, A , provided
that the rattlespace is constrained by

=

R ≤ D, (8.7)

where D is a prescribed positive quantity.

The initial conditions of Eq. (8.6) imply that the isolator starts acting
at the beginning instant of the crash pulse. A considerable decrease in the
peak magnitude of the absolute acceleration of the object can be provided
by engaging the control of the isolator in advance, before the crash pulse
has been applied to the vehicle. This leads to a revised statement of the
optimal control problem.

Problem 8.2 Pre-Acting Optimal Control

List of Variables for Problem 8.2

State variables x Displacement of the object relative to

the base

z Displacement of the base relative to the

inertial reference frame