OPTIMAL SHOCK ISOLATION FOR MULTI-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 125

Problem 4.3 Analogue of Problem 4.1 for Rigid System

List of Variables for Problem 4.3

State variables x Displacement of the object relative to

the base

z Displacement of the base relative to the

inertial reference frame

Control variable F Shock isolator force

External disturbance z(t) Acceleration of the base

Functions and
parameters of the model

m 1 Mass of body 1

m 2 Mass of body 2

J 1

Performance index Maximum magnitude of the

displacement of the object relative to the
base

J 2

Performance criterion
subjected to a constraint

Maximum magnitude of the force
applied to body 1

Constraint P Maximum allowable value for the

criterion

J 2

J 1

Optimal solution Optimal value of the performance

index

J 1

For Eq. (4.20) subject to the initial conditions of Eq. (4.21), find

J 1 J 1 (F ),

= inf

F

(4.24)

provided that

J 2 (F ) ≤ P,

(4.25)

where P is a prescribed positive magnitude. The greatest lower bound
in Eq. (4.24) is required to be found in the class of piecewise continuous
controls F(t) .

Note that in the case under consideration the constraint of Eq. (4.25)
implies the constraint

m 1 + m 2

| F |≤ P (4.26)

m 1

on the control force.