OPTIMAL SHOCK ISOLATION FOR MULTI-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 155

and the performance criteria of Eqs. (4.125) and (4.126) become

J 1 (u 2 ) =

t ∈ [0 , ∞ ) | x(t) | ,

max (4.132)

J 2 (u 2 ) =

t ∈ [0 , ∞ ) | u 2 (t) | .

max (4.133)

The criteria of Eqs. (4.132) and (4.133) coincide with those of Eqs. (3.5)
and (3.7), respectively. Problem 4.6 can be formulated as follows:

Problem 4.7 Another Formulation for the Auxiliary Problem

List of Variables for Problem 4.7

State variable x Displacement of the object relative to

the base

Control variable u 2 Force F 2 divided by the mass of the

object, u 2 = F 2 /m , absolute acceleration
of the object

External disturbance v The negative of the shock acceleration

pulse, v =− σ

J 1

Performance index Maximum absolute value of the

displacement of the object relative to the
base

Performance criterion
subjected to a constraint

J 2 Maximum magnitude of the absolute

acceleration of the object

Constraint D 2 Maximum allowable value for the

criterion J 2

J min
1

Optimal solution Optimal value of the performance

index

J 1

u 2 Optimal control u 2

For the system of Eq. (4.130) subject to the initial conditions of
Eq. (4.131) and a prescribed disturbance v(t) , find an open-loop control
u 2 (t) to minimize the criterion

J 1 , provided that the criterion J 2 is

constrained by J 2 ≤ D 2 .

Problem 4.7 coincides with Problem 3.1, except the variables u and U
in Problem 3.1 are replaced by u 2 and D 2 in Problem 4.7.