OPTIMAL SHOCK ISOLATION FOR MULTI-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 123

Performance index J 1 Maximum magnitude of the

displacement of body 2 relative to the
base

Constraint U Constraint on the control u :maximum

allowable value for the criterion J 2
divided by the mass of body 1,
U = P/m 1

Optimal solution J 1 (u 0 ) Optimal value of the performance index

J 1

u 0 Optimal control u

x 0 Optimal time history of displacement x

ξ 0 Optimal time history of displacement ξ

For the system governed by Eqs. (4.13) and (4.14) subject to the ini-
tial conditions of Eq. (4.15) and a prescribed external disturbance v(t) ,
find a piecewise continuous optimal control u = u 0 (t) that satisfies the

constraint

| u |≤ U (4.16)

and minimizes the functional

J 1 (u) =

t ∈ [0 , ∞ ) | y(t) |=

max

t ∈ [0 , ∞ ) | x(t) − ξ(t) | .

max (4.17)

The solution of Problem 4.2 allows one to determine the minimal value
of the performance index J 1 (F ) in Problem 4.1: J 1 (u 0 )

J 1 . In addition,
if the solution of Problem 4.2 is known, the optimal control F = F 0 (t)
for Problem 4.1 can be readily calculated. The solution of the system of
Eqs. (4.13) and (4.14) for the initial conditions of Eq. (4.15) and the con-
trol u

=

ξ 0 (t) . The superscript zero indicates
that this solution corresponds to the optimal control u 0 (t) . From Eq. (4.12),
y

= u 0 (t) gives x = x 0 (t) and ξ =

= y 0 (t) = x 0 (t) − ξ 0 (t) . By substituting x 0 (t) , y 0 (t) , z(t) ¨ =− v(t) ,and

=

W(t) m 1 u 0 (t) into Eq. (4.10), the optimal control F 0 (t) can be deter-

minedintheform

= ¨ − +

F 0 (t) m 2 ( y 0 (t) v(t)) m 1 u 0 (t). (4.18)

Note that at the instants when the optimal control u 0 (t) undergoes discon-
tinuities, the velocity

˙

y 0 (t) also undergoes jumps. Let v(t) be a piecewise