BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 61

When a<U , the magnitude of the acceleration of the base does not
exceed the maximum magnitude of the absolute acceleration allowed for
the object for all t ≥ 0, and Eqs. (3.29) - (3.32) do not apply. For this case,

the optimal control is

⎧
⎨

− a if 0 ≤ t ≤ τ,

u 0 (t) ≡− v(t) = (3.33)

⎩

0if t > τ,

and can be provided by attaching the object rigidly to the base. Accordingly,
J 1 (u 0 ) = 0.

Use Eqs. (3.11), (3.12), (3.31), and (3.32) to obtain the solution of the
reciprocal Problem 3.2. The function g(U) of Eq. (3.10) for this case has
the form

1

V τ .

V 2
2 U

U

g(U) = − (3.34)

Following the procedure defined in Section 3.1.4, fix the constraint param-
eter D and find g − 1 (D) . This can be performed either graphically as is
illustrated in Fig. 3.2 or analytically by solving the equation g(U) = D for
U . The solution gives

V 2

g − 1 (D) =

Vτ .

(3.35)

2 D +

Using Eq. (3.11), substitute the expression of Eq. (3.35) for U into

Eq. (3.32) to find the optimal control

⎧
⎨ V 2 2 D

V − τ,

− if 0 ≤ t ≤

2 D + Vτ

u 0 D (t)

=

(3.36)

⎩

if t > 2 D

0

V − τ.

The minimum value of the performance index J 2 , in accordance with
Eqs. (3.12) and (3.35), is determined from

V 2

J 2 (u 0 D ) =

Vτ .

(3.37)

2 D +