OPTIMAL SHOCK ISOLATION FOR MULTI-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 126

Introduce the notation

F P

m 1

u =

m 2 ,

v =−¨ z, U = (4.27)

m 1 +

to formulate Problem 4.3 as follows:

Problem 4.4 Optimum Shock Isolation Problem for
Single-Degree-of-Freedom System

List of Variables for Problem 4.4

State variable x Displacement of the object relative to the

base

Control variable u Shock isolator force F divided by the sum

of the masses of bodies 1 and 2,
u = F/(m 1 + m 2 )

External disturbance v Shock acceleration pulse, the negative of

the acceleration of the base, v =−¨ z

J 1

Performance index Maximum magnitude of the displacement

of the object relative to the base

Constraint U Constraint on the control u : the maximum

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

J 1 ( u 0 ) J 1

Optimal solution Optimal value of the performance index

u 0 Optimal control u

x 0 Optimal time history of displacement x

For the system

x ¨ = u + v (4.28)

subject to the initial conditions

x( 0 ) = 0 , x( 0 ) ˙ = 0 (4.29)

and a prescribed external disturbance v(t) , find a piecewise continuous
control u =˜ u 0 (t) such that

J 1 ( J 1 (u),

u 0 ) ˜ = min

u

(4.30)

provided that

| u |≤ U. (4.31)