BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 104

For the system governed by Eqs. (3.180) and (3.181), find the disturbance
function v 0 (t) belonging to the class W such that

J(v 0 ) = max

v ∈ W J(v).

(3.182)

The best disturbance problem is that of the determination of the lower
bound of the functional J(v) for the disturbances of the class W .

Problem 3.8 Best Disturbance Problem

List of Variables for Problem 3.8

State variable x Displacement of the object relative to

the base

External disturbance v Shock acceleration pulse, the negative of

the acceleration of the base

W Class of the external disturbances v

Functions and parameters of
the model

u Force of the shock isolator divided by

the mass of the object, absolute
acceleration of the object

Performance index J Performance criterion to be minimized

Optimal solution J ( v 0 ) Minimum value of the performance

index J

v 0 Best disturbance

For the system governed by Eqs. (3.180) and (3.181), find the disturbance
function v 0 (t) belonging to the class W such that

J(v 0 ) = min

v ∈ W J(v).

(3.183)

The worst-to-best ratio is defined by

J(v 0 )
J(v 0 ) .

R = (3.184)