BASIC OPTIMAL SHOCK ISOLATION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS - Injury Biomechanics and Control: Optimal Protection from Impact - page 107

Solution of the Optimal Control Problems: Computational
Technique Even for the linear system of Eq. (3.186), the solution of
the best and worst disturbance problems (Problems 3.7 and 3.8) requires
numerical methods. To this end, the continuous-time formulation of these
problems is replaced by a discrete-time approximation. The time axis
is discretized with step size h . On the time intervals (i − 1 )h ≤ t<

ih, i = 1 , 2 ,..., the function v(t) is assumed to be constant:

⎨

v i if (i − 1 )h < T ,

v(t) = (3.193)

⎩

0if (i − 1 )h ≥ T, (i − 1 )h ≤ t<ih.

1 ,...,N , play the role of design vari-
ables when solving optimization problems such as Problems 3.7 and 3.8.
The number of these design variables is defined as follows:

The constant parameters v i ,i =

⎨ T

h

T
h

=

if 0 ,

N = (3.194)

T
h

1 f T
h

⎩

+ = 0 ,

where the square and curly brackets denote the integer and fractional parts,
respectively, of the expression within these brackets.
The solution of Eq. (3.186) subject to the initial conditions of Eq. (3.181)
has the convolution form

t

x(t) = g(t − τ)v(τ) dτ,

0

(3.195)

t
0 g(t − τ)v(τ) dτ,

x(t) =

where

⎨

ω − D exp ( − ζω n t) sin ω D t if ζ< 1 ,

g(t) = t exp ( − ζω n t) if ζ = 1 ,

⎩

ω n ω − D exp ( ζ 2 .

(3.196)
The function g(t) is referred to as the impulse response function (fun-
damental solution) for Eq. (3.186). It satisfies the differential equation of

− ζω n t) sinh ω D t if ζ > 1 , D = | 1 − |