Engineering Optimization: Theory and Practice, Fourth Edition

18 Introduction to Optimization

or

xi−mig−k 1 vi=miai (E 1 )

where the massmican be expressed as

mi=mi− 1 −k 2 s (E 2 )

andk 1 andk 2 are constants. Equation (E 1 ) can be used to express the acceleration,ai,

as

ai=

xi

mi

−g−

k 1 vi

mi

(E 3 )

Iftidenotes the time taken by the rocket to travel from pointito pointi+1, the

distance traveled between the pointsiandi+1 can be expressed as

s=viti+^12 aiti^2

or

1

2

ti^2

(

xi

mi

−g−

k 1 vi

mi

)

+tivi−s= 0 (E 4 )

from whichtican be determined as

ti=

−vi±

√

vi^2 + 2 s

(

xi

mi

−g−

k 1 vi

mi

)

xi

mi

−g−

k 1 vi

mi

(E 5 )

Of the two values given by Eq. (E 5 ), the positive value has to be chosen forti. The

velocity of the rocket at pointi+ 1 , vi+ 1 , can be expressed in terms ofvi as (by

assuming the acceleration between pointsiandi+1 to be constant for simplicity)

vi+ 1 =vi+aiti (E 6 )

The substitution of Eqs. (E 3 ) and (E 5 ) into Eq. (E 6 ) leads to

vi+ 1 =

√

v^2 i+ 2 s

(

xi

mi

−g−

k 1 vi

mi

)

(E 7 )

Froman analysis of the problem, the control variables can be identified as the thrusts,

xi, and the state variables as the velocities,vi. Since the rocket starts at point 1 and

stops at point 13,

v 1 =v 13 = 0 (E 8 )