Decomposition Analysis of an Automotive Powertrain Design ...

τ dV
τ 0-60 = τ :⌡ ⌠ dt dt = 60 mph; initial conditions: V o = 0 (68)
0
τ
τ 5-20 = τ :⌡ ⌠ 0
dV
dt dt = 20 mph; initial conditions: V o = 5 mph (69)
4
S 0-4 = ⌡ ⌠ 0
4
( ⌡ ⌠ 0
dV
dt dt ) dt ; initial conditions: V o = 0. (70)
Computationally, each integral is discretized into sums where the integrand is computed for each
discrete time interval, ∆t i . For decompositon analysis, ∆t i can be viewed as a state variable, and in
the original formulation of this problem (Wagner 1993) it was considered in the FDT. However,
in the final implementation of a numerical integration scheme, it becomes a parameter in the sense
that it is prescribed by the integration scheme, and hence neglected in the construction of the FDT.
This treatment of the time parameter, makes partitioning results obtained with heuristic search
methods directly comparable to those obtained with global search methods.
The gradeability criteria are defined by steady-state formulations of the equations of
motion. Rearrangement yields the following expressions
and
sin α s = (T e (60 ξ fd ξ 1 V s /(2 r e π)) ξ fd ξ 1 η fd η gb R to - r e (F roll + F aero )) (r e M g) -1 (71)
sin (α c ) = (T e (60 ξ fd ξ ngears-1 V c /(2 r e π)) ξ fd ξ ngears-1 η fd η gb - r e (F roll + F aero )) (r e M g) -1
(72)
where, α c = 3.43 o (6% grade) and V s = 5 mph.
Equations (1) through (72) define the functional relationships among the variables
describing the system. Formulation of the optimal design problem by identifying objective function
and constraints follows.
The objective is to develop a vehicle that competes in a given market, meets emissions
requirements, and has the maximum fuel economy possible. Target acceleration characteristics are
specified to deem it competitive in its market. The U.S. Federal Clean Air Act constrains the NOx
emissions to no more than 0.4 g/mile; time to accelerate from 0 to 60 mph is no greater than the