  Autonomy and electrification in automotive control systems have made
modern-day powertrains one of the most complex cyber-physical systems. This
paper presents a benchmark algorithm to quantify the performance of complex
automotive systems exhibiting mechanical, electrical, and thermal interactions
at various time-scales. Traditionally Dynamic Programming has been used for
benchmarking the performance, however, it fails to deliver results for system
with higher number of states and control lever due to curse of dimensionality.
We propose "PS3", a three-step algorithm for mixed-integer nonlinear optimal
control problems with application to powertrain energy management. PS3 uses
pseudo-spectral collocation theory for highly accurate modeling of dynamics.
Based on the validated powertrain component models, we have addressed
simultaneous optimization of electrical (SOC), vehicular (eco-driving) and
thermal (after-treatment and battery temperatures) dynamics along with an
integer (gear and engine on/off) control and its corresponding (dwell-time)
constraints. PS3 is used to solve such large-scale powertrain problems having
fast and slow dynamic states, discontinuous behaviors, non-differentiable and
linearly interpolated 1-D and 2-D maps, as well as combinatorial constraints.
Five case study powertrain control problems are given to benchmark the accuracy
and computational effort against Dynamic Programming. Our analysis shows that
this algorithm does not scale computational burden as Dynamic Programming does,
and can handle highly complex interactions that occur in modern-day
powertrains, without compromising nonlinear and complex plant modeling.
