  The ``prediction + optimal control'' scheme has shown good performance in
many applications of automotive, traffic, robot, and building control. In
practice, the prediction results are simply considered correct in the optimal
control design process. However, in reality, these predictions may never be
perfect. Under a conventional stochastic optimal control formulation, it is
difficult to answer questions like ``what if the predictions are wrong''. This
paper presents an analysis framework for predictive optimal control where the
subjective belief about the future is no longer considered perfect. A novel
concept called the hidden prediction state is proposed to establish connections
among the predictors, the subjective beliefs, the control policies and the
objective control performance. Based on this framework, the predictor
evaluation problem is analyzed. Three commonly-used predictor evaluation
measures, including the mean squared error, the regret and the log-likelihood,
are considered. It is shown that neither using the mean square error nor using
the likelihood can guarantee a monotonic relationship between the predictor
error and the optimal control cost. To guarantee control cost improvement, it
is suggested the predictor should be evaluated with the control performance,
e.g., using the optimal control cost or the regret to evaluate predictors.
Numerical examples and examples from automotive applications with real-world
driving data are provided to illustrate the ideas and the results.
