  We investigate the suboptimality resulting from the application of nominal
model predictive control (MPC) to a nonlinear discrete time stochastic system.
The suboptimality is defined with respect to the corresponding stochastic
optimal control problem (OCP) that minimizes the expected cost of the closed
loop system. In this context, nominal MPC corresponds to a form of
certainty-equivalent control (CEC). We prove that, in a smooth and
unconstrained setting, the suboptimality growth is of fourth order with respect
to the level of uncertainty, a parameter which we can think of as a standard
deviation. This implies that the suboptimality does not grow very quickly as
the level of uncertainty is increased, providing further insight into the
practical success of nominal MPC. Similarly, the difference between the optimal
and suboptimal control inputs is of second order. We illustrate the result on a
simple numerical example, which we also use to show how the proven relationship
may cease to hold in the presence of state constraints.
