In this paper we propose a formulation for approximate constrained nonlinear
output-feedback stochastic model predictive control. Starting from the ideal
but intractable stochastic optimal control problem (OCP), which involves the
optimization over output-dependent policies, we use linearization with respect
to the uncertainty to derive a tractable approximation which includes knowledge
of the output model. This allows us to compute the expected value for the outer
functions of the OCP exactly. Crucially, the dual control effect is preserved
by this approximation. In consequence, the resulting controller is aware of how
the choice of inputs affects the information available in the future which in
turn influences subsequent controls. Thus, it can be classified as a form of
implicit dual control