  We propose and analyze a stabilizing iteration scheme for the algorithmic
implementation of model predictive control for linear discrete-time systems.
Polytopic input and state constraints are considered and handled by means of
so-called relaxed logarithmic barrier functions. The required on-line
optimization is based on warm starting and performs only a limited, possibly
small, number of optimization algorithm iterations between two consecutive
sampling instants. The optimization algorithm dynamics as well as the resulting
suboptimality of the applied control input are taken into account explicitly in
the stability analysis, and the origin of the resulting overall closed-loop
system, consisting of state and optimization algorithm dynamics, is proven to
be asymptotically stable. The corresponding constraint satisfaction properties
are also analyzed. The theoretical results and a presented numerical example
illustrate the fact that asymptotic stability as well as a satisfactory
closed-loop performance can be achieved by performing only a single
optimization algorithm iteration at each sampling step.
