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We propose and analyze a stabilizing iteration scheme for the algorithmicimplementation of model predictive control for linear discrete-time systems.Polytopic input and state constraints are considered and handled by means ofso-called relaxed logarithmic barrier functions. The required on-lineoptimization is based on warm starting and performs only a limited, possiblysmall, number of optimization algorithm iterations between two consecutivesampling instants. The optimization algorithm dynamics as well as the resultingsuboptimality of the applied control input are taken into account explicitly inthe stability analysis, and the origin of the resulting overall closed-loopsystem, consisting of state and optimization algorithm dynamics, is proven tobe asymptotically stable. The corresponding constraint satisfaction propertiesare also analyzed. The theoretical results and a presented numerical exampleillustrate the fact that asymptotic stability as well as a satisfactoryclosed-loop performance can be achieved by performing only a singleoptimization algorithm iteration at each sampling step.

A stabilizing iteration scheme for model predictive control based on relaxed barrier functions