We derive practical stability results for finite-control set and mixed-integer model predictive control. Thereby, we investigate the evolution of the closed-loop system in the presence of control rounding and draw conclusions about optimality. The paper integrates integral approximation strategies with the inherent robustness properties of conventional model predictive control with stabilizing terminal conditions. We propose an alternative Lyapunov function candidate and elaborate in detail the importance of the rounding history on the closed-loop performance. Finally, we embed sum-up rounding into our theoretical findings, which evaluates the rounding history and limits the integral approximation error. Numerical experiments illustrate the importance of an advanced rounding strategy in the context of mixed-inter model predictive control.
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