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Fast Switching in Mixed-Integer Model Predictive Control

Artemi Makarow, Christian Kirches

TL;DR

The paper tackles stabilizing mixed-integer model predictive control with finite actuation sets by relaxing to a continuous OCP via partial outer convexification and then reconstructing integer controls on an oversampling grid using sum-up rounding (SUR). It proves P-practical asymptotic stability by combining nominal MPC robustness with SUR error-bounds and a downstream oversampling strategy, showing that the rounding error can be made arbitrarily small with sufficiently fine oversampling. The method integrates stabilizing terminal conditions and input perturbation robustness (ISS/RAS) to ensure stability under rounding, and provides a clear path to practical implementation for fast-dynamics applications. A Van-der-Pol example demonstrates near-relaxed behavior and negligible online overhead for SUR, supporting its relevance to fast-switching domains such as power electronics.

Abstract

We deduce stability results for finite control set and mixed-integer model predictive control with a downstream oversampling phase. The presentation rests upon the inherent robustness of model predictive control with stabilizing terminal conditions and techniques for solving mixed-integer optimal control problems by continuous optimization. Partial outer convexification and binary relaxation transform mixed-integer problems into common optimal control problems. We deduce nominal asymptotic stability for the resulting relaxed system formulation and implement sum-up rounding to restore efficiently integer feasibility on an oversampling time grid. If fast control switching is technically possible and inexpensive, we can approximate the relaxed system behavior in the state space arbitrarily close. We integrate input perturbed model predictive control with practical asymptotic stability. Numerical experiments illustrate practical relevance of fast control switching.

Fast Switching in Mixed-Integer Model Predictive Control

TL;DR

The paper tackles stabilizing mixed-integer model predictive control with finite actuation sets by relaxing to a continuous OCP via partial outer convexification and then reconstructing integer controls on an oversampling grid using sum-up rounding (SUR). It proves P-practical asymptotic stability by combining nominal MPC robustness with SUR error-bounds and a downstream oversampling strategy, showing that the rounding error can be made arbitrarily small with sufficiently fine oversampling. The method integrates stabilizing terminal conditions and input perturbation robustness (ISS/RAS) to ensure stability under rounding, and provides a clear path to practical implementation for fast-dynamics applications. A Van-der-Pol example demonstrates near-relaxed behavior and negligible online overhead for SUR, supporting its relevance to fast-switching domains such as power electronics.

Abstract

We deduce stability results for finite control set and mixed-integer model predictive control with a downstream oversampling phase. The presentation rests upon the inherent robustness of model predictive control with stabilizing terminal conditions and techniques for solving mixed-integer optimal control problems by continuous optimization. Partial outer convexification and binary relaxation transform mixed-integer problems into common optimal control problems. We deduce nominal asymptotic stability for the resulting relaxed system formulation and implement sum-up rounding to restore efficiently integer feasibility on an oversampling time grid. If fast control switching is technically possible and inexpensive, we can approximate the relaxed system behavior in the state space arbitrarily close. We integrate input perturbed model predictive control with practical asymptotic stability. Numerical experiments illustrate practical relevance of fast control switching.

Paper Structure

This paper contains 18 sections, 5 theorems, 44 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Time Discretization
  4. Finite Control Set and Partial Outer Convexification
  5. Null Space of $R$
  6. Downstream Temporal Oversampling
  7. Practical Set-Point Stabilization
  8. Nominal Stability with Stabilizing Terminal Ingredients
  9. Inherent Robustness to Input Perturbations
  10. Control Input Rounding
  11. Numerical Experiments
  12. Conclusion
  13. Proof of Proposition \ref{['prop:nominal_stability']}
  14. Proof of Proposition \ref{['prop:practical_stability']}
  15. Terminal Constraint Satisfaction
  16. ...and 3 more sections

Key Result

Proposition 3.1

Suppose Assumptions assum:steady-state--assum:terminal_control_law hold. Then the optimal value function $V_{N} $ is a Lyapunov function on the feasible set $\mathcal{X}_{N}$: The origin is asymptotically stable in the positive invariant set $\mathcal{X}_{N}$ for the relaxed system eq:sys_cl. There is a function $\beta \in \mathcal{K} \mathcal{L}$ such that for all $x \in \mathcal{X}_{N}$ and a

Figures (1)

  • Figure 1: Robust closed-loop control with SUR. Set-point stabilization for systems \ref{['eq:sys_cl']} and \ref{['eq:sys_cl_d']}. Top left: Phase portrait. Top right: Close-up view of left plot. Bottom left: Closed-loop control trajectories of systems \ref{['eq:sys_cl']} and \ref{['eq:sys_cl_d']} with $\delta t = \Delta t$. Bottom right: Evolutions of the optimal value function $V_{N} $ along ${\phi}_{ \Delta t } $ and $\check{\phi}_{ \Delta t } $ for different oversampling times $\delta t$.

Theorems & Definitions (9)

  • Proposition 3.1: Asymptotic Stability After Relaxation
  • Proposition 3.2: Stability With Input Perturbations
  • Proposition 4.1: Upper Approximation Bound, see Sager2010
  • Theorem 4.2: Fast Switching in MI-MPC
  • proof
  • Remark 1: Quadratic Convex Multipliers
  • proof
  • Lemma A1: See Allan2017
  • proof