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Uniting Iteration Limits for Mixed-Integer Quadratic MPC

Luke Fina, Christopher Petersen

TL;DR

The paper tackles stability-preserving model predictive control when solver compute is limited by iteration budgets. It introduces a hybrid 'united' controller that switches between two MIQP-MPC solvers with high and low iteration limits, and provides an interpretable branch-and-bound algorithm to realize this switch. Through formal hybrid-systems analysis, it proves asymptotic stability and robustness of the unified control law and demonstrates, via spacecraft rendezvous simulations (switching thrusters and minimum thrust), that meaningful compute reductions are achievable without sacrificing stability. The work offers practical guidelines for tuning iteration-limit parameters and shows how to balance branch-and-bound and QP iterations in compute-constrained environments. Overall, it provides a complete framework—from theory to implementable algorithms and simulations—for suboptimal MIQP-MPC with controllable computational dynamics.

Abstract

Iteration limited model predictive control (MPC) can stabilize a feedback control system under sufficient conditions; this work explores combining a low iteration limit MPC with a high iteration limit MPC for mixed-integer quadratic programs (MIQPs) where the suboptimality is due to solver iteration limits. To combine the two MPCs a hybrid systems controller is developed that ``unites'' two MIQP-MPC solvers where the iteration limits of interest are the branch-and-bound and quadratic programming iteration limits. Asymptotic stability and robustness of the hybrid feedback control system are theoretically derived. Then an interpretable branch-and-bound algorithm and implementable uniting controller algorithm are developed. Finally, the developed algorithms and varying iteration limits are empirically evaluated in simulation for the switching thruster and minimum thrust spacecraft rendezvous problems.

Uniting Iteration Limits for Mixed-Integer Quadratic MPC

TL;DR

The paper tackles stability-preserving model predictive control when solver compute is limited by iteration budgets. It introduces a hybrid 'united' controller that switches between two MIQP-MPC solvers with high and low iteration limits, and provides an interpretable branch-and-bound algorithm to realize this switch. Through formal hybrid-systems analysis, it proves asymptotic stability and robustness of the unified control law and demonstrates, via spacecraft rendezvous simulations (switching thrusters and minimum thrust), that meaningful compute reductions are achievable without sacrificing stability. The work offers practical guidelines for tuning iteration-limit parameters and shows how to balance branch-and-bound and QP iterations in compute-constrained environments. Overall, it provides a complete framework—from theory to implementable algorithms and simulations—for suboptimal MIQP-MPC with controllable computational dynamics.

Abstract

Iteration limited model predictive control (MPC) can stabilize a feedback control system under sufficient conditions; this work explores combining a low iteration limit MPC with a high iteration limit MPC for mixed-integer quadratic programs (MIQPs) where the suboptimality is due to solver iteration limits. To combine the two MPCs a hybrid systems controller is developed that ``unites'' two MIQP-MPC solvers where the iteration limits of interest are the branch-and-bound and quadratic programming iteration limits. Asymptotic stability and robustness of the hybrid feedback control system are theoretically derived. Then an interpretable branch-and-bound algorithm and implementable uniting controller algorithm are developed. Finally, the developed algorithms and varying iteration limits are empirically evaluated in simulation for the switching thruster and minimum thrust spacecraft rendezvous problems.
Paper Structure (19 sections, 3 theorems, 21 equations, 13 figures, 4 tables, 3 algorithms)

This paper contains 19 sections, 3 theorems, 21 equations, 13 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Parametric MIQP-MPC Formulation
  4. Iteration Limited Constraint Mappings
  5. Hybrid Systems Model
  6. United Control Law
  7. Stability Analysis
  8. Algorithms
  9. Branch-and-Bound Preliminaries
  10. Simulation Results
  11. Algorithm \ref{['alg:uniting_control']} Compared to a Fixed Iteration Limit
  12. Results for Switching Control
  13. Results for Minimum Thrust
  14. A Closer Look at Algorithm \ref{['alg:uniting_control']} Near the Terminal Set
  15. Computational Improvements
  16. ...and 4 more sections

Key Result

Proposition 1

The feasible constraint set value mapping, $h(x_{k},b),$ is continuous on the feasible parameter set, where the feasible parameter set is defined as

Figures (13)

  • Figure 1: Feedback Loop for Hybrid Uniting Control Law
  • Figure 2: Supervisory Law Sets
  • Figure 3: Example of a branch-and-bound tree via Algorithm \ref{['alg:bnb']}.
  • Figure 4: Fixed Iteration Limits for Switching Thruster (Example \ref{['prob:switching_thrusters']}).
  • Figure 5: Varying Branch-and-Bound Iteration Limits for Switching Thruster (Example \ref{['prob:switching_thrusters']}).
  • ...and 8 more figures

Theorems & Definitions (7)

  • Proposition 1
  • Definition 1
  • Lemma 1
  • Proposition 2
  • Remark 1
  • Example 1: Spacecraft Rendezvous and Proximity with Switching Thrusters
  • Example 2: Spacecraft Rendezvous and Proximity with Minimum Thrust Control