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Minimal Constraint Violation Probability in Model Predictive Control for Linear Systems

Michael Fink, Tim Brüdigam, Dirk Wollherr, Marion Leibold

TL;DR

This work tackles the problem of enforcing linear state constraints under additive uncertainty in Model Predictive Control by minimizing the probability of constraint violation rather than enforcing hard constraints alone. It introduces CVPM-MPC, which first constructs an optimal CVPM input set that minimizes $ \Pr( x \notin \mathcal{X})$ and then uses this set to define admissible inputs for the MPC optimization, ensuring recursive feasibility. Theoretical results establish ISS of the closed-loop system under standard assumptions, and the method accommodates time-variant constraints with discussion of short- and long-term changes. A DC-DC converter example demonstrates that CVPM-MPC can achieve safer operation with significantly reduced violation probabilities, while offering two practical probability-approximation paths (Monte Carlo and a Gaussian-based QP). Overall, the approach provides a scalable, less-conservative alternative to RMPC/SMPC for linear systems, with potential extensions to joint probabilistic-robust formulations and time-varying constraint handling.

Abstract

Handling uncertainty in model predictive control comes with various challenges, especially when considering state constraints under uncertainty. Most methods focus on either the conservative approach of robustly accounting for uncertainty or allowing a small probability of constraint violation. In this work, we propose a linear model predictive control approach that minimizes the probability that linear state constraints are violated in the presence of additive uncertainty. This is achieved by first determining a set of inputs that minimize the probability of constraint violation. Then, this resulting set is used to define admissible inputs for the optimal control problem. Recursive feasibility is guaranteed and input-to-state stability is proved under assumptions. Numerical results illustrate the benefits of the proposed model predictive control approach.

Minimal Constraint Violation Probability in Model Predictive Control for Linear Systems

TL;DR

This work tackles the problem of enforcing linear state constraints under additive uncertainty in Model Predictive Control by minimizing the probability of constraint violation rather than enforcing hard constraints alone. It introduces CVPM-MPC, which first constructs an optimal CVPM input set that minimizes and then uses this set to define admissible inputs for the MPC optimization, ensuring recursive feasibility. Theoretical results establish ISS of the closed-loop system under standard assumptions, and the method accommodates time-variant constraints with discussion of short- and long-term changes. A DC-DC converter example demonstrates that CVPM-MPC can achieve safer operation with significantly reduced violation probabilities, while offering two practical probability-approximation paths (Monte Carlo and a Gaussian-based QP). Overall, the approach provides a scalable, less-conservative alternative to RMPC/SMPC for linear systems, with potential extensions to joint probabilistic-robust formulations and time-varying constraint handling.

Abstract

Handling uncertainty in model predictive control comes with various challenges, especially when considering state constraints under uncertainty. Most methods focus on either the conservative approach of robustly accounting for uncertainty or allowing a small probability of constraint violation. In this work, we propose a linear model predictive control approach that minimizes the probability that linear state constraints are violated in the presence of additive uncertainty. This is achieved by first determining a set of inputs that minimize the probability of constraint violation. Then, this resulting set is used to define admissible inputs for the optimal control problem. Recursive feasibility is guaranteed and input-to-state stability is proved under assumptions. Numerical results illustrate the benefits of the proposed model predictive control approach.
Paper Structure (29 sections, 6 theorems, 19 equations, 2 figures)

This paper contains 29 sections, 6 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contribution and Structure
  4. Notation
  5. Problem Setup
  6. System Dynamics
  7. Model Predictive Control
  8. Problem Statement
  9. Method
  10. CVPM-MPC
  11. CVPM-MPC Formulation
  12. Probability Optimization in
  13. Sampling-Based Probability Optimization
  14. Probability Optimization approximated as Quadratic Program
  15. Properties
  16. ...and 14 more sections

Key Result

Lemma 1

If Assumption ass:rci holds, then for all $\in$ there exists an input such that the state at the next time step is also in the set , i.e., is robust control invariant.

Figures (2)

  • Figure 1: Comparison of probability minimization methods; Left: trajectory of Monte Carlo Integration (red), Quadratic Program approximation (blue, Sec. \ref{['sec:approx']}); Right: constraint violation probability
  • Figure 2: Simulation of CVPM-MPC with an unmodeled disturbance: Left: convergence to the set $\mathcal{X}$; Right: convergence to the set $\mathcal{X}$ after unmodeled disturbance. (green: ; red: ; bright marker: current state)

Theorems & Definitions (22)

  • Definition 1: Optimal CVPM Input Set
  • Definition 2: CVPM
  • Definition 3: CVPM
  • Remark 1
  • Remark 2
  • Definition 4: Recursive Feasibility
  • Remark 3
  • Lemma 1
  • proof
  • Theorem 1
  • ...and 12 more