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Time-Optimal Model Predictive Control for Linear Systems with Multiplicative Uncertainties

Renato Quartullo, Andrea Garulli, Mirko Leomanni

TL;DR

The paper tackles time-optimal Model Predictive Control for linear discrete-time systems with multiplicative uncertainties modeled as interval matrices $[A\ B] \in \mathcal{I}_S$. It develops an offline matrix-zonotope bounding strategy via a bounding operator $\mathbb{T}_{\mathcal{I}}$ and an adaptive terminal set to guarantee recursive feasibility and finite-time convergence. By over-approximating the uncertain error dynamics with interval bounds $\mathcal{B}_k(j)$, the online optimization becomes tractable as a MILP, demonstrated on a satellite orbital rendezvous scenario. Results show reduced conservatism and real-time capability (sub-second solve times) with improved region of attraction and lower fuel consumption compared to additive-disturbance approaches.

Abstract

This paper presents a time-optimal Model Predictive Control (MPC) scheme for linear discrete-time systems subject to multiplicative uncertainties represented by interval matrices. To render the uncertainty propagation computationally tractable, the set-valued error system dynamics are approximated using a matrix-zonotope-based bounding operator. Recursive feasibility and finite-time convergence are ensured through an adaptive terminal constraint mechanism. A key advantage of the proposed approach is that all the necessary bounding sets can be computed offline, substantially reducing the online computational burden. The effectiveness of the method is illustrated via a numerical case study on an orbital rendezvous maneuver between two satellites.

Time-Optimal Model Predictive Control for Linear Systems with Multiplicative Uncertainties

TL;DR

The paper tackles time-optimal Model Predictive Control for linear discrete-time systems with multiplicative uncertainties modeled as interval matrices . It develops an offline matrix-zonotope bounding strategy via a bounding operator and an adaptive terminal set to guarantee recursive feasibility and finite-time convergence. By over-approximating the uncertain error dynamics with interval bounds , the online optimization becomes tractable as a MILP, demonstrated on a satellite orbital rendezvous scenario. Results show reduced conservatism and real-time capability (sub-second solve times) with improved region of attraction and lower fuel consumption compared to additive-disturbance approaches.

Abstract

This paper presents a time-optimal Model Predictive Control (MPC) scheme for linear discrete-time systems subject to multiplicative uncertainties represented by interval matrices. To render the uncertainty propagation computationally tractable, the set-valued error system dynamics are approximated using a matrix-zonotope-based bounding operator. Recursive feasibility and finite-time convergence are ensured through an adaptive terminal constraint mechanism. A key advantage of the proposed approach is that all the necessary bounding sets can be computed offline, substantially reducing the online computational burden. The effectiveness of the method is illustrated via a numerical case study on an orbital rendezvous maneuver between two satellites.

Paper Structure

This paper contains 15 sections, 7 theorems, 68 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Notation
  4. Properties of interval matrices and matrix zonotopes
  5. Problem Formulation
  6. Time-Optimal Robust MPC
  7. Adaptive Terminal Set Design for Recursive Feasibility
  8. Computational aspects
  9. Numerical example
  10. Conclusions
  11. Appendix
  12. Proof of Proposition \ref{['thm:overapprox']}
  13. Auxiliary lemmas
  14. Proof of Theorem \ref{['thm:RF']}
  15. Proof of Theorem \ref{['thm:convergence']}

Key Result

Proposition 1

Let $A\in\mathbb{R}^{n \times l}$ and $M \in \mathbb{R}^{l \times q}$, with $M \succeq 0$. Then,

Figures (3)

  • Figure 1: ROA comparison between TOR-MPC (blue) and quartullo2024robust (red).
  • Figure 2: Relative position trajectories obtained by applying TOR-MPC (blue) and quartullo2024robust (red). The green set is the visibility region $\mathcal{X}$.
  • Figure 3: Relative velocity (a) and input acceleration (b) profiles obtained by applying TOR-MPC in all tested initial conditions.

Theorems & Definitions (14)

  • Definition 1
  • Proposition 1
  • Proposition 2
  • Definition 2
  • Proposition 3
  • Definition 3
  • Remark 1
  • Definition 4
  • Theorem 1
  • Theorem 2
  • ...and 4 more