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Learning-based Rigid Tube Model Predictive Control

Yulong Gao, Shuhao Yan, Jian Zhou, Mark Cannon, Alessandro Abate, Karl H. Johansson

TL;DR

This work tackles robust MPC for discrete-time linear systems with mixed state-input constraints when the true disturbance set ${\mathbb{W}}_{true}$ is unknown. It introduces online learning to construct a quantified disturbance set ${\hat{\mathbb{W}}^{\star}_k}$ from a known conservative set ${\mathbb{W}}$ and disturbance realizations, enabling a LP-based rigid-tube MPC that updates at every step. Key contributions include a LP-equivalent reformulation using the quantified disturbance set, probabilistic risk bounds between ${\mathbb{W}}_{true}$ and ${\hat{\mathbb{W}}^{\star}_k}$, and an online tube update mechanism that avoids recomputing Minkowski sums, with numerical results showing enlarged feasible regions and reduced computation compared to robust and scenario MPC. The approach advances data-driven uncertainty refinement in safety-critical control by balancing robustness with reduced conservatism and computational load.

Abstract

This paper is concerned with model predictive control (MPC) of discrete-time linear systems subject to bounded additive disturbance and mixed constraints on the state and input, whereas the true disturbance set is unknown. Unlike most existing work on robust MPC, we propose an algorithm incorporating online learning that builds on prior knowledge of the disturbance, i.e., a known but conservative disturbance set. We approximate the true disturbance set at each time step with a parameterised set, which is referred to as a quantified disturbance set, using disturbance realisations. A key novelty is that the parameterisation of these quantified disturbance sets enjoys desirable properties such that the quantified disturbance set and its corresponding rigid tube bounding disturbance propagation can be efficiently updated online. We provide statistical gaps between the true and quantified disturbance sets, based on which, probabilistic recursive feasibility of MPC optimisation problems is discussed. Numerical simulations are provided to demonstrate the effectiveness of our proposed algorithm and compare with conventional robust MPC algorithms.

Learning-based Rigid Tube Model Predictive Control

TL;DR

This work tackles robust MPC for discrete-time linear systems with mixed state-input constraints when the true disturbance set is unknown. It introduces online learning to construct a quantified disturbance set from a known conservative set and disturbance realizations, enabling a LP-based rigid-tube MPC that updates at every step. Key contributions include a LP-equivalent reformulation using the quantified disturbance set, probabilistic risk bounds between and , and an online tube update mechanism that avoids recomputing Minkowski sums, with numerical results showing enlarged feasible regions and reduced computation compared to robust and scenario MPC. The approach advances data-driven uncertainty refinement in safety-critical control by balancing robustness with reduced conservatism and computational load.

Abstract

This paper is concerned with model predictive control (MPC) of discrete-time linear systems subject to bounded additive disturbance and mixed constraints on the state and input, whereas the true disturbance set is unknown. Unlike most existing work on robust MPC, we propose an algorithm incorporating online learning that builds on prior knowledge of the disturbance, i.e., a known but conservative disturbance set. We approximate the true disturbance set at each time step with a parameterised set, which is referred to as a quantified disturbance set, using disturbance realisations. A key novelty is that the parameterisation of these quantified disturbance sets enjoys desirable properties such that the quantified disturbance set and its corresponding rigid tube bounding disturbance propagation can be efficiently updated online. We provide statistical gaps between the true and quantified disturbance sets, based on which, probabilistic recursive feasibility of MPC optimisation problems is discussed. Numerical simulations are provided to demonstrate the effectiveness of our proposed algorithm and compare with conventional robust MPC algorithms.
Paper Structure (14 sections, 7 theorems, 22 equations, 3 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 7 theorems, 22 equations, 3 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Description
  3. Rigid Tube MPC based on Conservative Disturbance Set $\mathbb{W}$
  4. Problem under Study
  5. Learning Uncertainty Set
  6. Online Update of $\hat{\mathbb{W}}^{\star}_{k}$
  7. Learning-based Rigid Tube MPC
  8. Computing $\nu_k$
  9. Recursive Feasibility
  10. Case Study
  11. Comparison of the Feasible Regions
  12. Online Evaluation of UQ-RMPC
  13. Comparison with Scenario MPC
  14. Conclusion

Key Result

lemma 1

If Assumption Ass holds and $\Phi$ is strictly stable, there exist a finite integer $r$ and a scalar $\rho\in [0,1)$ such that (i) $\Phi^r\mathbb{W}\subseteq \rho \mathbb{W}$ and (ii) the set is a convex and compact set, satisfying $\Phi \mathbb{S} \oplus \mathbb{W}\subseteq \mathbb{S}$ and $\bigoplus_{i=0}^{\infty}\Phi^i\mathbb{W}\subseteq\mathbb{S}$.

Figures (3)

  • Figure 1: The framework.
  • Figure 2: Comparison of disturbance sets $\mathbb{W}$ [], $\mathbb{W}_{\rm true}$ [], $\hat{\mathbb{W}}_{\rm opt}$ [], and $\hat{\mathbb{W}}_0^{\star}$ [] and their corresponding feasible regions $\mathcal{F}_{\rm MPC}$ [], $\mathcal{F}_{\rm true}$ [], $\hat{\mathcal{F}}_{\rm opt}$ [], and $\hat{\mathcal{F}}_0$ []. (a) The disturbance sets with $|\mathcal{I}_0^w| = 50$; (b) The feasible regions with $|\mathcal{I}_0^w| = 50$; (c) The disturbance sets with $|\mathcal{I}_0^w| = 20000$; (d) The feasible regions with $|\mathcal{I}_0^w| = 20000$.
  • Figure 3: (a) Volume of $\hat{\mathbb{W}}_0^{\star}$ for MC simulations for different $\mathcal{I}^w_0$ (std. means standard deviation); (b) State trajectory by UQ-RMPC with $|\mathcal{I}_0^w|=100$; (c) State trajectory by UQ-RMPC with $|\mathcal{I}_0^w|=20000$; (d)Volume of $\hat{\mathbb{W}}_k^{\star}$ for MC simulations with $\mathcal{I}^w_k$.

Theorems & Definitions (13)

  • lemma 1: rakovic05
  • proposition 1
  • proof
  • theorem 1
  • proof
  • proposition 2
  • proof
  • proposition 3
  • proof
  • proposition 4
  • ...and 3 more