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Iterative Learning Predictive Control for Constrained Uncertain Systems

Riccardo Zuliani, Efe C. Balta, Alisa Rupenyan, John Lygeros

TL;DR

The paper develops Iterative Learning Predictive Control (ILPC) for constrained nonlinear systems with both state-dependent uncertainties and bounded additive noise. By embedding a set-membership disturbance learning mechanism into a binary mixed-integer ILC upper layer and a convex receding-horizon MPC lower layer, the approach achieves robust constraint satisfaction while progressively improving nominal performance and converging toward the optimal trajectory under imperfect model knowledge. Key contributions include disturbance-set updates $\mathcal{D}_{k|n}(t)$, a shrinking-horizon MPC, a binary relaxation that yields a convex program, and rigorous guarantees of robust feasibility, non-degradation of performance, and probabilistic convergence to the optimal solution. The methodology is validated via simulation, showing significant gains in tracking accuracy and constraint compliance, with practical implications for repetitive manufacturing and other constrained, uncertain processes. Future work aims to reduce conservatism through advanced uncertainty representations (polytopes/zonotopes) and probabilistic formulations.

Abstract

Iterative learning control (ILC) improves the performance of a repetitive system by learning from previous trials. ILC can be combined with Model Predictive Control (MPC) to mitigate non-repetitive disturbances, thus improving overall system performance. However, existing approaches either assume perfect model knowledge or fail to actively learn system uncertainties, leading to conservativeness. To address these limitations we propose a binary mixed-integer ILC scheme, combined with a convex MPC scheme, that ensures robust constraint satisfaction, non-increasing nominal cost, and convergence to optimal performance. Our scheme is designed for uncertain nonlinear systems subject to both bounded additive stochastic noise and additive uncertain components. We showcase the benefits of our scheme in simulation.

Iterative Learning Predictive Control for Constrained Uncertain Systems

TL;DR

The paper develops Iterative Learning Predictive Control (ILPC) for constrained nonlinear systems with both state-dependent uncertainties and bounded additive noise. By embedding a set-membership disturbance learning mechanism into a binary mixed-integer ILC upper layer and a convex receding-horizon MPC lower layer, the approach achieves robust constraint satisfaction while progressively improving nominal performance and converging toward the optimal trajectory under imperfect model knowledge. Key contributions include disturbance-set updates , a shrinking-horizon MPC, a binary relaxation that yields a convex program, and rigorous guarantees of robust feasibility, non-degradation of performance, and probabilistic convergence to the optimal solution. The methodology is validated via simulation, showing significant gains in tracking accuracy and constraint compliance, with practical implications for repetitive manufacturing and other constrained, uncertain processes. Future work aims to reduce conservatism through advanced uncertainty representations (polytopes/zonotopes) and probabilistic formulations.

Abstract

Iterative learning control (ILC) improves the performance of a repetitive system by learning from previous trials. ILC can be combined with Model Predictive Control (MPC) to mitigate non-repetitive disturbances, thus improving overall system performance. However, existing approaches either assume perfect model knowledge or fail to actively learn system uncertainties, leading to conservativeness. To address these limitations we propose a binary mixed-integer ILC scheme, combined with a convex MPC scheme, that ensures robust constraint satisfaction, non-increasing nominal cost, and convergence to optimal performance. Our scheme is designed for uncertain nonlinear systems subject to both bounded additive stochastic noise and additive uncertain components. We showcase the benefits of our scheme in simulation.

Paper Structure

This paper contains 18 sections, 16 theorems, 118 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Problem reformulation
  4. Approximating the disturbance
  5. Choosing the reference
  6. Computing set estimates
  7. State Error evolution
  8. Robust constraint satisfaction
  9. Iterative learning predictive control
  10. Analysis
  11. Robust constraint satisfaction
  12. Recursive feasibility
  13. Non-degrading performance
  14. Convergence to optimal performance
  15. Binary constraint relaxation
  16. ...and 3 more sections

Key Result

Lemma 1

Under ass:PF_disturbanceass:PF_noise, we have $d_k(t) \in \mathcal{D}_{k|n}(t)$ for all $k,n\in\mathbb{N}$, $k \leq n$, and $t\in\mathbb{Z}_{[0,T-1]}{}$, where Moreover

Figures (6)

  • Figure 1: A depiction of the operation in \ref{['eq:PREF_set_estimates']} to obtain the smallest set $\mathcal{D}_{1|0}(t)$ containing $d_1(t)$ (in red). The intersection is between two sets $\bar{\mathcal{D}}_1(t)$ (in yellow), and $\bar{\mathcal{D}}_0(t)$ (dashed blue line) with the addition of $\mathcal{B}(m(t)\|x_0(t)-x_1(t)\|)$ to account for the variation of the state.
  • Figure 2: Reference input signal $\bar{u}(t)$.
  • Figure 3: Reference signal $r(t)$.
  • Figure 4: State-independent disturbance $d(t)$.
  • Figure 5: Closed-loop state trajectories.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Proposition 1: Robust constraint satisfaction
  • proof
  • ...and 25 more