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Adaptive Stochastic Predictive Control from Noisy Data: A Sampling-based Approach

Johannes Teutsch, Christopher Narr, Sebastian Kerz, Dirk Wollherr, Marion Leibold

TL;DR

This work presents an online adaptive stochastic predictive control framework for LTI systems with unknown parameters and bounded disturbances. It maps disturbance information to a distribution over consistent system parameters and uses sampling-based probabilistic scaling to deterministically approximate chance constraints, enabling tractable online MPC. The method guarantees recursive feasibility through a robust first-step constraint and achieves closed-loop constraint satisfaction with high confidence, while adaptively refining constraints as new data arrives. Numerical experiments on a DC-DC converter illustrate performance gains over a non-adaptive approach, with improved tracking and reduced cost, at a manageable computational cost. The approach offers a practical, data-driven pathway to robust, high-performance predictive control in the presence of parametric uncertainty and stochastic disturbances.

Abstract

In this work, an adaptive predictive control scheme for linear systems with unknown parameters and bounded additive disturbances is proposed. In contrast to related adaptive control approaches that robustly consider the parametric uncertainty, the proposed method handles all uncertainties stochastically by employing an online adaptive sampling-based approximation of chance constraints. The approach requires initial data in the form of a short input-output trajectory and distributional knowledge of the disturbances. This prior knowledge is used to construct an initial set of data-consistent system parameters and a distribution that allows for sample generation. As new data stream in online, the set of consistent system parameters is adapted by exploiting set membership identification. Consequently, chance constraints are deterministically approximated using a probabilistic scaling approach by sampling from the set of system parameters. In combination with a robust constraint on the first predicted step, recursive feasibility of the proposed predictive controller and closed-loop constraint satisfaction are guaranteed. A numerical example demonstrates the efficacy of the proposed method.

Adaptive Stochastic Predictive Control from Noisy Data: A Sampling-based Approach

TL;DR

This work presents an online adaptive stochastic predictive control framework for LTI systems with unknown parameters and bounded disturbances. It maps disturbance information to a distribution over consistent system parameters and uses sampling-based probabilistic scaling to deterministically approximate chance constraints, enabling tractable online MPC. The method guarantees recursive feasibility through a robust first-step constraint and achieves closed-loop constraint satisfaction with high confidence, while adaptively refining constraints as new data arrives. Numerical experiments on a DC-DC converter illustrate performance gains over a non-adaptive approach, with improved tracking and reduced cost, at a manageable computational cost. The approach offers a practical, data-driven pathway to robust, high-performance predictive control in the presence of parametric uncertainty and stochastic disturbances.

Abstract

In this work, an adaptive predictive control scheme for linear systems with unknown parameters and bounded additive disturbances is proposed. In contrast to related adaptive control approaches that robustly consider the parametric uncertainty, the proposed method handles all uncertainties stochastically by employing an online adaptive sampling-based approximation of chance constraints. The approach requires initial data in the form of a short input-output trajectory and distributional knowledge of the disturbances. This prior knowledge is used to construct an initial set of data-consistent system parameters and a distribution that allows for sample generation. As new data stream in online, the set of consistent system parameters is adapted by exploiting set membership identification. Consequently, chance constraints are deterministically approximated using a probabilistic scaling approach by sampling from the set of system parameters. In combination with a robust constraint on the first predicted step, recursive feasibility of the proposed predictive controller and closed-loop constraint satisfaction are guaranteed. A numerical example demonstrates the efficacy of the proposed method.
Paper Structure (16 sections, 3 theorems, 33 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 16 sections, 3 theorems, 33 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup and Preliminaries
  3. Problem Setup
  4. Probabilistic Scaling
  5. Adaptive Stochastic Predictive Control
  6. System Data and Adaptive Set of System Matrices
  7. Constraint Sampling
  8. Reformulation of the Cost Function
  9. Control Algorithm
  10. Control-theoretic Properties
  11. Discussion on Computational Aspects
  12. Numerical Evaluation
  13. Simulation Setup
  14. Simulation Results
  15. Conclusion
  16. ...and 1 more sections

Key Result

Proposition 1

For a given candidate SAS $\mathbb{Z}^{\mathrm{S}}(\sigma)$ with center $\bm{\zeta}_{\mathrm{c}} \in \mathbb{Z}^{\mathrm{P}}$, risk parameter $\varepsilon \in (0,1)$, and confidence $\beta \in (0,1)$, let the sample complexity $N_{\mathrm{S}}$ be chosen as $N_{\mathrm{S}} \ge N_{\mathrm{PS}}(\vareps Furthermore, for $N_{\mathrm{S}}$ iid uncertainty samples $\bm{w}^{(i)}$, $i \in \mathbb{N}_1^{N_{\

Figures (2)

  • Figure 1: Closed-loop output trajectories for reference tracking. The output constraints are depicted as black dashed lines and the output reference $_{k}$ is depicted in red. (NA: Non-adaptive scheme; A: Adaptive scheme)
  • Figure 2: Mean evolution of the scaling factors $\sigma^*_{k,l}$ for output $\varepsilon$-CSS approximation $\tilde{\mathbb{Y}}_{k,l}$ following Algorithm \ref{['alg:scaling']}.

Theorems & Definitions (7)

  • Definition 1: Scaling Factor mammarella2022chance
  • Proposition 1: Probabilistic Scaling of SAS mammarella2022chance
  • Remark 1
  • Theorem 1: Recursive Feasibility
  • proof
  • Corollary 1: Closed-loop Constraint Satisfaction
  • proof