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Sampling-based Stochastic Data-driven Predictive Control under Data Uncertainty - Extended Version

Johannes Teutsch, Sebastian Kerz, Dirk Wollherr, Marion Leibold

TL;DR

The paper tackles data-driven predictive control for LTI systems with process disturbances and probabilistic output constraints by developing a sampling-based approach that does not require disturbance measurements. It extends Willems' fundamental lemma to construct data-driven predictors and introduces a disturbance-consistency parameterization that yields a set of feasible disturbance trajectories for sampling. By offline sampling and an ε-CCS reformulation, the method delivers a tractable OCP with recursive feasibility guaranteed via a robust first-step constraint and stability in expectation (RASiE) under probabilistic cost bounds. The approach demonstrates improved prediction accuracy and closed-loop performance when using disturbances sampled from the set of consistent trajectories, with practical benefits for systems where disturbance data are unavailable. The extended version provides additional material on verifiability, prior-model integration, and design procedures for stabilizing ingredients and terminal sets, supported by a numerical DC-DC converter example.

Abstract

We present a stochastic constrained output-feedback data-driven predictive control scheme for linear time-invariant systems subject to bounded additive disturbances. The approach uses data-driven predictors based on an extension of Willems' fundamental lemma and requires only a single persistently exciting input-output data trajectory. Compared to current state-of-the-art approaches, we do not rely on availability of exact disturbance data. Instead, we leverage a novel parameterization of the unknown disturbance data considering consistency with the measured data and the system class. This allows for deterministic approximation of the chance constraints in a sampling-based fashion. A robust constraint on the first predicted step enables recursive feasibility, closed-loop constraint satisfaction, and robust asymptotic stability in expectation under standard assumptions. A numerical example demonstrates the efficiency of the proposed control scheme.

Sampling-based Stochastic Data-driven Predictive Control under Data Uncertainty - Extended Version

TL;DR

The paper tackles data-driven predictive control for LTI systems with process disturbances and probabilistic output constraints by developing a sampling-based approach that does not require disturbance measurements. It extends Willems' fundamental lemma to construct data-driven predictors and introduces a disturbance-consistency parameterization that yields a set of feasible disturbance trajectories for sampling. By offline sampling and an ε-CCS reformulation, the method delivers a tractable OCP with recursive feasibility guaranteed via a robust first-step constraint and stability in expectation (RASiE) under probabilistic cost bounds. The approach demonstrates improved prediction accuracy and closed-loop performance when using disturbances sampled from the set of consistent trajectories, with practical benefits for systems where disturbance data are unavailable. The extended version provides additional material on verifiability, prior-model integration, and design procedures for stabilizing ingredients and terminal sets, supported by a numerical DC-DC converter example.

Abstract

We present a stochastic constrained output-feedback data-driven predictive control scheme for linear time-invariant systems subject to bounded additive disturbances. The approach uses data-driven predictors based on an extension of Willems' fundamental lemma and requires only a single persistently exciting input-output data trajectory. Compared to current state-of-the-art approaches, we do not rely on availability of exact disturbance data. Instead, we leverage a novel parameterization of the unknown disturbance data considering consistency with the measured data and the system class. This allows for deterministic approximation of the chance constraints in a sampling-based fashion. A robust constraint on the first predicted step enables recursive feasibility, closed-loop constraint satisfaction, and robust asymptotic stability in expectation under standard assumptions. A numerical example demonstrates the efficiency of the proposed control scheme.
Paper Structure (38 sections, 10 theorems, 77 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 38 sections, 10 theorems, 77 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup & Preliminaries
  3. Problem Setup & Preliminaries
  4. Problem Setup
  5. Data-driven System Representation
  6. Consistency of Disturbance Data
  7. Consistent Disturbance Data
  8. Set of Consistent Disturbance Data
  9. Sampling-based Stochastic DPC
  10. Data-driven Sample-based Predictions
  11. Constraint Sampling & Reformulation of Cost Function
  12. Control Algorithm
  13. Closed-loop Properties
  14. Recursive Feasibility
  15. Robust Asymptotic Stability in Expectation
  16. ...and 23 more sections

Key Result

Lemma 1

Consider a controllable LTI system $\Sigma$ of the form eq:system_minimal and measured data trajectories $\glsd{ud}_T$, $\glsd{dd}_T$, $\glsd{yd}_T$, and $\glsd{xid}_{T}$ where $T \ge +$, $\in \mathbb{N}$, and $\ge \mathrm{lag}\left(\Sigma\right)$.$\mathrm{lag}\left(\Sigma\right)$ of an LTI system with $_k$ and $\glsd{xid}_{T-+1} = \{_i\}_{i=1}^{T-+1}$ according to eq:extstate.

Figures (3)

  • Figure 1: Comparison of (sampled) open-loop output predictions.
  • Figure 2: Trajectories of 50 exemplary runs under the proposed controller subject to disturbances. Constraints are shown in dotted black lines.
  • Figure 3: Visualization of the consistency constraint on the disturbance data. The original disturbance bound is shown in black, while the true (unknown) disturbance data are given in blue. Bounds resulting from the set of consistent disturbance data \ref{['eq:distset_consistency']} are depicted in red. The bounds depicted in green additionally include prior model knowledge.

Theorems & Definitions (29)

  • Definition 1: Persistency of excitation willems2005note
  • Remark 1
  • Lemma 1: Extended fundamental lemma
  • proof
  • Proposition 1: Consistency constraint
  • Remark 2
  • Definition 2: Consistent disturbance data trajectories
  • Proposition 2: Consistency parameterization
  • proof
  • Remark 3
  • ...and 19 more