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Interpolation Conditions for Data Consistency and Prediction in Noisy Linear Systems

Martina Vanelli, Nima Monshizadeh, Julien M. Hendrickx

TL;DR

This paper presents an interpolation-based data-driven framework for noisy discrete-time LTI systems with known input matrix $B$ and unknown state matrix $A$, under a bound on energy growth $L$ and bounded process-noise energy. By deriving necessary and sufficient interpolation conditions via PSD completions, the authors obtain SDP formulations for data-consistency verification and for inferring minimal noise or energy amplification compatible with observed data. They show that the one-step-ahead feasible state set $\mathcal{X}_+(u)$ is an ellipsoid in the noise-free case and a union of ellipsoids when noise is present, enabling safety verification and worst-case energy minimization. The approach provides a principled path toward data-driven control without explicit system identification and suggests extensions to multi-step predictions and nonlinear dynamics.

Abstract

We develop an interpolation-based framework for noisy linear systems with unknown system matrix with bounded norm (implying bounded growth or non-increasing energy), and bounded process noise energy. The proposed approach characterizes all trajectories consistent with the measured data and these prior bounds in a purely data-driven manner. This characterization enables data-consistency verification, inference, and one-step ahead prediction, which can be leveraged for safety verification and cost minimization. Ultimately, this work represents a preliminary step toward exploiting interpolation conditions in data-driven control, offering a systematic way to characterize trajectories consistent with a dynamical system within a given class and enabling their use in control design.

Interpolation Conditions for Data Consistency and Prediction in Noisy Linear Systems

TL;DR

This paper presents an interpolation-based data-driven framework for noisy discrete-time LTI systems with known input matrix and unknown state matrix , under a bound on energy growth and bounded process-noise energy. By deriving necessary and sufficient interpolation conditions via PSD completions, the authors obtain SDP formulations for data-consistency verification and for inferring minimal noise or energy amplification compatible with observed data. They show that the one-step-ahead feasible state set is an ellipsoid in the noise-free case and a union of ellipsoids when noise is present, enabling safety verification and worst-case energy minimization. The approach provides a principled path toward data-driven control without explicit system identification and suggests extensions to multi-step predictions and nonlinear dynamics.

Abstract

We develop an interpolation-based framework for noisy linear systems with unknown system matrix with bounded norm (implying bounded growth or non-increasing energy), and bounded process noise energy. The proposed approach characterizes all trajectories consistent with the measured data and these prior bounds in a purely data-driven manner. This characterization enables data-consistency verification, inference, and one-step ahead prediction, which can be leveraged for safety verification and cost minimization. Ultimately, this work represents a preliminary step toward exploiting interpolation conditions in data-driven control, offering a systematic way to characterize trajectories consistent with a dynamical system within a given class and enabling their use in control design.

Paper Structure

This paper contains 17 sections, 8 theorems, 63 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Data consistency and inference
  3. One-step-ahead prediction, safety, and energy minimization
  4. Notation
  5. Problem setting
  6. Prior knowledge on system dynamics
  7. Prior knowledge on the noise
  8. Objectives
  9. Data consistency and inference
  10. One-step ahead prediction
  11. Interpolation conditions
  12. Data-consistency and prediction in LTI systems
  13. Data-consistency and inference
  14. One-step ahead prediction
  15. Ellipsoidal feasible sets
  16. ...and 2 more sections

Key Result

Proposition 1

Given the data$Z$ in $\mathbb{R}^{m\times t}$ and $Y$ in $\mathbb{R}^{n\times t}$ and the bound$G\in \mathbb{R}^{m \times m}$, $G\succeq 0$, there exists $M \in \mathbb{R}^{n\times m}$ such that if and only if

Figures (3)

  • Figure 1: Collected data-points.
  • Figure 2: Representation of Example \ref{['ex:ell_no_noise']}
  • Figure 3: Representation of Example \ref{['ex:un_ell']}

Theorems & Definitions (19)

  • Remark 1
  • Proposition 1: bisoffi2024controller
  • Proposition 2
  • Definition 1
  • Proposition 3: Multi-bound extension
  • proof
  • Corollary 1: Data-consistency
  • Example 1
  • Theorem 1: One-step ahead prediction
  • proof
  • ...and 9 more