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Data-Driven Superstabilizing Control under Quadratically-Bounded Errors-in-Variables Noise

Jared Miller, Tianyu Dai, Mario Sznaier

TL;DR

The paper addresses stabilizing all discrete-time LTI plants consistent with data under quadratically bounded EIV noise by formulating an infinite-dimensional LP that enforces $W$-superstability across the data-consistent plant set. It leverages a theorem of alternatives to eliminate noise variables and utilizes the moment-SOS hierarchy to obtain tractable finite relaxations, with full, dense, and sparse truncation schemes. An extended superstability variant introduces joint optimization over a diagonal $W$ to enlarge the stabilizable set, and numerical experiments illustrate the method's potential and computational limits under different truncation orders. The work advances robust, data-driven superstabilization in EIV settings by integrating set-membership ideas, SOS certificates, and rigorous complexity analyses. Practical impact lies in providing a principled, certifiable approach to stabilize systems from imperfect data where both state and input are contaminated, using scalable SOS-based relaxations and extensions to accommodate changing Lyapunov metrics.

Abstract

The Error-in-Variables model of system identification/control involves nontrivial input and measurement corruption of observed data, resulting in generically nonconvex optimization problems. This paper performs full-state-feedback stabilizing control of all discrete-time linear systems that are consistent with observed data for which the input and measurement noise obey quadratic bounds. Instances of such quadratic bounds include elementwise norm bounds (at each time sample), energy bounds (across the entire signal), and chance constraints arising from (sub)gaussian noise. Superstabilizing controllers are generated through the solution of a sum-of-squares hierarchy of semidefinite programs. A theorem of alternatives is employed to eliminate the input and measurement noise process, thus improving tractability.

Data-Driven Superstabilizing Control under Quadratically-Bounded Errors-in-Variables Noise

TL;DR

The paper addresses stabilizing all discrete-time LTI plants consistent with data under quadratically bounded EIV noise by formulating an infinite-dimensional LP that enforces -superstability across the data-consistent plant set. It leverages a theorem of alternatives to eliminate noise variables and utilizes the moment-SOS hierarchy to obtain tractable finite relaxations, with full, dense, and sparse truncation schemes. An extended superstability variant introduces joint optimization over a diagonal to enlarge the stabilizable set, and numerical experiments illustrate the method's potential and computational limits under different truncation orders. The work advances robust, data-driven superstabilization in EIV settings by integrating set-membership ideas, SOS certificates, and rigorous complexity analyses. Practical impact lies in providing a principled, certifiable approach to stabilize systems from imperfect data where both state and input are contaminated, using scalable SOS-based relaxations and extensions to accommodate changing Lyapunov metrics.

Abstract

The Error-in-Variables model of system identification/control involves nontrivial input and measurement corruption of observed data, resulting in generically nonconvex optimization problems. This paper performs full-state-feedback stabilizing control of all discrete-time linear systems that are consistent with observed data for which the input and measurement noise obey quadratic bounds. Instances of such quadratic bounds include elementwise norm bounds (at each time sample), energy bounds (across the entire signal), and chance constraints arising from (sub)gaussian noise. Superstabilizing controllers are generated through the solution of a sum-of-squares hierarchy of semidefinite programs. A theorem of alternatives is employed to eliminate the input and measurement noise process, thus improving tractability.
Paper Structure (22 sections, 3 theorems, 37 equations, 1 table)

This paper contains 22 sections, 3 theorems, 37 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Quadratic Noise Bounds
  5. Elementwise Norm Constraints
  6. Energy Constraints
  7. Superstabilizing Control
  8. Sum-of-Squares Matrices Background
  9. Quadratically-Bounded Linear Programs
  10. Consistency Set
  11. Full Linear Program
  12. Robustified Linear Program
  13. Truncated Sum-of-Squares Programs
  14. Quadratically-Bounded Truncations
  15. Full Program
  16. ...and 7 more sections

Key Result

Theorem III.1

$W$-Superstability through a common $K$ (for Problem prob:ess) can be ensured by solving the following infinite-dimensional LP in terms of a matrix $K \in \mathbb{R}^{m \times n}$ and a matrix-valued function $M(A, B): \mathcal{P} \rightarrow \mathbb{R}^{f \times f}$ (Equation (12) of miller2022eivl

Theorems & Definitions (9)

  • Theorem III.1
  • proof
  • Remark 1
  • Theorem III.2
  • proof
  • Theorem III.3
  • proof
  • Remark 2
  • Remark 3