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Input-Output Data-Driven Representation: Non-Minimality and Stability

Joowon Lee, Nam Hoon Jo, Hyungbo Shim, Florian Dörfler, Jinsung Kim

TL;DR

The paper analyzes a widely used data-driven representation for LTI systems built from finite-horizon input-output trajectories, revealing that the recursion of this non-minimal predictor introduces latent poles. It proves that, when the data are processed via Moore–Penrose inverses of the data matrices, all latent poles are Schur stable regardless of the underlying system's stability, and even in the presence of small noise. Leveraging this result, the authors construct a stabilizable and detectable non-minimal realization from data and design an output-feedback LQR controller that achieves model-based optimality under appropriate initialization. The study extends to data-driven inversion, showing that asymptotic unknown-input estimation is possible for minimum-phase systems, enabling robust data-driven disturbance-observer functionality (DD-DOB). The theoretical results are supported by numerical examples for SISO and MIMO systems, demonstrating improved stability and performance with longer data horizons, and providing practical guidance for data-driven control and estimation tasks.

Abstract

Many recent data-driven control approaches for linear time-invariant systems are based on finite-horizon prediction of output trajectories using input-output data matrices. When applied recursively, this predictor forms a dynamic system representation. This data-driven representation is generally non-minimal, containing latent poles in addition to the system's original poles. In this article, we show that these latent poles are guaranteed to be stable through the use of the Moore-Penrose inverses of the data matrices, regardless of the system's stability and even in the presence of small noise in data. This result obviates the need to eliminate the latent poles through procedures that resort to low-rank approximation in data-driven control and analysis. It is then applied to construct a stabilizable and detectable realization from data, from which we design an output feedback linear quadratic regulator (LQR) controller. Furthermore, we extend this principle to data-driven inversion, enabling asymptotic unknown input estimation for minimum-phase systems.

Input-Output Data-Driven Representation: Non-Minimality and Stability

TL;DR

The paper analyzes a widely used data-driven representation for LTI systems built from finite-horizon input-output trajectories, revealing that the recursion of this non-minimal predictor introduces latent poles. It proves that, when the data are processed via Moore–Penrose inverses of the data matrices, all latent poles are Schur stable regardless of the underlying system's stability, and even in the presence of small noise. Leveraging this result, the authors construct a stabilizable and detectable non-minimal realization from data and design an output-feedback LQR controller that achieves model-based optimality under appropriate initialization. The study extends to data-driven inversion, showing that asymptotic unknown-input estimation is possible for minimum-phase systems, enabling robust data-driven disturbance-observer functionality (DD-DOB). The theoretical results are supported by numerical examples for SISO and MIMO systems, demonstrating improved stability and performance with longer data horizons, and providing practical guidance for data-driven control and estimation tasks.

Abstract

Many recent data-driven control approaches for linear time-invariant systems are based on finite-horizon prediction of output trajectories using input-output data matrices. When applied recursively, this predictor forms a dynamic system representation. This data-driven representation is generally non-minimal, containing latent poles in addition to the system's original poles. In this article, we show that these latent poles are guaranteed to be stable through the use of the Moore-Penrose inverses of the data matrices, regardless of the system's stability and even in the presence of small noise in data. This result obviates the need to eliminate the latent poles through procedures that resort to low-rank approximation in data-driven control and analysis. It is then applied to construct a stabilizable and detectable realization from data, from which we design an output feedback linear quadratic regulator (LQR) controller. Furthermore, we extend this principle to data-driven inversion, enabling asymptotic unknown input estimation for minimum-phase systems.

Paper Structure

This paper contains 24 sections, 16 theorems, 101 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notations
  3. Problem Formulation and Preliminaries
  4. Problem Formulation
  5. Preliminaries
  6. Data-Driven Representation with Stable Latent Poles
  7. Single-Output Case
  8. General Result
  9. Effect of Noisy Data
  10. Numerical Examples
  11. Stable System
  12. Unstable System
  13. Application to Data-Driven Output Feedback Control
  14. Non-Minimal State-Space Realization
  15. Design of Output Feedback LQR Controller
  16. ...and 9 more sections

Key Result

Proposition 1

For $N\geq l$, $\mathop{\mathrm{image}}\nolimits =\mathcal{W}_{N+M}$ if and only if eq:rankcond holds. Under this condition, given $u(t-N:t+M-1)$ and $y(t-N:t-1)$, $y(t:t+M-1)$ is uniquely determined by eq:intro b for any $g(t)\in {\mathbb R}^T$ satisfying eq:intro a.

Figures (8)

  • Figure 1: The output of the system (the black solid line) and $y(t)$ of \ref{['eq:intro h']} when $h^\top=Y_\mathrm{f}\mathrm{H}^\dagger$ (the blue dashed line) and $h^\top=Y_\mathrm{f}\mathrm{H}^\mathrm{G}$ with some randomly selected $\mathrm{H}^\mathrm{G}$ (the red dotted line). More details on the simulation can be found in Section \ref{['subsec:pole ex']}.
  • Figure 2: Pole-zero plots of the system \ref{['eq:invpen']} (the orange marks) and its data-driven representation \ref{['eq:DDR MP']} from noise-free and noisy data (the blue marks). The black circle represents the unit circle.
  • Figure 3: Average SNR (defined by \ref{['eq:SNR']}) across the $10$ input-output trajectories by varying $N$.
  • Figure 4: Output of the system \ref{['eq:invpen']} with the controller \ref{['eq:LQRctr']} under online output measurement noise.
  • Figure 5: $\mathcal{H}_2$ norm of the closed-loop system \ref{['eq:H2norm']} with respect to the system \ref{['eq:invpen']} (the blue circle-marked line) and \ref{['eq:submarine']} (the black square-marked line) by varying $N$.
  • ...and 3 more figures

Theorems & Definitions (43)

  • Remark 1
  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 2
  • ...and 33 more