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On the sample complexity of stabilizing linear dynamical systems from data

Steffen W. R. Werner, Benjamin Peherstorfer

TL;DR

The paper addresses data-driven stabilization of linear time-invariant systems by showing that stabilization requires data corresponding to the system's intrinsic dimension $n$ (its McMillan degree) rather than the ambient observation dimension $N$. It develops a framework to infer stabilizing controllers directly from high-dimensional data by projecting onto a low-dimensional subspace and lifting the reduced controller via a left inverse $V^{\dagger}$, with the closed-loop spectrum determined by the reduced controller $\hat{K}$. The key contributions include a rigorous lifting result, an analysis for approximately low-dimensional systems, and a practical computational procedure (including re-projection) that achieves stabilization with far fewer samples than model learning would require. Numerical examples across synthetic and fluid-flow settings demonstrate substantial data-efficiency gains and improved stability relative to traditional two-step identification-based approaches.

Abstract

Learning controllers from data for stabilizing dynamical systems typically follows a two step process of first identifying a model and then constructing a controller based on the identified model. However, learning models means identifying generic descriptions of the dynamics of systems, which can require large amounts of data and extracting information that are unnecessary for the specific task of stabilization. The contribution of this work is to show that if a linear dynamical system has dimension (McMillan degree) $n$, then there always exist $n$ states from which a stabilizing feedback controller can be constructed, independent of the dimension of the representation of the observed states and the number of inputs. By building on previous work, this finding implies that any linear dynamical system can be stabilized from fewer observed states than the minimal number of states required for learning a model of the dynamics. The theoretical findings are demonstrated with numerical experiments that show the stabilization of the flow behind a cylinder from less data than necessary for learning a model.

On the sample complexity of stabilizing linear dynamical systems from data

TL;DR

The paper addresses data-driven stabilization of linear time-invariant systems by showing that stabilization requires data corresponding to the system's intrinsic dimension (its McMillan degree) rather than the ambient observation dimension . It develops a framework to infer stabilizing controllers directly from high-dimensional data by projecting onto a low-dimensional subspace and lifting the reduced controller via a left inverse , with the closed-loop spectrum determined by the reduced controller . The key contributions include a rigorous lifting result, an analysis for approximately low-dimensional systems, and a practical computational procedure (including re-projection) that achieves stabilization with far fewer samples than model learning would require. Numerical examples across synthetic and fluid-flow settings demonstrate substantial data-efficiency gains and improved stability relative to traditional two-step identification-based approaches.

Abstract

Learning controllers from data for stabilizing dynamical systems typically follows a two step process of first identifying a model and then constructing a controller based on the identified model. However, learning models means identifying generic descriptions of the dynamics of systems, which can require large amounts of data and extracting information that are unnecessary for the specific task of stabilization. The contribution of this work is to show that if a linear dynamical system has dimension (McMillan degree) , then there always exist states from which a stabilizing feedback controller can be constructed, independent of the dimension of the representation of the observed states and the number of inputs. By building on previous work, this finding implies that any linear dynamical system can be stabilized from fewer observed states than the minimal number of states required for learning a model of the dynamics. The theoretical findings are demonstrated with numerical experiments that show the stabilization of the flow behind a cylinder from less data than necessary for learning a model.
Paper Structure (20 sections, 9 theorems, 52 equations, 4 figures, 2 algorithms)

This paper contains 20 sections, 9 theorems, 52 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sampling data from dynamical systems
  4. Control via system identification
  5. Inferring controllers without system identification
  6. Inferring low-dimensional controllers from high-dimensional states
  7. Controller inference for stabilizing intrinsically low-dimensional systems
  8. Lifting controllers
  9. Inferring low-dimensional controllers
  10. Controller inference for stabilizing approximately low-dimensional systems
  11. Amplification of state approximation errors
  12. Perturbations of the closed-loop spectrum
  13. Computational procedure for controller inference
  14. Controller inference
  15. Data collection via re-projection
  16. ...and 5 more sections

Key Result

Proposition 1

Let $(U_{-}, X_{-}, X_{+})$ be a data triplet. The underlying state-space model eqn:dtsys (or eqn:ctsys) can be uniquely identified from the data triplet as if and only if where $$ is a right inverse in the sense of

Figures (4)

  • Figure 1: Number of data samples: In all experiments, the number of samples for control inference is lower than the number of samples required for learning a minimal model of the system. Furthermore, the number of samples is orders of magnitude lower than what would be required for learning traditional non-minimal models of the same dimension as the observed states. This is in agreement with \ref{['thm:lowdatainform']} and the discussion in \ref{['subsec:apprdata']}.
  • Figure 2: Synthetic example: The proposed inference approach leads to stabilizing controllers with data sets of only four samples in this example. In contrast, the classical two-step control procedure of first identifying a model and then constructing a controller leads to unstable dynamics in this example because of too few data samples in the data set, which is in agreement with \ref{['prp:sysident', 'thm:lowdatainform']}.
  • Figure 3: Heat flow: The controllers constructed via the inference approach stabilize the system. In contrast, applying the traditional two-step approach of first identifying a model and then controlling to the same data set only manages to decrease the growth of outputs but fails to stabilize the system in the discrete-time case. In case of the continuous-time system, the controller obtained from the identified model even accelerates the growth of the outputs and thus further destabilizes the system.
  • Figure 4: Flow behind cylinder: In each sub figure, the top plot shows the magnitude of the high-dimensional state at final time and the bottom plot shows the averaged velocity in horizontal direction at four probes. Controller inference is able to stabilize the system, whereas traditional data-driven control via system identification fails when applied to the same data set.

Theorems & Definitions (15)

  • Proposition 1: Identification of state-space model VanD96
  • Proposition 2: Data informativity in discrete time VanETetal20
  • Corollary 1: Data informativity in continuous time
  • proof
  • Lemma 1: Low-dimensional subspaces and state-space dimensions
  • proof
  • Theorem 1: Lifting controllers
  • proof : Proof of \ref{['thm:lowdimfeed']}.
  • Corollary 2: Spectrum of closed-loop matrices
  • proof : Proof.
  • ...and 5 more