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End-to-end guarantees for indirect data-driven control of bilinear systems with finite stochastic data

Nicolas Chatzikiriakos, Robin Strässer, Frank Allgöwer, Andrea Iannelli

TL;DR

The paper addresses end-to-end guarantees for indirect data-driven control of bilinear systems under finite, possibly unbounded, stochastic data. It develops both a priori and data-dependent finite-sample bounds for identifying the bilinear model from independent sub-Gaussian data, then couples these bounds with robust control designs—via LMI and SOS approaches—to achieve exponential stability of the closed loop with high probability. A key contribution is expressing the identification error as a quadratic residual bound that scales with state and input, enabling tractable controller synthesis. The work also draws connections to Koopman operator theory to extend the results to nonlinear systems and validates the framework through numerical examples, including a nonlinear inverted pendulum lifted to a bilinear surrogate. This end-to-end methodology provides practical, provable guarantees for data-driven control in settings with noisy, finite data and offers a pathway to broader nonlinear applications through lifted representations.

Abstract

In this paper we propose an end-to-end algorithm for indirect data-driven control for bilinear systems with stability guarantees. We consider the case where the collected i.i.d. data is affected by probabilistic noise with possibly unbounded support and leverage tools from statistical learning theory to derive finite sample identification error bounds. To this end, we solve the bilinear identification problem by solving a set of linear and affine identification problems, by a particular choice of a control input during the data collection phase. We provide a priori as well as data-dependent finite sample identification error bounds on the individual matrices as well as ellipsoidal bounds, both of which are structurally suitable for control. Further, we integrate the structure of the derived identification error bounds in a robust controller design to obtain an exponentially stable closed-loop. By means of an extensive numerical study we showcase the interplay between the controller design and the derived identification error bounds. Moreover, we note appealing connections of our results to indirect data-driven control of general nonlinear systems through Koopman operator theory and discuss how our results may be applied in this setup.

End-to-end guarantees for indirect data-driven control of bilinear systems with finite stochastic data

TL;DR

The paper addresses end-to-end guarantees for indirect data-driven control of bilinear systems under finite, possibly unbounded, stochastic data. It develops both a priori and data-dependent finite-sample bounds for identifying the bilinear model from independent sub-Gaussian data, then couples these bounds with robust control designs—via LMI and SOS approaches—to achieve exponential stability of the closed loop with high probability. A key contribution is expressing the identification error as a quadratic residual bound that scales with state and input, enabling tractable controller synthesis. The work also draws connections to Koopman operator theory to extend the results to nonlinear systems and validates the framework through numerical examples, including a nonlinear inverted pendulum lifted to a bilinear surrogate. This end-to-end methodology provides practical, provable guarantees for data-driven control in settings with noisy, finite data and offers a pathway to broader nonlinear applications through lifted representations.

Abstract

In this paper we propose an end-to-end algorithm for indirect data-driven control for bilinear systems with stability guarantees. We consider the case where the collected i.i.d. data is affected by probabilistic noise with possibly unbounded support and leverage tools from statistical learning theory to derive finite sample identification error bounds. To this end, we solve the bilinear identification problem by solving a set of linear and affine identification problems, by a particular choice of a control input during the data collection phase. We provide a priori as well as data-dependent finite sample identification error bounds on the individual matrices as well as ellipsoidal bounds, both of which are structurally suitable for control. Further, we integrate the structure of the derived identification error bounds in a robust controller design to obtain an exponentially stable closed-loop. By means of an extensive numerical study we showcase the interplay between the controller design and the derived identification error bounds. Moreover, we note appealing connections of our results to indirect data-driven control of general nonlinear systems through Koopman operator theory and discuss how our results may be applied in this setup.
Paper Structure (23 sections, 14 theorems, 90 equations, 3 figures, 1 table, 2 algorithms)

This paper contains 23 sections, 14 theorems, 90 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Finite sample identification error bounds
  4. A priori identification error bounds
  5. Data-dependent identification error bounds
  6. Sample complexity of identifying bilinear systems
  7. Implications for data-driven control of nonlinear systems
  8. Controller design for bilinear systems
  9. Individual identification error bounds
  10. Ellipsoidal identification error bounds
  11. Controller design
  12. Controller design via linear robust control
  13. Controller design via SOS optimization
  14. Numerical examples
  15. Error bounds
  16. ...and 8 more sections

Key Result

Theorem 4

Consider the autonomous system eq:BiLinSys1. Fix a failure probability $\delta\in (0,1)$ and let the data $\{x_+^{{(\ell)}}, x^{{(\ell)}} \}_{\ell = 1}^{T_0}$ be collected according to Assumption ass:sampling. If then the identification error eq:LQErrorII of the OLS estimate eq:LsSol1 is bounded by with probability at least $1-\frac{\delta}{2}$.

Figures (3)

  • Figure 1: Mean of the identification error \ref{['fig:Ex1_Monte']}, data-based bounds \ref{['fig:Ex1_data']}, and a priori error bounds \ref{['fig:Ex1_priori']} through $100$ Monte Carlo simulations. Shaded areas are respective $3\sigma$-bands.
  • Figure 2: RoA of the academic example for $\mathcal{X}=\{x\mid\|x\|^2\leq c\}$ and the minimum required data length for individual error bounds \ref{['fig:exmp-2D-RoA-individual-minimal']} and ellipsoidal error bounds \ref{['fig:exmp-2D-RoA-ellipsoidal-minimal']} as well as the RoA with ellipsoidal error bounds for the minimum data length required for individual error bounds \ref{['fig:exmp-2D-RoA-ellipsoidal-T-individual']}.
  • Figure 3: Closed-loop trajectories of the nonlinear inverted pendulum example in Section \ref{['sec:exmp-nonlinear-pendulum']} with the SOS controller $\kappa_\mathrm{SOS}$.

Theorems & Definitions (16)

  • Remark 1
  • Remark 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Lemma 7
  • Theorem 8
  • Theorem 9
  • Lemma 10
  • Proposition 11
  • ...and 6 more