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A Stochastic Fundamental Lemma with Reduced Disturbance Data Requirements

Ruchuan Ou, Guanru Pan, Timm Faulwasser

TL;DR

This work addresses the data demands of the stochastic fundamental lemma for LTI systems with process disturbances by introducing a causality-informed variant that removes the need for past disturbance data. The approach leverages Polynomial Chaos Expansions (PCE) to decompose the stochastic dynamics into a nominal deterministic part and disturbance-induced error subsystems, and recasts these in a joint disturbance basis with a causality-consistent structure. By converting disturbances into initial-condition effects and exploiting disturbance-free trajectories, the method enables data-driven propagation and stochastic optimal control with smaller Hankel matrices and reduced computational burden. The numerical aircraft example demonstrates substantial reductions in data and computation time while maintaining comparable performance, highlighting the practical impact for data-efficient, stochastic, data-driven control. Future work includes disturbance estimator error bounds, extension to nonlinear systems, and fast computation for stochastic OCPs.

Abstract

Recently, the fundamental lemma by Willems et. al has been extended towards stochastic LTI systems subject to process disturbances. Using this lemma requires previously recorded data of inputs, outputs, and disturbances. In this paper, we exploit causality concepts of stochastic control to propose a variant of the stochastic fundamental lemma that does not require past disturbance data in the Hankel matrices. Our developments rely on polynomial chaos expansions and on the knowledge of the disturbance distribution. Similar to our previous results, the proposed variant of the fundamental lemma allows to predict future input-output trajectories of stochastic LTI systems. We draw upon a numerical example to illustrate the proposed variant in data-driven control context.

A Stochastic Fundamental Lemma with Reduced Disturbance Data Requirements

TL;DR

This work addresses the data demands of the stochastic fundamental lemma for LTI systems with process disturbances by introducing a causality-informed variant that removes the need for past disturbance data. The approach leverages Polynomial Chaos Expansions (PCE) to decompose the stochastic dynamics into a nominal deterministic part and disturbance-induced error subsystems, and recasts these in a joint disturbance basis with a causality-consistent structure. By converting disturbances into initial-condition effects and exploiting disturbance-free trajectories, the method enables data-driven propagation and stochastic optimal control with smaller Hankel matrices and reduced computational burden. The numerical aircraft example demonstrates substantial reductions in data and computation time while maintaining comparable performance, highlighting the practical impact for data-efficient, stochastic, data-driven control. Future work includes disturbance estimator error bounds, extension to nonlinear systems, and fast computation for stochastic OCPs.

Abstract

Recently, the fundamental lemma by Willems et. al has been extended towards stochastic LTI systems subject to process disturbances. Using this lemma requires previously recorded data of inputs, outputs, and disturbances. In this paper, we exploit causality concepts of stochastic control to propose a variant of the stochastic fundamental lemma that does not require past disturbance data in the Hankel matrices. Our developments rely on polynomial chaos expansions and on the knowledge of the disturbance distribution. Similar to our previous results, the proposed variant of the fundamental lemma allows to predict future input-output trajectories of stochastic LTI systems. We draw upon a numerical example to illustrate the proposed variant in data-driven control context.

Paper Structure

This paper contains 13 sections, 6 theorems, 58 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Polynomial Chaos Expansion
  4. Stochastic Fundamental Lemma
  5. Forward Uncertainty Propagation Without Disturbance Data
  6. VARX model
  7. System decomposition
  8. PCE basis structure
  9. Causality in PCE
  10. Propagation without past disturbance data
  11. Estimation of Disturbance-Free Data
  12. Numerical example
  13. Conclusion

Key Result

Lemma 1

Consider system eq:Dyn and a $T$-length realization trajectory tuple $(u,y,w)_{[0,T-1]}^{\mathrm{d}}$ of realization dynamics eq:DynReal. We assume that system eq:DynReal is controllable and let $(u,w)_{[0,T-1]}^{\mathrm{d}}$ be persistently exciting of order $N+{n_x}$. Then $(U,Y,W)_{[0,N-1]}$ is a holds for all $(z, Z)\in \{ (u, U), (y, Y), (w, W)\}$.

Figures (2)

  • Figure 1: Evolution of the PDFs of the outputs $Y^1$ and $Y^2$ over horizon $N=25$. Deep blue-dashed line: Chance constraint.
  • Figure 2: Aircraft example with Gaussian disturbance. Red-solid line: Scheme I; blue-solid line with circle marker: Scheme II; black-dashed line: Scheme III; deep blue-dashed line: Chance constraint.

Theorems & Definitions (17)

  • Definition 1: Exact PCE representation
  • Remark 1: Generic affine PCE series
  • Definition 2: Persistency of excitation
  • Lemma 1: Stochastic fundamental lemma
  • Corollary 1
  • Remark 2
  • Example 1
  • Remark 3: Obtaining undisturbed system data
  • Lemma 2
  • proof
  • ...and 7 more