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Transformer-Enhanced Data-Driven Output Reachability with Conformal Coverage Guarantees

Zhen Zhang, Peng Xie, Wenyuan Wu, Yanliang Huang, Amr Alanwar

Abstract

This paper considers output reachability analysis for linear time-invariant systems with unknown state-space matrices and unknown observation map, given only noisy input-output measurements. The Cayley--Hamilton theorem is applied to eliminate the latent state algebraically, producing an autoregressive input-output model whose parameter uncertainty is enclosed in a matrix zonotope. Set-valued propagation of this model yields output reachable sets with deterministic containment guarantees under a bounded aggregated residual assumption. The conservatism inherent in the lifted matrix-zonotope product is then mitigated by a decoder-only Transformer trained on labels obtained through directional contraction of the formal envelope via an exterior non-reachability certificate. Split conformal prediction restores distribution-free coverage at both per-step and trajectory levels without access to the true reachable-set hull. The framework is validated on a five-dimensional system with multiple unknown observation matrices.

Transformer-Enhanced Data-Driven Output Reachability with Conformal Coverage Guarantees

Abstract

This paper considers output reachability analysis for linear time-invariant systems with unknown state-space matrices and unknown observation map, given only noisy input-output measurements. The Cayley--Hamilton theorem is applied to eliminate the latent state algebraically, producing an autoregressive input-output model whose parameter uncertainty is enclosed in a matrix zonotope. Set-valued propagation of this model yields output reachable sets with deterministic containment guarantees under a bounded aggregated residual assumption. The conservatism inherent in the lifted matrix-zonotope product is then mitigated by a decoder-only Transformer trained on labels obtained through directional contraction of the formal envelope via an exterior non-reachability certificate. Split conformal prediction restores distribution-free coverage at both per-step and trajectory levels without access to the true reachable-set hull. The framework is validated on a five-dimensional system with multiple unknown observation matrices.

Paper Structure

This paper contains 19 sections, 8 theorems, 31 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Notations
  4. Set Representations
  5. System Model and Assumptions
  6. Problem Statement
  7. Conformal Prediction
  8. Data-Driven Output Reachability
  9. Output Autoregressive Representation
  10. Data-Driven Model Set Construction
  11. Output Reachable Set Propagation
  12. Transformer-Enhanced Output Reachability
  13. Zonotope Fitting from Trajectory Samples
  14. Certificate-Assisted Label Refinement
  15. Zonotope Tokenization and Training Data
  16. ...and 4 more sections

Key Result

Lemma 1

Consider system eq:ss under Assumption ass:order. For all $k \geq n_o$ with $n_o = n_x$, the output satisfies where $a_1, \dots, a_{n_o}$ are the coefficients of the characteristic polynomial of $A$ with $a_0 = 1$, the input coefficients are $b_i = \sum_{j=0}^{i-1} a_j \, C A^{i-1-j} B$, and the aggregated residual is $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure E1: Data-driven output reachable sets under three sensor configurations ($n_y=2$).
  • Figure E2: Transformer-enhanced output reachable sets for three sensor configurations.

Theorems & Definitions (23)

  • Definition 1: Zonotope kuhn1998rigorously
  • Definition 2: Matrix Zonotope althoff2010reachability
  • Remark 1
  • Definition 3: Exact output reachable set
  • Lemma 1: Output autoregressive form hespanha2018linear
  • Remark 2
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • ...and 13 more