Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimal Information Control Invariance via Vector Quantization

Ege Yuceel, Teodor Tchalakov, Sayan Mitra

Abstract

Safety-critical autonomous systems must satisfy hard state constraints under tight computational and sensing budgets, yet learning-based controllers are often far more complex than safe operation requires. To formalize this gap, we study how many distinct control signals are needed to render a compact set forward invariant under sampled-data control, connecting the question to the information-theoretic notion of invariance entropy. We propose a vector-quantized autoencoder that jointly learns a state-space partition and a finite control codebook, and develop an iterative forward certification algorithm that uses Lipschitz-based reachable-set enclosures and sum-of-squares programming. On a 12-dimensional nonlinear quadrotor model, the learned controller achieves a $157\times$ reduction in codebook size over a uniform grid baseline while preserving invariance, and we empirically characterize the minimum sensing resolution compatible with safe operation.

Minimal Information Control Invariance via Vector Quantization

Abstract

Safety-critical autonomous systems must satisfy hard state constraints under tight computational and sensing budgets, yet learning-based controllers are often far more complex than safe operation requires. To formalize this gap, we study how many distinct control signals are needed to render a compact set forward invariant under sampled-data control, connecting the question to the information-theoretic notion of invariance entropy. We propose a vector-quantized autoencoder that jointly learns a state-space partition and a finite control codebook, and develop an iterative forward certification algorithm that uses Lipschitz-based reachable-set enclosures and sum-of-squares programming. On a 12-dimensional nonlinear quadrotor model, the learned controller achieves a reduction in codebook size over a uniform grid baseline while preserving invariance, and we empirically characterize the minimum sensing resolution compatible with safe operation.

Paper Structure

This paper contains 12 sections, 2 theorems, 22 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Preliminaries
  4. Minimal Controller Design via VQ-AEs
  5. Learning a Minimal Control Codebook via VQ-AE
  6. Correctness of the Learned Controller
  7. Sum-of-Squares Certification
  8. Empirical Results
  9. Codebook Compression
  10. Baseline Comparisons
  11. Sensing Resolution and Invariance
  12. Conclusion

Key Result

Proposition 1

$\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure A1: VQ-AE controller architecture. The quadrotor estimates its state $\hat{x}(t) \in \mathbb{R}^{12}$ from images of an ArUco marker. The encoder $E_\theta$ maps $\hat{x}(t)$ to a latent vector $z$, quantized to the nearest codebook entry $i = \arg\min_k \|z - e_k\|_2^2$, inducing a partition $\{A_i\}_{i=1}^M$ of $Q$. A 2D projection of the partition is shown; the full 12-dimensional partition contains additional regions. Arrows display the controlled vector field within each region, directing trajectories inward to maintain invariance of $Q$. The decoder $D_\phi$ maps index $i$ to an open-loop control applied over $[t_k, t_k + T_s)$.
  • Figure F1: Codebook size $|\mathcal{V}_{T_s}|$ and closed-loop invariance vs. codebook pressure $\lambda_{pr}$. Invariance holds at $100\%$ down to $|\mathcal{V}_{T_s}|=14$ codes, then collapses. Green checkmarks indicate IFC certification success; all controllers at or beyond the collapse point are unverified.
  • Figure F2: Outputs at different resolutions (indicated above). Closed-loop invariance and terminal safety over 10 s versus image resolution. Invariance remains above $99\%$ down to $\sim65$ px, then degrades with lower resolution.

Theorems & Definitions (6)

  • Definition 1: $T$-Spanning Set
  • Definition 2: Invariance Entropy
  • Proposition 1
  • proof
  • Theorem 1
  • proof