Minimal Information Control Invariance via Vector Quantization

Ege Yuceel; Teodor Tchalakov; Sayan Mitra

eess.SYApr 3, 2026

Minimal Information Control Invariance via Vector Quantization

Authors: Ege Yuceel, Teodor Tchalakov, Sayan Mitra

Organizations: Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA.

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.

Figures & tables

Fig. 1: 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 $[t_{k},t_{k}+T_{s})$ .

Fig. 2: 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.

Method	$\|\mathcal{V}_{T_{s}}\|$	CL Inv. (%)
VQ-AE (ours, pruned)	26	99.979
VQ-AE (ours, unpruned)	171	99.989
VQ-AE partition + OL LQR	171	0.000
Uniform grid + NN	4096	99.975
Uniform grid + CL LQR	2.2M	76.870

TABLE I: Comparison of codebook size and closed-loop invariance rate across baselines, evaluated over

10^{5}

trials for 10 s.

Fig. 3: Outputs at different resolutions (indicated above). Closed-loop invariance and terminal safety over 10 s versus image resolution. Invariance remains above $99\%$ down to $\sim 65$ px, then degrades with lower resolution.

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Real-Time Synthesis of Robust Controlled Invariant Sets for Monotone Systems

Yasin Sonmez, Mahmoud Khaled, Majid Zamani, Murat Arcak

University of California, Berkeley, USA · Ludwig-Maximilian University, Germany · University of Colorado Boulder, USA

Dynamical Systems Safety-Critical Control Robust Control

Open paper AskPDF arXiv

Safety-critical control of autonomous systems requires formal safety certificates, such as controlled invariant sets, that must be computed online as conditions change. Although standard synthesis algorithms scale poorly with state dimension, monotone dynamical systems with lower-closed safety specifications allow for accelerated computation of controlled invariant sets. In particular, lazy fixed-point algorithms exploit monotonicity and track only the antichain basis of the set. However, membership tests and redundancy checks against an evolving basis remain major bottlenecks. We introduce a threshold-function reformulation in which a lower-closed set on a d-dimensional grid is represented by its column heights along a designated axis. This reformulates the greatest-fixed-point iteration as independent one-dimensional binary searches, one per grid column, yielding an embarrassingly parallel iteration with asymptotically lower computational complexity than the lazy fixed-point algorithm. Experiments synthesize invariant sets on 3D grids with 10^9 cells in under 50 ms and 10^14 cells in under two minutes. We further demonstrate online re-synthesis in a safety-informed model predictive controller example.

Abstract

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