Safety Controller Synthesis for Stochastic Polynomial Time-Delayed Systems
Omid Akbarzadeh, MohammadHossein Ashoori, Amy Nejati, Abolfazl Lavaei
TL;DR
This work addresses safety controller synthesis for discrete-time stochastic nonlinear polynomial systems with time-invariant delays by extending control barrier certificates through the Krasovskii framework. It introduces two barrier classes, Krasovskii Quadratic CBC (K-QCBC) and Krasovskii Polynomial CBC (K-PCBC), capable of capturing the joint influence of current and delayed states and providing probabilistic safety guarantees under input constraints via SOS optimization. The authors formulate tractable SOS programs to jointly compute the barrier certificates and their associated safety controllers, and validate the approach on three case studies (academic system, jet engine compressor, spacecraft) demonstrating robustness to delays and quantifiable safety risks. A key trade-off is highlighted between the computational cost and the expressiveness of the barrier (quadratic versus polynomial) and whether input constraints are enforced. The framework thus offers a principled, scalable path to provably safe operation of delayed stochastic systems with polynomial dynamics, with potential extensions to broader dynamics and noise distributions.
Abstract
This work develops a theoretical framework for safety controller synthesis in discrete-time stochastic nonlinear polynomial systems subject to time-invariant delays (dt-SNPS-td). While safety analysis of stochastic systems using control barrier certificates (CBC) has been widely studied, developing safety controllers for stochastic systems with time delays remains largely unexplored. The main challenge arises from the need to account for the influence of delayed components when formulating and enforcing safety conditions. To address this, we employ Krasovskii control barrier certificates, which extend the conventional CBC framework by augmenting it with an additional summation term that captures the influence of delayed states. This formulation integrates both the current and delayed components into a unified barrier structure, enabling safety synthesis for stochastic systems with time delays. The proposed approach synthesizes safety controllers under input constraints, offering probabilistic safety guarantees robust to such delays: it ensures that all trajectories of the dt-SNPS-td remain within the prescribed safe region while fulfilling a quantified probabilistic bound. To achieve this, our method reformulates the safety constraints as a sum-of-squares optimization program, enabling the systematic construction of Krasovskii CBC together with their associated safety controllers. We validate the proposed framework through three case studies, including two physical systems, demonstrating its effectiveness and practical applicability.