Safety Controller Synthesis for Stochastic Polynomial Time-Delayed Systems

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.

💡 Research Summary

The paper tackles the challenging problem of synthesizing safety‑guaranteed controllers for discrete‑time stochastic nonlinear polynomial systems that include time‑invariant state delays (denoted dt‑SNPS‑td). While control barrier certificates (CBC) have been widely used for safety verification and controller synthesis in stochastic systems without delays, extending these techniques to delayed systems is non‑trivial because the delayed states influence the system dynamics and must be accounted for in the safety conditions.

To address this, the authors introduce a Krasovskii‑based extension of CBC. The key idea is to augment the state with its h‑step history, forming an extended vector ξₖ =