Emergency-Brake Simplex: Toward A Verifiably Safe Control-CPS Architecture for Abrupt Runtime Reachability Constraint Changes

TL;DR

The paper tackles safety guarantees in control-CPS when runtime reachability constraints change abruptly, which can invalidate the Lyapunov-based safety region. It introduces Online Reference State Optimization (ORSOP) to adjust the reference state $\vec{x}_{o}$ instead of redesigning the controller, thereby preserving reachability safety with minimal delay. ORSOP combines analytical solutions (Case 1 and Case 2) with a fall-back Interior Point Method (IPM) Newton’s method, delivering substantial computational speedups over Online-Controller-Redesign (OCR) and improving success rates under tight deadlines. The method demonstrates that reconfiguring the reference state within the feasible set can maintain safety while enabling rapid recovery, offering a practical and verifiable mechanism for fault-tolerant control-CPS in the presence of evolving safety constraints. Technical innovations include a 3-step procedure (KKT-based Step 1, feasibility check Step 2, and IPM-based Step 3), explicit gradient/Hessian expressions for IPM, and a relaxation of structural assumptions via a state-space transform that converts ellipsoidal safety sets to spheres for tractable optimization.

Abstract

When a system's constraints change abruptly, the system's reachability safety does no longer sustain. Thus, the system can reach a forbidden/dangerous value. Conventional remedy practically involves online controller redesign (OCR) to re-establish the reachability's compliance with the new constraints, which, however, is usually too slow. There is a need for an online strategy capable of managing runtime changes in reachability constraints. However, to the best of the authors' knowledge, this topic has not been addressed in the existing literature. In this paper, we propose a fast fault tolerance strategy to recover the system's reachability safety in runtime. Instead of redesigning the system's controller, we propose to change the system's reference state to modify the system's reachability to comply with the new constraints. We frame the reference state search as an optimization problem and employ the Karush-Kuhn-Tucker (KKT) method as well as the Interior Point Method (IPM) based Newton's method (as a fallback for the KKT method) for fast solution derivation. The optimization also allows more future fault tolerance. Numerical simulations demonstrate that our method outperforms the conventional OCR method in terms of computational efficiency and success rate. Specifically, the results show that the proposed method finds a solution $10^{2}$ (with the IPM based Newton's method) $\sim 10^{4}$ (with the KKT method) times faster than the OCR method. Additionally, the improvement rate of the success rate of our method over the OCR method is $40.81\%$ without considering the deadline of run time. The success rate remains at $49.44\%$ for the proposed method, while it becomes $0\%$ for the OCR method when a deadline of $1.5 \; seconds$ is imposed.

Figures (2)

• Figure 1: Illustration of Lyapunov ellipsoid, forbidden region, operational region, initial state, state trajectory, and reference state.
• Figure 2: Illustration of an original Lyapunov ellipsoid $\mathcal{E}$ (delineated by the black dash-dot line) violating the new reachability constraints (delineated by the black solid lines) in 2D space. We intend to find a new reference state $\vec{x}'_{\sf o}$ (marked by the red star) in the feasible region of the reference state $\mathcal{R}_{\sf o}$ (the red area delineated by the red solid lines), so that the new Lyapunov ellipsoid $\mathcal{E}"$ (delineated by the red dash-dot line) does not overlap with the new forbidden region $\mathcal{F}'$ (the gray area delineated by the black solid lines).

Theorems & Definitions (17)

• Definition 1
• Lemma 1
• Lemma 2
• Theorem 1
• proof
• Corollary 1: Validity of ORSOP
• proof
• Remark 1
• Proposition 1
• Theorem 2