Bridging fixed-time stability and safety reachability: A critic-only hierarchical framework for faulty euler-lagrange systems

Ensuring safety in Euler–Lagrange (EL) systems under actuator faults and state constraints is challenging when computational limitations hinder standard reachability and reinforcement learning (RL) methods. This paper presents a hierarchical framework combining fixed-time stability with real-time Hamilton–Jacobi (HJ) reachability learning. To overcome the high computational cost of actor-critic schemes, we develop an efficient critic-only RL architecture to compute optimal safe control actions online. A fixed-time neural observer supports this by simultaneously estimating unmeasurable states and reconstructing complex actuator faults. To integrate the safety critic with nominal tracking, we propose a Safety-Aware Adaptive Prescribed Performance Control (PPC) scheme that resolves a fundamental obstacle evasion paradox. Standard PPC enforces strictly shrinking boundaries, so physical evasion inevitably violates these limits, triggering Barrier Lyapunov Function (BLF) singularities. We solve this using an axis-specific dynamic boundary acting as a mathematical repeller. Activated by the safety critic, the boundary expands strictly faster than the evasion error grows. Embedding this expansion into the nominal controller smoothly relaxes restorative tracking forces during evasive maneuvers. The framework guarantees safe set forward invariance, maintains performance bounds without singularities, and ensures fixed-time convergence of estimation and tracking errors.