A Resilience Quantitative Assessment Framework for Cyber–Physical Systems: Mathematical Modeling and Simulation

As cyber threats continue to grow in complexity and persistence, resilience has become a critical requirement for cyber–physical systems (CPSs). Resilience quantitative assessment is essential for supporting secure system design and ensuring reliable operation. Although various methods have been proposed for evaluating CPS resilience, major challenges remain in accurately modeling the interaction between cyber and physical domains and in providing structured guidance for resilience-oriented design. This study proposes an integrated CPS resilience assessment framework that combines cyber-layer anomaly modeling based on Markov chains with mathematical modeling of performance degradation and recovery in the physical domain. The framework establishes a structured evaluation process through parameter normalization and cyber–physical coupling, enabling the generation of resilience curves that clearly represent system performance changes under adverse conditions. A case study involving an industrial controller equipped with a diversity-redundancy architecture is conducted to demonstrate the applicability of the proposed method. Modeling and simulation results indicate that the framework effectively reveals key resilience characteristics and supports performance-informed design optimization.