Mathematical Model for Cyber Security

Cyber-physical systems (CPS), such as smart grids, industrial control systems, and autonomous infrastructure, are increasingly targeted by sophisticated cyberattacks that exploit sensor and actuator vulnerabilities. To ensure resilient operation under adversarial conditions, this paper presents a mathematical framework based on dynamical systems theory and secure control strategies. The system dynamics are modeled using linear state-space equations with stochastic disturbances, incorporating adversarial inputs to represent cyberattacks on sensors and actuators. State estimation is performed using a Kalman filter, and residual analysis is employed for real-time attack detection. When residuals exceed a predefined threshold, indicating abnormal behavior, the system triggers an alarm and switches to a conservative fallback control policy. Two representative case studies are simulated: a temperature regulation system under sensor attack, and a water tank level control system under actuator manipulation. In both scenarios, the secure control framework effectively detects and mitigates the impact of the attacks, demonstrating the robustness and practicality of the proposed methodology. This approach provides a foundation for developing resilient control architectures in safety-critical CPS environments.