Resilient Extremum Seeking Control for Cyber-Physical Systems Under Denial-of-Service Attacks

Resilient Extremum Seeking Control (ESC) for cyber-physical systems under Denial-of-Service (DoS) attacks utilizes a hold-input strategy where the most recently successfully transmitted signals are retained during communication blackouts. This approach prevents the harmful correlation between DoS attacks and the ESC dither signal, which can otherwise create significant optimization bias and displace the system equilibrium.

The proposed architecture ensures that successful transmissions maintain an exponentially contracting mode in the averaged error dynamics, while DoS intervals create a neutral mode that preserves recent information. Consequently, increasingly severe DoS attacks primarily slow the convergence rate of the optimization process rather than destroying system stability, provided that communication is not permanently blocked.

Key findings indicate that: Zero-input strategies are vulnerable to dither-synchronized attacks, introducing an $\mathcal{O}(1/a)$ bias in the averaged dynamics. The hold-input mechanism structurally decouples the attack from the probing signal, ensuring resilience against such correlated attacks. * Stability is guaranteed for any attack fraction strictly smaller than one, with convergence rates explicitly quantified based on attack duration and success probability.

Summary

This work addresses extremum seeking control (ESC) in cyber-physical systems where communication channels are subject to denial-of-service (DoS) attacks. ESC is a model-free optimization method that uses persistent excitation and feedback to drive a system toward an unknown optimum, making it attractive for adaptive control and real-time optimization in settings where a precise plant model is unavailable. The paper focuses on a discrete-time formulation, which is relevant to digital controllers and networked systems, and considers the practical failure mode in which DoS attacks interrupt the transmission of control-relevant signals. Instead of assuming reliable communication, the proposed architecture explicitly accommodates intermittent packet loss by holding the most recently successfully transmitted signals during attack intervals.

The central contribution is a resilient ESC architecture that remains functional despite communication disruptions. The key insight is that, under DoS, the controller can continue operating on the last valid information rather than resetting or failing when new packets are blocked. This “hold-last-signal” strategy effectively bounds the information staleness experienced by the closed loop and allows the controller to preserve the essential dithering and feedback dynamics needed for extremum seeking. The result is a discrete-time design that links the admissible frequency or duration of DoS attacks to the stability and convergence properties of the closed-loop system, providing a clear resilience condition rather than relying on idealized communication assumptions.

This matters because many cyber-physical applications—such as mobile robotics, power-system optimization, and networked process control—must perform online optimization over unreliable or adversarial communication links. DoS attacks are particularly damaging to adaptive and optimization-based controllers because they can corrupt the delicate interplay between excitation, estimation, and update laws. By showing that ESC can be made robust through a simple yet principled holding mechanism, the work bridges adaptive control, networked control, and cyber resilience. It suggests a practical path toward deploying optimization-driven control in realistic cyber-physical environments where perfect communication cannot be guaranteed.