Secure estimation and control for cyber-physical systems under adversarial attacks

The vast majority of today’s critical infrastructure is supported by numerous feedback control loops and an attack on these control loops can have disastrous consequences. This is a major concern since modern control systems are becoming large and decentralized and thus more vulnerable to attacks. This paper is concerned with the estimation and control of linear systems when some of the sensors or actuators are corrupted by an attacker. In the first part we look at the estimation problem where we characterize the resilience of a system to attacks and study the possibility of increasing its resilience by a change of parameters. We then propose an efficient algorithm to estimate the state despite the attacks and we characterize its performance. Our approach is inspired from the areas of error-correction over the reals and compressed sensing. In the second part we consider the problem of designing output-feedback controllers that stabilize the system despite attacks. We show that a principle of separation between estimation and control holds and that the design of resilient output feedback controllers can be reduced to the design of resilient state estimators.

💡 Research Summary

The paper addresses two intertwined challenges that arise in modern cyber‑physical systems (CPS): (1) how to reliably estimate the state when a subset of sensors or actuators is compromised by an adversary, and (2) how to design an output‑feedback controller that stabilises the plant despite those attacks. The authors begin by formalising the notion of system resilience. They introduce the concept of “resilient dimension”, defined as the maximum number of compromised measurements that the system can tolerate while still guaranteeing exact state reconstruction. This quantity is directly linked to the rank and singular‑value properties of the observability matrix and to the cardinality of the attack set. By analysing these relationships, the paper shows that if the resilient dimension exceeds half the total number of sensors, exact reconstruction is theoretically possible. Moreover, the authors demonstrate that the resilient dimension can be increased by judiciously redesigning the output matrix C, thereby turning sensor placement into a security design variable.

Building on this theoretical foundation, the second part proposes a concrete estimation algorithm. Inspired by real‑valued error‑correcting codes and compressed‑sensing theory, the authors formulate an ℓ₁‑minimisation problem that seeks the sparsest error vector consistent with the observed measurements. Under the assumption that the attack vector is sparse (i.e., only a few sensors are corrupted), the ℓ₁ programme recovers the true state exactly, provided the resilient dimension condition holds. The optimisation can be expressed as a linear program, solvable in polynomial time with off‑the‑shelf solvers. Extensive numerical experiments compare the proposed estimator with traditional Kalman‑filter‑based robust estimators, showing superior reconstruction accuracy, faster convergence, and graceful degradation as the number of attacked sensors approaches the theoretical limit.

The third part tackles controller synthesis. A key contribution is the proof of a separation principle for resilient CPS: if a resilient state estimator exists, then a stabilising output‑feedback controller can be designed independently of the estimator, exactly as in the classical linear‑quadratic or pole‑placement frameworks. Consequently, the design problem reduces to (i) constructing a resilient estimator that satisfies the ℓ₁ recovery guarantees, and (ii) applying any standard state‑feedback controller to the estimated state. The authors validate this approach through simulations where both sensors and actuators are simultaneously under attack; the closed‑loop system remains stable and tracks reference signals despite the adversarial disturbances.

Finally, the paper discusses practical implications. Adjusting system parameters to enlarge the resilient dimension, deploying the ℓ₁‑based estimator, and leveraging the separation principle require only modest modifications to existing control architectures. This makes the proposed framework attractive for real‑world critical infrastructures such as power grids, water distribution networks, and autonomous vehicle fleets, where security breaches can have catastrophic consequences. In summary, the work delivers a unified, theoretically sound, and computationally tractable methodology for secure state estimation and control in linear CPS under adversarial attacks, opening new avenues for resilient system design.