An H-infinity Approach to Stealth-resilient Control Design Shaunak D. Bopardikar Alberto Speranzon Jo˜ ao P. Hespanha Abstract—We revisit the problem of stealth attack – a coordinated attack through several possible entry points – on a closed loop linear time invariant dynamical system. We propose a notion of the impact of such an attack on the system and consider a novel metric related to the H-infinity norm of the inverse of the system dynamics (assuming it exists) as a measure of the security of the system. We show that the problem can be cast as a linear matrix inequality optimization with the system parameters (observer gain) as the variable. This formulation allows a user to re-design the system from the perspective of minimizing the impact of any stealth attack. Numerical results on simulated data illustrate the security and performance tradeoff using the proposed approach. Index Terms— Stealth attacks, resilient control, H- infinity design I. I NTRODUCTION Cyber-physical systems (CPSs) have become ubiq- uitous in modern times. A CPS is composed of a set of devices directly interacting with the physical layer of the system, such as sensors, actuators, controllers, monitors, etc., and can exchange data by cyber means, such as the intra or internet. CPSs are used in both safety critical systems, such as power networks, water purification systems, airplanes, and in less critical in- frastructures, such as heating and lighting systems. As CPSs are inherently interconnected, mostly using IP- based protocols, attacks on such system are growing and they can be rather elaborate such as STUXNET [5] or the latest attack on Ukraine’s power grid [1]. Since an attack on a CPS can have major physical consequences, it is highly desirable to develop design tools to assess the impact of security measures on the performance of such systems: clearly there are tradeoffs between control performance and security. In the CPSs security research literature, authors have been considering various models of attacks. In [2] and [6], the authors have considered false data injections on static estimators. This attack is modeled as corruption of measurements that are used for state estimation. Shaunak D. Bopardikar (email: bopardsd@utrc.utc.com) is with the United Technologies Research Center, Berkeley, CA, USA. Alberto Speranzon (email: alberto.speranzon@gmail.com) was with United Technologies Research Center, East Hartford, CT, USA when this work was performed. He is presently with Honey- well Aerospace – Advanced Technologies. Jo˜ ao P. Hespanha (email: hespanha@ece.ucsb.edu) is with the Electrical and Computer Engineering Department at University of California Santa Barbara, CA, USA. This work was supported in part by United Technologies Research Center, under the Cyber-Physical Systems Security Initiative. Conditions on the systems properties that prevent these attacks are derived in [8], where the authors show that an attack exists if and only if the system dynamics have an unstable mode and the associated eigenvector satisfies a technical assumption. More recent work, [12] and [3], provide methods to change the system model so that a class of stealthy attacks on the actuators or sensors can be detects. In [10] the authors provide a more general framework to analyze several types of attacks on power systems and networks. In particular, general conditions for attack detection and identifiability for descriptor linear time-invariant systems are defined. The problem of trading off security and control per- formance has been researched in recent years. One of the first papers to explore such a tradeoff is [7], where an additive Gaussian noise with zero mean and known covariance is added to the control input. The addition of such noise deteriorates the control performance – measured with respect to an LQG cost – but enables the detectability of the attack. More recently, in [9], the authors have extended the results considering such noise signature (watermarking) to be the output of a hidden Markov model. In a recent paper [11] the authors propose a framework to design secure and computa- tionally efficient cyber-physical systems. In particular, they explore a tradeoff between the sampling period in a control system and the probability of an attacker to be able to decode an encrypted sensor message as a function of the number of bits in the encryption key and the importance such sensor has in the observability of the system. In [14], the authors leverage such framework to develop a control/security design method – they also consider schedulability of processes in a CPU as part of the cost – and formulate an optimization problem from where, for various security levels, one can obtain the control performance vs security Pareto curve. In a similar spirit [13] considers such tradeoffs, where the control performance is related to the tracking error and the security level is associated to the number of bits used to encrypt the sensor and actuator signals. In this paper, we revisit the problem of stealth attack – a coordinated attack through several possible entry points – on a closed loop linear time invariant dynamical system. The main difference with respect to our previous work [3] is that we now consider the case of noise in the system, and thereby focus on minimizing the conse- quences of a stealth attack on the system. We propose a