Opacity and Structural Resilience in Cyberphysical Systems

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## Abstract

Cyberphysical systems (CPSs) integrate communication, control, and computation with physical processes. Examples include power systems, water distribution networks, and on a smaller scale, medical devices and home control systems. Since these systems are often controlled over a network, the sharing of information among systems and across geographies makes them vulnerable to attacks carried out (possibly remotely) by malicious adversaries. An attack could be carried out on the physical system, on the computer(s) controlling the system, or on the communication links between the system and the computer. Thus, significant material damage can be caused by an attacker who is able to gain access to the system, and such attacks will often have the consequence of causing widespread disruption to everyday life. Therefore, ensuring the safety of information critical to nominal operation of the system is of utmost importance. This dissertation addresses two problems in the broad area of the Control and Security of Cyberphysical Systems.

First, we present a framework for opacity in CPSs modeled as a discrete-time linear time-invariant (DT-LTI) system. The current state-of-the-art in this field studies opacity for discrete event systems (DESs) described by regular languages. However, the states in a DES are discrete; in many practical systems, it is common for states (and other system variables) to take continuous values. We define a notion of opacity called k-initial state opacity (k-ISO) for such systems. A set of secret states is said to be k-ISO with respect to a set of nonsecret states if the outputs at time k of every trajectory starting from the set of secret states is indistinguishable from the output at time k of some trajectory starting from the set of nonsecret states. Necessary and sufficient conditions to establish k-ISO are presented in terms of sets of reachable states. Opacity of a given DT-LTI system is shown to be equivalent to the output controllability of a system obeying the same dynamics, but with different initial conditions.

We then study the case where there is more than one adversarial observer, and define several notions of decentralized opacity. These notions of decentralized opacity will depend on whether there is a centralized coordinator or not, and the presence or absence of collusion among the adversaries. We establish conditions for decentralized opacity in terms of sets of reachable states. In the case of colluding adversaries, we present a condition for non-opacity in terms of the structure of the communication graph.

We extend this work to formulate notions of opacity for discrete-time switched linear systems. A switched system consists of a finite number of subsystems and a rule that orchestrates switching among them. We distinguish between the cases when the secret is specified as a set of initial modes, a set of initial states, or a combination of the two. The novelty of our schemes is in the fact that we place restrictions on: i) the allowed transitions between modes (specified by a directed graph), ii) the number of allowed changes of modes (specified by lengths of paths in the directed graph), and iii) the dwell times in each mode. Each notion of opacity is characterized in terms of allowed switching sequences and sets of reachable states and/ or modes. Finally we present algorithmic procedures to verify these notions, and provide bounds on their computational complexity.

Second, we study the resilience of CPSs to denial-of-service (DoS) and integrity attacks. The CPS is modeled as a linear structured system, and its resilience to an attack is interpreted in a graph-theoretic framework. The structural systems approach presumes knowledge of only the positions of zero and nonzero entries in the system matrices to infer system properties. This approach is attractive due to the fact that these properties will hold for almost every admissible numerical realization of the system. The structural resilience of the system is characterized in terms of unmatched vertices in maximum matchings of the bipartite graph and connected components of directed graph representations of the system under attack. Further, we establish a condition based on the zero structure of an input matrix that will ensure that the system is structurally resilient to a state feedback integrity attack if it is also resilient to a DoS attack.

Finally, we formulate an extension to the case of switched structured systems, and derive conditions for such systems to be structurally resilient to a DoS attack.