# Biblio

We present a controller synthesis algorithm for a discrete time reach-avoid problem in the presence of adversaries. Our model of the adversary captures typical malicious attacks en- visioned on cyber-physical systems such as sensor spoofing, controller corruption, and actuator intrusion. After formu- lating the problem in a general setting, we present a sound and complete algorithm for the case with linear dynamics and an adversary with a budget on the total L2-norm of its actions. The algorithm relies on a result from linear control theory that enables us to decompose and precisely compute the reachable states of the system in terms of a symbolic simulation of the adversary-free dynamics and the total uncertainty induced by the adversary. With this de- composition, the synthesis problem eliminates the universal quantifier on the adversary’s choices and the symbolic con- troller actions can be effectively solved using an SMT solver. The constraints induced by the adversary are computed by solving second-order cone programmings. The algorithm is later extended to synthesize state-dependent controller and to generate attacks for the adversary. We present prelimi- nary experimental results that show the effectiveness of this approach on several example problems.

Presented at the NSA Science of Security Quarterly Meeting, July 2016.

Presented at the NSA Science of Security Quarterly Meeting, July 2016.

Presented as part of the Illinois Science of Security Lablet Bi-Weekly Meetings, September 2014.

Presented at the NSA Science of Security Quarterly Lablet Meeting, October 2015.

Presented at the Illinois Lablet Science of Security Bi-weekly Meeting, March 2015.

Presented at the NSA Science of Security Quarterly Meeting, October 2014.

Presented as part of the Illinois Science of Security Lablet Bi-Weekly Meeting, September 2014.

This article describes our recent progress on the development of rigorous analytical metrics for assessing the threat-performance trade-off in control systems. Computing systems that monitor and control physical processes are now pervasive, yet their security is frequently an afterthought rather than a first-order design consideration. We investigate a rational basis for deciding—at the design level—how much investment should be made to secure the system.

We present a technique for bounded invariant verification of nonlinear networked dynamical systems with delayed interconnections. The underlying problem in precise boundedtime verification lies with computing bounds on the sensitivity of trajectories (or solutions) to changes in initial states and inputs of the system. For large networks, computing this sensitivity

with precision guarantees is challenging. We introduce the notion of input-to-state (IS) discrepancy of each module or subsystem in a larger nonlinear networked dynamical system. The IS discrepancy bounds the distance between two solutions or trajectories of a module in terms of their initial states and their inputs. Given the IS discrepancy functions of the modules, we show that it is possible to effectively construct a reduced (low dimensional) time-delayed dynamical system, such that the trajectory of this reduced model precisely bounds the distance between the trajectories of the complete network with changed initial states. Using the above results we develop a sound and relatively complete algorithm for bounded invariant verification of networked dynamical systems consisting of nonlinear modules interacting through possibly delayed signals. Finally, we introduce a local version of IS discrepancy and show that it is possible to compute them using only the Lipschitz constant and the Jacobian of the dynamic function of the modules.

In distributed control systems with shared resources, participating agents can improve the overall performance of the system by sharing data about their personal preferences. In this paper, we formulate and study a natural tradeoff arising in these problems between the privacy of the agent’s data and the performance of the control system.We formalize privacy in terms of differential privacy of agents’ preference vectors. The overall control system consists of N agents with linear discrete-time coupled dynamics, each controlled to track its preference vector. Performance of the system is measured by the mean squared tracking error.We present a mechanism that achieves differential privacy by adding Laplace noise to the shared information in a way that depends on the sensitivity of the control system to the private data. We show that for stable systems the performance cost of using this type of privacy preserving mechanism grows as O(T 3/Nε2 ), where T is the time horizon and ε is the privacy parameter. For unstable systems, the cost grows exponentially with time. From an estimation point of view, we establish a lower-bound for the entropy of any unbiased estimator of the private data from any noise-adding mechanism that gives ε-differential privacy.We show that the mechanism achieving this lower-bound is a randomized mechanism that also uses Laplace noise.

In distributed control systems with shared resources, participating agents can improve the overall performance of the system by sharing data about their personal references. In this paper, we formulate and study a natural tradeoff arising in these problems between the privacy of the agent’s data and the performance of the control system.We formalize privacy in terms of differential privacy of agents’ preference vectors. The overall control system consists of *N* agents with linear discrete-time coupled dynamics, each controlled to track its preference vector. Performance of the system is measured by the mean squared tracking error. We present a mechanism that achieves differential privacy by adding Laplace noise to the shared information in a way that depends on the sensitivity of the control system to the private data. We show that for stable systems the performance cost of using this type of privacy preserving mechanism grows as *O*(*T*^{3 }/*N*ε^{2}), where T is the time horizon and ε is the privacy parameter. For unstable systems, the cost grows exponentially with time. From an estimation point of view, we establish a lower-bound for the entropy of any unbiased estimator of the private data from any noise-adding mechanism that gives ε-differential privacy. We show that the mechanism achieving this lower-bound is a randomized mechanism that also uses Laplace noise.

Design and testing of pacemaker is challenging because of the need to capture the interaction between the physical processes (e.g. voltage signal in cardiac tissue) and the embedded software (e.g. a pacemaker). At the same time, there is a growing need for design and certification methodologies that can provide quality assurance for the embedded software. We describe recent progress in simulation-based techniques that are capable of ensuring guaranteed coverage. Our methods employ discrep- ancy functions, which impose bounds on system dynamics, and proceed through iteratively constructing over-approximations of the reachable set of states. We are able to prove time bounded safety or produce counterexamples. We illustrate the techniques by analyzing a family of pacemaker designs against time duration requirements and synthesize safe parameter ranges. We conclude by outlining the potential uses of this technology to improve the safety of medical device designs.

Individuals sharing information can improve the cost or performance of a distributed control system. But, sharing may also violate privacy. We develop a general framework for studying the cost of diﬀerential privacy in systems where a collection of agents, with coupled dynamics, communicate for sensing their shared environment while pursuing individ- ual preferences. First, we propose a communication strategy that relies on adding carefully chosen random noise to agent states and show that it preserves diﬀerential privacy. Of course, the higher the standard deviation of the noise, the higher the cost of privacy. For linear distributed control systems with quadratic cost functions, the standard deviation becomes independent of the number agents and it decays with the maximum eigenvalue of the dynamics matrix. Furthermore, for stable dynamics, the noise to be added is independent of the number of agents as well as the time horizon up to which privacy is desired.

In a discrete-time linear multi-agent control system, where the agents are coupled via an environmental state, knowledge of the environmental state is desirable to control the agents locally. However, since the environmental state depends on the behavior of the agents, sharing it directly among these agents jeopardizes the privacy of the agents' pro les, de ned as the combination of the agents' initial states and the sequence of local control inputs over time. A commonly used solution is to randomize the environmental state before sharing { this leads to a natural trade-o between the privacy of the agents' pro les and the variance of estimating the environmental state. By treating the multi-agent system as a probabilistic model of the environmental state parametrized by the agents' pro les, we show that when the agents' pro les is "-di erentially private, there is a lower bound on the `1 induced norm of the covariance matrix of the minimum-variance unbiased estimator of the environmental state. This lower bound is achieved by a randomized mechanism that uses Laplace noise.

In distributed optimization and iterative consensus literature, a standard problem is for *N *agents to minimize a function* f* over a subset of R^{n}, where the cost function is expressed as Σ fi . In this paper, we study the private distributed optimization (PDOP) problem with the additional requirement that the cost function of the individual agents should remain differentially private. The adversary attempts to infer information about the private cost functions from the messages that the agents exchange. Achieving differential privacy requires that any change of an individual’s cost function only results in unsubstantial changes in the statistics of the messages. We propose a class of iterative algorithms for solving PDOP, which achieves differential privacy and convergence to the optimal value. Our analysis reveals the dependence of the achieved accuracy and the privacy levels on the the parameters of the algorithm.

The iterative consensus problem requires a set of processes or agents with diﬀerent initial values, to interact and update their states to eventually converge to a common value. Pro- tocols solving iterative consensus serve as building blocks in a variety of systems where distributed coordination is re- quired for load balancing, data aggregation, sensor fusion, ﬁltering, and synchronization. In this paper, we introduce the private iterative consensus problem where agents are re- quired to converge while protecting the privacy of their ini- tial values from honest but curious adversaries. Protecting the initial states, in many applications, suﬃce to protect all subsequent states of the individual participants.

We adapt the notion of diﬀerential privacy in this setting of iterative computation. Next, we present (i) a server-based and (ii) a completely distributed randomized mechanism for solving diﬀerentially private iterative consensus with adver- saries who can observe the messages as well as the internal states of the server and a subset of the clients. Our analysis establishes the tradeoﬀ between privacy and the accuracy.

The concept of differential privacy stems from the study of private query of datasets. In this work, we apply this concept to metric spaces to study a mechanism that randomizes a deterministic query by adding mean-zero noise to keep differential privacy.

Veriﬁcation algorithms for networks of nonlinear hybrid automata (HA) can aid us understand and control biological processes such as cardiac arrhythmia, formation of memory, and genetic regulation. We present an algorithm for over-approximating reach sets of networks of nonlinear HA which can be used for sound and relatively complete invariant checking. First, it uses automatically computed input-to-state discrepancy functions for the individual automata modules in the network *A* for constructing a low-dimensional model *M*. Simulations of both *A* and *M* are then used to compute the reach tubes for *A*. These techniques enable us to handle a challenging veriﬁcation problem involving a network of cardiac cells, where each cell has four continuous variables and 29 locations. Our prototype tool can check bounded-time invariants for networks with 5 cells (20 continuous variables, 29^{5} locations) typically in less than 15 minutes for up to reasonable time horizons. From the computed reach tubes we can infer biologically relevant properties of the network from a set of initial states.

State estimation is a fundamental problem for monitoring and controlling systems. Engineering systems interconnect sensing and computing devices over a shared bandwidth-limited channels, and therefore, estimation algorithms should strive to use bandwidth optimally. We present a notion of entropy for state estimation of switched nonlinear dynamical systems, an upper bound for it and a state estimation algorithm for the case when the switching signal is unobservable. Our approach relies on the notion of topological entropy and uses techniques from the theory for control under limited information. We show that the average bit rate used is optimal in the sense that, the eciency gap of the algorithm is within an additive constant of the gap between estimation entropy of the system and its known upper-bound. We apply the algorithm to two system models and discuss the performance implications of the number of tracked modes.

State estimation is a fundamental problem for monitoring and controlling systems. Engineering systems interconnect sensing and computing devices over a shared bandwidth-limited channels, and therefore, estimation algorithms should strive to use bandwidth optimally. We present a notion of entropy for state estimation of switched nonlinear dynamical systems, an upper bound for it and a state estimation algorithm for the case when the switching signal is unobservable. Our approach relies on the notion of topological entropy and uses techniques from the theory for control under limited information. We show that the average bit rate used is optimal in the sense that, the efficiency gap of the algorithm is within an additive constant of the gap between estimation entropy of the system and its known upper-bound. We apply the algorithm to two system models and discuss the performance implications of the number of tracked modes.

We present a modular technique for simulation-based bounded verification for nonlinear dynamical systems. We introduce the notion of input-to-state discrepancy of each subsystem Ai in a larger nonlinear dynamical system A which bounds the distance between two (possibly diverging) trajectories of Ai in terms of their initial states and inputs. Using the IS discrepancy functions, we construct a low dimensional deter- ministic dynamical system M (δ). For any two trajectories of A starting δ distance apart, we show that one of them bloated by a factor determined by the trajectory of M con- tains the other. Further, by choosing appropriately small δ’s the overapproximations computed by the above method can be made arbitrarily precise. Using the above results we de- velop a sound and relatively complete algorithm for bounded safety verification of nonlinear ODEs. Our preliminary ex- periments with a prototype implementation of the algorithm show that the approach can be effective for verification of nonlinear models.