Shaunak D. Bopardikar

Shaunak D. Bopardikar, Dhagash Mehta, Jonathan D. Hauenstein

We revisit the static coverage control problem for placement of vehicles with simple motion on the real line, under the assumption that the cost is a polynomial function of the locations of the vehicles. The main contribution of this paper is to demonstrate the use of tools from numerical algebraic geometry, in particular, a numerical polynomial homotopy continuation method that guarantees to find all solutions of polynomial equations, in order to characterize the \emph{global minima} for the coverage control problem. The results are then compared against a classic distributed approach involving the use of Lloyd descent, which is known to converge only to a local minimum under certain technical conditions.

Shaunak D. Bopardikar

Given a linear dynamical system, we consider the problem of constructing an approximate system using only a subset of the sensors out of the total set such that the observability Gramian of the new system is approximately equal to that of the original system. Our contributions are as follows. First, we present a randomized algorithm that samples the sensors with replacement as per specified distributions. For specific metrics of the observability Gramian such as the trace or the maximum eigenvalue, we derive novel bounds on the number of samples required to yield a high probability lower bound on the metric evaluated on the approximate Gramian. Second, with a different distribution, we derive high probability bounds on other standard metrics used in sensor selection, including the minimum eigenvalue or the trace of the Gramian inverse. This distribution requires a number of samples which is larger than the one required for the trace and the maximum eigenvalue, but guarantees non-singularity of the approximate Gramian if the original system is observable with high probability. Third, we demonstrate how the randomized procedure can be used for recursive state estimation using fewer sensors than the original system and provide a high probability upper bound on the initial error covariance. We supplement the theoretical results with several insightful numerical studies and comparisons with competing greedy approaches.

Prajakta Surve, Shaunak D. Bopardikar, Daigo Shishika et al.

We consider a hide-and-seek game between a Hider and a Seeker over a finite set of locations. The Hider chooses one location to conceal a stationary treasure, while the Seeker visits the locations sequentially along a route. As the search progresses, the Hider observes a prefix of the Seeker's route. After observing this information, the Hider has the option to relocate the treasure at most once to another unvisited location by paying a switching cost. We study two seeker models. In the first, the Seeker is unaware of the fact that the Hider can relocate. In the second, the Seeker select its route while accounting for the possibility that the Hider observes its path and reallocates. For the restricted case, we define the value-of-information created by the reveal and derive upper bounds in terms of the switching cost using a worst-case evaluation over routes. We also show that seeker awareness reduces the game value, with the difference between the restricted and feedback models bounded by the entry-wise gap between the corresponding payoff matrices. Numerical examples show how this benefit decreases as the switching cost increases and as the reveal occurs later along the route.

Dinesh Patra, Prajakta Surve, Ashish R. Hota et al.

We consider a pursuit-evasion scenario involving a group of pursuers and a single evader in a two-dimensional unbounded environment. The pursuers aim to capture the evader in finite time while ensuring the evader remains enclosed within the convex hull of their positions until capture, without knowledge of the evader's heading angle. Prior works have addressed the problem of encirclement and capture separately in different contexts. In this paper, we present a class of strategies for the pursuers that guarantee capture in finite time while maintaining encirclement, irrespective of the evader's strategy. Furthermore, we derive an upper bound on the time to capture. Numerical results highlight the effectiveness of the proposed framework against a range of evader strategies.

Arnab Bhattacharya, Thiagarajan Ramachandran, Sandeep Banik et al.

Adversary emulation is an offensive exercise that provides a comprehensive assessment of a system's resilience against cyber attacks. However, adversary emulation is typically a manual process, making it costly and hard to deploy in cyber-physical systems (CPS) with complex dynamics, vulnerabilities, and operational uncertainties. In this paper, we develop an automated, domain-aware approach to adversary emulation for CPS. We formulate a Markov Decision Process (MDP) model to determine an optimal attack sequence over a hybrid attack graph with cyber (discrete) and physical (continuous) components and related physical dynamics. We apply model-based and model-free reinforcement learning (RL) methods to solve the discrete-continuous MDP in a tractable fashion. As a baseline, we also develop a greedy attack algorithm and compare it with the RL procedures. We summarize our findings through a numerical study on sensor deception attacks in buildings to compare the performance and solution quality of the proposed algorithms.

Sandeep Banik, Shaunak D. Bopardikar

We consider the classic motion planning problem defined over a roadmap in which a vehicle seeks to find an optimal path from a source to a destination in presence of an attacker who can launch attacks on the vehicle over any edge of the roadmap. The vehicle (defender) has the capability to switch on/off a countermeasure that can detect and permanently disable the attack if it occurs concurrently. We model the problem of traveling along en edge using the framework of a simultaneous zero-sum dynamic game (edge-game) with a stopping state played between an attacker and defender. We characterize the Nash equiliria of an edge-game and provide closed form expressions for two actions per player. We further provide an analytic and approximate expression on the value of an edge-game and characterize conditions under which it grows sub-linearly with the number of stages. We study the sensitivity of Nash equilibrium to the (i) cost of using the countermeasure, (ii) cost of motion and (iii) benefit of disabling the attack. The solution of an edge-game is used to formulate and solve for the secure planning problem known as a meta-game. We design an efficient heuristic by converting the problem to a shortest path problem using the edge cost as the solution of corresponding edge-games. We illustrate our findings through several insightful simulations.

Shaunak D. Bopardikar, George S. Eskander Ekladious

This paper presents a sequential randomized lowrank matrix factorization approach for incrementally predicting values of an unknown function at test points using the Gaussian Processes framework. It is well-known that in the Gaussian processes framework, the computational bottlenecks are the inversion of the (regularized) kernel matrix and the computation of the hyper-parameters defining the kernel. The main contributions of this paper are two-fold. First, we formalize an approach to compute the inverse of the kernel matrix using randomized matrix factorization algorithms in a streaming scenario, i.e., data is generated incrementally over time. The metrics of accuracy and computational efficiency of the proposed method are compared against a batch approach based on use of randomized matrix factorization and an existing streaming approach based on approximating the Gaussian process by a finite set of basis vectors. Second, we extend the sequential factorization approach to a class of kernel functions for which the hyperparameters can be efficiently optimized. All results are demonstrated on two publicly available datasets.

Alberto Speranzon, Shaunak D. Bopardikar

In this paper, we cast the classic problem of achieving k-anonymity for a given database as a problem in algebraic topology. Using techniques from this field of mathematics, we propose a framework for k-anonymity that brings new insights and algorithms to anonymize a database. We begin by addressing the simpler case when the data lies in a metric space. This case is instrumental to introduce the main ideas and notation. Specifically, by mapping a database to the Euclidean space and by considering the distance between datapoints, we introduce a simplicial representation of the data and show how concepts from algebraic topology, such as the nerve complex and persistent homology, can be applied to efficiently obtain the entire spectrum of k-anonymity of the database for various values of k and levels of generalization. For this representation, we provide an analytic characterization of conditions under which a given representation of the dataset is k-anonymous. We introduce a weighted barcode diagram which, in this context, becomes a computational tool to tradeoff data anonymity with data loss expressed as level of generalization. Some simulations results are used to illustrate the main idea of the paper. We conclude the paper with a discussion on how to extend this method to address the general case of a mix of categorical and metric data.

Shaunak D. Bopardikar, Brendan J. Englot, Alberto Speranzon

In this paper, we extend the recent body of work on planning under uncertainty to include the fact that sensors may not provide any measurement owing to misdetection. This is caused either by adverse environmental conditions that prevent the sensors from making measurements or by the fundamental limitations of the sensors. Examples include RF-based ranging devices that intermittently do not receive the signal from beacons because of obstacles; the misdetection of features by a camera system in detrimental lighting conditions; a LIDAR sensor that is pointed at a glass-based material such as a window, etc. The main contribution of this paper is twofold. We first show that it is possible to obtain an analytical bound on the performance of a state estimator under sensor misdetection occurring stochastically over time in the environment. We then show how this bound can be used in a sample-based path planning algorithm to produce a path that trades off accuracy and robustness. Computational results demonstrate the benefit of the approach and comparisons are made with the state of the art in path planning under state uncertainty.