Robust Distributed Averaging: When are Potential-Theoretic Strategies Optimal?

We study the interaction between a network designer and an adversary over a dynamical network. The network consists of nodes performing continuous-time distributed averaging. The adversary strategically disconnects a set of links to prevent the nodes from reaching consensus. Meanwhile, the network designer assists the nodes in reaching consensus by changing the weights of a limited number of links in the network. We formulate two Stackelberg games to describe this competition where the order in which the players act is reversed in the two problems. Although the canonical equations provided by the Pontryagin's maximum principle seem to be intractable, we provide an alternative characterization for the optimal strategies that makes connection to potential theory. Finally, we provide a sufficient condition for the existence of a saddle-point equilibrium for the underlying zero-sum game.Comment: 32 pages, 1 figure, submitted to IEEE Transactions on Automatic Control. arXiv admin note: text overlap with arXiv:1304.005

Secure Sensor Design Against Undetected Infiltration: Minimum Impact-Minimum Damage

We propose a new defense mechanism against undetected infiltration into controllers in cyber-physical systems. To this end, we cautiously design the outputs of the sensors that monitor the state of the system. Different from the defense mechanisms that seek to detect infiltration, the proposed approach seeks to minimize the damage of possible attacks before they have been detected. Controller of a cyber-physical system could have been infiltrated into by an undetected attacker at any time of the operation. Disregarding such a possibility and disclosing system's state without caution benefits the attacker in his/her malicious objective. Therefore, secure sensor design can improve the security of cyber-physical systems further when incorporated along with other defense mechanisms. We, specifically, consider a controlled Gauss-Markov process, where the controller could have been infiltrated into at any time within the system's operation. In the sense of game-theoretic hierarchical equilibrium, we provide a semi-definite programming based algorithm to compute the optimal linear secure sensor outputs and analyze the performance for various scenarios numerically.Comment: Submitted to the IEEE Transactions on Automatic Contro

Stability Structures of Conjunctive Boolean Networks

A Boolean network is a finite dynamical system, whose variables take values from a binary set. The value update rule for each variable is a Boolean function, depending on a selected subset of variables. Boolean networks have been widely used in modeling gene regulatory networks. We focus in this paper on a special class of Boolean networks, termed as conjunctive Boolean networks. A Boolean network is conjunctive if the associated value update rule is comprised of only AND operations. It is known that any trajectory of a finite dynamical system will enter a periodic orbit. We characterize in this paper all periodic orbits of a conjunctive Boolean network whose underlying graph is strongly connected. In particular, we establish a bijection between the set of periodic orbits and the set of binary necklaces of a certain length. We further investigate the stability of a periodic orbit. Specifically, we perturb a state in the periodic orbit by changing the value of a single entry of the state. The trajectory, with the perturbed state being the initial condition, will enter another (possibly the same) periodic orbit in finite time steps. We then provide a complete characterization of all such transitions from one periodic orbit to another. In particular, we construct a digraph, with the vertices being the periodic orbits, and the (directed) edges representing the transitions among the orbits. We call such a digraph the stability structure of the conjunctive Boolean network

Controllability of Conjunctive Boolean Networks with Application to Gene Regulation

A Boolean network is a finite state discrete time dynamical system. At each step, each variable takes a value from a binary set. The value update rule for each variable is a local function which depends only on a selected subset of variables. Boolean networks have been used in modeling gene regulatory networks. We focus in this paper on a special class of Boolean networks, namely the conjunctive Boolean networks (CBNs), whose value update rule is comprised of only logic AND operations. It is known that any trajectory of a Boolean network will enter a periodic orbit. Periodic orbits of a CBN have been completely understood. In this paper, we investigate the orbit-controllability and state-controllability of a CBN: We ask the question of how one can steer a CBN to enter any periodic orbit or to reach any final state, from any initial state. We establish necessary and sufficient conditions for a CBN to be orbit-controllable and state-controllable. Furthermore, explicit control laws are presented along the analysis

Distributed Discrete-time Optimization in Multi-agent Networks Using only Sign of Relative State

This paper proposes distributed discrete-time algorithms to cooperatively solve an additive cost optimization problem in multi-agent networks. The striking feature lies in the use of only the sign of relative state information between neighbors, which substantially differentiates our algorithms from others in the existing literature. We first interpret the proposed algorithms in terms of the penalty method in optimization theory and then perform non-asymptotic analysis to study convergence for static network graphs. Compared with the celebrated distributed subgradient algorithms, which however use the exact relative state information, the convergence speed is essentially not affected by the loss of information. We also study how introducing noise into the relative state information and randomly activated graphs affect the performance of our algorithms. Finally, we validate the theoretical results on a class of distributed quantile regression problems.Comment: Part of this work has been presented in American Control Conference (ACC) 2018, first version posted on arxiv on Sep. 2017, IEEE Transactions on Automatic Control, 201

Centralized Volatility Reduction for Electricity Markets

Increased penetration of wind energy will make electricity market prices more volatile. As a result, market participants will bear increased financial risks, which impacts investment decisions and in turn, makes it harder to achieve sustainable energy goals. As a remedy, in this paper, we propose an insurance market that complements any wholesale market design. Our mechanism can be run by any suitable financial entity such as an independent system operator, with the aim of reducing the financial effects of volatile prices. We provide theoretical guarantees, analytically characterize the outcomes over a copperplate power system example, and numerically explore the same for a modified IEEE 14-bus test system.Comment: arXiv admin note: text overlap with arXiv:1704.0036

Asymptotic Behavior of Conjunctive Boolean Networks Over Weakly Connected Digraphs

A conjunctive Boolean network (CBN) is a finite state dynamical system, whose variables take values from a binary set, and the value update rule for each variable is a Boolean function consisting only of logic AND operations. We investigate the asymptotic behavior of CBNs by computing their periodic orbits. When the underlying digraph is strongly connected, the periodic orbits of the associated CBN has been completely understood, one-to-one corresponding to binary necklaces of a certain length given by the loop number of the graph. We characterize in the paper the periodic orbits of CBNs over an arbitrary weakly connected digraphs. We establish, among other things, a new method to investigate their asymptotic behavior. Specifically, we introduce a graphical-approach, termed system reduction, which turns the underlying digraph into a special weakly connected digraph whose strongly connected components are all cycles. We show that the reduced system uniquely determines the asymptotic behavior of the original system. Moreover, we provide a constructive method for computing the periodic orbit of the reduced system, which the system will enter for a given but arbitrary initial condition

Graph-Theoretic Framework for Unified Analysis of Observability and Data Injection Attacks in the Smart Grid

In this paper, a novel graph-theoretic framework is proposed to generalize the analysis of a broad set of security attacks, including observability and data injection attacks, that target the state estimator of a smart grid. First, the notion of observability attacks is defined based on a proposed graph-theoretic construct. In this respect, a structured approach is proposed to characterize critical sets, whose removal renders the system unobservable. It is then shown that, for the system to be observable, these critical sets must be part of a maximum matching over a proposed bipartite graph. In addition, it is shown that stealthy data injection attacks (SDIAs) constitute a special case of these observability attacks. Then, various attack strategies and defense policies, for observability and data injection attacks, are shown to be amenable to analysis using the introduced graph-theoretic framework. The proposed framework is then shown to provide a unified basis for analysis of four key security problems (among others), pertaining to the characterization of: 1) The sparsest SDIA; 2) the sparsest SDIA including a certain measurement; 3) a set of measurements which must be defended to thwart all potential SDIAs; and 4) the set of measurements, which when protected, can thwart any SDIA whose cardinality is below a certain threshold. A case study using the IEEE 14-bus system with a set of 17 measurements is used to support the theoretical findings

Cash-settled options for wholesale electricity markets

Wholesale electricity market designs in practice do not provide the market participants with adequate mechanisms to hedge their financial risks. Demanders and suppliers will likely face even greater risks with the deepening penetration of variable renewable resources like wind and solar. This paper explores the design of a centralized cash-settled call option market to mitigate such risks. A cash-settled call option is a financial instrument that allows its holder the right to claim a monetary reward equal to the positive difference between the real-time price of an underlying commodity and a pre-negotiated strike price for an upfront fee. Through an example, we illustrate that a bilateral call option can reduce the payment volatility of market participants. Then, we design a centralized clearing mechanism for call options that generalizes the bilateral trade. We illustrate through an example how the centralized clearing mechanism generalizes the bilateral trade. Finally, the effect of risk preference of the market participants, as well as some generalizations are discussed.Comment: Proc. 20th IFAC World Congress (IFAC WC 2017), Toulouse, France, July 9-14, 2017 (accepted

Robust Distributed Averaging in Networks

In this work, we consider two types of adversarial attacks on a network of nodes seeking to reach consensus. The first type involves an adversary that is capable of breaking a specific number of links at each time instant. In the second attack, the adversary is capable of corrupting the values of the nodes by adding a noise signal. In this latter case, we assume that the adversary is constrained by a power budget. We consider the optimization problem of the adversary and fully characterize its optimum strategy for each scenario.Comment: This is an elaborated version of our paper: "Consensus in the presence of an adversary," in Proceedings of the 3rd IFAC Workshop on Distributed Estimation and Control in Networked Systems (NecSys), 201