arXiv subjects

Francesco Bullo

Publications and source records attributed to Francesco Bullo.

At least 127 records · Page 7Linked to original sources

arXiv2018

Synchronization of Coupled Oscillators: The Taylor Expansion of the Inverse Kuramoto Map

Synchronization in the networks of coupled oscillators is a widely studied topic in different areas. It is well-known that synchronization occurs if the connectivity of the network dominates heterogeneity of the oscillators. Despite extensive study on this topic, the quest for sharp closed-form synchronization tests is still in vain. In this paper, we present an algorithm for finding the Taylor expansion of the inverse Kuramoto map. We show that this Taylor series can be used to obtain a hierarchy of increasingly accurate approximate tests with low computational complexity. These approximate tests are then used to estimate the threshold of synchronization as well as the position of the synchronization manifold of the network.

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arXiv2018

Linear Stochastic Approximation Algorithms and Group Consensus over Random Signed Networks: A Technical Report with All Proofs

This paper studies linear stochastic approximation (SA) algorithms and their application to multi-agent systems in engineering and sociology. As main contribution, we provide necessary and sufficient conditions for convergence of linear SA algorithms to a deterministic or random final vector. We also characterize the system convergence rate, when the system is convergent. Moreover, differing from non-negative gain functions in traditional SA algorithms, this paper considers also the case when the gain functions are allowed to take arbitrary real numbers. Using our general treatment, we provide necessary and sufficient conditions to reach consensus and group consensus for first-order discrete-time multi-agent system over random signed networks and with state-dependent noise. Finally, we extend our results to the setting of multi-dimensional linear SA algorithms and characterize the behavior of the multi-dimensional Friedkin-Johnsen model over random interaction networks.

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arXiv2018

Synchronization of Kuramoto Oscillators via Cutset Projections

Synchronization in coupled oscillators networks is a remarkable phenomenon of relevance in numerous fields. For Kuramoto oscillators the loss of synchronization is determined by a trade-off between coupling strength and oscillator heterogeneity. Despite extensive prior work, the existing sufficient conditions for synchronization are either very conservative or heuristic and approximate. Using a novel cutset projection operator, we propose a new family of sufficient synchronization conditions; these conditions rigorously identify the correct functional form of the trade-off between coupling strength and oscillator heterogeneity. To overcome the need to solve a nonconvex optimization problem, we then provide two explicit bounding methods, thereby obtaining (i) the best-known sufficient condition for unweighted graphs based on the 2-norm, and (ii) the first-known generally-applicable sufficient condition based on the $\infty$-norm. We conclude with a comparative study of our novel $\infty$-norm condition for specific topologies and IEEE test cases; for most IEEE test cases our new sufficient condition is one to two orders of magnitude more accurate than previous rigorous tests.

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arXiv2018

The Meeting Time of Multiple Random Walks

This article rigorously analyzes the meeting time between pursuers and evaders performing random walks on digraphs. There exist several bounds on the expected meeting time between random walkers on graphs in the literature, however, closed-form expressions are limited in scope. By utilizing the notion that multiple random walks on a common graph can be understood as a single random walk on the Kronecker product graph, we are able to provide the first analytic expression for the meeting time in terms of the transition matrices of the random walkers when modeled by either discrete-time Markov chains or continuous-time Markov chains. We further extend the results to the case of multiple pursuers and multiple evaders performing independent random walks. We present various sufficient conditions for pairs (or tuples) of transition matrices that satisfy certain conditions on the absorbing classes for which finite meeting times are guaranteed to exist.

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arXiv2018

Markov Chains with Maximum Return Time Entropy for Robotic Surveillance

Motivated by robotic surveillance applications, this paper studies the novel problem of maximizing the return time entropy of a Markov chain, subject to a graph topology with travel times and stationary distribution. The return time entropy is the weighted average, over all graph nodes, of the entropy of the first return times of the Markov chain; this objective function is a function series that does not admit in general a closed form. The paper features theoretical and computational contributions. First, we obtain a discrete-time delayed linear system for the return time probability distribution and establish its convergence properties. We show that the objective function is continuous over a compact set and therefore admits a global maximum; a unique globally-optimal solution is known only for complete graphs with unitary travel times. We then establish upper and lower bounds between the return time entropy and the well-known entropy rate of the Markov chain. To compute the optimal Markov chain numerically, we establish the asymptotic equality between entropy, conditional entropy and truncated entropy, and propose an iteration to compute the gradient of the truncated entropy. Finally, we apply these results to the robotic surveillance problem. Our numerical results show that, for a model of rational intruder over prototypical graph topologies and test cases, the maximum return time entropy chain performs better than several existing Markov chains.

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arXiv2017

Opinion Dynamics and Social Power Evolution: A Single-Timescale Model

This paper studies the evolution of self-appraisal and social power, for a group of individuals who discuss and form opinions. We consider a modification of the recently proposed DeGroot-Friedkin (DF) model, in which the opinion formation process takes place on the same timescale as the reflected appraisal process; we call this new model the single-timescale DF model. We provide a comprehensive analysis of the equilibria and convergence properties of the model for the settings of irreducible and reducible influence networks. For the setting of irreducible influence networks, the single-timescale DF model has the same behavior as the original DF model, that is, it predicts among other things that the social power ranking among individuals is asymptotically equal to their centrality ranking, that social power tends to accumulate at the top of the centrality ranking hierarchy, and that an autocratic (resp., democratic) power structure arises when the centrality scores are maximally nonuniform (resp., uniform). For the setting of reducible influence networks, the single-timescale DF model behaves differently from the original DF model in two ways. First, an individual, who corresponds to a reducible node in a reducible influence network, can keep all social power in the single-timescale DF model if the initial condition does so, whereas its social power asymptotically vanishes in the original DF model. Second, when the associated network has multiple sinks, the two models behave very differently: the original DF model has a single globally-attractive equilibrium, whereas any partition of social power among the sinks is allowable at equilibrium in the single-timescale DF model.

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arXiv2017

Asynchronous and Dynamic Coverage Control Scheme for Persistent Surveillance Missions

A decomposition-based coverage control scheme is proposed for multi-agent, persistent surveillance missions operating in a communication-constrained, dynamic environment. The proposed approach decouples high-level task assignment from low-level motion planning in a modular framework. Coverage assignments and surveillance parameters are managed by a central base station, and transmitted to mobile agents via unplanned and asynchronous exchanges. Coverage updates promote load balancing, while maintaining geometric and temporal characteristics that allow effective pairing with generic path planners. Namely, the proposed scheme guarantees that (i) coverage regions are connected and collectively cover the environment, (ii) subregions may only go uncovered for bounded periods of time, (iii) collisions (or sensing overlaps) are inherently avoided, and (iv) under static event likelihoods, the collective coverage regions converge to a Pareto-optimal configuration. This management scheme is then paired with a generic path planner satisfying loose assumptions. The scheme is illustrated through simulated surveillance missions.

cs.MA↗

arXiv2017

On the Dynamics of Deterministic Epidemic Propagation over Networks

In this work we review a class of deterministic nonlinear models for the propagation of infectious diseases over contact networks with strongly-connected topologies. We consider network models for susceptible-infected (SI), susceptible-infected-susceptible (SIS), and susceptible-infected-recovered (SIR) settings. In each setting, we provide a comprehensive nonlinear analysis of equilibria, stability properties, convergence, monotonicity, positivity, and threshold conditions. For the network SI setting, specific contributions include establishing its equilibria, stability, and positivity properties. For the network SIS setting, we review a well-known deterministic model, provide novel results on the computation and characterization of the endemic state (when the system is above the epidemic threshold), and present alternative proofs for some of its properties. Finally, for the network SIR setting, we propose novel results for transient behavior, threshold conditions, stability properties, and asymptotic convergence. These results are analogous to those well-known for the scalar case. In addition, we provide a novel iterative algorithm to compute the asymptotic state of the network SIR system.

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arXiv2016

Dynamic Models of Appraisal Networks Explaining Collective Learning

This paper proposes models of learning process in teams of individuals who collectively execute a sequence of tasks and whose actions are determined by individual skill levels and networks of interpersonal appraisals and influence. The closely-related proposed models have increasing complexity, starting with a centralized manager-based assignment and learning model, and finishing with a social model of interpersonal appraisal, assignments, learning, and influences. We show how rational optimal behavior arises along the task sequence for each model, and discuss conditions of suboptimality. Our models are grounded in replicator dynamics from evolutionary games, influence networks from mathematical sociology, and transactive memory systems from organization science.

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arXiv2016

Competitive Propagation: Models, Asymptotic Behavior and Multi-stage Games

In this paper we propose a class of propagation models for multiple competing products over a social network. We consider two propagation mechanisms: social conversion and self conversion, corresponding, respectively, to endogenous and exogenous factors. A novel concept, the product-conversion graph, is proposed to characterize the interplay among competing products. According to the chronological order of social and self conversions, we develop two Markov-chain models and, based on the independence approximation, we approximate them with two respective difference equations systems. Theoretical analysis on these two approximation models reveals the dependency of the systems' asymptotic behavior on the structures of both the product-conversion graph and the social network, as well as the initial condition. In addition to the theoretical work, accuracy of the independence approximation and the asymptotic behavior of the Markov-chain model are investigated via numerical analysis, for the case where social conversion occurs before self conversion. Finally, we propose a class of multi-player and multi-stage competitive propagation games and discuss the seeding-quality trade-off, as well as the allocation of seeding resources among the individuals. We investigate the unique Nash equilibrium at each stage and analyze the system's behavior when every player is adopting the policy at the Nash equilibrium.

cs.SI↗

arXiv2016

Distributed Monitoring of Voltage Collapse Sensitivity Indices

The assessment of voltage stability margins is a promising direction for wide-area monitoring systems. Accurate monitoring architectures for long-term voltage instability are typically centralized and lack scalability, while completely decentralized approaches relying on local measurements tend towards inaccuracy. Here we present distributed linear algorithms for the online computation of voltage collapse sensitivity indices. The computations are collectively performed by processors embedded at each bus in the smart grid, using synchronized phasor measurements and communication of voltage phasors between neighboring buses. Our algorithms provably converge to the proper index values, as would be calculated using centralized information, but but do not require any central decision maker for coordination. Modifications of the algorithms to account for generator reactive power limits are discussed. We illustrate the effectiveness of our designs with a case study of the New England 39 bus system.

math.OC↗

arXiv2016

Voltage stress minimization by optimal reactive power control

A standard operational requirement in power systems is that the voltage magnitudes lie within prespecified bounds. Conventional engineering wisdom suggests that such a tightly-regulated profile, imposed for system design purposes and good operation of the network, should also guarantee a secure system, operating far from static bifurcation instabilities such as voltage collapse. In general however, these two objectives are distinct and must be separately enforced. We formulate an optimization problem which maximizes the distance to voltage collapse through injections of reactive power, subject to power flow and operational voltage constraints. By exploiting a linear approximation of the power flow equations we arrive at a convex reformulation which can be efficiently solved for the optimal injections. We also address the planning problem of allocating the resources by recasting our problem in a sparsity-promoting framework that allows us to choose a desired trade-off between optimality of injections and the number of required actuators. Finally, we present a distributed algorithm to solve the optimization problem, showing that it can be implemented on-line as a feedback controller. We illustrate the performance of our results with the IEEE30 bus network.

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arXiv2016

Voltage Stabilization in Microgrids via Quadratic Droop Control

We consider the problem of voltage stability and reactive power balancing in islanded small-scale electrical networks outfitted with DC/AC inverters ("microgrids"). A droop-like voltage feedback controller is proposed which is quadratic in the local voltage magnitude, allowing for the application of circuit-theoretic analysis techniques to the closed-loop system. The operating points of the closed-loop microgrid are in exact correspondence with the solutions of a reduced power flow equation, and we provide explicit solutions and small-signal stability analyses under several static and dynamic load models. Controller optimality is characterized as follows: we show a one-to-one correspondence between the high-voltage equilibrium of the microgrid under quadratic droop control, and the solution of an optimization problem which minimizes a trade-off between reactive power dissipation and voltage deviations. Power sharing performance of the controller is characterized as a function of the controller gains, network topology, and parameters. Perhaps surprisingly, proportional sharing of the total load between inverters is achieved in the low-gain limit, independent of the circuit topology or reactances. All results hold for arbitrary grid topologies, with arbitrary numbers of inverters and loads. Numerical results confirm the robustness of the controller to unmodeled dynamics.

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arXiv2015

Secondary Frequency and Voltage Control of Islanded Microgrids via Distributed Averaging

In this work we present new distributed controllers for secondary frequency and voltage control in islanded microgrids. Inspired by techniques from cooperative control, the proposed controllers use localized information and nearest-neighbor communication to collectively perform secondary control actions. The frequency controller rapidly regulates the microgrid frequency to its nominal value while maintaining active power sharing among the distributed generators. Tuning of the voltage controller provides a simple and intuitive trade-off between the conflicting goals of voltage regulation and reactive power sharing. Our designs require no knowledge of the microgrid topology, impedances or loads. The distributed architecture allows for flexibility and redundancy, and eliminates the need for a central microgrid controller. We provide a voltage stability analysis and present extensive experimental results validating our designs, verifying robust performance under communication failure and during plug-and-play operation.

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arXiv2015

On Resistive Networks of Constant Power Devices

This brief examines the behavior of DC circuits comprised of resistively interconnected constant power devices, as may arise in DC microgrids containing micro-sources and constant power loads. We derive a sufficient condition for all operating points of the circuit to lie in a desirable set, where the average nodal voltage level is high and nodal voltages are tightly clustered near one another. Our condition has the elegant physical interpretation that the ratio of resistive losses to total injected power should be small compared to a measure of network heterogeneity, as quantified by a ratio of conductance matrix eigenvalues. Perhaps surprisingly, the interplay between the circuit topology, branch conductances and the constant power devices implicitly defines a nominal voltage level for the circuit, despite the explicit absence of voltage-regulated nodes.

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arXiv2014

Breaking the Hierarchy: Distributed Control & Economic Optimality in Microgrids

Modeled after the hierarchical control architecture of power transmission systems, a layering of primary, secondary, and tertiary control has become the standard operation paradigm for islanded microgrids. Despite this superficial similarity, the control objectives in microgrids across these three layers are varied and ambitious, and they must be achieved while allowing for robust plug-and-play operation and maximal flexibility, without hierarchical decision making and time-scale separations. In this work, we explore control strategies for these three layers and illuminate some possibly-unexpected connections and dependencies among them. Building from a first-principle analysis of decentralized primary droop control, we study centralized, decentralized, and distributed architectures for secondary frequency regulation. We find that averaging-based distributed controllers using communication among the generation units offer the best combination of flexibility and performance. We further leverage these results to study constrained AC economic dispatch in a tertiary control layer. Surprisingly, we show that the minimizers of the economic dispatch problem are in one-to-one correspondence with the set of steady-states reachable by droop control. In other words, the adoption of droop control is necessary and sufficient to achieve economic optimization. This equivalence results in simple guidelines to select the droop coefficients, which include the known criteria for power sharing. We illustrate the performance and robustness of our designs through simulations.

math.OC↗

arXiv2014

Controllability Metrics, Limitations and Algorithms for Complex Networks

This paper studies the problem of controlling complex networks, that is, the joint problem of selecting a set of control nodes and of designing a control input to steer a network to a target state. For this problem (i) we propose a metric to quantify the difficulty of the control problem as a function of the required control energy, (ii) we derive bounds based on the system dynamics (network topology and weights) to characterize the tradeoff between the control energy and the number of control nodes, and (iii) we propose an open-loop control strategy with performance guarantees. In our strategy we select control nodes by relying on network partitioning, and we design the control input by leveraging optimal and distributed control techniques. Our findings show several control limitations and properties. For instance, for Schur stable and symmetric networks: (i) if the number of control nodes is constant, then the control energy increases exponentially with the number of network nodes, (ii) if the number of control nodes is a fixed fraction of the network nodes, then certain networks can be controlled with constant energy independently of the network dimension, and (iii) clustered networks may be easier to control because, for sufficiently many control nodes, the control energy depends only on the controllability properties of the clusters and on their coupling strength. We validate our results with examples from power networks, social networks, and epidemics spreading.

eess.SY↗

arXiv2013

Sparsity-Promoting Optimal Wide-Area Control of Power Networks

Inter-area oscillations in bulk power systems are typically poorly controllable by means of local decentralized control. Recent research efforts have been aimed at developing wide- area control strategies that involve communication of remote signals. In conventional wide-area control, the control structure is fixed a priori typically based on modal criteria. In contrast, here we employ the recently-introduced paradigm of sparsity- promoting optimal control to simultaneously identify the optimal control structure and optimize the closed-loop performance. To induce a sparse control architecture, we regularize the standard quadratic performance index with an l1-penalty on the feedback matrix. The quadratic objective functions are inspired by the classic slow coherency theory and are aimed at imitating homogeneous networks without inter-area oscillations. We use the New England power grid model to demonstrate that the proposed combination of the sparsity-promoting control design with the slow coherency objectives performs almost as well as the optimal centralized control while only making use of a single wide-area communication link. In addition to this nominal performance, we also demonstrate that our control strategy yields favorable robustness margins and that it can be used to identify a sparse control architecture for control design via alternative means.

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