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Synchronization of coupled oscillators: The taylor expansion of the inverse kuramoto map

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arXiv 2018年

作者： Huang, Elizabeth Y. Jafarpour, Saber Bullo, Francesco Department of Mechanical Engineering Center of Control Dynamical Systems and Computation University of California Santa Barbara93106-5070 United States

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. Copyright © 2018, The Authors. All rights reserved.

关键词： Synchronization

Graph-theoretic small-gain theorems for metzler matrices and monotone systems

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arXiv 2019年

作者： Duan, Xiaoming Jafarpour, Saber Bullo, Francesco Mechanical Engineering Department Center of Control Dynamical Systems and Computation UC Santa BarbaraCA93106-5070 United States

This paper studies the graph-theoretic conditions for stability of positive monotone systems. Using concepts from the input-to-state stability and network small-gain theory, we first establish necessary and sufficient conditions for the stability of linear positive systems described by Metzler matrices. Specifically, we define and compute two forms of input-to-state stability gains for Metzler systems, namely max-interconnection gains and sum-interconnection gains. Then, based on the max-interconnection gains, we show that the cyclic small-gain theorem becomes necessary and sufficient for the stability of Metzler systems;based on the sum-interconnection gains, we obtain novel graph-theoretic conditions for the stability of Metzler systems. All these conditions highlight the role of cycles in the interconnection graph and unveil how the structural properties of the graph affect stability. Finally, we extend our results to the nonlinear monotone system and obtain similar sufficient conditions for global asymptotic stability. Copyright © 2019, The Authors. All rights reserved.

关键词： Graph theory

Synchronization of Kuramoto Oscillators via Cutset Projections

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arXiv 2017年

作者： Jafarpour, Saber Bullo, Francesco Mechanical Engineering Department Center of Control Dynamical-Systems and Computation UC Santa BarbaraCA93106-5070 United States

—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 ∞-norm. We conclude with a comparative study of our novel ∞-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. Copyright © 2017, The Authors. All rights reserved.

关键词： Synchronization

A Tale of Two Doses: Model Identification and Optimal Vaccination for COVID-19

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A Tale of Two Doses: Model Identification and Optimal Vaccin...

IEEE Conference on Decision and control

作者： Raphael Chinchilla Guosong Yang Murat K. Erdal Ramon R. Costa João P. Hespanha Center for Control Dynamical Systems and Computation University of California Santa Barbara USA Federal University of Rio de Janeiro Brazil

ISBN: (纸本)9781665436601

We address the model identification and the computation of optimal vaccination policies for the coronavirus disease 2019 (COVID-19). We consider a stochastic Susceptible– Infected–Removed (SIR) model that captures the effect of multiple vaccine treatments, each requiring a different number of doses and providing different levels of protection against the disease. We show that the inclusion of vaccination data enables the estimation of the state of the model and key model parameters that are otherwise not identifiable. This estimates can, in turn, be used to design strategic approaches to vaccination that aim at minimizing the number of deaths and the economic cost of the disease. We illustrate these results with numerical examples.

关键词： COVID-19 Costs computational modeling Conferences Stochastic processes Estimation Data models

Non-Euclidean Contraction Theory for Robust Nonlinear Stability

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arXiv 2021年

作者： Davydov, Alexander Jafarpour, Saber Bullo, Francesco The Department of Mechanical Engineering The Center for Control Dynamical Systems and Computation University of California Santa Barbara93106-5070 United States

We study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and characterize their properties. We introduce and study the sign and max pairings for the `1 and `∞ norms, respectively. Using weak pairings, we establish five equivalent characterizations for contraction, including the one-sided Lipschitz condition for the vector field as well as logarithmic norm and Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to an equilibrium. Finally, as an application, we provide (i) incremental input-to-state stability and finite input-state gain properties for contracting systems, and (ii) a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system. Copyright © 2021, The Authors. All rights reserved.

关键词： dynamical systems

Markov chains with maximum return time entropy for robotic surveillance

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arXiv 2018年

作者： Duan, Xiaoming George, Mishel Bullo, Francesco Mechanical Engineering Department Center of Control Dynamical Systems and Computation UC Santa BarbaraCA93106-5070 United States

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. Copyright © 2018, The Authors. All rights reserved.

关键词： Linear systems

Topology inference with multivariate cumulants: The Möbius Inference Algorithm

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arXiv 2020年

作者： Smith, Kevin D. Jafarpour, Saber Swami, Ananthram Bullo, Francesco Center of Control Dynamical Systems and Computation UC Santa Barbara CA93106-5070 United States Army Research Laboratory

Many tasks regarding the monitoring, management, and design of communication networks rely on knowledge of the routing topology. However, the standard approach to topology mapping-namely, active probing with traceroutes-relies on cooperation from increasingly non-cooperative routers, leading to missing information. Network tomography, which uses end-to-end measurements of additive link metrics (like delays or log packet loss rates) across monitor paths, is a possible remedy. Network tomography does not require that routers cooperate with traceroute probes, and it has already been used to infer the structure of multicast trees. This paper goes a step further. We provide a tomographic method to infer the underlying routing topology of an arbitrary set of monitor paths using the joint distribution of end-to-end measurements, without making any assumptions on routing behavior. Our approach, called the Möbius Inference Algorithm (MIA), uses cumulants of this distribution to quantify high-order interactions among monitor paths, and it applies Möbius inversion to "disentangle" these interactions. In addition to MIA, we provide a more practical variant called Sparse Möbius Inference, which uses various sparsity heuristics to reduce the number and order of cumulants required to be estimated. We show the viability of our approach using synthetic case studies based on real-world ISP topologies. Copyright © 2020, The Authors. All rights reserved.

关键词： Topology

Voltage Stabilization in Microgrids via Quadratic Droop control

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Voltage Stabilization in Microgrids via Quadratic Droop Cont...

IEEE Annual Conference on Decision and control

作者： John W. Simpson-Porco Florian Dorfler Francesco Bullo Center for Control Dynamical Systems and Computation University of California Santa Barbara Department of Electrical Engineering University of California Los Angeles

ISBN: (纸本)9781467357159

Motivated by the growing interest in energy technology and smart grid architectures, we consider the problem of voltage stability and reactive power balancing in low-voltage electrical networks equipped with DC/AC inverters ("microgrids"). It is generally believed that high-voltage equilibria of such networks are stable, but the locations of these equilibria are unknown, as is the critical network load where stability is lost. Inspired by the "control by interconnection" paradigm developed for port-Hamiltonian systems, we propose a novel droop-like inverter controller which is quadratic in the local voltage magnitude. Remarkably, under this controller the closed-loop network is again a well-posed electrical circuit. We find that the equilibria of the quadratic droop-controlled network are in exact correspondence with the solutions of a reduced power flow equation. For general network topologies, we study some simple yet insightful solutions of this equation, and for the frequently-encountered case of a parallel microgrid, we present a concise and closed-form condition for the existence of an exponentially stable high-voltage network equilibrium. Our condition establishes the existence of a critical inductive load for the network, which depends only on the network topology, admittances, and controller gains. We compare and contrast our design with the conventional droop controller, investigate the relationship between the two, and validate the robustness of our design through simulation.

关键词： Inverters Microgrids Equations Voltage control Mathematical model Vectors Reactive power

Stochastic strategies for robotic surveillance as stackelberg games

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arXiv 2020年

作者： Duan, Xiaoming Paccagnan, Dario Bullo, Francesco Mechanical Engineering Department the Center of Control Dynamical Systems and Computation UC Santa BarbaraCA93106-5070 United States

This paper studies a stochastic robotic surveillance problem where a mobile robot moves randomly on a graph to capture a potential intruder that strategically attacks a location on the graph. The intruder is assumed to be omniscient: it knows the current location of the mobile agent and can learn the surveillance strategy. The goal for the mobile robot is to design a stochastic strategy so as to maximize the probability of capturing the intruder. We model the strategic interactions between the surveillance robot and the intruder as a Stackelberg game, and optimal and suboptimal Markov chain based surveillance strategies in star, complete and line graphs are studied. We first derive a universal upper bound on the capture probability, i.e., the performance limit for the surveillance agent. We show that this upper bound is tight in the complete graph and further provide suboptimality guarantees for a natural design. For the star and line graphs, we first characterize dominant strategies for the surveillance agent and the intruder. Then, we rigorously prove the optimal strategy for the surveillance agent. Copyright © 2020, The Authors. All rights reserved.

关键词： Stochastic systems