## Research ProjectsMy research centers around the design and analysis of algorithms that facilitate our understanding and ability to influence the behavior of network systems. I design algorithms that make use of network structure and the local interactions between components in order to predict, control, and guarantee of the safe and reliable operation of the aggregate system in a distributed way that requires no centralized coordination. Spatially distributed models are fundamental to the study of complex systems. Diffusively coupled networks, in which different components of a network adjust their behavior according to the local differences between themselves and their neighbors, are a ubiquitous class of spatially-distributed models. Employing the tools of convex optimization, nonlinear dynamical systems theory, machine learning, and numerical computation allows us to characterize and control network systems in a variety of contexts, including multi-aircraft formation control, microelectronics, cellular reaction-diffusion networks, and social networks. ## Adaptive synchronization in networks
Our approach is well-suited to several applications. For example, it can be used as an efficient distributed way to determine community structure in dynamic networks. Another potential application is image segmentation through identification of clusters in the graph structure underlying the image. Our approach can furthermore be applied to multi-vehicle synchronized path following and oscillator clock locking in electro-mechanical circuits, among other cooperative control problems. ## ReferencesS. Y. Shafi and M. Arcak. An adaptive algorithm for synchronization in diffusively-coupled systems, ## Synchronization of diffusively coupled limit cycle oscillators
## ReferencesS. Y. Shafi, M. Arcak, M. Jovanovic, and A. Packard. Synchronization of diffusively coupled limit cycle oscillators, Accepted, S. Y. Shafi, M. Arcak, and M. Jovanovic. “Synchronization of limit cycle oscillations in diffusively-coupled systems,” in ## Spatial uniformity in diffusively-coupled systems using weighted L_2 norm contractions
## ReferencesS. Y. Shafi*, Z. Aminzare*, M. Arcak, and E. Sontag. “Weighted L_2 norm contractions in diffusively-coupled systems,” S. Y. Shafi*, Z. Aminzare*, M. Arcak, and E. Sontag. “Spatial uniformity in diffusively-coupled systems using weighted L_2 norm contractions,” in Z. Aminzare*, S. Y. Shafi*, M. Arcak, and E. Sontag. Guaranteeing spatial uniformity in reaction-diffusion systems using weighted L_2 norm contractions, S. Y. Shafi. Guaranteeing spatial uniformity in diffusively-coupled systems, preprint on arXiv, 2012. ## Distributed Control of Multiagent Systems
## ReferencesS. Y. Shafi, M. Arcak, and L. El Ghaoui, Graph weight allocation to meet Laplacian spectral constraints, in S. Y. Shafi, M. Arcak, and L. El Ghaoui, “Graph weight design for Laplacian eigenvalue constraints with multi-agent systems applications,” in S. Y. Shafi, M. Arcak, and L. El Ghaoui, “Designing node and edge weights of a graph to meet laplacian eigenvalue constraints,” in ## Selected Course Projects## Distributed Approaches to Cooperative and Non-Cooperative Network Resource Sharing Problems using Simultaneous Primal-Dual DescentWe study network resource sharing problems. The aim of this project is twofold, and spans distributed optimization and game theory. First, it generalizes the network utility maximization problem to arbitrary inequality-constrained convex sets, and addresses the situation where individual agent costs are directly affected by the actions of others. We assume that the system objective is to minimize the sum of the individual loss functions. Each agent solves a decoupled problem to determine its resource use given the prices determined by the network. We show that the problem can be decomposed and expressed as continuous-time dynamical subsystems that share a common input-output structure. We show how the input output structure can be exploited using Lyapunov stability theory to prove asymptotic convergence to the equilibrium under use of primal-dual gradient descent methods. Our approach is similar to the classical Arrow-Hurwicz-Uzawa algorithm for resource allocation using saddle point formulations. ## Numerical Schemes from the Perspective of Consensus: Exploring Connections between Agreement Problems and PDEsWe examine the connections between the modeling and control of partial differential equations and distributed control. In particular, the discussion focuses on parabolic and elliptic partial differential equations, which can be characterized from the point of view of graph theory. For example, when the Laplace operator is discretized over a spatial domain subject to certain boundary conditions, the resulting discrete operator can be analyzed through the lens of spectral graph theory. The structure and spectrum of the discretized graph Laplacian matrix can be used to characterize behavior of the continuous PDE such as synchronization and consensus as well as stability. |