Spatial Uniformity in Diffusively Coupled Systems

We are studying spatially distributed models, which arise in the study biological reaction-diffusion networks, voltage-controlled coupled oscillator circuits, and multiagent distributed control. The models we study fall under the category of diffusively coupled systems, wherein interaction between different parts of a system can be represented by diffusion. In the continuous case (PDEs), the coupling is expressed through elliptic operators, whereas in the discrete case (compartmental ODEs), the coupling is expressed through weighted graph Laplacians. We develop analytical and computational tools that certify spatial synchrony under different operating conditions and levels of coupling. In particular, we consider systems with stable steady state behavior, but whose higher order spatial modes are destabilized by large diffusion, and describe techniques that either certify or rule out spatial synchrony in solutions. Our methods apply to spatially distributed over large and sparse networks in which the nature and strength of coupling is highly varying depending on location in the network. We make use of tools from convex optimization, robustness analysis, and nonlinear systems theory.

Y. Shafi, M. Arcak, and M. Jovanovic. Synchronization of Limit Cycle Oscillations in Diffusively-Coupled Systems, in preparation, 2012.

Y. Shafi. Guaranteeing Spatial Uniformity in Diffusively-Coupled Systems, preprint on arXiv, 2012.

Distributed Control of Multiagent Systems

We are broadly interested in network design problems for cooperative control of multiagent systems. The systems consist of homogeneous plants with a network of feedback interconnections. We assume that the plant dynamics are stabilizable by local state feedback, and we want the agents to converge quickly to formation. However, it is often the case that the interconnection topology is inherently unstable or introduces undesirable properties such as unbounded error propagation as the number of agents in the system grows. We seek to eliminate or mitigate these problems by applying node and edge weighting to the interconnection structures. We derive linear matrix inequality constraints that can be incorporated into convex optimization problems to solve graph design problems to improve the dynamics of multiagent systems. We have demonstrated the applicability of the method to achieve formation and mitigate error propagation and instability in vehicle strings and PVTOL aircraft.

Y. Shafi, M. Arcak, and L. El Ghaoui, Graph Weight Allocation to meet Laplacian Spectral Constraints, in IEEE Transactions on Automatic Control, 2012.

Y. Shafi, M. Arcak, and L. El Ghaoui, “Graph weight design for Laplacian eigenvalue constraints with multi-agent systems applications,” in Proc. Conference on Decision and Control, 2011.

Y. Shafi, M. Arcak, and L. El Ghaoui, “Designing node and edge weights of a graph to meet laplacian eigenvalue constraints,” in Proc. Allerton Conference, 2010.

Selected Course Projects

Distributed Approaches to Cooperative and Non-Cooperative Network Resource Sharing Problems using Simultaneous Primal-Dual Descent

We 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 PDEs

We 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.