S. Yusef Shafi

Designing node and edge weights of a graph to meet Laplacian eigenvalue constraints

We consider agents connected over a network, and propose a method to design an optimal interconnection such that the gap between the largest and smallest Laplacian eigenvalues of the graph representing the network is minimized. We study ways of imposing constraints that may arise in physical systems, such as enforcing lower bounds on connectivity and upper bounds on gain as well as network cost. In particular, we show that node and edge weights of a given graph can be simultaneously adjusted via convex optimization to achieve improvements in its Laplacian spectrum.

Graph Weight Allocation to Meet Laplacian Spectral Constraints

We adjust the node and edge weightings of graphs using convex optimization to impose bounds on their Laplacian spectra. First, we derive necessary and sufficient conditions that characterize the feasibility of spectral bounds given positive node and edge weightings. Synthesizing these conditions leads naturally to algorithms that exploit convexity to achieve several eigenvalue bounds simultaneously. The algorithms we propose apply to many graph design problems as well as multi-agent systems control. Finally, we suggest efficient ways to accommodate larger graphs, and show that dual formulations lead to substantial improvement in the size of graphs that can be addressed.

Synchronization of diffusively-coupled limit cycle oscillators

We develop analytical and numerical conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems. We examine two classes of systems: reaction-diffusion PDEs with Neumann boundary conditions, and compartmental ODEs, where compartments are interconnected through diffusion terms with adjacent compartments. In both cases the uncoupled dynamics are governed by a nonlinear system that admits an asymptotically stable limit cycle. We provide two-time scale averaging methods for certifying stability of spatially homogeneous time-periodic trajectories in the presence of sufficiently small or large diffusion and develop methods using the structured singular value for the case of intermediate diffusion. We highlight cases where diffusion stabilizes or destabilizes such trajectories.

An adaptive algorithm for synchronization in diffusively-coupled systems

We present an adaptive algorithm that guarantees synchronization in diffusively-coupled systems. We first consider compartmental systems of ODEs where variables in each compartment are interconnected through diffusion terms with like variables in other compartments. Each set of variables may have its own weighted undirected graph describing the topology of the interconnection between compartments. The link weights are updated adaptively according to the magnitude of the difference between neighboring agents connected by the link. We next consider reaction-diffusion PDEs with Neumann boundary conditions and derive an analogous algorithm guaranteeing spatial homogenization of the solutions. We provide several numerical examples demonstrating the results.

Graph weight design for Laplacian eigenvalue constraints with multi-agent systems applications

We adjust the node and edge weightings of graphs using convex optimization to impose bounds on their Laplacian spectra. We first derive necessary and sufficient conditions that characterize the feasibility of spectral bounds given positive node and edge weightings. Next, we propose algorithms that exploit convexity to achieve these bounds. The design and analysis tools are useful for a variety of stability and control problems in multi-agent systems.

Synchronization under space and time-dependent heterogeneities

We study output synchronization of nonlinear systems subject to a class of spatially and temporally-varying heterogeneities. We begin by addressing reaction-diffusion PDEs, and show that an incremental passivity property in the nominal reaction dynamics is fundamental in guaranteeing spatial homogenization of trajectories. Our distributed control achieves homogenization by defining an internal model subsystem for each output corresponding to the respective heterogeneity. We also design a similar distributed internal model control that achieves output synchronization of a network of incrementally passive nonlinear ODE systems in the presence of heterogeneities. We prove that our control laws effectively compensate for desynchronizing effects due to the differences between the heterogeneous inputs. We illustrate our algorithms on ring oscillators and bistable systems. The main contribution of the paper is a constructive and unified approach to guaranteeing homogenization under heterogeneities for both reaction-diffusion PDE and diffusively-coupled ODE systems.

Adaptive Synchronization of Diffusively Coupled Systems

We present an adaptive algorithm that guarantees synchronization in diffusively coupled systems. We first consider compartmental systems of ODEs where variables in each compartment are interconnected through diffusion terms with like variables in other compartments. Each set of variables may have its own weighted undirected graph describing the topology of the interconnection between compartments. The link weights are updated adaptively according to the magnitude of the difference between neighboring agents connected by each link. We show that an incremental passivity property is fundamental in guaranteeing output synchronization. We next consider reaction-diffusion PDEs with Neumann boundary conditions and derive an analogous algorithm guaranteeing spatial homogenization of the solutions. We provide several numerical examples demonstrating the results.

Spatial uniformity in diffusively-coupled systems using weighted L 2 norm contractions

We present conditions that guarantee spatial uniformity in diffusively-coupled systems. Diffusive coupling is a ubiquitous form of local interaction, arising in diverse areas including multiagent coordination and pattern formation in biochemical networks. The conditions we derive make use of the Jacobian matrix and Neumann eigenvalues of elliptic operators, and generalize and unify existing theory about asymptotic convergence of trajectories of reaction-diffusion partial differential equations as well as compartmental ordinary differential equations. We present numerical tests making use of linear matrix inequalities that may be used to certify these conditions. We discuss an example pertaining to electromechanical oscillators. The papers main contributions are unified verifiable relaxed conditions that guarantee synchrony.

Synchronization of limit cycle oscillations in diffusively-coupled systems

We present analytical and numerical conditions to verify whether limit cycle oscillations synchronize in diffusively coupled systems. We consider both compartmental ODE models, where each compartment represents a spatial domain of components interconnected through diffusion terms with like components in different compartments, and reaction-diffusion PDEs with Neumann boundary conditions. In both the discrete and continuous spatial domains, we assume the uncoupled dynamics are determined by a nonlinear system which admits an asymptotically stable limit cycle. The main contribution of the paper is a method to certify when the stable oscillatory trajectories of a diffusively coupled system are robust to diffusion, and to highlight cases where diffusion in fact leads to loss of spatial synchrony. We illustrate our results with a relaxation oscillator example.

Homogenization in reaction-diffusion PDEs under space and time-dependent heterogeneities

We study spatial homogenization of nonlinear reaction-diffusion PDEs subject to a class of spatially and temporally-varying heterogeneities. We show that an incremental passivity property in the nominal reaction dynamics is fundamental in guaranteeing homogenization of trajectories. Our distributed control achieves homogenization by defining an internal model subsystem for each output corresponding to the respective heterogeneity. We illustrate our algorithm on a system with bistable dynamics. The main contribution of the paper is a constructive approach to guarantee spatial homogenization under heterogeneities.