Graph-Based Control Design for Network Dynamics with Time Delays

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When complex systems interact with other complex systems, unexpected behavior may result, sometimes with severe consequences. A few examples include stock trading, disease epidemics, vehicle traffic, and opinion propagation in social media. A great deal of progress has been made towards the analysis and control of complex interdependent systems by modeling them as interconnected networks of dynamic nodes, and by making extensive use of the mathematical tools of graph theory. However this framework to date lacks good methods for incorporating the dynamics of the network itself. Yet, in all real systems the time required for communication and decision-making causes delays, and neglecting these can significantly degrade the accuracy of the analytical result. This project will demonstrate new tools to relax the current standard simplifying assumptions -- for example, identical dynamic nodes and uniform delay times -- and allow more realistic models. The project will also show how these new tools may be scaled to address extremely large and complex systems.Lacking a strong analytical framework, it is common for control engineers to design control systems for complex networked systems while ignoring time delay. Subsequently stable behavior is verified under a highly simplified delay model, with control redesign if the system is sufficiently robust. This project will provide the missing systematic control-design framework, specifically by considering three key ingredients, namely, (i) dynamical systems, (ii) network structure as a parameter, and (iii) the effects of delays on dynamical behavior. Development of such a framework is, however, challenging as network dynamics are large scale, and by the very nature of their design, are impacted by multiple time delays. Moreover a well-established connection between algebraic networks and dynamics affected by time delays does not exist. Recent theoretical results in connecting network graph Laplacian eigenvalues to the eigenvalues associated with the dynamical behavior (~ stability, performance) of a class of network systems with delays show promise in addressing the challenging problem at hand. The intellectual merit of this project lies in expanding such theories to multiple delays, heterogeneous dynamics, and large-scale networks, as well as developing tools to study invariance features between eigenvalues, stability, and performance, and advanced decomposition techniques to simplify the understanding and control design of such networks. The successful completion of this project will lead to the analysis, optimization, and control of large-scale network dynamics affected by time delays, as arising in a broad spectrum of applications.