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Creating Realistic Models with Time Delay

August 13, 2015

MIE Associate Professor Rifat Sipahi was awarded a $293K NSF grant for his "Graph-Based Control Design for Network Dynamics with Time Delays".

A broad spectrum of engineered and natural systems surrounding us comprise of networks and multiple interacting agents in a dynamical system setting. In robotic networks for example, robots are the agents, the network pertains to which robot interacts with which others, and dynamical system setting arises because the behavior of robots can evolve based on their decision making and how they interact with each other as well as the environment. Across different disciplines, dynamical networks present similarities. For example in traffic flow, each vehicle is an agent whose behavior is affected by its neighboring vehicles, and in disease epidemics, healthy and infected individuals interact with each other, which ultimately determine how a virus spreads.

Time delays/after effects naturally arise in many network settings. Delays, large or small, can be deceiving, detrimental, and limiting. For example, remote surgery technology could be strongly affected by time delays because delayed information sent over the Internet from a remote site would confuse the surgeons when trying to synchronize their actions with the remote surgery robot. In virus epidemics, an individual may be infected by a virus but he/she will not present any symptoms before the virus incubation time elapses. Seemingly healthy, this individual may then interact with and infect healthy individuals, and such interactions may keep occurring. This “hidden” spreading dynamics makes the containment of any virus extremely difficult. In a different context, human reaction delays affect driving patterns in traffic which eventually contribute to traffic-jam formations.

Understanding of dynamical networks affected by time delays is key to better designing and controlling such networks, or influencing their behavior in desirable ways. In this project, we will develop a mathematical framework to study the behavior of large scale networks with agents of different characteristics and time delays. Given a particular network dynamics with time delays, the main objective is to reveal the rules by which one can determine how to intervene and manipulate the network, or influence the agents in a strategic way to create an overall desirable behavior from the network. Developed theoretical tools will be implemented on various simulation- and experiment-based platforms including a robotic network.

Abstract Source: NSF

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.