Graph-Based Control Design for Network Dynamics with Time Delays

当复杂系统与其他复杂系统相互作用时，可能会产生意想不到的行为，有时会带来严重的后果。一些例子包括股票交易、疾病流行、车辆交通和社交媒体上的意见传播。通过将复杂的相互依赖的系统建模为动态节点的相互连接的网络，并通过广泛使用图论的数学工具，在分析和控制复杂系统方面取得了很大的进展。然而，到目前为止，这个框架缺乏整合网络本身动态的好方法。然而，在所有实际系统中，通信和决策所需的时间会导致延迟，忽略这些会显著降低分析结果的准确性。该项目将展示新的工具来放松当前的标准简化假设-例如，相同的动态节点和统一的延迟时间-并允许更现实的模型。该项目还将展示如何将这些新工具扩展到处理极其大型和复杂的系统。由于缺乏强大的分析框架，控制工程师在设计复杂网络系统的控制系统时往往忽略了时间延迟。随后，在高度简化的延迟模型下验证了稳定行为，如果系统足够鲁棒，则重新设计控制。该项目将提供缺失的系统控制设计框架，特别是通过考虑三个关键因素，即(i)动力系统，（ii）网络结构作为参数，以及（iii）延迟对动力行为的影响。然而，开发这样一个框架是具有挑战性的，因为网络动态是大规模的，并且由于其设计的本质，受到多个时间延迟的影响。此外，代数网络和受时滞影响的动力学之间的良好联系并不存在。最近将网络图拉普拉斯特征值与一类具有延迟的网络系统的动态行为（~稳定性，性能）相关的特征值联系起来的理论结果显示，有希望解决手头的挑战性问题。本项目的智力优势在于将这些理论扩展到多延迟，异构动力学和大规模网络，以及开发研究特征值，稳定性和性能之间的不变性特征的工具，以及简化此类网络的理解和控制设计的先进分解技术。该项目的成功完成将导致受时间延迟影响的大规模网络动态的分析，优化和控制，如在广泛的应用中出现的。

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.