Fundamental limitations and complex dynamics in nonlinear network systems with uncertainty

本文的目标是发现和理解具有不确定性的非线性网络系统稳定性和估计性能的基本限制。具有不确定性的网络系统，如电网、互联网、基因调控网络、通信网络等，在科学与工程中无处不在。不确定性的各种来源包括系统参数的不确定性，互连，信道延迟和脱落，以及外部噪声。尽管它们无处不在，但对于构建健壮的网络系统的理解很少，也缺乏系统的分析和设计工具。由非线性元素组成的网络系统由于可能以复杂的非平衡动力学形式出现自涌现行为，使得网络系统的情况更加复杂。本提案中提出的研究将有助于我们了解这种具有不确定性的网络系统设计的基本限制。智力优势：本提案中用于分析具有不确定性的非线性网络系统的方法和工具受到随机动力系统遍历理论的启发。该提案的变革思想在于将随机动力系统的遍历理论的方法和工具扩展和发展到随机控制动力系统，以了解具有不确定性的非线性网络控制系统的估计和控制中的基本性能限制。推广的形式主义对具有不确定性的非线性镇定的基本性能限制给出了新的结果。新结果的主要亮点是它指出了非线性系统的全局非平衡动力学在获得稳定性能限制方面所起的重要作用。这一新结果的全局性使它们与现有的无不确定性非线性稳定性能限制结果有根本的不同，其中的限制是用线性化的局部特征值表示的。本文提出的研究具有不确定性的非线性估计的基本性能限制的动机是基于在有限的传感器资源下建立一个鲁棒监测网络的应用问题。这个问题是联合技术公司的应用兴趣，PI正在与联合技术研究中心（UTRC）合作解决这个问题。此外，基于遍历理论的随机控制动力系统的形式也将用于解释非线性网络控制系统中由于链路故障和互连不确定性而产生的复杂动力学。更广泛的影响：由于不确定性而出现的复杂动力学的结果将有助于预测网络系统的不稳定性和故障。与UTRC的研究合作将为研究生提供机会，研究行业感兴趣的研究问题，并在行业中度过时间。这种合作还将导致学术界和工业界之间的知识交流，反之亦然。该项目的跨学科性质将为培养研究生在跨多个学科的前沿研究方面提供充足的机会。

The objective of this proposal is to discover and understand the fundamental limitations on the stability and estimation performance of nonlinear network system with uncertainty. Network systems with uncertainty, such as the power grid, Internet, gene regulatory networks, and communication networks are ubiquitous in science and engineering. The various sources of uncertainty include uncertainty in system parameters, interconnection, channel delays and drop-outs, and external noise. In spite of their ubiquitous presence, there is very little understanding and a lack of systematic analysis and design tools for building robust networks systems. The situation of network systems consisting of nonlinear elements is further complicated because of the possibility of self-emergent behavior in the form of complex non-equilibrium dynamics. The proposed research in this proposal will help us understand the fundamental limitations in the design of such network systems with uncertainty.Intellectual Merit: The methods and tools to be developed in this proposal for the analysis of nonlinear network systems with uncertainty are inspired from the ergodic theory of random dynamical systems. The transformative idea of this proposal is in the extension and development of methods and tools from ergodic theory of random dynamical system to random control dynamical system for the purpose of understanding the fundamental performance limitations in the estimation and control of nonlinear network control systems with uncertainty. The extended formalism has lead to the new results on the fundamental performance limitations on nonlinear stabilization with uncertainty. The main highlight of the new results is that it point to the important role-played by the global nonequilibrium dynamics of nonlinear systems in obtaining performance limitations on stabilization. The global nature of this new result make them fundamentally different than the existing performance limitation results on nonlinear stabilization without uncertainty, where the limitations are expressed in terms of the local eigenvalues of the linearization. The proposed research on fundamental performance limitations in nonlinear estimation with uncertainty is motivated from the applied problem of interest of building a robust surveillance network with limited sensor resources. This problem is of applied interest to United Technologies Company and the PI is collaborating with United Technologies Research Center (UTRC) on this problem. Furthermore the developed ergodic theory-based formalism for random control dynamical systems will also be used to explain emergence of complex dynamics that arise due to link failures and uncertainty in interconnection, in the nonlinear network control system.Broader Impact: The results on the emergence of complex dynamics due to uncertainty will help predict the instabilities and failures of the network systems. The research collaboration with UTRC, will provide opportunity to graduate students to work on research problems of industry interest and spend time at industry. This collaboration will also lead to exchange of know-how between academia and industry and vice versa. The interdisciplinary nature of this project will provide ample opportunities to train graduate students in leading edge research that cuts across multiple disciplines.