Fundamental limitations and complex dynamics in nonlinear network systems with uncertainty

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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.