Quantifying Complex Behavior in Large-Scale Systems through Structured Uncertainty Analysis
Large-scale systems of interacting dynamical elements are ubiquitous in nature and are increasingly common in engineered systems. Examples of such systems include large networks such as the power grid, vehicular traffic, social networks and many more. Such networks exhibit many complex phenomena arising from dynamic interaction between individual components. Qualitative understanding and quantitative prediction of these phenomena is a major current scientific and engineering challenge. This award supports research to develop quantitative analysis methods based on the techniques of structured stochastic uncertainty. These techniques model large-scale systems operating in uncertain environments, which include external forcing and attacks as well as internal disorder in interaction dynamics. The main observation is that even though individual uncertainty sources may be very small, their aggregate network effect may be large and even catastrophic, leading to loss of network stability or performance. Structured uncertainty analysis aims at quantifying exactly which type of small-scale uncertainty could lead to large-scale network-wide phenomena.
Uncertainties in large-scale dynamic systems can occur in system parameters or structure as well as in uncertain forcing and disturbances. The theoretical aims of the research program are twofold. The first is to develop a unified approach to the seemingly distinct problems of large-scale networked systems on the one hand, and systems over continuum space described by partial differential equations on the other. These two sets of problems will both be treated as spatially distributed systems, but with discrete space being described by the network structure in the former, and continuum space in the latter. This theoretical synthesis enables a comparative development of results and methods in both settings in a manner similar to, but significantly richer, than analogies between discrete- and continuous-time systems. The second aim is to develop highly-scalable, simulation-free performance and stability tests for linear systems with both additive and multiplicative noise. Application areas for which specialized techniques will be developed are network/cooperative control where both the network structure and conditions are stochastically varying, and fluid flow with stochastic field coefficients. This methodological synthesis contributes to the foundations of Control Theory in the area of additive and multiplicative noise. It enables the solution of several problems in uncertain systems that are typically investigated through expensive stochastic simulations, and are therefore not scalable to large-scale systems. The ability to address discrete and continuum space in a unified manner will allow for a new synthesis of concepts and results between uncertain networks and systems described by partial differential equations.
