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dc.contributorMehta, Prashant G.
dc.contributorShanbhag, Vinayak V.
dc.contributorMehta, Prashant G.
dc.contributorShanbhag, Vinayak V.
dc.contributorBasar, Tamer
dc.contributorCox, Charles L.
dc.contributorMeyn, Sean P.
dc.contributorVakakis, Alexander F.
dc.contributor.authorYin, Huibing
dc.date.accessioned2019-10-29T08:08:13Z
dc.date.available2019-10-29T08:08:13Z
dc.date.created2018-09-05 01:21
dc.date.issued2012-05-22
dc.identifieroai:www.ideals.illinois.edu:2142/30922
dc.identifierhttp://hdl.handle.net/2142/30922
dc.identifier.urihttp://hdl.handle.net/20.500.12424/2583821
dc.description.abstractCompared to linear systems, nonlinear generalizations may exhibit both non-equilibrium and equilibrium
 behavior in the long run. The characterization of such behavior is challenging, particularly when overlaid by
 an optimization or control layer, and is of relevance in a range of applications, e.g., neuroscience, biology,
 economics, communication networks and power systems. The objective of this thesis is to consider these
 questions for two prototypical applications of nonlinear multi-agent systems: (1) large population of coupled
 oscillators and (2) communication networks. The research is divided into the following three parts:
 Synchronization of oscillators: The purpose of this part is to understand phase transition in noncoop-
 erative dynamic games with a large number of agents. The focus of analysis is on a variation of the large
 population linear quadratic Gaussian (LQG) model proposed by Huang et. al. 2007 [1], comprised here
 of a controlled N-dimensional stochastic differential equation model, coupled only through a cost function.
 The states are interpreted as phase angles for a collection of non-homogeneous oscillators, and in this way
 the model may be regarded as an extension of the classical coupled oscillator model of Kuramoto.
 A deterministic PDE model is proposed, which is shown to approximate the stochastic system as the
 population size approaches infinity. Key to the analysis of the PDE model is the existence of a particular
 Nash equilibrium in which the agents ???opt out??? of the game, setting their controls to zero, resulting in the
 ???incoherence??? equilibrium.
 Next we introduce approximate dynamic programming (ADP) techniques for the design and adaptation
 (learning) of approximately optimal control laws for this model. For this purpose, a parameterization is
 proposed, based on analysis of the mean-field PDE model for the game. In an offline setting, a Galerkin
 procedure is introduced to choose the optimal parameters. In an online setting, a steepest descent algorithm
 is proposed. We provide detailed analysis of the optimal parameter values as well as the Bellman error with
 both the Galerkin approximation and the online algorithm.
 Methods from dynamical systems theory are used in a bifurcation analysis, based on a linearization of
 the PDE model about the incoherence equilibrium. A critical value of the control cost parameter is identified:
 Above this value, the oscillators are incoherent; and below this value (when control is sufficiently cheap)
 the oscillators synchronize. Then we simplify the analysis by relating the solutions of the PDE model to the
 solutions of a certain nonlinear eigenvalue problem. Both analysis and computation are significantly easier
 for the nonlinear eigenvalue problem. Apart from the bifurcation analysis that shows existence of a phase
 transition, we also describe a Lyapunov-Schmidt perturbation method to obtain asymptotic formulae for the
 small amplitude bifurcated solutions.
 
 A key question in the design of engineered competitive systems has been that of the efficiency of the
 associated equilibria. Yet, there is little known in this regard in the context of stochastic dynamic games in a
 large population regime. Here, we examine the efficiency of the associated mean-field equilibria with respect
 to a related welfare optimization problem. We construct variational problems both for the noncooperative
 game and its centralized counterpart and employ these problems as a vehicle for conducting this analysis.
 Using a bifurcation analysis, we analyze the variational solutions and the associated efficiency loss. An
 expression for the local bound of efficiency loss is obtained for the homogeneous population.
 All the conclusions are illustrated with results from numerical experiments.
 Nash games with coupled strategy sets: Generalized Nash equilibria (GNE) represent extensions of the
 Nash solution concept when the strategy sets are coupled across agents. We consider a restricted class of
 such games, referred to as generalized Nash games, in which the agents contend with shared or common
 constraints and their payoff functions are further linked via a scaled congestion cost metric. When strategy
 sets are continuous and the metric is an increasing convex function, a solution to a related variational in-
 equality provides a set of equilibria characterized by common Lagrange multipliers for shared constraints.
 In general, this variational inequality problem is non-monotone. However, we show that under mild con-
 ditions, it admits solutions, even in the absence of restrictive compactness assumptions on strategy sets.
 Additionally, we show that the equilibrium is locally unique both in the primal space as well as in the larger
 primal-dual space. The existence statements can be generalized to accommodate a piecewise-smooth metric
 while affine restrictions, surprisingly, lead to both existence and uniqueness guarantees. The second half of
 the part provides a brief discussion of distributed computation of such equilibria in monotone regimes via
 a distributed iterative Tikhonov regularization (ITR) scheme. Notably, such schemes are single-timescale
 counterparts of standard Tikhonov regularization methods and involve updating the regularization parameter
 after every gradient step. Application of such techniques to a class of network flow rate allocation games
 suggests that the ITR schemes perform better than their two-timescale counterparts.
 Nonlinear network flow control with AQM feedback: The last part of this thesis investigates stability,
 bifurcation and oscillations arising in a communication network model with a large number of heteroge-
 neous users adopting a Transmission Control Protocol (TCP)-like rate control scheme with an Active Queue
 Management (AQM) router. The heterogeneity in the system is due to different user delays that are known
 and fixed but taken from a given distribution. It is shown that for any given distribution of delays, there
 exists a critical amount of feedback (due to AQM) at which the equilibrium loses stability and a limit cy-
 cling solution develops via a Hopf bifurcation. The nature (criticality) of the bifurcation is investigated with
 the aid of Lyapunov-Schmidt perturbation method. The results of the analysis are numerically verified and
 provide valuable insights into dynamics of the AQM control system.
dc.languageen
dc.language.isoeng
dc.rightsCopyright 2012 Huibing Yin
dc.subjectmean-field game
dc.subjectoscillators
dc.subjectsynchronization of oscillators
dc.subjectstatic and dynamic game
dc.subjectphase transition
dc.subjectflow control
dc.subjectmean-field approximation
dc.subjectoptimal control
dc.subjectnonlinear system
dc.subjectHamilton-Jacobi-Bellman (HJB) equation
dc.subjectFokker-Planck-Kolmogorov (FPK) equation
dc.subjectApproximate Dynamic Programming (ADP)
dc.subjectNash equilibrium
dc.subjectactive queue management
dc.titleNoncooperative static and dynamic games: addressing shared constraints and phase transitions
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ge.dataimportlabelOAI metadata object
ge.identifier.legacyglobethics:15237379
ge.identifier.permalinkhttps://www.globethics.net/gel/15237379
ge.lastmodificationdate2018-09-05 01:21
ge.lastmodificationuseradmin@pointsoftware.ch (import)
ge.submissions0
ge.oai.exportid149801
ge.oai.repositoryid4412
ge.oai.setnameGraduate Dissertations and Theses at Illinois
ge.oai.setnameDept. of Mechanical Science and Engineering
ge.oai.setnameCollege of Engineering
ge.oai.setnameGraduate Dissertations and Theses at Illinois
ge.oai.setnameDissertations and Theses - Mechanical Science and Engineering
ge.oai.setspeccom_2142_5130
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ge.oai.setspeccom_2142_234
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ge.linkhttp://hdl.handle.net/2142/30922


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