Mean Field, Distributed and Hybrid Control Systems

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The overall objective of the proposed research program is to advance the fundamental analysis and design of control systems for (i) distributed large scale systems (i.e. systems without centralized control or information which are of high state cardinality) and (ii) systems with both continuous and discrete states and dynamics. Such systems are ubiquitous in the contemporary networked information and computation based environment and so their enhanced control and optimization would bring great benefits. Mean Field Game (MFG) theory constitutes a fundamental new methodology for the analysis and design of distributed large scale systems which originated in NSERC supported work by the proposer together with his student Minyi Huang and the co-supervisor Roland Malhame' during 2003 – 2007, and, independently, by J.-M. Lasry and P.- L. Lions during 2006 – 2007. Since then the field has undergone vigorous development due to the continuing contributions of these and other researchers world wide; moreover, potential applications of MFG methodology and new, challenging problems are continually being discovered in a vast range of domains. MFG theory establishes the existence of approximate Nash equilibria generated solely by local feedback for stochastic dynamical system agents in games involving large numbers of agents. The equilibria and the feedback laws applied in the finite population case are obtained via the far simpler solutions to the fundamental infinite population MFG Hamilton-Jacobi-Bellman and (McKean-Vlasov)-Fokker-Planck-Kolmogorov PDEs which are linked by the state distribution of a generic agent, otherwise known as the system's mean field. As the number of agents goes to infinity the approximation error goes to zero. This proposal maps a five year research program on the following themes: (a) Estimation in MF Systems: a central problem for MF system theory is that partially observed systems with major agents (i.e. non-asymptotically vanishing with population size) have stochastic mean fields which must be recursively estimated along with the major agents' partially observed states. A state estimation theory for non-linear partially observed MFG systems with major and minor agents, and hence stochastic mean fields, will be of great significance and will be exploited in the development of (b) Spatially Dynamic MF Systems, (c) Coalition Formation in MF Systems, (d) Locally Infinite and Large Scale Topology MF Systems, and (e) MF Market and Auction Models.In large areas of systems and control engineering hybrid control systems are synonymous with advanced control methodology. Recent advances have generated both a sophisticated Hybrid Minimum Principle (HMP) theory and computational methods, and a parallel theory of Hybrid Dynamic Programming (HDP). Building on the proposer's work, it is planned to generate (i) a unified HMP - HDP theory, (ii) a theory and associated algorithms for the fundamental class of HDP problems where the systems are subject to state switching constraints, and (iii) an integrated HDP - Model Predictive Control methodology.Success in the overall program will yield 8 PhDs and 4 MEng degrees, plus 4 PDFs, and will contribute to the emergence of new techniques enabling networks of distributed adaptive agents to control large scale systems evolving in random environments; prime examples (already initiated in Canada) include the integration of smart meters, electric vehicle charging systems and localized generation, where the vehicles and generators are themselves optimized using hybrid system control algorithms.