State Estimation for the Individual and the Population in Mean Field Control with Application to Demand Dispatch

This paper concerns state estimation problems in a mean field control setting. In a finite population model, the goal is to estimate the joint distribution of the population state and the state of a typical individual. The observation equations are a noisy measurement of the population. The general results are applied to demand dispatch for regulation of the power grid, based on randomized local control algorithms. In prior work by the authors it has been shown that local control can be carefully designed so that the aggregate of loads behaves as a controllable resource with accuracy matching or exceeding traditional sources of frequency regulation. The operational cost is nearly zero in many cases. The information exchange between grid and load is minimal, but it is assumed in the overall control architecture that the aggregate power consumption of loads is available to the grid operator. It is shown that the Kalman filter can be constructed to reduce these communication requirements,Comment: To appear, IEEE Trans. Auto. Control. Preliminary version appeared in the 54rd IEEE Conference on Decision and Control, 201

A Class of Networked Multi-Agent Control Systems: Interference Induced Games, Filtering, Nash Equilibria

RÉSUMÉ: Nous considérons une classe de systèmes de contrôle stochastiques linéaires scalaires en réseau dans lesquels un grand nombre d'agents contrôlés envoient leurs états à un concentrateur central qui, à son tour, envoie des commandes de contrôle silencieuses basées sur ses observations et vise à minimiser un coût quadratique donné. La technologie de communication est l'accès multiple par répartition en code (CDMA) et, par conséquent, les signaux reçus sur le concentrateur central sont corrompus par des interférences. Les niveaux des signaux envoyés par les agents sont considérés proportionnels à leur état, et le traitement des signaux basés sur le CDMA réduit l'interférence d'autres agents d'un facteur de 1/N où N est le nombre d'agents. L'interférence existante crée par inadvertance une situation de jeu dans laquelle les actions d'un agent affectent son état et donc par interférence, la capacité d'autres agents à estimer les leurs, influençant à leur tour leur capacité à contrôler leur état. Ceci conduit à des problèmes d'estimation fortement couplés. Cela conduit également à une situation de contrôle dual puisque les contrôles individuels contrôlent l'état mais affectent également le potentiel d'estimation de cet état. La thèse comporte trois parties principales. Dans la première partie, nous montrons que le fait d'ignorer le terme d'interférence et d'utiliser un principe de séparation pour le contrôle mène à des équilibres de Nash asymptotiques en N, pourvu que la dynamique individuelle soit stable ou “pas excessivement” instable. Que pour certaines classes de coût et de paramètres dynamiques, les lois de contrôle séparées optimales obtenues en ignorant le couplage interférentiel, sont asymptotiquement optimales lorsque le nombre d'agents passe à l'infini, formant ainsi pour un nombre de joueurs fini N, un équilibre �-Nash. Plus généralement, les lois de contrôle séparées optimales peuvent ne pas être asymptotiquement optimales et peuvent en fait conduire à un comportement global instable. Nous considérons donc une classe de lois de contrôle centralisées paramétrées selon lesquelles le gain séparé de Kalman est traité comme le gain arbitraire d'un observateur analogue à un observateur de Luenberger. Les régions de stabilité du système sont caractérisées et la nature des politiques optimales de contrôle coopératif au sein de la classe considérée est explorée. La deuxième partie concerne l'extension du travail dans la première partie au-delà du seuil d'instabilité des contrôles coopératifs. Il est alors observé que les contrôles linéaires invariants dans le temps basés sur les sorties des filtres de dimension croissante semblent toujours maintenir la stabilité du système et d'intrigantes propriétés sur les estimations des états sont observées numériquement. ABSTRACT: We consider a class of networked linear scalar stochastic control systems whereby a large number of controlled agents send their states to a central hub, which in turn sends back noiseless control commands based on its observations, and aimed at minimizing a given quadratic cost. The communication technology is code division multiple access (CDMA), and as a result signals received at the central hub are corrupted by interference. The signals sent by agents are considered proportional to their state, and CDMA based signal processing reduces other agents' interference by a factor of 1/N where N is the number of agents. The existing interference inadvertently creates a game situation whereby the actions of one agent affect its state and thus through interference, the ability of other agents to estimate theirs, in turn influencing their ability to control their state. This leads to highly coupled estimation problems. It also leads to a dual control situation as individual controls both steer the state and affect the estimation potential of that state. The thesis is presented in three main parts. In the first part, we show that ignoring the interference term and using a separation principle for control provably leads to Nash equilibria asymptotic in N, as long as individual dynamics are stable or “not exceedingly” unstable. In particular, we establish that for certain classes of cost and dynamic parameters, optimal separated control laws obtained by ignoring the interference coupling are asymptotically optimal when the number of agents goes to infinity, thus forming for finite N an �-Nash equilibrium. More generally though, optimal separated control laws may not be asymptotically optimal, and can in fact result in unstable overall behavior. Thus we consider a class of parameterized decentralized control laws whereby the separated Kalman gain is treated as the arbitrary gain of a Luenberger like observer. System stability regions are characterized and the nature of optimal cooperative control policies within the considered class is explored. The second part is concerned with the extension of the work in the first part past the instability threshold for the previous cooperative Luenberger like observers. It is observed that time invariant linear controls based on the outputs of growing dimension filters appear to always maintain system stability, and intriguing state estimate properties are numerically observed. More specifically, we tackle the case of exact decentralized filtering under a class of time invariant certainty equivalent feedback controllers, and numerically investigate both stabilization ability and performance of such controllers as the state estimate feedback gain varies. While the optimum filters have memory requirements which become infinite over time, the stabilization ability of their finite memory approximation is also tested

Causality and aggregation in economics: the use of high dimensional panel data in micro-econometrics and macro-econometrics

This study proposes one plausible procedure to address two methodological issues, which are common in micro- and macro- econometric analyses, for the full realization of research potential brought by recently available high dimensional data. To address the issue of how to infer the causal structure from empirical regularities, graphical causal models are proposed to inductively infer causal structure from non-temporal and non-experimental data. However, the (probabilistic) stability condition for the graphical causal models can be violated for high dimensional data, given that close co-movements and thus near deterministic relations are oftentimes observed among variables in high dimensional data. Aggregation methods are proposed as one possible way to address this matter, allowing one to infer causal relationships among disaggregated variables based on aggregated variables. Aggregation methods also are helpful to address the issue of how to incorporate a large information set into an empirical model, given that econometric considerations, such as degrees-of-freedom and multicollinearity, require an economy of parameters in empirical models. However, actual aggregation requires legitimate classifications for interpretable and consistent aggregation. Based on the generalized condition for the consistent and interpretable aggregation derived from aggregation theory and statistical dimensional methods, we propose plausible methodological procedure to consistently address the two related issues of causal inference and actual aggregation procedures. Additional issues for empirical studies of micro-economics and macro-economics are also discussed. The proposed procedure provides an inductive guidance for the specification issues among the direct, inverse, and mixed demand systems and an inverse demand system, which is statistically supported, is identified for the consumer behavior of soft drink consumption. The proposed procedure also provides ways to incorporate large information set into an empirical model with allowing structural understanding of U.S. macro-economy, which was difficult to obtain based on the previously used factor augmented vector autoregressive (FAVAR) framework. The empirical results suggest the plausibility of the proposed method to incorporate large information sets into empirical studies by inductively addressing multicollinearity problem in high dimensional data

Another look at the transactions demand for money in Nigeria

This study sets out to model non-bank public's desired holdings of five different measures of money in the Nigerian economy. These are currency outside banks (COB), demand deposits (DD), narrow money (Ml), quasi money (QM), and broad money (M2).The study addresses many of the pitfalls in

Econometricmodel of industry, profits, and tatonnement adjustment

This study presents a quantitative analysis of one of the main forces in an economy, disaggregated short term profits, and of the process whereby the system adjusts itself to the temporary equilibrium indicated by such forces, a generalised tatonnement. Quarterly ten-equation econometric models explaining industry behaviour and profits are developed from a basic industry model for ten mutually exclusive and exhaustive industries. These models are connected with each other and with the whole by a number of linkages and by being embedded in a skeletal economy model. The system is solved at two levels. Firstly the industry models are solved individually for given values of the linking variables; the results are used to choose between alternative specifications of the models and to assess the adequacy of the formulation adopted. Secondly the whole system is solved iteratively by solving the industry models for some given trial values of the linking variables, using these solutions to derive new trial values, and repeating the process until these values converge; the results are used to assess the efficacy of the tatonnement process. The results indicate that the models proposed are good predictors of disaggregated short term profits and that the tatonnement process used produces rapid convergence to a consistent equilibrium. It is also suggested from the discrepancy between the tatonnement (quasi-competitive) and actual (imperfectly competitive) solutions that the capitalist system is inefficient in that it produces too much

Optimisation, Optimal Control and Nonlinear Dynamics in Electrical Power, Energy Storage and Renewable Energy Systems

The electrical power system is undergoing a revolution enabled by advances in telecommunications, computer hardware and software, measurement, metering systems, IoT, and power electronics. Furthermore, the increasing integration of intermittent renewable energy sources, energy storage devices, and electric vehicles and the drive for energy efficiency have pushed power systems to modernise and adopt new technologies. The resulting smart grid is characterised, in part, by a bi-directional flow of energy and information. The evolution of the power grid, as well as its interconnection with energy storage systems and renewable energy sources, has created new opportunities for optimising not only their techno-economic aspects at the planning stages but also their control and operation. However, new challenges emerge in the optimization of these systems due to their complexity and nonlinear dynamic behaviour as well as the uncertainties involved.This volume is a selection of 20 papers carefully made by the editors from the MDPI topic “Optimisation, Optimal Control and Nonlinear Dynamics in Electrical Power, Energy Storage and Renewable Energy Systems”, which was closed in April 2022. The selected papers address the above challenges and exemplify the significant benefits that optimisation and nonlinear control techniques can bring to modern power and energy systems