Convergence Results for Ergodic Control of Ensembles via Iterated Function Systems

This PhD thesis studies the ergodic properties of a class of ensemble systems described and modelled by closed-loop feedback and iterated function systems (IFS), a discrete-time Markov chain. An extensive literature survey of IFS is provided in chapter 2, emphasizing some open questions from work by Fioravanti et al. In Chapter 3, an abstract result from Fioravanti et al. is validated numerically in the context of power system application. Distributed energy resources (DERs) that may be incorporated into power networks whose aggregate demand must be regulated include controllable loads and battery energy storage systems. To apply concepts of predictability and fairness to power system applications, the long-term averages of pricing or incentives provided should be independent of the initial state of the operators of the DER, the aggregator, and the power grid. In addition, with incrementally input-to-state stable (ISS) controllers, this concept of predictability and fairness may be assured even when accounting for the non-linearity of the alternating-current model. The probability of sampling the response functions of the agent has, up until in the literature, either been constant or, at the very least, a stated set of probability functions has been present. However, it is possible that the population, and therefore the agent's responses, may change over time even if the number of agents that make up the population remains the same. In this context, chapter 4 investigates the IFS with time-varying and state (or place) dependent probabilities and demonstrates that such a time-varying system has the existence and the uniqueness of piece-wise invariant measures. When the ensemble changes over time, a question to investigate is whether the subsequent invariant distributions differ significantly. We estimate the distance between two successive invariant probability measures by assuming that the overall probabilities of a fluctuating agent's responses are limited by some constant. In addition, we demonstrate that if stochastic difference equations are used to realize the states of a time-varying IFS system, and if all of the matrices involved in the affine transformations are Schur, then the system forgets its initial condition. In some applications involving smart cities, it is reasonable to assume that the probabilistic model is not affected by time. On the other hand, the vast majority of actual scenarios include a population that changes over time. From a theoretical point of view, the control of an ensemble with time-varying components could be difficult. In chapter 5 with IFS, we provide a model that ensures predictability and fairness in social sensing to develop a strategy to guarantee predictability and fairness even though probabilistic models might differ over time. This particular instance was not considered in any of our earlier chapters. Next, in chapter 6, we extend and generalize the results from Fioravanti et al. regarding a closed-loop feedback model of a discrete-time dynamical system consisting of an ensemble system to two ensemble populations, especially when these populations are coupled. We use iterated function systems to explain the interconnection of two populations and verify the ergodicity result. We extend this notion to a large-scale interconnection and verify ergodicity. The previous-mentioned closed-loop feedback system had a linear controller and a filter. In chapter 7, we assume they are non-linear and prove the ergodicity conclusion while making certain regularity assumptions about the agent's response functions. In addition, we put forth some unique contraction conditions that might be of broad interest in the literature on control theory. The last topic of the thesis examines an AIMD algorithm difficulty when numerous AIMD networks are connected (often by some non-linearity). Chapter 8 shows that such systems inherit the ergodic features of individual AIMD networks.