Online Machine Learning for Graph Topology Identification from Multiple Time Series

High dimensional time series data are observed in many complex systems. In networked data, some of the time series are influenced by other time series. Identifying these relations encoded in a graph structure or topology among the time series is of paramount interest in certain applications since the identified structure can provide insights about the underlying system and can assist in inference tasks. In practice, the underlying topology is usually sparse, that is, not all the participating time series in influence each other. The goal of this dissertation pertains to study the problem of sparse topology identification under various settings. Topology identification from time series is a challenging task. The first major challenge in topology identification is that the assumption of static topology does not hold always in practice since most of the practical systems are evolving with time. For instance, in econometrics, social networks, etc., the relations among the time series can change over time. Identifying the topologies of such dynamic networks is a major challenge. The second major challenge is that in most practical scenarios, the data is not available at once - it is coming in a streaming fashion. Hence, batch approaches are either not applicable or they become computationally expensive since a batch algorithm is needed to be run when a new datum becomes available. The third challenge is that the multi-dimensional time series data can contain missing values due faulty sensors, privacy and security reasons, or due to saving energy. We address the aforementioned challenges in this dissertation by proposing online/-batch algorithms to solve the problem of time-varying topology identification. A model based on vector autoregressive (VAR) process is adopted initially. The parameters of the VAR model reveal the topology of the underlying network. First, two online algorithms are proposed for the case of streaming data. Next, using the same VAR model, two online algorithms under the framework of online optimization are presented to track the time-varying topologies. To evaluate the performance of propose online algorithms, we show that both the proposed algorithms incur a sublinear static regret. To characterize the performance theoretically in time-varying scenarios, a bound on the dynamic regret for one of the proposed algorithms (TIRSO) is derived. Next, using a structural equation model (SEM) for topology identification, an online algorithm for tracking time-varying topologies is proposed, and a bound on the dynamic regret is also derived for the proposed algorithm. Moreover, using a non-stationary VAR model, an algorithm for dynamic topology identification and breakpoint detection is also proposed, where the notion of local structural breakpoint is introduced to accommodate the concept of breakpoint where instead of the whole topology, only a few edges vary. Finally, the problem of tracking VAR-based time-varying topologies with missing data is investigated. Online algorithms are proposed where the joint signal and topology estimation is carried out. Dynamic regret analysis is also presented for the proposed algorithm. For all the previously mentioned works, simulation tests about the proposed algorithms are also presented and discussed in this dissertation. The numerical results of the proposed algorithms corroborate with the theoretical analysis presented in this dissertation.publishedVersio

Estimation and stability of nonlinear control systems under intermittent information with applications to multi-agent robotics

This dissertation investigates the role of intermittent information in estimation and control problems and applies the obtained results to multi-agent tasks in robotics. First, we develop a stochastic hybrid model of mobile networks able to capture a large variety of heterogeneous multi-agent problems and phenomena. This model is applied to a case study where a heterogeneous mobile sensor network cooperatively detects and tracks mobile targets based on intermittent observations. When these observations form a satisfactory target trajectory, a mobile sensor is switched to the pursuit mode and deployed to capture the target. The cost of operating the sensors is determined from the geometric properties of the network, environment and probability of target detection. The above case study is motivated by the Marco Polo game played by children in swimming pools. Second, we develop adaptive sampling of targets positions in order to minimize energy consumption, while satisfying performance guarantees such as increased probability of detection over time, and no-escape conditions. A parsimonious predictor-corrector tracking filter, that uses geometrical properties of targets\u27 tracks to estimate their positions using imperfect and intermittent measurements, is presented. It is shown that this filter requires substantially less information and processing power than the Unscented Kalman Filter and Sampling Importance Resampling Particle Filter, while providing comparable estimation performance in the presence of intermittent information. Third, we investigate stability of nonlinear control systems under intermittent information. We replace the traditional periodic paradigm, where the up-to-date information is transmitted and control laws are executed in a periodic fashion, with the event-triggered paradigm. Building on the small gain theorem, we develop input-output triggered control algorithms yielding stable closed-loop systems. In other words, based on the currently available (but outdated) measurements of the outputs and external inputs of a plant, a mechanism triggering when to obtain new measurements and update the control inputs is provided. Depending on the noise environment, the developed algorithm yields stable, asymptotically stable, and Lp-stable (with bias) closed-loop systems. Control loops are modeled as interconnections of hybrid systems for which novel results on Lp-stability are presented. Prediction of a triggering event is achieved by employing Lp-gains over a finite horizon in the small gain theorem. By resorting to convex programming, a method to compute Lp-gains over a finite horizon is devised. Next, we investigate optimal intermittent feedback for nonlinear control systems. Using the currently available measurements from a plant, we develop a methodology that outputs when to update the control law with new measurements such that a given cost function is minimized. Our cost function captures trade-offs between the performance and energy consumption of the control system. The optimization problem is formulated as a Dynamic Programming problem, and Approximate Dynamic Programming is employed to solve it. Instead of advocating a particular approximation architecture for Approximate Dynamic Programming, we formulate properties that successful approximation architectures satisfy. In addition, we consider problems with partially observable states, and propose Particle Filtering to deal with partially observable states and intermittent feedback. Finally, we investigate a decentralized output synchronization problem of heterogeneous linear systems. We develop a self-triggered output broadcasting policy for the interconnected systems. Broadcasting time instants adapt to the current communication topology. For a fixed topology, our broadcasting policy yields global exponential output synchronization, and Lp-stable output synchronization in the presence of disturbances. Employing a converse Lyapunov theorem for impulsive systems, we provide an average dwell time condition that yields disturbance-to-state stable output synchronization in case of switching topology. Our approach is applicable to directed and unbalanced communication topologies.\u2

Distributed estimation over a low-cost sensor network: a review of state-of-the-art

Proliferation of low-cost, lightweight, and power efficient sensors and advances in networked systems enable the employment of multiple sensors. Distributed estimation provides a scalable and fault-robust fusion framework with a peer-to-peer communication architecture. For this reason, there seems to be a real need for a critical review of existing and, more importantly, recent advances in the domain of distributed estimation over a low-cost sensor network. This paper presents a comprehensive review of the state-of-the-art solutions in this research area, exploring their characteristics, advantages, and challenging issues. Additionally, several open problems and future avenues of research are highlighted

Stability and Control in Complex Networks of Dynamical Systems

Stability analysis of networked dynamical systems has been of interest in many disciplines such as biology and physics and chemistry with applications such as LASER cooling and plasma stability. These large networks are often modeled to have a completely random (Erdös-Rényi) or semi-random (Small-World) topologies. The former model is often used due to mathematical tractability while the latter has been shown to be a better model for most real life networks. The recent emergence of cyber physical systems, and in particular the smart grid, has given rise to a number of engineering questions regarding the control and optimization of such networks. Some of the these questions are: How can the stability of a random network be characterized in probabilistic terms? Can the effects of network topology and system dynamics be separated? What does it take to control a large random network? Can decentralized (pinning) control be effective? If not, how large does the control network needs to be? How can decentralized or distributed controllers be designed? How the size of control network would scale with the size of networked system? Motivated by these questions, we began by studying the probability of stability of synchronization in random networks of oscillators. We developed a stability condition separating the effects of topology and node dynamics and evaluated bounds on the probability of stability for both Erdös-Rényi (ER) and Small-World (SW) network topology models. We then turned our attention to the more realistic scenario where the dynamics of the nodes and couplings are mismatched. Utilizing the concept of ε-synchronization, we have studied the probability of synchronization and showed that the synchronization error, ε, can be arbitrarily reduced using linear controllers. We have also considered the decentralized approach of pinning control to ensure stability in such complex networks. In the pinning method, decentralized controllers are used to control a fraction of the nodes in the network. This is different from traditional decentralized approaches where all the nodes have their own controllers. While the problem of selecting the minimum number of pinning nodes is known to be NP-hard and grows exponentially with the number of nodes in the network we have devised a suboptimal algorithm to select the pinning nodes which converges linearly with network size. We have also analyzed the effectiveness of the pinning approach for the synchronization of oscillators in the networks with fast switching, where the network links disconnect and reconnect quickly relative to the node dynamics. To address the scaling problem in the design of distributed control networks, we have employed a random control network to stabilize a random plant network. Our results show that for an ER plant network, the control network needs to grow linearly with the size of the plant network