This paper presents a continuous-time coverage algorithm based on the singu- lar perturbations method. The algorithm generates a control for a network of autonomous agents. The algorithm is distributed in the sense that the agents need only information about their neighbors. The singular perturbation method is applied as a means to update the control at a faster time scale to approach performance of a centralized approach. The paper concludes with results from simulations and experiment.

Network optimization problems arise naturally as a way of encoding the coordination task entrusted to multi-agent systems deployed in many areas of engineering, including power, communication, transportation, and swarm robotics. The large-scale nature of these systems coupled with the intrinsic modularity in their structure due to the technological advances in communication, embedded computing, and parallel processing requires a shift from the traditional paradigm of centralized decision-making to a distributed one. This transition, which is essential to harness the true capabilities of modern cyberphysical systems, raises a number of noteworthy challenges as well as opportunities, and has sparked the development of solutions that scale with the number of agents, provide plug-and-play capabilities, and are resilient against single points of failure. Motivated by these considerations, this thesis is a contribution to the growing body of work that deals with the synthesis and analysis of provably correct algorithmic solutions to structured network problems.

Specifically, the thesis is divided into two parts. The first part focuses on synthesizing algorithmic solutions for application-agnostic large-scale network problems. We consider constrained optimization problems where the global objective function is the aggregate of local objectives of the participating agents; the collective goal of the agents and the underlying interaction pattern among them define the constraints. Using continuously differentiable exact penalty functions and globally projected dynamical systems, we then propose privacy-preserving, scalable, accelerated and anytime algorithms to solve these optimization problems. The second part is application-oriented and deals with constrained optimization problems in the context of power systems. In particular, we focus on the utilization of distributed energy resources for frequency regulation in the modern grid. We design distributed time-invariant controllers stabilizing the time-varying power dynamics for primary frequency control, and develop meaningful abstractions for groups of distributed energy resources to participate in the secondary frequency control market.

Multi-robot systems can accomplish a variety of tasks through the power of coordination.There are mutliple benefits. These systems have many advantages over a single very complex robot in term of scalability, versatility, and adaptability. In many cases, the robots cannot accomplish much by itself, but coordination empowers them the ability to complete various objectives. Even when the individuals robots are very capable, coordination can increase robot efficiency by allocating robots with fitting tasks. In both scenarios, the problem of balancing the system objectives arise naturally, and properly addressing it can lead to better overall performance. Motivated by this observation, this dissertation seek to understand how different objectives can be put together and how to strike a balance between them. We consider control objectives at the most fundamental level to control systems, such as stability, system safety, smoothness of the controller, performance, and resources spent for accomplishing tasks.

This dissertation is divided into two parts. The first part deals with control laws that considerboth stability and safety objectives. We design controllers that can satisfy simultaneously conditions given by control Lyapunov functions and control barrier functions. Depending on the smoothness properties of the given functions, we guarantee the continuity or smoothness of the controller. In particular, we design a continuous controller for connectivity maintenance, and also design a universal formula for smooth safe stabilization. In the second part, we study the resource-efficient implementation of control laws using event-triggered control. We improve the existing event-triggered control framework for stabilization by incorporating prescribed performance into the design. The resulting framework further enhances the advantage of resource conservation characteristic of event-triggered control. We build on the proposed framework to design an intrinsically Zeno-free distributed triggering mechanisms for network systems. In addition, this dissertation also explores unconventional ways to utilize the event-triggered control framework. In one way, we deviate ourselves from trigger conditions that use Lyapunov functions replacing it instead with barrier certificate and develop an event-triggered control framework for safety objectives. Another interesting way we explore to use event-triggered control is in the context of human supervised multiobjective optimization. In this setting, we consider the human as a valuable resource, which should be used sparingly, and use event-triggered control to accommodate various models of human performance, such as constraints on the response time and the interaction frequency.

Autonomous unmanned vehicles (UxVs) can be useful in many scenarios

including disaster relief, production and manufacturing, as well as

carrying out Naval missions such as surveillance, mapping of unknown

regions and pursuit of other hostile vehicles. When considering

these scenarios, one of the most difficult challenges is determining

which actions or tasks the vehicles should take in order to most

efficiently satisfy the objectives. This challenge becomes more

difficult with the inclusion of multiple vehicles, because the

action and state space scale exponentially with the number of

agents. Many planning algorithms suffer from the curse of

dimensionality as more agents are included, sampling for

suitable actions in the joint action space becomes infeasible within

a reasonable amount of time. To enable autonomy, methods that can be

applied to a variety of scenarios are invaluable because they reduce

human involvement and time.

Recently, advances in technology enable algorithms that require more

computational power to be effective but work in broader

frameworks. We offer three main approaches to multi-agent planning

which are all inspired by model-based reinforcement learning.

First, we address the curse of dimensionality and investigate how to

spatially reduce the state space of massive environments where

agents are deployed. We do this in a hierarchical fashion by

searching subspaces of the environment, called sub-environments, and

creating plans to optimally take actions in those sub-environments.

Next, we utilize game-theoretic techniques paired with simulated

annealing as an approach for agent cooperation when planning in a

finite time horizon. One problem with this approach is that agents

are capable of breaking promises with other agents right before

execution. To address this, we propose several variations that

discourage agents from changing plans in the near future and

encourages joint planning in the long term. Lastly, we propose a

tree-search algorithm that is aided by a convolutional neural

network. The convolutional neural network takes advantage of

spatial features that are natural in UxV deployment and offers

recommendations for action selection during tree search. In

addition, we propose some design features for the tree search that

target multi-agent deployment applications.

Ranging from natural phenomena such as biological and chemical systems to artificial technologies such as mechanical and electronic devices, dynamical systems form an inseparable part of the world surrounding us. Understanding, modeling, predicting, and controlling such systems have always been the leading goals of science and engineering. While in the past centuries, the most advances in the field of dynamical systems were mainly analytical and based on limited observations, in the last decade, we have witnessed a rapid growth in our ability to gather, store, and process data. This data-driven revolution has imposed a high demand for new viewpoints and systematic structures that can effectively utilize the available modern tools. The Koopman operator theory for dynamical systems has recently gained widespread attention. Unlike the traditional state space methods, which explain the evolution of the states according to the dynamics, the Koopman operator characterizes the effect of the dynamics on functions in a linear function space. Despite the system being linear or nonlinear, its associated Koopman operator is always linear. This linearity proves to be extremely useful for algorithmic computations.

This dissertation is focused on the data-driven analysis of dynamical systems based on their associated Koopman operator. Given the infinite-dimensional nature of the Koopman operator, we aim to find finite-dimensional spaces on which the operator's action can be captured accurately. Such finite-dimensional spaces must be close to being invariant under the action of the Koopman operator; otherwise, the approximation will be erroneous. We provide data-driven algebraic methods to provably find the exact maximal Koopman-invariant subspace and all Koopman eigenfunctions in any arbitrary finite-dimensional space of functions. We also offer equivalent algorithms tailored for working with large and streaming data sets, as well as parallel computing hardware. Keeping in mind that an exact finite-dimensional linear model might not capture complete information for some systems, we also consider approximating subspaces to capture more information about the dynamics. We provide data-driven measures to quantify how close a linear space of functions is to being invariant under the Koopman operator. In addition, we provide an algebraic procedure that can approximate Koopman-invariant subspaces with tunable accuracy.

Many natural and man-made systems, ranging from the

nervous system to power and transportation grids to societies, exhibit

dynamic behaviors that evolve over a sparse and complex network.

This networked aspect raises significant challenges and opportunities for the identification, analysis, and control of such dynamic behaviors. While some of these challenges emanate from the networked aspect \emph{per se} (such as the sparsity of connections between system components and the interplay between nodal \emph{communication} and network dynamics), various challenges arise from the specific application areas (such as privacy concerns in cyber-physical systems or the need for \emph{scalable} algorithm designs due to the large size of various biological and engineered networks). On the other hand, networked systems provide significant opportunities and allow for performance and robustness levels that are far beyond reach for centralized systems, with examples ranging from the Internet (of Things) to the smart grid and the brain. This dissertation aims to address several of these challenges and harness these opportunities.

The dissertation is divided into three parts. In the first part, we study privacy concerns whose resolution is vital for the utility of networked cyber-physical systems. We study the problems of average consensus and convex optimization as two principal distributed computations occurring over networks and design algorithm with rigorous privacy guarantees that provide a \emph{best achievable} tradeoff between network utility and privacy. In the second part, we analyze networks with resource constraints. More specifically, we study three problems of stabilization under communication (bandwidth and latency) limitations in sensing and actuation, optimal time-varying control scheduling problem under limited number of actuators and control energy, and the structure identification problem of under-sensed networks (i.e., networks with latent nodes). Finally in the last part, we focus on the intersection of networked dynamical systems and neuroscience and draw connections between brain network dynamics and two extensively studied but yet not fully understood neuro-cognitive phenomena: goal-driven selective attention and neural oscillations. Using a novel axiomatic approach, we establish these connections in the form of necessary and/or sufficient conditions on the network structure that match the network output trajectories with experimentally observed brain activity.

From the electric power grid, to social networks, to the human brain, many systems of engineering and scientific interest are obtained by interconnecting simpler subsystems. This interconnection can be as simple as a feedback loop, or have a complicated network structure. However, in each case, the dynamic coupling results in complicated behaviors that cannot be explained simply by looking at the constituent components in isolation. This poses two major challenges: first, in terms of developing mathematical models to gain insight into the behavior of complex systems, and second, in terms of optimizing their operation or controlling them. The goal of this thesis is to develop a mathematical framework to tackle these challenges using tools from nonlinear dynamics, control theory, optimization, and network science.

This thesis is divided into two parts, each focusing on a specific class of interconnected system arising in real world applications. In the first part, we focus on developing a ``systems theory'' of optimization algorithms, in order to understand their properties and study their interconnection with physical processes. We demonstrate that tools from safety-critical control can be used to synthesize flows solving constrained nonlinear optimization problems, with safety, stability and robustness guarantees that make them ideal for online implementation when interconnected with physical processes. The second part of this thesis discusses mathematical modeling of interconnected neurological systems, in order to understand and control epileptic seizures. We model the epileptic brain with Linear Threshold Networks (LTNs), and analyze the interplay between the network structure and the dynamical properties they exhibit. We characterize conditions on the network structure under which oscillations spread in LTNs, and develop strategies to optimally modify networks to prevent the spread of epileptic seizures.

Electric power systems safety is a fundamental aspect of the operation and management of the grid. In order to maintain safety, the power system is operated around a nominal frequency. In fact, large frequency fluctuations can trigger generator relay-protection mechanisms and load shedding, which may further jeopardize network integrity, leading to cascading failures. Without appropriate estimations on the possible consequences resulting from contingency, operational architectures, and control safeguards in place, the likelihood of such events is not negligible, given that the high penetration of non-rotational renewable resources provides less inertia, possibly inducing higher frequency excursions. These observations motivate us in this thesis to develop approximation and control schemes to efficiently estimate the transient-state evolution subject to disturbances and contingencies and further actively mitigate undesired transient frequency deviations.

This thesis first develops methods to efficiently compute the set of disturbances on a power network that do not tip the frequency of each bus and the power flow in each transmission line beyond their respective bounds. For a linearized power network model, we propose a sampling method to provide superset and subset approximations with a desired accuracy of the set of feasible disturbances. We also introduce an error metric to measure the approximation gap and design an algorithm that is able to reduce its value without impacting the complexity of the resulting set approximations.

As a natural follow-up to our on approximating feasible disturbances, we seek to further regulate transient frequency via novel control schemes. With regard to this, this thesis proposes three control strategies that all achieve local stabilization of power networks characterized by nonlinear swing equations and, at the same time, delimit the transient frequencies of targeted buses to a desired safe interval. To handle the coordination of large numbers of resources in an adaptive and scalable fashion, all three controllers can be implemented in an either partially or fully distributed fashion. Specifically, we synthesize the first transient frequency controller by having it satisfy a transient frequency constraint and an asymptotic stability constraint. Benefitting from its structural simplicity, the controller can be implemented in a distributed fashion by merely allowing each controlled bus physically measure the states of neighbors. To reduce the control effort, the second MPC-based controller enables control command cooperation by communication; however, the coordination is limited within a designed range, and the control algorithm is only partially distributed, potentially non-Lipschitz, and not as computationally efficient. The third controller successfully addresses all these issues via a bilayered structure and information exchange with up to 2-hop neighbors.

The brain, composed of billions of interconnected neurons, forms a complex network that exhibits an incredibly wide array of behaviors. Treating this structure as a dynamical system provides a multitude of tools to use to model and understand the relationship between structure and function in the brain. As subnetworks exist at all levels in the brain, ranging from networks of individual neurons to networks of entire regions, a diverse set of models, each with different properties have been considered to study different dynamic brain behaviors. One such class of models are firing rate models, which monitor the average spike rate of populations of neurons. A particular firing rate model is the linear-threshold network model, which exhibits a wide range of rich behaviors based on the underlying network structure and inputs. This ability to exhibit a variety of behaviors motivates the use of this model to study a variety of dynamical behaviors observed in the brain.

This dissertation considers three problems within the realm of modeling dynamical brain behaviors with the linear-threshold model. First, motivated by the appearance of oscillatory behavior when observing brain activity, we study the existence of oscillations in the linear-threshold dynamics. In order to provide sufficient conditions for oscillations in specific network topologies we also provide conditions for the stability of equilibrium points that maintain a specific support. Second, we discuss the dynamical brain behavior of selective inhibition and recruitment. We consider thalamocortical networks with both hierarchical and star-connected topologies, and focus on how the inclusion of the thalamus can improve the stabilizability properties of the linear-threshold dynamics relevant to the application. We finish by investigating the problem of reference tracking for the linear-threshold dynamics, which can be used to frame many brain behaviors. We approach this both analytically and with a data-driven approach to better match observations on how the brain processes information.