Large (> 1000kg), expensive (> $1bn), monolithic, flagship-class satellites (e.g., Hubble, Chandra, James Webb telescopes) may soon be overshadowed by constellations of many, smaller, cheaper, more agile spacecraft working together as a coordinated system. In this work, we study several important attitude and orbit control problems relevant to the formation and operation of small spacecraft constellations. Our example problems include agile attitude maneuvering for multi-target acquisition in an Earth observation application, a centralized approach to constellation formation in low Earth orbit, a distributed approach to constellation formation in Mars areostationary orbit, and orbital rendezvous with objects in the inner Solar System. Our main tool is trajectory optimization, an approach that is appropriate for spacecraft applications where we often seek solutions that are optimal (e.g., minimum-time, minimum-fuel, minimum-control-effort) in the presence of mission or spacecraft constraints. We employ recent advances in sequential convex programming to efficiently handle the nonlinear dynamics and non-convex constraints that exist in our problem formulations.

Important control applications in the fields of robotics and transportation frequently involve high-dimensional systems; for example, robots with many degrees of freedom, or groups of connected automated vehicles (CAVs). In safety-critical applications, the system of interest must also meet a complex set of requirements formalized in a control specification. Formal methods for control synthesis are useful in such applications since they partially automate the controller design process, while also providing guarantees that the control specification will be satisfied. However, when the system dimension becomes too large, such techniques cannot be directly applied due to their computational complexity.

This dissertation focuses on addressing this challenge using hierarchical and distributed control techniques, with an emphasis on applications in transportation. To enable control synthesis for high-dimensional systems, in Chapter 2 we propose techniques for constructing approximate abstractions of a class of interconnected control systems, to be used in a hierarchical control framework. On the application side, in Chapters 3 and 4 we explore the potential for CAVs to improve the efficiency and safety of traffic flows at intersections. For improving efficiency, we discuss how traffic throughput at intersections can be increased by forming vehicle platoons. In particular, this is accomplished using a distributed model-predictive control approach, which is enabled by vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communication. For improving safety, we show how safe behaviors can be accurately characterized for certain challenging driving maneuvers, such as unprotected left turns.

In this dissertation we propose a control synthesis and analysis framework for nonlinear, and neural network (NN) controlled systems with robustness guarantees. We quantify systems' robustness against external disturbances and perturbations using the following measures: (i) the forward reachable set; (ii) the backward reachable set; (iii) the tracking error bound; (iv) the region of attraction. These measures are all characterized by sublevel sets of storage functions satisfying appropriate dissipation inequalities that account for external disturbances and perturbations. Integral quadratic constraints (IQCs) are used to describe perturbations, allowing for a variety of perturbations including parametric uncertainty, unmodeled dynamics, and nonlinear activation functions in NNs. We formulate sum-of-squares (SOS) constraints and Linear matrix inequality conditions by merging dissipation inequalities with IQCs to compute controllers and their associated robustness measures.

We start off by focusing on the finite time horizon robustness of uncertain nonlinear (polynomial) systems, which are modeled as interconnections of nominal polynomial systems and perturbations. We propose a method of outer-approximating the forward reachable set on finite horizons for the uncertain nonlinear systems with controllers given. Then we move from analysis to control synthesis, and present a method for synthesizing a polynomial control law that steers the system to the target set with the goal of maximizing inner-approximations to the backward reachable set. The approximations to both the forward reachable set and backward reachable set are characterized by time dependent polynomial storage functions, and are computed using SOS programming. IQCs with both hard and soft factorizations are used to describe perturbations.

Furthermore, we address robust trajectory planning and control design for nonlinear systems. A hierarchical trajectory planning and control framework is proposed, where a low-fidelity model is used to plan trajectories satisfying planning constraints, and a high-fidelity model is used for synthesizing tracking controllers guaranteeing the boundedness of the error state between the low- and high-fidelity models. We formulate SOS optimizations for computing the tracking controllers and their associated tracking error bound, with the goal of minimizing the volume of the tracking error bound. The tracking error bound is then used to redesign the planning constraints to guarantee safety of the system.

Finally, we move to NN controlled systems. We propose two theorems to prove local stability of NN controlled linear time invariant systems, and to compute inner-approximations to the region of attraction. The first theorem merges dissipation inequalities with local sector quadratic constraints (QCs) to bound the nonlinear activation functions in the NN. The second theorem includes IQCs to allow for perturbations, and further refine the description of activation functions by capturing their slope information. Then we move from analysis to control synthesis. Loop transformation is used to derive stability and safety conditions that are jointly convex in the Lyapunov function, weights of the NN controller, and the Lagrange multipliers for including QCs. These convex conditions are incorporated in the imitation learning process, which trades off between imitation learning accuracy, and size of the region of attraction inner-approximations, to learn robust NN controllers. We propose an alternating direction method of multipliers based algorithm to solve the constrained imitation learning problem.

we demonstrate several techniques to prove safety guarantees for robust control problems with statistical structure; that is, for data-driven dynamical modeling or verification problems where uncertainty is modeled by probability. These guarantees are probabilistic in nature, in accordance with the statistical nature of the uncertainty, and can be derived with limited model assumptions. Indeed, some of the techniques require no more than measurability. We focus on two data-driven control problems: estimation of forward reachable sets from data, and robust control of time- and frequency-domain models defined by a Gaussian process regression model. In the former, we apply scenario optimization and statistical learning theory to obtain probabilistic guarantees of accuracy and confidence with minimal system knowledge. In the latter, we apply the theory of suprema of Gaussian processes to establish high-probability regions of attraction, L2 gain bounds, and integral quadratic constraints for the uncertain system.

Complex systems, such as autonomous vehicles and missile guidance systems, use hierarchical control schemes where each control layer employs a different system model. This approach enhances computational efficiency because using a simpler model in the higher-level control layer reduces computation times, enabling real-time control strategies. This dissertation presents a framework in which a lower-fidelity planning model is employed for online planning, and a tracking controller, synthesized offline, keeps the tracking error between the high-fidelity (“tracking”) model and the planning model within a bounded set. To ensure safety, the error that arises from the different models in each control layer is rigorously accounted for through augmentation of the planner safety constraints with the tracking error bound. Accommodating more sources of real-world uncertainty enhances the safety and usefulness of the control scheme. We next describe a robust extension which utilizes integral quadratic constraints to accommodate input uncertainties such as unknown delays or unmodeled actuator dynamics in the tracking model. Finally, through a case study of shared vehicle control between a human driver and a supervisory autonomous system in longitudinal driving scenarios, we present a novel method called Driver-in-the-Loop Contingency MPC that leverages simplified dynamics to compute invariant sets that guarantee safety with respect to other vehicles. This contribution can be viewed as adding robustness to other agents in the planning layer.

With a growing number of vehicles and an increasing complexity of transportation systems, traffic management and traffic optimization become more and more crucial in mitigating congestion, reducing travel delay and improving traffic state. The concept of a “Smart” city that incorporates various intelligent systems related to infrastructure modification, wireless communication, networking and centralized/decentralized controllers is considered to be the next evolutionary stage of the modern urban world. This dissertation focuses on optimization models and algorithms leveraging traffic management apps (navigation and reservation) and vehicle-to-everything (V2X) communication for mitigating congestion level, reducing fuel consumption and minimizing travel delay for vehicles in urban areas. An ability to share data, receive and send requests and directions allows traffic agents, both on-road (moving) and off-road (parked), to significantly improve utilization of transportation resources. To demonstrate the impact network-level control policies have on a system’s social delay, in our work [1], presented in Chapter 2, we propose a greedy optimization algorithm that eliminates “Braess” routes and derives a paradox-free subnetwork to be implemented in navigation apps. Prior literature that studied the Braess paradox was not able to provide efficient tools for improving the equilibrium state. Topology analysis methods could only predict the occurrence of the paradox but could not deal with it. Results that focused on a single link or route removal were ineffective for large networks. Other methods that discussed tolling or closing roads were too restrictive and required significant infrastructure modifications. Our approach, on the other hand, is more flexible and can be effectively applied to real-world systems to completely eliminate the inefficiency and momentarily reduce total travel time. In addition, we address the challenging task of incorporating queue delay into the network representation by introducing “phantom links”. The following chapters focus on link-level models dealing with local traffic inefficiencies. In Chapters 3 (corresponding work [2]) and 4 (extension to the work [3]) we explore moving traffic management methods and benefits their implementation has with respect to traffic throughput, travel time and fuel consumption. We demonstrate how the optimal platoon formation algorithm can improve traffic progression on urban streets and freeways. Earlier methods either focused on the Ad-hoc protocols, which have limited application in mixed traffic due to its short range, or tried to reduce the travel delay at the cost of the increased traffic disturbance, which was both ineffective and potentially harmful. Our approach, aimed at travel time minimization, takes on the local clustering method and proposes an intelligent platoon merging system that addresses a major part of its common complications and difficulties. In particular, minor infrastructure modifications accompanied by V2I communication protocols enable efficient localization and coordination with minimal traffic disruption and make it possible to artificially increase road capacity and improve congested regions. For fuel consumption, we discuss a novel queue estimation procedure, built upon the vehicle labeling system presented in [3], to be used in a prediction-based model that ties together Speed Advisory System and actuated traffic lights to reduce idling at intersections and smooth driving patterns. Moving on from thru-traffic optimization, in Chapter 5 we focus on curb management issues and propose a delivery vehicles’ operation hours partitioning model to be implemented in a reservation app. Previous works that discussed parking reservations did not consider delivery vehicles and their specific arrival patterns and focused mostly on general cars. Another commonly used parking policy, dynamic parking pricing, extensively studied in the past, is not effective when dealing with delivery vehicles, since they do not usually pay for parking. Our system, on the other hand, focuses on implementing parking equilibrium by physically constraining delivery trucks’ parking choices. In particular, this model is aimed at eliminating double-parking. In Chapter 6 we start looking at practical implementation of curb management techniques. More specifically, we investigate a new delivery vehicle activity monitoring method via dashcam video footage and data analysis algorithms. Switching from the non-scalable and costly data collection techniques, such as static surveillance, we propose a more flexible, low-cost and effective model for parking patterns recognition and curb management implementation.

Diffusively coupled networks, in which different components of a network adjust their behavior according to the local sum of differences between their own and neighbors' states, are a ubiquitous class of spatially-distributed models appearing in engineered and biological systems, and produce surprisingly rich dynamics. In this dissertation, we develop analysis methods and distributed algorithms that exploit the local structure of the network as well as individual component dynamics in order to guarantee the desirable operation of the aggregate network system in the absence of centralized coordination.

We begin by formulating network design problems to guarantee coordination of multi-agent systems by imposing constraints on the underlying structure of the diffusive coupling graph linking agents. The resulting constraint-derived linear matrix inequalities may then be iteratively solved using convex semidefinite programming, resulting in significant performance gains in several multi-agent systems problems. Our approach furthermore identifies critical nodes and edges in a network, and aids in developing strategies to enhance connectivity and robustness.

Next considering the case where desired steady-state trajectories may be time-varying, we derive conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems. Conversely, we highlight cases of diffusion-driven instability, a phenomenon widely hypothesized as a mechanism behind pattern formation in biological systems, in which sufficiently large diffusive coupling may destabilize a spatially homogeneous periodic orbit. The analytical and numerical conditions we derive lend insight to designing distributed laws where the local behavior about a specific attractor is of interest.

We then turn to the problem of developing distributed laws that guarantee spatially uniform behavior globally in diffusively coupled systems. We first study systems with spatially-dependent diffusive coupling, and subsequently examine the case where the network itself has dynamics and adapts according to the agents' states. We develop adaptive laws to guarantee synchronization in systems with spatially-dependent coupling, and apply these results to a number of problems of interest arising from multi-agent systems and cooperative control. We finally address the case where the dynamics of the agents may be subject to spatially-varying input disturbances, and derive distributed laws to guarantee synchronization in the presence of such heterogeneities.