Distributed Optimal Control of Cyber-Physical Systems: Controller Synthesis, Architecture Design and System Identification

The centralized paradigm of a single controller and a single plant upon which modern control theory is built is no longer applicable to modern cyber-physical systems of interest, such as the power-grid, software defined networks or automated highways systems, as these are all large-scale and spatially distributed. Both the scale and the distributed nature of these systems has motivated the decentralization of control schemes into local sub-controllers that measure, exchange and act on locally available subsets of the globally available system information. This decentralization of control logic leads to different decision makers acting on asymmetric information sets, introduces the need for coordination between them, and perhaps not surprisingly makes the resulting optimal control problem much harder to solve. In fact, shortly after such questions were posed, it was realized that seemingly simple decentralized optimal control problems are computationally intractable to solve, with the Wistenhausen counterexample being a famous instance of this phenomenon. Spurred on by this perhaps discouraging result, a concerted 40 year effort to identify tractable classes of distributed optimal control problems culminated in the notion of quadratic invariance, which loosely states that if sub-controllers can exchange information with each other at least as quickly as the effect of their control actions propagates through the plant, then the resulting distributed optimal control problem admits a convex formulation.

The identification of quadratic invariance as an appropriate means of "convexifying" distributed optimal control problems led to a renewed enthusiasm in the controller synthesis community, resulting in a rich set of results over the past decade. The contributions of this thesis can be seen as being a part of this broader family of results, with a particular focus on closing the gap between theory and practice by relaxing or removing assumptions made in the traditional distributed optimal control framework. Our contributions are to the foundational theory of distributed optimal control, and fall under three broad categories, namely controller synthesis, architecture design and system identification.

We begin by providing two novel controller synthesis algorithms. The first is a solution to the distributed H-infinity optimal control problem subject to delay constraints, and provides the only known exact characterization of delay-constrained distributed controllers satisfying an H-infinity norm bound. The second is an explicit dynamic programming solution to a two player LQR state-feedback problem with varying delays. Accommodating varying delays represents an important first step in combining distributed optimal control theory with the area of Networked Control Systems that considers lossy channels in the feedback loop. Our next set of results are concerned with controller architecture design. When designing controllers for large-scale systems, the architectural aspects of the controller such as the placement of actuators, sensors, and the communication links between them can no longer be taken as given -- indeed the task of designing this architecture is now as important as the design of the control laws themselves. To address this task, we formulate the Regularization for Design (RFD) framework, which is a unifying computationally tractable approach, based on the model matching framework and atomic norm regularization, for the simultaneous co-design of a structured optimal controller and the architecture needed to implement it. Our final result is a contribution to distributed system identification. Traditional system identification techniques such as subspace identification are not computationally scalable, and destroy rather than leverage any a priori information about the system's interconnection structure. We argue that in the context of system identification, an essential building block of any scalable algorithm is the ability to estimate local dynamics within a large interconnected system. To that end we propose a promising heuristic for identifying the dynamics of a subsystem that is still connected to a large system. We exploit the fact that the transfer function of the local dynamics is low-order, but full-rank, while the transfer function of the global dynamics is high-order, but low-rank, to formulate this separation task as a nuclear norm minimization problem. Finally, we conclude with a brief discussion of future research directions, with a particular emphasis on how to incorporate the results of this thesis, and those of optimal control theory in general, into a broader theory of dynamics, control and optimization in layered architectures.

Efficient methods for stochastic optimal control

The Hamilton Jacobi Bellman (HJB) equation is central to stochastic optimal control (SOC) theory, yielding the optimal solution to general problems specified by known dynamics and a specified cost functional. Given the assumption of quadratic cost on the control input, it is well known that the HJB reduces to a particular partial differential equation (PDE). While powerful, this reduction is not commonly used as the PDE is of second order, is nonlinear, and examples exist where the problem may not have a solution in a classical sense. Furthermore, each state of the system appears as another dimension of the PDE, giving rise to the curse of dimensionality. Since the number of degrees of freedom required to solve the optimal control problem grows exponentially with dimension, the problem becomes intractable for systems with all but modest dimension.

In the last decade researchers have found that under certain, fairly non-restrictive structural assumptions, the HJB may be transformed into a linear PDE, with an interesting analogue in the discretized domain of Markov Decision Processes (MDP). The work presented in this thesis uses the linearity of this particular form of the HJB PDE to push the computational boundaries of stochastic optimal control.

This is done by crafting together previously disjoint lines of research in computation. The first of these is the use of Sum of Squares (SOS) techniques for synthesis of control policies. A candidate polynomial with variable coefficients is proposed as the solution to the stochastic optimal control problem. An SOS relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function. It is shown that these results extend to arbitrary parabolic and elliptic PDEs, yielding a novel method for Uncertainty Quantification (UQ) of systems governed by partial differential constraints. Domain decomposition techniques are also made available, allowing for such problems to be solved via parallelization and low-order polynomials.

The optimization-based SOS technique is then contrasted with the Separated Representation (SR) approach from the applied mathematics community. The technique allows for systems of equations to be solved through a low-rank decomposition that results in algorithms that scale linearly with dimensionality. Its application in stochastic optimal control allows for previously uncomputable problems to be solved quickly, scaling to such complex systems as the Quadcopter and VTOL aircraft. This technique may be combined with the SOS approach, yielding not only a numerical technique, but also an analytical one that allows for entirely new classes of systems to be studied and for stability properties to be guaranteed.

The analysis of the linear HJB is completed by the study of its implications in application. It is shown that the HJB and a popular technique in robotics, the use of navigation functions, sit on opposite ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. Analytical solutions to the HJB in these settings are available in simplified domains, yielding guidance towards optimality for approximation schemes. Finally, the use of HJB equations in temporal multi-task planning problems is investigated. It is demonstrated that such problems are reducible to a sequence of SOC problems linked via boundary conditions. The linearity of the PDE allows us to pre-compute control policy primitives and then compose them, at essentially zero cost, to satisfy a complex temporal logic specification.

Stochastic optimal control

H. J. Kushner has obtained the differential equation satisfied by the optimal feedback control law for a stochastic control system in which the plant dynamics and observations are perturbed by independent additive Gaussian white noise processes. However, the differentiation includes the first and second functional derivatives and, except for a restricted set of systems, is too complex to solve with present techniques.

This investigation studies the optimal control law for the open loop system and incorporates it in a sub-optimal feedback control law. This suboptimal control law's performance is at least as good as that of the optimal control function and satisfies a differential equation involving only the first functional derivative. The solution of this equation is equivalent to solving two two-point boundary valued integro-partial differential equations. An approximate solution has advantages over the conventional approximate solution of Kushner's equation.

As a result of this study, well known results of deterministic optimal control are deduced from the analysis of optimal open loop control.

Control of dynamical systems with temporal logic specifications

This thesis is motivated by safety-critical applications involving autonomous air, ground, and space vehicles carrying out complex tasks in uncertain and adversarial environments. We use temporal logic as a language to formally specify complex tasks and system properties. Temporal logic specifications generalize the classical notions of stability and reachability that are studied in the control and hybrid systems communities. Given a system model and a formal task specification, the goal is to automatically synthesize a control policy for the system that ensures that the system satisfies the specification. This thesis presents novel control policy synthesis algorithms for optimal and robust control of dynamical systems with temporal logic specifications. Furthermore, it introduces algorithms that are efficient and extend to high-dimensional dynamical systems.

The first contribution of this thesis is the generalization of a classical linear temporal logic (LTL) control synthesis approach to optimal and robust control. We show how we can extend automata-based synthesis techniques for discrete abstractions of dynamical systems to create optimal and robust controllers that are guaranteed to satisfy an LTL specification. Such optimal and robust controllers can be computed at little extra computational cost compared to computing a feasible controller.

The second contribution of this thesis addresses the scalability of control synthesis with LTL specifications. A major limitation of the standard automaton-based approach for control with LTL specifications is that the automaton might be doubly-exponential in the size of the LTL specification. We introduce a fragment of LTL for which one can compute feasible control policies in time polynomial in the size of the system and specification. Additionally, we show how to compute optimal control policies for a variety of cost functions, and identify interesting cases when this can be done in polynomial time. These techniques are particularly relevant for online control, as one can guarantee that a feasible solution can be found quickly, and then iteratively improve on the quality as time permits.

The final contribution of this thesis is a set of algorithms for computing feasible trajectories for high-dimensional, nonlinear systems with LTL specifications. These algorithms avoid a potentially computationally-expensive process of computing a discrete abstraction, and instead compute directly on the system's continuous state space. The first method uses an automaton representing the specification to directly encode a series of constrained-reachability subproblems, which can be solved in a modular fashion by using standard techniques. The second method encodes an LTL formula as mixed-integer linear programming constraints on the dynamical system. We demonstrate these approaches with numerical experiments on temporal logic motion planning problems with high-dimensional (10+ states) continuous systems.

Optimization and Control of Power Flow in Distribution Networks

Climate change is arguably the most critical issue facing our generation and the next. As we move towards a sustainable future, the grid is rapidly evolving with the integration of more and more renewable energy resources and the emergence of electric vehicles. In particular, large scale adoption of residential and commercial solar photovoltaics (PV) plants is completely changing the traditional slowly-varying unidirectional power flow nature of distribution systems. High share of intermittent renewables pose several technical challenges, including voltage and frequency control. But along with these challenges, renewable generators also bring with them millions of new DC-AC inverter controllers each year. These fast power electronic devices can provide an unprecedented opportunity to increase energy efficiency and improve power quality, if combined with well-designed inverter control algorithms. The main goal of this dissertation is to develop scalable power flow optimization and control methods that achieve system-wide efficiency, reliability, and robustness for power distribution networks of future with high penetration of distributed inverter-based renewable generators.

Proposed solutions to power flow control problems in the literature range from fully centralized to fully local ones. In this thesis, we will focus on the two ends of this spectrum. In the first half of this thesis (chapters 2 and 3), we seek optimal solutions to voltage control problems provided a centralized architecture with complete information. These solutions are particularly important for better understanding the overall system behavior and can serve as a benchmark to compare the performance of other control methods against. To this end, we first propose a branch flow model (BFM) for the analysis and optimization of radial and meshed networks. This model leads to a new approach to solve optimal power flow (OPF) problems using a two step relaxation procedure, which has proven to be both reliable and computationally efficient in dealing with the non-convexity of power flow equations in radial and weakly-meshed distribution networks. We will then apply the results to fast time- scale inverter var control problem and evaluate the performance on real-world circuits in Southern California Edison’s service territory.

The second half (chapters 4 and 5), however, is dedicated to study local control approaches, as they are the only options available for immediate implementation on today’s distribution networks that lack sufficient monitoring and communication infrastructure. In particular, we will follow a reverse and forward engineering approach to study the recently proposed piecewise linear volt/var control curves. It is the aim of this dissertation to tackle some key problems in these two areas and contribute by providing rigorous theoretical basis for future work.

Optimal guidance of low-thrust interplanetary space vehicles

The low-thrust guidance problem is defined as the minimum terminal variance (MTV) control of a space vehicle subjected to random perturbations of its trajectory. To accomplish this control task, only bounded thrust level and thrust angle deviations are allowed, and these must be calculated based solely on the information gained from noisy, partial observations of the state. In order to establish the validity of various approximations, the problem is first investigated under the idealized conditions of perfect state information and negligible dynamic errors. To check each approximate model, an algorithm is developed to facilitate the computation of the open loop trajectories for the nonlinear bang-bang system. Using the results of this phase in conjunction with the Ornstein-Uhlenbeck process as a model for the random inputs to the system, the MTV guidance problem is reformulated as a stochastic, bang-bang, optimal control problem. Since a complete analytic solution seems to be unattainable, asymptotic solutions are developed by numerical methods. However, it is shown analytically that a Kalman filter in cascade with an appropriate nonlinear MTV controller is an optimal configuration. The resulting system is simulated using the Monte Carlo technique and is compared to other guidance schemes of current interest.

Robust Control of Evolutionary Dynamics

The application of principles from evolutionary biology has long been used to gain new insights into the progression and clinical control of both infectious diseases and neoplasms. This iterative evolutionary process consists of expansion, diversification and selection within an adaptive landscape - species are subject to random genetic or epigenetic alterations that result in variations; genetic information is inherited through asexual reproduction and strong selective pressures such as therapeutic intervention can lead to the adaptation and expansion of resistant variants. These principles lie at the center of modern evolutionary synthesis and constitute the primary reasons for the development of resistance and therapeutic failure, but also provide a framework that allows for more effective control.

A model system for studying the evolution of resistance and control of therapeutic failure is the treatment of chronic HIV-1 infection by broadly neutralizing antibody (bNAb) therapy. A relatively recent discovery is that a minority of HIV-infected individuals can produce broadly neutralizing antibodies, that is, antibodies that inhibit infection by many strains of HIV. Passive transfer of human antibodies for the prevention and treatment of HIV-1 infection is increasingly being considered as an alternative to a conventional vaccine. However, recent evolution studies have uncovered that antibody treatment can exert selective pressure on virus that results in the rapid evolution of resistance. In certain cases, complete resistance to an antibody is conferred with a single amino acid substitution on the viral envelope of HIV.

The challenges in uncovering resistance mechanisms and designing effective combination strategies to control evolutionary processes and prevent therapeutic failure apply more broadly. We are motivated by two questions: Can we predict the evolution to resistance by characterizing genetic alterations that contribute to modified phenotypic fitness? Given an evolutionary landscape and a set of candidate therapies, can we computationally synthesize treatment strategies that control evolution to resistance?

To address the first question, we propose a mathematical framework to reason about evolutionary dynamics of HIV from computationally derived Gibbs energy fitness landscapes -- expanding the theoretical concept of an evolutionary landscape originally conceived by Sewall Wright to a computable, quantifiable, multidimensional, structurally defined fitness surface upon which to study complex HIV evolutionary outcomes.

To design combination treatment strategies that control evolution to resistance, we propose a methodology that solves for optimal combinations and concentrations of candidate therapies, and allows for the ability to quantifiably explore tradeoffs in treatment design, such as limiting the number of candidate therapies in the combination, dosage constraints and robustness to error. Our algorithm is based on the application of recent results in optimal control to an HIV evolutionary dynamics model and is constructed from experimentally derived antibody resistant phenotypes and their single antibody pharmacodynamics. This method represents a first step towards integrating principled engineering techniques with an experimentally based mathematical model in the rational design of combination treatment strategies and offers predictive understanding of the effects of combination therapies of evolutionary dynamics and resistance of HIV. Preliminary in vitro studies suggest that the combination antibody therapies predicted by our algorithm can neutralize heterogeneous viral populations despite containing resistant mutations.

I. Foehn winds of southern California. II. Foehn wind cyclo-genesis. III. Weather conditions associated with the Akron disaster. IV. The Los Angeles storm of December 30, 1933 to January 1, 1934

I. Foehn winds of southern California.
An investigation of the hot, dry and dust laden winds occurring in the late fall and early winter in the Los Angeles Basin and attributed in the past to the influences of the desert regions to the north revealed that these currents were of a foehn nature. Their properties were found to be entirely due to dynamical heating produced in the descent from the high level areas in the interior to the lower Los Angeles Basin. Any dust associated with the phenomenon was found to be acquired from the Los Angeles area rather than transported from the desert. It was found that the frequency of occurrence of a mild type foehn of this nature during this season was sufficient to warrant its classification as a winter monsoon. This results from the topography of the Los Angeles region which allows an easy entrance to the air from the interior by virtue of the low level mountain passes north of the area. This monsoon provides the mild winter climate of southern California since temperatures associated with the foehn currents are far higher than those experienced when maritime air from the adjacent Pacific Ocean occupies the region.

II. Foehn wind cyclo-genesis.
Intense anticyclones frequently build up over the high level regions of the Great Basin and Columbia Plateau which lie between the Sierra Nevada and Cascade Mountains to the west and the Rocky Mountains to the east. The outflow from these anticyclones produce extensive foehns east of the Rockies in the comparatively low level areas of the middle west and the Canadian provinces of Alberta and Saskatchewan. Normally at this season of the year very cold polar continental air masses are present over this territory and with the occurrence of these foehns marked discontinuity surfaces arise between the warm foehn current, which is obliged to slide over a colder mass, and the Pc air to the east. Cyclones are easily produced from this phenomenon and take the form of unstable waves which propagate along the discontinuity surface between the two dissimilar masses. A continual series of such cyclones was found to occur as long as the Great Basin anticyclone is maintained with undiminished intensity.

III. Weather conditions associated with the Akron disaster.
This situation illustrates the speedy development and propagation of young disturbances in the eastern United States during the spring of the year under the influence of the conditionally unstable tropical maritime air masses which characterise the region. It also furnishes an excellent example of the superiority of air mass and frontal methods of weather prediction for aircraft operation over the older methods based upon pressure distribution.

IV. The Los Angeles storm of December 30, 1933 to January 1, 1934.
This discussion points out some of the fundamental interactions occurring between air masses of the North Pacific Ocean in connection with Pacific Coast storms and the value of topographic and aerological considerations in predicting them. Estimates of rainfall intensity and duration from analyses of this type may be made and would prove very valuable in the Los Angeles area in connection with flood control problems.

Nonlinear oscillations in discrete and continuous systems

The first thesis topic is a perturbation method for resonantly coupled nonlinear oscillators. By successive near-identity transformations of the original equations, one obtains new equations with simple structure that describe the long time evolution of the motion. This technique is related to two-timing in that secular terms are suppressed in the transformation equations. The method has some important advantages. Appropriate time scalings are generated naturally by the method, and don't need to be guessed as in two-timing. Furthermore, by continuing the procedure to higher order, one extends (formally) the time scale of valid approximation. Examples illustrate these claims. Using this method, we investigate resonance in conservative, non-conservative and time dependent problems. Each example is chosen to highlight a certain aspect of the method.

The second thesis topic concerns the coupling of nonlinear chemical oscillators. The first problem is the propagation of chemical waves of an oscillating reaction in a diffusive medium. Using two-timing, we derive a nonlinear equation that determines how spatial variations in the phase of the oscillations evolves in time. This result is the key to understanding the propagation of chemical waves. In particular, we use it to account for certain experimental observations on the Belusov-Zhabotinskii reaction.

Next, we analyse the interaction between a pair of coupled chemical oscillators. This time, we derive an equation for the phase shift, which measures how much the oscillators are out of phase. This result is the key to understanding M. Marek's and I. Stuchl's results on coupled reactor systems. In particular, our model accounts for synchronization and its bifurcation into rhythm splitting.

Finally, we analyse large systems of coupled chemical oscillators. Using a continuum approximation, we demonstrate mechanisms that cause auto-synchronization in such systems.

Optimal design of building structures using genetic algorithms

A general framework for multi-criteria optimal design is presented which is well-suited for automated design of structural systems. A systematic computer-aided optimal design decision process is developed which allows the designer to rapidly evaluate and improve a proposed design by taking into account the major factors of interest related to different aspects such as design, construction, and operation.

The proposed optimal design process requires the selection of the most promising choice of design parameters taken from a large design space, based on an evaluation using specified criteria. The design parameters specify a particular design, and so they relate to member sizes, structural configuration, etc. The evaluation of the design uses performance parameters which may include structural response parameters, risks due to uncertain loads and modeling errors, construction and operating costs, etc. Preference functions are used to implement the design criteria in a "soft" form. These preference functions give a measure of the degree of satisfaction of each design criterion. The overall evaluation measure for a design is built up from the individual measures for each criterion through a preference combination rule. The goal of the optimal design process is to obtain a design that has the highest overall evaluation measure - an optimization problem.

Genetic algorithms are stochastic optimization methods that are based on evolutionary theory. They provide the exploration power necessary to explore high-dimensional search spaces to seek these optimal solutions. Two special genetic algorithms, hGA and vGA, are presented here for continuous and discrete optimization problems, respectively.

The methodology is demonstrated with several examples involving the design of truss and frame systems. These examples are solved by using the proposed hGA and vGA.

Formal methods for control synthesis in partially observed environments : application to autonomous robotic manipulation

Modern robots are increasingly expected to function in uncertain and dynamically challenging environments, often in proximity with humans. In addition, wide scale adoption of robots requires on-the-fly adaptability of software for diverse application. These requirements strongly suggest the need to adopt formal representations of high level goals and safety specifications, especially as temporal logic formulas. This approach allows for the use of formal verification techniques for controller synthesis that can give guarantees for safety and performance. Robots operating in unstructured environments also face limited sensing capability. Correctly inferring a robot's progress toward high level goal can be challenging.

This thesis develops new algorithms for synthesizing discrete controllers in partially known environments under specifications represented as linear temporal logic (LTL) formulas. It is inspired by recent developments in finite abstraction techniques for hybrid systems and motion planning problems. The robot and its environment is assumed to have a finite abstraction as a Partially Observable Markov Decision Process (POMDP), which is a powerful model class capable of representing a wide variety of problems. However, synthesizing controllers that satisfy LTL goals over POMDPs is a challenging problem which has received only limited attention.

This thesis proposes tractable, approximate algorithms for the control synthesis problem using Finite State Controllers (FSCs). The use of FSCs to control finite POMDPs allows for the closed system to be analyzed as finite global Markov chain. The thesis explicitly shows how transient and steady state behavior of the global Markov chains can be related to two different criteria with respect to satisfaction of LTL formulas. First, the maximization of the probability of LTL satisfaction is related to an optimization problem over a parametrization of the FSC. Analytic computation of gradients are derived which allows the use of first order optimization techniques.

The second criterion encourages rapid and frequent visits to a restricted set of states over infinite executions. It is formulated as a constrained optimization problem with a discounted long term reward objective by the novel utilization of a fundamental equation for Markov chains - the Poisson equation. A new constrained policy iteration technique is proposed to solve the resulting dynamic program, which also provides a way to escape local maxima.

The algorithms proposed in the thesis are applied to the task planning and execution challenges faced during the DARPA Autonomous Robotic Manipulation - Software challenge.

Optimal Data Distributions in Machine Learning

In the first part of the thesis we explore three fundamental questions that arise naturally when we conceive a machine learning scenario where the training and test distributions can differ. Contrary to conventional wisdom, we show that in fact mismatched training and test distribution can yield better out-of-sample performance. This optimal performance can be obtained by training with the dual distribution. This optimal training distribution depends on the test distribution set by the problem, but not on the target function that we want to learn. We show how to obtain this distribution in both discrete and continuous input spaces, as well as how to approximate it in a practical scenario. Benefits of using this distribution are exemplified in both synthetic and real data sets.

In order to apply the dual distribution in the supervised learning scenario where the training data set is fixed, it is necessary to use weights to make the sample appear as if it came from the dual distribution. We explore the negative effect that weighting a sample can have. The theoretical decomposition of the use of weights regarding its effect on the out-of-sample error is easy to understand but not actionable in practice, as the quantities involved cannot be computed. Hence, we propose the Targeted Weighting algorithm that determines if, for a given set of weights, the out-of-sample performance will improve or not in a practical setting. This is necessary as the setting assumes there are no labeled points distributed according to the test distribution, only unlabeled samples.

Finally, we propose a new class of matching algorithms that can be used to match the training set to a desired distribution, such as the dual distribution (or the test distribution). These algorithms can be applied to very large datasets, and we show how they lead to improved performance in a large real dataset such as the Netflix dataset. Their computational complexity is the main reason for their advantage over previous algorithms proposed in the covariate shift literature.

In the second part of the thesis we apply Machine Learning to the problem of behavior recognition. We develop a specific behavior classifier to study fly aggression, and we develop a system that allows analyzing behavior in videos of animals, with minimal supervision. The system, which we call CUBA (Caltech Unsupervised Behavior Analysis), allows detecting movemes, actions, and stories from time series describing the position of animals in videos. The method summarizes the data, as well as it provides biologists with a mathematical tool to test new hypotheses. Other benefits of CUBA include finding classifiers for specific behaviors without the need for annotation, as well as providing means to discriminate groups of animals, for example, according to their genetic line.

Incremental Control Synthesis for Robotics in the Presence of Temporal Logic Specifications

This thesis presents methods for incrementally constructing controllers in the presence of uncertainty and nonlinear dynamics. The basic setting is motion planning subject to temporal logic specifications. Broadly, two categories of problems are treated. The first is reactive formal synthesis when so-called discrete abstractions are available. The fragment of linear-time temporal logic (LTL) known as GR(1) is used to express assumptions about an adversarial environment and requirements of the controller. Two problems of changes to a specification are posed that concern the two major aspects of GR(1): safety and liveness. Algorithms providing incremental updates to strategies are presented as solutions. In support of these, an annotation of strategies is developed that facilitates repeated modifications. A variety of properties are proven about it, including necessity of existence and sufficiency for a strategy to be winning. The second category of problems considered is non-reactive (open-loop) synthesis in the absence of a discrete abstraction. Instead, the presented stochastic optimization methods directly construct a control input sequence that achieves low cost and satisfies a LTL formula. Several relaxations are considered as heuristics to address the rarity of sampling trajectories that satisfy an LTL formula and demonstrated to improve convergence rates for Dubins car and single-integrators subject to a recurrence task.

Distributed Control and Optimization for Communication and Power Systems

We are at the cusp of a historic transformation of both communication system and electricity system. This creates challenges as well as opportunities for the study of networked systems. Problems of these systems typically involve a huge number of end points that require intelligent coordination in a distributed manner. In this thesis, we develop models, theories, and scalable distributed optimization and control algorithms to overcome these challenges.

This thesis focuses on two specific areas: multi-path TCP (Transmission Control Protocol) and electricity distribution system operation and control. Multi-path TCP (MP-TCP) is a TCP extension that allows a single data stream to be split across multiple paths. MP-TCP has the potential to greatly improve reliability as well as efficiency of communication devices. We propose a fluid model for a large class of MP-TCP algorithms and identify design criteria that guarantee the existence, uniqueness, and stability of system equilibrium. We clarify how algorithm parameters impact TCP-friendliness, responsiveness, and window oscillation and demonstrate an inevitable tradeoff among these properties. We discuss the implications of these properties on the behavior of existing algorithms and motivate a new algorithm Balia (balanced linked adaptation) which generalizes existing algorithms and strikes a good balance among TCP-friendliness, responsiveness, and window oscillation. We have implemented Balia in the Linux kernel. We use our prototype to compare the new proposed algorithm Balia with existing MP-TCP algorithms.

Our second focus is on designing computationally efficient algorithms for electricity distribution system operation and control. First, we develop efficient algorithms for feeder reconfiguration in distribution networks. The feeder reconfiguration problem chooses the on/off status of the switches in a distribution network in order to minimize a certain cost such as power loss. It is a mixed integer nonlinear program and hence hard to solve. We propose a heuristic algorithm that is based on the recently developed convex relaxation of the optimal power flow problem. The algorithm is efficient and can successfully computes an optimal configuration on all networks that we have tested. Moreover we prove that the algorithm solves the feeder reconfiguration problem optimally under certain conditions. We also propose a more efficient algorithm and it incurs a loss in optimality of less than 3% on the test networks.

Second, we develop efficient distributed algorithms that solve the optimal power flow (OPF) problem on distribution networks. The OPF problem determines a network operating point that minimizes a certain objective such as generation cost or power loss. Traditionally OPF is solved in a centralized manner. With increasing penetration of volatile renewable energy resources in distribution systems, we need faster and distributed solutions for real-time feedback control. This is difficult because power flow equations are nonlinear and kirchhoff's law is global. We propose solutions for both balanced and unbalanced radial distribution networks. They exploit recent results that suggest solving for a globally optimal solution of OPF over a radial network through a second-order cone program (SOCP) or semi-definite program (SDP) relaxation. Our distributed algorithms are based on the alternating direction method of multiplier (ADMM), but unlike standard ADMM-based distributed OPF algorithms that require solving optimization subproblems using iterative methods, the proposed solutions exploit the problem structure that greatly reduce the computation time. Specifically, for balanced networks, our decomposition allows us to derive closed form solutions for these subproblems and it speeds up the convergence by 1000x times in simulations. For unbalanced networks, the subproblems reduce to either closed form solutions or eigenvalue problems whose size remains constant as the network scales up and computation time is reduced by 100x compared with iterative methods.