Robust control theory was developed in the late twentieth century as a mathematical framework to enable the principled incorporation of uncertainty into engineering design in applications like aerospace. However, engineered technologies that interface with living systems in applications like medicine and ecology must accommodate uncertainties and unmodeled dynamics far beyond what robust control theory has historically achieved. This thesis develops a robust control foundation for overcoming large-scale uncertainty and designing interfaces with living systems, through formal theory and three case studies: neural control of movement, immune control of viruses, and homeostatic control of neoplasia in the moon jellyfish.

The central argument of this thesis is that these three systems, along with many others, have two key properties that enable new approaches to the uncertainty intrinsic to their study: they are themselves control systems, and they are multicellular systems. These properties motivate new work in control theory, blending recent results in localized and distributed control with older results from robust and modern control. The resulting theory framework answers domain-specific questions, guides the design of new experiments and technologies, and enables a conceptual synthesis. By leveraging the fact that these are *multicellular control* systems, we are able to make progress in theory, basic science, and engineering.

One eventual goal of bioengineering is to build complex biological machines that fully realize the unique potential of biotechnology, namely adaptation, survival, growth, and dominance. In order to do so, not only do we need theoretical understanding and reliable manufacturing of biological parts and components, we also need a systems theory that captures fundamental structures to obtain insight about the space of all possible behaviors when parts are put together. This enables us to understand what can and cannot be achieved. Examples from other engineering disciplines are Turing machines for computers, information channels for communication networks, linear input output systems for electrical circuits, and thermodynamics for heat engines. This work is an attempt at developing a systems theory tailored to biomolecular systems in cells. The results form the following statements.

Biomolecular systems are binding and catalysis reactions. Catalysis determines the direction of change, while binding regulates how the catalysis rates vary with reactant concentrations. Given a binding reaction network, the full range of regulatory profiles can be captured by the reaction orders of catalysis, which in turn is constrained in polyhedral sets determined by the stoichiometry of binding. This constitute a rule, that since cells control catalysis by binding, cells control catalysis rates by regulating reaction orders constrained in polyhedral sets. This rule has ramifications in several directions. On metabolism, by incorporating the constraint that reaction orders of metabolic fluxes, not the fluxes themselves, are controlled, we can predict metabolism dynamics directly from network stoichiometry, e.g. glycolytic oscillations and growth arrests. This is a fully dynamic upgrade of flux balance analysis, a popular constraint-based method to model metabolism. On systems biology, this rule derives a method of biocircuit analysis based on the full range of values that reaction orders can take. This allows discovery of necessary and sufficient conditions for a circuit to achieve a certain function, thus revealing regimes hidden by traditional methods of analysis. It also promotes holistic comparisons of different circuit implementations, e.g. activating versus repressing, thur enabling biocircuit design where we know when a design will work, and when a design will fail. On dynamics and control of biocircuits, reaction order can work as a robust basis for stability, perfect adaptation, multistability, and oscillations. Lyapunov functions and dissipative control theory tailored for biomolecular systems are constructed based on reaction orders. On the mathematics of biology, it relates bioregulation to convex polyhedra, log derivative operator decompositions, and fundamental rules of calculus for positive variables.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/DGPY-1679, author = {Olsman, Noah Andrew}, title = {Architecture, Design, and Tradeoffs in Biomolecular Feedback Systems}, school = {California Institute of Technology}, year = {2019}, doi = {10.7907/DGPY-1679}, url = {https://resolver.caltech.edu/CaltechTHESIS:11262018-100422941}, abstract = {A core pursuit in systems and synthetic biology is the analysis of the connection between the low-level structure and parameters of a biomolecular network and its high-level function and performance. Elucidating this mapping has become increasingly feasible as precise measurements of both input parameters and output dynamics become abundant. At the same time, cross-pollination between biology and engineering has led to the realization that many of the mathematical tools from control theory are well-suited to analyze biological processes.

The goal of this thesis is to use tools from control theory to analyze a variety of biomolecular systems from both natural and synthetic settings, and subsequently yield insight into the architecture, tradeoffs, and limitations of biological network. In Chapter 2, I demonstrate how allosteric proteins can be used to respond logarithmically to changes in signal. In Chapter 3, I show how control theoretic techniques can be used to inform the design of synthetic integral feedback networks that implement feedback with a sequestration mechanism. Finally, in Chapter 4 I present a novel simplified model of the *E. coli* heat shock response system and show how the the mapping of circuit parameters to function depends on the network’s architecture.

The unifying theme of this research is that the conceptual framework used to study engineered systems is remarkably well-suited to biology. That being said, it is important to apply these tools in a way that is informed by the molecular details of biological processes. By combining structural and biochemical data with the functional perspective of engineering, it is possible to understand the architectural principles that underlie living systems.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, } @phdthesis{10.7907/VYQY-DF47, author = {Nakahira, Yorie}, title = {Connecting the Speed-Accuracy Trade-Offs in Sensorimotor Control and Neurophysiology Reveals Diversity Sweet Spots in Layered Control Architectures}, school = {California Institute of Technology}, year = {2019}, doi = {10.7907/VYQY-DF47}, url = {https://resolver.caltech.edu/CaltechTHESIS:06072019-083145024}, abstract = {Nervous systems sense, communicate, compute, and actuate movement using distributed components with trade-offs in speed, accuracy, sparsity, noise, and saturation. Nevertheless, the resulting control can achieve remarkably fast, accurate, and robust performance due to a highly effective layered control architecture. However, this architecture has received little attention from the existing research. This is in part because of the lack of theory that connects speed-accuracy trade-offs (SATs) in the components neurophysiology with system-level sensorimotor control and characterizes the overall system performance when different layers (planning vs. reflex layer) act work jointly. In thesis, we present a theoretical framework that provides a synthetic perspective of both levels and layers. We then use this framework to clarify the properties of effective layered architectures and explain why there exists extreme diversity across layers (planning vs. reflex layers) and within levels (sensorimotor versus neural/muscle hardware levels). The framework characterizes how the sensorimotor SATs are constrained by the component SATs of neurons communicating with spikes and their sensory and muscle endpoints, in both stochastic and deterministic models. The theoretical predictions are also verified using driving experiments. Our results lead to a novel concept, termed ``diversity sweet spots (DSSs)’’: the appropriate diversity in the properties of neurons and muscles across layers and within levels help create systems that are both fast and accurate despite being built from components that are individually slow or inaccurate. At the component level, this concept explains why there are extreme heterogeneities in the neural or muscle composition. At the system level, DSSs explain the benefits of layering to allow extreme heterogeneities in speed and accuracy in different sensorimotor loops. Similar issues and properties also extend down to the cellular level in biology and outward to our most advanced network technologies from smart grid to the Internet of Things. We present our initial step in expanding our framework to that area and widely-open area of research for future direction.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/Z9TX3CK8, author = {Leong, Yoke Peng}, title = {Optimal Controller Synthesis for Nonlinear Systems}, school = {California Institute of Technology}, year = {2018}, doi = {10.7907/Z9TX3CK8}, url = {https://resolver.caltech.edu/CaltechTHESIS:12162017-121220572}, abstract = {Optimal controller synthesis is a challenging problem to solve. However, in many applications such as robotics, nonlinearity is unavoidable. Apart from optimality, correctness of the system behaviors with respect to system specifications such as stability and obstacle avoidance is vital for engineering applications. Many existing techniques consider either the optimality or the correctness of system behavior. Rarely, a tool exists that considers both. Furthermore, most existing optimal controller synthesis techniques are not scalable because they either require ad-hoc design or they suffer from the curse of dimensionality.

This thesis aims to close these gaps by proposing optimal controller synthesis techniques for two classes of nonlinear systems: linearly solvable nonlinear systems and hybrid nonlinear systems. Linearly solvable systems have associated Hamilton- Jacobi-Bellman (HJB) equations that can be transformed from the original nonlinear partial differential equation (PDE) into a linear PDE through a logarithmic transformation. The first part of this thesis presets two methods to synthesize optimal controller for linearly solvable nonlinear systems. The first technique uses a hierarchy of sums-of-square programs to compute a sequence of suboptimal controllers that have non-increasing suboptimality for first exit and finite horizon problems. This technique is the first systematic approach to provide stability and suboptimal performance guarantees for stochastic nonlinear systems in one framework. The second technique uses the low rank tensor decomposition framework to solve the linear HJB equation for first exit, finite horizon, and infinite horizon problems. This technique scale linearly with dimensions, alleviating the curse of dimensionality and enabling us to solve the linear HJB equation for a quadcopter model that is a twelve-dimensional system on a personal laptop. A new algorithm is proposed for a key step in the controller synthesis algorithm to solve the ill-conditioning issue that arises in the original algorithm. A MATLAB toolbox that implements the algorithms is developed, and the performance of these algorithms is illustrated by a few engineering examples.

Apart from stability, in many applications, more complex specifications such as obstacle avoidance, reachability, and surveillance are required. The second part of the thesis describes methods to synthesize optimal controllers for hybrid nonlinear systems with quantitative objectives (i.e., minimizing cost) and qualitative objectives (i.e., satisfying specifications). This thesis focuses on two types of qualitative objectives, regular objectives, and ω-regular objectives. Regular objectives capture bounded time behavior such as reachability, and ω-regular objectives capture long term behavior such as surveillance. For both types of objectives, an abstraction-refinement procedure that preserves the cost is developed. A two-player game is solved on the product of the abstract system and the given objectives to synthesize the suboptimal controller for the hybrid nonlinear system. By refining the abstract system, the algorithms are guaranteed to converge to the optimal cost and return the optimal controller if the original systems are robust with respect to the initial states and the optimal controller inputs. The proposed technique is the first abstraction-refinement based technique to combine both quantitative and qualitative objectives into one framework. A Python implementation of the algorithms are developed, and a few engineering examples are presented to illustrate the performance of these algorithms.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, } @phdthesis{10.7907/Z95M63PF, author = {Wang, Yuh-Shyang}, title = {A System Level Approach to Optimal Controller Design for Large-Scale Distributed Systems}, school = {California Institute of Technology}, year = {2017}, doi = {10.7907/Z95M63PF}, url = {https://resolver.caltech.edu/CaltechTHESIS:12122016-113630092}, abstract = {Modern cyber-physical systems, such as the smart grid, software-defined networks, and automated highway systems, are large-scale, physically distributed, and interconnected. The scale of these systems poses fundamental challenges for controller design: the traditional optimal control methods are globally centralized, which require solving a large-scale optimization problem with the knowledge of the global plant model, and collecting global measurement instantaneously during implementation. The ultimate goal of distributed control design is to provide a local, distributed, scalable, and coordinated control scheme to achieve centralized control objectives with nearly global transient optimality.

This dissertation provides a novel theoretical and computational contribution to the area of constrained linear optimal control, with a particular emphasis on addressing the scalability of controller design and implementation for large-scale distributed systems. Our approach provides a fundamental rethinking of controller design: we extend a control design problem to a system level design problem, where we directly optimize the desired closed loop behavior of the feedback system. We show that many traditional topics in the optimal control literature, including the parameterization of stabilizing controller and the synthesis of centralized and distributed controller, can all be cast as a special case of a system level design problem. The system level approach therefore unifies many existing results in the field of distributed optimal control, and solves many previously open problems.

Our system level approach has at least the following four technical merits. First, we characterize the broadest known class of constrained linear optimal control problem that admits a convex formulation. Specifically, we show that the set of convex system level design problems is a strict superset of those that can be parameterized using quadratic invariance. Second, we identify a class of system level design problems, which we called the localized optimal control problems, that are scalable to arbitrary large-scale systems. In particular, the parallel synthesis and implementation complexity of the localized optimal controller are O(1) compared to the size of the networked system. Third, we provide a unified framework to simultaneously incorporate user-specified design specification on the closed loop and the hardware implementation constraints on the controller into the optimal controller design process. Lastly, we provide a system level approach that supports the co-design of optimal controller and its sensing and actuating architecture.

We demonstrate the effectiveness of our method on a 51200-state randomized heterogeneous power network model, and show that the system level approach provides superior scalability over the centralized and distributed method. For such a large-scale example, the theoretical computation time for the centralized scheme is more than 200 days, and the distributed optimal control scheme is intractable. In contrast, it only takes 38 minutes to synthesize a localized optimal controller that achieves at least 99% global optimality guarantee.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/Z9X34VDV, author = {You, Seungil}, title = {A Direct Approach to Robustness Optimization}, school = {California Institute of Technology}, year = {2016}, doi = {10.7907/Z9X34VDV}, url = {https://resolver.caltech.edu/CaltechTHESIS:08122015-172710296}, abstract = {This dissertation reformulates and streamlines the core tools of robustness analysis for linear time invariant systems using now-standard methods in convex optimization. In particular, robust performance analysis can be formulated as a primal convex optimization in the form of a semidefinite program using a semidefinite representation of a set of Gramians. The same approach with semidefinite programming duality is applied to develop a linear matrix inequality test for well-connectedness analysis, and many existing results such as the Kalman-Yakubovich–Popov lemma and various scaled small gain tests are derived in an elegant fashion. More importantly, unlike the classical approach, a decision variable in this novel optimization framework contains all inner products of signals in a system, and an algorithm for constructing an input and state pair of a system corresponding to the optimal solution of robustness optimization is presented based on this information. This insight may open up new research directions, and as one such example, this dissertation proposes a semidefinite programming relaxation of a cardinality constrained variant of the H ∞ norm, which we term sparse H ∞ analysis, where an adversarial disturbance can use only a limited number of channels. Finally, sparse H ∞ analysis is applied to the linearized swing dynamics in order to detect potential vulnerable spots in power networks.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/Z99884Z0, author = {Matni, Nikolai}, title = {Distributed Optimal Control of Cyber-Physical Systems: Controller Synthesis, Architecture Design and System Identification}, school = {California Institute of Technology}, year = {2016}, doi = {10.7907/Z99884Z0}, url = {https://resolver.caltech.edu/CaltechTHESIS:03312016-100604768}, abstract = {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*_{∞} optimal control problem subject to delay constraints, and provides the only known exact characterization of delay-constrained distributed controllers satisfying an *H*_{∞} 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.

While some of the deepest results in nature are those that give explicit bounds between important physical quantities, some of the most intriguing and celebrated of such bounds come from fields where there is still a great deal of disagreement and confusion regarding even the most fundamental aspects of the theories. For example, in quantum mechanics, there is still no complete consensus as to whether the limitations associated with Heisenberg’s Uncertainty Principle derive from an inherent randomness in physics, or rather from limitations in the measurement process itself, resulting from phenomena like back action. Likewise, the second law of thermodynamics makes a statement regarding the increase in entropy of closed systems, yet the theory itself has neither a universally-accepted definition of equilibrium, nor an adequate explanation of how a system with underlying microscopically Hamiltonian dynamics (reversible) settles into a fixed distribution.

Motivated by these physical theories, and perhaps their inconsistencies, in this thesis we use dynamical systems theory to investigate how the very simplest of systems, even with no physical constraints, are characterized by bounds that give limits to the ability to make measurements on them. Using an existing interpretation, we start by examining how dissipative systems can be viewed as high-dimensional lossless systems, and how taking this view necessarily implies the existence of a noise process that results from the uncertainty in the initial system state. This fluctuation-dissipation result plays a central role in a measurement model that we examine, in particular describing how noise is inevitably injected into a system during a measurement, noise that can be viewed as originating either from the randomness of the many degrees of freedom of the measurement device, or of the environment. This noise constitutes one component of measurement back action, and ultimately imposes limits on measurement uncertainty. Depending on the assumptions we make about active devices, and their limitations, this back action can be offset to varying degrees via control. It turns out that using active devices to reduce measurement back action leads to estimation problems that have non-zero uncertainty lower bounds, the most interesting of which arise when the observed system is lossless. One such lower bound, a main contribution of this work, can be viewed as a classical version of a Heisenberg uncertainty relation between the system’s position and momentum. We finally also revisit the murky question of how macroscopic dissipation appears from lossless dynamics, and propose alternative approaches for framing the question using existing systematic methods of model reduction.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/Z1BQ-ZX85, author = {Chandra, Fiona Adriani}, title = {Limits and Tradeoffs in the Control of Autocatalytic Systems}, school = {California Institute of Technology}, year = {2013}, doi = {10.7907/Z1BQ-ZX85}, url = {https://resolver.caltech.edu/CaltechTHESIS:06032013-143158080}, abstract = {Despite the complexity of biological networks, we find that certain common architectures govern network structures. These architectures impose fundamental constraints on system performance and create tradeoffs that the system must balance in the face of uncertainty in the environment. This means that while a system may be optimized for a specific function through evolution, the optimal achievable state must follow these constraints. One such constraining architecture is autocatalysis, as seen in many biological networks including glycolysis and ribosomal protein synthesis. Using a minimal model, we show that ATP autocatalysis in glycolysis imposes stability and performance constraints and that the experimentally well-studied glycolytic oscillations are in fact a consequence of a tradeoff between error minimization and stability. We also show that additional complexity in the network results in increased robustness. Ribosome synthesis is also autocatalytic where ribosomes must be used to make more ribosomal proteins. When ribosomes have higher protein content, the autocatalysis is increased. We show that this autocatalysis destabilizes the system, slows down response, and also constrains the system’s performance. On a larger scale, transcriptional regulation of whole organisms also follows architectural constraints and this can be seen in the differences between bacterial and yeast transcription networks. We show that the degree distributions of bacterial transcription network follow a power law distribution while the yeast network follows an exponential distribution. We then explored the evolutionary models that have previously been proposed and show that neither the preferential linking model nor the duplication-divergence model of network evolution generates the power-law, hierarchical structure found in bacteria. However, in real biological systems, the generation of new nodes occurs through both duplication and horizontal gene transfers, and we show that a biologically reasonable combination of the two mechanisms generates the desired network.}, address = {1200 East California Boulevard, Pasadena, California 91125}, month = {June}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/NHVJ-FX37, author = {Li, Na (Lina)}, title = {Distributed Optimization in Power Networks and General Multi-agent Systems}, school = {California Institute of Technology}, year = {2013}, doi = {10.7907/NHVJ-FX37}, url = {https://resolver.caltech.edu/CaltechTHESIS:05312013-122007615}, abstract = {The dissertation studies the general area of complex networked systems that consist of interconnected and active heterogeneous components and usually operate in uncertain environments and with incomplete information. Problems associated with those systems are typically large-scale and computationally intractable, yet they are also very well-structured and have features that can be exploited by appropriate modeling and computational methods. The goal of this thesis is to develop foundational theories and tools to exploit those structures that can lead to computationally-efficient and distributed solutions, and apply them to improve systems operations and architecture.

Specifically, the thesis focuses on two concrete areas. The first one is to design distributed rules to manage distributed energy resources in the power network. The power network is undergoing a fundamental transformation. The future smart grid, especially on the distribution system, will be a large-scale network of distributed energy resources (DERs), each introducing random and rapid fluctuations in power supply, demand, voltage and frequency. These DERs provide a tremendous opportunity for sustainability, efficiency, and power reliability. However, there are daunting technical challenges in managing these DERs and optimizing their operation. The focus of this dissertation is to develop scalable, distributed, and real-time control and optimization to achieve system-wide efficiency, reliability, and robustness for the future power grid. In particular, we will present how to explore the power network structure to design efficient and distributed market and algorithms for the energy management. We will also show how to connect the algorithms with physical dynamics and existing control mechanisms for real-time control in power networks.

The second focus is to develop distributed optimization rules for general multi-agent engineering systems. A central goal in multiagent systems is to design local control laws for the individual agents to ensure that the emergent global behavior is desirable with respect to the given system level objective. Ideally, a system designer seeks to satisfy this goal while conditioning each agent’s control on the least amount of information possible. Our work focused on achieving this goal using the framework of game theory. In particular, we derived a systematic methodology for designing local agent objective functions that guarantees (i) an equivalence between the resulting game-theoretic equilibria and the system level design objective and (ii) that the resulting game possesses an inherent structure that can be exploited for distributed learning, e.g., potential games. The control design can then be completed by applying any distributed learning algorithm that guarantees convergence to the game-theoretic equilibrium. One main advantage of this game theoretic approach is that it provides a hierarchical decomposition between the decomposition of the systemic objective (game design) and the specific local decision rules (distributed learning algorithms). This decomposition provides the system designer with tremendous flexibility to meet the design objectives and constraints inherent in a broad class of multiagent systems. Furthermore, in many settings the resulting controllers will be inherently robust to a host of uncertainties including asynchronous clock rates, delays in information, and component failures.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/E750-2M74, author = {Sojoudi, Somayeh}, title = {Mathematical Study of Complex Networks: Brain, Internet, and Power Grid}, school = {California Institute of Technology}, year = {2013}, doi = {10.7907/E750-2M74}, url = {https://resolver.caltech.edu/CaltechTHESIS:05252013-081655550}, abstract = {The dissertation is concerned with the mathematical study of various network problems. First, three real-world networks are considered: (i) the human brain network (ii) communication networks, (iii) electric power networks. Although these networks perform very different tasks, they share similar mathematical foundations. The high-level goal is to analyze and/or synthesis each of these systems from a “control and optimization” point of view. After studying these three real-world networks, two abstract network problems are also explored, which are motivated by power systems. The first one is “flow optimization over a flow network” and the second one is “nonlinear optimization over a generalized weighted graph”. The results derived in this dissertation are summarized below.

Brain Networks: Neuroimaging data reveals the coordinated activity of spatially distinct brain regions, which may be represented mathematically as a network of nodes (brain regions) and links (interdependencies). To obtain the brain connectivity network, the graphs associated with the correlation matrix and the inverse covariance matrix—describing marginal and conditional dependencies between brain regions—have been proposed in the literature. A question arises as to whether any of these graphs provides useful information about the brain connectivity. Due to the electrical properties of the brain, this problem will be investigated in the context of electrical circuits. First, we consider an electric circuit model and show that the inverse covariance matrix of the node voltages reveals the topology of the circuit. Second, we study the problem of finding the topology of the circuit based on only measurement. In this case, by assuming that the circuit is hidden inside a black box and only the nodal signals are available for measurement, the aim is to find the topology of the circuit when a limited number of samples are available. For this purpose, we deploy the graphical lasso technique to estimate a sparse inverse covariance matrix. It is shown that the graphical lasso may find most of the circuit topology if the exact covariance matrix is well-conditioned. However, it may fail to work well when this matrix is ill-conditioned. To deal with ill-conditioned matrices, we propose a small modification to the graphical lasso algorithm and demonstrate its performance. Finally, the technique developed in this work will be applied to the resting-state fMRI data of a number of healthy subjects.

Communication Networks: Congestion control techniques aim to adjust the transmission rates of competing users in the Internet in such a way that the network resources are shared efficiently. Despite the progress in the analysis and synthesis of the Internet congestion control, almost all existing fluid models of congestion control assume that every link in the path of a flow observes the original source rate. To address this issue, a more accurate model is derived in this work for the behavior of the network under an arbitrary congestion controller, which takes into account of the effect of buffering (queueing) on data flows. Using this model, it is proved that the well-known Internet congestion control algorithms may no longer be stable for the common pricing schemes, unless a sufficient condition is satisfied. It is also shown that these algorithms are guaranteed to be stable if a new pricing mechanism is used.

Electrical Power Networks: Optimal power flow (OPF) has been one of the most studied problems for power systems since its introduction by Carpentier in 1962. This problem is concerned with finding an optimal operating point of a power network minimizing the total power generation cost subject to network and physical constraints. It is well known that OPF is computationally hard to solve due to the nonlinear interrelation among the optimization variables. The objective is to identify a large class of networks over which every OPF problem can be solved in polynomial time. To this end, a convex relaxation is proposed, which solves the OPF problem exactly for every radial network and every meshed network with a sufficient number of phase shifters, provided power over-delivery is allowed. The concept of “power over-delivery” is equivalent to relaxing the power balance equations to inequality constraints.

Flow Networks: In this part of the dissertation, the minimum-cost flow problem over an arbitrary flow network is considered. In this problem, each node is associated with some possibly unknown injection, each line has two unknown flows at its ends related to each other via a nonlinear function, and all injections and flows need to satisfy certain box constraints. This problem, named generalized network flow (GNF), is highly non-convex due to its nonlinear equality constraints. Under the assumption of monotonicity and convexity of the flow and cost functions, a convex relaxation is proposed, which always finds the optimal injections. A primary application of this work is in the OPF problem. The results of this work on GNF prove that the relaxation on power balance equations (i.e., load over-delivery) is not needed in practice under a very mild angle assumption.

Generalized Weighted Graphs: Motivated by power optimizations, this part aims to find a global optimization technique for a nonlinear optimization defined over a generalized weighted graph. Every edge of this type of graph is associated with a weight set corresponding to the known parameters of the optimization (e.g., the coefficients). The motivation behind this problem is to investigate how the (hidden) structure of a given real/complex valued optimization makes the problem easy to solve, and indeed the generalized weighted graph is introduced to capture the structure of an optimization. Various sufficient conditions are derived, which relate the polynomial-time solvability of different classes of optimization problems to weak properties of the generalized weighted graph such as its topology and the sign definiteness of its weight sets. As an application, it is proved that a broad class of real and complex optimizations over power networks are polynomial-time solvable due to the passivity of transmission lines and transformers.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/0ANZ-8209, author = {Lamperski, Andrew G.}, title = {Hierarchies, Spikes, and Hybrid Systems: Physiologically Inspired Control Problems}, school = {California Institute of Technology}, year = {2011}, doi = {10.7907/0ANZ-8209}, url = {https://resolver.caltech.edu/CaltechTHESIS:06022011-110025485}, abstract = {In animal motor control and locomotion, neurons process information, muscles are the actuators, and the body is the plant. Control theory is the standard mathematical language for describing motor control and locomotion, but many phenomena in physiological control remain outside of the scope of control theoretic reasoning. Unlike traditional engineering control systems, nearly all the components of physiological control systems have complex dynamics. Instead of a fast centralized computer, an animal implements controllers using a distributed network of slow, nonlinear, and noisy neurons. Rather than having linear plants and actuators, the animal must control limbs with nonlinear and hybrid dynamics.

This dissertation develops basic control theory motivated by physiological systems. Dynamical phenomena that arise in physiology but remain outside the scope of mathematical methods are isolated and studied in general control theoretic frameworks. In particular, three problems are discussed: distributed linear quadratic Gaussian (LQG) control with communication delays, control over communication channels modeled after spiking neurons, and Zeno stability of hybrid systems.

Motivated by the presence of delays in the human motor system, Chapter 2 explores the architecture of distributed LQG controllers when communication between subsystems is limited by delays. Sensory and motor command information is processed in several different regions throughout the nervous system. Since processing speed in neurons is limited, information from different sensory and motor regions can only be integrated after a time delay. In spite of this difficulty, humans make efficient and reliable motions that are well-described by optimal feedback control. Optimal delay compensation is studied in a distributed LQG framework. The structure that emerges as the result of optimization resembles a management hierarchy, bearing similarities with the organization of the motor system.

Networked control systems, in which communication between the controller and the plant occurs over a special neuron-inspired channel, are analyzed in Chapter 3. In addition to being the basic computing elements, neurons are the long-range communication channels of the body. Neurons transmit information in the form of short-lived voltage spikes, called action potentials. Sufficient conditions for stable control over the spiking channel are presented, along with bounds on tracking error and data rates.

The final technical chapter studies the connections between Zeno behavior and Lyapunov stability. Zeno behavior occurs in a hybrid system when an infinite number of discrete transitions occurs in a finite amount of time. While Zeno behavior results from modeling abstractions, it is commonly observed in models of mechanical systems undergoing impacts, including models important for locomotion. Often, Zeno behavior is associated with dynamical mode transitions, such as knee locking and the transition between bouncing and sliding. To reason about such transitions without modifying the models, the chapter on hybrid systems gives Lyapunov-like sufficient conditions for Zeno behavior. A technique for constructing the Lyapunov-like certificates is presented for a general class of mechanical systems undergoing impacts.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/2ZYV-FF66, author = {Nahmad Bensusan, Marcos}, title = {Interpretation and Scaling of Positional Information During Development}, school = {California Institute of Technology}, year = {2011}, doi = {10.7907/2ZYV-FF66}, url = {https://resolver.caltech.edu/CaltechTHESIS:10212010-000757213}, abstract = {

Cells in a developing animal require information about their relative position in order to function and differentiate appropriately. In the classical view, cellular positional information is interpreted from the concentration of chemical signals known as morphogens. However, recent studies have questioned the ability of morphogens to establish gene expression patterns in a concentration-dependent manner. Here we combine theoretical tools and experimental work in Drosophila melanogaster to investigate the mechanisms by which positional information is interpreted from a morphogen gradient and the ability of patterns to scale with respect to the size of the system.

First, we study how a concentration gradient of the signaling molecule Hedgehog establishes multiple patterns of gene expression along the anterior-posterior axis of the Drosophila wing disc. Using mathematical modeling as a hypotheses-generating tool, we predicted that positional information cannot be explained by different concentration thresholds from a static Hedgehog gradient. Instead, we propose that cells take into account their history of Hedgehog signaling exposure to determine patterns of gene expression. We provide experimental evidence that supports our model and conclude that gradient dynamics, resulting from the gene network architecture of the Hedgehog signaling pathway, determine pattern formation in the wing disc.

Second, we introduce a theoretical formalism to study the role of morphogen gradient dynamics in developmental patterning. Given a mathematical model of pattern formation, we define and compute parameter perturbations that leave invariant the steady-state distribution of the relevant morphogen. We propose that this approach can be used as a tool to design genetic experiments that assay the function of morphogen dynamics.

Lastly, we use dorsal-ventral patterning of the early Drosophila embryo as a model to study scaling of gene expression patterns with respect to natural variations in axis length, that is, the ability to establish positional information relative to the size of the system. We provide evidence that gene expression patterns that depend on the maternal factor Dorsal, scale along the dorsal-ventral axis. Our data suggest that scaling in this system is a gene-dependent rather than a position-dependent property. We propose that the mechanisms for scaling depend on feedback interactions downstream of Dorsal.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Stathopoulos, Angelike and Doyle, John Comstock}, } @phdthesis{10.7907/CM46-5R54, author = {Lavaei, Javad}, title = {Large-Scale Complex Systems: From Antenna Circuits to Power Grids}, school = {California Institute of Technology}, year = {2011}, doi = {10.7907/CM46-5R54}, url = {https://resolver.caltech.edu/CaltechTHESIS:05132011-113642762}, abstract = {This dissertation is motivated by the lack of scalable methods for the analysis and synthesis of different large-scale complex systems appearing in electrical and computer engineering. The systems of interest in this work are power networks, analog circuits, antenna systems, communication networks and distributed control systems. By combining theories from control and optimization, the high-level objective is to develop new design tools and algorithms that explicitly exploit the physical properties of these practical systems (e.g., passivity of electrical elements or sparsity of network topology). To this end, the aforementioned systems are categorized intro three classes of systems, and then studied in Parts I, II, and III of this dissertation, as explained below:

Power networks: In Part I of this work, the operation planning of power networks using efficient algorithms is studied. The primary focus is on the optimal power flow (OPF) problem, which has been studied by the operations research and power communities in the past 50 years with little success. In this part, it is shown that there exists an efficient method to solve a practical OPF problem along with many other energy-related optimization problems such as dynamic OPF or security-constrained OPF. The main reason for the successful convexification of these optimization problems is also identified to be the physical properties of a power circuit, especially the passivity of transmission lines.

Circuits and systems: Motivated by different applications in power networks, electromagnetics and optics, Part II of this work studies the fundamental limits associated with the synthesis of a particular type of linear circuit. It is shown that the optimal design of the parameters of this type of circuit can be performed in polynomial time if the circuit is passive and there are sufficient number of controllable (unknown) parameters. This result introduces a trade-off between the design simplicity and the implementation complexity for an important class of linear circuits. As an application of this methodology, the design of smart antennas is also studied; the goal is to devise an intelligent wireless communication device in order to avoid co-channel interference, power consumption in undesired directions and security issues. Since the existing smart antennas are either hard to program or hard to implement, a new type of smart antenna is synthesized by utilizing tools from algebraic geometry, control, communications, and circuits, which is both easy to program and easy to implement.

Distributed computation: The first problem tackled in Part III of this work is a very simple type of distributed computation, referred to as quantized consensus, which aims to compute the average of a set of numbers using a distributed algorithm subject to a quantization error. It is shown that quantized consensus is reached by means of a recently proposed gossip algorithm, and the convergence time of the algorithm is also derived. The second problem studied in Part III is a more advanced type of distributed computation, which is the distributed resource allocation problem for the Internet. The existing distributed resource allocation algorithms aim to maximize the utility of the network only at the equilibrium point and ignore the transient behavior of the network. To address this issue, it is shown that optimal control theory provides powerful tools for designing distributed resource allocation algorithms with a guaranteed real-time performance.

The results of this work can all be integrated to address real-world interdisciplinary problems, such as the design of the next generation of the electrical power grid, named the Smart Grid.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/P1DS-Q379, author = {Gayme, Dennice F. Maynard}, title = {A Robust Control Approach to Understanding Nonlinear Mechanisms in Shear Flow Turbulence}, school = {California Institute of Technology}, year = {2010}, doi = {10.7907/P1DS-Q379}, url = {https://resolver.caltech.edu/CaltechTHESIS:05272010-195149679}, abstract = {

A robust control framework is used to investigate a streamwise constant projection of the Navier Stokes equations for plane Couette flow. Study of this streamwise constant model is motivated by both numerical and experimental observations that suggest the prevalence and importance of streamwise and quasi-streamwise elongated structures. Small-amplitude Gaussian noise forcing is applied to a two-dimensional, three-velocity component (2D/3C) model to describe its response in the presence of disturbances, uncertainty and modeling errors. A comparison of the results with Direct Numerical Simulation (DNS) data demonstrates that the simulations capture salient features of fully developed turbulence. In particular, the change in mean velocity profile from the nominal laminar to the characteristic “S” shaped turbulent profile. The application of Taylor’s hypothesis shows that the model can also reproduce downstream information in the form of large-scale coherence resembling numerically and experimentally observed flow features. The 2D/3C model is able to generate “turbulent-like” behavior under small-amplitude stochastic noise. The laminar flow solution is globally stable, therefore transition to turbulence in this model is likely a consequence of the laminar flow solution’s lack of robustness in the presence of disturbances and uncertainty. In fact, large disturbance amplification is common in both this model and the linearized Navier Stokes equations.

Periodic spanwise/wall-normal (z–y) plane stream functions are used as input to develop a forced 2D/3C streamwise velocity equation. The resulting steady-state solution is qualitatively similar to a fully turbulent spatial field of DNS data. Both numerical methods and a perturbation analysis confirm that the momentum transfer that produces a “turbulent-like” mean profile requires a nonlinear streamwise velocity equation.

A system theoretic approach is used to study the amplification mechanisms that develop through the 2D/3C nonlinear coupling in the streamwise velocity equation. The spanwise/wall-normal plane forcing required to produce each stream function is computedand used to define an induced norm from this forcing input to the streamwise velocity. This input-output response is used to determine the energy optimal spanwise wavelength (i.e.,the preferential spacing) over a range of Reynolds numbers and forcing amplitudes.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/MHMV-9M59, author = {Buzi, Gentian}, title = {Control Theoretic Analysis of Autocatalytic Networks in Biology with Applications to Glycolysis}, school = {California Institute of Technology}, year = {2010}, doi = {10.7907/MHMV-9M59}, url = {https://resolver.caltech.edu/CaltechTHESIS:02262010-130618704}, abstract = {Metabolic networks in the cell break down food and resources to create useful energy and components. At the same time they use those same components and energy in the process, thus making autocatalysis an unavoidable part of core metabolism. The simplest and most widely studied autocatalytic network is the glycolytic pathway. It is common to every cell of living organisms, from bacteria to humans. Its special autocatalytic structure, like the structure of many similar autocatalytic networks, makes the pathway hard to control and can lead to instabilities.

In this thesis, we study autocatalytic metabolic networks, specifically glycolysis, to investigate fundamental performance tradeoffs in these network topologies. We hypothesize that instabilities in glycolysis are a result of performance tradeoffs that stem from the structure of the pathways and a conservation law, mathematically described by a special form of the Bode Sensitivity Integral. We show that pathway size and intermediate metabolite consumption adversely affect the performance of the pathway, while reversibility of chemical reactions improves performance. We establish tight bounds for the feedback control gains that guarantee stability of pathways of arbitrary size and arbitrary parameter values for the intermediate reactions.

In addition, we investigate effects of perturbations in metabolite concentrations through the estimation of invariant subsets of the region of attraction around nominal operating conditions. To this end we use a numerical procedure composed of system theoretic characterizations and optimization-based formulations. For large, computationally intractable systems we employ a different technique based on the underlying biological structure, which offers a natural decomposition of the system into a feedback interconnection of two input-output subsystems. This decomposition simplifies the analysis and leads to analytical construction of Lyapunov functions for a large family of autocatalytic pathways.

The results of our analysis reveal fundamental tradeoffs between performance and robustness, energy efficiency, pathway evolvability and computational complexity in these networks.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/WXN5-9A47, author = {Martinez Estrada, Alfredo}, title = {A Treatise on Econometric Forecasting}, school = {California Institute of Technology}, year = {2007}, doi = {10.7907/WXN5-9A47}, url = {https://resolver.caltech.edu/CaltechETD:etd-05222007-101946}, abstract = {We investigate the effects of model misspecification and stochastic dynamics in the problem of forecasting. In economics and many fields of engineering, many researchers are guilty of the dangerous practice of treating their mathematical models as the true data generating mechanisms responsible for the observed phenomena and downplaying or omitting all together the important step of model verification. In recent years, econometricians have acknowledged the need to account for model misspecification in the problems of estimation and forecasting. In particular, a large body of work has emerged to address properties of estimators under model misspecification, along with a plethora of misspecification testing methodologies. In this work, we investigate the combined effects of model misspecification and various types of stochastic dynamics on forecasts based on linear regression models. The data generating process (DGP) is assumed unknown to the forecaster except for the nature of process dependencies, i.e., independent identically distributed, covariance stationary, or nonstationary. Estimation is carried out by means of ordinary least squares, and forecasts are evaluated with the mean squared forecast error (MSFE) or mean square error of prediction. We investigate the sample size dependence of the MSFE. For this purpose, we develop an algorithm to approximate the MSFE by an expression depending only on the sample size n and moments of the processes. The approximation is constructed by Taylor series expansions of the squared forecast error which do not require knowledge of the functional form of the DGP. The approximation can be used to determine the existence of optimal observation windows which result in the minimum MSFE. We assess the accuracy of the approximating algorithm with Monte Carlo experiments.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/JZX4-QN41, author = {Liu, Xin}, title = {Robustness, Complexity, Validation and Risk}, school = {California Institute of Technology}, year = {2007}, doi = {10.7907/JZX4-QN41}, url = {https://resolver.caltech.edu/CaltechETD:etd-05272007-214755}, abstract = {A robust design process starts with modeling of the physical system and the uncertainty it faces. Robust design tools are then applied to achieve specified performance criteria. Verification of system properties is crucial as improvements on the modeling and design practices can be made based on results of such verification. In this thesis, we discuss three aspects of this closed-loop process.

First and the most important aspect is the possibility of the feedback from verification to system modeling and design. When verification is hard, what does it tell us about our system? When the system is robust, would it be easy to verify so? We study the relation between robustness of a system property posed as a decision problem and the proof complexity of verifying such property. We examine this relation in two classes of problems: percolation lattices and linear programming problems, and show complexity is upper-bounded by the reciprocal of robustness, i.e. fragility.

The second aspect we study is model validation. More precisely, when given a candidate model and experiment data, how do we rigorously refute the model or gain information about the consistent parameter set? Different methods for model invalidation and parameter inference are demonstrated with the G-protein signaling system in yeast to show the advantages and hurdles in their applications.

While quantification of robustness requirements has been well-studied in engineering, it is just emerging in the field of finance. Robustness specification in finance is closely related to the availability of proper risk measures. We study the estimation of a coherent risk measure, Expected Shortfall (ES). A consistent and asymptotically normal estimator for ES based on empirical likelihood is proposed. Although empirical likelihood based estimators usually involve numerically solving optimization problems that are not necessarily convex, computation of our estimator can be carried out in a sequential manner, avoiding solving non-convex optimization problems.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/9G3P-7F13, author = {Li, Lun}, title = {Topologies of Complex Networks: Functions and Structures}, school = {California Institute of Technology}, year = {2007}, doi = {10.7907/9G3P-7F13}, url = {https://resolver.caltech.edu/CaltechETD:etd-05282007-223415}, abstract = {During the last decade, significant efforts have been made toward improving our understanding of the topological structures underlying complex networks and illuminating some of the intriguing large-scale properties exhibited by these systems. The dominant theme of these efforts has been on studying the graph-theoretic properties of the corresponding connectivity structures and on developing universal theories and models that transcend system-specific details and describe the different systems well in a statistical sense.

However, in this thesis we argue that these efforts have had limited success and are in need of substantial correction. Using a highly engineered system, the Internet, as a case study we demonstrate that networks are designed for a purpose, and ignoring that aspect or obscuring it with the use of some generic but random mechanism can result in models that misrepresent what matters for system functions. By accounting in a minimal manner for both the functional requirements and structural features inherent in the design of an engineered system, we propose an alternative, optimization-based modeling approach that highlights the necessary trade-offs between system performance and the technological and economic constraints that are crucial when designing the system. We show that our proposed approach yields network models that not only match the large-scale graph-theoretic properties of measured router-level topologies well but are also fully consistent with engineering intuition and networking reality, especially as far as their performance aspects and robustness properties are concerned. In fact, we show that our design-inspired network models can be easily distinguished from previously considered probabilistic network models and efficiently achieve the level of performance for which they were designed in the first place.

While this thesis focuses on the Internet, it has much broader implications for complex networks and graph theory generally. To better differentiate between different graphs that are identical in certain graph statistics, we introduce a structural metric, the s-metric, and demonstrate that it provides insights into the diversity of graphs constrained by certain common properties and sheds new light on many classic graph concepts such as the various notions of self-similarity, likelihood, and assortativity. Our s-metric clarifies much of the confusion surrounding the sensational qualitative claims in the current graph theory literature for complex networks and offers a rigorous and quantitative alternative.

Moreover, to examine the space of graphs that satisfy certain common properties, we propose a new approach that is based on establishing a link between two graphs if and only if one can be obtained from the other via a local transformation. Exploring the resulting connected space of graphs by dividing it into countable subspaces provides a much clearer picture on the whole space. We also show that this space of graphs has a rich and interesting structure and that some properties of the latter can be related to features of the individual graphs in this space (e.g., degree variability of a node *g* in the space of graphs and the s-metric for g).

Optimization theory and game theory provide a suite of tools that are flexible in modelling various network systems, and a rich series of equilibrium solution concepts and convergent algorithms. In this thesis, we view network protocols as distributed algorithms achieving the corresponding network equilibria, and study wireless network design and control in optimization and game-theoretic frameworks.

Specifically, we first take a holistic approach and design an overall framework for the protocol architecture in ad hoc wireless networks. The goal is to integrate various protocol layers into a unified framework, by regarding them as distributed computations over the network to solve some optimization problem. Our current theory integrates three functions–congestion control, routing and scheduling–in transport, network and link layers into a coherent framework. These three functions interact through and are regulated by congestion price so as to achieve a global optimality, even in a time-varying environment. This framework is promising to be extended to provide a mathematical theory for network architecture, and to allow us to systematically carry out cross-layer design.

We then develop a general game-theoretic framework for contention control. We define a general game-theoretic model, called random access game, to study the contention/interaction among wireless nodes, and propose a novel medium access method derived from carrier sensing multiple access with collision avoidance in which each node estimates its conditional collision probability and adjusts its persistence probability or contention window, according to a distributed strategy update mechanism achieving the Nash equilibrium of random access game. This results in simple dynamics, controllable performance objectives, good short-term fairness, low collision, and high throughput. As wireless nodes can estimate conditional collision probabilities by observing consecutive idle slots between transmissions, we can decouple contention control from handling failed transmissions. This also opens up other opportunities such as rate adaptation to channel variations. In addition to providing a general and systematic design methodology for medium access control, the random access game model also provides an analytical framework to understand the equilibrium properties such as throughput, loss and fairness, and dynamic properties of different medium access protocols and their interactions.

Finally, we conclude this work with some suggestions for future research.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock and Low, Steven H.}, } @phdthesis{10.7907/4DQ0-GA49, author = {Wang, Jiantao}, title = {A Theoretical Study of Internet Congestion Control: Equilibrium and Dynamics}, school = {California Institute of Technology}, year = {2006}, doi = {10.7907/4DQ0-GA49}, url = {https://resolver.caltech.edu/CaltechETD:etd-11122005-082753}, abstract = {In the last several years, significant progress has been made in modelling the Internet congestion control using theories from convex optimization and feedback control. In this dissertation, the equilibrium and dynamics of various congestion control schemes are rigorously studied using these mathematical frameworks.

First, we study the dynamics of TCP/AQM systems. We demonstrate that the dynamics of queue and average window in Reno/RED networks are determined predominantly by the protocol stability, not by AIMD probing nor noise traffic. Our study shows that Reno/RED becomes unstable when delay increases and more strikingly, when link capacity increases. Therefore, TCP Reno is ill suited for the future high-speed network, which has motivated the design of FAST TCP. Using a continuous-time model, we prove that FAST TCP is globally stable without feedback delays and provide a sufficient condition for local stability when feedback delays are present. We also introduce a discrete-time model for FAST TCP that fully captures the effect of self-clocking and derive the local stability condition for general networks with feedback delays.

Second, the equilibrium properties (i.e., fairness, throughput, and capacity) of TCP/AQM systems are studied using the utility maximization framework. We quantitatively capture the variations in network throughput with changes in link capacity and allocation fairness. We clarify the open conjecture of whether a fairer allocation is always more efficient. The effects of changes in routing are studied using a joint optimization problem over both source rates and their routes. We investigate whether minimal-cost routing with proper link costs can solve this joint optimization problem in a distributed way. We also identify the tradeoff between achievable utility and routing stability.

At the end, two other related projects are briefly described.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock and Low, Steven H.}, } @phdthesis{10.7907/Y6B1-ZZ55, author = {Gregory, Irene Michelle}, title = {Design and Stability Analysis of an Integrated Controller for Highly Flexible Advanced Aircraft Utilizing the Novel Nonlinear Dynamic Inversion}, school = {California Institute of Technology}, year = {2005}, doi = {10.7907/Y6B1-ZZ55}, url = {https://resolver.caltech.edu/CaltechETD:etd-07232004-075729}, abstract = {High performance aircraft of the future will be designed to be lighter, more maneuverable, and operate over an ever expanding flight envelope. This set of conditions will necessarily mean highly flexible vehicles operating in nonlinear regimes. A methodology proposed to better optimize their responses to both pilot input and external disturbances, as well as to decrease the cost of vehicle design is the novel dynamic inversion. The attractiveness of this methodology lies in the fact that the inherent nonlinearities of the problem and the coupled nature of flexible dynamics are explicitly considered.

The contribution of this work to the state of the art is predicated on the development and application of the novel dynamic inversion methodology to handle highly flexible aircraft in an integrated flight/structural mode control manner. The unprecedented small separation between rigid body and flexible dynamics as well as the reciprocal interaction between them due to flight control action are the key elements of the aircraft model. The novel approach to the nonlinear dynamic inversion allows the methodology to more intelligently handle flexible dynamics in the context of the dual objectives of integrated flight/SMC control by altering flexible mode damping without cancellation; thus, improving disturbance response and avoiding the potentially destabilizing effect of pole cancellation close to the j-omega-axis in case of modeling uncertainty. The necessary level of model complexity for design has been established with particular attention given to understanding physics. The effect of uncertainty in the structural mode dynamics has been addressed.

Further contribution of this work addresses the issue of stability of the dynamic systems driven by nonlinear controllers. One result shows how assessing stability of an n-dimensional system can be reduced to checking stability of a two-dimensional one using algebraic expressions that are based on the vehicle characteristics such as aerodynamic coefficients. This reduces a complicated dynamical problem to something purely algebraic and manageably complex. Another approach is based on algorithmically finding a local Lyapunov function using sum of squares. The presented results are the first to address the question of stability for the nonlinear dynamic inversion in the presence of flexible dynamics.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/S3BJ-4M47, author = {Prajna, Stephen}, title = {Optimization-Based Methods for Nonlinear and Hybrid Systems Verification}, school = {California Institute of Technology}, year = {2005}, doi = {10.7907/S3BJ-4M47}, url = {https://resolver.caltech.edu/CaltechETD:etd-05272005-144358}, abstract = {

Complex behaviors that can be exhibited by hybrid systems make the verification of such systems both important and challenging. Due to the infinite number of possibilities taken by the continuous state and the uncertainties in the system, exhaustive simulation is impossible, and also computing the set of reachable states is generally intractable. Nevertheless, the ever-increasing presence of hybrid systems in safety critical applications makes it evident that verification is an issue that has to be addressed.

In this thesis, we develop a unified methodology for verifying temporal properties of continuous and hybrid systems. Our framework does not require explicit computation of reachable states. Instead, functions of state termed barrier certificates and density functions are used in conjunction with deductive inference to prove properties such as safety, reachability, eventuality, and their combinations. As a consequence, the proposed methods are directly applicable to systems with nonlinearity, uncertainty, and constraints. Moreover, it is possible to treat safety verification of stochastic systems in a similar fashion, by computing an upper-bound on the probability of reaching the unsafe states.

We formulate verification using barrier certificates and density functions as convex programming problems. For systems with polynomial descriptions, sum of squares optimization can be used to construct polynomial barrier certificates and density functions in a computationally scalable manner. Some examples are presented to illustrate the use of the methods. At the end, the convexity of the problem formulation is also exploited to prove a converse theorem in safety verification using barrier certificates.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, } @phdthesis{10.7907/5YG6-JG32, author = {Papachristodoulou, Antonis}, title = {Scalable Analysis of Nonlinear Systems Using Convex Optimization}, school = {California Institute of Technology}, year = {2005}, doi = {10.7907/5YG6-JG32}, url = {https://resolver.caltech.edu/CaltechETD:etd-05082005-100243}, abstract = {In this thesis, we investigate how convex optimization can be used to analyze different classes of nonlinear systems at various scales algorithmically. The methodology is based on the construction of appropriate Lyapunov-type certificates using sum of squares techniques.

After a brief introduction on the mathematical tools that we will be using, we turn our attention to robust stability and performance analysis of systems described by Ordinary Differential Equations. A general framework for constrained systems analysis is developed, under which stability of systems with polynomial, non polynomial vector fields and switching systems, as well as estimating the region of attraction and the L_{2} gain can be treated in a unified manner. Examples from biology and aerospace illustrate our methodology.

We then consider systems described by Functional Differential Equations (FDEs), i.e., time-delay systems. Their main characteristic is that they are infinite dimensional, which complicates their analysis. We first show how the complete Lyapunov-Krasovskii functional can be constructed algorithmically for linear time delay systems. Then, we concentrate on delay-independent and delay-dependent stability analysis of nonlinear FDEs using sum of squares techniques. An example from ecology is given.

The scalable stability analysis of congestion control algorithms for the Internet is investigated next. The models we use result in an arbitrary interconnection of FDE subsystems, for which we require that stability holds for arbitrary delays, network topologies and link capacities. Through a constructive proof, we develop a Lyapunov functional for FAST - a recently developed network congestion control scheme - so that the Lyapunov stability properties scale with the system size. We also show how other network congestion control schemes can be analyzed in the same way.

Finally, we concentrate on systems described by Partial Differential Equations. We show that axially constant perturbations of the Navier-Stokes equations for Hagen-Poiseuille flow are globally stable, even though the background noise is amplified as R^{3} where R is the Reynolds number, giving a ‘robust yet fragile’ interpretation. We also propose a sum of squares methodology for the analysis of systems described by parabolic PDEs.

We conclude this work with an account for future research.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/QYN9-0Z55, author = {Jing, Yindi}, title = {Space-Time Code Design and Its Applications in Wireless Networks}, school = {California Institute of Technology}, year = {2005}, doi = {10.7907/QYN9-0Z55}, url = {https://resolver.caltech.edu/CaltechETD:etd-09072004-204814}, abstract = {This thesis has two main contributions: the designs of differential/non-differential unitary space-time codes for multiple-antenna systems and the analysis of the diversity gain when using space-time coding among nodes in wireless networks.

Capacity has long been a bottleneck in wireless communications. Recently, multiple-antenna techniques have been used in wireless communications to combat the fading effect, which improves both the channel capacity and performance greatly. A recently proposed method for communicating with multiple antennas over block-fading channels is unitary space-time modulation, which can achieve the channel capacity at high SNR. However, it is not clear how to generate well performing unitary space-time codes that lend themselves to efficient encoding and decoding. In this thesis, the design of unitary space-time codes using Cayley transform is proposed. The codes are designed based on an information-theoretic criterion and have a polynomial-time near-maximum-likelihood decoding algorithm. Simulations suggest that the resulting codes allow for effective high-rate data transmissions in multiple-antenna communication systems without knowing the channel. Another well-known transmission scheme for multiple-antenna systems with unknown channel information at both the transmitter and the receiver is differential unitary space-time modulation. It can be regarded as a generalization of DPSK and is suitable for continuous fading. In differential unitary space-time modulation, fully diverse constellations, i.e., sets of unitary matrices whose pairwise differences are non-singular, are wanted for their good pairwise error properties. In this thesis, Lie groups and their representations are used in solving the design problem. Fully diverse differential unitary space-time codes for systems with four and three transmit antennas are constructed based on the Lie groups Sp(2) and SU(3). The designed codes have high diversity products, lend themselves to a fast maximum-likelihood decoding algorithm, and simulation results show that they outperform other existing codes, especially at high SNR.

Then the idea of space-time coding devised for multiple-antenna systems is applied to communications over wireless networks. In wireless relay networks, the relay nodes encode the signals they receive from the transmit node into a distributed space-time code and transmit the encoded signals to the receive node. It is shown in this thesis that at very high SNR, the diversity gain achieved by this scheme is almost the same as that of a multiple-antenna system whose number of transmit antennas is the same as the number of relay nodes in the network, which means that the relay nodes work as if they can cooperate fully and have full knowledge of the message. However, at moderate SNR, the diversity gain of the wireless network is inferior to that of the multiple-antenna system. It is further shown that for a fixed total power consumed in the network, the optimal power allocation is that the transmitter uses half the power and the relays share the other half fairly. This result addresses the question of what performance a relay network can achieve. Both it and its extensions have many applications to wireless ad hoc and sensory network communications.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/0J1D-1B18, author = {Bobba, Kumar Manoj}, title = {Robust Flow Stability: Theory, Computations and Experiments in Near Wall Turbulence}, school = {California Institute of Technology}, year = {2004}, doi = {10.7907/0J1D-1B18}, url = {https://resolver.caltech.edu/CaltechETD:etd-05282004-143324}, abstract = {Helmholtz established the field of hydrodynamic stability with his pioneering work in 1868. From then on, hydrodynamic stability became an important tool in understanding various fundamental fluid flow phenomena in engineering (mechanical, aeronautics, chemical, materials, civil, etc.) and science (astrophysics, geophysics, biophysics, etc.), and turbulence in particular. However, there are many discrepancies between classical hydrodynamic stability theory and experiments. In this thesis, the limitations of traditional hydrodynamic stability theory are shown and a framework for robust flow stability theory is formulated. A host of new techniques like gramians, singular values, operator norms, etc. are introduced to understand the role of various kinds of uncertainty. An interesting feature of this framework is the close interplay between theory and computations. It is shown that a subset of Navier-Stokes equations are globally, non-nonlinearly stable for all Reynolds number. Yet, invoking this new theory, it is shown that these equations produce structures (vortices and streaks) as seen in the experiments. The experiments are done in zero pressure gradient transiting boundary layer on a flat plate in free surface tunnel. Digital particle image velocimetry, and MEMS based laser Doppler velocimeter and shear stress sensors have been used to make quantitative measurements of the flow. Various theoretical and computational predictions are in excellent agreement with the experimental data. A closely related topic of modeling, simulation and complexity reduction of large mechanics problems with multiple spatial and temporal scales is also studied. A nice method that rigorously quantifies the important scales and automatically gives models of the problem to various levels of accuracy is introduced. Computations done using spectral methods are presented.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock and Gharib, Morteza}, } @phdthesis{10.7907/CRHD-3202, author = {Jadbabaie, Ali}, title = {Receding horizon control of nonlinear systems: a control Lyapunov function approach}, school = {California Institute of Technology}, year = {2001}, doi = {10.7907/CRHD-3202}, url = {https://resolver.caltech.edu/CaltechTHESIS:10262010-112027161}, abstract = {With the advent of faster and cheaper computers, optimization based control methodologies have become a viable candidate for control of nonlinear systems. Over the past twenty years, a group of such control schemes have been successfully used in the process control industry where the processes are either intrinsically stable or have very large time constants.

The purpose of this thesis is to provide a theoretical framework for synthesis of a class of optimization based control schemes, known as receding horizon control techniques for nonlinear systems such as unmanned aerial vehicles.

It is well known that unconstrained infinite horizon optimal control may be used to construct a stabilizing controller for a nonlinear system. In this thesis, we show that similar stabilization results may be achieved using unconstrained finite horizon optimal control. The key idea is to approximate the tail of the infinite horizon cost-to-go using, as terminal cost, an appropriate control Lyapunov function (CLF). A CLF can be thought of as generalization of the concept of a Lyapunov function to systems with inputs.

Roughly speaking, the terminal CLF should provide an (incremental) upper bound on the cost. In this fashion, important stability characteristics may be retained without the use of terminal constraints such as those employed by a number of other researchers. The absence of constraints allows a significant speedup in computation.

Furthermore, it is shown that in order to guarantee stability, it suffices to satisfy an improvement property, thereby relaxing the requirement that truly optimal trajectories be found.

We provide a complete analysis of the stability and region of attraction/operation properties of receding horizon control strategies that utilize finite horizon approximations in the proposed class. It is shown that the guaranteed region of operation contains that of the CLF controller and may be made as large as desired by increasing the optimization horizon (restricted, of course, to the infinite horizon domain). Moreover, it is easily seen that both CLF and infinite horizon optimal control approaches are limiting cases of our receding horizon strategy. The key results are illustrated using a familiar example, the inverted pendulum, as well as models of the Caltech ducted fan at hover and forward flight, where significant improvements in guaranteed region of operation and cost are noted.

We also develop an optimization based scheme for generation of aggressive trajectories for hover and forward flight models of the Caltech ducted fan experiment, using a technique known as trajectory morphing. The main idea behind trajectory morphing is to develop a simplified model of the nonlinear system and solve the trajectory generation problem for that model. The resulting trajectory is then used as a reference in a receding horizon optimization scheme to generate trajectories of the original nonlinear system. Several aggressive trajectories are obtained in this fashion for the forward flight model of the Caltech ducted fan experiment.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock and Murray, Richard M.}, } @phdthesis{10.7907/1f3r-va82, author = {Zhu, Xiaoyun}, title = {Hard vs. soft bounds in probablilistic robustness analysis and generalized source coding and optimal web layout design}, school = {California Institute of Technology}, year = {2000}, doi = {10.7907/1f3r-va82}, url = {https://resolver.caltech.edu/CaltechETD:etd-05042006-131410}, abstract = {NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document. Part I: The relationship between hard vs. soft bounds and probabilistic vs. worst-case problem formulations for robustness analysis has been a source of some apparent confusion in the control community, and this thesis attempts to clarify some of these issues. Essentially, worst-case analysis involves computing the maximum of a function which measures performance over some set of uncertainty. Probabilistic analysis assumes some distribution on the uncertainty and computes the resulting probability measure on performance. Exact computation in each case is intractable in general. In the past most research focused on computing hard bounds on worst-case performance. This thesis explores the use of both hard and soft bounds in probabilistic robustness analysis, and investigates the computational complexity of the problems through extensive numerical experimentation. We focus on the simplest possible problem formulations that we believe reveal the difficulties associated with more general probabilistic analysis. By extending the standard structured singular value […] framework to allow for probabilistic descriptions of uncertainty, probabilistic […] is defined, which characterizes the probability distribution of some performance function. The computation of probabilistic […] involves approximating the level surface of the function in the parameter space, which is even more complex than the worst-case […] computation, a well-known NP-hard problem. In particular, providing sufficiently tight bounds in the tail of the distribution is extremely difficult. This thesis proposes three different methods for computing a hard upper bound on probabilistic […] whose tightness can be tested by comparison with the soft bound provided by Monte-Carlo simulations. At the same time, the efficiency of the soft bounds can be significantly improved with the information from the hard bound computation. Among the three algorithms proposed, the LC-BNB algorithm is proven by numerical experiments to provide the best average performance on random examples. One particular example is shown in the end to demonstrate the effectiveness of the method. Part II: The design of robust and reliable networks and network services has become an increasingly challenging task in today’s Internet world. To achieve this goal, understanding the characteristics of Internet traffic plays a more and more critical role. Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network traffic. Moreover, a direct link has been established between the self-similar nature of measured aggregate network traffic and the underlying heavy-tailed distributions of the Web traffic at the source level. This thesis provides a natural and plausible explanation for the origin of heavy tails in Web traffic by introducing a series of simplified models for optimal Web layout design with varying levels of realism and analytic tractability. The basic approach is to view the minimization of the average file download time as a generalization of standard source coding for data compression, but with the design of the Web layout rather than the codewords. The results, however, are quite different from standard source coding, as all assumptions produce power law distributions for a wide variety of user behavior models. In addition, a simulation model of more complex Web site layouts is proposed, with more detailed hyperlinks and user behavior. The throughput of a Web site can be maximized by taking advantage of information on user access patterns and rearranging (splitting or merging) files on the Web site accordingly, with a constraint on available resources. A heuristic optimization on random graphs is formulated, with user navigation modeled as Markov Chains. Simulations on different classes of graphs as well as more realistic models with simple geometries in individual Web pages all produce power law tails in the resulting size distributions of the files transferred from the Web sites. This again verifies our conjecture that heavy-tailed distributions result naturally from the tradeoff between the design objective and limited resources, and suggests a methodology for aiding in the design of high-throughput Web sites.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/2K6Y-CH43, author = {Parrilo, Pablo A.}, title = {Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization}, school = {California Institute of Technology}, year = {2000}, doi = {10.7907/2K6Y-CH43}, url = {https://resolver.caltech.edu/CaltechETD:etd-05062004-055516}, abstract = {In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the associated linear operator maps the cone of positive semidefinite matrices onto itself. In this case, the optimal value equals the spectral radius of the operator. It is shown that some rank minimization problems, as well as generalizations of the structured singular value (*m**u*) LMIs, have exactly this property.

In the same spirit of exploiting structure to achieve computational efficiency, an algorithm for the numerical solution of a special class of frequency-dependent LMIs is presented. These optimization problems arise from robustness analysis questions, via the Kalman-Yakubovich-Popov lemma. The procedure is an outer approximation method based on the algorithms used in the computation of hinf norms for linear, time invariant systems. The result is especially useful for systems with large state dimension.

The other main contribution in this thesis is the formulation of a convex optimization framework for semialgebraic problems, i.e., those that can be expressed by polynomial equalities and inequalities. The key element is the interaction of concepts in real algebraic geometry (Positivstellensatz) and semidefinite programming.

To this end, an LMI formulation for the sums of squares decomposition for multivariable polynomials is presented. Based on this, it is shown how to construct sufficient Positivstellensatz-based convex tests to prove that certain sets are empty. Among other applications, this leads to a nonlinear extension of many LMI based results in uncertain linear system analysis.

Within the same framework, we develop stronger criteria for matrix copositivity, and generalizations of the well-known standard semidefinite relaxations for quadratic programming.

Some applications to new and previously studied problems are presented. A few examples are Lyapunov function computation, robust bifurcation analysis, structured singular values, etc. It is shown that the proposed methods allow for improved solutions for very diverse questions in continuous and combinatorial optimization.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/aw0t-7881, author = {Khatri, Sven H.}, title = {Extensions to the structured singular value}, school = {California Institute of Technology}, year = {1999}, doi = {10.7907/aw0t-7881}, url = {https://resolver.caltech.edu/CaltechETD:etd-02082008-161357}, abstract = {NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

There are two basic approaches to robustness analysis. The first is Monte Carlo analysis which randomly samples parameter space to generate a profile for the typical behavior of the system. The other approach is fundamentally worst case, where the objective is to determine the worst behavior in a set of models. The structured singular value, […], is a powerful frame work for worst case analysis. Where […] is a measure of the distance to singularity using the co-norm.

Under the appropriate projection, the uncertainty sets in the standard […] framework that admit analysis are hypercubes. In this work, […] and the computation of the bounds is extended to spherical sets or equivalently measuring the distance to singularity using the 2-norm. The upper bound is constructed by converting the spherical set of operators into a quadratic form relating the input and output vectors. Using a separating hyperplane or the S-procedure, a linear matrix inequality (LMI) upper bound can be constructed which is tighter than and consistent with the standard p upper bound. This new upper bound has special structure that can be exploited for efficient computation and the standard power algorithm is extended to compute lower bounds for spherical […]. The upper bound construction is further generalized to more exotic regions like arbitrary ellipsoids, the Cartesian product of ellipsoids, and the intersection of ellipsoids. These generalizations are unified with the standard structures. These new tools enable the analysis of more exotic descriptions of uncertain models.

For many real world problems, the worst case paradigm leads to overly pessimistic answers and Monte Carlo methods are computationally expensive to obtain reasonable probabilistic descriptions for rare events.

A few natural probabilistic robustness analysis questions are posed within the […] framework. The proper formulation is as a mixed probabilistic and worst case uncertainty structure. Using branch and bound algorithms, an upper bound can be computed for probabilistic robustness. Motivated by this approach, a purely probabilistic […] problem is posed and bounds are computed. Using the existing machinery, the branch and bound computation cost grows exponentially in the average case for questions of probabilistic robustness. This growth is due to gridding an n-dimensional surface with hypercubes.

A motivation for the extensions of […] to other uncertainty descriptions which admit analysis is to enable more efficient gridding techniques than just hypercubes. The desired fundamental region is a hypercube with a linear constraint. The motivation for this choice is the rank one problem. For rank one, the boundary of singularity is a hyperplane, but the conventional branch and bound tools still result in exponential gridding growth.

The generalization of the […] framework is used to formulate an LMI upper bound for […] with the linear constraint on uncertainty space. This is done by constructing the upper bound for the intersection of an eccentric ellipsoid with the standard uncertainty set. A more promising approach to this computation is the construction of an implicit […] problem where the linear constraints on the uncertainty can be generically rewritten as an algebraic constraint on signals. This may lead to improvements on average to the branch and bound algorithms for probabilistic robustness analysis.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/4AD2-0T48, author = {Primbs, James A.}, title = {Nonlinear optimal control: a receding horizon appoach}, school = {California Institute of Technology}, year = {1999}, doi = {10.7907/4AD2-0T48}, url = {https://resolver.caltech.edu/CaltechETD:etd-10172005-103315}, abstract = {As advances in computing power forge ahead at an unparalleled rate, an increasingly compelling question that spans nearly every discipline is how best to exploit these advances. At one extreme, a tempting approach is to throw as much computational power at a problem as possible. Unfortunately, this is rarely a justifiable approach unless one has some theoretical guarantee of the efficacy of the computations. At the other extreme, not taking advantage of available computing power is unnecessarily limiting. In general, it is only through a careful inspection of the strengths and weaknesses of all available approaches that an optimal balance between analysis and computation is achieved. This thesis addresses the delicate interaction between theory and computation in the context of optimal control.

An exact solution to the nonlinear optimal control problem is known to be prohibitively difficult, both analytically and computationally. Nevertheless, a number of alternative (suboptimal) approaches have been developed. Many of these techniques approach the problem from an off-line, analytical point of view, designing a controller based on a detailed analysis of the system dynamics. A concept particularly amenable to this point of view is that of a control Lyapunov function. These techniques extend the Lyapunov methodology to control systems. In contrast, so-called receding horizon techniques rely purely on on-line computation to determine a control law. While offering an alternative method of attacking the optimal control problem, receding horizon implementations often lack solid theoretical stability guarantees.

In this thesis, we uncover a synergistic relationship that holds between control Lyapunov function based schemes and on-line receding horizon style computation. These connections derive from the classical Hamilton-Jacobi-Bellman and Euler-Lagrange approaches to optimal control. By returning to these roots, a broad class of control Lyapunov schemes are shown to admit natural extensions to receding horizon schemes, benefiting from the performance advantages of on-line computation. From the receding horizon point of view, the use of a control Lyapunov function is a convenient solution to not only the theoretical properties that receding horizon control typically lacks, but also unexpectedly eases many of the difficult implementation requirements associated with on-line computation. After developing these schemes for the unconstrained nonlinear optimal control problem, the entire design methodology is illustrated on a simple model of a longitudinal flight control system. They are then extended to time-varying and input constrained nonlinear systems, offering a promising new paradigm for nonlinear optimal control design.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/5VNR-GF60, author = {Huang, Yun}, title = {Nonlinear optimal control: an enhanced quasi-LPV approach}, school = {California Institute of Technology}, year = {1999}, doi = {10.7907/5VNR-GF60}, url = {https://resolver.caltech.edu/CaltechETD:etd-05212007-082553}, abstract = {Realistic models of physical systems are often nonlinear. Our objective is to synthesize controllers for nonlinear systems that not only provide stability, but also deliver good closed-loop performance. The frozen Riccati equation approach is thoroughly examined. Although it suffers fundamental deficiencies due to its pointwise nature, it is proven that optimality is always possible under a certain assumption on the optimal value of the performance index. This is a consequence of the non-uniqueness of the pointwise linear model of the nonlinear dynamics. However, one cannot assess a priori the guaranteed global performance for a particular model choice. An alternative to the pointwise design is to treat nonlinear plants as linear parameter varying systems with the underlying parameters being functions of the state variables. By exploiting the variation rate bounds of the parameters, a controller that smoothly schedules on the parameters can be synthesized by solving a convex optimization problem. Depending upon the choice of the variation rate bounds, the resulting controller can range from replicating the pointwise design result, which comes with no guarantee on performance, to providing quadratic stability, in which case it can withstand arbitrarily fast parameter variation. Under the above quasi-LPV framework, we present a new scheme that incorporates the freedom of choosing the state-dependent linear representation into the control design process. It is shown that the L2-gain analysis can be reformulated as an infinite dimensional convex optimization problem, and an approximate solution can be obtained by solving a collection of linear matrix inequalities. The synthesis problem is cast as a minimization over an infinite dimensional bilinear matrix inequality constraint. An iterative algorithm, similar to the “D - K iteration” for µ synthesis, is proposed to compute the best achievable performance. It is demonstrated through several examples that this approach can effectively reduce conservatism of the overall design.}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/jh14-eg48, author = {Glavaski, Sonja}, title = {Robust system analysis and nonlinear system model reduction}, school = {California Institute of Technology}, year = {1998}, doi = {10.7907/jh14-eg48}, url = {https://resolver.caltech.edu/CaltechETD:etd-08122005-094404}, abstract = {NOTE: Text or symbols not renderable in plain ASCII are indicated by […]. Abstract is included in .pdf document.

The aim of the first part of this thesis is to broaden the classes of linear systems and performance measures that numerical tools for robustness analysis can be used for. First, we consider robustness problems involving uncertain real parameters and present several new approaches to computing an improved structured singular value […] lower bound. We combine these algorithms to yield a substantially improved power algorithm.

Then, we show that both the worst case […] performance and the worst case […] performance of uncertain systems subject to norm bounded structured LTI perturbations can be written exactly in terms of the skewed […]. The algorithm for the structured singular value lower bound computation, can be extended to computing skewed […] lower bound without significant loss of performance or accuracy.

We also demonstrate how a power algorithm can be used to compute a necessary condition for disturbance rejection of both discrete and continuous time nonlinear systems. For the general case of a system with a non-optimal controller this algorithm can provide us with knowledge of the worst case disturbance.

In the second part of this thesis we explore different approaches to the model reduction of systems. First, we show that the balancing transforma and Galerkin projection commute. We also demonstrate that if the balancing transformation matrix is orthogonal, balanced truncation and Galerkin projection commute.

Next, we pursue model reduction of nonlinear systems with rotational symmetry. We separate the movement of the wave from the evolution of the wave shape using the “centering procedure,” and accurately approximate the shape of the wave with just few modes. The method may be viewed as a way of implementing the Karhunen-Loeve expansion on the space of solutions of the given PDE modulo a given symmetry group. The methodology is quite general and therefore should be useful in a variety of problems.}, address = {1200 East California Boulevard, Pasadena, California 91125}, } @phdthesis{10.7907/4R8P-RR02, author = {D’Andrea, Raffaello}, title = {Generalizations of H-infinity optimization. Control of rotating stall}, school = {California Institute of Technology}, year = {1997}, doi = {10.7907/4R8P-RR02}, url = {https://resolver.caltech.edu/CaltechETD:etd-01092008-110959}, abstract = {Arguably one of the most significant contributions to the field of optimal control has been the formulation and eventual solution of the H∞ design problem. Armed with this mathematical tool, designs which are robust to plant uncertainty and insensitive to plant parameters can be performed in a systematic and rigorous fashion.

The H∞ methodology, however, typically leads to conservative designs. The reasons are twofold. The first is that the plant uncertainty can only be accounted for in an approximate manner, with the result that designs are performed for a set of allowable systems which is larger than what is being modeled; thus the resulting control strategy is forced to guard against non-realizable situations, potentially sacrificing system performance. The second has to do with the physical interpretation of H∞ optimization: the minimization of a system’s power to power gain. Thus it is implicitly assumed in the design process that the worst case disturbance is allowed to be an arbitrary power signal, such as a sinusoid. This is clearly a poor modeling choice for many types of physical disturbances, such as sensor or thermal noise, wind gusts, and impulsive forces.

The main contribution of this dissertation is the extension of H∞ optimization to allow for general closed loop design objectives which address the two limitations outlined above. In particular, non-conservative, computationally tractable, linear matrix inequality based methods for control design are developed for a certain class of physically motivated uncertain systems. In addition, these new techniques can accommodate constraints on the allowable disturbances, excluding unrealistic disturbances from the design process.

Another contribution of this dissertation is an attempt to view control in the broader context of system design. Typically, a control algorithm is only sought after the system to be controlled has already been designed, and the type and location of the actuators and sensors has been determined. For most applications, however, the level of performance which can be attained by any control strategy is dictated by the dynamics of the plant. Thus from a system level, the above methodology is not optimal, since the control design process is decoupled from the design of the rest of the system. By adopting the behavioral framework for systems, an optimization problem where the given system is not treated as an input-output operator, a natural assumption when considering first principles models, is formulated and solved. The interpretation of the above extension of H∞ optimization is that of designing optimal systems.

In contrast to the general purpose tools developed in the first part of the dissertation and summarized above, the second part deals with an actual experimental problem, that of controlling rotating stall using pulsed air injection in a low-speed, axial flow compressor. By modeling the injection of air as an unsteady shift in the compressor characteristic, the viability of various air injection orientations are established. A control strategy is developed which controls the pulsing of air in front of the rotor face based on unsteady pressure measurements near the rotor face. Experimental results show that this technique eliminates the hysteresis loop normally associated with rotating stall. A parametric study is used to determine the optimal control parameters for suppression of stall. The resulting control strategy is also shown to suppress surge when a plenum is present. Using a high fidelity model, the main features of the experimental results are duplicated via simulations. The main contributions of this part of the dissertation are a simple control scheme which has the potential of greatly increasing the operability of compressors, and a low-order modeling mechanism which captures the essential features of air injection, facilitating subsequent analyses and control designs which make use of air injectors.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/MPV7-2Q79, author = {Beck, Carolyn Louise}, title = {Model Reduction and Minimality for Uncertain Systems}, school = {California Institute of Technology}, year = {1997}, doi = {10.7907/MPV7-2Q79}, url = {https://resolver.caltech.edu/CaltechETD:etd-01042008-091550}, abstract = {

The emphasis of this thesis is on the development of systematic methods for reducing the size and complexity of uncertain system models. Given a model for a large complex system, the objective of these methods is to find a simplified model which accurately describes the physical system, thus facilitating subsequent control design and analysis.

Model reduction methods and realization theory are presented for uncertain systems represented by Linear Fractional Transformations (LFTs) on a block diagonal uncertainty structure. A complete generalization of balanced realizations, balanced Gramians and balanced truncation model reduction with guaranteed error bounds is given, which is based on computing solutions to a pair of Linear Matrix Inequalities (LMIs). A necessary and sufficient condition for exact reducibility of uncertain systems, the converse of minimality, is also presented. This condition further generalizes the role of controllability and observability Gramians, and is expressed in terms of singular solutions to the same LMIs. These reduction methods provide a systematic means for both uncertainty simplification and state order reduction in the case of uncertain systems, but also may be interpreted as state order reduction for multi-dimensional systems.

LFTs also provide a convenient way of obtaining realizations for systems described by rational functions of several noncommuting indeterminates. Such functions arise naturally in robust control when studying systems with structured uncertainty, but also may be viewed as a particular type of description for a formal power series. This thesis establishes connections between minimal LFT realizations and minimal linear representations of formal power series, which have been studied extensively in a variety of disciplines, including nonlinear system realization theory. The result is a fairly complete development of minimal realization theory for LFT systems.

General LMI problems and solutions are discussed with the aim of providing sufficient background and references for the construction of computational procedures to reduce uncertain systems. A simple algorithm for computing balanced reduced models of uncertain systems is presented, followed by a discussion of the application of this procedure to a pressurized water reactor for a nuclear power plant.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/4txk-p492, author = {Tierno, Jorge E.}, title = {A computational approach to nonlinear system analysis}, school = {California Institute of Technology}, year = {1996}, doi = {10.7907/4txk-p492}, url = {https://resolver.caltech.edu/CaltechETD:etd-01072008-082023}, abstract = {

Most practical control systems have significant nonlinear components. However, these systems are typically analyzed either through robustness analysis of their linearizations, or through extensive simulation of their nonlinear models. Other forms of analysis of nonlinear systems have not as yet led to computationally tractable solutions. The aim of this thesis is to extend the analysis methodology for linear systems given by the structured singular value framework to nonlinear systems. We study the question: Given an uncertain nonlinear system, driven by a nominal command signal over a finite time horizon, and subject to bounded noise, norm bounded feedback components, and uncertain parameters, how far from the nominal trajectory will the actual trajectory be? In order to inherit the properties of the structured singular value, we will use the 2-norm as measure for noise signals and undermodeled feedback components. As is the case for robustness analysis of linear systems, we can only find efficient computation algorithms for upper and lower bounds to the answer to this question.

To compute the lower bound we develop a power algorithm similar to the one developed for the structured singular value. Since, as was the case for linear systems, the algorithm is not guaranteed to converge in general, its analysis has to be done empirically. We test this algorithm by applying it to simulations of real systems and show that it performs better than other available optimization methods. To develop an upper bound, we study a class of rational nonlinear systems. We show that for problems in this class, an uncertain, constrained linear system can be constructed that achieves the same performance level. Upper bounds on the performance of these systems can be computed by solving linear matrix inequalities. Finally, we study extensions that can be obtained to these analysis methods when the system is linear but time varying.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/sdad-de68, author = {Morris, John Christopher}, title = {Experimental Control: a Helicopter Case Study}, school = {California Institute of Technology}, year = {1996}, doi = {10.7907/sdad-de68}, url = {https://resolver.caltech.edu/CaltechETD:etd-12202007-115753}, abstract = {Robust control has not been used as widely as it could because modelling tools have not advanced as far as analysis and synthesis tools. This becomes readily apparent when applying robust control theory to real problems. With this in mind, an experimental platform was designed and built to study the application of robust control. This platform consists of a real-time computer and a radio-controlled model helicopter mounted on a six degree-of-freedom stand. Experimental systems provide the opportunity not only to verify the applicability of new control theory but also to highlight potential deficiencies.

Traditional system identification and control techniques were used to construct hover controllers for the model helicopter. These techniques are not suitable for the construction of robust models for a system of this complexity. In particular, there was no systematic way to augment nominal identified models with uncertainty suitable for the construction of robust controllers.

To address this issue, frequency-domain model validation algorithms and software were developed. These algorithms provide a methodology for verifying the applicability and consistency between experimental data and robust models. Additionally, they provide a method whereby nominal model parameters can be tuned in a robust setting. This is the first set of software tools which provide this capability for general linear uncertain systems.

Using these new software tools, a systematic design process was developed which incorporated frequency-domain model validation analysis, µ-analysis and µ-synthesis, simulation, and implementation. This design process proved to be a valuable new tool for constructing robust models and designing robust control systems. In particular, by applying this design process to the helicopter, the size of uncertainty in the robust model was substantially reduced without sacrificing the ability of the model to “cover” experimental data and the first controller implemented performed well. This was strikingly different from the results obtained when using standard robust control techniques, where several controllers destabilized the helicopter when implemented, even though they performed well under simulation.

The model validation software and design process provide a consistent methodology and systematic framework which connects system identification, the construction of robust models, and controller synthesis with experimental data. For the first time the control engineer can compute measures on the validity of a robust model, with respect to all observed data on the actual physical system, which are directly related to the robustness measures resulting from µ-analysis and µ-synthesis.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/nvwd-hw47, author = {Newlin, Matthew Philip}, title = {Model Validation, Control, and Computation}, school = {California Institute of Technology}, year = {1996}, doi = {10.7907/nvwd-hw47}, url = {https://resolver.caltech.edu/CaltechETD:etd-01032008-090000}, abstract = {

Engineering in general is concerned with controlling and predicting future behavior with some certainty despite having only imperfect information. Although feedback can be an exceptionally effective engineering tool and is often easy to apply, the behavior of a system under feedback can be extremely sensitive to model mismatch, which is always present. The potential for unpredictable behavior is a major drawback to the engineering application of feedback. Robust control theory addresses this difficulty by parametrizing a family of feedback controllers that are less sensitive to model mismatch.

Despite encouraging early applications, robust control theory has so far been deficient in analysis of systems, synthesis of controllers, connection to real problems, and applicability to nonlinear problems. Further, results on the computational complexity of robust control problems that necessitate either bounds computation or a restricted class of problems have cast doubts about the potential utility of the area.

Initial work in robust control focused on complex uncertainty in the frequency domain. A perceived deficiency is that such model sets are unrealistic: uncertainty in mass, stiffness, aero-coefficients, and the like are naturally modeled as real variations. This thesis includes initial work on practical upper bound computation and substantially improved lower bound computation for moderately large robust control analysis problems that include such real parametric uncertainty, despite the computational complexity of the problems. Although better upper bound computation than that described here is now available for small problems, such is not the case for large problems. The improved lower bound computation chronicled here is desirable because the initial lower bound computation for problems with real parametric uncertainty is not as reliable as in the complex case. Additionally, this thesis shows that branch and bound is a limited but critical tool for better computation, a fact that previously has gone unrecognized.

Together, these contributions allow for the practical computation of robust control problems of engineering interest and provide the basis not only for applications that may ultimately determine the utility of the robust control paradigm but also for the computation of various outgrowths of the [mu] framework, which is the basis for computational robust control.

One such outgrowth is the model validation problem. Model validation tests whether a robust control model in the [mu] framework is consistent with experimentally determined time histories quite a different problem than standard system identification. This thesis shows that the model validation problem is indeed closely related to the standard [mu] problem and its computation.

The practical computation of the model validation problem, which should follow naturally from the work presented here, provides the basis for the connection between robust control theory and practical applications. Future work along these lines should elevate the application of robust control theory from chance and intuition to a standard engineering tool.

Further, the techniques that render the model validation problem similar to the standard [mu] problem are applicable to a great variety of systems analysis and design problems. This newly perceived generality of the [mu] paradigm may ultimately provide a unifying framework for the many seemingly disparate aspects of systems and control design.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/3X73-5F28, author = {Paganini-Herrera, Fernando}, title = {Sets and Constraints in the Analysis of Uncertain Systems}, school = {California Institute of Technology}, year = {1996}, doi = {10.7907/3X73-5F28}, url = {https://resolver.caltech.edu/CaltechETD:etd-09172007-080812}, abstract = {This thesis is concerned with the analysis of dynamical systems in the presence of model uncertainty. The approach of robust control theory has been to describe uncertainty in terms of a structured set of models, and has proven successful for questions, like stability, which call for a worst-case evaluation over this set. In this respect, a first contribution of this thesis is to provide robust stability tests for the situation of combined time varying, time invariant and parametric uncertainties.

The worst-case setting has not been so attractive for questions of disturbance rejection, since the resulting performance criteria (e.g., H_{∞}) treat the disturbance as an adversary and ignore important spectral structure, usually better characterized by the theory of stochastic processes. The main contribution of this thesis is to show that the set-based methodology can indeed be extended to the modeling of white noise, by employing standard statistical tests in order to identify a typical set, and performing subsequent analysis in a worst-case setting. Particularly attractive sets are those described by quadratic signal constraints, which have proven to be very powerful for the characterization of unmodeled dynamics. The combination of white noise and unmodeled dynamics constitutes the Robust H_{2} performance problem, which is rooted in the origins of robust control theory. By extending the scope of the quadratic constraint methodology we obtain a solution to this problem in terms of a convex condition for robustness analysis, which for the first time places it on an equal footing with the H_{∞} performance measure.

A separate contribution of this thesis is the development of a framework for analysis of uncertain systems in implicit form, in terms of equations rather than input-output maps. This formulation is motivated from first principles modeling, and provides an extension of the standard input-output robustness theory. In particular, we obtain in this way a standard form for robustness analysis problems with constraints, which also provides a common setting for robustness analysis and questions of model validation and system identification.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/dnxg-nz58, author = {Lu, Wei-Min}, title = {Control of Uncertain Systems: State-Space Characterizations}, school = {California Institute of Technology}, year = {1995}, doi = {10.7907/dnxg-nz58}, url = {https://resolver.caltech.edu/CaltechETD:etd-03022006-131646}, abstract = {A central issue in control system design has been to deal with uncertainty and nonlinearity in the systems. In this dissertation, an integrated treatment for both uncertainty and nonlinearity is proposed. This dissertation consists of two relatively independent parts. The first part deals with uncertain linear systems, while the second part treats uncertain nonlinear systems.

In the first part, the problem of control synthesis of uncertain linear systems is considered. A linear fractional transformation (LFT) framework is proposed for robust control design of uncertain linear control systems with structured uncertainty. Linear parameter-varying systems whose coefficients depend on some time-invariant unknown parameters are treated in a general algebraic framework; both the stabilization and the H_{∞}-control problems are considered. For uncertain linear systems under structured perturbations, robustness synthesis problems are characterized in terms of linear matrix inequalities (LMIs) in the LFT framework. A generalized PBH test is also used to characterize the robustness synthesis problems. Moreover, a separation principle for the control synthesis of uncertain linear systems is revealed. The machinery also streamlines a number of results concerning the analysis and synthesis of multidimensional systems.

In the second part, the problem of control synthesis for nonlinear systems is addressed; stabilization, L^{1}-control, H_{∞}-control, robustness analysis, and robustness synthesis problems for nonlinear systems are examined in detail. In particular, locally and globally stabilizing controller parameterizations for nonlinear systems are derived; the formulae generalize the celebrated Youla-parameterization for linear systems. Both nonlinear L^{1}-control and nonlinear H_{∞}-control are also considered for dealing with disturbance attenuation problems for nonlinear systems. The L^{1}-performance and L^{1}-control of nonlinear systems are characterized in terms of certain invariance sets of the state space; in addition, the relation between the L^{1}-control of a continuous-time system and the ℓ^{1}-control of the related Euler approximated discrete-time systems is established. A systematic treatment for H_{∞}-control synthesis of nonlinear systems is provided; the nonlinear H_{∞}-control problem is characterized in terms of Hamilton-Jacobi Inequalities (HJIs) and nonlinear matrix inequalities (NLMIs); a class of H_{∞}-controllers are parameterized as a fractional transformation of contractive stable parameters. Finally, the problems of stability and performance robustness analysis and synthesis for uncertain nonlinear systems subject to structured perturbations with bounded L_{2}-gains are introduced; they are characterized in terms of HJIs and NLMIs as well. Computational issues are also addressed; it is confirmed that the computation needed for robustness analysis and synthesis of nonlinear systems is of equivalent difficulty to that for checking Lyapunov stability.

This dissertation applies recent theoretical developments in control to two practical examples. The first example is control of the primary circuit of a pressurized water nuclear reactor. This is an interesting example because the plant is complex and its dynamics vary greatly over the operating range of interest. The second example is a thrust-vectored ducted fan engine, a nonlinear flight control experiment at Caltech.

The main part of this dissertation is the application of linear parameter-dependent control techniques to the examples. The synthesis technique is based on the solution of linear matrix inequalities (LMIs) and produces a controller which achieves specified performance against the worst-case time variation of measurable parameters entering the plant in a linear fractional manner. Thus the plant can have widely varying dynamics over the operating range, a quality possessed by both examples. The controllers designed with these methods perform extremely well and are compared to H_{∞}, gain-scheduled, and nonlinear controllers.

Additionally, an in-depth examination of the model of the ducted fan is performed, including system identification. From this work, we proceed to apply various techniques to examine what they can tell us in the context of a practical example. The primary technique is LMI-based model validation.

The contribution this dissertation makes is to show that parameter-dependent control techniques can be applied with great effectiveness to practical applications. Moreover, the trade-off between modelling and controller performance is examined in some detail. Finally, we demonstrate the applicability of recent model validation techniques in practice, and discuss stabilizability issues.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/ZMMS-XA57, author = {Cortelezzi, Luca}, title = {A theoretical and computational study on active wake control}, school = {California Institute of Technology}, year = {1993}, doi = {10.7907/ZMMS-XA57}, url = {https://resolver.caltech.edu/CaltechETD:etd-09302005-111117}, abstract = {In the first part of this dissertation a two-dimensional unsteady separated flow past a semi-infinite plate with transverse motion is considered. The flow is assumed incompressible and at high Reynolds number. The rolling-up of the separated shear-layer is modelled by a point vortex whose time dependent circulation is predicted by an unsteady Kutta condition. A power-law starting flow is assumed along with a power-law for the transverse motion. The effects of the motion of the plate on the starting vortex circulation and trajectory are presented. A suitable vortex shedding mechanism is introduced and a class of flows involving several vortices is presented. Subsequently, a control strategy able to maintain constant circulation when a vortex is present is derived. An exact solution for the non-linear controller is then obtained. Dynamical system analysis is used to explore the performance of the controlled system. Finally, the control strategy is applied to a class of flows and the results are discussed.

In the second part of this dissertation the previous results are extended to the case of a two-dimensional unsteady separated flow past a plate of variable length. Again the rolling-up of the separated shear-layer is modelled by a vortex pair whose time dependent circulation is predicted by an unsteady Kutta condition. A power-law starting flow is assumed while the plate length is kept constant. The results of the simulations are presented and the model validated. A time-dependent scaling which unveils the universality of the phenomenon is discussed. The previous vortex shedding mechanism is implemented and a vortex merging scheme is tested in a class of flows involving several vortices and is shown to be highly accurate. Subsequently, a control strategy able to maintain constant circulation when a vortex pair is present is derived. An exact solution for the non-linear controller is obtained in the form of an ordinary differential equation. Dynamical system analysis is used to explore the performance of the controlled system and the existence of a controllability region is discussed. Finally, the control strategy is applied to two classes of flows and the results are presented.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Leonard, Anthony and Doyle, John Comstock}, } @phdthesis{10.7907/HJF8-J281, author = {Young, Peter Michael}, title = {Robustness with Parametric and Dynamic Uncertainty}, school = {California Institute of Technology}, year = {1993}, doi = {10.7907/HJF8-J281}, url = {https://resolver.caltech.edu/CaltechETD:etd-11302007-075425}, abstract = {In many disciplines of engineering it is often convenient, for analysis and design purposes, to approximate the real behavior of physical systems by mathematical models. For some applications however, and in particular when one wishes to design a high performance controller, the differences between the behavior of the mathematical model and the physical system can be crucial to the performance of the final design. The theory of robust control attempts to take into account these inherent inaccuracies in the model, and provide systematic analysis and design techniques in the face of this “uncertainty.”

These goals can be restated as formal mathematical problems. In order to handle more realistic descriptions of physical systems, one has to allow more sophisticated models, and this leads to more difficult mathematical problems. In this thesis we will consider both the theoretical and computational aspects of such problems. In particular we will consider robustness in the presence of both real (e. g., parametric) and complex (e. g., dynamic) structured uncertainty.

This leads to a consideration of the general mixed µ analysis and synthesis problems. Some special cases of the analysis problem can be solved exactly, but the general problem is in fact NP hard, so that in order to develop solutions for large problems with reasonable computational requirements, we will adopt a scheme of computing and refining upper and lower bounds. By exploiting the theoretical properties of the problem, we are able to develop practical algorithms, capable of handling mixed µ analysis problems with tens of parameters, in computation times that are typically of the order of minutes. This is despite the fact that the mixed µ problem appears to have inherently combinatoric worst-case behavior.

For the synthesis problem a new “D,G-K iteration” procedure is developed to design a stabilizing controller which attempts to minimize the peak value across frequency of mixed µ. The scheme utilizes a combination of some new results from the mixed µ upper bound problem with the H_{∞} optimal control solution. The theoretical results developed here have already been successfully applied to a number of real engineering problems, and some of these applications are briefly reviewed, to illustrate the advantages offered by the new analysis and synthesis techniques.

Stringent requirements envisioned for the pointing and shape accuracy of future space missions necessitate advances in the control of large flexible structures. These structures will be extremely flexible, with little natural damping and modes densely packed throughout the frequency domain. Due to their size and complexity, testing of these structures will lead to system models that are inaccurate for control purposes. Therefore, control design methods must be developed to account for model inaccuracies or *uncertainties*. Such methods should optimize the robustness and performance characteristics of control laws based on the accuracy of the design model.

This thesis focuses on incorporating knowledge of the mismatch between the physical system and its mathematical models into the control design process. Control design models are developed to fit into the structured singular value (µ) framework that is used in the analysis and synthesis of control laws. To validate and verify theoretical developments, a flexible structure experiment is developed to investigate large flexible control problems in a laboratory environment. The Caltech experiment has a number of their attributes: closely spaced, lightly damped modes, collocated and noncollocated sensors and actuators combined with numerous modes in the controller crossover (roll off) region.

The experimental structure is used to investigate several important issues related to control of flexible structures: tradeoffs between robustness and performance associated with uncertainty modeling for flexible structures, robust control of flexible modes in the controller crossover region and benefits and limitations of collocated versus noncollocated control design. A consistent trend in the results indicates that an accurate description of the flexible structure and model errors is required to synthesize high performance, robust control laws for flexible structures.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, } @phdthesis{10.7907/gz0q-x789, author = {Dailey, Russell Lane}, title = {Conic Sector Analysis for Digital Control Systems with Structured Uncertainty}, school = {California Institute of Technology}, year = {1987}, doi = {10.7907/gz0q-x789}, url = {https://resolver.caltech.edu/CaltechETD:etd-03042008-093526}, abstract = {This thesis presents a method which greatly reduces the conservativeness of conic sector analysis for sampled data feedback systems. The new method evaluates the stability and closed-loop performance of systems with structured uncertainty in the plant transfer function, including MIMO systems and those with multiple sampling rates. In contrast to most multirate analysis techniques, the sampling rates need not be related by rational numbers; this allows analysis when samplers are not strobed to a common clock.

The method is based on a theorem from P. M. Thompson which shows how to construct a conic sector containing a hybrid operator. Combining this theorem with the Structured Singular Value approach of J. C. Doyle, with its heavy use of diagonal scaling, provides an analysis framework for systems with multiple structured plant perturbations. Chapter 3 presents a theorem for the optimal conic sector radius in the SISO case; a MIMO extension of the the theorem completes the development of the new method. Chapter 5 gives three examples.

Chapter 6 presents a new method, based on the complex cepstrum, for synthesis of SISO rational functions to match given “target” transfer functions. The method offers complete control over stability and right half plane zeros. It solves directly for poles and zeros, avoiding the numerical sensitivity of methods which solve for polynomial coefficients. It can synthesize minimum phase functions to match a given magnitude or phase curve. In an example, it is used to synthesize a low- order digital replacement for an analog compensator which gives no degradation of stability margin or step response.

This thesis also presents a method for Kranc vector switch decomposition in state space; this is for stability analysis and input-output simulation of perturbed multirate systems. Moving the 30-year-old Kranc technique from the frequency domain to the state-space domain simplifies the analysis tremendously. Because the number of states is preserved, the dimensionality problems long associated with the Kranc method disappear. The new method is also useful for simulating intersample ripple behavior.

}, address = {1200 East California Boulevard, Pasadena, California 91125}, advisor = {Doyle, John Comstock}, }