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

The centralized paradigm of a single controller and a single plant upon which modern control theory is built is no longer applicable to modern cyber-physical systems of interest, such as the power-grid, software defined networks or automated highways systems, as these are all large-scale and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

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

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

Synthetic molecular machines for active self-assembly : prototype algorithms, designs, and experimental study

Computer science and electrical engineering have been the great success story of the twentieth century. The neat modularity and mapping of a language onto circuits has led to robots on Mars, desktop computers and smartphones. But these devices are not yet able to do some of the things that life takes for granted: repair a scratch, reproduce, regenerate, or grow exponentially fast–all while remaining functional.

This thesis explores and develops algorithms, molecular implementations, and theoretical proofs in the context of “active self-assembly” of molecular systems. The long-term vision of active self-assembly is the theoretical and physical implementation of materials that are composed of reconfigurable units with the programmability and adaptability of biology’s numerous molecular machines. En route to this goal, we must first find a way to overcome the memory limitations of molecular systems, and to discover the limits of complexity that can be achieved with individual molecules.

One of the main thrusts in molecular programming is to use computer science as a tool for figuring out what can be achieved. While molecular systems that are Turing-complete have been demonstrated [Winfree, 1996], these systems still cannot achieve some of the feats biology has achieved.

One might think that because a system is Turing-complete, capable of computing “anything,” that it can do any arbitrary task. But while it can simulate any digital computational problem, there are many behaviors that are not “computations” in a classical sense, and cannot be directly implemented. Examples include exponential growth and molecular motion relative to a surface.

Passive self-assembly systems cannot implement these behaviors because (a) molecular motion relative to a surface requires a source of fuel that is external to the system, and (b) passive systems are too slow to assemble exponentially-fast-growing structures. We call these behaviors “energetically incomplete” programmable behaviors. This class of behaviors includes any behavior where a passive physical system simply does not have enough physical energy to perform the specified tasks in the requisite amount of time.

As we will demonstrate and prove, a sufficiently expressive implementation of an “active” molecular self-assembly approach can achieve these behaviors. Using an external source of fuel solves part of the the problem, so the system is not “energetically incomplete.” But the programmable system also needs to have sufficient expressive power to achieve the specified behaviors. Perhaps surprisingly, some of these systems do not even require Turing completeness to be sufficiently expressive.

Building on a large variety of work by other scientists in the fields of DNA nanotechnology, chemistry and reconfigurable robotics, this thesis introduces several research contributions in the context of active self-assembly.

We show that simple primitives such as insertion and deletion are able to generate complex and interesting results such as the growth of a linear polymer in logarithmic time and the ability of a linear polymer to treadmill. To this end we developed a formal model for active-self assembly that is directly implementable with DNA molecules. We show that this model is computationally equivalent to a machine capable of producing strings that are stronger than regular languages and, at most, as strong as context-free grammars. This is a great advance in the theory of active self- assembly as prior models were either entirely theoretical or only implementable in the context of macro-scale robotics.

We developed a chain reaction method for the autonomous exponential growth of a linear DNA polymer. Our method is based on the insertion of molecules into the assembly, which generates two new insertion sites for every initial one employed. The building of a line in logarithmic time is a first step toward building a shape in logarithmic time. We demonstrate the first construction of a synthetic linear polymer that grows exponentially fast via insertion. We show that monomer molecules are converted into the polymer in logarithmic time via spectrofluorimetry and gel electrophoresis experiments. We also demonstrate the division of these polymers via the addition of a single DNA complex that competes with the insertion mechanism. This shows the growth of a population of polymers in logarithmic time. We characterize the DNA insertion mechanism that we utilize in Chapter 4. We experimentally demonstrate that we can control the kinetics of this re- action over at least seven orders of magnitude, by programming the sequences of DNA that initiate the reaction.

In addition, we review co-authored work on programming molecular robots using prescriptive landscapes of DNA origami; this was the first microscopic demonstration of programming a molec- ular robot to walk on a 2-dimensional surface. We developed a snapshot method for imaging these random walking molecular robots and a CAPTCHA-like analysis method for difficult-to-interpret imaging data.

Design, specification, and synthesis of aircraft electric power systems control logic

Cyber-physical systems integrate computation, networking, and physical processes. Substantial research challenges exist in the design and verification of such large-scale, distributed sensing, ac- tuation, and control systems. Rapidly improving technology and recent advances in control theory, networked systems, and computer science give us the opportunity to drastically improve our approach to integrated flow of information and cooperative behavior. Current systems rely on text-based spec- ifications and manual design. Using new technology advances, we can create easier, more efficient, and cheaper ways of developing these control systems. This thesis will focus on design considera- tions for system topologies, ways to formally and automatically specify requirements, and methods to synthesize reactive control protocols, all within the context of an aircraft electric power system as a representative application area.

This thesis consists of three complementary parts: synthesis, specification, and design. The first section focuses on the synthesis of central and distributed reactive controllers for an aircraft elec- tric power system. This approach incorporates methodologies from computer science and control. The resulting controllers are correct by construction with respect to system requirements, which are formulated using the specification language of linear temporal logic (LTL). The second section addresses how to formally specify requirements and introduces a domain-specific language for electric power systems. A software tool automatically converts high-level requirements into LTL and synthesizes a controller.

The final sections focus on design space exploration. A design methodology is proposed that uses mixed-integer linear programming to obtain candidate topologies, which are then used to synthesize controllers. The discrete-time control logic is then verified in real-time by two methods: hardware and simulation. Finally, the problem of partial observability and dynamic state estimation is ex- plored. Given a set placement of sensors on an electric power system, measurements from these sensors can be used in conjunction with control logic to infer the state of the system.

Relating thermodynamics to information theory : the equality of free energy and mutual information

In this thesis we uncover a new relation which links thermodynamics and information theory. We consider time as a channel and the detailed state of a physical system as a message. As the system evolves with time, ever present noise insures that the "message" is corrupted. Thermodynamic free energy measures the approach of the system toward equilibrium. Information theoretical mutual information measures the loss of memory of initial state. We regard the free energy and the mutual information as operators which map probability distributions over state space to real numbers. In the limit of long times, we show how the free energy operator and the mutual information operator asymptotically attain a very simple relationship to one another. This relationship is founded on the common appearance of entropy in the two operators and on an identity between internal energy and conditional entropy. The use of conditional entropy is what distinguishes our approach from previous efforts to relate thermodynamics and information theory.

Physical investigations of small particles : (I) aerosol particle charging and flux enhancement and (II) whispering gallery mode sensing

Part I

Particles are a key feature of planetary atmospheres. On Earth they represent the greatest source of uncertainty in the global energy budget. This uncertainty can be addressed by making more measurement, by improving the theoretical analysis of measurements, and by better modeling basic particle nucleation and initial particle growth within an atmosphere. This work will focus on the latter two methods of improvement.

Uncertainty in measurements is largely due to particle charging. Accurate descriptions of particle charging are challenging because one deals with particles in a gas as opposed to a vacuum, so different length scales come into play. Previous studies have considered the effects of transition between the continuum and kinetic regime and the effects of two and three body interactions within the kinetic regime. These studies, however, use questionable assumptions about the charging process which resulted in skewed observations, and bias in the proposed dynamics of aerosol particles. These assumptions affect both the ions and particles in the system. Ions are assumed to be point monopoles that have a single characteristic speed rather than follow a distribution. Particles are assumed to be perfect conductors that have up to five elementary charges on them. The effects of three body interaction, ion-molecule-particle, are also overestimated. By revising this theory so that the basic physical attributes of both ions and particles and their interactions are better represented, we are able to make more accurate predictions of particle charging in both the kinetic and continuum regimes.

The same revised theory that was used above to model ion charging can also be applied to the flux of neutral vapor phase molecules to a particle or initial cluster. Using these results we can model the vapor flux to a neutral or charged particle due to diffusion and electromagnetic interactions. In many classical theories currently applied to these models, the finite size of the molecule and the electromagnetic interaction between the molecule and particle, especially for the neutral particle case, are completely ignored, or, as is often the case for a permanent dipole vapor species, strongly underestimated. Comparing our model to these classical models we determine an “enhancement factor” to characterize how important the addition of these physical parameters and processes is to the understanding of particle nucleation and growth.

Part II

Whispering gallery mode (WGM) optical biosensors are capable of extraordinarily sensitive specific and non-specific detection of species suspended in a gas or fluid. Recent experimental results suggest that these devices may attain single-molecule sensitivity to protein solutions in the form of stepwise shifts in their resonance wavelength, \lambda_{R}, but present sensor models predict much smaller steps than were reported. This study examines the physical interaction between a WGM sensor and a molecule adsorbed to its surface, exploring assumptions made in previous efforts to model WGM sensor behavior, and describing computational schemes that model the experiments for which single protein sensitivity was reported. The resulting model is used to simulate sensor performance, within constraints imposed by the limited material property data. On this basis, we conclude that nonlinear optical effects would be needed to attain the reported sensitivity, and that, in the experiments for which extreme sensitivity was reported, a bound protein experiences optical energy fluxes too high for such effects to be ignored.

Part I. Energy straggling of ^4He below 2.0 MeV in Al, Ni, Au and Pt. Part II. Studies of the Ti-W metallization system on Si

Part I.

In recent years, backscattering spectrometry has become an important tool for the analysis of thin films. An inherent limitation, though, is the loss of depth resolution due to energy straggling of the beam. To investigate this, energy straggling of ⁴He has been measured in thin films of Ni, Al, Au and Pt. Straggling is roughly proportional to square root of thickness, appears to have a slight energy dependence and generally decreases with decreasing atomic number of the adsorber. The results are compared with predictions of theory and with previous measurements. While Ni measurements are in fair agreement with Bohr's theory, Al measurements are 30% above and Au measurements are 40% below predicted values. The Au and Pt measurements give straggling values which are close to one another.

Part II.

MeV backscattering spectrometry and X-ray diffraction are used to investigate the behavior of sputter-deposited Ti-W mixed films on Si substrates. During vacuum anneals at temperatures near 700°C for several hours, the metallization layer reacts with the substrate. Backscattering analysis shows that the resulting compound layer is uniform in composition and contains Ti, Wand Si. The Ti:W ratio in the compound corresponds to that of the deposited metal film. X-ray analyses with Reed and Guinier cameras reveal the presence of the ternary Ti_xW_(1-x)Si₂ compound. Its composition is unaffected by oxygen contamination during annealing, but the reaction rate is affected. The rate measured on samples with about 15% oxygen contamination after annealing is linear, of the order of 0.5 Å per second at 725°C, and depends on the crystallographic orientation of the substrate and the dc bias during sputter-deposition of the Ti-W film.

Au layers of about 1000 Å thickness were deposited onto unreacted Ti-W films on Si. When annealed at 400°C these samples underwent a color change,and SEM micrographs of the samples showed that an intricate pattern of fissures which were typically 3µm wide had evolved. Analysis by electron microprobe revealed that Au had segregated preferentially into the fissures. This result suggests that Ti-W is not a barrier to Au-Si intermixing at 400°C.

A study of a new electromechanical energy conversion process using superconducting frequency modulated resonators

Experimental demonstrations and theoretical analyses of a new electromechanical energy conversion process which is made feasible only by the unique properties of superconductors are presented in this dissertation. This energy conversion process is characterized by a highly efficient direct energy transformation from microwave energy into mechanical energy or vice versa and can be achieved at high power level. It is an application of a well established physical principle known as the adiabatic theorem (Boltzmann-Ehrenfest theorem) and in this case time dependent superconducting boundaries provide the necessary interface between the microwave energy on one hand and the mechanical work on the other. The mechanism which brings about the conversion is another known phenomenon - the Doppler effect. The resonant frequency of a superconducting resonator undergoes continuous infinitesimal shifts when the resonator boundaries are adiabatically changed in time by an external mechanical mechanism. These small frequency shifts can accumulate coherently over an extended period of time to produce a macroscopic shift when the resonator remains resonantly excited throughout this process. In addition, the electromagnetic energy in s ide the resonator which is proportional to the oscillation frequency is al so accordingly changed so that a direct conversion between electromagnetic and mechanical energies takes place. The intrinsically high efficiency of this process is due to the electromechanical interactions involved in the conversion rather than a process of thermodynamic nature and therefore is not limited by the thermodynamic value.

A highly reentrant superconducting resonator resonating in the range of 90 to 160 MHz was used for demonstrating this new conversion technique. The resonant frequency was mechanically modulated at a rate of two kilohertz. Experimental results showed that the time evolution of the electromagnetic energy inside this frequency modulated (FM) superconducting resonator indeed behaved as predicted and thus demonstrated the unique features of this process. A proposed usage of FM superconducting resonators as electromechanical energy conversion devices is given along with some practical design considerations. This device seems to be very promising in producing high power (~10W/cm^3) microwave energy at 10 - 30 GHz.

Weakly coupled FM resonator system is also analytically studied for its potential applications. This system shows an interesting switching characteristic with which the spatial distribution of microwave energies can be manipulated by external means. It was found that if the modulation was properly applied, a high degree (>95%) of unidirectional energy transfer from one resonator to the other could be accomplished. Applications of this characteristic to fabricate high efficiency energy switching devices and high power microwave pulse generators are also found feasible with present superconducting technology.

Solar System Small-Body Demographics with the Palomar Transient Factory Survey

Observational studies of our solar system's small-body populations (asteroids and comets) offer insight into the history of our planetary system, as these minor planets represent the left-over building blocks from its formation. The Palomar Transient Factory (PTF) survey began in 2009 as the latest wide-field sky-survey program to be conducted on the 1.2-meter Samuel Oschin telescope at Palomar Observatory. Though its main science program has been the discovery of high-energy extragalactic sources (such as supernovae), during its first five years PTF has collected nearly five million observations of over half a million unique solar system small bodies. This thesis begins to analyze this vast data set to address key population-level science topics, including: the detection rates of rare main-belt comets and small near-Earth asteroids, the spin and shape properties of asteroids as inferred from their lightcurves, the applicability of this visible light data to the interpretation of ultraviolet asteroid observations, and a comparison of the physical properties of main-belt and Jovian Trojan asteroids. Future sky-surveys would benefit from application of the analytical techniques presented herein, which include novel modeling methods and unique applications of machine-learning classification. The PTF asteroid small-body data produced in the course of this thesis work should remain a fertile source of solar system science and discovery for years to come.