Effects of density differences on lateral mixing in open-channel flows

This study investigates lateral mixing of tracer fluids in turbulent open-channel flows when the tracer and ambient fluids have different densities. Longitudinal dispersion in flows with longitudinal density gradients is investigated also.

Lateral mixing was studied in a laboratory flume by introducing fluid tracers at the ambient flow velocity continuously and uniformly across a fraction of the flume width and over the entire depth of the ambient flow. Fluid samples were taken to obtain concentration distributions in cross-sections at various distances, x, downstream from the tracer source. The data were used to calculate variances of the lateral distributions of the depth-averaged concentration. When there was a difference in density between the tracer and the ambient fluids, lateral mixing close to the source was enhanced by density-induced secondary flows; however, far downstream where the density gradients were small, lateral mixing rates were independent of the initial density difference. A dimensional analysis of the problem and the data show that the normalized variance is a function of only three dimensionless numbers, which represent: (1) the x-coordinate, (2) the source width, and (3) the buoyancy flux from the source.

A simplified set of equations of motion for a fluid with a horizontal density gradient was integrated to give an expression for the density-induced velocity distribution. The dispersion coefficient due to this velocity distribution was also obtained. Using this dispersion coefficient in an analysis for predicting lateral mixing rates in the experiments of this investigation gave only qualitative agreement with the data. However, predicted longitudinal salinity distributions in an idealized laboratory estuary agree well with published data.

Spectral density of first order Piecewise linear systems excited by white noise

The Fokker-Planck (FP) equation is used to develop a general method for finding the spectral density for a class of randomly excited first order systems. This class consists of systems satisfying stochastic differential equations of form ẋ + f(x) = m/Ʃ/j = 1 h_j(x)n_j(t) where f and the h_j are piecewise linear functions (not necessarily continuous), and the n_j are stationary Gaussian white noise. For such systems, it is shown how the Laplace-transformed FP equation can be solved for the transformed transition probability density. By manipulation of the FP equation and its adjoint, a formula is derived for the transformed autocorrelation function in terms of the transformed transition density. From this, the spectral density is readily obtained. The method generalizes that of Caughey and Dienes, J. Appl. Phys., 32.11.

This method is applied to 4 subclasses: (1) m = 1, h₁ = const. (forcing function excitation); (2) m = 1, h₁ = f (parametric excitation); (3) m = 2, h₁ = const., h₂ = f, n₁ and n₂ correlated; (4) the same, uncorrelated. Many special cases, especially in subclass (1), are worked through to obtain explicit formulas for the spectral density, most of which have not been obtained before. Some results are graphed.

Dealing with parametrically excited first order systems leads to two complications. There is some controversy concerning the form of the FP equation involved (see Gray and Caughey, J. Math. Phys., 44.3); and the conditions which apply at irregular points, where the second order coefficient of the FP equation vanishes, are not obvious but require use of the mathematical theory of diffusion processes developed by Feller and others. These points are discussed in the first chapter, relevant results from various sources being summarized and applied. Also discussed is the steady-state density (the limit of the transition density as t → ∞).

I. The structure and properties of closed circular duplex DNA. II. Chemically interacting systems at equilibrium in a buoyant density gradient

I. The binding of the intercalating dye ethidium bromide to closed circular SV 40 DNA causes an unwinding of the duplex structure and a simultaneous and quantitatively equivalent unwinding of the superhelices. The buoyant densities and sedimentation velocities of both intact (I) and singly nicked (II) SV 40 DNAs were measured as a function of free dye concentration. The buoyant density data were used to determine the binding isotherms over a dye concentration range extending from 0 to 600 µg/m1 in 5.8 M CsCl. At high dye concentrations all of the binding sites in II, but not in I, are saturated. At free dye concentrations less than 5.4 µg/ml, I has a greater affinity for dye than II. At a critical amount of dye bound I and II have equal affinities, and at higher dye concentration I has a lower affinity than II. The number of superhelical turns, τ, present in I is calculated at each dye concentration using Fuller and Waring's (1964) estimate of the angle of duplex unwinding per intercalation. The results reveal that SV 40 DNA I contains about -13 superhelical turns in concentrated salt solutions.

The free energy of superhelix formation is calculated as a function of τ from a consideration of the effect of the superhelical turns upon the binding isotherm of ethidium bromide to SV 40 DNA I. The value of the free energy is about 100 kcal/mole DNA in the native molecule. The free energy estimates are used to calculate the pitch and radius of the superhelix as a function of the number of superhelical turns. The pitch and radius of the native I superhelix are 430 Å and 135 Å, respectively.

A buoyant density method for the isolation and detection of closed circular DNA is described. The method is based upon the reduced binding of the intercalating dye, ethidium bromide, by closed circular DNA. In an application of this method it is found that HeLa cells contain in addition to closed circular mitochondrial DNA of mean length 4.81 microns, a heterogeneous group of smaller DNA molecules which vary in size from 0.2 to 3.5 microns and a paucidisperse group of multiples of the mitochondrial length.

II. The general theory is presented for the sedimentation equilibrium of a macromolecule in a concentrated binary solvent in the presence of an additional reacting small molecule. Equations are derived for the calculation of the buoyant density of the complex and for the determination of the binding isotherm of the reagent to the macrospecies. The standard buoyant density, a thermodynamic function, is defined and the density gradients which characterize the four component system are derived. The theory is applied to the specific cases of the binding of ethidium bromide to SV 40 DNA and of the binding of mercury and silver to DNA.

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.