The phenomenon of drag reduction by polymers in turbulent flow has beenstudied over the last 60 years. New insight have been recently gained by meansof numerical simulation of dilute polymer solution at moderate values of theturbulent Reynolds number and elasticity. In this thesis, we track elastic parti-cles in Lagrangian frame in turbulent channel flow at Reτ = 180, by tracking,where the single particle obeys the FENE (finite extendible nonlinear elastic)formulation for dumbbel model. The feedback from polymers to the flow is notconsidered, while the Lagrangian approach enables us to consider high valuesof polymer elasticity. In addition, the finite time Lyapunov exponent (FTLE)of the flow is computed tracking infinitesimal material elements advected bythe flow. Following the large deviation theory, the Cramer’s function of theprobability density function of the FTLE for large values of time intervals isstudied at different wall-normal positions. The one-way effect of the turbulentflow on polymers is investigated by looking at the elongation and orientation ofthe polymers, with different relaxation times, across the channel. The confor-mation tensor of the polymers deformation which is an important contributionin the momentum balance equation is calculated by averaging in wall-parallelplanes and compared to theories available in the literature. / QC 20100706

Last decade witnessed the explosive growth in mobile devices and their traffic demand, and hence the significant increase in the energy cost of the cellular service providers. One major component of energy expenditure comes from the operation of base stations. How to reduce energy cost of base stations while satisfying users’ soaring demands has become an imperative yet challenging problem. In this dissertation, we investigate the minimization of the long-term time-averaged expected energy cost while guaranteeing network strong stability. Specifically, considering flow routing, link scheduling, and energy constraints, we formulate a time-coupling stochastic Mixed-Integer Non-Linear Programming (MINLP) problem, which is prohibitively expensive to solve. We reformulate the problem by employing Lyapunov optimization theory and develop a decomposition based algorithm which ensures network strong stability. We obtain the bounds on the optimal result of the original problem and demonstrate the tightness of the bounds and the efficacy of the proposed scheme.

This thesis develops Lyapunov-based control techniques for nonlinear process systems subject to input constraints and stochastic uncertainty. The problems considered include those which focus on the null-controllable region (NCR) for unstable systems. The NCR is the set of states in the state-space from where controllability to desired equilibrium point is possible. For unstable systems, the presence of input constraints induces bounds on the NCR and thereby limits the ability of any controller to steer the system at will. Common approaches for applying control to such systems utilize Control Lyapunov Functions (CLFs) . Such functions can be used for both designing controllers and also preforming closed--loop stability analysis. Existing CLF-based controllers result in closed--loop stability regions that are subsets of the NCR and do not guarantee closed--loop stability from the entire NCR. In effort to mitigate this shortcoming, we introduce a special type of CLF known as a Constrained Control Lyapunov Function (CCLF) which accounts for the presence of input constraints in its definition. CCLFs result in closed--loop stability regions which correspond to the NCR. We demonstrate how CCLFs can be constructed using a function defined by the NCR boundary trajectories for varying values of the available control capacity. We first consider linear systems and utilize available explicit characterization of the NCR to construct CCLFs. We then develop a Model Predictive Control (MPC) design which utilizes this CCLF to achieve stability from the entire NCR for linear anti-stable systems. We then consider the problem of nonlinear systems where explicit characterizations of the NCR boundary are not available. To do so, the problem of boundary construction is considered and an algorithm which is computationally tractable is developed and results in the construction of the boundary trajectories. This algorithm utilizes properties of the boundary pertaining to control equilibrium points to initialize the controllability minimum principle. We then turn to the problem of closed--loop stabilization from the entire NCR for nonlinear systems. Following a similar development as the CCLF construction for linear systems, we establish the validity of the use of the NCR as a CCLF for nonlinear systems. This development involves relaxing the conditions which define a classical CLF and results in CCLF-based control achieving stability to an to an equilibrium manifold. To achieve stabilization from the entire NCR, the CCLF-based control design is coupled with a classical CLF-based controller in a hybrid control framework. In the final part of this thesis, we consider nonlinear systems subject to stochastic uncertainty. Here we design a Lyapunov-based model predictive controller (LMPC) which provides an explicitly characterized region from where stability can be probabilistically obtained. The design exploits the constraint-handling ability of model predictive controllers in order to inherent the stabilization in probability characterization of a Lyapunov-based feedback controller. All the proposed control designs along with the NCR boundary computation are illustrated using simulation results. / Thesis / Doctor of Philosophy (PhD)

We study the Lyapunov exponents and their associated invariant subspaces for infinite dimensional random dynamical systems in a Banach space, which are generated by, for example, stochastic or random partial differential equations. We prove a multiplicative ergodic theorem. Then, we use this theorem to establish the stable and unstable manifold theorem for nonuniformly hyperbolic random invariant sets.

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No description available.

Power systems are extremely non-linear systems which require substantial modeling and control efforts to run continuously. The movement of the power system in parameter and state space is often not well understood, thus making it difficult or impossible to determine whether the system is nearing instability. This dissertation demonstrates several ways in which the power system stability boundary can be calculated. The power system movements evaluated here address the effects of inter-area oscillations on the system which occur in the seconds to minutes time period. The first uses gain scheduling techniques through creation of a set of linear parameter varying (LPV) systems for many operating points of the non-linear system. In the case presented, load and line reactance are used as parameters. The scheduling variables are the power flows in tie lines of the system due to the useful information they provide about the power system state in addition to being available for measurement. A linear controller is developed for the LPV model using H₂/H∞ with pole placement objectives. When the control is applied to the non-linear system, the proposed algorithm predicts the response of the non-linear system to the control by determining if the current system state is located within the domain of attraction of the equilibrium. If the stability domain contains a convex combination of the two points, the control will aid the system in moving towards the equilibrium. The second contribution of this thesis is through the development and implementation of a pseudo non-linear evaluation of a power system as it moves through state space. A system linearization occurs first to compute a multi-objective state space controller. For each contingency definition, many variations of the power system example are created and assigned to the particular contingency class. The powerflow variations and contingency controls are combined to run sets of time series analysis in which the Lyapunov function is tracked over three time steps. This data is utilized for a classification analysis which identifies and classifies the data by the contingency type. The goal is that whenever a new event occurs on the system, real time data can be fed into the trained tree to provide a control for application to increase system damping. / Ph. D.

There are still many open questions regarding spatiotemporal chaos although many well developed theories exist for chaos in time. Rayleigh-B'enard convection is a paradigmatic example of spatiotemporal chaos that is also experimentally accessible. Discoveries uncovered using numerics can often be compared with experiments which can provide new physical insights. Lyapunov diagnostics can provide important information about the dynamics of small perturbations for chaotic systems. Covariant Lyapunov vectors reveal the true direction of perturbation growth and decay. The degree of hyperbolicity can also be quantified by the covariant Lyapunov vectors. To know whether a dynamical system is hyperbolic is important for the development of a theoretical understanding. In this thesis, the degree of hyperbolicity is calculated for chaotic Rayleigh-B'enard convection. For the values of the Rayleigh number explored, it is shown that the dynamics are non-hyperbolic. The spatial distribution of the covariant Lyapunov vectors is different for the different Lyapunov vectors. Localization is used to quantify this variation. The spatial localization of the covariant Lyapunov vectors has a decreasing trend as the order of the Lyapunov vector increases. The spatial localization of the covariant Lyapunov vectors are found to be related to the instantaneous Lyapunov exponents. The correlation is stronger as the order of the Lyapunov vector decreases. The covariant Lyapunov vectors are also computed using a spectral element approach. This allows an exploration of the covariant Lyapunov vectors in larger domains and for experimental conditions. The finite conductivity and finite thickness of the lateral boundaries of an experimental convection domain is also studied. Results are presented for the variation of the Nusselt number and fractal dimension for different boundary conditions. The fractal dimension changes dramatically with the variation of the finite conductivity. / Ph. D.

An important phenomenon of systems driven far-from-equilibrium is spatiotemporal chaos where the dynamics are aperiodic in both time and space. We explored this numerically for three systems: the Lorenz-96 model, the Swift-Hohenberg equation, and Rayleigh-Bénard convection. The Lorenz-96 model is a continuous in time and discrete in space phenomenological model that captures important features of atmosphere dynamics. We computed the fractal dimension as a function of system size and external forcing to estimate characteristic length and time scales describing the chaotic dynamics. We found extensive chaos with significant deviations from extensivity for small changes in system size and also the power-law growth of the dimension with increasing forcing. The Swift-Hohenberg equation is a partial differential equation for a scalar field, which has been widely used as a model for the study of pattern formation. We found that the magnitude of the mean flow in this model must be sufficiently large for spiral defect chaos to occur. We also explored the spatiotemporal chaos in experimentally accessible Rayleigh-Bénard convection using large-scale numerical simulations of the Boussinesq equations and the corresponding tangent space equations. We performed a careful study analyzing the impact of variations in the domain size, Rayleigh number, and Prandtl number on the system dynamics and fractal dimension. In addition, we quantified the dynamics of the spectrum of Lyapunov exponents and the leading order Lyapunov vector in an effort to connect directly with the dynamics of the flow field patterns. Further, we numerically studied the synchronization of chaos in convective flows by imposing time-dependent boundary conditions from a principal domain onto an initially quiescent target domain. We identified a synchronization length scale to quantify the size of a chaotic element using only information from the pattern dynamics. We also explored the relationship of this length scale with the pattern wavelength. Finally, we analyzed bioconvection which occurs as the result of the collective behavior of a suspension of swimming microorganisms. We developed a series of simulations to capture the gyrotactic pattern formation of the swimming algae. The results can be compared with the corresponding trend of pattern instabilities observed in the experimental studies. / Ph. D.

Powered exoskeletons have the potential to revolutionize the labor workplace across many disciplines, from manufacturing to agriculture. However, there are still many barriers to adoption and widespread implementation of exoskeletons. One major research gap of powered exoskeletons currently is the development of a control framework to best cooperate with the user. This limitation is first in understanding the physical and cognitive interaction between the user and exoskeleton, and then in designing a controller that addresses this interaction in a way that provides both physical assistance towards completing a task, and a decrease in the cognitive demand of operating the device. This work demonstrates that multi-objective, optimization-based control can be used to provide a coincident implementation of autonomous robot control, and human-input driven control. A parameter called 'acceptance' can be added to the weights of the cost functions to allow for an automatic trade-off in control priority between the user and robot objectives. This is paired with an update function that allows for the exoskeleton control objectives to track the user objectives over time. This results in a cooperative, powered exoskeleton controller that is responsive to user input, dynamically adjusting control autonomy to allow the user to act to complete a task, learn the control objective, and then offload all effort required to complete the task to the autonomous controller. This reduction in effort is physical assistance directly towards completing the task, and should reduce the cognitive load the user experiences when completing the task. To test the hypothesis of whether high task assistance lowers the cognitive load of the user, a study is designed and conducted to test the effect of the shared autonomy controller on the user's experience operating the robot. The user operates the robot under zero-, full-, and shared-autonomy control cases. Physical workload, measured through the force they exert to complete the task, and cognitive workload, measured through pupil dilation, are evaluated to significantly show that high-assistance operation can lower the cognitive load experienced by a user alongside the physical assistance provided. Automatic adjustment in autonomy works to allow this assistance while allowing the user to be responsive to changing objectives and disturbances. The controller does not remove all mental effort from operation, but shows that high acceptance does lead to less mental effort. When implementing this control beyond the simple reaching task used in the study, however, the controller must be able to both track to the user's desired objective and converge to a high-assistance state to lead to the reduction in cognitive load. To achieve this behavior, first is presented a method to design and enforce Lyapunov stability conditions of individual tasks within a multi-objective controller. Then, with an assumption on the form of the input the user will provide to accomplish their intended task, it is shown that the exoskeleton can stably track an acceptance-weighted combination of the user and robot desired objectives. This guarantee of following the proper trajectory at corresponding autonomy levels results in comparable accuracy in tracking a simulated objective as the base shared autonomy approach, but with a much higher acceptance level, indicating a better match between the user and exoskeleton control objectives, as well as a greater decrease in cognitive load. This process of enforcing stability conditions to shape human-exoskeleton system behavior is shown to be applicable to more tasks, and is in preparation for validation with further user studies. / Doctor of Philosophy / Powered exoskeletons are robots that can be worn by users to physically aid them in accomplishing tasks. These robots differ in scale, from single-joint devices like powered ankle supports or lower-back braces for lifting, to large, multi-joint devices with a broad range of capabilities and potential applications. These multi-joint exoskeletons have been used in many applications such as medical rehabilitation robots, and labor-assisting devices for enhancing strength and avoiding injury. Broader use and adoption in industry could have a great positive impact on the experience of workers performing any heavy-labor tasks. There are still barriers to widespread adoption, however. When closely interacting with machinery like a powered exoskeleton, workers want guarantees of saftey, trust, and cooperation that current exoskeletons have not been able to provide. In fact, studies have shown that industrial devices capable of providing significant assistive force when accomplishing a task, also tend to impart additional, uncomfortable disturbance forces on the user. For example, a lower-body exoskeleton meant to help in lifting tasks might make the simple act of walking more difficult, both physically and mentally. There is a need for exoskeletons that are intuitively cooperative, and can provide both physical assistance towards completing a task and cognitive assistance that makes coordinating with the human user easier. In this dissertation we examine the control problem of powered exoskeletons. In the past, many powered exoskeleton controllers are direct, scripted controllers with exact objectives, or actions tied only to human input. To go beyond this, we leverage "multi-objective-control", originally designed for humanoid robots, which is capable of controlling the robot to accomplish multiple goals at the same time. This approach is the base on which a more complex controller can be created. We show first that the multi-objective control can be used to achieve human desired actions and robot autonomous control tasks at the same time, with a parameter to trade-off which actor, the human or the robot, has the priority control at that time. This framework has the capacity to allow the human to instruct the robot in tasks to accomplish, and then robot can fully mimic the user, offloading the physical effort required to accomplish the task. It is proposed that this offloading of effort from the user will also lower the cognitive load the user is under when actively commanding the exoskeleton. To test this hypothesis, a user study is conducted where human operators work with an upper-body powered exoskeleton to complete a simple reaching task. This study shows that on average, the more assistance the exoskeleton provides to the user, the lower their mental demand is. Additionally, when responding to new challenges or sudden disturbances, the robot can easily cooperate, balancing its own autonomy with the user's to allow the user to respond as they need to their changing environment, then resume active assistance when the change is resolved. Finally, to guarantee that the exoskeleton responds quickly and accurately to the user's intentions, a new strategy is derived to update the robot's internal objectives to match the users' goals. This strategy is based on the assumption that the exoskeleton knows what type of task the user is trying to complete. If this is true, then the exoskeleton can estimate the users objectives from the actions they task, and ensure assistance towards completing the task. This control design is proven in simulation, and in preparation for followup studies to evaluate the user experience of this improved strategy.