Few model reduction techniques exist for dynamical systems whose parameters vary with time. We have particular interest here in linear time-periodic dynamical systems; we seek a structure-preserving algorithm for model reduction of linear time-periodic (LTP) dynamical systems of large scale that generalizes from the linear time-invariant (LTI) model reduction problem. We extend the familiar LTI system theory to analogous concepts in the LTP setting. First, we represent the LTP system as a convolution operator of a bivariate periodic kernel function. The kernel suggests a representation of the system as a frequency operator, called the Harmonic Transfer Function. Second, we exploit the Hilbert space structure of the family of LTP systems to develop necessary conditions for optimal approximations. Additionally, we show an a posteriori error bound written in terms of the $\\mathcal H_2$ norm of related LTI multiple input/multiple output system. This bound inspires an algorithm to construct approximations of reduced order. To verify the efficacy of this algorithm we apply it to three models: (1) fluid flow around a cylinder by a finite element discretization of the Navier-Stokes equations, (2) thermal diffusion through a plate modeled by the heat equation, and (3) structural model of component 1r of the Russian service module of the International Space Station. / Master of Science

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In adaptive control the goal is to deal with systems that have unknown and/or time-varying parameters. Adaptive control techniques have been developed since 1950’s and most results were proven in the cases when the time-variations were non-existent or slow. However the results pertaining to systems with fast time-variations are still limited, in particular, when it comes to plants with unstable zero dynamics. In this work we adopt the controller design technique from the area of gain scheduling, where the time-varying parameter is assumed to be measurable. We propose the design of a nonlinear periodic controller, where in each period the state and parameter values are estimated and an appropriate stabilizing control signal is applied. It is shown that the closed loop system is stable under fast parameter variations with persistent jumps: the trajectory of the closed loop state in response to the initial condition is bounded by a decaying exponential plus a gain times the size of the noise. Our approach imposes several constraints on the plant; however, we show that there exists at least one interesting class of systems, which includes plants with unstable zero dynamics, that can be stabilized by our controller.

Nonlinear optics represents a significant area of research and technology concerned with the modification of material optical properties using light. The interaction between light and such materials gives rise to a multitude of nonlinear optical effects, including second har-monic generation, third harmonic generation, high harmonic generation, and sum frequency generation. This thesis focuses on a specific and relevant nonlinear phenomenon within this field, namely the nonlinear Kerr effect, which involves the modification of a material’s re-fractive index through the exposure to an intense beam of light. The nonlinear Kerr effect holds promise for various applications, such as self-phase modulation in laser technology and the utilization of optical solitons in telecommunications. However, the limited availability of materials with sufficiently strong Kerr effects often restricts the practical application of this effect across different industries. Concurrently, optical time-varying systems play crucial roles in modern technologies, in-cluding optical modulators, LiDAR systems, and adaptive cameras. These systems involve the dynamic modification of optical properties. To achieve ultra-fast modulation of light properties, it is beneficial to explore materials with ultra-fast modulation speeds of the op-tical refractive index for integration into time-varying systems. While electro-optical effects represent the most common methods for achieving high-speed modulation of the effective refractive index, the utilization of all-optical methods, such as the nonlinear Kerr effect, presents an alternative approach. Nevertheless, the absence of simultaneous high speed and large nonlinear Kerr response in the majority of well-established materials restricts the utilization of the Kerr effect in time-varying systems.This thesis focuses on the study of a group of materials known as epsilon-near-zero (ENZ) materials, where the real part of the permittivity vanishes at a specific wavelength referred to as the ENZ wavelength. Specifically, indium-tin-oxide (ITO), a transparent conducting oxide, is investigated, with its ENZ wavelength falling within the infrared region of the elec-tromagnetic spectrum. ITO has been shown to possess a record-breaking large nonlinear Kerr effect with sub-picosecond response times, making it an excellent candidate for all-optical time-varying systems. The primary objective of this research is to investigate the applications of this large, fast nonlinear response and, where possible, enhance its effective-ness. One notable application of rapid and substantial modifications in the refractive index of a material is adiabatic wavelength conversion of light. In one project, a thin layer of ITO is subjected to a pump-probe setup, where an intense pump beam of light triggers the nonlinear response of ITO, causing the refractive index to rapidly change while a probe beam passes through the modulated system. Consequently, the wavelength of the probe beam undergoes conversion. Furthermore, it has been demonstrated that the nonlinear response of ITO can be sig-nificantly enhanced in the presence of a plasmonic metasurface. Metasurfaces consist of two-dimensional arrays of sub-wavelength scattering objects capable of manipulating the vectorial properties of light. In another project, we design a gradient metasurface composed of gold placed over ITO, enabling the diffraction of incident light into various diffraction orders depending on the ratio between the wavelength of light and the periodicity of the metasurface. This unique property is utilized to dynamically steer the diffraction orders of the probe beam, achieving wavelength conversion by exciting the nonlinear response of the ITO substrate with a second pump beam. Additionally, we investigate the interaction of resonance modes in an amorphous silicon metasurface, known as Mie modes, with an inherently dark mode in a thin layer of ITO known as the ENZ mode. Through experimental and analytical approaches, we demonstrate that two fundamental Mie modes, electric dipole resonance and magnetic dipole resonance, can strongly couple with the ENZ mode. This strong coupling creates a highly complex system with a large and rapid nonlinear response, enabling the manipulation of light on sub-picosecond timescales. In our final main project, we delve into investigating the nonlinear response of ITO nanoparticles. To accomplish this, we put forth a numerical recursive approach that allows us to incorporate the significant nonlinear Kerr effect of ITO into inherently linear simulation environments. Subsequently, we employ this proposed method to extract the scattering pattern of sub-wavelength antennas fabricated from ITO in both linear and nonlinear optical regimes. Our objective is to explore the potential applications of ITO nanoantennas in various fields. Moreover, this thesis encompasses other projects related to ENZ materials. We investi-gate the nonlinear response of an artificially created ENZ medium by stacking subsequent layers of materials with negative and positive permittivities within the visible range of the electromagnetic spectrum. Additionally, we explore the nonlinear response of nanoparticles made of ITO. Lastly, we present our investigations into the strong coupling of the ENZ mode in a thin layer of ITO with surface plasmon polaritons in a layer of gold in contact with ITO.

This dissertation addresses fundamental theoretical problems relevant to flight control for aerial vehicles and weapons in highly uncertain dynamical environment. The approach taken in this dissertation is the L1 adaptive control, which is elaborated from its design perspective for output feedback solution and is extended to time-varying reference systems to support augmentation of gain-scheduled baseline controllers. Compared to conventional adaptive controllers, L1 control has the following advantages: i) it has guaranteed uniformly bounded transient response for system's both signals, input and output; ii) it enables fast adaptation while maintains a bounded away from zero time-delay margin. The proposed adaptive control approach can recover the nominal performance of the flight control systems in the presence of rapid variation of uncertainties. Furthermore, the benefit of L1 adaptive control is its promise for development of theoretically justified tools for Verification and Validation (V&V) of adaptive systems. Adaptive control for uncertain systems usually needs to handle two types of uncertainties: matched and unmatched uncertainties. Both of these two uncertainties will appear in practical flight control problems. In this dissertation, adaptive approaches which can compensate for these two types of uncertainties will be discussed respectively. Two architectures of L1 adaptive control, namely L1 state feedback adaptive control and L1 output feedback adaptive control, are studied. The state feedback adaptive control is applied for compensation of matched uncertainties. Although the state feedback scheme is capable of handling certain type of unmatched uncertainties, such approach is not explored in this dissertation. On the other hand, the output feedback approach is mainly aimed to solve problems in the presence of unmatched uncertainties. The dissertation first discusses the state feedback L1 adaptive control for time-invariant reference systems. The adaptive controller is designed to augment an existing baseline controller. The closed loop system of the plant and the baseline controller is time-invariant. This closed loop system, which is a Linear Time Invariant (LTI) system, determines the dynamics of the reference system. The adaptive feedback can compensate for nonlinear state- and time-dependent uncertainty with uniformly bounded transient response. In this dissertation we discuss the Multi-Input Multi-Output (MIMO) extension of the method. Two flight control examples,Unmanned Combat Aerial Vehicle (UCAV) and Aerial Refueling Autopilot, are considered in the presence of nonlinear uncertainties and control surface failures. The L1 adaptive controller without any redesign leads to scaled response for system's both signals, input and output, dependent upon changes in the initial conditions, system parameters and uncertainties. The time-delay margin analysis for these two examples verifies the theoretical claims. Next, the output feedback approach is studied. The adaptive output feedback controller can be applied to reference systems that do not verify the Strict Positive Real (SPR) condition for their input-output transfer function. In this dissertation, specific design guidelines are presented that render the approach suitable for practical applications. A missile autopilot design example is given to demonstrate the benefits of the design approach. Finally, the L1 state feedback adaptive controller is extended to time-varying reference systems. The adaptive controller intends to augment a gain-scheduled baseline controller. The reference system, which is determined by the closed loop system of the plant and the baseline gain-scheduled controller, is time-varying. The adaptive controller with time-varying reference system is proved to have guaranteed performance bounds similar to those obtained for the case of linear time-invariant reference systems. With this result, the aerial refueling application can be extended to a complete scenario, which includes a racetrack maneuver for an aircraft. The concluding chapter discusses the challenging issues for future research. / Ph. D.

The thesis treats model reduction for linear time-varying systems. Time-varying models appear in many fields, including power systems, chemical engineering, aeronautics, and computational science. They can also be used for approximation of time-invariant nonlinear models. Model reduction is a topic that deals with simplification of complex models. This is important since it facilitates analysis and synthesis of controllers. The thesis consists of two parts. The first part provides an introduction to the topics of time-varying systems and model reduction. Here, notation, standard results, examples, and some results from the second part of the thesis are presented. The second part of the thesis consists of four papers. In the first paper, we study the balanced truncation method for linear time-varying state-space models. We derive error bounds for the simplified models. These bounds are generalizations of well-known time-invariant results, derived with other methods. In the second paper, we apply balanced truncation to a high-order model of a diesel exhaust catalyst. Furthermore, we discuss practical issues of balanced truncation and approximative discretization. In the third paper, we look at frequency-domain analysis of linear time-periodic impulse-response models. By decomposing the models into Taylor and Fourier series, we can analyze convergence properties of different truncated representations. In the fourth paper, we use the frequency-domain representation developed in the third paper, the harmonic transfer function, to generalize Bode's sensitivity integral. This result quantifies limitations for feedback control of linear time-periodic systems. / QC 20120206

<p>This dissertation extends the theory of persistent homology to time varying systems. Most of the previous work has been dedicated to using this powerful tool in topological data analysis to study static point clouds. In particular, given a point cloud, we can construct its persistence diagram. Since the diagram varies continuously as the point cloud varies continuously, we study the space of time varying persistence diagrams, called vineyards when they were introduced by Cohen-Steiner, Edelsbrunner, and Morozov.</p><p>We will first show that with a good choice of metric, these vineyards are stable for small perturbations of their associated point clouds. We will also define a new mean for a set of persistence diagrams based on the work of Mileyko et al. which, unlike the previously defined mean, is continuous for geodesic vineyards. </p><p>Next, we study the sensor network problem posed by Ghrist and de Silva, and their application of persistent homology to understand when a set of sensors covers a given region. Giving each of these sensors a probability of failure over time, we show that an exact computation of the probability of failure of the whole system is NP-hard, but give an algorithm which can predict failure in the case of a monitored system.</p><p>Finally, we apply these methods to an automated system which can cluster agents moving in aerial images by their behaviors. We build a data structure for storing and querying the information in real-time, and define behavior vectors which quantify behaviors of interest. This clustering by behavior can be used to find groups of interest, for which we can also quantify behaviors in order to determine whether the group is working together to achieve a common goal, and we speculate that this work can be extended to improving tracking algorithms as well as behavioral predictors.</p> / Dissertation

Novel methods of system identification are developed in this dissertation. First set of methods are designed to realize time varying linear dynamical system models from input-output experimental data. The preliminary results obtained in a recent paper by the author are extended to establish a new algorithm called the Time Varying Eigensystem Realization Algorithm (TVERA). The central aim of this algorithm is to obtain a linear, time varying, discrete time model sequence of the dynamic system directly from the input-output data. Important results relating to concepts concerning coordinate systems for linear time varying systems are developed (discrete time theory) and an intuitive understanding of equivalent realizations is provided. A procedure to develop first few time step models is detailed, providing a unified solution to the time varying identification problem. The practical problem of identifying the time varying generalized Markov parameters required for TVERA is presented as the next result. In the process, we generalize the classical time invariant input output AutoRegressive model with an eXogenous input (ARX) models to the time varying case and realize an asymptotically stable observer as a byproduct of the calculations. It is further found that the choice of the generalized time varying ARX model (GTV-ARX) can be set to realize a time varying dead beat observer. Methods to use the developed algorithm(s) in this research are then considered for application to the identification of system models that are bilinear in nature. The fact that bilinear plant models become linear for constant inputs is used in the development of an algorithm that generalizes the classical developments of Juang. An intercept problem is considered as a candidate for application of the time varying identification scheme, where departure motion dynamics model sequence is calculated about a nominal trajectory with suboptimal performance owing to the presence of unstructured perturbations. Control application is subsequently demonstrated. The dynamics of a particle in a rotating tube is considered next for identification using the time varying eigensystem realization algorithm. Continuous time bilinear system identification method is demonstrated using the particle example and the identification of an automobile brake model.

In this dissertation, neural networks are designed to effectively model static non-linear aeroelastic problems in adaptive structures and linear dynamic aeroelastic systems with time varying stiffness. The use of adaptive materials in aircraft wings allows for the change of the contour or the configuration of a wing (morphing) in flight. The use of smart materials, to accomplish these deformations, can imply that the stiffness of the wing with a morphing contour changes as the contour changes. For a rapidly oscillating body in a fluid field, continuously adapting structural parameters may render the wing to behave as a time variant system. Even the internal spars/ribs of the aircraft wing which define the wing stiffness can be made adaptive, that is, their stiffness can be made to vary with time. The immediate effect on the structural dynamics of the wing, is that, the wing motion is governed by a differential equation with time varying coefficients. The study of this concept of a time varying torsional stiffness, made possible by the use of active materials and adaptive spars, in the dynamic aeroelastic behavior of an adaptable airfoil is performed here. A time marching technique is developed for solving linear structural dynamic problems with time-varying parameters. This time-marching technique borrows from the concept of Time-Finite Elements in the sense that for each time interval considered in the time-marching, an analytical solution is obtained. The analytical solution for each time interval is in the form of a matrix exponential and hence this technique is termed as Matrix Exponential time marching. Using this time marching technique, Artificial Neural Networks can be trained to represent the dynamic behavior of any linearly time varying system. In order to extend this methodology to dynamic aeroelasticity, it is also necessary to model the unsteady aerodynamic loads over an airfoil. Accordingly, an unsteady aerodynamic panel method is developed using a distributed set of doublet panels over the surface of the airfoil and along its wake. When the aerodynamic loads predicted by this panel method are made available to the Matrix Exponential time marching scheme for every time interval, a dynamic aeroelastic solver for a time varying aeroelastic system is obtained. This solver is now used to train an array of neural networks to represent the response of this two dimensional aeroelastic system with a time varying torsional stiffness. These neural networks are developed into a control system for flutter suppression. Another type of aeroelastic problem of an adaptive structure that is investigated here is the shape control of an adaptive bump situated on the leading edge of an airfoil. Such a bump is useful in achieving flow separation control for lateral directional maneuverability of the aircraft. Since actuators are being used to create this bump on the wing surface, the energy required to do so needs to be minimized. The adverse pressure drag as a result of this bump needs to be controlled so that the loss in lift over the wing is made minimal. The design of such a "spoiler bump" on the surface of the airfoil is an optimization problem of maximizing pressure drag due to flow separation while minimizing the loss in lift and energy required to deform the bump. One neural network is trained using the CFD code FLUENT to represent the aerodynamic loading over the bump. A second neural network is trained for calculating the actuator loads, bump displacement and lift, drag forces over the airfoil using the finite element solver, ANSYS and the previously trained neural network. This non-linear aeroelastic model of the deforming bump on an airfoil surface using neural networks can serve as a fore-runner for other non-linear aeroelastic problems. This work enhances the traditional aeroelastic modeling by introducing time varying parameters in the differential equations of motion. It investigates the calculation of non-conservative aerodynamic loads on morphing contours and the resulting structural deformation for non-linear aeroelastic problems through the use of neural networks. Geometric modeling of morphing contours is also addressed. / Ph. D.

Although the finite element method is often applied to analyze the dynamics of structures, its application to large, complex structures can be time-consuming and errors in the modeling process may negatively affect the accuracy of analyses based on the model. System identification techniques attempt to circumvent these problems by using experimental response data to characterize or identify a system. However, identification of structures that are time-varying or nonlinear is problematic because the available methods generally require prior understanding about the equations of motion for the system. Nonlinear system identification techniques are generally only applicable to nonlinearities where the functional form of the nonlinearity is known and a general nonlinear system identification theory is not available as is the case with linear theory. Linear time-varying identification methods have been proposed for application to nonlinear systems, but methods for general time-varying systems where the form of the time variance is unknown have only been available for single-input single-output models. This dissertation presents several general linear time-varying methods for multiple-input multiple-output systems where the form of the time variance is entirely unknown. The methods use the proper orthogonal decomposition of measured response data combined with linear system theory to construct a model for predicting the response of an arbitrary linear or nonlinear system without any knowledge of the equations of motion. Separate methods are derived for predicting responses to initial displacements, initial velocities, and forcing functions. Some methods require only one data set but only promise accurate solutions for linear, time-invariant systems that are lightly damped and have a mass matrix proportional to the identity matrix. Other methods use multiple data sets and are valid for general time-varying systems. The proposed methods are applied to linear time-invariant, time-varying, and nonlinear systems via numerical examples and experiments and the factors affecting the accuracy of the methods are discussed. / Ph. D.