No description available.

The development of navigation filters that make use of robust estimation techniques is important due to the sensitivity of the typical minimum L2 norm techniques, such as the Kalman filter, to deviations in the assumed underlying probability distribution. In particular, those distributions with thicker tails than the Gaussian distribution can give rise to erratic filter performance and inconsistency of results. This dissertation discusses the development of an adaptive discrete-time robust nonlinear filtering technique based on a recursive form of Huber's mixed minimum L1/L2 norm approach to estimation, which is robust with respect to deviations from the assumed Gaussian error probability distributions inherent to the Kalman filter. This mixed norm approach is applied to a type of Sigma-Point Kalman filter, known as the Divided Difference Filter, which can capture second-order effects of nonlinearities in the system and measurement dynamics. Additionally, if these assumed parameters of the distribution differ greatly from the true parameters, then the filter can exhibit large errors and possibly divergence in nonlinear problems. This behavior is possible even if the true error distributions are Gaussian. To remedy these problems, adaptive filtering techniques have been introduced in order to automatically tune the Kalman filter by estimating the measurement and process noise covariances, however these techniques can also be highly sensitive to the nature of the underlying error distributions. The Huber-based formulations of the filtering problem also make some assumptions regarding the distribution, namely the approach considers a class of contaminated densities in the neighborhood of the Gaussian density. Essentially the method assumes that the statistics of the main Gaussian density are known, as well as the ratio or percentage of the contamination. The technique can be improved upon by the introduction of a method to adaptively estimate the noise statistics along with the state and state error covariance matrix. One technique in common use for adaptively estimating the noise statistics in real-time filtering applications is known as covariance matching. The covariance matching technique is an intuitively appealing approach in which the measurement noise and process noise covariances are determined in such a way that the true residual covariance matches the theoretically predicted covariance. The true residual covariance is approximated in real time using the sample covariance, over some finite buffer of stored residuals. The drawback to this approach is that the presence of outliers and non-Gaussianity can create problems of robustness with the use of the covariance matching technique. Therefore some additional steps must be taken to identify the outliers before forming the covariance estimates. In this dissertation, an adaptive scheme is proposed whereby the filter can estimate the process noise and measurement noise covariance matrices along with the state estimate and state estimate error covariance matrix. The adaptation technique adopts a robust approach to estimating these covariances that can resist the effects of outliers. The particular outlier identification method employed in this paper is based on quantities known as projection statistics, which utilize the sample median and median absolute deviation, and as a result are highly effective technique for multivariate outlier identification. These projection statistics are then employed as weights in the covariance matching procedure in order to reduce the influence of the outliers. The hybrid robust/adaptive nonlinear filtering methods introduced in this dissertation are applied to the problem of 6-DOF rendezvous navigation in elliptical orbit. The full nonlinear equations of relative motion are formulated in spherical coordinates centered on the target orbit. A relatively simple control law based on feedback linearization is used to track a desired rendezvous trajectory. The attitude dynamics are parameterized using Modified Rodrigues Parameters, which are advantageous for both control law development and estimation since they constitute a minimal 3-parameter attitude description. A switching technique which exploits the stereographic projection properties of the MRP coordinate is utilized to avoid singularities which inevitably arise in minimal attitude descriptions. This dissertation also introduces the proper covariance transformations associated with the singularity avoidance switching technique. An attitude control law based on backstepping is employed to track the target vehicle. A sensor suite consisting of a generic lidar or optical sensor, an Inertial Measurement Unit, consisting of accelerometers and gyroscopes, a star tracker, and a horizon sensor are utilized to provide measurement data to the navigation filters so that the chaser vehicle can estimate its relative state during the rendezvous maneuver. Several filters are implemented for comparison, including the Extended Kalman Filter, First and Second-Order Divided Difference Filters and Huber-based generalizations of these filters that include adaptive techniques for estimating the noise covariances. Monte-Carlo simulations are presented which include both Gaussian and non-Gaussian errors, including mismatches in the assumed noise covariances in the navigation filters in order to illustrate the benefits of the robust/adaptive nonlinear filters. Additionally, computational burdens of the various filters is compared. / Ph. D.

For over a century, physicians have prescribed x-ray radiation to destroy or impede the growth of cancerous tumours. Modern radiation therapy machines shape the radiation beam to balance the competing goals of maximizing irradiation of cancerous tissue and minimizing irradiation of healthy tissue, an objective complicated by tumour motion during the treatment and errors positioning the patient to align the tumour with the radiation beam. Recent medical imaging advances have motivated interest in using feedback during radiation therapy to track the tumour in real time and mitigate these complications. This thesis investigates how real-time feedback control can be used to track the tumour and focus the radiation beam tightly around the tumour. Improving on these results, a feedback control system is proposed for intensity modulated radiation therapy which allows a non-uniform radiation dose to be applied to the tumour. Motivated by the results of the proposed control systems, this thesis also examines two theoretical control problems: estimating the output of an unknown system when a sensor delay prevents its direct measurement, and designing a controller to provide an arbitrarily fast nonovershooting step response.

A bulk of the research on piezoelectric systems in recent years can be classified into two categories, 1) studies of linear piezoelectric oscillator arrays, 2) studies of nonlinear piezoelectric oscillators. This dissertation derives novel linear and nonlinear modeling and estimation methods for such piezoelectric systems. In the first part, this work develops modeling and design methods for Piezoelectric Subordinate Oscillator Arrays (PSOAs) for the wideband vibration attenuation problem. PSOAs offer a straightforward and low mass ratio solution to cancel out the resonant peaks in a host structure's frequency domain. Further, they provide adaptability through shunt tuning, which gives them the ability to recover performance losses because of structural parameter errors. This dissertation studies the derivation of governing equations that result in a closed-form expression for the frequency response function. It also analyzes systematic approaches to assign distributions to the nondimensional parameters in the frequency response function to achieve the desired flat-band frequency response. Finally, the effectiveness of PSOAs under ideal and nonideal conditions are demonstrated in this dissertation through extensive numerical and experimental studies. The concept of performance recovery, introduced in empirical studies, gives a measure of the PSOA's effectiveness in the presence of disorder before and after capacitive tuning. The second part of this dissertation introduces novel modeling and estimation methods for nonlinear piezoelectric oscillators. Traditional modeling techniques require knowledge of the structure as well as the source of nonlinearity. Data-driven modeling techniques used extensively in recent times build approximations. An adaptive estimation method, that uses reproducing kernel Hilbert space (RKHS) embedding methods, can estimate the underlying nonlinear function that governs the system's dynamics. A model built by such a method can overcome some of the limitations of the modeling approaches mentioned above. This dissertation discusses (i) how to construct the RKHS based estimator for the piezoelectric oscillator problem, (ii) how to choose kernel centers for approximating the RKHS, and (iii) derives sufficient conditions for convergence of the function estimate to the actual function. In each of these discussions, numerical studies are used to show the RKHS based adaptive estimator's effectiveness for identifying linearities in piezoelectric oscillators. / Doctor of Philosophy / Piezoelectric materials are materials that generate an electric charge when mechanical stress is applied, and vice versa, in a lossless transformation. Engineers have used piezoelectric materials for a variety of applications, including vibration control and energy harvesting. This dissertation introduces (1) novel methods for vibration attenuation using an array of piezoelectric oscillators, and (2) methods to model and estimate the nonlinear behavior exhibited by piezoelectric materials at very high mechanical forces or electric charges. Arrays of piezoelectric oscillators attached to a host structure are termed piezoelectric subordinate oscillator arrays (PSOAs). With the careful design of PSOAs, we show that we can reduce the vibration of the host structure. This dissertation analyzes methodologies for designing PSOAs and illustrates their vibration attenuation capabilities numerically and experimentally. The numerical and experimental studies also illustrate the robustness of PSOAs. In the second part of this dissertation, we analyze reproducing kernel Hilbert space embedding methods for adaptive estimation of nonlinearities in piezoelectric systems. Kernel methods are extensively used in machine learning, and control theorists have studied adaptive estimation of functions in finite-dimensional spaces. In this work, we adapt kernel methods for adaptive estimation of functions in infinite-dimensional spaces that appear while modeling piezoelectric systems. We derive theorems that ensure convergence of function estimates to the actual function and develop algorithms for careful selection of the kernel basis functions.

This work investigates modeling, approximation, estimation, and control for classes of nonlinear systems whose state evolves in space $mathbb{R}^n times H$, where $mathbb{R}^n$ is a n-dimensional Euclidean space and $H$ is a infinite dimensional Hilbert space. Specifically, two classes of nonlinear systems are studied in this dissertation. The first topic develops a novel framework for adaptive estimation of nonlinear systems using reproducing kernel Hilbert spaces. A nonlinear adaptive estimation problem is cast as a time-varying estimation problem in $mathbb{R}^d times H$. In contrast to most conventional strategies for ODEs, the approach here embeds the estimate of the unknown nonlinear function appearing in the plant in a reproducing kernel Hilbert space (RKHS), $H$. Furthermore, the well-posedness of the framework in the new formulation is established. We derive the sufficient conditions for existence, uniqueness, and stability of an infinite dimensional adaptive estimation problem. A condition for persistence of excitation in a RKHS in terms of an evaluation functional is introduced to establish the convergence of finite dimensional approximations of the unknown function in RKHS. Lastly, a numerical validation of this framework is presented, which could have potential applications in terrain mapping algorithms. The second topic delves into estimation and control of history dependent differential equations. This study is motivated by the increasing interest in estimation and control techniques for robotic systems whose governing equations include history dependent nonlinearities. The governing dynamics are modeled using a specific form of functional differential equations. The class of history dependent differential equations in this work is constructed using integral operators that depend on distributed parameters. Consequently, the resulting estimation and control equations define a distributed parameter system whose state, and distributed parameters evolve in finite and infinite dimensional spaces, respectively. The well-posedness of the governing equations is established by deriving sufficient conditions for existence, uniqueness and stability for the class of functional differential equations. The error estimates for multiwavelet approximation of such history dependent operators are derived. These estimates help determine the rate of convergence of finite dimensional approximations of the online estimation equations to the infinite dimensional solution of distributed parameter system. At last, we present the adaptive sliding mode control strategy developed for the history dependent functional differential equations and numerically validate the results on a simplified pitch-plunge wing model. / Ph. D.

Design of nonlinear observers has received considerable attention since the early development of methods for linear state estimation. The most popular approach is the extended Kalman filter (EKF), that goes through significant degradation in the presence of nonlinearities, particularly if unmodeled dynamics are coupled to the process and the measurement. For uncertain nonlinear systems, adaptive observers have been introduced to estimate the unknown state variables where no priori information about the unknown parameters is available. While establishing global results, these approaches are applicable only to systems transformable to output feedback form. Over the recent years, neural network (NN) based identification and estimation schemes have been proposed that relax the assumptions on the system at the price of sacrificing on the global nature of the results. However, most of the NN based adaptive observer approaches in the literature require knowledge of the full dimension of the system, therefore may not be suitable for systems with unmodeled dynamics. We first propose a novel approach to nonlinear state estimation from the perspective of augmenting a linear time invariant observer with an adaptive element. The class of nonlinear systems treated here are finite but of otherwise unknown dimension. The objective is to improve the performance of the linear observer when applied to a nonlinear system. The approach relies on the ability of the NNs to approximate the unknown dynamics from finite time histories of available measurements. Next we investigate nonlinear state estimation from the perspective of adaptively augmenting an existing time varying observer, such as an EKF. EKFs find their applications mostly in target tracking problems. The proposed approaches are robust to unmodeled dynamics, including unmodeled disturbances. Lastly, we consider the problem of adaptive estimation in the presence of feedback control for a class of uncertain nonlinear systems with unmodeled dynamics and disturbances coupled to the process. The states from the adaptive EKF are used as inputs to the control law, which in target tracking usually takes the form of a guidance law. The applications of this approach lie in the areas of missile-target tracking, formation flight control and obstacle avoidance.

For over a century, physicians have prescribed x-ray radiation to destroy or impede the growth of cancerous tumours. Modern radiation therapy machines shape the radiation beam to balance the competing goals of maximizing irradiation of cancerous tissue and minimizing irradiation of healthy tissue, an objective complicated by tumour motion during the treatment and errors positioning the patient to align the tumour with the radiation beam. Recent medical imaging advances have motivated interest in using feedback during radiation therapy to track the tumour in real time and mitigate these complications. This thesis investigates how real-time feedback control can be used to track the tumour and focus the radiation beam tightly around the tumour. Improving on these results, a feedback control system is proposed for intensity modulated radiation therapy which allows a non-uniform radiation dose to be applied to the tumour. Motivated by the results of the proposed control systems, this thesis also examines two theoretical control problems: estimating the output of an unknown system when a sensor delay prevents its direct measurement, and designing a controller to provide an arbitrarily fast nonovershooting step response.

As data-driven modeling becomes more prevalent for representing the uncertain dynamical systems, concerns also arise regarding the reliability of these methods. Recent developments in approximation theory provide a new perspective to studying these problems. This dissertation analyzes the convergence of two kernel-based, data-driven modeling methods, the reproducing kernel Hilbert space (RKHS) embedding method and the empirical-analytical Lagrangian (EAL) model. RKHS embedding is a non-parametric extension of the classical adaptive estimation method that embeds the uncertain function in an RKHS, an infinite-dimensional function space. As a result the original uncertain system of ordinary differential equations are understood as components of a distributed parameter system. Similarly to the classical approach for adaptive estimation, a novel definition of persistent excitation (PE) is introduced, which is proven to guarantee the pointwise convergence of the estimate of function over the PE domain. The finite-dimensional approximation of the RKHS embedding method is based on approximant spaces that consist of kernel basis functions centered at samples in the state space. This dissertation shows that explicit rate of convergence of the RKHS embedding method can be derived by choosing specific types of native spaces. In particular, when the RKHS is continuously embedded in a Sobolev space, the approximation error is proven to decrease at a rate determined by the fill distance of the samples in the PE domain. This dissertation initially studies scalar-valued RKHS, and subsequently the RKHS embedding method is extended for the estimation of vector-valued uncertain functions. Like the scalar-valued case, the formulation of vector-valued RKHS embedding is proven to be well-posed. The notion of partial PE is also generalized, and it is shown that the rate of convergence derived for the scalar-valued approximation still holds true for certain separable operator-valued kernels. The second part of this dissertation studies the EAL modeling method, which is a hybrid mechanical model for Lagrangian systems with uncertain holonomic constraints. For the singular perturbed form of the system, the kernel method is applied to approximate a penalty potential that is introduced to approximately enforce constraints. In this dissertation, the accuracy confidence function is introduced to characterize the constraint violation of an approximate trajectory. We prove that the confidence function can be decomposed into a term representing the bias and another term representing the variation. Numerical simulations are conducted to examine the factors that affect the error, including the spectral filtering, the number of samples, and the accumulation of integration error. / Doctor of Philosophy / As data-driven modeling is becoming more prevalent for representing uncertain dynamical systems, concerns also arise regarding the reliability of these methods. This dissertation employs recent developments in approximation theory to provide rigorous error analysis for two certain kernel-based approaches for modeling dynamical systems. The reproducing kernel Hilbert space (RKHS) embedding method is a non-parametric extension of the classical adaptive estimation for identifying uncertain functions in nonlinear systems. By embedding the uncertain function in a properly selected RKHS, the nonlinear state equation in Euclidean space is transformed into a linear evolution in an infinite-dimensional RKHS, where the function estimation error can be characterized directly and precisely. Pointwise convergence of the function estimate is proven over the domain that is persistently excited (PE). And a finite-dimensional approximation can be constructed within an arbitrarily small error bound. The empirical-analytical Lagrangian (EAL) model is developed to approximate the trajectory of Lagrangian systems with uncertain configuration manifold. Employing the kernel method, a penalty potential is constructed from the observation data to ``push'' the trajectory towards the actual configuration manifold. A probabilistic error bound is derived for the distance of the approximated trajectory away from the actual manifold. The error bound is proven to contain a bias term and a variance term, both of which are determined by the parameters of the kernel method.

No description available.

As resident space object populations grow, and satellite propulsion capabilities improve, it will become increasingly challenging for space-reliant nations to maintain space situational awareness using current human-in-the-loop methods. This dissertation develops several real-time adaptive approaches to autonomous sensor network management for tracking multiple maneuvering and non-maneuvering satellites with a diversely populated Space Object Surveillance and Identification network. The proposed methods integrate suboptimal Partially Observed Markov Decision Processes (POMDPs) with covariance inflation or multiple model adaptive estimation techniques to task sensors and maintain viable orbit estimates for all targets. The POMDPs developed in this dissertation use information-based and system-based metrics to determine the rewards and costs associated with tasking a specific sensor to track a particular satellite. Like in real-world situations, the population of target satellites vastly outnumbers the available set of sensors. Robust and adaptable tasking algorithms are needed in this scenario to determine how and when sensors should be tasked. The strategies developed in this dissertation successfully track 207 non-maneuvering and maneuvering spacecraft using only 24 ground and space-based sensors. The results show that multiple model adaptive estimation coupled with a multi-metric, suboptimal POMDP can effectively and efficiently task a diverse network of sensors to track multiple maneuvering spacecraft, while simultaneously monitoring a large number of non-maneuvering objects. Overall, this dissertation demonstrates the potential for autonomous and adaptable sensor network command and control for real-world space situational awareness. / Ph. D. / As the number of spacecraft in orbit increase, and satellite propulsion capabilities improve, it will become increasingly difficult for space-reliant nations to keep track of every object orbiting earth using human-in-the-loop methods. Already, the population of target satellites vastly outnumbers the available set of sensors. At any given time, a given network of sensors cannot observe every satellite in orbit, and must manage the available sensors effectively to keep track of every object of interest. The ability to maintain actionable knowledge of every orbiting object of interest is known as space situational awareness. Conventional tracking processes have generally not changed for decades, and were designed when there were far fewer satellites in orbit with little or no ability to maneuver. These methods involve large numbers of operators and engineers who schedule a network of sensors under the assumption that the satellites will not unexpectedly change their orbits for long periods of time. In the near future, traditional space surveillance approaches will become insufficient at maintaining space situational awareness, particularly if more satellites conduct unanticipated maneuvers. This dissertation develops several real-time approaches for controlling a diverse network of ground and space-based sensors that remove the need for human intervention. These fully computer-based command and control processes adapt to dynamic situations and automatically task sensors to rapidly track multiple maneuvering and non-maneuvering satellites. The decision processes used to determine which sensors should be tasked to observe a particular spacecraft compare the amount of information that can be collected in a single observation and the workload a sensor must execute to collect the observation. The command and control strategies developed in this dissertation successfully track 207 non-maneuvering and maneuvering spacecraft using only 24 ground and space-based sensors. The results show that adaptive, fully autonomous sensor network control processes can effectively and efficiently task a diverse set of sensors to track multiple maneuvering spacecraft, while simultaneously monitoring a large number of non-maneuvering objects. Overall, this dissertation demonstrates the potential for adaptive, computer-based sensor network command and control for real-world space situational awareness. This research was supported by the Virginia Tech New Horizons Graduate Scholar Program, the Ted and Karyn Hume Center for National Security and Technology, the DARPA Hallmark program, and the U.S. Joint Warfare Analysis Center.