An Optimal Family of Exponentially Accurate One-Bit Sigma-Delta Quantization Schemes

Sigma-Delta modulation is a popular method for analog-to-digital conversion of bandlimited signals that employs coarse quantization coupled with oversampling. The standard mathematical model for the error analysis of the method measures the performance of a given scheme by the rate at which the associated reconstruction error decays as a function of the oversampling ratio $\lambda$. It was recently shown that exponential accuracy of the form $O(2^{-r\lambda})$ can be achieved by appropriate one-bit Sigma-Delta modulation schemes. By general information-entropy arguments $r$ must be less than 1. The current best known value for $r$ is approximately 0.088. The schemes that were designed to achieve this accuracy employ the "greedy" quantization rule coupled with feedback filters that fall into a class we call "minimally supported". In this paper, we study the minimization problem that corresponds to optimizing the error decay rate for this class of feedback filters. We solve a relaxed version of this problem exactly and provide explicit asymptotics of the solutions. From these relaxed solutions, we find asymptotically optimal solutions of the original problem, which improve the best known exponential error decay rate to $r \approx 0.102$. Our method draws from the theory of orthogonal polynomials; in particular, it relates the optimal filters to the zero sets of Chebyshev polynomials of the second kind.Comment: 35 pages, 3 figure

Control by model error estimation

Modern control theory relies upon the fidelity of the mathematical model of the system. Truncated modes, external disturbances, and parameter errors in linear system models are corrected by augmenting to the original system of equations an 'error system' which is designed to approximate the effects of such model errors. A Chebyshev error system is developed for application to the Large Space Telescope (LST)

A Novel Experimental Approach Using A Reconfigurable Test Setup For Complex Nonlinear Dynamic Systems

Experimental nonlinear dynamics is an important area of study in the modern engineering field, with engineering applications in structural dynamics, structural control, and structural health monitoring. As a result, the discipline has experienced a great influx of research efforts to develop a versatile and reliable experimental methodology. A technical challenge in many experimental studies is the procurement of a device that exhibits the desired nonlinear behavior. As a result, many researchers have longed for a versatile, but accurate, testing methodology that has complete freedom to simulate a wide range of nonlinearities and stochastic behaviors. The objective of this study is to develop a reconfigurable test setup as a tool to be used in a wide range of nonlinear dynamic studies. The main components include a moving mass whose restoring force can accurately be controlled and reprogrammed (with software) based upon measured displacement and velocity readings at each time step. The device offers control over nonlinear characteristics and the equation of dynamic motion. The advantage of having such an experimental setup is the ability to simulate various types of nonlinearities with the same test setup. As a result, the data collected can be used to help validate nonlinear modeling, system identification, and stochastic analysis studies. A physical test apparatus was developed, and various mechanical, electrical, and programming calibrations were performed for reliable experimental studies. To display potential uses for the reconfigurable approach, examples are presented where the device has been used to create physical data for use in change detection and deterioration studies. In addition, a demonstration is presented of the device’s ability to physically simulate a large-scale orifice viscous damper, commonly used in vibration mitigation in bridges and buildings. For a large-scale viscous damper, physical testing is required to ensure structural design properties. However, due to the large scale of the dampers, expensive dynamic loading tests can be carried out at a very iii limited number of facilities. Using the reconfigurable test setup, the dynamic signature of the large-scale viscous damper can accurately be simulated with pre-collected data. The development of a system capable of emulating the restoring force of a nonlinear device with software is a novel approach and requires further calibration for increased reliability and accuracy. A discussion regarding the challenges faced when developing the methodology is presented and possible solutions are recommended. The methodology introduced by this apparatus is very promising. The device is a valuable experimental tool for researchers and designers, allowing for physical data collection, modeling, analysis, and validation of a wide class of nonlinear phenomena that commonly occur in a wide variety of engineering applications