Smart structure control systems commonly use piezoceramic sensors or accelerometers as vibration measurement devices. These measurement devices often produce noisy and/or low-precision signals, which makes it difficult to measure small-amplitude vibrations. Laser interferometry devices pose as an alternative high-precision position measurement method, capable of nanometer-scale resolution. The aim of this research is to utilize a model-based design approach to develop and implement a real-time Linear Quadratic Gaussian (LQG) regulator for a piezoelectric actuated smart structure using a high-precision laser interferometry measurement system to suppress the excitation of vibratory modes. The analytical model of the smart structure is derived using the extended Hamilton Principle and Euler-Bernoulli beam theory, and the equations of motion for the system are constructed using the assumed-modes method. The analytical model is organized in state-space form, in which the effects of a low-pass filter and sampling of the digital control system are also accounted for. The analytical model is subsequently validated against a finite-element model in Abaqus, a lumped parameter model in Simscape Multibody, and experimental modal analysis using the physical system. A discrete-time proportional-derivative (PD) controller is designed in a heuristic fashion to serve as a baseline performance criterion for the LQG regulator. The Kalman Filter observer and Linear Quadratic Regulator (LQR) components of the LQG regulator are also derived from the state-space model. It is found that the behavior of the analytical model closely matches that of the physical system, and the performance of the LQG regulator exceeds that of the PD controller. The LQG regulator demonstrated quality estimation of the state variables of the system and further constitutes an exceptional closed-loop control system for active vibration control and disturbance rejection of the smart structure.

Der weltraumgestützte Gravitationswellendetektor LISA (Laser Interferometer Space Antenna) besteht aus drei identischen Satelliten an Bord derer sich jeweils zwei frei schwebende Testmassen befinden. Die Lage der einzelnen Testmassen in Bezug auf die zugehörige optische Bank muss mit einer Genauigkeit besser 1 pm/sqrt(Hz) in der Abstands- und besser 10 nrad/sqrt(Hz) in der Winkelmessung erfolgen. In der vorliegenden Arbeit wird ein kompaktes optisches Auslesesystem präsentiert, welches als Prototyp für diese Abstands- und Winkelmetrologie dient. Das dafür entwickelte polarisierende Heterodyn-Interferometer mit räumlich getrennten Frequenzen basiert auf einem hoch-symmetrischen Design, bei dem zur optimalen Gleichtakt-Unterdrückung Mess- und Referenzarm die gleiche Polarisation und Frequenz sowie annähernd gleiche optische Pfade haben. Für die Winkelmessung wird die Methode der differentiellen Wellenfrontmessung eingesetzt. In einem ersten Prototyp-Aufbau wird ein Rauschniveau von weniger als 100 pm/sqrt(Hz) in der Translations- und von weniger als 100 nrad/sqrt(Hz) in der Winkelmessung (beides für Frequenzen oberhalb 0.1 Hz) demonstriert. In einem zweiten Prototyp-Aufbau werden zusätzlich eine Intensitätsstabilisierung und ein Phasenlock der beiden Frequenzen implementiert. Die analoge Phasenmessung ist durch eine digitale, FPGA basierte, ersetzt. Mit diesem Aufbau wird ein Rauschen kleiner 5 pm/sqrt(Hz) in der Translationsmessung und kleiner 10 nrad/sqrt(Hz) in der Winkelmessung, beides für Frequenzen größer 0.01 Hz, erreicht. Eine Rausch-Analyse wurde durchgeführt und die Nichtlinearitäten des Interferometers bestimmt. Das Interferometer wurde im Hinblick auf die LISA Mission entwickelt, findet seine Anwendung aber auch bei der Charakterisierung der dimensionalen Stabilität von ultra-stabilen Materialien sowie in der optischen Profilometrie. Die Adaptierung des Interferometers dazu sowie erste Resultate zu beiden Anwendungen werden in dieser Arbeit präsentiert. / The space-based gravitational wave detector LISA (Laser Interferometer Space Antenna) consists of three identical satellites. Each satellite accommodates two free-flying proof masses whose distance and tilt with respect to its corresponding optical bench must be measured with at least 1 pm/sqrt(Hz) sensitivity in translation and at least 10 nrad/sqrt(Hz) sensitivity in tilt measurement. In this thesis, a compact optical readout system is presented, which serves as a prototype for the LISA proof mass attitude metrology. We developed a polarizing heterodyne interferometer with spatially separated frequencies. For optimum common mode rejection, it is based on a highly symmetric design, where measurement and reference beam have the same frequency and polarization, and similar optical pathlengths. The method of differential wavefront sensing (DWS) is utilized for the tilt measurement. In a first prototype setup noise levels below 100 pm/sqrt(Hz) in translation and below 100 nrad/sqrt(Hz) in tilt measurement (both for frequencies above 0.1 Hz) are achieved. A second prototype was developed with additional intensity stabilization and phaselock of the two heterodyne frequencies. The analog phase measurement is replaced by a digital one, based on a Field Programmable Gate Array (FPGA). With this setup, noise levels below 5 pm/sqrt(Hz) in translation measurement and below 10 nrad/sqrt(Hz) in tilt measurement, both for frequencies above 0.01Hz, are demonstrated. A noise analysis was carried out and the nonlinearities of the interferometer were measured. The interferometer was developed for the LISA mission, but it also finds its application in characterizing the dimensional stability of ultra-stable materials such as carbon-fiber reinforced plastic (CFRP) and in optical profilometry. The adaptation of the interferometer and first results in both applications are presented in this work.

Ultraprecision single point diamond turning is required to produce parts with sub-nanometer surface roughness and sub-micrometer surface profiles tolerances. These parts have applications in the optics industry, where tight form accuracy is required while achieving high surface finish quality. Generally, parts can be polished to achieve the desired finish, but then the form accuracy can easily be lost in the process rendering the part unusable. Currently, most mid to low spatial frequency surface finish errors are inspected offline. This is done by physically removing the workpiece from the machining fixture and mounting the part in a laser interferometer. This action introduces errors in itself through minute differences in the support conditions of the over constrained part on a machine as compared to the mounting conditions used for part measurement. Once removed, the fixture induced stresses and the part’s internal residual stresses relax and change the shape of the generally thin parts machined in these applications. Thereby, the offline inspection provides an erroneous description of the performance of the machine. This research explores the use of a single, high resolution, capacitance sensor to quickly and qualitatively measure the low to mid spatial frequencies on the workpiece surface, while it is mounted in a fixture on a standard ultraprecision single point diamond turning machine after a standard facing operation. Following initial testing, a strong qualitative correlation exists between the surface profiling on a standard offline system and this online measuring system. Despite environmental effects and the effects of the machine on the measurement system, the capacitive system with some modifications and awareness of its measurement method is a viable option for measuring mid to low spatial frequencies on a workpiece surface mounted on an ultraprecision machine with a resolution of 1nm with an error band of ±5nm with a 20kHz bandwidth. / Thesis / Master of Applied Science (MASc)