| The ability to image and analyze fluorescent molecules both in vitro and in vivo is of great interest in molecular biology. Tracking systems to enable such imaging continue to be developed based on a variety of approaches. Existing tracking techniques generally require complicated and expensive experimental setups or are limited in their capability. This dissertation describes a system for tracking multiple fluorescent particles in a standard confocal microscope with a piezoactuated nanopositioning stage. A position estimation algorithm, fluoroBancroft, is utilized to analytically estimate particle position from a collection of measurements taken at discrete locations around the particle. This estimate is then used in a linear quadratic Gaussian (LQG) controller to regulate the tracking error. The technique relies on a standard confocal setup, making it easier to implement than other tracking schemes.;The experimental results indicated that the system can track single and multiple particles successfully in both two and three dimensions. To verify tracking and characterize tracking performance in these experiments, a CCD camera was introduced into the physical setup and synchronized to capture an image at every measurement location. In two dimensions, the overall tracking error was approximated by the standard deviation of the position estimates derived from each of the images. We find the tracking error increases as the square root of the diffusion coefficient plus an additional error that comes from position estimation error, digital-to-analog or analog-to-digital error and controller parameters mismatch. While the CCD-based estimates of the 3-D position of the particle were not accurate enough to quantify tracking performance, they did provide independent confirmation of tracking.;Because the performance of the estimation depends strongly on the choice of measurement pattern, we also describe work on optimizing that pattern to minimize the variance of the estimate subject to an unbiasedness constraint The analysis takes ad- vantage of the fact that the natural logarithm of a Poisson random variable with large rate can be approximated as a random variable with a Gaussian distribution. A sufficient condition for an unbiased measurement constellation and the optimal radius of a given constellation geometry with six measurements are then derived. The results are illustrated through both numerical simulation and experiments.;In addition to optimizing the measurement pattern, this dissertation also describes new results on the time-optimal control of second-order systems and an application of that theory to increase the throughput of the tracking system. Two linear affine mappings are derived to transfer the system to the normal coordinates. Based on the switching curve for a holdable equilibrium target state constructed in the normal coordinates, the switching number and switching time of the bang-bang control are numerically calculated, and a feedback time-optimal control law is designed too. The switching curves for both non-equilibrium and non-holdable equilibrium target state are also discussed. |
