Tomographic reconstruction using multiple laser guide stars (MLGS) will be required by the next generation of extremely large (30 m class) telescopes (ELT). Modal decomposition of wavefront phase using Zernike polynomials is a widely used technique in adaptive optics (AO) research. However, this approach breaks down with the large number of degrees of freedom required by ELTs.This research proposes the use of an alternative basis, the disk harmonic functions, to overcome the disadvantages of the Zernike basis at high spatial resolution. A method of fast, analytic, modal tomographic modeling is developed and used for fast calculation of reconstruction matrices used on-sky at the MMT telescope.The specific reconstruction techniques of ground layer adaptive optics and laser tomography adaptive optics using MLGS are presented along with the results of on-sky experiments at the MMT. In addition to developing a laser AO instrument for the MMT, these experiments provide a test bed for validating the reconstruction techniques that will be critical to the success of ELTs.An approach to using real-time wavefront sensor and deformable mirror telemetry from the MLGS system to estimate the vertical distribution of turbulence in the atmosphere is also presented.

We present the results of a survey of 45 young (<250>Myr), close (<50>pc) stars with the Simultaneous Differential Imager (SDI) implemented at the VLT and the MMT for the direct detection of extrasolar planets. Our SDI devices use a double Wollaston prism and a quad filter to take images simultaneously at three wavelengths surrounding the 1.62 um methane absorption bandhead found in the spectrum of cool brown dwarfs and extrasolar giant planets. By performing a difference of adaptive optics corrected images in these filters, speckle noise from the primary star can be significantly attenuated, resulting in photon (and flat-field) noise limited data. In our VLT data, we achieved H band contrasts > 10 mag (5 sigma) at a separation of 0.5" from the primary star on 45% of our targets and H band contrasts of > 9 mag at a separation of 0.5'' on 80% of our targets. With this degree of attenuation, we should be able to image (5 sigma detection) a 7 MJup planet 15 AU from a 70 Myr K1 star at 15 pc or a 7.8 MJup planet at 2 AU from a 12 Myr M star at 10 pc. Using the capabilities of the unique SDI device, we also discovered a methane-rich substellar companion to SCR 1845-6357 (a recently discovered (Hambly et al., 2004) M8.5 star just 3.85 pc from the Sun (Henry et al., 2006) at a separation of 4.5 AU (1.170''+-0.003'' on the sky) and fainter by 3.57$\pm$0.057 mag in the 1.575 um SDI filter.We also present high resolution (~0.1''), very high Strehl ratio (0.97+-0.03) mid-infrared (IR) adaptive optics (AO) images of the AGB star RV Boo utilizing the MMT adaptive secondary AO system. RV Boo was observed at a number of wavelengths over two epochs and appeared slightly extended at all wavelengths. With such high Strehls we can achieve super-resolutions of 0.1'' by deconvolving RV Boo with a point-spread function (PSF) derived from an unresolved star.SDI on ground based telescopes provides significant speckle attenuations down to star-planet contrasts of ~1-3x10^4. To test the classical SDI technique at contrasts of 10^6-9, we implemented a similar multiwavelength differential imaging scheme for the JPL High Contrast Imaging Testbed.

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Markov chain Monte Carlo algorithms (MCMC) and Adaptive Markov chain Monte Carlo algorithms (AMCMC) are most important methods of approximately sampling from complicated probability distributions and are widely used in statistics, computer science, chemistry, physics, etc. The core problem to use these algorithms is to build up asymptotic theories for them. In this thesis, we show the Central Limit Theorem (CLT) for the uniformly ergodic Markov chain using the regeneration method. We exploit the weakest uniform drift conditions to ensure the ergodicity and WLLN of AMCMC. Further we answer the open problem 21 in Roberts and Rosenthal [48] through constructing a counter example and finding out some stronger condition which indicates the ergodic property of AMCMC. We find that the conditions (a) and (b) in [46] are not sufficient for WLLN holds when the functional is unbounded. We also prove the WLLN for unbounded functions with some stronger conditions. Finally we consider the practical aspects of adaptive MCMC (AMCMC). We try some toy examples to explain that the general adaptive random walk Metropolis is not efficient for sampling from multi-model targets. Therefore we discuss the mixed regional adaptation (MRAPT) on the compact state space and the modified mixed regional adaptation on the general state space in which the regional proposal distributions are optimal and the switches between different models are very efficient. The theoretical proof is to show that the algorithms proposed here fall within the scope of general theorems that are used to validate AMCMC. As an application of our theoretical results, we analyze the real data about the ``Loss of Heterozygosity" (LOH) using MRAPT.

Output feedback control design techniques are required in practice due to the limited number of sensors/measurements available for feedback. This thesis focuses on output feedback controller design techniques for nonlinear systems subject to different system restrictions. The problem of controlling the heart dynamics in a real time manner is formulated as an adaptive learning output-tracking problem. For a class of nonlinear dynamic systems with unknown nonlinearities and non-affine control input , a Lyapunov-based technique is used to develop a control law. An adaptive learning algorithm is exploited that guarantees the stability of the closed-loop system and convergence of the output tracking error to an adjustable neighborhood of the origin. In addition, good approximation of the unknown nonlinearities is also achieved by incorporating a per- sistent exciting signal in the parameter update law. The effectiveness of the proposed method is demonstrated by an application to a cardiac conduction system modelled by two coupled driven oscillators. An output feedback design technique is developed to achieve semi-global practical stabilization for a class of non-minimum phase nonlinear systems, subject to param- eter uncertainties. This work provides a constructive controller design method for an auxiliary system, whose existence is crucial, but is only assumed in (Isidori, 2000). The control design technique is used to regulate the benchmark van de Vusse reactor. Simulation results demonstrate satisfactory controller performance. The output feedback control design for a class of non-minimum phase nonlinear systems with unknown nonlinearities is studied. The proposed approach is able to combine the two previous design methods and provide a stabilizing output feedback control law. The performance of the proposed method is demonstrated by simulation results. / Thesis (Ph.D, Chemical Engineering) -- Queen's University, 2009-12-22 15:47:05.884

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