In this dissertation, we focus on solving Nonlinear Least Squares problems using a supervised approach. In particular, we developed a Supervised Descent Method (SDM), performed thorough theoretical analysis, and demonstrated its effectiveness on optimizing analytic functions, and four other real-world applications: Inverse Kinematics, Rigid Tracking, Face Alignment (frontal and multi-view), and 3D Object Pose Estimation. In Rigid Tracking, SDM was able to take advantage of more robust features, such as, HoG and SIFT. Those non-differentiable image features were out of consideration of previous work because they relied on gradient-based methods for optimization. In Inverse Kinematics where we minimize a non-convex function, SDM achieved significantly better convergence than gradient-based approaches. In Face Alignment, SDM achieved state-of-the-arts results. Moreover, it was extremely computationally efficient, which makes it applicable for many mobile applications. In addition, we provided a unified view of several popular methods including SDM on sequential prediction, and reformulated them as a sequence of function compositions. Finally, we suggested some future research directions on SDM and sequential prediction.

<p>PolyTech AB in Malmköping manufactures gyro stabilized camera systems or helicopter applications. In this Master´s Thesis a shaker test rig for vibration testing of these systems is designed, implemented and evaluated. The shaker is required to have an adjustable frequency and displacement and different shakers that meet these requirements are treated in a literature study.</p><p>The shaker chosen in the test rig is based on a mechanical solution that is described in detail. Additionally all components used in the test rig are described and modelled. The test rig is identified and evaluated from different experiments carried out at PolyTech, where the major part of the identification is based on data collected from accelerometers.</p><p>The test rig model is used to develop a controller that controls the frequency and the displacement of the shaker. A three-phase motor is used to control the frequency of the shaker and a linear actuator with a servo is used to control the displacement. The servo controller is designed using observer and state feedback techniques.</p><p>Additionally, the mount in which the camera system is hanging is modelled and identified, where the identification method is based on nonlinear least squares (NLS) curve fitting technique.</p>

PolyTech AB in Malmköping manufactures gyro stabilized camera systems or helicopter applications. In this Master´s Thesis a shaker test rig for vibration testing of these systems is designed, implemented and evaluated. The shaker is required to have an adjustable frequency and displacement and different shakers that meet these requirements are treated in a literature study. The shaker chosen in the test rig is based on a mechanical solution that is described in detail. Additionally all components used in the test rig are described and modelled. The test rig is identified and evaluated from different experiments carried out at PolyTech, where the major part of the identification is based on data collected from accelerometers. The test rig model is used to develop a controller that controls the frequency and the displacement of the shaker. A three-phase motor is used to control the frequency of the shaker and a linear actuator with a servo is used to control the displacement. The servo controller is designed using observer and state feedback techniques. Additionally, the mount in which the camera system is hanging is modelled and identified, where the identification method is based on nonlinear least squares (NLS) curve fitting technique.

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Recent developments in smart grid technology have spawned interest in the use of phasor measurement units to help create a reliable power system transmission and distribution infrastructure. Wide-area monitoring systems (WAMSs) utilizing synchrophasor measurements can help with understanding, forecasting, or even controlling the status of power grid stability in real-time. A power system Frequency Monitoring Network (FNET) was first proposed in 2001 and was established in 2004. As a pioneering WAMS, it serves the entire North American power grid through advanced situational awareness techniques, such as real-time event alerts, accurate event location estimation, animated event visualization, and post event analysis. Traditionally, Phasor Measurement Units (PMUs) have utilized signals obtained from current transformers (CTs) to compute current phasors. Unfortunately, this requires that CTs must be directly connected with buses, transformers or power lines. Chapters 2, 3 will introduce an innovative phasor measurement instrument, the Non-contact Frequency Disturbance Recorder (NFDR), which uses the magnetic and electric fields generated by power transmission lines to obtain current phasor measurements. The NFDR is developed on the same hardware platform as the Frequency Disturbance Recorder (FDR), which is actually a single phase PMU. Prototype testing of the NFDR in both the laboratory and the field environments were performed. Testing results show that measurement accuracy of the NFDR satisfies the requirements for power system dynamics observation. Researchers have been developing various techniques in power system phasor measurement and frequency estimation, due to their importance in reflecting system health. Each method has its own pros and cons regarding accuracy and speed. The DFT (Discrete Fourier Transform) based algorithm that is adopted by the FDR device is particularly suitable for tracking system dynamic changes and is immune to harmonic distortions, but it has not proven to be very robust when the input signal is polluted by random noise. Chapter 4 will discuss the Least Mean Squares-based methods for power system frequency tracking, compared with a DFT-based algorithm. Wide-area monitoring systems based on real time PMU measurements can provide great visibility to the angle instability conditions. Chapter 5 focuses on developing an early warning algorithm on the FNET platform. / Ph. D.

<p>In computational science it is common to describe dynamic systems by mathematical models in forms of differential or integral equations. These models may contain parameters that have to be computed for the model to be complete. For the special type of ordinary differential equations studied in this thesis, the resulting parameter estimation problem is a separable nonlinear least squares problem with equality constraints. This problem can be solved by iteration, but due to complicated computations of derivatives and the existence of several local minima, so called short-cut methods may be an alternative. These methods are based on simplified versions of the original problem. An algorithm, called the modified Kaufman algorithm, is proposed and it takes the separability into account. Moreover, different kinds of discretizations and formulations of the optimization problem are discussed as well as the effect of ill-conditioning.</p><p>Computation of parameters often includes as a part solution of linear system of equations <i>Ax = b</i>. The corresponding pseudoinverse solution depends on the properties of the matrix <i>A</i> and vector <i>b</i>. The singular value decomposition of <i>A</i> can then be used to construct error propagation matrices and by use of these it is possible to investigate how changes in the input data affect the solution <i>x</i>. Theoretical error bounds based on condition numbers indicate the worst case but the use of experimental error analysis makes it possible to also have information about the effect of a more limited amount of perturbations and in that sense be more realistic. It is shown how the effect of perturbations can be analyzed by a semi-experimental analysis. The analysis combines the theory of the error propagation matrices with an experimental error analysis based on randomly generated perturbations that takes the structure of <i>A</i> into account</p>

In computational science it is common to describe dynamic systems by mathematical models in forms of differential or integral equations. These models may contain parameters that have to be computed for the model to be complete. For the special type of ordinary differential equations studied in this thesis, the resulting parameter estimation problem is a separable nonlinear least squares problem with equality constraints. This problem can be solved by iteration, but due to complicated computations of derivatives and the existence of several local minima, so called short-cut methods may be an alternative. These methods are based on simplified versions of the original problem. An algorithm, called the modified Kaufman algorithm, is proposed and it takes the separability into account. Moreover, different kinds of discretizations and formulations of the optimization problem are discussed as well as the effect of ill-conditioning. Computation of parameters often includes as a part solution of linear system of equations Ax = b. The corresponding pseudoinverse solution depends on the properties of the matrix A and vector b. The singular value decomposition of A can then be used to construct error propagation matrices and by use of these it is possible to investigate how changes in the input data affect the solution x. Theoretical error bounds based on condition numbers indicate the worst case but the use of experimental error analysis makes it possible to also have information about the effect of a more limited amount of perturbations and in that sense be more realistic. It is shown how the effect of perturbations can be analyzed by a semi-experimental analysis. The analysis combines the theory of the error propagation matrices with an experimental error analysis based on randomly generated perturbations that takes the structure of A into account

A dimensionally compatible system of equations for stand basal area, volume, and basal area and volume growth was derived using dimensional analysis. These equations are analytically and numerically consistent with dimensionally compatible individual tree volume and taper equations and share parameters with them. Parameters for the system can be estimated by fitting individual tree taper and volume equations or by fitting stand level basal area and volume equations. In either case the parameters are nearly identical. Therefore, parameters for the system can be estimated at the tree or stand level without changing the results. Data from a thinning study in loblolly pine (Pinus taeda L.) plantations established on cutover site-prepared lands were used to estimate the parameters. However, the developed system of equations is general and can be applied to other tree species in other locales. / Ph. D.

This thesis considers the problem of tracking an object in world coordinates using measurements obtained from multiple uncalibrated cameras. A general approach to track the location of a target involves different phases including calibrating the camera, detecting the object's feature points over frames, tracking the object over frames and analyzing object's motion and behavior. The approach contains two stages. First, the problem of camera calibration using a calibration object is studied. This approach retrieves the camera parameters from the known locations of ground data in 3D and their corresponding image coordinates. The next important part of this work is to develop an automated system to estimate the trajectory of the object in 3D from image sequences. This is achieved by combining, adapting and integrating several state-of-the-art algorithms. Synthetic data based on a nearly constant velocity object motion model is used to evaluate the performance of camera calibration and state estimation algorithms.

In visually guided control of a robot, a large residual problem occurs when the robot configuration is not in the neighborhood of the target acquisition configuration. Most existing uncalibrated visual servoing algorithms use quasi-Gauss-Newton methods which are effective for small residual problems. The solution used in this study switches between a full quasi-Newton method for large residual case and the quasi-Gauss-Newton methods for the small case. Visual servoing to handle large residual problems for tracking a moving target has not previously appeared in the literature. For large residual problems various Hessian approximations are introduced including an approximation of the entire Hessian matrix, the dynamic BFGS (DBFGS) algorithm, and two distinct approximations of the residual term, the modified BFGS (MBFGS) algorithm and the dynamic full Newton method with BFGS (DFN-BFGS) algorithm. Due to the fact that the quasi-Gauss-Newton method has the advantage of fast convergence, the quasi-Gauss-Newton step is used as the iteration is sufficiently near the desired solution. A switching algorithm combines a full quasi-Newton method and a quasi-Gauss-Newton method. Switching occurs if the image error norm is less than the switching criterion, which is heuristically selected. An adaptive forgetting factor called the dynamic adaptive forgetting factor (DAFF) is presented. The DAFF method is a heuristic scheme to determine the forgetting factor value based on the image error norm. Compared to other existing adaptive forgetting factor schemes, the DAFF method yields the best performance for both convergence time and the RMS error. Simulation results verify validity of the proposed switching algorithms with the DAFF method for large residual problems. The switching MBFGS algorithm with the DAFF method significantly improves tracking performance in the presence of noise. This work is the first successfully developed model independent, vision-guided control for large residual with capability to stably track a moving target with a robot.