A System of Subroutines For Iteratively Reweighted Least Squares Computations

A description of a system of subroutines to compute solutions to the iteratively reweighted least squares problem is presented. The weights are determined from the data and linear fit and are computed as functions of the scaled residuals. Iteratively reweighted least squares is a part of robust statistics where "robustness" means relative insensitivity to moderate departures from assumptions. The software for iteratively reweighted least squares is cast as semi-portable Fortran code whose performance is unaffected (in the sense that performance will not be degraded) by the computer or operating-system environment in which it is used. An [ell sub1] start and an [ell sub2] start are provided. Eight weight functions, a numerical rank determination, convergence criterion, and a stem-and-leaf display are included.

Convergence analysis of generalized iteratively reweighted least squares algorithms on convex function spaces

The computation of robust regression estimates often relies on minimization of a convex functional on a convex set. In this paper we discuss a general technique for a large class of convex functionals to compute the minimizers iteratively which is closely related to majorization-minimization algorithms. Our approach is based on a quadratic approximation of the functional to be minimized and includes the iteratively reweighted least squares algorithm as a special case. We prove convergence on convex function spaces for general coercive and convex functionals F and derive geometric convergence in certain unconstrained settings. The algorithm is applied to TV penalized quantile regression and is compared with a step size corrected Newton-Raphson algorithm. It is found that typically in the first steps the iteratively reweighted least squares algorithm performs significantly better, whereas the Newton type method outpaces the former only after many iterations. Finally, in the setting of bivariate regression with unimodality constraints we illustrate how this algorithm allows to utilize highly efficient algorithms for special quadratic programs in more complex settings. --regression analysis,monotone regression,quantile regression,shape constraints,L1 regression,nonparametric regression,total variation semi-norm,reweighted least squares,Fermat's problem,convex approximation,quadratic approximation,pool adjacent violators algorithm

Linear multivariable control : numerical considerations

Bibliography: p. 31-32.Grant ERDA-E(49-18)-2087.by Alan J. Laub

Robust estimation of excitations in mechanical systems using M-estimators –Experimental applications

This second part of the study presents some experimentalapplications to mechanicalsystems in which the results of excitationestimation, obtained using traditional least squares and M-estimate, are compared. The first case presented is a single input–multiple outputs system: a simple test-rig for the study of the vibrations of a two-degrees of freedom system is employed to identify the constraint displacement that causes the measured mass vibrations in presence of heavy noise. The second case is a multiple inputs–multiple outputs system: a rotor test-rig is used to identify the positions, the amplitudes and the phases of two unbalances using the vibrations measured in the bearings. In this case, also an additional theoretical part is introduced about the basics of model-based identification in the frequency domain applied to rotor dynamics. The last case is again a single input–multiple outputs system, but in an industrial application: experimental vibrations of a 320 MW steam turbo-generator are used to identify position and amount of a known balancing mass in an on-field real case. Moreover, whilst in the numerical examples presented in the first part the knowledge of the system was perfect, in these cases some uncertainties are present also in the system model. Finally, the paper introduces the use of the M-estimate technique to evaluate the adequacy the model of the system, by means of the analysis of the weights attributed to the measures as a function of the frequency of the excitation

Increasing the robustness of fault identification in rotor dynamics by means of M-estimators

partially_open3One of the most common problems in rotordynamics is the identification of faults and model-based methods are often used for this purpose. In some applications, the least-squares (LS) estimate is used to find out the position and the severity of impending faults on the basis of experimental vibration data of rotating machinery. Anyhow LS are not very robust with respect to possible outliers (noise and gross errors) in the experimental data and to inaccuracies in the model. The introduction of weights in the LS algorithm has proven to be effective in increasing the robustness and successful experimental cases, both on test rigs and on real machines, are reported in literature. However, the arbitrary choice of the weights is normally based on operators’ experience. In this paper, an improvement is presented by introducing a method that is robust in itself, the M-estimate, which allows defining automatically the weights. This method is general and can be applied in every problem of regression or estimation, not necessarily related to rotordynamics. The fundamental theoretical aspects are introduced in the first part, while several experimental test cases are presented by means of faultidentification on a test rig and on a gas turbo generator in the second part of the paper. The obtained results highlight the increasing of the accuracy allowed by M-estimate.P. Pennacchi; A. Vania; N. BachschmidPennacchi, PAOLO EMILIO LINO MARIA; Vania, ANDREA TOMMASO; Bachschmid, Nicolo

Robust estimate of excitations in mechanical systems using M-estimators - Theoretical background and numerical applications

This second part of the study presents some experimental applications to mechanicalsystems in which the results of excitation estimation, obtained using traditional least squares and M-estimate, are compared. The first case presented is a single input–multiple outputs system: a simple test-rig for the study of the vibrations of a two-degrees of freedom system is employed to identify the constraint displacement that causes the measured mass vibrations in presence of heavy noise. The second case is a multiple inputs–multiple outputs system: a rotor test-rig is used to identify the positions, the amplitudes and the phases of two unbalances using the vibrations measured in the bearings. In this case, also an additional theoretical part is introduced about the basics of model-based identification in the frequency domain applied to rotor dynamics. The last case is again a single input–multiple outputs system, but in an industrial application: experimental vibrations of a 320 MW steam turbo-generator are used to identify position and amount of a known balancing mass in an on-field real case. Moreover, whilst in the numerical examples presented in the first part the knowledge of the system was perfect, in these cases some uncertainties are present also in the system model. Finally, the paper introduces the use of the M-estimate technique to evaluate the adequacy the model of the system, by means of the analysis of the weights attributed to the measures as a function of the frequency of the excitation

A Simplified Phase Display System for 3D Surface Measurement and Abnormal Surface Pattern Detection

Today’s engineering products demand increasingly strict tolerances. The shape of a machined surface plays a critical role to the desired functionality of a product. Even a small error can be the difference between a successful product launch and a major delay. It is important to develop tools that confirm the quality and accuracy of manufactured products. The key to assessing the quality is robust measurement and inspection tools combined with advanced analysis. This research is motivated by the goals of 1) developing an advanced optical metrology system that provides accurate 3D profiles of target objects with curvature and irregular texture and 2) developing algorithms that can recognize and extract meaningful surface features with the consideration of machining process information. A new low cost measurement system with a simple coherent interferometric fringe projection system is developed. Comparing with existing optical measurement systems, the developed system generates fringe patterns on object surface through a pair of fiber optics that have a relatively simple and flexible configuration. Three-dimensional measurements of a variety of surfaces with curvatures demonstrate the applicability and flexibility of the developed system. An improved phase unwrapping algorithm based on a flood fill method is developed to enhance the performance of image processing. The developed algorithm performs phase unwrapping under the guidance of a hybrid quality map that is generated by considering the quality of both acquired original intensity images and the calculated wrapped phase map. Advances in metrology systems enable engineers to obtain a large amount of surface information. A systematic framework for surface shape characterization and abnormal pattern detection is proposed to take the advantage of the availability of high definition surface measurements through advanced metrology systems. The proposed framework evaluates a measured surface in two stages. The first step focuses on the extraction of general shape (e.g., surface form) from measurement for surface functionality evaluation and process monitoring. The second step focuses on the extraction of application specific surface details with the consideration of process information (e.g., surface waviness). Applications of automatic abnormal surface pattern detection have been demonstrated. In summary, this research focuses on two core areas: 1) developing metrology system that is capable of measuring engineered surfaces accurately; 2) proposing a methodology that can extract meaningful information from high definition measurements with consideration of process information and product functionality.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/136999/1/xinweng_1.pd