Minimum Condition Number Research Articles

Sensitivity and reliability analysis to MSBAS regularization for the estimation of surface deformation over a mine

To systematically and thoroughly analyze the sensitivity and reliability of the MSBAS regularization for the estimation of surface deformation over a mine in combination with an application example, this study processed 101 Sentinel-1A/B SAR images, constructed and solved the 2D deformation models using SVD and Tikhonov regularization methods with different orders and parameters, and estimated the vertical and east-west surface deformation time series in a mine of China. Then, this study collected the leveling-monitoring vertical surface deformation data on three leveling points, and compared and analyzed the sensitivity and reliability of the MSBAS regularization methods for estimating vertical surface deformation. The results indicate that different regularization orders and parameters can lead to thousands of times differences in condition numbers and significant differences in ill-posed degree of the deformation models. The zero-order Tikhonov regularized deformation model with regularization parameter of 0.1 has the minimum condition number and the equation is not ill-posed. The first-order Tikhonov regularized deformation model with regularization parameter of 0.001 has the maximum condition number and the equation is seriously ill-posed. As a result, the estimates of 2D surface deformation models with different parameters and orders are also different in terms of numerical values and accuracy. Compared with the leveling-monitoring data, the first- and second-order MSBAS regularization methods with parameter 0.1 have the minimum fluctuation and the maximum correlation coefficients between the estimated values and the leveling-monitoring values, and are also closest to the leveling-monitoring results with the highest accuracy.

Research on selection method of aero-engine health parameters based on correlation and condition number

Aiming at the problem of how to select aero-engine health parameters when the number of sensors is limited, a method of selecting aero-engine health parameters based on correlation and condition number is proposed. Firstly, the engine health parameters are preliminarily selected based on correlation. When the influence of two health parameters on engine output parameters is strongly correlated, only one of them needs to be selected. Then health parameters are further selected based on condition number of sensors parameters degradation matrixes. The larger the condition number is, the more ill conditioned the matrix is and the worse the estimation effect of the on-board model is. According to this, the combination of health parameters with the minimum condition number is selected. The proposed method can quickly select optimal health parameters and improve the accuracy of engine adaptive estimation. The results demonstrate that the stronger the correlation between the two health parameters, the greater the impact on the accuracy of the on-board model. Compared with the singular value decomposition-Kalman filter (SVD-KF) method and other combinations of health parameters, the on-board model accuracy of the optimal combination selected by this method is greatly improved, and has the best state parameter tracking effect.

Reconstruction Optimization Algorithm of 3D Temperature Distribution Based on Tucker Decomposition

For the purpose of solving the large temperature field reconstruction error caused by different measuring point arrangements and the problem that the prior dataset cannot be built due to data loss or distortion in actual measurement, a three-dimensional temperature profile reconstruction optimization algorithm is proposed to repair the empirical dataset and optimize the arrangement of temperature measuring points based on Tucker decomposition, the minimum condition number method, the greedy algorithm, and the hill climbing algorithm. We used the Tucker decomposition algorithm to repair the missing data and obtain the complete prior dataset and the core tensor. By optimizing the dimension of the core tensor and the number and position of the measuring points calculated by the minimum condition number method, the greedy algorithm, and the mountain climbing algorithm, the real-time three-dimensional distribution of the temperature field is reconstructed. The results show that the Tucker decomposition optimization algorithm could accurately complete the prior dataset, and compared with the original algorithm, the proposed optimal placement algorithm improves the reconstruction accuracy by more than 20%. At the same time, the algorithm has strong robustness and anti-noise, and the relative error is less than 4.0% and 6.0% with different signal-to-noise ratios. It indicates that the proposed method can solve the problem of building an empirical dataset and 3D temperature distribution reconstruction more accurately and stably in industry.

Primary Decomposition with Differential Operators

Abstract We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential primary decompositions are unique up to change of bases. Our results generalize the construction of Noetherian operators for primary ideals in the analytic theory of Ehrenpreis–Palamodov, and they offer a concise method for representing affine schemes. The case of modules is also addressed. We implemented an algorithm in Macaulay2 that computes the minimal decomposition for an ideal in a polynomial ring.

Full-Stokes Retrieving and Configuration Optimization in a Time-Integration Imaging Polarimeter.

A time-integration imaging polarimeter with continuous rotating retarder is presented, and its full-Stokes retrieving and configuration optimization are also demonstrated. The mathematical expression between the full-Stokes vector and the time-integration light intensities is derived. As a result, the state of polarization of incident light can be retrieved by only one matrix calculation. However, the modulation matrix deviates from the initial well-conditioned status due to time integration. Thus, we re-optimize the nominal angles for the special retardance of 132° and 90° with an exposure angle of 30°, which results in a reduction of 31.8% and 16.8% of condition numbers comparing to the original configuration, respectively. We also give global optimization results under different exposure angles and retardance of retarder; as a result, the 137.7° of retardance achieves a minimal condition number of 2.0, which indicates a well-conditioned polarimeter configuration. Besides, the frame-by-frame algorithm ensures the dynamic performance of the presented polarimeter. For a general brushless DC motor with a rotating speed of over 2000 rounds per minute, the speed of polarization imaging will achieve up to 270 frames per second. High precision and excellent dynamic performance, together with features of compactness, simplicity, and low cost, may give this traditional imaging polarimeter new life and attractive prospects.

A linear and nonlinear analysis of the shallow water equations and its impact on boundary conditions

We derive boundary conditions and estimates based on the energy and entropy analysis of systems of the nonlinear shallow water equations in two spatial dimensions. It is shown that the energy method provides more details, but is fully consistent with the entropy analysis. The details brought forward by the nonlinear energy analysis allow us to pinpoint where the difference between the linear and nonlinear analysis originate. We find that the result from the linear analysis does not necessarily hold in the nonlinear case. The nonlinear analysis leads in general to a different minimal number of boundary conditions compared with the linear analysis. In particular, and contrary to the linear case, the magnitude of the flow does not influence the number of required boundary conditions.

Optimal and two-step adaptive quantum detector tomography

Quantum detector tomography is a fundamental technique for calibrating quantum devices and performing quantum engineering tasks. In this paper, we design optimal probe states for detector estimation based on the minimum upper bound of the mean squared error (UMSE) and the maximum robustness. We establish the minimum UMSE and the minimum condition number for quantum detectors and provide concrete examples that can achieve optimal detector tomography. In order to enhance the estimation precision, we also propose a two-step adaptive detector tomography algorithm to optimize the probe states adaptively based on a modified fidelity index. We present a sufficient condition on when the estimation error of our two-step strategy scales inversely proportional to the number of state copies. Moreover, the superposition of coherent states is used as probe states for quantum detector tomography and the estimation error is analyzed. Numerical results demonstrate the effectiveness of both the proposed optimal and adaptive quantum detector tomography methods.

Optimal elliptical retarder in rotating compensator imaging polarimetry.

In this Letter, we present a new, to the best of our knowledge, design for elliptical retarders based on two quarter-wave retarders with an angle offset, which is particularly well suited for polarimetric measurements based on rotating compensator systems. We show that this simple elliptical retarder design offers the minimum condition number achieved by traditionally optimal polarimeter setups and can be used to further improve the accuracy of polarimetric imaging measurements in the presence of error sources, with the advantage that it only requires common quarter-wave retarders.

Investigation of multi-objective optimization for integrating design and control of ionic liquid-based extractive distillation

Extractive distillation is widely accepted and commercialized for separating azeotrope mixtures, but using traditional organic solvents leads to intrinsic obstacles such as the high energy requirement and solvent-related environmental problems. Benefiting from the development of green solvents, using ionic liquids instead of traditional solvents as entrainers for extractive distillation has attracted wide attention. This paper presents a multi-objective optimization strategy to integrate the design and control of the ionic liquid-based extractive distillation processes. In the optimization processes, the condition number (CN) and total annual cost (TAC) are minimized as the objective functions to evaluate the control properties and design of three different separation systems. The optimization results are shown by the Pareto front, which is based on trade-offs between controllability and economics. The environmental impact is evaluated by the environmental impact potential of energy for each system in a posterior stage during the optimization processes. Designs with minimum TAC (PT), trade-off designs between TAC and CN (PO) and designs with minimum CN (PC) on Pareto front are optimized and compared. The results show that the increase of reflux ratio can improve the control properties for the ionic liquid-based extractive distillation processes. Furthermore, the dynamic simulation of PT and PO are carried out to compare their dynamic responses based on the same dynamic control structure, and PO shows better performance in the aspects which include maximum deviation and residual error of dynamic response.

Comparing the prevalence of multimorbidity using different operational definitions in primary care in Singapore based on a cross-sectional study using retrospective, large administrative data

ObjectivesMultimorbidity is a norm in primary care. A consensus on its operational definition remains lacking especially in the list of chronic conditions considered. This study aimed to compare six different...

Optimizing Diffusion Imaging Protocols for Structural Connectomics in Mouse Models of Neurological Conditions.

Network approaches provide sensitive biomarkers for neurological conditions, such as Alzheimer’s disease (AD). Mouse models can help advance our understanding of underlying pathologies, by dissecting vulnerable circuits. While the mouse brain contains less white matter compared to the human brain, axonal diameters compare relatively well (e.g., ~0.6 μm in the mouse and ~0.65–1.05 μm in the human corpus callosum). This makes the mouse an attractive test bed for novel diffusion models and imaging protocols. Remaining questions on the accuracy and uncertainty of connectomes have prompted us to evaluate diffusion imaging protocols with various spatial and angular resolutions. We have derived structural connectomes by extracting gradient subsets from a high-spatial, high-angular resolution diffusion acquisition (120 directions, 43-μm-size voxels). We have simulated protocols with 12, 15, 20, 30, 45, 60, 80, 100, and 120 angles and at 43, 86, or 172-μm voxel sizes. The rotational stability of these schemes increased with angular resolution. The minimum condition number was achieved for 120 directions, followed by 60 and 45 directions. The percentage of voxels containing one dyad was exceeded by those with two dyads after 45 directions, and for the highest spatial resolution protocols. For the 86- or 172-μm resolutions, these ratios converged toward 55% for one and 39% for two dyads, respectively, with <7% from voxels with three dyads. Tractography errors, estimated through dyad dispersion, decreased most with angular resolution. Spatial resolution effects became noticeable at 172 μm. Smaller tracts, e.g., the fornix, were affected more than larger ones, e.g., the fimbria. We observed an inflection point for 45 directions, and an asymptotic behavior after 60 directions, corresponding to similar projection density maps. Spatially downsampling to 86 μm, while maintaining the angular resolution, achieved a subgraph similarity of 96% relative to the reference. Using 60 directions with 86- or 172-μm voxels resulted in 94% similarity. Node similarity metrics indicated that major white matter tracts were more robust to downsampling relative to cortical regions. Our study provides guidelines for new protocols in mouse models of neurological conditions, so as to achieve similar connectomes, while increasing efficiency.

Deceptive Targets Generation Simulation Against Multichannel SAR

Traditional synthetic aperture radar (SAR) deceptive jamming can effectively generate deceptive scenes or false targets in SAR images. However, these false targets or scenes can be easily distinguished or eliminated by the multichannel SAR system. To interfere with the multichannel SAR, we first analyzed the results of SAR deceptive jamming generated by one transponder and two transponders against three-channel SAR- ground moving target indication (GMTI). Then, we propose a new deceptive jamming method against three-channel SAR-GMTI by using three synergetic transponders. By modulating each transponder with a complex coefficient, three synergetic transponders can generate false moving targets with the controllable radial velocity and located azimuth position in three-channel SAR-GMTI. Besides, in this paper, we also introduce an algorithm to deploy three transponders reasonably by utilizing the minimum condition number. In the end, a general architecture of multiple transponders deceiving multichannel SAR is given. The proposed method can not only generate deceptive false targets against multichannel SAR-GMTI, but also guide the production of a deceptive digital elevation model (DEM) against multichannel interferometric SAR (InSAR). Simulations verify the effectiveness of the proposed method.

Vibration load and transfer path identification of vehicle using inverse matrix method based on singular value decomposition

Based on the theory of inverse transfer matrix, a novel method for simultaneous load identification of vehicle vibration is presented in this paper. Some response, excitation, reference points (called key points) and their transfer paths, which have severe effects on the vibration of a whole vehicle, are defined. The transfer functions among the key points are measured by experiments, and thereby a transfer function matrix of vehicle vibration is established. To solve ill-conditioning problem in the transfer function matrix, the methodology of singular value decomposition is introduced into matrix inversion in the excitation load identification. To reduce the identification error, four transfer function matrices with different reference points and condition numbers are selected and discussed. The results show that the more the reference points are, the smaller the condition number of transfer function matrix is, the higher the accuracy of excitation load identification. The transfer function matrix with minimum condition number is used to identify the excitation loads at the vehicle key points. Experimental verifications suggest that the newly proposed method is effective and feasible for excitation load identification of vehicle vibration. Using the identified excitation loads, furthermore, the vibration causes of the steering wheel and seat rail are obtained, which is helpful for improving vibration performance of the sample vehicle. In applications, the excitation load identification method proposed in this paper may be applied not only to other types of vehicle, but also to other complex electromechanical products for load identification in engineering.

On the Accuracy of Inertial Parameter Estimation of a Free-Flying Robot While Grasping an Object

When model-based controllers are used for carrying out precise tasks, the estimation of model parameters is key for a better trajectory tracking performance. We consider the scenario of a free-flying space robot with limited actuation that has grasped an object of unknown properties. The inertial parameters - mass, centre of mass and the inertia tensor - of the robot-object system are to be determined. In our previous works, we used a parameter estimation algorithm where truncated Fourier series were used to represent both the reference excitation trajectory and the executed one. That algorithm is the focus of this paper, along with a study of the factors that could contribute to a loss of accuracy in the obtained estimates. Simulation results with the Space CoBot free-flyer robot are used to find the causes of the latter: first, a modified version of the minimum condition number criterion is used to generate feasible excitatory trajectories. The results are compared with those given by the maximum information criterion to check for better excitation. Next, an appropriate number of harmonics has to be found for the Fourier series, which will be used to fit the measured data. This is done autonomously, based on the least squares residual. Finally, the relation between input saturation while executing the excitation trajectory and the errors in the resulting parameter estimates is studied.

Jones pupil metrology of lithographic projection lens and its optimal configuration in the presence of error sources.

Mueller matrix imaging polarimeter (MMIP) can be used to measure the polarization aberration (PA) of a lithographic projector in the form of the Mueller pupil, while the Jones pupil is required for lithographic imaging simulations, projection lens design and PA evaluation. In this paper, a Jones pupil measurement method of lithographic projection lens is proposed. The measurement device of the method is the same as an MMIP, but a new polarimetric measurement equation is derived to solve the Jones pupil directly from the Kronecker product of the Jones matrix and the measured intensities. Two new polarimeter configurations with the minimum condition number are designed to further improve the accuracy in the presence of error sources. The performance of the method is evaluated by measurement errors of a typical Jones pupil in the presence of error sources. Comparisons between the proposed method and the conventional method, in which the Jones pupil is converted from the Mueller pupil measured by MMIP, are given. The results validate that the measurement accuracy of the Jones pupil is significantly improved without increasing the complexity of existing measurement systems.

Construction and Layout of Coordinated Multirobots System Based on Parallel Thought

Aiming at the problems that are the construction and the layout optimization of the coordinated multirobots system, the concept of an equivalent parallel robot (EPR) is proposed based on parallel thought in this paper. That is, each robot is regarded as a branched chain of the EPR in coordination. Thus, it can be converted into the problem for the type synthesis and the layout of the parallel robot. Firstly, the definition of robotic characteristic (C) set and the corresponding operational rule are given. Then, the minimum occupied area, cycle time, and condition number are used as the optimized objectives. The weight ratio is used to express the importance of the three objectives to meet the requirements of different scenes. Afterwards, based on the theory of C set, the three translational (3T) operational requirements that can be extended to other degree of freedom (DOF) are analyzed. Finally, the feasibility of the proposed methods is verified by the DOF analysis and simulation experiment. Moreover, the multiobjective layout problem is optimized based on the artificial bee colony (ABC) algorithm. The results show that the constructed robot combinations and the layout optimization method are satisfactory.

Minimum condition number by orthogonal projection row selection of artificial boundary conditions for finite element model update and damage detection

The method of artificial boundary conditions provides a way of significantly expanding the amount of useable data for performing model update or damage detection. It accomplishes this by artificially applying boundary conditions directly to the measured frequency response functions, which can then be curve fit to obtain the modal parameters. A critical task in the application of this method is the selection of modes from a large available set of candidate modes generated by the application of various artificial boundary conditions. An orthogonal projection procedure is used to select modes from which to build a composite sensitivity matrix of minimum condition number for update and damage detection. Both simulation and experiment are used to investigate this method. The experimental study makes use of two beams, identical except for damage installed in one beam. Issues regarding the number and type of artificial boundary conditions, the synthesis of boundary conditions, and test frequency resolution are studied. Both the simulation and the experiment reveal that model updating and damage detection can be accomplished with high accuracy.

Permitted experimental errors for optimized variable-retarder Mueller-matrix polarimeters.

An optimized Mueller-matrix polarimeter is simulated. The polarimeter is optimized by finding the configurations of the polarization state generator and polarization state analyzer that give the minimum condition number. Noise is included in the measurement of the polarimeter intensities, and the eigenvalue calibration procedure is used to reduce the errors in the final Mueller matrix. Controlled errors are introduced to the polarimeter configuration, and the error in the final measured Mueller matrix is calculated as a function of these configuration errors. It is found that the alignment of the retarder axes in the polarimeter is much more important than the use of the ideal, optimized retardance values. In particular, the misalignment of the retarders farthest from the sample is the error source with the highest impact in the precision of the polarimeter.

Omnidirectional Aerial Vehicles With Unidirectional Thrusters: Theory, Optimal Design, and Control

This paper presents a theoretical study on omni-directional aerial vehicles with body-frame fixed unidirectional thrusters. Omniplus multi-rotor designs are defined as the ones that allow to exert a total wrench in any direction using positive-only lift force and drag moment (i.e., positive rotational speed) for each rotor blade. Algebraic conditions for a design to be omniplus are derived, a simple necessary condition being the fact that at least seven propellers have to be used. An energy optimal design strategy is then defined as the one minimizing the maximum norm of the input set needed to span a certain wrench ellipsoid for the adopted input allocation strategy. Two corresponding major design criteria are then introduced: firstly, a minimum allocation-matrix condition number aims at an equal sharing of the effort needed to generate wrenches in any direction; secondly, imposing a balanced design guarantees an equal sharing of the extra effort needed to keep the input in the non-negative orthant. We propose a numerical algorithm to solve such optimal design problem and a control algorithm to control any omnidirectional platform. The work is concluded with informative simulation results in non-ideal conditions.

The design of a 3-CPS parallel robot for maximum dexterity

The optimum design of a 3-CPS parallel-kinematics machine (PKM), dubbed the SDelta Robot, is investigated in this paper based on a dexterity index. The six-dof PKM described here bears a simple three-limb structure with all its motors mounted on the base, thereby avoiding limb-interference while offering a low inertial load, which makes it quite suitable for high-speed operations. In this paper we find the inverse of the robot forward-kinematics Jacobian matrix symbolically, to be used in singularity analysis, design for isotropy and optimal control; in fact, the inverse derived here applies to all PKMs whose six active-wrench axes intersect pairwise. The inverse of a 6 × 6 matrix is usually difficult to derive or evaluate in symbolic form. We find the symbolic inverse of a Jacobian matrix that is applicable to the large class of PKMs mentioned above, which, to the authors’ knowledge, is an original contribution. Next, we formulate an optimization problem of the robot, based on its condition number, for maximum dexterity. Drawing from the optimization results, we offer some guidelines on choosing the optimum design parameters. It is shown that the SDelta can achieve a local minimum condition number close to unity.