Linear Error Estimator Research Articles

Piecewise reconstruction of membership function approximation errors for Takagi–Sugeno fuzzy control

We propose a conservative relaxed control method to stabilize Takagi–Sugeno (TS) fuzzy systems. First, a conventional piecewise linear function approximation method is adopted to consider the membership functions’ information in the closed-loop system’s stability analysis. Subsequently, a model reconstruction method is proposed to reformulate linear approximation error terms in the TS fuzzy form, which can effectively introduce more information into stability conditions and achieve better robust control effects. Then, based on a fuzzy Lyapunov function approach, control design conditions are derived in terms of linear matrix inequalities, with less conservatism for related results in the literature. Finally, via a benchmark experiment, it is verified that the proposed method can show better robustness with fewer decision variables in robust convex conditions.

LINEAR AND NON-LINEAR WAVELET APPROXIMATIONS OF FUNCTIONS OF LIPSCHITZ CLASS AND RELATED CLASSES USING THE HAAR WAVELET SERIES

In this paper, linear wavelet approximation of a functions f belonging to Lipξ, and Lip(ξ, p), 1 ≤ p < ∞, have been determined by using Haar scaling function and Haar wavelet series. A non-linear wavelet approximation of function belonging L2(R) and Lipα class has been obtained. The comparison of the order of linear and non-linear approximations have been studied. It is observed that a non-linear approximation error is better and sharper than linear approximation error.

Lifted Bilinear Model-based Linear Model Predictive Control with Scalability

We propose a novel linear model predictive control (MPC) using a lifted bilinear model based on Koopman theory, which is computationally scalable against the dimension of the target system and the prediction horizon length. In MPC, the accuracy of the prediction model determines control performance, but it is a challenge to reduce the computational cost especially when considering nonlinearity of the model. To address this, a method has been proposed which represents the nonlinear input affine system as a lifted bilinear model and utilizes linear approximation and prediction error correction regarding the lifted state to achieve a low-computational-cost linear MPC with equivalent performance to nonlinear MPC. However, although the previous studies have shown its effectiveness for relatively low-order systems, it has not been applied to practical systems with higher dimensions. In this study, we extend the conventional method and propose a scalable linear MPC using a lifted bilinear model and apply it to higher-order nonlinear systems. In this paper, a quadrotor system operating in three-dimensional space is considered and its analytical lifted bilinear model is derived. In the formulation of linear MPC using the lifted bilinear model, an error correction method is newly introduced to feedback error for adjustment of the numerical relationships among the elements in the lifted state. The effectiveness of the proposed method is demonstrated through numerical simulations.

Two-sided estimation of linear approximation error second - order hypersurfaces

The following aspects of multidimensional space are considered in the article. The parametrizations of the hypersurface of the second order are considered, which allow visualizing them in the process of discretization. The development of the principle of vector modeling for displaying objects of multidimensional space on a two-dimensional plane made it possible to visually display the solution to the problem of linear approximation of a second-order hypersurface. The proposed method for discretizing a second-order hypersurface makes it possible to create a simple algorithm for its linear ash-approximation, which is the basis for solving the linear programming problem in automatic mode. The developed method of two-sided estimation of the error in the case of linear approximation of the second-order hypersurface allows us to present the problem of determining the bearing capacity of structures as a linear programming problem.

Nullspace-Based Control Allocation of Overactuated UAV Platforms

Multirotor copters with full six Degree of Freedom (DoF) maneuvering are often overactuated. The control allocation of overactuated UAV platforms can have an infinite number of solutions due to their redundancy. The most common allocation framework is based on Force Decomposition (FD), which provides a robust least-square solution and is easy to implement, but it does not account for input constraints. A control allocation framework based on Quadratic Programming (QP) can take input constraints; however, it will introduce linear approximation error and it has not been formulated and implemented on physical systems. In this letter, we propose a nullspace-based allocation framework for overactuated UAV platforms that combines the advantages of the FD-based framework and QP-based framework. Our proposed nullspace-based framework can take input constraints while eliminating the linear approximation error in the control allocation. It is also robust against controller sampling frequency and measurement noise. These three allocation frameworks are implemented and compared on our novel overactuated UAV platform in simulation to evaluate the robustness of the proposed method and in experiment for a vertical rotation trajectory to demonstrate its performance.

A Mixed-Integer Programming Model for Effective Distribution of Relief Supplies in Disaster

The topic of this study is the field of humanitarian logistics for disaster response. Many existing studies have revealed that compliance with the golden time in response to a disaster determines the success or failure of relief activities, and logistics costs account for 80% of the disaster response cost. Besides, the agility, responsiveness, and effectiveness of the humanitarian logistics system are emphasized in consideration of the disaster situation’s characteristics, such as the urgency of life-saving and rapid environmental changes. In other words, they emphasize the importance of logistics activities in disaster response, which includes the effective and efficient distribution of relief supplies. This study proposes a mathematical model for establishing a transport plan to distribute relief supplies in a disaster situation. To determine vehicles’ route and the amount of relief for cities suffering a disaster, it mainly considers the urgency, effectiveness (restoration rate), and uncertainty in the logistics system. The model is initially developed as a mixed-integer nonlinear programming (MINLP) model containing some nonlinear functions and transform into a Mixed-integer linear programming (MILP) model using a logarithmic transformation and piecewise linear approximation method. Furthermore, a minimax problem is suggested to search for breakpoints and slopes to define a piecewise linear function that minimizes the linear approximation error. A numerical experiment is performed to verify the MILP model, and linear approximation error is also analyzed in the experiment.

Spectroscopic uncertainty impacts on OCO-2/3 retrievals of XCO2

Estimates of atmospheric carbon dioxide concentration from satellites now span multiple years and are providing information on key processes in the carbon cycle. Methodology for retrieving CO2 from satellite spectra continues to change as understanding of instrument calibration, other atmospheric constituents, and trace gas spectroscopy improve. In this work, we provide an update to linear error estimates for retrieval of carbon dioxide for the Orbiting Carbon Observatory-2/3 (OCO-2/3) due spectroscopic uncertainties. Linear error analysis is used to propagate uncertainties in laboratory conditions and spectroscopic model assumptions to sensitivity in retrieved XCO2 and surface pressure from OCO-2/3. Error estimates due to modeling assumptions for line mixing, collision-induced absorption, and water vapor exhibit noticeable variability across locations and seasons. On the other hand, error estimates due to perturbations in laboratory conditions are nearly constant in space and time.

Experimental investigation of neural network estimator and transfer learning techniques for K-edge spectral CT imaging.

Spectral computed tomography (CT) material decomposition algorithms require accurate physics-based models or empirically derived models. This study investigates a machine learning algorithm and transfer learning techniques for Spectral CTimaging of K-edge contrast agents using simulated and experimental measurements. A feed forward multilayer perceptron was implemented and trained on data acquired using a step wedge phantom containing acrylic, aluminum, and gadolinium materials. The neural network estimator was evaluated by scanning a rod phantom with varying dilutions of gadolinium oxide nanoparticles and by scanning a rat leg specimen with injected nanoparticles on a bench-top photon-counting computed tomography system. The algorithm decomposed each spectral projection measurement into path lengths of acrylic and aluminum and mass lengths of gadolinium. Each basis material sinogram was reconstructed into basis material images using filtered backprojection. Machine learning techniques of data standardization, transfer learning from aggregated pixel data, and transfer learning from simulations were investigated to improve image quality. The algorithm was compared to a previously published empirical method based on a linear approximation and calibration error look-up tables. The combined transfer learning techniques did not improve quantification in the rod phantom and provided only a small qualitative improvement in ring artifacts. Transfer learning from aggregated pixel data and from simulations improved the qualitative image quality of the rat specimen, for which the calibration data were limited. Transfer learning from aggregated pixel data andsimulations estimated 3.26, 6.26, and 12.45mg/mL Gd concentrations compared to true 2.72, 5.44, and 10.88mg/mL concentrations in the rod phantom. Additionally, the neural networks were able to separate the soft tissue, bone, and gadolinium nanoparticles of the ex vivo rat leg specimen into the different basis images. The results demonstrate that empirical K-edge imaging from calibration measurements using machine learning and transfer learning is possible without explicit models of material attenuations, incident spectra, or the detector response.

A Discriminative Projection and Representation-Based Classification Framework for Face Recognition

The sparse representation-based classifier (SRC) has been developed and verified as having great potential for real-world face recognition. In this paper, we propose a discriminative projection and representation-based classification (DPRC) method to enhance the discriminant ability of the SRC. The proposed method first obtains a discriminative projection matrix not only maximizing the ratio of the distance within interclass over the distance within intraclass, but also minimizing the linear approximation error within intraclass. Then it maps the original data onto the discriminative space, and adopts an SRC method to obtain the final solution. An inexact augmented Lagrangian method of multiplier is proposed for finding the optimal representation vector in our framework, and a proximal alternating minimization method is adopted to the iteration subproblems of the proposed method. The proposed method is proven to have the subsequence convergence property. Experimental results on Yale, ORL, and AR face image databases demonstrate that, compared with some existing feature extraction methods based on the SRC, the proposed DPRC method is more efficient.

Validation of OCO-2 error analysis using simulated retrievals

Abstract. Characterization of errors and sensitivity in remotely sensed observations of greenhouse gases is necessary for their use in estimating regional-scale fluxes. We analyze 15 orbits of the simulated Orbiting Carbon Observatory-2 (OCO-2) with the Atmospheric Carbon Observations from Space (ACOS) retrieval, which utilizes an optimal estimation approach, to compare predicted versus actual errors in the retrieved CO2 state. We find that the nonlinearity in the retrieval system results in XCO2 errors of ∼0.9 ppm. The predicted measurement error (resulting from radiance measurement error), about 0.2 ppm, is accurate, and an upper bound on the smoothing error (resulting from imperfect sensitivity) is not more than 0.3 ppm greater than predicted. However, the predicted XCO2 interferent error (resulting from jointly retrieved parameters) is a factor of 4 larger than predicted. This results from some interferent parameter errors that are larger than predicted, as well as some interferent parameter errors that are more strongly correlated with XCO2 error than predicted by linear error estimation. Variations in the magnitude of CO2 Jacobians at different retrieved states, which vary similarly for the upper and lower partial columns, could explain the higher interferent errors. A related finding is that the error correlation within the CO2 profiles is less negative than predicted and that reducing the magnitude of the negative correlation between the upper and lower partial columns from −0.9 to −0.5 results in agreement between the predicted and actual XCO2 error. We additionally study how the postprocessing bias correction affects errors. The bias-corrected results found in the operational OCO-2 Lite product consist of linear modification of XCO2 based on specific retrieved values, such as the CO2 grad del (δ∇CO2), (“grad del” is a measure of the change in the profile shape versus the prior) and dP (the retrieved surface pressure minus the prior). We find similar linear relationships between XCO2 error and dP or δ∇CO2 but see a very complex pattern of errors throughout the entire state vector. Possibilities for mitigating biases are proposed, though additional study is needed.

Iterative method for frequency domain identification of continuous processes with delay time

AbstractDelay time that affects the performances of many control synthesis techniques in controlled systems is common in most chemical industries. Estimating the delay time is also a difficult problem in identification fields. In this paper, aiming at continuous processes with delay time, an iterative least square identification method is proposed in the frequency domain. By introducing a truncated first‐order Taylor expansion, the delay time term is linearized. Linear regression equation for least squares (LS) is directly derived from the transfer function whose nonlinear term is replaced by a linear one. For reducing the linear approximation error and getting more accurate estimations of the model parameters, an iterative algorithm is developed based on the LS. Moreover, the proposed method can be easily extended to a closed‐loop system without increasing the order of the identified model. Simulations verify the effectiveness, fast convergence rates, and robustness of the proposed identification algorithm.

Solution of inverse thermal problem for assessment of thermal parameters of engineered H2 storage materials

New materials for hydrogen storage applications require accurate thermal parameters for the design of the thermal management system. The task of material characterization has been formulated as an inverse thermal problem of parameter estimation from thermal conductivity measurements in orthogonal directions. The three-dimensional (3D) forward problem that models the transient thermal conductivity measurement experiment has been implemented in COMSOL®. The backward problem has been solved using the MATLAB®’s Global Optimization Toolbox™, augmented with an original acceleration strategy that is based on a smooth parametric reconstruction of the objective function manifold. The error in the parameter estimates is dominated by the inaccuracy of the physical forward problem, with the statistical variation of the measurements playing a secondary role. The traditional method of linear error estimation based on the computation of a sensitivity matrix in the vicinity of the optimal solution has been shown to overestimate the errors in the parameter estimates due to the low values of the sensitivity matrix norm in the vicinity of the optimal solution. The error was therefore re-estimated from the reconstruction of the objective function manifold close to the optimal parameter values.

Optimal Recovery of Functions of the Class <i>H<sub>p,p</sub></i> (1≤ <i>p ≤ ∞) </i>in the Unit Circle

In the below work the problem of optimal recovery of functions in Hardy class is covered. Namely, by the values ​​of these functions in a finite number of points lying in the unit circle determined their value at a given point. Coefficients of the linear best approximation method and error of the best method are calculated. The functions are considered with some given weight function.

Critical dimension in profile semiparametric estimation

This paper revisits the classical inference results for profile quasi maximum likelihood estimators (profile MLE) in semiparametric models. We mainly focus on two prominent theorems: the Wilks phenomenon and Fisher expansion for the profile MLE are stated in a new fashion allowing finite samples and model misspecification. The method of study is also essentially different from the usual analysis of the semiparametric problem based on the notion of the hardest parametric submodel. Instead we derive finite sample deviation bounds for the linear approximation error for the gradient of the loglikelihood. This novel approach particularly allows to address the impact of the effective target and nuisance dimension on the accuracy of the results. The obtained nonasymptotic results are surprisingly sharp and yield the classical asymptotic statements including the asymptotic normality and efficiency of the profile MLE. The general results are specified for the important special case of an i.i.d. sample and the analysis is exemplified with a single index model.

Study on Error in Interpolation Algorithm of the-5-DOF-Hybrid Robot

In order to improve the accuracy of the-5-DOF-hybrid robot motion control, it is necessary to establish a mathematical model of the error in the interpolation process. On the basis of the structure characteristics of five degrees of freedom hybrid robot and the algorithm of linear interpolation and circular interpolation, the error calculation patterns about the concave surface machining and convex surface machining are set up respectively in the paper. Then, examples are analyzed. The results show that the nonlinear error caused by using the straight-line interpolation method is much smaller than the normal vector rotation error and linear approximation error; when conducting surface machining, the error caused by the application of circular interpolation algorithm is less than linear interpolation error. By the comparison of the error analysis of different interpolation method, the control methods for reducing errors are proposed.

Efficient reliability estimate of passive thermal hydraulic safety system with automatic differentiation

An approach for efficient estimation of passive safety system functional reliability has been developed and applied to a simplified model of the passive residual heat transport system typical of sodium cooled fast reactors to demonstrate the reduction in computational time. The method is based on generating linear approximations to the best estimate computer code, using the technique of automatic reverse differentiation. This technique enables determination of linear approximation to the code in a few runs independent of the number of input variables for each response variable. The likely error due to linear approximation is reduced by augmented sampling through best estimate code in the neighborhood of the linear failure surface but in a sub domain where linear approximation error is relatively more. The efficiency of this new approach is compared with importance sampling MCS which uses the linear approximation near the failure region and with Direct Monte-Carlo Simulation. In the importance sampling MCS, variants employing random sampling with Box–Muller algorithm and Markov Chain algorithm are inter-compared. The significance of the results with respect to system reliability is also discussed.

A Modified BP Algorithm for LDPC Decoding Based on Minimum Mean Square Error Criterion

In this paper, a modified Belief Propagation (BP) decoding algorithm for low-density parity check (LDPC) codes based on minimum mean square error (MMSE) criterion is proposed. This modified algorithm uses linear equation to replace the hyperbolic function in the original BP algorithm and optimizes the linear approximation error based on MMSE criterion. As a result, compared with the standard BP algorithm the computational complexity is reduced significantly as the modified algorithm requires only addition operations to implement. Besides that simulation results show our modified algorithm can achieve an error performance very close to the BP algorithm on the additive white Gaussian noise channel.

On window size selection in the windowed Fourier ridges algorithm

The windowed Fourier ridges (WFR) algorithm can be used to extract the frequency and phase information from an exponential or carrier fringe pattern. The selection of the window size for this algorithm is investigated in this paper. For exponential phase fringe patterns, the window size does not need to be adjusted according to the signal frequency. Hence the scaling strategy of the wavelet transform is not necessary. Instead, the window size should be selected according to the balance between the linear phase approximation error and the noise level. For carrier fringe patterns, the influence of the conjugate component should also be considered. However it can be ignored when the lowest frequency of the fringe pattern is higher than 1.5/ σ ( σ is used to represent the window size).

On window size selection in the windowed Fourier ridges algorithm: Addendum

The windowed Fourier transform is a useful technique for fringe pattern analysis. It has been shown that the proper selection of the window size is a balance between the linear phase approximation error and the influence of noise [Q. Kemao, On widow size selection in windowed Fourier ridges algorithm. Opt Lasers Eng, 2007, accepted for publication]. Since the fringe intensity and noise level usually vary spatially, the window size should also be spatially adaptive in order to reach a good balance for each pixel of the fringe pattern. This addendum first shows that the windowed Fourier transform with a spatially fixed window size (SFWS) is still practically useful and then discusses the window size competition strategies for the windowed Fourier transform with a spatially adaptive window size (SAWS). The windowed Fourier transform with a SAWS is theoretically better than that with a SFWS but it is also more challenging in use. The windowed Fourier ridges algorithm is used for analysis throughout this paper. This analysis is also applicable to the windowed Fourier filtering algorithm.

Adaptive Kalman Filtering and Smoothing for Tracking Vocal Tract Resonances Using a Continuous-Valued Hidden Dynamic Model

A novel Kalman filtering/smoothing algorithm is presented for efficient and accurate estimation of vocal tract resonances or formants, which are natural frequencies and bandwidths of the resonator from larynx to lips, in fluent speech. The algorithm uses a dynamic model, with a state-space formulation, where the resonance frequency and bandwidth values are treated as continuous-valued state variables. The observation equation of the model is constructed by an analytical predictive function from the resonance frequencies and bandwidths to LPC cepstra as the observation vectors. This nonlinear function is adaptively linearized, and a residual or bias term, which is adaptively trained, is added to the nonlinear function to represent the iteratively reduced piecewise linear approximation error. Details of the piecewise linearization design process are described. An iterative tracking algorithm is presented, which embeds both the adaptive residual training and piecewise linearization design in the Kalman filtering/smoothing framework. Experiments on estimating resonances in Switchboard speech data show accurate estimation results. In particular, the effectiveness of the adaptive residual training is demonstrated. Our approach provides a solution to the traditional hidden formant problem, and produces meaningful results even during consonantal closures when the supra-laryngeal source may cause no spectral prominences in speech acoustics