First-order Taylor Series Expansion Research Articles

Design of Air-Breathing Hypersonic Vehicle Control System

The focus of this paper is uncertainty modeling and controller designing of air-breathing hypersonic vehicle. First, the hypersonic vehicle longitudinal dynamics model is made on the base of the fitting of aerodynamic parameters. As to the uncertainty of model parameters, a linear model of the uncertain part is established and the LQR controller parameters are thereupon designed with state transition and a first-order Taylor series expansion method, based on feedback linearization of the system. The simulation results that the hypersonic flight control system stated in this paper can well realize the tracking of speed and trajectory angle input command under the situation of floating model parameters.

Explicit reliability sensitivities of linear structures with interval uncertainties under stationary stochastic excitation

Abstract Reliability sensitivity evaluation of randomly excited linear structures with uncertain parameters is addressed. The excitation is modeled as a stationary Gaussian random process. Uncertainty affecting structural parameters is handled within a non-probabilistic framework by applying the interval model. Under the assumption of independent up-crossings of a specified threshold, a procedure for the analytical derivation of interval reliability sensitivity is presented. The key idea is to perform stochastic analysis in the frequency domain by applying the improved interval analysis via extra unitary interval in conjunction with a novel series expansion of the inverse of an interval matrix with modifications, herein called Interval Rational Series Expansion (IRSE). This approach yields approximate explicit expressions of interval response statistics and structural reliability along with their sensitivities with respect to the uncertain parameters. Reliability sensitivities provide useful information to increase the safety level and can be also exploited in the context of first-order interval Taylor series expansion to estimate the bounds of interval reliability when slight uncertainties are involved. A wind-excited truss structure with interval stiffness properties is analyzed to show the effectiveness of the proposed procedure.

Robust Calibration Algorithm for Multiplicative Modeling Errors Against Location Deviations of Auxiliary Sources

It is well known that the azimuth deviations of the auxiliary sources severely degrade the performance of classical subspace-based calibration methods that assume the direction-of-arrivals of calibration sources are perfectly measured. Therefore, aiming at the effects of source location deviations, the estimation variance of the multiplicative modeling errors for the subspace-based calibration method is first derived by applying matrix eigen-perturbation theory and first-order perturbation analysis approach. The theoretical analysis is undertaken under the assumption that the azimuth deviations are small enough for the first-order perturbation analysis to be valid. In addition, to mitigate the effects of the location errors, a structured total least squares optimization model is established using first-order Taylor series expansion method. Then, the corresponding numerical algorithm is presented to provide a robust estimate for multiplicative modeling errors. The exact Cramer---Rao bound expressions for the unknowns are also deduced in the presence of the azimuth deviations. Simulation results confirm the effectiveness of the theoretical analysis and demonstrate the desirable behavior of the robust calibration algorithm in comparison with the subspace-based calibration methods.

Adaptive Monte Carlo and GUM methods for the evaluation of measurement uncertainty of cylindricity error

Abstract Measurement uncertainty is one of the most important concepts in geometrical product specification (GPS). The “Guide to the expression of uncertainty in measurement (GUM)” is the internationally accepted master document for the evaluation of uncertainty. The GUM method (GUMM) requires the use of a first-order Taylor series expansion for propagating uncertainties. However, when the mathematical model of measurand is strongly non-linear the use of this linear approximation may be inadequate. Supplement 1 to GUM (GUM S1) has recently been proposed based on the basis of probability density functions (PDFs) using the Monte Carlo method (MCM). In order to solve the problem that the number of Monte Carlo trials needs to be selected priori, adaptive Monte Carlo method (AMCM) described in GUM S1 is recommended to control over the quality of the numerical results provided by MCM. The measurement and evaluation of cylindricity errors are essential to ensure proper assembly and good performance. In this paper, the mathematical model of cylindricity error based on the minimum zone condition is established and a quasi particle swarm optimization algorithm (QPSO) is proposed for searching the cylindricity error. Because the model is non-linear, it is necessary to verify whether GUMM is valid for the evaluation of measurement uncertainty of cylindricity error. Then, AMCM and GUMM are developed to estimate the uncertainty. The procedure of AMCM scheme and the validation of GUMM using AMCM are given in detail. Practical example is illustrated and the result shows that GUMM is not completely valid for high-precision evaluation of the measurement uncertainty of cylindricity error if only the first-order terms in the Taylor series approximation are taken into account. Compared with conventional methods, not only the proposed QPSO method can search the minimum zone cylindricity error precisely and rapidly, but also the Monte Carlo simulation is adaptive and AMCM can provide control variables (i.e. expected value, standard uncertainty and lower and higher coverage interval endpoints) with an expected numerical tolerance. The methods can be extended to the evaluation of measurement uncertainty of other form errors such as roundness and sphericity errors.

Optimal Two-Impulse Cotangent Rendezvous Between Coplanar Elliptical Orbits

This paper studies the two-impulse cotangent rendezvous problem between two coplanar elliptical orbits. This problem requires the same flight time for two spacecraft and a cotangent transfer between the initial and final orbits. For two coplanar circular orbits, the closed-form solution is obtained and its total cost is equal to that of the Hohmann transfer. However, for two coplanar elliptical orbits, the solutions are obtained only by a numerical iterative algorithm. There are many solutions for the multiple-revolution case. Moreover, the minimum-fuel two-impulse cotangent transfer can be expressed as the true anomaly of final orbit. With the minimum-fuel transfer, a simple method for the optimal revolution numbers is proposed based on the first-order Taylor series expansion of the flight-time equation. Then, the minimum-fuel two-impulse cotangent rendezvous is obtained by calculating and comparing two or four candidates. Two numerical examples are provided to apply the proposed technique for all solutions and the minimum-fuel solution to the two-impulse cotangent rendezvous problem.

Non-probabilistic information fusion technique for structural damage identification based on measured dynamic data with uncertainty

Based on measured natural frequencies and acceleration responses, a non-probabilistic information fusion technique is proposed for the structural damage detection by adopting the set-membership identification (SMI) and two-step model updating procedure. Due to the insufficiency and uncertainty of information obtained from measurements, the uncertain problem of damage identification is addressed with interval variables in this paper. Based on the first-order Taylor series expansion, the interval bounds of the elemental stiffness parameters in undamaged and damaged models are estimated, respectively. The possibility of damage existence (PoDE) in elements is proposed as the quantitative measure of structural damage probability, which is more reasonable in the condition of insufficient measurement data. In comparison with the identification method based on a single kind of information, the SMI method will improve the accuracy in damage identification, which reflects the information fusion concept based on the non-probabilistic set. A numerical example is performed to demonstrate the feasibility and effectiveness of the proposed technique.

Robust Capon beamformer based on direction search

In the presence of pointing errors, a direction search based robust Capon beamformer is proposed. The direction of the desired signal is first estimated by searching for the local maximum of the array output power using the first-order Taylor series expansion of the array steering vector, and then the robust Capon beamformer is developed with the estimated direction as the presumed one.

Membership-set identification method for structural damage based on measured natural frequencies and static displacements

Based on measured natural frequencies and static displacements, an improved interval analysis technique is proposed for structural damage detection by adopting membership-set identification and two-step model updating procedures. Due to the scarcity of uncertain information, the uncertainties are considered as interval numbers in this article. Via the first-order Taylor series expansion, the interval bounds of the elemental stiffness parameters of undamaged and damaged structures are obtained. The structural damage is detected by the quantitative measure of the possibility of damage existence in elements, which is more reasonable than the probability of damage existence in the condition of less measurement data. In this study, the conversation of the interval analysis method is remarkably reduced by the membership-set identification technique. The present method is applied to a truss structure and a steel cantilever plate for damage identification, and the damage identification results obtained by the interval analysis method and probabilistic method are compared. This article also discusses the effects of damage level and uncertainty level on detection results. The numerical examples show that the wide intervals resulting from the interval operation can be narrowed by the proposed non-probabilistic approach, and the feasibility and applicability of the present method are validated.

Non-probabilistic wear reliability analysis of swash-plate/slipper of water hydraulic piston motor based on convex model

Water hydraulic piston motor is one of the important transmission components in a water hydraulic system. For lack of enough experimental information to analyze the reliability in water hydraulic motor design, the convex model is chosen as wear reliability model for hydraulic friction pairs, which is more economical and reasonable than the interval model. The non-probabilistic reliability index based on the convex model is established using first-order Taylor series expansion, and the non-probabilistic reliability model of wear reliability analysis for the friction pair of swash-plate/slipper in water hydraulic piston motor is introduced based on the convex model. For the non-probabilistic reliability model, only the bounds on uncertain parameters are needed, instead of probabilistic density distribution or statistical quantities. Numerical examples are used to illustrate the validity and feasibility of the presented method.

기준신호원을 이용한 배열센서의 위치, 이득, 위상 보정기법

The geometrical and electrical errors of array sensors can severely degrade the performance of array sensor system. Various calibration techniques are developed to alleviate this problem. In this paper, two different calibration methods with respect to location, gain and phase of array sensors are presented. One method applies the first-order Taylor series expansion to approximate the true steering vector from the nominal values of array sensors. Then a set of equations is formed by using the null characteristics of the MUSIC spectrum to estimate errors of location, gain and phase of array sensors. Another method estimates these errors based on the data covariance matrix of pilot sources. From the simulations, it is demonstrated that two calibration algorithms calibrated an array system successfully. In addition to that, Fistas and Manikas's algorithm is more robust against noise than Ng and Lie's one when SNR is from 10dB to 50dB.

Modified particle swarm optimization structure approach to direction of arrival estimation

This study considers the problem of estimating the direction-of-arrival (DOA) for code-division multiple access (CDMA) signals. In this type of problem, the associated cost function of the DOA estimation is generally a computationally-expensive and highly-nonlinear optimization problem. A fast convergence of the global optimization algorithm is therefore required to attain results within a short amount of time. In this paper, we propose a new application of the modify particle swarm optimization (MPSO) structure to achieve a global optimal solution with a fast convergence rate for this type of DOA estimation problem. The MPSO uses a first-order Taylor series expansion of the objective function to address the issue of enhanced PSO search capacity for finding the global optimum leads to increased performance. The first-order Taylor series approximates the spatial scanning vector in terms of estimating deviation results in and reducing to a simple one-dimensional optimization problem and the estimating deviation has the tendency to fly toward a better search area. Thus, the estimating deviation can be used to update the velocity of the PSO. Finally, several numerical examples are presented to illustrate the design procedure and to confirm the performance of the proposed method.

Interval Analysis Method for Structural Damage Identification Based on Multiple Load Cases

Based on the measured static displacements, an improved interval analysis technique was proposed for the structural damage identification. Due to the scarcity of uncertain information, the uncertainties were considered as interval numbers in this paper. Via the first-order Taylor series expansion, the interval bounds of the elemental stiffness parameters of undamaged and damaged structures are respectively obtained. The structural damage was detected by the quantitative measure of the possibility of damage existence in elements, which was more reasonable than the probability of damage existence in the condition of less sample points for the measurement data. Furthermore, the classic interval analysis method was improved by adopting the membership-set identification and two-step model updating procedure to make identification results more accurate. An uncertain truss structure was employed for damage identification, the damage identification results obtained by interval analysis method and probabilistic method, respectively, were compared. Moreover, the effects on the detection results of the damage level and uncertainty level subjected to single or multiple load cases were studied as well. The numerical example shows that the wide intervals resulting from the interval operation can be narrowed by the improved nonprobabilistic approach, and the feasibility and effectiveness of the present method were validated.

Disaggregation of SMOS Soil Moisture in Southeastern Australia

Disaggregation based on Physical And Theoretical scale Change (DisPATCh) is an algorithm dedicated to the disaggregation of soil moisture observations using high-resolution soil temperature data. DisPATCh converts soil temperature fields into soil moisture fields given a semi-empirical soil evaporative efficiency model and a first-order Taylor series expansion around the field-mean soil moisture. In this study, the disaggregation approach is applied to Soil Moisture and Ocean Salinity (SMOS) satellite data over the 500 km by 100 km Australian Airborne Calibration/validation Experiments for SMOS (AACES) area. The 40-km resolution SMOS surface soil moisture pixels are disaggregated at 1-km resolution using the soil skin temperature derived from Moderate Resolution Imaging Spectroradiometer (MODIS) data, and subsequently compared with the AACES intensive ground measurements aggregated at 1-km resolution. The objective is to test DisPATCh under various surface and atmospheric conditions. It is found that the accuracy of disaggregation products varies greatly according to season: while the correlation coefficient between disaggregated and in situ soil moisture is about 0.7 during the summer AACES, it is approximately zero during the winter AACES, consistent with a weaker coupling between evaporation and surface soil moisture in temperate than in semi-arid climate. Moreover, during the summer AACES, the correlation coefficient between disaggregated and in situ soil moisture is increased from 0.70 to 0.85, by separating the 1-km pixels where MODIS temperature is mainly controlled by soil evaporation, from those where MODIS temperature is controlled by both soil evaporation and vegetation transpiration. It is also found that the 5-km resolution atmospheric correction of the official MODIS temperature data has a significant impact on DisPATCh output. An alternative atmospheric correction at 40-km resolution increases the correlation coefficient between disaggregated and in situ soil moisture from 0.72 to 0.82 during the summer AACES. Results indicate that DisPATCh has a strong potential in low-vegetated semi-arid areas where it can be used as a tool to evaluate SMOS data (by reducing the mismatch in spatial extent between SMOS observations and localized in situ measurements), and as a further step, to derive a 1-km resolution soil moisture product adapted for large-scale hydrological studies.

An Improved Array-Error Estimation Method for Constellation SAR Systems

In this letter, we consider the problem of estimating gain-phase and position errors for constellation synthetic aperture radar (SAR) systems. In the conventional method, the position error estimation is based on the first-order Taylor series expansion of the position-error exponential function. However, the first-order Taylor series expansion causes an approximation error, resulting in the inaccuracy of the estimation by the conventional method. In this letter, an improved method is developed to overcome this problem, based on the fact that the aforementioned approximation error decreases with the reduction in position errors. In the improved method, we first compensate the position error estimates obtained at the kth iteration in order to reduce the remaining position errors at the (k + 1) th iteration. Then, the position error estimates obtained at all iterations are summed as the estimates of the true position errors. In this way, the improved method removes the aforementioned approximation error, leading to estimates with high accuracy. Simulation results verify that the estimates by the improved method are closer to the true array errors than those by the conventional method. In addition, simulation results show that the improved method is more robust to position errors than the conventional method. Furthermore, the increase in the computational load of the improved method is negligible.

Efficient Propagation of Error Through System Models for Functions Common in Engineering

System modeling can help designers make and verify design decisions early in the design process if the model’s accuracy can be determined. The formula typically used to analytically propagate error is based on a first-order Taylor series expansion. Consequently, this formula can be wrong by one or more orders of magnitude for nonlinear systems. Clearly, adding higher-order terms increases the accuracy of the approximation but it also requires higher computational cost. This paper shows that truncation error can be reduced and accuracy increased without additional computational cost by applying a predictable correction factor to lower-order approximations. The efficiency of this method is demonstrated in the kinematic model of a flapping wing. While Taylor series error propagation is typically applicable only to closed-form equations, the procedure followed in this paper may be used with other types of models, provided that model outputs can be determined from model inputs, derivatives can be calculated, and truncation error is predictable.

RESPONSE STATISTICS OF LINEAR STRUCTURES WITH UNCERTAIN-BUT-BOUNDED PARAMETERS UNDER GAUSSIAN STOCHASTIC INPUT

Uncertainty plays a fundamental role in structural engineering since it may affect both external excitations and structural parameters. In this study, the analysis of linear structures with slight variations of the structural parameters subjected to stochastic excitation is addressed. It is realistically assumed that sufficient data are available to model the external excitation as a Gaussian random process, while only fragmentary or incomplete information about the structural parameters are known. Under this assumption, a nonprobabilistic approach is pursued and the fluctuating properties are modeled as uncertain-but-bounded parameters via interval analysis. A method for evaluating the lower and upper bounds of the second-order statistics of the response is presented. The proposed procedure basically consists in combining random vibration theory with first-order interval Taylor series expansion of the mean-value and covariance vectors of the response. After some algebra, the sets of first-order ordinary differential equations ruling the nominal and first-order sensitivity vectors of response statistics are derived. Once such equations are solved, the bounds of the mean-value and covariance vectors of the response can be evaluated by handy formulas. To validate the procedure, numerical results concerning two different structures with uncertain-but-bounded stiffness properties under seismic excitation are presented.

Generalized expressions of second and third order for the evaluation of standard measurement uncertainty

Abstract The Guide to the Expression of Uncertainty in Measurement (GUM) requires the use of a first-order Taylor series expansion for propagating uncertainties. However, when the measurement function is strongly non-linear the use of this linear approximation may be inadequate and therefore higher order terms from the Taylor series cannot be neglected. The present paper aims to derive generalized expressions of second and third order for the evaluation of the estimate of a measurand and its associated standard uncertainty. A case study is given to illustrate an application of the proposed methods and the results obtained with the GUM method are compared to the corresponding ones when applying the method proposed in GUM Supplement 1.

Accurate phase expansion on reference planes in grating projection profilometry

A novel phase expansion technology based on a binary quadratic rational polynomial model is proposed. Firstly, the unwrapped phase distribution on reference planes parallel with a zero reference plane in a different grating project profilometry measurement system is analyzed and synthesized. Then a binary quadratic rational polynomial model is proposed to express the uneven phase distribution on every reference plane. The least-squares iterative estimation method based on a first-order Taylor series expansion is employed to obtain the polynomial coefficients. Finally, accurate phase expansion of reference planes is implemented according to the rational polynomial model. The results of the experiments proved that the method proposed in this paper can recover the phase distribution on the full field from parts reference planes, and the measurement errors such as the geometrical parameter error, system structure error, real reference plane profile error and random error can be decreased.

Adjusted intraclass correlation coefficients for binary data: methods and estimates from a cluster-randomized trial in primary care

Accurate estimates of the intraclass correlation coefficient (ICC) are important for calculating appropriate sample sizes for cluster-randomized trials. The ICC and hence the sample size may be reduced through adjustment for baseline covariates. A method exists for calculating adjusted ICCs for binary outcomes based on the logit link, used to calculate odds ratios. Recent interest in presenting relative risks rather than odds ratios indicates that a method based on the log link is needed. To determine and evaluate a method for calculating adjusted ICCs based on the log link, and to provide and compare unadjusted and adjusted ICCs from a cluster-randomized trial in primary care based on the logit and log link. Two methods are proposed for calculating adjusted ICCs for the log link based on a first-order Taylor series expansion and properties of the lognormal distribution. The methods are evaluated by simulation. Unadjusted and adjusted ICCs are calculated for binary outcomes from the Point of Care Testing (PoCT) Trial using the logit and log link. The methods for calculating adjusted ICCs for the log link produced similar results unless the between cluster variance was large. Unadjusted ICCs for the PoCT Trial ranged from 0.001 to 0.048. The impact of adjustment on the ICC varied between outcomes and link functions, ranging from a 59% reduction to an 89% increase. The true ICC was unknown for the simulation study. Adjustment was made for age and gender only for the PoCT Trial. The method for calculating adjusted ICCs for binary outcomes depends on the link function. For the log link, the method based on the lognormal distribution is recommended. This method will be useful for cluster-randomized trials where the relative risk, rather than the odds ratio, is the effect measure of interest.

Continuous- and discrete-time fixed-gain controller designs for the control of vehicle lateral dynamics

In this paper, fixed-gain feedback linearization controls are presented to stabilize the vehicle lateral dynamics at bifurcation points for both continuous-time and discrete-time cases. Based on the assumption of constant driving speed, a second-order nonlinear lateral dynamics model is adopted for controller design. Via the feedback linearization scheme and the first-order Taylor series expansion, a time-invariant feedback linearization control is proposed as a fixed-gain linear version of the previously proposed nonlinear one. Furthermore, the conventional linear quadratic regulator (LQR) design is applied to facilitate the choice of the fixed-gain matrix. Refined controls to compensate the model uncertainty and their local stability analysis are provided. Extension of the continuous-time design results to discrete-time cases is also addressed. Numerical simulations for an example model demonstrate the effectiveness of the proposed continuous-time and discrete-time design results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society