Interval Analysis Method Research Articles

Load Identification Method Based on Interval Analysis and Tikhonov Regularization and Its Application

In order to study the dynamic force identification method of an end-plate pick of shearer, a dynamic force identification technique based on interval theory was proposed. The dynamic force interval identification model is established by describing and quantifying the identified parameters. By using the interval analysis method of the first-order Taylor expansion, the dynamic force identification is transformed into two kinds of deterministic inverse problems at the midpoint of the uncertain parameter and its gradient identification. The Tikhonov regularization method is used to solve two kinds of deterministic problems, and the upper and lower boundaries of dynamic force of the end-plate pick are determined. The results show that the deviations between the identified dynamic force and the actual dynamic force are basically within 2% and 5%, and the average uncertainties are up to 7% and 10%. Therefore, the proposed method can effectively determine the upper and lower boundaries of dynamic force of the end-plate pick, improve the solving efficiency, and provide a new research method for studying the coal rock mechanism of the pick cutting load.

Differentiation of Heart Failure Patients by the Ratio of the Scaling Exponents of Cardiac Interbeat Intervals.

Heart failure (HF) is one of the most frequent heart diseases. It is usually characterized with structural and functional cardiac abnormalities followed by dysfunction of autonomic cardiac control. Current methods of heartbeat interval analysis are not capable to differentiate HF patients and some new differentiation of HF patients could be useful in the determination of the direction of their treatment. In this study, we examined potential of the ratio of the short-term and long-term scaling exponents (α1 and α2) to separate HF patients with similar level of reduced cardiac autonomic nervous system control and with no significant difference in age, left ventricular ejection fraction (LVEF) and NYHA class. Thirty-five healthy control subjects and 46 HF patients underwent 20 min of continuous supine resting ECG recording. The interbeat interval time series were analyzed using standardized power spectrum analysis, detrended fluctuation analysis method and standard Poincaré plot (PP) analysis with measures of asymmetry of the PP. Compared with healthy control group, in HF patients linear measures of autonomic cardiac control were statistically significantly reduced (p < 0.05), heart rate asymmetry was preserved (Cup > Cdown, p < 0.01), and long-term scaling exponent α2 was significantly higher. Cluster analysis of the ratio of short- and long-term scaling exponents showed capability of this parameter to separate four clusters of HF patients. Clusters were determined by interplay of presence of short-term and long-term correlations in interbeat intervals. Complementary measure, commonly accepted ratio of the PP descriptors, SD2/SD1, showed tendency toward statistical significance to separate HF patients in obtained clusters. Also, heart rate asymmetry was preserved only in two clusters. Finally, a multiple regression analysis showed that the ratio α1/α2 could be used as an integrated measure of cardiac dynamic with complex physiological background which, besides spectral components as measures of autonomic cardiac control, also involves breathing frequency and mechanical cardiac parameter, left ventricular ejection fraction.

Error Space Estimation of Three Degrees of Freedom Planar Parallel Mechanisms

Abstract This paper develops a geometric method to estimate the error space of 3-DOF planar mechanisms with the Minimum Volume Ellipsoid Enclosing (MVEE) approach. Both the joint clearances and actuator errors are considered in this method. Three typical planar parallel mechanisms are used to demonstrate. Error spaces of their serial limbs are analyzed. Thereafter, limb-error-space-constrained mobility of the manipulator, namely, the manipulator error space is analyzed. The MVEE method has been applied to simplify the constraint modeling. A closed-form expression for the manipulator error space is derived. The volume of the manipulator error space is numerically estimated. The approach in this paper is to develop a geometric error analysis method of parallel mechanisms with clear algebraic expressions. Moreover, no forward kinematics computations have been performed in the proposed method, in contrast to the widely used interval analysis method. Although the estimated error space is larger than the actual one, because the enclosing ellipses enlarge the regions of limb error space, the method has an attractive advantage of high computational efficiency.

An inverse method for distributed dynamic load identification of structures with interval uncertainties

A time-domain based method for distributed dynamic load identification is proposed in this study considering unknown-but-bounded uncertainties in structural systems. The spatiotemporal dynamic load is further approximated by Chebyshev orthogonal polynomial in time history. Thus, the problem of distributed load reconstruction may be converted into the issue of polynomial coefficient calculation at each sampling time by utilizing a series of dynamic analysis. In accordance with the practical engineering, the acceleration response is used as the system input. In terms of the uncertainty quantification problems, the interval analysis method based on Taylor expansion (IAMBTE) is systematically developed to accomplish the envelope interval of identified load. To facilitate the analysis, two kinds of load are required to be identified, among which the nominal value of the identified load may be straightforwardly achieved through the inverse system, whereas the interval boundaries should be settled by the interval propagation analysis. Eventually, two numerical examples are investigated to demonstrate the efficiency and precision of the developed methodology.

Robust design methodologies to the adaptive cycle engine system performance: preliminary analysis

Abstract Uncertainty is a common and unavoidable problem for the design of a new energy system. As a potential power system for the next generation aircraft, the Adaptive Cycle Engine (ACE) has attracted attention in many aero-engine developed countries. However, the introduction of new technology and the requirement of wider adaptability bring more uncertainties to the ACE system performance design process. To better utilize the advantage of the ACE performance, a robust design methodology should be applied. This paper analyzes the source of uncertainties, studies the method of addressing these uncertainties, and presents a design architecture to direct the robust design. The results show that the increase of knowledge and application of other universal design theories are helpful to eliminate the epistemic uncertainties. The uncertainty quantification methods, such as probabilistic method and interval analysis method, can be applied to address the aleatory uncertainties. Under the direction of a wheel-shaped design architecture, the robust design of ACE system performance can be completed through the further study at different design levels and the iteration between them. This design methodology can be applied to the design process of other energy systems with many new technical characteristics.

Robust kinematics design of MacPherson suspension based on a double-loop multi-objective particle swarm optimization algorithm

In the traditional design of a vehicle suspension system, mechanical parameters, such as spring stiffness and tire radial stiffness, are determined before optimizing the coordinates of suspension hard-points based on handling and stability. However, the mechanical parameters of vehicle suspension systems vary with repeated changes in ambient temperature and loading conditions. Consequently, the original hard-point coordinates cannot guarantee satisfactory handling and stability after long-term vehicle use. To address this issue, an A0 class vehicle with a MacPherson suspension was modeled in ADAMS/Car and was validated against the results of field tests. The relationships between the maximum absolute values of front wheel alignment parameters during parallel wheel travel analysis and the coordinates of suspension hard-points, spring stiffness, and tire radial stiffness were determined using support vector regression. Next, a multi-objective optimization function was formulated based on the interval analysis method. Finally, a novel double-loop multi-objective particle swarm optimization algorithm was designed for global optimization of hard-point coordinates. The ADAMS simulation results indicated that, compared with the vehicle with the original hard-point coordinates, the double-loop multi-objective particle swarm optimization algorithm, the traditional multi-objective particle swarm optimization algorithm, and the genetic algorithm can effectively reduce the variation ranges of front wheel alignment parameters, regardless of the values of the mechanical parameters. It also showed that the double-loop multi-objective particle swarm optimization algorithm performs better than the traditional multi-objective particle swarm optimization and the genetic algorithm with both the original and changed mechanical parameters. Except for an increase in the variation range of the caster angle by 9.24–10.69%, the variation ranges of the toe angle, camber angle, and kingpin inclination angle under various mechanical parameters using the double-loop multi-objective particle swarm optimization algorithm were observed to be reduced by 50.45–79.39%, 1.84–4.24%, and 2.11–2.96%, respectively. Last but not least, the proposed double-loop multi-objective particle swarm optimization algorithm provided better handling, stability, and ride comfort values than the traditional multi-objective particle swarm optimization algorithm and the genetic algorithm.

Study on the small-scale effect on wave propagation characteristics of fluid-filled carbon nanotubes based on nonlocal fluid theory

In this article, Timoshenko’s beam model is established to investigate the wave propagation behaviors for a fluid-conveying carbon nanotube when employing the nonlocal stress–strain gradient coupled theory and nonlocal fluid theory. The governing equations of motion for the carbon nanotube are derived. The small-scale influences induced by the nanotube are simulated by nonlocal and strain gradient effects, and the scale effect induced by fluid flow is first investigated applying nonlocal fluid theory. Numerical results obtained by solving the governing equations indicate that the nonlocal effect induced by the nanotube leads to wave damping and a decrease in stiffness, while the strain gradient effect contributes to wave promotion and an enhancement in stiffness. The scale effect caused by the inner fluid only leads to a decay for a high-mode wave since there is no influence from fluid flow on the low-mode wave. The numerical solution is validated by comparing with Monte Carlo simulation and interval analysis method.

Fault Tree Analysis of an Airborne Refrigeration System Based on a Hyperellipsoidal Model

The airborne refrigeration system is an important subsystem in aircraft environmental control. The increasing complexity of this subsystem means that cabin temperature issues are sufficiently complex to exhibit various faults during operation. In this study, a fault tree analysis is applied to establish a model based on the fault mechanisms of an airborne refrigeration system to improve its reliability. Considering that the probability of occurrence of all basic event faults in the interval of operation is extremely low and that the probability decreases as the number of basic events increases, a new method of fault tree interval analysis is proposed. Based on the hyperellipsoidal description of uncertain variables, the reliability evaluation of the system can be realized using a two-layer Monte Carlo sampling method. Accordingly, the failure modes that are most likely to occur are determined by importance analysis, providing conclusions that can help to improve system reliability and performance.

Сравнение методов интервального анализа и стандартных статистических методов в задаче оценивания экспериментальных данных с неопределённостью

Сравнение методов интервального анализа и стандартных статистических методов в задаче оценивания экспериментальных данных с неопределённостью

Interval differential evolution with dimension-reduction interval analysis method for uncertain optimization problems

• A new optimization method is proposed to directly solve uncertain problems with interval parameters. • Dimension-reduction interval analysis calculates intervals of objective function and constraints caused by uncertainty. • The satisfaction value of interval possibility degree model utilizes to handle the uncertain constraints. • The interval preferential rule directly realizes the ranking of different design vectors. • The computational accuracy and efficiency of proposed method is verified. A constrained nonlinear interval optimization method under the framework of differential evolution algorithm is developed to solve the uncertain structural optimization problems with interval uncertainties. The proposed method is a direct optimization method based on the interval differential evolution and dimension-reduction interval analysis. The interval preferential rule based on the satisfaction value of interval possibility degree model is used to realize the direct interval ranking of different design vectors. At each evolutionary generation, the outer optimizer by differential evolution optimizer searches for the best solution within the design space. The dimension-reduction interval analysis is employed to calculate the intervals of objective and constraints for each design vector in the inner layer. This operation transforms the original nesting optimization problem into a single loop one which improves the computational efficiency of the proposed method. Finally, the effectiveness of the presented direct method is verified by two numerical examples and an engineering application.

ОПРЕДЕЛЕНИЕ ПРЕДЕЛЬНОЙ АБСОЛЮТНОЙ И ОТНОСИТЕЛЬНОЙ ПОГРЕШНОСТИ ВЫЧИСЛЕНИЯ ОСНОВНЫХ ТИПОВ ИНДЕКСОВ ПОТРЕБИТЕЛЬСКИХ ЦЕН

Цель работы: определение предельной абсолютной и относительной погрешности, возникающей при вычислении основных типов индексов потребительских цен: индекса Ласпейреса, индекса Фишера, индекса Джевонса, индекса Уолша, индекса Лоу, индекса Пааше. Использованные методы: линеаризация функции случайных величин, методы теории погрешностей приближенных вычислений, методы интервального анализа. Результаты работы: Получены, используя методы теории погрешностей приближенных вычислений, выражения для вычисления значений предельной абсолютной и относительной погрешности, возникающей при вычислении основных типов индексов потребительских цен. Приведены выражения для вычисления значений основных типов индексов потребительских цен методами интервального анализа с представлением чисел в системе центр-радиус. Для условного примера выполнено вычисление интервальных значений анализируемых индексов и определена ширина интервала неопределённости, возникающего при их вычислении. Показано, что при вычислении на одних и тех же данных имеет место следующее упорядочение ИПЦ по возрастанию значений центра интервала неопределенности: индекс Ласпейреса, индекс Фишера, индекс Джевонса, индекс Уолша, индекс Лоу, индекс Пааше. Показано, что при вычислении на одних и тех же данных имеет место следующее упорядочение ИПЦ по возрастанию значений относительной ширины доверительного интервала: индекс Джевонса, индекс Фишера, индекс Паше, индекс Ласпейреса, индекс Уолша, индекс Лоу. Тема дальнейших исследований: разработка и согласование способов вычисления погрешностей, возникающих при вычислении ИПЦ, является важной задачей для дальнейших исследований. Отсутствие её общепринятого решения усложняет получение корректных выводов об изменениях в состоянии изучаемых экономических систем.

Multiobjective Optimization Approach for Coordinating Different DG from Distribution Network Operator

Integrating with analysis of uncertainties, this paper presented a multiobjective optimization approach for coordinating different DG from the perspective of Distribution Network Operator (DISOPER). Aiming to three uncertain factors including fuzzy variable, random variable, and interval variable, the information entropy and interval analysis methods are adopted to construct multistate models of multisource uncertainty. The information entropy method is to convert fuzzy variable into equivalent random variable. Interval analysis method is to transform random variables into interval variables by setting a confidence level. Then plenty of simulation analysis based on the small probability event and expectation are investigated to reduce the computational burden and eliminate invalid computation. Subsequently, multiobjective formulations based on multistate are built by analyzing systematical power loss, voltage quality, reliability, and environment change provide some reference for DISOPER in dealing with access of privately owned DG units. Furthermore, based on network topology analysis and modified nondominated sorting genetic algorithm (NSGA), a combinatorial optimization method is proposed to reduce search space and solve the constructed formulations efficiently. Simulations are carried out on IEEE 37-bus systems and results are presented and discussed.

КОМПЛЕКСНЕ ОЦІНЮВАННЯ ЕФЕКТИВНОСТІ ВСТАНОВЛЕННЯ ДОДАТКОВИХ ДЖЕРЕЛ РЕАКТИВНОЇ ПОТУЖНОСТІ У РОЗПОДІЛЬНИХ ЕЛЕКТРИЧНИХ МЕРЕЖАХ

To develop a method for assessing the effectiveness of reactive power sources (RPS) in electrical grids. The efficiency indicator was obtained as a result of combining the method of assessing the quality of the functioning of electrical grids on the basis of Markov networks and the method of interval analysis of energy losses. The efficiency index improves the formulation of the problem of optimizing the connection of the RPS to electrical networks. It improves the quality of this task. A new method of forming an indicator of the guaranteed effectiveness of the installing the RPS is obtained. It provides an unambiguous comprehensive evaluation of efficiency, takes into account the reliability of the electrical network, the quality of the voltage in its nodes, the accuracy of calculating the released and lost electricity. The method makes it possible to construct more efficient algorithms for optimizing the connection of RPS. They allow excluding from the search area optimal solutions "indefinite" fragments of the network without the help of an analyst. Estimation of the real effect is difficult or impossible for such fragments due to frequent failures or lack of monitoring devices. This correction of the search area improves the quality of the solution.

Thermo-electro-mechanical vibration analysis of nonlocal piezoelectric nanoplates involving material uncertainties

This study aims to detect the influence of material uncertainties on the free vibration of nonlocal piezoelectric nanoplates under thermo-electro-mechanical loadings. The size-dependent governing equations are derived based on the nonlocal theory and Hamilton’s principle, and Navier method is used for the solution. Considering inadequate experimental data, uncertain-but-bounded parameters are used to quantify the uncertain nanomaterial properties. Based on the interval analysis theory, a novel hull iterative algorithm (HIA) is proposed to evaluate the thermo-electro-mechanical vibration behavior of piezoelectric nanoplates. The presented method is compared with Monte Carlo method and sensitivity based interval analysis method, and well agreements are achieved. A sensitivity analysis is performed to identify the most dominant uncertain parameters. Using the HIA, parametric studies are carried out to explore the combined effects of material uncertainties and temperature change, external electric voltage, biaxial force as well as nonlocal parameter on the natural frequency of piezoelectric nanoplates.

Integer Interval Value of Milne’s Predictor and Milne’s Corrector Method for First Order ODE

In this paper A new approaches to solve the approximate solution of the initial value problem for the first order ordinarydifferential equations and the solution can be used to compute y numerically specified the value of near to in the interval analysis method and also used Milne’s predictor and corrector for interval. In interval method gives a more accurate theapproximate solution of life situation and numerical illustration are given

An interval analysis method for fatigue crack growth life prediction with uncertainty

This paper presents an interval analysis method for the fatigue crack growth (FCG) life prediction with uncertain-but-bounded parameters. The method can effectively address the difficulty in the uncertainty modeling of the FCG life due to the lack of experimental samples. The uncertain parameters are quantified by interval variables which are classified into correlated and uncorrelated ones. All the intervals are uniformly transformed into an independent standard space, in which the standard intervals are then divided into subintervals to deal with the large uncertainty problem involved in the calculation of the FCG life. The subinterval analysis method is then employed to predict the upper and lower bounds of the FCG life. The proposed method is well validated by the numerical examples and experimental test. It is demonstrated that this method possesses an acceptable computational accuracy even there exist large uncertainty and correlation in parameters. Moreover, the correlation coefficient between crack propagation rate constants shows a considerable effect on the predicted FCG life.

An Iterative Dimension-Wise Approach to the Structural Analysis with Interval Uncertainties

The structural analysis is inevitably surrounded with uncertainties and the interval analysis is a favorable method if insufficient data is available on uncertainties. The accuracy of current interval analysis methods including the interval perturbation method (IPM), subinterval perturbation method (SIPM) and dimension-wise approach (DWA) depends on a reference point (RP), e.g., the expansion point in IPM, for some problems due to ignoring the co-operative effects of multiple interval inputs on the response. To this end, an iterative dimension-wise approach (IDWA) is proposed. Either the minimal or maximal input vector of the response is identified as an RP by a global update in which a novel RP is dimension-wisely assembled by the minimal or maximal points of all sectional curves of the response surface at a previous RP through a local update. The interval response is calculated by deterministic solvers at the minimal and maximal input vectors. An acoustic analysis problem is studied eventually to validate the effectiveness of the proposed method, from which conclusions are drawn.

Application of Interval Analysis Methods for NPP Power Units Safety and Reliability Assessment

The study analizes the requirements for instrumentation and control systems of NPP units, which are currently under modernization in accordance with the state program for long-term operation of Ukrainian NPP units. The requirements established in the documents that regulate reliability and quality of performing functions by these systems emphasize the importance of developing methods to improve the accuracy of parameter measurements, methods to increase reliability of assessing safety and reliability indexes of NPP unit operation, as well as the accuracy of personnel decisions in the presence of many uncertainties and inaccuracies in the description of initial data that affect implementing necessary management actions by personnel. The paper presents the examples of the impact of initial data uncertainty on solving the tasks of safety and reliability assessment for NPP units. The study analyzes existing methods to account and describe initial data uncertainty and modeling uncertainty in the tasks of safety and operational reliability assessment for NPP units. The possibility of applying mathematical methods and tools of interval analysis to improve reliability of solving the issues of assessing safety, reliability and efficiency of NPP unit operation under conditions of initial data uncertainty was justified. The study presents the basic principles of interval calculations, which allow considering inaccuracies when specifying the initial data, parameter uncertainties and multi-mode nature of functioning of such complex technical systems as NPP units. The prospects of applying the main theoretical provisions of interval analysis and their implementation in the form of numerical methods and models to solve current issues of assessing safety, reliability and efficiency of accident-free operation of NPP units in compliance with the state program for long-term operation were specified.

Error estimation and sensitivity analysis of the 3-UPU translational parallel robot due to design parameter uncertainties

This work deals with the estimation and the sensitivity analysis of the 3-UPU parallel robot error. Based on the Newton–Euler formalism, the robot dynamic model is given in a closed form. This model is validated by the software ADAMS. Using the interval analysis method, a new algorithm is proposed, which estimates the errors in the motion of the end-effector and the errors in the actuator forces as a function of the design parameters uncertainties. The obtained results show that the kinematic errors are minimal at the workspace center. Moreover, these errors increase as the platform moves along the vertical axis. It is also shown that kinematic errors in the actuator joints are the most influential parameters on the manipulator accuracy. Therefore, using actuators with a higher accuracy can highly reduce the errors in motion of the platform.

Uncertainty analysis for wave dispersion behavior of carbon nanotubes embedded in Pasternak-type elastic medium

At nano-scale, the molecular defects and manufacturing tolerances of carbon nanotubes (CNTs) both would introduce inevitable uncertainties. In this paper, a theoretical framework based on interval analysis method is presented to study the wave-dispersion behavior of CNTs embedded in a Pasternak-type elastic medium with uncertain material properties. Different from probabilistic analysis approach, the material properties of CNTs are considered as uncertain-but-bounded variables, which can overcome the massive information required to set up the probabilistic distribution function. Based on the nonlocal Timoshenko beam theory, the wave-dispersion relationship of the embedded CNTs is derived. The upper and lower bounds of the wave frequencies are predicted under different elastic foundation and size scale parameters. It is demonstrated that the uncertain material properties have considerable effect on the wave-dispersion behavior of CNTs, and the size scale and elastic medium will also affect the wave-dispersion response of CNTs significantly. The Monte Carlo simulation is also presented to validate the proposed interval analysis method. The numerical results presented in this paper provide useful guidelines for employing CNTs as reinforced fibers to enhance the conventional composite materials.