Interval Analysis Method Research Articles

Experimental Validation of a Guaranteed Nonlinear Model Predictive Control

This paper combines the interval analysis tools with the nonlinear model predictive control (NMPC). The NMPC strategy is formulated based on an uncertain dynamic model expressed as nonlinear ordinary differential equations (ODEs). All the dynamic parameters are identified in a guaranteed way considering the various uncertainties on the embedded sensors and the system’s design. The NMPC problem is solved at each time step using validated simulation and interval analysis methods to compute the optimal and safe control inputs over a finite prediction horizon. This approach considers several constraints which are crucial for the system’s safety and stability, namely the state and the control limits. The proposed controller consists of two steps: filtering and branching procedures enabling to find the input intervals that fulfill the state constraints and ensure the convergence to the reference set. Then, the optimization procedure allows for computing the optimal and punctual control input that must be sent to the system’s actuators for the pendulum stabilization. The validated NMPC capabilities are illustrated through several simulations under the DynIbex library and experiments using an inverted pendulum.

Uncertainty optimization of pure electric vehicle interior tire/road noise comfort based on data-driven

Without the masking effect of engine noise, tire/road (TR) noise is increasingly becoming noticeable in pure electric vehicles (PEVs) and represents a primary concern for drivers and passengers. Currently, numerous works have studied PEV motor and powertrain noises, but few studies have investigated the tire/road structure-borne (TRS) noise of PEVs. Therefore, in this paper, the sound quality of TRS noise is studied through objective and subjective evaluations, and the group paired comparison method (GPCM) is proposed to evaluate a large noise sample set. The correlation between sound quality metrics and the subjective annoyance of TRS noise is analyzed, and the contribution of chassis dynamic parameters to the TRS noise of PEVs is quantified. In addition, because of the nonlinear transfer and complex characteristics of TRS noise, the expected design results will easily be affected by material, processing and manufacturing uncertainties, which are difficult to process with conventional optimization methods. Therefore, to overcome the uncertainty problem, an improved interval analysis method (IIAM) is proposed. This method is used to optimize the interior TR sound quality of PEVs while treating riding comfort as a constraint. The optimized result of the IIAM is compared with that of the advanced genetic algorithm (GA) optimization method. Through real vehicle verification, the proposed IIAM outperforms the GA method in terms of accuracy and robustness.

Research on multi-interval coupling optimization of ship main dimensions for minimum EEDI

In order to improve energy-saving effect of the YUKUN ship, minimize the value of the Energy Efficiency Design Index (EEDI), while considering the impact of uncertain factors. In this research, interval analysis method is used in uncertain optimization, and main dimensions of interval optimization research for EEDI are carried out. First, multi-island genetic algorithm (MIGA), particle swarm optimization (PSO) and adaptive simulated annealing algorithm (ASA) optimization methods are used for deterministic optimization, to analyze the influence of different optimization algorithms on the optimization results and to obtain the best deterministic optimization scheme. Second, considering uncertain factors—flow velocity VW, ship's wet surface area S and total resistance coefficient Ct—the interval analysis method is used to determine the main dimensions of the optimization problem and to analyze the influence of single interval and multi-interval coupling on the EEDI. Analysis shows that an increased number of uncertain factors will increase EEDI while decrease the energy saving effect, as will the coupling effect between uncertain factors. Final value of EEDI obtained by multi-interval coupling optimization, however, can be more realistic and guarantees the reliability of the scheme of the ship's main dimensions, and also be feasible as is still below the required baseline.

Computation of controlled invariant sets for discrete-time switched nonlinear systems with time-delay

We consider the concept of set invariance for discrete-time switched nonlinear systems with time-delays, referred to here as D-invariance. Specifically, the existence of controlled invariant sets for nonlinear switched systems with time-delay is proved by defining a suitable predecessor operator. As the main contribution, we use the interval analysis method to iteratively computing the inner-approximation of a controlled D-invariant set, which is a key step towards invariance control of switched nonlinear systems with time-delay. The sequence of the approximated controlled D-invariant sets are obtained by considering all possible delays within a given bounded region of the state space, so that a D-invariant set and controller that are independent of time-delay are obtained. Finally, two examples are given to show the performance of this method.

A Newton iteration‐based interval analysis method for nonlinear structural systems with uncertain‐but‐bounded parameters

AbstractIn this article, the propagating effect of uncertainties in nonlinear structural systems is investigated. Via considering uncertain‐but‐bounded parameters as interval variables, a Newton iteration‐based interval analysis method (NI‐IAM) is proposed to quantify the uncertainty of the nonlinear structural response, namely predicting the upper and lower bounds of the response. In the proposed method, a nonlinear structural system with uncertain interval parameters is described as a series of interval nonlinear equations. With Taylor series expansion, an interval iteration scheme is established to successively approximate the upper and lower bounds of the response based on the Newton method. The structural response bounds in every iteration step are updated by solving linear interval increment determination equations via the Lagrangian multiplier method. Thereout, an uncertain nonlinear problem is simplified to a series of uncertain linear issues. The convergence of the proposed method is further discussed in this article. Numerical examples are provided to demonstrate the effectiveness and applicability of the proposed method by contrast with several existing methods. Moreover, the compatibility between numerical results of the perturbation‐based probabilistic method and the proposed method is then studied and verified.

FATIGUE LIFE ANALYSIS OF PANELS UNDER WIND LOAD BASED ON NON-PROBABILISTIC SET THEORY

In the existing fatigue analysis methods, structural material parameters and geometric dimensions are usually considered to be deterministic variables. For real structures, however, the structural parameters are bounded but uncertain. It is unsafe to estimate the fatigue life of a structure according to the nominal values. In this paper, we apply the interval analysis method based on non-probabilistic set theory to estimating the roof fatigue life under fluctuation wind load. Considering the thickness and elastic modulus of large-span roofs under bounded-but-uncertain circumstances, we calculate the dynamic response range of long-span roof structures under fluctuating wind load by the interval finite element method and interval dynamic response analysis method. The time history curve of the interval stress of large-span roofs is obtained through the constitutive equation and stress. The amplitudes of the interval stress are calculated by the rain-flow counting method. Considering the S-N curve of the roof material, we estimate the fatigue life interval of large-span roofs by Miner's law.

Dynamics of a flexible multi-tethered satellite formation in a Halo orbit with uncertain parameters

The nonlinear dynamics of a flexible rotating multi-tethered satellite formation in a Halo orbit with uncertain-but-bounded parameters are examined in this paper via an interval analysis method. In Hill's problem, equations of motion for the focused system are derived with the flexibility and extensibility of tethers involved. Not limited to the system of concern, the developed dynamic model could be generally applied to any N-body multi-tethered satellite system of arbitrary configurations. In addition, the space environmental perturbation of the thermal effect is considered in the dynamics modeling to improve accuracy. Based on interval descriptions, a bounds estimation problem is formulated for the station keeping phase of the system with uncertain interval parameters. Furthermore, an interval method of high efficiency, which utilizes the Chebyshev series for approximation, is presented to solve this problem by only using the lower and upper bounds of the uncertain parameters and avoiding a global search in numerical results. Finally, numerical simulations have been performed to demonstrate the effectiveness and feasibility of the proposed method and to evaluate the influences of the uncertain interval parameters on the dynamic behaviors of the system.

A Proposed Bearing Load Identification Method to Uncertain Rotor Systems

Bearings are considered as important mechanical components in rotating machines. Bearing load is used as an indication of monitoring rotor system health, but there are interval and probability uncertain parameters in the process of obtaining bearing load from the rotor system. A bearing load strip enclosed by two bounding distributions is then formed, rather than a single distribution that we usually obtain through the load identification method for a deterministic rotor system. In this paper, a computational inverse approach that combines the interval and perturbation analysis method with regularization is presented to stably identify bearing load strip. Using an interval analysis method, a calculated transient response of the rotor structure only subjecting to the bearing load can be approximated as a linear function of the interval parameters in the rotor system. The perturbation analysis method based on Taylor expansion is used to transform the problem of the bearing load identification involving in probability parameters into two kinds of certain inverse problem, namely, the bearing load identification combining the mean value of uncertain parameters with calculated transient response function and the sensitivity identification of bearing load to each probability parameter. Regularization is used to overcome ill-posedness of bearing load identification arising from the noise-contaminated observed response. A rotor system with two bearings is investigated to demonstrate the effectiveness and accuracy of the presented method.

Mooring Damage Identification of Floating Wind Turbine Using a Non-Probabilistic Approach Under Different Environmental Conditions

This paper discusses the damage identification in the mooring line system of a floating wind turbine (FWT) exposed to various environmental loads. The proposed method incorporates a non-probabilistic method into artificial neural networks (ANNs). The non-probabilistic method is used to overcome the problem of uncertainties. For this purpose, the interval analysis method is used to calculate the lower and upper bounds of ANNs input data. This data contains some of the natural frequencies utilized to train two different ANNs and predict the output data which is the interval bounds of mooring line stiffness. Additionally, in order to reduce computational time and more importantly, identify damage in various conditions, the proposed method is trained using constant loads (CL) case (deterministic loads, including constant wind speed and airy wave model) and is tested using random loads (RL) case (including Kaimal wind model and JONSWAP wave theory). The superiority of this method is assessed by applying the deterministic method for damage identification. The results demonstrate that the proposed non-probabilistic method identifies the location and severity of damage more accurately compared to a deterministic one. This superiority is getting more remarkable as the difference in uncertainty levels between training and testing data is increasing.

Uncertainty Handling in Structural Damage Detection via Non-Probabilistic Meta-Models and Interval Mathematics, a Data-Analytics Approach

Recent advancements in sensor technology have resulted in the collection of massive amounts of measured data from the structures that are being monitored. However, these data include inherent measurement errors that often cause the assessment of quantitative damage to be ill-conditioned. Attempts to incorporate a probabilistic method into a model have provided promising solutions to this problem by considering the uncertainties as random variables, mostly modeled with Gaussian probability distribution. However, the success of probabilistic methods is limited due the lack of adequate information required to obtain an unbiased probabilistic distribution of uncertainties. Moreover, the probabilistic surrogate models involve complicated and expensive computations, especially when generating output data. In this study, a non-probabilistic surrogate model based on wavelet weighted least squares support vector machine (WWLS-SVM) is proposed to address the problem of uncertainty in vibration-based damage detection. The input data for WWLS-SVM consists of selected wavelet packet decomposition (WPD) features of the structural response signals, and the output is the Young’s modulus of structural elements. This method calculates the changes in the lower and upper boundaries of Young’s modulus based on an interval analysis method. Considering the uncertainties in the input parameters, the surrogate model is used to predict this interval-bound output. The proposed approach is applied to detect simulated damage in the four-story benchmark structure of the IASC-ASCE SHM group. The results show that the performance of the proposed method is superior to that of the direct finite element model in the uncertainty-based damage detection of structures and requires less computational effort.

Taylor Expansion and Matrix-Based Interval Analysis of Linear Arrays With Patch Element Pattern Tolerance

In this study, a novel interval analysis (IA) approach for linear arrays with element pattern tolerance is proposed. This study differentiates itself from the classic interval analysis (CIA) method because it focuses on the effects of the array element (e.g., patch) pattern tolerance on the antenna array pattern; the tolerance may be caused by fabrication errors in the element. The closed-form pattern expressions of the element (e.g., patch) and array are deduced by the interval arithmetic method. To mitigate the interval extension (also referred to as the overestimation problem) caused by the dependency problem, Taylor expansion and matrix-based interval analysis (TMIA) methods are proposed and implemented in this study. A set of numerical examples is reported and analyzed to indicate the effectiveness of the proposed TMIA approach with the results of Monte Carlo (MC) and CIA methods, as well as to indicate its potential capabilities and advantages in the actual application of industrial antenna arrays.

The influence of mechanical uncertainties on the free vibration of functionally graded graphene-reinforced porous nanocomposite shells of revolution

This paper is concerned with the analysis of uncertainty propagation in the free vibration response of functionally graded porous (FGP) nanocomposite shells of revolution reinforced with graphene platelet (GPL). In this regard, the effects of uncertainties in the material properties as well as the geometry of the reinforcements on the free vibration responses of the FGP-GPL shells with different types of Gaussian curvature are assessed. The propagation of uncertainty and frequency sensitivity of the shells of revolution are examined for various GPL dispersions and porosity distributions. For this purpose, the interval analysis method which is an appropriate technique for the uncertainty analysis of systems with bounded uncertainties is utilized. In order to solve the equations of motion based on the higher-order shear deformation theory and to obtain the free vibration responses of the shells of revolution, the Fourier Differential Quadrature (FDQ) technique is employed. According to the analysis results, the shells’ type of curvature has a significant effect on the trend of uncertainty propagation and the sensitivity of the vibration frequencies in terms of various sources of uncertainty. Moreover, the results show that the uncertainty propagation and sensitivity are different in the frequencies corresponding to different modes of vibration.

The propagation of uncertainty in the geometrically nonlinear responses of smart sandwich porous cylindrical shells reinforced with graphene platelets

This paper is concerned with the analysis of uncertainty propagation in the nonlinear dynamic responses of smart sandwich cylindrical shells under rectangular, sine, and exponential loads. The sandwich shells are composed of a functionally graded porous (FGP) core reinforced with nanocomposite graphene platelets (GPLs) and surrounded by two piezoelectric layers. Various sources of uncertainties related to the material properties, the geometry of the reinforcements, and piezoelectric parameters are considered in this study. The uncertainty propagations in the nonlinear dynamic responses of the shells are examined for different variations of GPL dispersions and porosity distributions in the core. For this purpose, three different distributions for porosity and three different dispersions for graphene platelets have been considered in the thickness direction. In order to investigate the uncertainty propagation in the responses of the shells, the interval analysis method which is an appropriate technique for the uncertainty analysis of systems with bounded uncertainties is utilized. The Halpin-Tsai model is used to find the effective properties of the GPL-reinforced materials. The equations of motion are derived based on the higher-order shear deformation theory and Sanders’ nonlinear theory. In order to solve the equations of motion and to obtain the dynamic responses of the shells, the Fourier differential quadrature (FDQ) technique is employed. According to the analysis results, the sensitivities of the responses of internal moment, displacement, and internal force have the highest values relative to the uncertainty sources, respectively, whereas the uncertainty percentages for the responses of displacement, internal moment, and internal force respectively have the highest values for the examined loads.

Bounds for uncertain structural problems with large-range interval parameters

A novel bivariate interval function decomposition method is proposed and applied to predict the bounds of structural response with large-range interval parameters. When the existing interval methods solve large uncertainty problems, either the calculation accuracy is poor or better accuracy is often achieved at the cost of more computational effort. To overcome this drawback, the bivariate interval function decomposition (BIFD) is first constructed for the approximation of the original response function. The univariate and the bivariate points are substituted into the second-order Taylor expansion to derive BIFD; thus, the expression of BIFD contains only the one- and two-dimensional functions. Particularly, the response function is decomposed into the sum of multiple low-dimensional functions, and solving the bounds of multi-dimensional original response can be transformed into solving those of low-dimensional interval functions. Then, the sensitivity information of structural response with respect to uncertain parameters is utilized to save computational consumption. Finally, the precision and effectiveness of the method are validated by comparing it with the other six existing interval analysis methods through several numerical examples and engineering applications.

Methods of quantitative estimation and reduction of uncertainty of the accidental risk in fire explosive facilities

Abstract. Currently, ensuring the industrial safety of hazardous industrial facilities involves – along with conventional oversight – the risk-oriented approach that is significantly more flexible. The procedure of quantitative estimation of an accidental risk for hazardous industrial facilities is essentially one of the procedures of conformity assessment, as it includes the comparison of the risk indicators obtained by means of calculation (or expert assessment) with their standard values. The Aim of the paper is to define the problem of uncertainty that is associated with all the stages of quantitative estimation of an accidental risk, make a brief historical account, analyze its types and sources, describe the approaches employed as part of quantitative estimation of this uncertainty. Currently, it is accepted to identify the terminological, parametric and model types of uncertainty, whose examples are provided in the paper. Analysis shows that a fourth – computational – type should be added, whose contribution in many cases may be considerable. It is shown that, due to a number of circumstances, scalar numbers that are normally used for defining parameter values of the physical-mathematical models of failure processes are in reality mere indicators of the ranges of their value variation. Currently, uncertainties in the values of accidental risk parameters are accounted for using probabilistic and deterministic approaches, as well as fuzzy numbers. Methods. For the purpose of quantitative estimation of uncertainty, the paper employs the method of interval analysis. In the most general case, without using the hypothesis on the behaviour of a parameter value within the range of its possible variation, the parametric uncertainty can be defined with an interval number. In that case, all the required calculations are performed using interval methods. The natural (naive) version of interval analysis has a serious drawback that consists in an unjustified increase of the width of the interval number deduced by means of interval calculations, if one or more input parameters of the model enter into the calculation formula more than once, or the input parameters are functionally interdependent. Modern interval analysis employs methods allowing to alleviate this effect. They are briefly described in this paper. It is shown that if statistical information is available on the behaviour of parameter values within their variation intervals, the results of interval calculations of the accidental risk indicators can be significantly improved. The suggested method of reducing the computational uncertainty of quantitative estimation of the accidental risk in the interval setting is illustrated with a numerical example of risk indicator calculation for the “fireball” accident scenario. The paper sets forth the results of interval calculation of an individual accidental risk for an explosion and fire hazardous facility “reservoir with a flammable liquid” in three ways: a) naive; b) accounting for the effect of parameter correlation; c) additionally, accounting for available statistical information. Conclusions. Interval methods allow not only taking into consideration the presence of uncertainty in the accidental risk parameters, but evaluating it quantitatively. There are efficient methods of alleviating the negative.

Dynamic analysis of uncertain spur gear systems

Gear transmission systems are widely used in mechanical equipment. Due to manufacturing and processing errors, material defects, wear and external environments, there are uncertain parameters such as mass, mesh stiffness and support stiffness. Parametric uncertainty plays an important role in response predictions of the gear transmission systems. In order to evaluate reasonably the dynamic characteristics of the gear transmission systems, a dynamic model of an uncertain single-stage gear system is established. The dynamic responses of the bounded uncertain gear transmission system are investigated by an interval analysis method based on the Chebyshev inclusion function. A matrix form of the least square method is applied to improve the calculation efficiency of the interval approach. The investigation reveals the effects of different uncertain parameters on amplitude-frequency responses of the gear system. It is the first time to design and to implement a test rig of an uncertain single-stage spur gear transmission system. The experimental results verify the accuracy and the effectiveness of the theoretical analysis results. The results demonstrate that the parametric uncertainty may propagate in the gear transmission system and the propagation results in significant variations of the dynamic responses.

Approximations With Polynomial, Trigonometric, Exponential Splines of the Third Order and Boundary Value Problem

This paper is devoted to the construction of localapproximations of functions of one and two variables using thepolynomial, the trigonometric, and the exponential splines. Thesesplines are useful for visualizing flows of graphic information.Here, we also discuss the parallelization of computations. Someattention is paid to obtaining two-sided estimates of theapproximations using interval analysis methods. Particularattention is paid to solving the boundary value problem by usingthe polynomial splines and the trigonometric splines of the thirdand fourth order approximation. Using the considered splines,formulas for a numerical differentiation are constructed. Theseformulas are used to construct computational schemes for solvinga parabolic problem. Questions of approximation and stability ofthe obtained schemes are considered. Numerical examples arepresented.

Simultaneous determination of nitrate, chemical oxygen demand and turbidity in water based on UV-Vis absorption spectrometry combined with interval analysis.

In this paper, a new method for simultaneous determination of nitrate, COD and turbidity in water based on UV-Vis absorption spectrometry combined with interval analysis was studied. By analyzing the spectral absorption characteristics of nitrate, COD, and turbidity standard solutions and the mixtures of them, the absorption spectra in the range of 225-260nm, 260-320nm and 320-700nm were selected as the characteristic spectra of nitrate, COD and turbidity, respectively. Multiplicative scatter correction was employed to compensate turbidity of the absorption spectra of the mixture solutions in the wavelength range of 225-320nm. Then, the spectra after turbidity compensation in the range of 225-260nm was compensated for COD using the method of spectral difference. The original spectra in the range of 320-700nm, the turbidity compensated spectra in the range of 260-320nm, and the COD compensated spectra in the range of 225-260nm were analyzed by PLS algorithm in order to calculate the concentrations of nitrate, COD and turbidity in the mixture solutions. The results showed that this method could simultaneously and accurately determine the concentrations of nitrate, COD and turbidity. After interval analysis, all the correlation coefficients (R2) between the predicted values and the true values of nitrate, COD and turbidity were higher than 0.9, and root mean square error (RMSE) of predicted values were between 0.696 and 2.337.

Non-probabilistic method to consider uncertainties in structural damage identification based on Hybrid Jaya and Tree Seeds Algorithm

This paper proposes a novel non-probabilistic structural damage identification approach by developing a hybrid swarm intelligence technique based on Jaya and Tree Seeds Algorithm (TSA), taking into account the high-level uncertainties in the measurements and finite element modelling. The damage in structure is simulated as reduction of elemental stiffness, and structural damage identification is formulated as an optimization problem. To overcome the challenge for structural damage identification with a limited number of measurement data, an objective function based on the modal data and sparse regularization technique is defined. To make the optimization algorithm more powerful and robust, a hybridization of the K-means clustering based Jaya and TSA is proposed. Jaya algorithm is taken as the core in the hybridization. The clustering strategy is employed to replace solutions with low-quality objective values in the Jaya algorithm. Then the search strategy of the TSA is introduced into the best-so-far solution of each cycle. The proposed hybridization algorithm is termed as “ C-Jaya-TSA”. To enhance the capacity of the proposed algorithm to consider uncertainties, a non-probabilistic method is also integrated to calculate the interval bound (lower and upper bounds) of the elemental stiffness changes by using the interval analysis method. To better quantify the structural damage extents, Damage Measure Index (DMI) values are introduced for representing structural damage states. The DMI value can be viewed as a combination of deterministic stiffness reduction and the Possibility of Damage Existence (PoDE). Numerical benchmark functions, numerical studies and experimental investigations are conducted to verify the accuracy and performance of the proposed method. The identification results show that the developed C-Jaya-TSA integrated with the non-probabilistic interval analysis method is a promising tool to accurately identify the structural damage, even high-level uncertainties exist.

Interval analysis of rotor dynamic response based on Chebyshev polynomials

Uncertainty is extensively involved in the rotor systems of rotating machinery, which may cause an unstable vibrational response. To take the uncertainty into consideration for the uncertain rotor-bearing system, an improved unified interval analysis method based on the Chebyshev expansion is established in this paper. Firstly, the Chebyshev Interval Method (CIM) to calculate not only the critical speeds but also the dynamic response of rotor with uncertain parameters is introduced. Then, the numerical investigation is carried out based on the developed double disk rotor model and computation procedure, and the results demonstrate the validity. But when the uncertainty is sufficiently large to influence critical speeds, the upper and lower bounds are far from the actual bounds. In order to overcome the defects, a Bound Correction Interval analysis Method (BCIM) is proposed based on the Chebyshev expansion and the modal superposition. In use of the improved method, the bounds of the interval responses, especially the upper bound, are corrected, and the comparison with other methods demonstrates that the higher accuracy and a wider application range.