Interval Arithmetic Methods Research Articles

Estimation of the Resultant Expanded Uncertainty of the Output Quantities of the Measurement Chain Using the Discrete Wavelet Transform Algorithm

This paper discusses the role of the discrete wavelet transform algorithm in processing error signals present in the input quantities of the algorithm. In considering the error model of the measurement chain, the parameters of the error signals in the input quantities of the wavelet transform algorithm are estimated. Subsequently, in accounting for the algorithm’s properties, the parameters of its output values are determined, and the resulting uncertainty values of the output quantities of the measurement chain are estimated. The interval reduction arithmetic method is employed in the calculations for estimating the expanded uncertainty. All findings were validated through measurements conducted using the implemented measurement chain.

Calculation of harrington function (desirability function) values under interval determination of its arguments

Purpose of work. Development of proposals for desirability function calculation by interval arithmetic method. Obtained results. A connection was shown between multiple-criterial (vector) optimization problem and calculation of Harrington function (desirability function) values. Generalized desirability function was shown to be treated as a multiplicative convolution of partial desirabilities. Using golden section properties, a connection was discerned between hierarchy analysis methods and calculation of desirability function values. In this case the distribution of probable desirability function values among intervals determined in accordance with golden mean rule does coincide with values obtained by expert method. Application of golden proportion enables increasing the number of intervals; thus, it may prove convenient in improvement of Harrington function sensitivity. Conclusions. For partial desirability function in its explicit form such geometric characteristics were determined as values of tangent, curvature of curve and center of curvature coordinates. For a necessary averaged desirability system obtaining process control, such characteristics were determined in general and explicit form as partial desirability elasticity, generalized desirability elasticity relative to partial desirability, generalized desirability elasticity relative to a variable that has physical sense and respective dimension depending on particular subject area and affects partial desirability value. To determine the boundary value of one partial desirability being substituted for another partial desirability, the rate of substitution function and boundary rate of substitution elasticity function were established. Due to errors emerging in determination of desirability function values application of interval analysis methods was proposed. Application of interval numbers as presented in hyperbolic form was shown for calcutaion of partial and generalized desirability, A partial power series sum was determined relative to a variable presented in hyperbolic form. Suggestions. The method of calculation of Harrington function (desirability function) values at interval determination of arguments was shown to be eventually used in development of specifications and preliminary design of complicated engineering аnd organizational systems.

Dispatching optimization of city gas station district energy systems with multiple uncertainties based on an improved affine arithmetic method

According to the energy consumption characteristics of gas stations, the district energy system (DES) based on natural gas and renewable energy is a suitable form of energy supply. In the process of DES operation and dispatching, there are various uncertainties brought by the strong coupling of equipment and the fluctuation of internal and external surroundings, and these uncertain parameters are often correlated. In this paper, an improved affine arithmetic (AA) based method is proposed to solve the economic dispatching problem of DES with correlated uncertainties. Firstly, an optimal dispatching model considering economic and environmental factors is constructed. An affine algorithm is introduced and interval arithmetic (IA) is used to improve the problem of interval extension and high-order noise element. Taking a classical DES system as an example, the optimal dispatching of the system is analyzed using deterministic method, interval method and the proposed method. The application effect of the proposed method under different uncertainties scale and amplitude is also discussed. The calculation results in different scenarios show that compared with the traditional IA method, the proposed method can effectively reduce the width of the target value interval by about 13%, which can further contribute to the optimal operation decision.

Error analysis between two different fuzzy multiplication operations on Triangular Fuzzy Number

One of the basic contents of fuzzy mathematics is the mathematical operations of fuzzy numbers. But still some of the fuzzy mathematical operations are not determined. There are some operations between fuzzy numbers are exact and there are approximated operations. In this paper, a comparison has been presented between two different fuzzy multiplication operations. One of these two fuzzy multiplication operations is exact with using interval arithmetic cut method and the other one is an approximation method which uses a standard approximation. In the two different methods, the fuzzy numbers will be defined as triangular fuzzy numbers because the triangular fuzzy number is the most popular one between the other shapes of fuzzy numbers. A numerical example has been presented with graphical result to show the difference between them and to analyze the error.

A novel approach for transient stability using interval arithmetic and optimization methods

In this article, a new approach is proposed to include parametric uncertainties in the analysis of the transient stability of power systems by using intervals. Two methodologies based on the interval uncertainties were developed and implemented: (i) a robust extension of the potential energy boundary surface method combined with interval arithmetic and (ii) an optimization formulation for the transient stability problem. The interval uncertainties enable one to obtain closed-form expressions for stability assessment of uncertain systems, allowing a faster analysis. The results show that both methodologies can be used to find an accurate estimate of the critical clearing time for uncertain power systems more quickly than using the classical Monte Carlo approach.

A Profit Maximisation Solid Transportation Problem Using Genetic Algorithm in Fuzzy Environment

In this paper, a solid transportation problem has been considered in which the transportation is accomplished in two stages – firstly, from the origin(s) to the near by station(s) of the destination(s) and secondly, from the near by station(s) to the exact destination(s). Here, a fuzzy AUD (all unit discount) policy has been introduced based upon the amount of transportation along with a fuzzy fixed charge. In addition, a budget constraint has been incorporated taking fuzzy unit transportation cost. Then the proposed model has been converted into a single objective optimisation problem using interval arithmetic method. To solve the model, Genetic Algorithm (GA) has been used depending on roulette wheel selection, crossover and mutation. Finally, a numerical example has been illustrated to study the feasibility of the model.

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.

Verified computations for closed hyperbolic 3‐manifolds

Extending methods first used by Casson, we show how to verify a hyperbolic structure on a finite triangulation of a closed 3-manifold using interval arithmetic methods. A key ingredient is a new theoretical result (akin to a theorem by Neumann-Zagier and Moser for ideal triangulations upon which HIKMOT is based) showing that there is a redundancy among the edge equations if the edges avoid "gimbal lock". We successfully test the algorithm on known examples such as the orientable closed manifolds in the Hodgson-Weeks census and the bundle census by Bell. We also tackle a previously unsolved problem and determine all knots and links with up to 14 crossings that have a hyperbolic branched double cover.

Dynamic mode of the mathematical model of an electric multipole with memresistive branches in conditions of interval uncertainty

This article discusses a method for obtaining a mathematical model of an electric multipole with memresistive branches in finite parameter increments for a large signal mode in time domain calculation when calculating a direct current based on a generalized model obtained earlier. A computational algorithm for calculating nanoelectronic circuits based on methods of interval arithmetic is also given.

Static mode of the mathematical model of an electric multipole with memresistive branches in conditions of interval uncertainty

This article discusses a method for obtaining a mathematical model of an electric multipole with memresistive branches in finite parameter increments for a large signal mode when calculating a direct current based on a generalized model obtained earlier. A computational algorithm for calculating nanoelectronic circuits based on methods of interval arithmetic is also given.

Domain Theoretic Second-Order Euler's Method for Solving Initial Value Problems

A domain-theoretic method for solving initial value problems (IVPs) is presented, together with proofs of soundness, completeness, and some results on the algebraic complexity of the method. While the common fixed-precision interval arithmetic methods are restricted by the precision of the underlying machine architecture, domain-theoretic methods may be complete, i.e., the result may be obtained to any degree of accuracy. Furthermore, unlike methods based on interval arithmetic which require access to the syntactic representation of the vector field, domain-theoretic methods only deal with the semantics of the field, in the sense that the field is assumed to be given via finitely-representable approximations, to within any required accuracy.In contrast to the domain-theoretic first-order Euler method, the second-order method uses the local Lipschitz properties of the field. This is achieved by using a domain for Lipschitz functions, whose elements are consistent pairs that provide approximations of the field and its local Lipschitz properties. In the special case where the field is differentiable, the local Lipschitz properties are exactly the local differential properties of the field. In solving IVPs, Lipschitz continuity of the field is a common assumption, as a sufficient condition for uniqueness of the solution. While the validated methods for solving IVPs commonly impose further restrictions on the vector field, the second-order Euler method requires no further condition. In this sense, the method may be seen as the most general of its kind.To avoid complicated notations and lengthy arguments, the results of the paper are stated for the second-order Euler method. Nonetheless, the framework, and the results, may be extended to any higher-order Euler method, in a straightforward way.

A polynomial chaos expansion approach for nonlinear dynamic systems with interval uncertainty

This paper proposes a non-intrusive interval uncertainty analysis method for estimation of the dynamic response bounds of nonlinear systems with uncertain-but-bounded parameters using polynomial chaos expansion. The conventional interval arithmetic and Taylor series methods usually lead to large overestimation because of the intrinsic wrapping effect, especially for the multidimensional and non-monotonic problems. To overcome this drawback, a novel polynomial chaos inclusion function, based on the truncated polynomial chaos expansion, is proposed in the present work to evaluate interval functions. In this method, the Legendre polynomial in interval space is employed as the trial basis to expand the interval processes, and the polynomial coefficients are calculated through the collocation method. Two examples show that the polynomial chaos inclusion function is capable of determining tighter enclosures of the true solutions and effectively dealing with the wrapping effect. The response of nonlinear systems with respect to interval variables is approximated by the polynomial chaos inclusion function, through which the supremum and infimum of the dynamic responses over all time iteration steps can be easily estimated by an appropriate numerical solver. Four dynamics examples described by ordinary differential equations demonstrate the effectiveness, feasibility, and efficiency of the proposed interval uncertainty analysis method compared with other methods.

Treatment of uncertainties in green engineering

Uncertainties occurring in engineering are usually dealt with by means of statistical validation. Most value assignments are of statistical nature and thus only valid in a specific statistical way. This fact is, in most cases, only implicitly apparent but nevertheless almost always present. In green engineering this is not true. The calculation e.g. of life cycle assessments (LCAs) is generally based on a vast number of values. These data either cannot be validated in a statistical sense at all or they are too insufficient to be usable for statistical analyses. This means that final results in this field cannot be declared as trustworthy in a way that is common in engineering. In this paper, the method of interval arithmetic is proposed as a way to circumvent this severe epistemological problem. The consequences of the suggested approach are illustrated utilizing two typical examples: (1) Different reliabilities concerning theories for greenhouse effect influences of various chemical emissions on the greenhouse effect and thus on the global warming potential and (2) Situations, where one specific production site differs heavily from the average of the dedicated industrial sector. Computational implementation of the considered concept is briefly discussed. It is shown that misleading results, which may be quite common in this field, can possibly be avoided in the proposed way.

Matrix-Based Interval Arithmetic for Linear Array Tolerance Analysis With Excitation Amplitude Errors

This communication proposes a novel matrix-based interval arithmetic (mIA) method for linear antenna array pattern tolerance analysis with excitation amplitude errors. It derives the closed-form array power pattern bounds in an analytical fashion, which can predict the maximum deviations with respect to the referenced value efficiently. The array power pattern upper bound is its midpoint value summing two error items, the first-order derivative and the array error pattern, while the lower bound is its midpoint value subtracting the first-order derivative. The new expression derived by the mIA method only involves matrix manipulation and requires no condition judgment, which inherits simple coding and easy implementation. Representative numerical examples are presented to validate and demonstrate the effectiveness of the proposed mIA method.

Comparison of Classical Method, Extension Principle and α-Cuts and Interval Arithmetic Method in Solving System of Fuzzy Linear Equations

The system of linear equations plays a vital role in real life problems such as optimization, economics, and engineering. The parameters of the system of linear equations are modeled by taking the experimental or observation data. So the parameters of the system actually contain uncertainty rather than the crisp one. The uncertainties may be considered in term of interval or fuzzy numbers. In this paper, a detailed study of three solution techniques namely Classical Method, Extension Principle method and α-cuts and interval Arithmetic Method to solve the system of fuzzy linear equations has been done. Appropriate applications are given to illustrate each technique. Then we discuss the comparison of the different methods numerically and graphically.

Variation modeling and analysis with interval approach for the assembly of compliant aeronautical structures

Due to small production volume in aircraft industry, the available information of variation sources is often not enough to make assumptions on their probabilistic characteristics, especially in the stage of prototype manufacturing. To deal with the problem, an assembly variation modeling and analysis method based on the elasticity mechanics and interval approach is proposed for aircraft assembly. First, variation sources are modeled as bounded convex sets, which are defined as interval structural parameters in interval arithmetic. Then, variation modeling and analysis are successively implemented using the method of influence coefficient and interval arithmetic. After that, a uniform-splitting method is applied to achieve the refinement of the interval extension in variation analysis. To reduce the complexity of the finite element analysis and assembly variation computation, part deformation forms including warpage and torsion are concisely characterized with angle instead of the deviations of isolated key points on the part. The comparison of the assembly variations estimated with the proposed variation analysis method and actual experiment results verifies the effectiveness of the constructed assembly variation model and the proposed method. The interval approach–based assembly variation analysis method is a good complement to traditional probabilistic approach–based methods for compliant assembly systems, which is suited for linear and linearized nonlinear assembly systems. The proposed method provides an improved understanding of the application of compliant assembly variation analysis methods in aircraft manufacturing.

Reliable power flow and short circuit analysis of systems with uncertain data

This paper addresses the problem of uncertainties in the input parameters by specifying them as compact intervals, taking into consideration the errors in modelling and measurement of transmission line parameters and also the continuous influence of load measurement errors and fluctuations in the load demand. The power flow equations are modelled as a set of nonlinear algebraic equations which are first linearised using Taylor series expansion and the solution is obtained by the Krawczyk's method of interval arithmetic. For the short circuit analysis the prefault conditions are obtained from power flow analysis and the faulty network is then solved using Thevenin's equivalent network as seen from the fault point. The proposed method is applied to 3 bus, 14 bus and 30 bus IEEE test systems where load currents and fault currents for each relay are obtained in bounded form and thus well-defined relay coordination pairs are available.

Rigorous Analysis of Chua’s Circuit With a Smooth Nonlinearity

Dynamics of Chua’s circuit with a smooth nonlinearity is studied by means of interval arithmetic methods. We analyze behaviors of the system for several parameter values in the periodic and chaotic regions. For parameter values in the chaotic region, we find lower bounds for the topological entropy of a corresponding return map and prove that the system is chaotic in the topological sense. We find low-period orbits embedded in chaotic attractors and estimate the true value of the topological entropy. We construct a trapping region enclosing the spiral attractor and discuss how to prove the existence of a trapping region for the case of the double-scroll attractor.

Interval arithmetic operations for uncertainty analysis with correlated interval variables

A new interval arithmetic method is proposed to solve interval functions with correlated intervals through which the overestimation problem existing in interval analysis could be significantly alleviated. The correlation between interval parameters is defined by the multidimensional parallelepiped model which is convenient to describe the correlative and independent interval variables in a unified framework. The original interval variables with correlation are transformed into the standard space without correlation, and then the relationship between the original variables and the standard interval variables is obtained. The expressions of four basic interval arithmetic operations, namely addition, subtraction, multiplication, and division, are given in the standard space. Finally, several numerical examples and a two-step bar are used to demonstrate the effectiveness of the proposed method.

Solving Interval Linear Equations with Modified Interval Arithmetic

Gaussian elimination method is one of the widely used methods for solving linear equations. An interval version of Gaussian elimination method has been used by simply replacing each real arithmetic step by the corresponding interval arithmetic step. Two interval arithmetics technique has been considered for modified interval arithmetics as well as several existing interval arithmetics. In this paper, modified interval arithmetic has been introduced based on two interval arithmetics technique. If we solve interval linear system of equations by existing interval arithmetic method the replacing solution in interval system of equations, the interval width is more than the interval width of right hand side intervals. On the other hand, applying modified interval arithmetic the interval width is less than interval width than previously obtained by existing interval arithmetic. Moreover, the closeness of interval width in system of equations to the right hand side is important so modified interval arithmetic is more effective and efficient for solving interval linear system of equations.