A Limitation for Underestimation Via Twin Arithmetic

Computing an enclosure for the range of a rational function over an interval is one of the main goals of interval analysis. One way to obtain such an enclosure is to use interval arithmetic evaluation of a formula for the function. Often one would like to check how close the overestimation is to the correct range. Kreinovich, Nesterov, and Zheludeva (Reliable Computing 2(2) (1996)) suggested a new kind of twin arithmetic which produces a twin of intervals at the same time: the usual enclosure, i.e., an interval which is an overestimation for the range, and an interval which is contained in the range, i.e. an interval which is an underestimation for the range. We show in this paper that in certain cases the computed inner interval is much smaller than the correct range. For example, if the function has the same value in the two endpoints of the interval then the inner interval is either empty or contains only one point.     

A Right-Preconditioning Process for the Formal–Algebraic Approach to Inner and Outer Estimation of AE-Solution Sets

Aright-preconditioning process for linear interval systems has been presented by Neumaier in 1987. It allows the construction of an outer estimate of the united solution set of a square linear interval system in the form of a parallelepiped. The denomination “right-preconditioning” is used to describe the preconditioning processes which involve the matrix product AC in contrast to the (usual) left-preconditioning processes which involve the matrix product AC, where A and C are respectively the interval matrix of the studied linear interval system and the preconditioning matrix.The present paper presents a new right-preconditioning process similar to the one presented by Neumaier in 1987 but in the more general context of the inner and outer estimations of linear AEsolution sets. Following the spirit of the formal-algebraic approach to AE-solution sets estimation, summarized by Shary in 2002, the new right-preconditioning process is presented in the form of two new auxiliary interval equations. Then, the resolution of these auxiliary interval equations leads to inner and outer estimates of AE-solution sets in the form of parallelepipeds. This right-preconditioning process has two advantages: on one hand, the parallelepipeds estimates are often more precise than the interval vectors estimates computed by Shary. On the other hand, in many situations, it simplifies the formal algebraic approach to inner estimation of AE-solution sets. Therefore, some AE-solution sets which were almost impossible to inner estimate with interval vectors, become simple to inner estimate using parallelepipeds. These benefits are supported by theoretical results and by some experimentations on academic examples of linear interval systems.     

Interval Arithmetic on Multimedia Architectures

In this paper, we show how to implement interval arithmetic on multimedia architecture extensions. Due to the difficulties that current architectures have with the directed rounding modes the speed-up is rather modest. The study, nevertheless, may be helpful in future design of hardware extensions for interval arithmetic.     

Specification of hardware for interval arithmetic

Interval arithmetic, as it is standardized by the IEEE working group P1788 can be implemented by using floating point arithmetic units with directed rounding modes. The easiest way to represent an interval is by its two bounds. Simple formulas for the arithmetic operations can be applied. Our goal is to perform interval operations as fast as their floating point counterparts. Hence, we provide at least two units per operation. We also specify the operation for reverse multiplication (Neumaier in Vienna proposal for interval standardization, 2008) which can be implemented with the division unit. In this paper we do not care about optimization. Our primary intention is to give an easily understandable specification of hardware for interval arithmetic.     

Interval analytic treatment of convex programming problems

A nonlinear convex programming problem is solved by methods of interval arithmetic which take into account the input errors and the round-off errors. The problem is reduced to the solution of a nonlinear parameter dependent system of equations. Moreover error estimations are developed for special problems with uniformly convex cost functions.     

On Maximal Inner Estimation of the Solution Sets of Linear Systems with Interval Parameters

The purpose of this paper is to inquire the connection between maximal inner interval estimates of the solution sets to interval linear system and solutions of the dualization equation in Kaucher interval arithmetic. The results of our work are as follows: 1) a criterion of inner interval estimate of the solution set, 2) a criterion for a solution of dualization equation to be a maximal inner interval estimate of the solution set, 3) a criterion for multiplication by an interval matrix to be upper strictly isotone. This revised version was published online in August 2006 with corrections to the Cover Date.     

Two layered Genetic Programming for mixed-attribute data classification

The important problem of data classification spans numerous real life applications. The classification problem has been tackled by using Genetic Programming in many successful ways. Most approaches focus on classification of only one type of data. However, most of the real-world data contain a mixture of categorical and continuous attributes. In this paper, we present an approach to classify mixed attribute data using Two Layered Genetic Programming (L2GP). The presented approach does not transform data into any other type and combines the properties of arithmetic expressions (using numerical data) and logical expressions (using categorical data). The outer layer contains logical functions and some nodes. These nodes contain the inner layer and are either logical or arithmetic expressions. Logical expressions give their Boolean output to the outer tree. The arithmetic expressions give a real value as their output. Positive real value is considered true and a negative value is considered false. These outputs of inner layers are used to evaluate the outer layer which determines the classification decision. The proposed classification technique has been applied on various heterogeneous data classification problems and found successful.     

Enhancing numerical constraint propagation using multiple inclusion representations

Building tight and conservative enclosures of the solution set is of crucial importance in the design of efficient complete solvers for numerical constraint satisfaction problems (NCSPs). This paper proposes a novel generic algorithm enabling the cooperative use, during constraint propagation, of multiple enclosure techniques. The new algorithm brings into the constraint propagation framework the strength of techniques coming from different areas such as interval arithmetic, affine arithmetic, and mathematical programming. It is based on the directed acyclic graph (DAG) representation of NCSPs whose flexibility and expressiveness facilitates the design of fine-grained combination strategies for general factorable systems. The paper presents several possible combination strategies for creating practical instances of the generic algorithm. The experiments reported on a particular instance using interval constraint propagation, interval arithmetic, affine arithmetic, and linear programming illustrate the flexibility and efficiency of the approach.     

A New Technique in Systems Analysis Under Interval Uncertainty and Ambiguity

The main subject of this work is mathematical and computational aspects of modeling of static systems under interval uncertainty and/or ambiguity. A cornerstone of the new approach we are advancing in the present paper is, first, the rigorous and consistent use of the logical quantifiers to characterize and distinguish different kinds of interval uncertainty that occur in the course of modeling, and, second, the systematic use of Kaucher complete interval arithmetic for the solution of problems that are minimax by their nature. As a formalization of the mathematical problem statement, concepts of generalized solution sets and AE-solution sets to an interval system of equations, inequalities, etc., are introduced. The major practical result of our paper is the development of a number of techniques for inner and outer estimation of the so-called AE-solution sets to interval systems of equations. We work out, among others, formal approach, generalized interval Gauss-Seidel iteration, generalized preconditioning and PPS-methods. Along with the general nonlinear case, the linear systems are treated more thoroughly.     

Interval bounds for square roots and cube roots

The smallest machine representable interval containing the square root of a given machine representable number is sought. Assuming binary computers with optimal upward directed rounding, it is shown bya priori methods of error analysis that this interval may be obtained via Newton's method without using interval arithmetic. Less sharp but still useful results are obtained for the cube root.     

Outer Estimation of Generalized Solution Sets to Interval Linear Systems

The work advances a numerical technique for computing enclosures of generalized AE-solution sets to interval linear systems of equations. We develop an approach (called algebraic) in which the outer estimation problem reduces to a problem of computing algebraic solutions of an auxiliary interval equation in Kaucher complete interval arithmetic.     

具有JTIDS接口的无人机仿真系统

建立了一个具有联合战术信息分发系统接口的无人机仿真系统.在该系统中建立了某型无人机的动力学仿真模型,其气动系数均通过风洞试验获得,并采用单纯形调优法求取无人机初始状态;设计了控制无人机姿态和控制飞行轨迹的内外环控制系统模型;根据联合战术信息分发系统特性,采用模块化方法建立了联合战术信息分发系统接口,使操作员能通过仿真数据链对无人机动力学模型进行任务控制和状态监视.最后进行了虚拟任务仿真试验,结果表明该系统可以满足一定的战术任务需求.     

Data-driven dual-rate cascade control and application to pitch angle control of UAV

A data-driven design method for a cascade control system is proposed. The cascade control system consists of inner and outer loops, where the control interval of the outer loop is an integer multiple of the inner loop; hence, the system is a dual-rate system. In the proposed method, controllers in the inner and outer loops are designed based on one-shot data. In such a dual-rate cascade system, since the controllers are designed using different data-rate signals, the lifting technique is applied to align the dual-rate data. To show its effectiveness, the proposed method is compared with a conventional single-rate cascade control method, and numerical simulations and experiments are presented to examine servo and regulation performance.     

PROFIL/BIAS—A fast interval library

The interval data type is currently not supported in common programming languages. Therefore the implementation of algorithms using interval arithmetic requires special programming environments or at least special libraries. In this paper we present the C++ class library PROFIL which provides a user friendly environment for implementing interval algorithms. The main goals in the design of PROFIL were speed and portability. Therefore all interval operations in PROFIL use BIAS (Basic Interval Arithmetic Subroutines) [16]. BIAS defines a concise and portable interface for the basic scalar, vector, and matrix operations. The interface is independent of a specific interval representation or computation but permits machine specific and fast implementations. Based on this general specification we present an implementation in C using a lower/upper bound representation of intervals and directed roundings. By using few assembler instructions for switching the rounding modes and avoiding sign tests and rounding mode switches wherever possible, the computational costs of the interval operations were reduced significantly. This is especially important for RISC machines, where floating point instructions can be executed in few machine cycles. Comparisons with other interval arithmetic packages show an improvement in speed of about one order of magnitude.     

A surrogate model based nested optimization framework for inverse problem considering interval uncertainty

The inverse problem is a kind of engineering problem that estimates the input through the given output. In this paper, the given output described as interval uncertainty parameter is concerned, and the interval is formed by the interval midpoint and the interval radius. The two-step framework that estimates the midpoint and the radius separately is used. A novel nested optimization framework is proposed to estimate the input interval radius with more inputs than outputs. The nested framework has two loops: (i) the inner loop quantifies the lower and upper bounds of the output with given interval radiuses of inputs from the outer loop by two optimizations, and the results will be feedback to the outer loop as constraint values of outer loop; (ii) the outer loop maximizes the input interval radiuses to reduce the cost while meeting the constraints transformed from the given interval. The nested framework induces a high number of forward model computations in the loops and may lead to an unacceptable computational burden for most engineering applications. Therefore, the surrogate model is suggested. The Radial Basis Functions (RBF) surrogate model is used to relieve the computational burden. The effectiveness and the accuracy of the framework are verified through a mathematical example, a cantilever tube example and an airfoil example.     

The exact dot product as basic tool for long interval arithmetic

Computing with guarantees is based on two arithmetical features. One is fixed (double) precision interval arithmetic. The other one is dynamic precision interval arithmetic, here also called long interval arithmetic. The basic tool to achieve high speed dynamic precision arithmetic for real and interval data is an exact multiply and accumulate operation and with it an exact dot product. Pipelining allows to compute it at the same high speed as vector operations on conventional vector processors. Long interval arithmetic fully benefits from such high speed. Exactitude brings very high accuracy, and thereby stability into computation. This document, which has been incorporated into the draft standard for interval arithmetic being developed by IEEE P1788, specifies the implementation of an exact multiply and accumulate operation.     

A Novel Method Of Controller Design For Simultaneous Stabilization And Performance Improvement Of An Electromagnetic Levitation System

This paper describes design and implementation of a control philosophy for simultaneous stabilization and performance improvement of an electromagnetic levitation system. An electromagnetic levitation system is an inherently unstable and strongly nonlinear system. To determine the overall closed loop stability for such a system, cascade lead‐lag compensation has mostly been reported [1,2]. However, a single lead controller can not satisfy both stability and performance for such unstable systems [3]. Performance enhancement to satisfy the conflicting requirements of fast response with almost zero overshoot and zero steady state error has been successfully achieved by using a two loop controller configuration. The lead controller in the inner loop is designed to ensure stability while the outer loop PI controller is designed for performance enhancement. This approach decouples the twin requirements of stabilization (by the inner loop) and performance achievement (by the outer PI loop). The outermost PI controller has been designed using the ‘Approximate Model Matching’ technique [4]. The proposed control strategy has been implemented and the experimentation has been demonstrated successfully. Different experimental results have been included for verification.     

Hardware Support for Interval Arithmetic

A hardware unit for interval arithmetic (including division by an interval that contains zero) is described in this paper. After a brief introduction an instruction set for interval arithmetic is defined which is attractive from the mathematical point of view. These instructions consist of the basic arithmetic operations and comparisons for intervals including the relevant lattice operations. To enable high speed, the case selections for interval multiplication (9 cases) and interval division (14 cases) are done in hardware. The lower bound of the result is computed with rounding downwards and the upper bound with rounding upwards by parallel units simultaneously. The rounding mode must be an integral part of the arithmetic operation. Also the basic comparisons for intervals together with the corresponding lattice operations and the result selection in more complicated cases of multiplication and division are done in hardware. There they are executed by parallel units simultaneously. The circuits described in this paper show that with modest additional hardware costs interval arithmetic can be made almost as fast as simple floating-point arithmetic.     

A twin shear beam model for double chord RHS joints

A finite element elasto-plastic analysis of a twin shear beam model, developed to shed light on the behavior of separated double chord gap joints of rectangular hollow sections (RHS), is presented. The model treats a single beam member as composed of a plate simulating its inner web and a channel representing top and bottom flanges and the outer web. The twelve degrees of freedom (DOF) nonconforming plate bending element and the eight DOF plane stress elements are used to form a plate element for the inner web, while the channel consists of a grillage of beam elements. The analysis, extended beyond the elastic limit, accounts for strain hardening and material nonlinearity. A comparison of theoretical predictions with the experimental results of 24 specimens shows generally good agreement. The mean ratios of theoretical to experimental values are 0.97 and 0.87 for elastic stiffness and carrying capacity measured at a post elastic deformation limit.     

A generic lazy evaluation scheme for exact geometric computations

We present a generic C++ design to perform exact geometric computations efficiently using lazy evaluations. Exact geometric computations are critical for the robustness of geometric algorithms. Their efficiency is also important for many applications, hence the need for delaying the costly exact computations at run time until they are actually needed, if at all. Our approach is generic and extensible in the sense that it is possible to make it a library that users can apply to their own geometric objects and primitives. It involves techniques such as generic functor-adaptors, static and dynamic polymorphism, reference counting for the management of directed acyclic graphs, and exception handling for triggering exact computations when needed. It also relies on multi-precision arithmetic as well as interval arithmetic. We apply our approach to the whole geometry kernel of Cgal.