Interval Linear Systems Research Articles

On preconditioning and solving an extended class of interval parametric linear systems

We deal with interval parametric systems of linear equations and the goal is to solve such systems, which basically comes down to finding an enclosure for a parametric solution set. Obviously, we want this enclosure to be tight and cheap to compute; unfortunately, these two objectives are conflicting. The review of the available literature shows that in order to make a system more tractable, most of the solution methods use left preconditioning of the system by the midpoint inverse. Surprisingly, and in contrast to standard interval linear systems, our investigations have shown that double preconditioning can be more efficient than a single one, both in terms of checking the regularity of the system matrix and enclosing the solution set, which is demonstrated by numerical examples. Consequently, right (which was hitherto mentioned in the context of checking regularity of interval parametric matrices) and double preconditioning together with the p-solution concept enable us to solve a larger class of interval parametric linear systems than most existing methods. The applicability of the proposed approach to solving interval parametric linear systems is illustrated by several numerical examples.

New Farkas-type inequalities of mixed interval systems for AE solvability

Interval Farkas type inequality is an important tool to prove the optimality condition of interval optimization model. This research gives a Farkas-type inequality for AE solvability of a specific mixed interval linear system including both equality and inequality by using logical quantifiers. Some Farkas-type inequalities for different solvability established in previous literatures are special cases of the main result of this paper.

ANN Based Solution of Uncertain Linear Systems of Equations

Linear systems of equations have many applications in the area of engineering sciences, mathematics, operations research and statistics. It is worth mentioning that the coefficient matrix of the linear systems of equations may not be always crisp due to various uncertainties. These uncertainties may be in the form of interval. Likewise, the solution set and right hand side vector may also be in interval. In this respect, a fully interval linear system of equations \( (\tilde{P}\tilde{z} = \tilde{q}) \) is one where the coefficient matrix, the unknown vector and the right hand side vector all are in the form of interval. Although various authors proposed different methods to handle the fully linear systems of equations but those are sometimes problem specific etc. As such, in this paper \( n \times n \) fully interval linear systems of equations has been solved based on artificial neural network (ANN) model. In this regard, step by step algorithm has been included. Further, a convergence theorem has also been discussed for choosing suitable learning parameter. Few numerical examples and an application problem related to electrical circuit have been solved using the proposed method. Detail procedure has been discussed with numerical results to show the efficacy and powerfulness of the method.

Asymptotic Analysis of Linear and Interval Linear Fractional-Order Neutral Delay Differential Systems Described by the Caputo-Fabrizio Derivative

Asymptotic stability of linear and interval linear fractional-order neutral delay differential systems described by the Caputo-Fabrizio (CF) fractional derivatives is investigated. Using Laplace transform, a novel characteristic equation is derived. Stability criteria are established based on an algebraic approach and norm-based criteria are also presented. It is shown that asymptotic stability is ensured for linear fractional-order neutral delay differential systems provided that the underlying stability criterion holds for any delay parameter. In addition, sufficient conditions are derived to ensure the asymptotic stability of interval linear fractional order neutral delay differential systems. Examples are provided to illustrate the effectiveness and applicability of the theoretical results.

EA solutions and EA solvability to general interval linear systems

ABSTRACT S. Shary introduced the notion of quantifier solutions to interval systems, in which the uncertainty on the coefficients of a system is modelled by a selecting predicate, allowing an alternation in an arbitrary order of the logical quantifiers (∀ and ∃). AE solutions, a particular type of quantifier solutions, in which all the occurrences of the universal quantifier ∀ precede the occurrences of the existential quantifier ∃, have been studied by many authors. In this work we deal with EA solutions, another type of quantifier solutions to general interval linear systems, such that, in the selecting predicate, all the occurrences of the existential quantifier ∃ precede the occurrences of the universal quantifier ∀. The EA solutions can be considered as a dual form of AE solutions. We provide a full characterization of EA solutions. Moreover, we develop a necessary condition for EA solvability of the general interval linear systems, as well as a necessary and sufficient condition for a certain subcase.

A new soft computing approach for order diminution of interval system

Order diminution plays a significant role in the study of complex dynamical linear interval systems. Numerous strategies have been suggested in diverse field for order diminution of higher dimensional models. Nowadays, the optimization algorithms are commonly used for order diminution based on a specific error performance criterion known as cost or objective function. This study suggests a novel approach for order diminution of interval system using cuckoo search algorithm. The proposed technique utilises the Kharitonov’s polynomials to ensure a stable model. Typical examples have been considered to showcase the efficiency of the presented approach. Comparative study is also demonstrated with other available techniques in terms of different performance measures. The suggested approach has also been applied for order diminution of multi input multi output interval system.

Enclosing the solution set of the parametric generalised Sylvester matrix equation A(p)XB(p) + C(p)XD(p) = F(p)

In this paper, we investigate the parametric generalised Sylvester matrix equation , whose elements are linear functions of uncertain parameters varying within intervals. This model generalises both interval matrix equations and parametric interval linear systems, so it is a quite general model. First, we give some sufficient conditions under which the solution set of this parametric equation is bounded. We then propose several approaches for enclosing the solution set that acquire tighter enclosures than those obtained by relaxing the parametric system to an interval system. Some special cases of interval systems that inherently are parametric (have dependent structure) are considered, too. Finally, numerical experiments are given to illustrate the effectiveness of the proposed methods.

AE solutions to interval linear systems over max-plus algebra

This paper introduces a concept of AE solutions to interval max-plus linear systems, a rather general concept which includes many known concepts of solutions to interval systems: weak solutions, strong solutions, tolerance solutions and control solutions, as its special cases. We state full characterizations of AE solutions for the interval max-plus systems, including both linear inequalities and linear equations. Moreover, a kind of dependency characterized by double appearance of some sub-matrix of the constraint matrix is discussed, and we prove that which can be relaxed with no change in AE solvability.

Optimal value bounds in interval fractional linear programming and revenue efficiency measuring

This paper deals with the fractional linear programming problem in which input data can vary in some given real compact intervals. The aim is to compute the exact range of the optimal value function. A method is provided for the situation in which the feasible set is described by a linear interval system. Moreover, certain dependencies between the coefficients in the nominators and denominators can be involved. Also, we extend this approach for situations in which the same vector appears in different terms in nominators and denominators. The applicability of the approaches developed is illustrated in the context of the analysis of hospital performance.

Rank one interval enclosure of the parametric united solution set

Consider linear algebraic systems $$A(p)x=b(p)$$ , where the matrix and the right-hand side vector depend linearly on a number of parameters $$p=(p_1,\ldots , p_K)^{\top }$$ that vary within given intervals. For each interval parameter, the structure of the dependencies can be presented by a finite sum of rank one matrices. This representation implies new more general and powerful sufficient conditions for regularity of a parametric interval matrix and a flexible general methodology for solving parametric interval linear systems with an expanded scope of applicability.

AE solutions to two-sided interval linear systems over max-plus algebra

This paper introduces a concept of AE solutions to two-sided interval max-plus linear systems, a rather general concept which includes many known concepts of solutions to interval systems, in particular, weak, strong, tolerance and control solutions as its special cases. We state full characterizations of AE solutions for the two-sided interval max-plus systems, including both linear inequalities and linear equations. Moreover, we provide a specific example to illustrate an efficient method of finding the AE solution set.

Two Iterative Methods for Solving Linear Interval Systems

Conjugate gradient is an iterative method that solves a linear system Ax=b, where A is a positive definite matrix. We present this new iterative method for solving linear interval systems Ãx̃=b̃, where à is a diagonally dominant interval matrix, as defined in this paper. Our method is based on conjugate gradient algorithm in the context view of interval numbers. Numerical experiments show that the new interval modified conjugate gradient method minimizes the norm of the difference of Ãx̃ and b̃ at every step while the norm is sufficiently small. In addition, we present another iterative method that solves Ãx̃=b̃, where à is a diagonally dominant interval matrix. This method, using the idea of steepest descent, finds exact solution x̃ for linear interval systems, where Ãx̃=b̃; we present a proof that indicates that this iterative method is convergent. Also, our numerical experiments illustrate the efficiency of the proposed methods.

Qualitative Analysis of Stable Reduced Order Models for Interval Systems Using Mixed Methods

ABSTRACT A large number of problems are available for complex higher order systems. These systems are difficult to analyze and synthesize. Therefore it is needful to approximate higher order models to lower order models. Interval systems have constant coefficients which are uncertain within a finite range. In this paper, the frequency domain reduction methods are applied for reducing the order of interval systems. The three mixed methods are considered for reducing the linear dynamic interval systems. In these mixed methods, denominators of the reduced models are obtained by using differentiation method while the numerators of the reduced models are obtained by using differentiation, factor division, and pade approximation methods. The novel feature of the proposed methods is that it guarantees the stability of reduced order models if the original interval systems are stable. In addition, the mixed methods are compared qualitatively in terms of integral square error and integral absolute error to know about the best method among three mixed methods. Numerical examples are given to show the effectiveness of the mixed methods.

Solving interval linear programming problems with equality constraints using extended interval enclosure solutions

This paper focuses on solving systems of interval linear equations and interval linear programming in a computationally efficient way. Since the computational complexity of most interval enclosure numerical methods is often prohibitive, a procedure to obtain a relaxation of the interval enclosure solution that is computationally tractable is proposed. We show that our approach unifies the four standard interval solutions—the weak, strong, control and tolerance solutions. The interval linear system methods require $$n\cdot 2^{n}$$ linear solutions. However, in the case of linear programming problems, we show that this requires just two optimization problem of the size of the problem itself. Numerical examples illustrate our results.

A New Method for Analyzing Three-phase Faults under Data Uncertainties

Fault analysis uses a given set of data regarding generations and line branches. The main objective is to calculate voltages and currents as a specific fault location is considered. As the input data are imprecise, this paper incorporates the interval mathematics into fault analysis problem in order to take into account data uncertainties. Therefore, this paper proposes a new method for modeling an interval three-phase fault analysis. At each iteration, the resulting interval linear system of equations is solved using Krawczyk method. The proposed method is implemented in the Matlab environment using the Intlab toolbox. The 14-bus test system is used. Results are later compared with the Monte Carlo Simulation.

A novel reduced order modeling of interval system using soft computing optimization approach

This research article presents a novel algorithm for the model order reduction of higher order linear time interval systems using soft computing optimization approach. In the proposed method, a new recursive formula for alpha parameters is developed for determining reduced order interval model without formulating alpha and beta tables. The denominator and numerator polynomials of reduced order interval model are determined based on minimization of a multi-objective function comprising integral squared error and impulse response energy error using particle swarm optimization technique. The proposed algorithm has several advantageous features such as reduced computational complexity and stability preservation property. The efficacy of the proposed algorithm is illustrated through typical numerical examples available in the literature, and the results are successfully compared with the other familiar methods.

An analytical model for solving generalized interval eigenvalue problem

• Theorem and corollaries are derived for generalized interval eigenvalue problems . • An effective way to discriminate the sign invariance of eigenvector is presented. • The analytical model solves the exact bounds of eigenvalue with high efficiency. • Inapplicability of existing methods and superiority of proposed model are revealed. Previous algorithms on calculating the eigenvalue bounds of generalized interval eigenvalue problems usually require preconditions difficult to be met or their efficiency and accuracy are unguaranteed. To overcome these defects, an exact analytical model for obtaining the interval solution set of generalized eigenvalue problem is proposed by rigorous derivations. The eigenvalue set of generalized interval eigenvalue problem is firstly characterized according to the linear interval system and interval arithmetic. Then by introducing the Rayleigh quotient the formulation of mathematical programming on maximizing and minimizing the eigenvalue is derived for the generalized interval eigenvalue problem. The Kuhn-Tucker theorem is subsequently applied to deduce the theorem on computing the upper and lower bounds of interval solution set. By applying the proposed model in real engineering problems, it is demonstrated that the positive definiteness or non-negative decomposition of the matrix pair required by the existing analytical methods cannot be satisfied in some engineering situations where the proposed model is applicable.

A novel order reduction procedure for linear time invarient interval systems using SGO algorithm

In this paper, the authors presented a new algorithm for the reduction of high order linear time interval systems. In the proposed method, the Reduced Order Interval Model (ROIM) denominator and numerator polynomials are determined based on minimization ofobjective function comprising of Integral Squared Error r using Social Group Optimization (SGO). The SGO technique is found to be simple, easy in implementation and provides the optimal solution. Applicability and effectiveness of the proposed method are illustrated through a DC motor speed control system and a typical Seventh order system taken from the literature.

Solving the interval linear programming problem: A new algorithm for a general case

Based on the binding constraint indices of the optimal solution to the linear programming (LP) model, a feasible system of linear equations can be formed. Because an interval linear programming (ILP) model is the union of numerous LP models, an interval linear equations system (ILES) can be formed, which is the union of these conventional systems. Hence, a new algorithm is introduced in which an arbitrary characteristic model of the ILP model is chosen and solved. The set of indices of its binding constraints is then obtained. This set is used to form and solve an ILES using the enclosure method. If all the components of the interval solutions to this system are strictly non-negative, the optimal solution set (OSS) of the ILP model is determined as the subscription of the zone created by reversing the signs of the binding constraints of the worst model and the binding constraints of the best model. The solutions to several problems obtained by the new algorithm and a Monte Carlo simulation are compared. The proposed algorithm is applicable to large-scale problems. To this end, an ILP model with 270 constraints and 270 variables is solved.

On the Solutions of Interval Systems for Under-Constrained and Redundant Parallel Manipulators

For under-constrained and redundant parallel manipulators, the actuator inputs are studied with bounded variations in parameters and data. Problem is formulated within the context of force analysis. Discrete and analytical methods for interval linear systems are presented, categorized, and implemented to identify the solution set, as well as the minimum 2-norm least-squares solution set. The notions of parameter dependency and solution subsets are considered. The hyperplanes that bound the solution in each orthant characterize the solution set of manipulators. While the parameterized form of the interval entries of the Jacobian matrix and wrench produce the minimum 2-norm least-squares solution for the under-constrained and over-constrained systems of real matrices and vectors within the interval Jacobian matrix and wrench vector, respectively. Example manipulators are used to present the application of methods for identifying the solution and minimum norm solution sets for actuator forces/torques.