Combined method for correction interval systems of linear algebraic equations

The possibility of application of the interval analysis for data processing in the field of spectral analysis is considered. It is assumed that the data have interval uncertainty; therefore the problem of finding unknown concentrations is posed as a linear interval tolerance problem. The incompatibility of the interval system of linear algebraic equations is shown for the initial data using the apparatus of the recognizing functional. The relevance of the topic is due to the need for regularization of inconsistent interval systems of linear equations. The idea of S. P. Shary of a combined method for correcting a linear tolerance problem has been implemented. A new method for managing the solution by changing the linear algebraic equations interval system matrix elements radii has been developed. The research results can be used for example, to calculate the substance’s concentrations by measurement of the characteristic X-ray radiation.

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability.

Ellipsoidal Approximation of the Set of Solutions of Interval Systems of Linear Algebraic Equations

The article presents the results of the approximation of the set of solutions of interval systems of linear algebraic equations. These systems are used in the problems of modeling linear deterministic processes. It is assumed that the modeled process is described by an output variable and a set of input variables, the measurement errors of which are assumed to be set by known intervals symmetric with respect to the zero value. Traditionally, the sets of solutions of interval systems of linear algebraic equations in applied problems are approximated by a hyper-rectangular whose sides are parallel to the axes of the selected coordinate system. In this paper, we propose to use an ellipsoidal approximation of these sets, which is more efficient. The main results of the work include the substantiation of assumptions about the properties of the modeled process, the choice of a mathematical method for constructing an approximating ellipsoid, the proposed method for forming boundary points, and a numerical method for solving the problem. A computer simulation of the problem of estimating the parameters of a linear process is performed in Excel, which is used for a comparative study of approximations of solutions of interval systems of linear algebraic equations by a hyper-rectangular and an ellipse.

Reserve of characteristic inclusion for interval linear systems of relations

В работе рассматриваются интервальные линейные включения Cx ⊆ d в полной интервальной арифметике Каухера. Вводится количественная мера выполнения этого включения, названная “резервом включения”, исследуются ее свойства и приложения. Показано, что резерв включения оказывается полезным инструментом при изучении АЕ-решений и кванторных решений интервальных линейных систем уравнений и неравенств. В частности, использование резерва включения помогает при определении положения точки относительно множества решений, при исследовании пустоты множества решений или его внутренности и т.п In this paper, we consider interval linear inclusions Cx ⊆ d in the Kaucher complete interval arithmetic. These inclusions are important both on their own and because they provide equivalent and useful descriptions for the so-called quantifier solutions and AE-solutions to interval systems of linear algebraic relations of the form Ax σ b , where A is an interval m × n -matrix, x ∈ R , b is an interval m -vector, and σ ∈ {= , ≤ , ≥} . In other words, these are interval systems in which equations and non-strict inequalities can be mixed. Considering the inclusion Cx ⊆ d in the Kaucher complete interval arithmetic allows studing simultaneously and in a uniform way all the different special cases of quantifier solutions and AE-solutions of interval systems of linear relations, as well as using interval analysis methods. A quantitative measure, called the “inclusion reserve”, is introduced to characterize how strong the inclusion Cx ⊆ d is fulfilled. In our work, we investigate its properties and applications. It is shown that the inclusion reserve turns out to be a useful tool in the study of AE-solutions and quantifier solutions of interval linear systems of equations and inequalities. In particular, the use of the inclusion reserve helps to determine the position of a point relative to a solution set, in investigating whether the solution set is empty or not, whether a point is in the interior of the solution set, etc

Robust fractional-order fixed-structure controller design for uncertain non-commensurate fractional plants using fractional Kharitonov theorem

This article deals with the problem of robust fractional-order fixed-structure controller design for commensurate and non-commensurate fractional-order interval systems using fractional Kharitonov theorem. The contribution of this study is to develop a simple control methodology to stabilize the fractional-order Kharitonov-defined vortex polynomials. Using the idea of robust stability testing function and extending it to the systems under study, the straightforward graphical and systematic procedures are proposed to investigate the robust stability of the system by searching for a non-conservative fractional-order Kharitonov region in the controller parameters plane. This region can establish all the fractional-order controllers that make the uncertain fractional-order systems stable. The relation between the fractional-order Kharitonov region and the parameters of the stabilizing controller is also found. Finally, comparison results with three relevant works are given to illustrate the feasibility of the proposed method.