An Approach for Solving Multi-Objective Linear Fractional Programming Problem with fully Rough Interval Coefficients

In this paper, a multi-objective linear fractional programming (MOLFP) problem is considered where all of its coefficients in the objective function and constraints are rough intervals (RIs). At first, to solve this problem, we will construct two MOLFP problems with interval coefficients. One of these problems is a MOLFP where all of its coefficients are upper approximations of RIs and the other is a MOLFP where all of its coefficients are lower approximations of RIs. Second, the MOLFP problems are transformed into a single objective linear programming (LP) problem using a proposal given by Nuran Guzel. Finally, the single objective LP problem is solved by a regular simplex method which yields an efficient solution of the original MOLFP problem. A numerical example is given to demonstrate the results.

Investigation of the Robust Absolute Stability of the Tunnel Kiln Control System with Delay

The paper considers the system of automatic control of the tunnel kiln temperature conditions. The investigation of a delay influence on the transition process has been carried on. The transfer function of the object under control with interval coefficients taking into account possible effects of the parametric uncertainty has been obtained. A graphical method of representing the obtained results in the form of displaying the modified amplitude-phase characteristics on a complex plane has been applied which obviously demonstrates a robust absolute stability of the system under investigation. The simulation performed in the Matlab environment has proved the correctness of the results obtained.

Robust stability analysis of uncertain incommensurate fractional order quasi- polynomials in the presence of interval fractional orders and interval coefficients

This paper deals with the robust stability analysis of a class of incommensurate fractional order quasi-polynomials with a general type of interval uncertainties. The concept of the general type of interval uncertainties means that all the coefficients and orders of the fractional order quasi-polynomials have interval uncertainties. Generally, the computational complexity of specifying the robust stability of such a quasi-polynomial is shown in this paper. To this end, the robust stability is studied by Principle of Argument theorem. In fact, by presenting two theorems and three lemmas it is shown that the robust stability of a fractional order quasi-polynomial involving the general type of uncertainty can be simply investigated without needing to plot its value set by heavy computations. Examples are attested the validity of the paper results.

Solvability of a Bounded Parametric System in Max-Łukasiewicz Algebra

The max-Łukasiewicz algebra describes fuzzy systems working in discrete time which are based on two binary operations: the maximum and the Łukasiewicz triangular norm. The behavior of such a system in time depends on the solvability of the corresponding bounded parametric max-linear system. The aim of this study is to describe an algorithm recognizing for which values of the parameter the given bounded parametric max-linear system has a solution—represented by an appropriate state of the fuzzy system in consideration. Necessary and sufficient conditions of the solvability have been found and a polynomial recognition algorithm has been described. The correctness of the algorithm has been verified. The presented polynomial algorithm consists of three parts depending on the entries of the transition matrix and the required state vector. The results are illustrated by numerical examples. The presented results can be also applied in the study of the max-Łukasiewicz systems with interval coefficients. Furthermore, Łukasiewicz arithmetical conjunction can be used in various types of models, for example, in cash-flow system.

EA/AE-Eigenvectors of Interval Max-Min Matrices

Systems working in discrete time (discrete event systems, in short: DES)—based on binary operations: the maximum and the minimum—are studied in so-called max–min (fuzzy) algebra. The steady states of a DES correspond to eigenvectors of its transition matrix. In reality, the matrix (vector) entries are usually not exact numbers and they can instead be considered as values in some intervals. The aim of this paper is to investigate the eigenvectors for max–min matrices (vectors) with interval coefficients. This topic is closely related to the research of fuzzy DES in which the entries of state vectors and transition matrices are kept between 0 and 1, in order to describe uncertain and vague values. Such approach has many various applications, especially for decision-making support in biomedical research. On the other side, the interval data obtained as a result of impreciseness, or data errors, play important role in practise, and allow to model similar concepts. The interval approach in this paper is applied in combination with forall–exists quantification of the values. It is assumed that the set of indices is divided into two disjoint subsets: the E-indices correspond to those components of a DES, in which the existence of one entry in the assigned interval is only required, while the A-indices correspond to the universal quantifier, where all entries in the corresponding interval must be considered. In this paper, the properties of EA/AE-interval eigenvectors have been studied and characterized by equivalent conditions. Furthermore, numerical recognition algorithms working in polynomial time have been described. Finally, the results are illustrated by numerical examples.

Parametric Synthesis of a Robust Controller Based on the Method of Dominant Poles

In the paper a linear control system described by its characteristic polynomial with interval coefficients including parameters of controller linearly is considered. Problem of the research is finding parameters of a controller guaranteeing dynamic characteristics of a system despite interval parametric uncertainty of its object. It is proposed to base a controller synthesis on root quality indices: minimal stability degree and maximal oscillability degree. Desired values of these indices will be provided with the help of dominant poles method. Applying this method consists in placing a pair of complex-conjugate dominant poles; all other poles — unrestricted poles — will be placed by defining a right border of their allocation area on a complex plane. To apply dominant poles method, a feature of stability degree and oscillability degree to be determined by images of certain vertices of a parametric polytope was used. To synthesize a controller, it is proposed to divide its parameters in two groups: dependent ones and unrestricted ones. The first group of controller parameters is to provide desired allocation of dominant poles in one of vertices of parametric polytope (a dominant vertex). Unrestricted parameters of a controller are to provide desired distance between dominant poles and allocation area of unrestricted poles. To find coordinates of a dominant vertex and verifying vertices providing unrestricted poles allocation, an interval extension of basic phase equation of a root locus theory was developed. This resulted in interval phase inequalities, whose solution allows finding coordinates of desired vertices of characteristic polynomials coeffi cients polytope. Knowing a dominant vertex polynomial and dominant poles allows expressing dependent parameters of a controller from unrestricted ones. Obtained expressions allow placing unrestricted poles in a desired area of a complex plane. To do this, a D-partition by unrestricted parameters of a controller is performed in all verifying vertices of parametric polytope of a system. After choosing values of unrestricted parameters from intersection of all stability domains obtain during D-partition, dependent parameters of a controller can be calculated. An example of synthesizing a PID-controller guaranteeing desired values of dynamics characteristics for an interval control system of the fourth order is provided.