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The iterative decreasing dimension method (IDDM) is an iterative method used to solve the linear algebraic system A x = f . Such systems are important in modeling many problems in applied sciences. For a number of reasons, such as estimated measurements made for modeling, errors arising from floating point calculations, and approximation methods used for solutions, it becomes necessary to study intervals in the solutions of systems of linear equations. The objective of this paper is to utilize IDDM to achieve resolution in the interval linear system (ILS). During the calculations, the Kaucher space is considered an extended classical interval space. The solutions of Barth-Nuding and Hansen interval linear systems, which are commonly used in the literature to test the solutions of ILSs, are obtained with the interval iterative decreasing dimension method for interval linear systems (I-IDDM). Since IDDM is a variation method of Gaussian elimination, a comparative analysis of the results with the interval Gaussian elimination method (I-GEM) is performed. It has been demonstrated that our approach, I-IDDM, produces better outcomes than I-GEM. I-IDDM is also used to investigate the analog circuit problem, where interval analysis is crucial. [ABSTRACT FROM AUTHOR]

This paper provides an overview of the necessary and sufficient conditions for guaranteeing the unique solvability of absolute value equations. In addition to discussing the basic form of these equations, we also address several generalizations, including generalized absolute value equations and matrix absolute value equations. Our survey encompasses known results as well as novel characterizations proposed in this study. [ABSTRACT FROM AUTHOR]

In this study, Schur stability, sensitivity and continuity theorems have been mentioned. In addition, matrix families, interval matrix and extend of the intervals also have been mentioned. The IL and IC intervals of the matrix families have been determined so that the linear sums family L and convex combination family C are Schur stable. Samely, the I*L and I*C intervals have been determined and L and C are ω*-Schur stable. Afterwards, the methods which based on continuity theorems and the algorithms which based on the methods have been given. Extended intervals have been obtained with the help of the methods and the algorithms. All definitions are supported by examples. [ABSTRACT FROM AUTHOR]

Summary: Well‐known results on robust Hurwitz and Schur stability of interval polynomials and interval matrices are discussed. We fill in some gap in this theory. It is proved that, for n=3$$ n=3 $$, Schur stability of vertex matrices is not sufficient for robust Schur stability of the interval matrix. A certain general method for constructing examples of this kind is presented. [ABSTRACT FROM AUTHOR]

The iterative decreasing dimension method (IDDM) is an iterative method used to solve the linear algebraic system Ax=f. Such systems are important in modeling many problems in applied sciences. For a number of reasons, such as estimated measurements made for modeling, errors arising from floating point calculations, and approximation methods used for solutions, it becomes necessary to study intervals in the solutions of systems of linear equations. The objective of this paper is to utilize IDDM to achieve resolution in the interval linear system (ILS). During the calculations, the Kaucher space is considered an extended classical interval space. The solutions of Barth-Nuding and Hansen interval linear systems, which are commonly used in the literature to test the solutions of ILSs, are obtained with the interval iterative decreasing dimension method for interval linear systems (I-IDDM). Since IDDM is a variation method of Gaussian elimination, a comparative analysis of the results with the interval Gaussian elimination method (I-GEM) is performed. It has been demonstrated that our approach, I-IDDM, produces better outcomes than I-GEM. I-IDDM is also used to investigate the analog circuit problem, where interval analysis is crucial.

In this paper, the generalized eigenvalue problem for a pair of interval matrices is studied. First, the interval enclosures of every generalized eigenvalue and corresponding eigenvector for a pair of symmetric interval matrices are determined. One of these matrices is positive definite and nonnegative invertible. Next, the enclosure of all the generalized eigenvalues for a pair of interval matrices is derived under some assumptions, in which both of the interval matrices are not necessarily symmetric. [ABSTRACT FROM AUTHOR]

One of the main missions of a certain company is to predict its future for reasons of continuity, which reflect the balance of its long term, in various aspects. In this work, we propose the use of Markov Interval Chain models to help business leaders to make better decisions. The proposed model consists in considering the numbers of customers declared by each company, which are discrete values as centers of symmetric intervals. By this, we have avoided the problem of increase and decrease in the number of customers for each company. As an example, we applied this model to predict the distribution of market shares in the later period as a probability distribution intervals, which provides information's for companies to make decisions, and it gave satisfactory results. [ABSTRACT FROM AUTHOR]

Due to their relation to the linear complementarity problem, absolute value equations have been intensively studied recently. In this paper, we present error bound conditions for absolute value equations. Along with the error bounds, we introduce a condition number. We consider general scaled matrix p-norms, as well as particular p-norms. We discuss basic properties of the condition number, including its computational complexity. We present various bounds on the condition number, and we give exact formulae for special classes of matrices. Moreover, we consider matrices that appear based on the transformation from the linear complementarity problem. Finally, we apply the error bound to convergence analysis of two methods for solving absolute value equations. [ABSTRACT FROM AUTHOR]

This study outlines the necessary and sufficient criteria for swarm stability asymptotically, meaning consensus in a class of fractional-order multi-agent systems (FOMAS) with interval uncertainties for both fractional orders 0 < α < 1 and 1 < α < 2. The constraints are determined by the graph topology, agent dynamics, and neighbor interactions. It is demonstrated that the fractional-order interval multi-agent system achieves consensus if and only if there are some Hermitian matrices that satisfy a particular kind of complex Lyapunov inequality for all of the system vertex matrices. This is done by using the existence condition of the Hermitian matrices in a Lyapunov inequality. To do this, at first it is shown under which conditions a multi-agent system with unstable agents can still achieve consensus. Then, using a lemma and a theory, the Lyapunov inequality regarding the negativity of the maximum eigenvalue of an augmented matrix of a FOMAS is used to find some Hermitian matrices by checking only a limited number of system vertex matrices. As a result, the necessary and sufficient conditions to reach consensus in a FOMAS in the presence of internal uncertainties are obtained according to the Lyapunov inequalities. Using the main theory of the current paper, instead of countless matrices, only a limited number of vertex matrices need to be used in Lyapunov inequalities to find some Hermitian matrices. As a confirmation of the notion, some instances from numerical simulation are also provided at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we focus on generalizing Bπ R -matrices into the interval setting, but we prove that, in a sense, they are one. We derive mainly recognition methods for this interval matrix class, such as characterizations, necessary conditions, and sufficient ones. Next, we also take a look at interval B-matrices and interval doubly B-matrices, which were introduced recently, and we present characterizations through reduction for them and for Bπ R -matrices into the interval setting, but we prove that, in a sense, they are one. We derive mainly recognition methods for this interval matrix class, such as characterizations, necessary conditions, and sufficient ones. Next, we also take a look at interval B-matrices and interval doubly B-matrices, which were introduced recently, and we present characterizations through reduction for them and for Bπ R -matrices. [ABSTRACT FROM AUTHOR]

Motivated by recently published papers, we discuss the problem of solving absolute value equations of the form A x + | x | = b . Specifically, we show the equivalence of two sufficient conditions for the unsolvability of such equations which are based on linear programming. Furthermore, we prove that two generalizations of sufficient conditions for unique solvability do not extend the class at all, that is, they are equivalent to the original formulations. The conditions based on M-matrices and H-matrices also turn out to be special cases of the known statements. Eventually, we also point out two incorrect statements, and for one of them, we propose some corrected reformulations. [ABSTRACT FROM AUTHOR]

The purpose of this article is to put the concept of interval linear algebra on a sound algebraic setting, by judiciously defining a Field and a Vector Space over a field involving equivalence classes. Classifying mathematical objects by bringing them into canonical forms, is an important subject of study in every branch of Mathematics. The concepts of eigenvalues and eigenvectors are used in several fields including machine learning, quantum computing, communication system design, construction design, electrical and mechanical engineering etc. The study of interval matrices leads to canonical forms of interval matrices, which will help us classify and study interval matrices more effectively and more easily.

In this paper, a novel comprehensive model for the two-motor web-winding system is proposed, which combines the classical web-winding system model with the driven-motor model. Based on the proposed model, the state feedback robust control (SFRC) strategy is designed to handle systematic uncertainty and time-varying parameters. And then in order to reduce the control cost of the web-winding system, an optimal robust guaranteed cost control (ORGCC) approach is studied. First, the reference control inputs and error dynamic model are derived based on the proposed comprehensive model. Then, the SFRC strategy and ORGCC approach are designed to calculate control compensation. By regarding some suitable parameters as interval variables, the proposed controllers have good robustness for parameter variations and environmental disturbance. Finally, the simulation results illustrate the effectiveness of the proposed model and control. [ABSTRACT FROM AUTHOR]

The paper considers robust stability study of continuous and discrete interval dynamic systems by algebraic method. The original robustness results obtained for continuous and discrete linear interval dynamic systems within the algebraic direction of robustness stability are presented. The author formulated and proved the basic theorem on the robustness of linear continuous dynamic system with interval elements of the right-hand part matrix, which is determined through the separate angular coefficients of characteristic polynomial of the system. The basic theorem is proved on the basis of a lemma on the separative coefficients of the characteristic polynomial obtained by optimization methods of nonlinear programming on multiple interval elements of the system matrix. Their possible values can be the upper or lower limits of the corresponding interval or zero. A clarification note to the basic theorem for continuous systems is formulated. The idea lies in the need for a complete set of four angular polynomials for the robustness stability of the system, excluding multiple cases of the characteristic polynomial, when the set of Kharitonov polynomials degenerates and will consist of the less required four different polynomials. The theorem is obtained on the necessary and sufficient conditions of robustness stability for the polyhedron of interval matrices. A discrete analogue of the Kharitonov theorem is obtained for discrete systems. The algorithm of robustness stability determination for discrete interval dynamic systems is presented. Comparative characteristics of the results obtained in the works of well-known authors having studied the algebraic trend of robust stability problem are considered. They show the distinctive feature of this method, which consists in onsideration of interval matrices of general type. The validity of the method is tested on the known counterexamples to Bialas’s theorem, as well as the other researchers studying robustness problems of interval dynamic systems.

This study outlines the necessary and sufficient criteria for swarm stability asymptotically, meaning consensus in a class of fractional-order multi-agent systems (FOMAS) with interval uncertainties for both fractional orders 0 < α < 1 and 1 < α < 2. The constraints are determined by the graph topology, agent dynamics, and neighbor interactions. It is demonstrated that the fractional-order interval multi-agent system achieves consensus if and only if there are some Hermitian matrices that satisfy a particular kind of complex Lyapunov inequality for all of the system vertex matrices. This is done by using the existence condition of the Hermitian matrices in a Lyapunov inequality. To do this, at first it is shown under which conditions a multi-agent system with unstable agents can still achieve consensus. Then, using a lemma and a theory, the Lyapunov inequality regarding the negativity of the maximum eigenvalue of an augmented matrix of a FOMAS is used to find some Hermitian matrices by checking only a limited number of system vertex matrices. As a result, the necessary and sufficient conditions to reach consensus in a FOMAS in the presence of internal uncertainties are obtained according to the Lyapunov inequalities. Using the main theory of the current paper, instead of countless matrices, only a limited number of vertex matrices need to be used in Lyapunov inequalities to find some Hermitian matrices. As a confirmation of the notion, some instances from numerical simulation are also provided at the end of the paper.

In max-min algebra the standard pair of operations plus and times is replaced by the pair of operations maximum and minimum, respectively. A max-min matrix A is called strongly robust if the orbit x;A x;A2 x;: :: reaches the greatest eigenvector with any starting vector. We study a special type of the strong robustness called the strong X-robustness, the case that a starting vector is limited by a lower bound vector and an upper bound vector. The equivalent condition for the strong X-robustness is introduced and efficient algorithms for verifying the strong X-robustness is described. The strong X-robustness of a max-min matrix is extended to interval vectors X and interval matrices A using for-all-exists quantification of their interval and matrix entries. A complete characterization of AE/EA strong X-robustness of interval circulant matrices is presented. [ABSTRACT FROM AUTHOR]

This article investigates a coordinated optimal guaranteed cost control (DCOGCC) for a multi-motor roll-to-roll web machine with time-varying and uncertain parameters. The state–space oriented DCOGCC law is designed to ensure that each subsystem of the web machine is stable and the upper bound of performance index is minimum. Linear matrix inequality (LMI) conditions to guarantee the stability of the systems with DCOGCC are derived. A decentralized model is adopted by regarding the entire machine as a synthetic system, then the original decentralized model is transformed into an equivalent one. By choosing appropriate information of consecutive subsystems as coordination variables, the DCOGCC scheme may attenuate the interaction effects between subsystems. The results of some simulations and experiments are given to verify the feasibility and effectiveness of the DCOGCC. • A decentralized coordinated optimal guaranteed cost control (DCOGCC) is proposed for a web machine. • Local state-feedback and interaction feedback information are used to design DCOGCC law. • The interval matrix is introduced to deal with the parametric uncertainties. • Extensive comparative simulations and experiments are conducted to show the effectiveness of the DCOGCC. [ABSTRACT FROM AUTHOR]

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To deal with the nonlinear coupling, strong disturbances and parametric uncertainties existing in the multi-motor web-winding system, a robust decentralized H∞ control method is proposed in this paper. First, the whole web-winding system is considered as a synthetic system composed of several subsystems. Giving that some parameters are time dependent and set points are modified during the winding process, each subsystem can be seen as a dynamic interval system and interval matrix is introduced. Then, a robust decentralized H∞ controller is designed to attenuate tension fluctuations introduced by the external disturbances and interaction between two consecutive subsystems. Sufficient condition for the existence of robust decentralized H∞ control law is presented in terms of linear matrix inequality. Finally, some simulations and experiments are conducted with the conventional decentralized controller and the proposed controller. The comparative results show that the proposed control scheme greatly improves the control performance of the web tensions.

In max-min fuzzy algebra, the study of eigenvectors is important because it can help us to recognize the steady states of systems working in discrete steps. This paper investigates the properties of steady states described by max-min matrices and vectors with interval coefficients. The characteristics of the eigenspace structure and polynomial-time algorithms for recognition of tolerable and weakly-tolerable interval eigenvectors in max-min algebra are described. This research is a continuation of an earlier investigation concerning strongly-tolerable interval eigenvectors. [ABSTRACT FROM AUTHOR]

The Wiener index is classic and well-known topological index for the characterization of molecular graphs. In this paper, we study interval weighted graphs such as graphs where the edge weights are interval or interval matrices. In this study firstly we define interval weighted graph. Later we define Wiener index of interval weighted graph. From this definition, we give algorithms for finding Wiener index of interval weighted directed and undirected graph. [ABSTRACT FROM AUTHOR]

In this paper, we first give an overview of doubly stochastic interval matrices. Then, we present some theories about the interval linear system whose coefficient matrix is doubly stochastic interval matrix. Also, we give an outer estimation for the solution set of these systems. [ABSTRACT FROM AUTHOR]

Computing powers of interval matrices is a computationally hard problem. Indeed, it is NP-hard even when the exponent is 3 and the matrices only have interval components in one row and one column. Motivated by this result, we consider special types of interval matrices where the interval components occupy specific positions. We show that computing the third power of matrices with only one column occupied by interval components can be solved in cubic time; so the asymptotic time complexity is the same as for the real case (considering the textbook matrix product method). We further show that for a fixed exponent k and for each interval matrix (of an arbitrary size) whose kth power has components that can be expressed as polynomials in a fixed number of interval variables, the computation of the kth power is polynomial up to a given accuracy. Polynomiality is shown by using the Tarski method of quantifier elimination. This result is used to show the polynomiality of computing the cube of interval band matrices, among others. Additionally, we study parametric matrices and prove NP-hardness already for their squares. We also describe one specific class of interval parametric matrices that can be squared by a polynomial algorithm. [ABSTRACT FROM AUTHOR]

We generalize the Monge property of real matrices for interval matrices. We define two classes of interval matrices with the Monge property—in a strong and a weak sense. We study the fundamental properties of both types. We show several different characterizations of the strong Monge property. For the weak Monge property, we give a polynomial description and several sufficient and necessary conditions. For both classes, we study closure properties. We further propose a generalization of an algorithm by Deineko and Filonenko which for a given matrix returns row and column permutations such that the permuted matrix is Monge if the permutations exist. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

There are several methods to compute the optimal bounds of the objective function for interval quadratic programming (IQP) problems, but no method has yet been suggested to calculate a set of optimal solutions of IQP problems. This paper presents an accurate set of optimal solutions for the interval quadratic programming problems. The optimal solution of the quadratic programming problem is not essentially an extreme point. We first propose conditions that make the optimal solutions of the IQP to extreme points and then, using these conditions, we compute the exact set of optimal solutions for the IQP problem. Under these conditions, we show the intersection of two regions is equal to a set of optimal solutions of IQP. [ABSTRACT FROM AUTHOR]

Max-min algebra is an algebraic structure in which classical addition and multiplication are replaced by maximum and minimum, respectively. The notation A ⊗ X ⊗ C = B , where A , B , and C are given interval matrices, represents an interval max-min matrix equation. The paper deals with the solvability of interval matrix equations in max-min algebra. We define three types of solvability of interval max-min matrix equations, namely the strongly universal, universal and weakly universal solvability. We provide the equivalent conditions for each type of solvability that can be verified in polynomial times. [ABSTRACT FROM AUTHOR]

This paper is concerned with quasi-synchronization of delayed memristive neural networks (MNNs) with switching jumps mismatches via aperiodically intermittent control. The issue is presented for three reasons: 1) the existing controllers for synchronization may be too complicated and not economical; 2) under the influence of switching jumps mismatches, synchronization of MNNs may fail to achieve; and 3) matrix measure method is less conservative but cannot be applied directly to synchronization of MNNs. To overcome these difficulties, the concept of asynchronously switching time interval is proposed to describe the phenomenon when the drive-response MNNs switch their connection weights asynchronously. Then, aperiodically intermittent control is designed and quasi-synchronization analysis is carried out based on a combined method that compromises the merits of interval matrix method and matrix measure method. A quasi-synchronization criterion, expressed in terms of the mixture of ${p}$ -norm and matrix measure of the memristive connection weights, is established. Meanwhile, the fundamental reason for the failure of complete synchronization is revealed. Moreover, an explicit expression of the error level is obtained and the design of the controller under a predetermined error level is presented. The obtained results in this paper reduce the conservativeness and provide a novel insight into the research of synchronization of MNNs. [ABSTRACT FROM AUTHOR]

In max-min algebra the standard pair of operations: plus and times is substituted by pair of operations ⊕ and ⊗, where a ⊕ b = max ⁡ { a , b } , a ⊗ b = min ⁡ { a , b }. A square matrix A is X − robust if its eigenspace is reached starting at each vector of a given interval vector X = { x ; x _ ≤ x ≤ x ‾ }. Various versions of the reachability for an interval vector X and an interval matrix A = { A ; A _ ≤ A ≤ A ‾ } depending on the used quantifiers and their order are studied. The universal, possible, tolerance and weak tolerance A -robustness of X and X -robustness of A over max-min algebra are defined and polynomial algorithms for its checking in the case of circulant matrices are described. [ABSTRACT FROM AUTHOR]

In max–min fuzzy algebra, the standard pair of operations—plus and times—is substituted by a different pair of operations—maximum and minimum—which are involved in many optimization problems. Investigation of the properties of eigenvectors is important for these applications. The values of vector or matrix inputs in practice are usually not exact numbers and some intervals can instead be considered as values. This paper investigates the properties of matrices and vectors with interval coefficients. In addition, a complete solution of the strong tolerance interval eigenproblem in max–min algebra is presented. • The eigenproblem in max–min algebra for matrices and vectors with interval coefficients is investigated. • The strong tolerance interval eigenproblem has been completely solved. • Preserving the interval security level in computer net with data storage units and logical units is described. [ABSTRACT FROM AUTHOR]

Traditionally, game theory problems were considered for exact data, and the decisions were based on known payoffs. However, this assumption is rarely true in practice. Uncertainty in measurements and imprecise information must be taken into account. The interval-based approach for handling such uncertainties assumes that one has lower and upper bounds on payoffs. In this paper, interval bimatrix games are studied. Especially, we focus on three kinds of support set invariancy. Support of a mixed strategy consists of that pure strategies having positive probabilities. Given an interval-valued bimatrix game and supports for both players, the question states as follows: Does every bimatrix game instance have an equilibrium with the prescribed support? The other two kinds of invariancies are slight modifications: Has every bimatrix game instance an equilibrium being a subset/superset of the prescribed support? It is computationally difficult to answer these questions: the first case costs solving a large number of linear programs or mixed integer programs. For the remaining two cases a sufficient condition and a necessary condition are proposed, respectively. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the attitude tracking control of quadrotor unmanned aerial vehicles (UAV) with uncertainties and external disturbances. The control system is expressed on the special orthogonal group, SO(3), to avoid singularities and ambiguities associated with Euler angles and quaternions respectively. It is also the basis for achieving large angle tracking and complex UAV maneuver. Meanwhile, it greatly reducing the complexity of the controller design via variation-based linearization. And the approach of interval matrix is introduced to solve problems of controller online calculation and reduced control accuracy caused by uncertain parameters in model. The proposed robust H ∞ control system can exponential asymptotically follow a desired attitude trajectory in the presence of unstructured bounded disturbances. Finally, several simulation results are provided to validate the effectiveness, advantages and robustness of the proposed method. Highlights • Take disturbances into consideration and put forward a variation-based linearized attitude error system for quadrotor UAV on SO(3). • The interval matrix approach is introduced to describe the parameters changing within a certain range, including time-varying parameters and uncertain parameters. • The robust H ∞ attitude tracking control scheme for quadrotor on SO(3) is proposed. • The LMI exponential stability sufficient conditions are derived. • Simulations are done to demonstrate the effectiveness, advantages and robustness when implementing large angle flip and complex flight maneuvers. [ABSTRACT FROM AUTHOR]

In this paper, we give a necessary and sufficient condition for the powers of an interval matrix to be nilpotent. We show an interval matrix A is nilpotent if and only if p(B) = 0, where B is a point matrix, introduced by Mayer (Linear Algebra Appl. 58 (1984) 201-216), constructed by the (*) property. We observed that the spectral radius, determinant, and trace of a nilpotent interval matrix equal zero but in general its converse is not true. Some properties of nonnegative nilpotent interval matrices are derived. We also show that an irreducible interval matrix A is nilpotent if and only if |A| is nilpotent. [ABSTRACT FROM AUTHOR]

This study is concerned with the distributed fault tolerant tracking control for large‐scale multi‐motor web‐winding systems. Firstly, the web‐winding system is considered a synthetic system with several dynamic subsystems subject to multiple disturbances and actuator faults. Then, a disturbance compensation‐based distributed fault diagnosis scheme is developed to estimate the system actuator faults. By setting the reference outputs, a method of obtaining equilibrium control inputs and equilibrium states is formulated to achieve the distributed variation dynamic model at equilibrium points by Taylor expansion. Based on the estimated actuator faults, an effective distributed fault tolerant control strategy is proposed by the techniques of interval matrix and disturbance local compensation. Sufficient conditions of asymptotic stability of the estimation error systems and the closed‐loop systems are derived based on the Lyapunov theory. A fault diagnosis observer and a fault tolerant controller gain matrices are obtained by solving the linear matrix inequalities. Finally, simulations and analysis are performed on the three‐motor web‐winding system to verify the effectiveness of the proposed distributed fault tolerant control strategy. [ABSTRACT FROM AUTHOR]

Of central importance to the αBB algorithm is the calculation of the α values that guarantee the convexity of the underestimator. Improvement (reduction) of these values can result in tighter underestimators and thus increase the performance of the algorithm. For instance, it was shown by Wechsung et al. (J Glob Optim 58(3):429-438, 2014) that the emergence of the cluster effect can depend on the magnitude of the α values. Motivated by this, we present a refinement method that can improve (reduce) the magnitude of α values given by the scaled Gerschgorin method and thus create tighter convex underestimators for the αBB algorithm. We apply the new method and compare it with the scaled Gerschgorin on randomly generated interval symmetric matrices as well as interval Hessians taken from test functions. As a measure of comparison, we use the maximal separation distance between the original function and the underestimator. Based on the results obtained, we conclude that the proposed refinement method can significantly reduce the maximal separation distance when compared to the scaled Gerschgorin method. This approach therefore has the potential to improve the performance of the αBB algorithm. [ABSTRACT FROM AUTHOR]

Industrial robotic and mechatronic systems in real machinery are quadratically connected dynamical systems with parametric uncertainties, whose mathematical models can be extremely complex. Program motion stabilization of such systems, especially those described in quasi-coordinates, is a topical scientific problem, which does not yet have an acceptable for engineering developments comprehensive solution. The article proposes a possible concept of such a solution, based on the Lyapunov's direct method, extended to the interval dynamic systems domain with time derivatives of state variables written in the interval cubic forms. The authors conceptual approach is based on new concepts introduced in the article: covering form, form compensator, modular variables form. According to this approach, first of all, the transition to new complex system state variables formed with signatures is carried out. The mathematical apparatus is based on special interval matrix algebra transformations, which could not be included in the article because of the volume, and therefore are only stated. As a result, the authors obtain a new class of quasi-relay type controls with system state dependent modulated impact jumps, and therefore combining the properties of both soft smooth and hard forcing types. [ABSTRACT FROM AUTHOR]

The Łukasiewicz conjunction (sometimes also considered to be a logic of absolute comparison), which is used in multivalued logic and in fuzzy set theory, is one of the most important t-norms. In combination with the binary operation ‘maximum’, the Łukasiewicz t-norm forms the basis for the so-called max-Łuk algebra, with applications to the investigation of systems working in discrete steps (discrete events systems; DES, in short). Similar algebras describing the work of DES’s are based on other pairs of operations, such as max-min algebra, max-plus algebra, or max-T algebra (with a given t-norm, T). The investigation of the steady states in a DES leads to the study of the eigenvectors of the transition matrix in the corresponding max-algebra. In real systems, the input values are usually taken to be in some interval. Various types of interval eigenvectors of interval matrices in max-min and max-plus algebras have been described. This paper is oriented to the investigation of strong, strongly tolerable, and strongly universal interval eigenvectors in a max-Łuk algebra. The main method used in this paper is based on max-Ł linear combinations of matrices and vectors. Necessary and sufficient conditions for the recognition of strong, strongly tolerable, and strongly universal eigenvectors have been found. The theoretical results are illustrated by numerical examples.