Interval Matrices Research Articles

An interval extension of SMS method for computing weighted Moore–Penrose inverse

An interval extension of successive matrix squaring (SMS) method for computing the weighted Moore---Penrose inverse $$A^{\dagger }_{MN}$$AMN? along with its rigorous error bounds is proposed for given full rank $$m \times n$$m?n complex matrices A, where M and N be two Hermitian positive definite matrices of orders m and n, respectively. Starting with a suitably chosen complex interval matrix containing $$A^{\dagger }_{MN}$$AMN?, this method generates a sequence of complex interval matrices each enclosing $$A^{\dagger }_{MN}$$AMN? and converging to it. A new method is developed for constructing initial complex interval matrix containing $$A^{\dagger }_{MN}$$AMN?. Convergence theorems are established. The R-order convergence is shown to be equal to at least l, where $$l \ge 2$$l?2. A number of numerical examples are worked out to demonstrate its efficiency and effectiveness. Graphs are plotted to show variations of the number of iterations and computational times compared to matrix dimensions. It is observed that ISMS is more stable compared to SMS.

On the Interval Generalized Coupled Matrix Equations

In this work, the interval generalized coupled matrix equations \begin{equation*} \sum_{j=1}^{p}{{\bf{A}}_{ij}X_{j}}+\sum_{k=1}^{q}{Y_{k}{\bf{B}}_{ik}}={\bf{C}}_{i}, \qquad i=1,\ldots,p+q, \end{equation*} are studied in which ${\bf{A}}_{ij}$, ${\bf{B}}_{ik}$ and ${\bf{C}}_{i}$ are known real interval matrices, while $X_{j}$ and $Y_{k}$ are the unknown matrices for $j=1,\ldots,p$, $k=1,\ldots,q$ and $i=1,\ldots,p+q$. This paper discusses the so-called AE-solution sets for this system. In these types of solution sets, the elements of the involved interval matrices are quantified and all occurrences of the universal quantifier $\forall$ (if any) precede the occurrences of the existential quantifier $\exists$. The AE-solution sets are characterized and some sufficient conditions under which these types of solution sets are bounded are given. Also some approaches are proposed which include a numerical technique and an algebraic approach for enclosing some types of the AE-solution sets.

On the power sequence of a fuzzy interval matrix with max-min operation

An $$n \times n $$ interval matrix $${\mathcal {A}}= [\underline{A},\overline{A}]$$ is called to be a fuzzy interval matrix if $$0 \le \underline{A}_{ij} \le \overline{A}_{ij}\le 1$$ for all $$1 \le i, j \le n$$ . In this paper, we proposed the notion of max-min algebra of fuzzy interval matrices. We show that the max-min powers of a fuzzy interval matrix either converge or oscillate with a finite period. Conditions for limiting behavior of powers of a fuzzy interval matrix are established. Some properties of fuzzy interval matrices in max-min algebra are derived. Necessary and sufficient conditions for the powers of a fuzzy interval matrices in max-min algebra to be nilpotent are proposed as well.

Universal solvability of interval max-plus matrix equations

This paper deals with the solvability of interval matrix equations in max-plus algebra. Max-plus algebra is the algebraic structure in which classical addition and multiplication are replaced by ⊕ and ⊗, where a⊕b=max{a,b} and a⊗b=a+b.The notation A⊗X⊗C=B, where A, B, and C are given interval matrices, represents an interval max-plus matrix equation. We define three types of solvability of interval max-plus matrix equations, namely the strong universal , universal , and weak universal solvability. We derive the necessary and sufficient conditions which can be verified in polynomial times.

The united stable solution set of interval continuous-time algebraic Riccati equation and verified numerical computation of its outer estimation

This paper introduces the interval continuous-time algebraic Riccati equation $\mathbf{A}^* X + X\mathbf{A} + \mathbf{Q} -X \mathbf{G} X=0$, where $\mathbf{A}, \mathbf{G}$, and $\mathbf{Q}$ are known $n \times n$ complex interval matrices, $\mathbf{G}$ and $\mathbf{Q}$ are Hermitian, and $X$ is an unknown matrix of the same size, and develops two approaches for enclosing the united stable solution set of this interval equation. We first discuss the united stable solution set and then derive a nonlinear programming method in order to find an enclosure for the united stable solution set. We also advance an efficient technique for enclosing the united stable solution set based on a variant of the Krawczyk method together with some modifications. These modifications enable us to reduce the computational complexity significantly. Various numerical experiments established upon a number of standard benchmark examples are also given to show the efficiency of this modified Krawczyk technique.

Analytical aspects of the interval unilateral quadratic matrix equations and their united solution sets

This paper introduces the emph{interval unilateral quadratic matrix equation}, $IUQe$ and attempts to find various analytical results on its AE-solution sets in which $A,B$ and $CCC$ are known real interval matrices, while $X$ is an unknown matrix. These results are derived from a generalization of some results of Shary. We also give sufficient conditions for non-emptiness of some quasi-solution sets, provided that $A$ is regular. As the most common case, the united solution set has been studied and two direct methods for computing an outer estimation and an inner estimation of the united solution set of an interval unilateral quadratic matrix equation are proposed. The suggested techniques are based on nonlinear programming as well as sensitivity analysis.

Interval matrices: Regularity generates singularity

It is proved that regularity of an interval matrix implies singularity of four related interval matrices. The result is used to prove that for each nonsingular point matrix A, either A or A−1 can be brought to a singular matrix by perturbing only the diagonal entries by an amount of at most 1 each. As a consequence, the notion of a diagonally singularizable matrix is introduced.

A general unified framework for interval pairwise comparison matrices

Interval Pairwise Comparison Matrices have been widely used to account for uncertain statements concerning the preferences of decision makers. Several approaches have been proposed in the literature, such as multiplicative and fuzzy interval matrices. In this paper, we propose a general unified approach to Interval Pairwise Comparison Matrices, based on Abelian linearly ordered groups. In this framework, we generalize some consistency conditions provided for multiplicative and/or fuzzy interval pairwise comparison matrices and provide inclusion relations between them. Then, we provide a concept of distance between intervals that, together with a notion of mean defined over real continuous Abelian linearly ordered groups, allows us to provide a consistency index and an indeterminacy index. In this way, by means of suitable isomorphisms between Abelian linearly ordered groups, we will be able to compare the inconsistency and the indeterminacy of different kinds of Interval Pairwise Comparison Matrices, e.g. multiplicative, additive, and fuzzy, on a unique Cartesian coordinate system.

Sufficient regularity conditions for complex interval matrices and approximations of eigenvalues sets

In this paper, two approaches are described to establish verifiable sufficient regularity conditions of complex interval matrices. In the first approach, a complex interval matrix is mapped to a real block interval matrix and then its sufficient regularity conditions are obtained. In the second approach, a necessary condition for the singularity of a complex interval matrix is derived and used to get its sufficient regularity conditions. As an application, the above derived sufficient regularity conditions are used to investigate the location of the outer approximations of individual eigenvalue sets of complex interval matrices. Two algorithms are proposed and results obtained are compared with those obtained by earlier methods and Monte Carlo simulations. The advantages of these algorithms are that they can detect gaps in between the approximations of the whole eigenvalue sets. The second algorithm is very effective compared to the first algorithm from the computational time point of view. Several numerical examples and statistical experiments are worked out to validate and demonstrate the efficacy of our work.

State estimation for neural networks with jumping interval weight matrices and transmission delays

Abstract This work addresses the state estimation problem for the neural networks possessing Markovian jump weight matrices and transmission delays. Markovian jump interval matrices are introduced to model the uncertainty of the connections among neurons, and the polytopic model is used to describe the uncertainty of the Markov transition matrix. The mode-dependent transmission delays are introduced to describe an unideal communication channel. A sufficient condition of the stochastic stability and the strict-(Q, S, R) dissipative performance is derived for the augmented system. Mode-dependent estimator gains are then designed. At last, an example is employed to illustrate the obtained results.

Model Validation of PWM DC–DC Converters

This paper presents hybrid automaton modeling, comparative model validation, and formal verification of stability through reachability analysis of pulse width modulation (PWM) dc–dc converters. Conformance degree provides a measure of closeness between the proposed hybrid automata models and experimental data. Nondeterminism due to variations in circuit parameters is modeled using interval matrices. In direct contrast to the unsound and computationally-intensive Monte Carlo simulation, reachability analysis is introduced to overapproximate the set of reachable states and ensure stable operation of PWM dc–dc converters. Using a 200 W experimental prototype of a buck converter, hybrid automata models of open-loop, and hysteresis-controlled converters are first validated against experimental data using their conformance degrees. Next, converter stability is formally verified through reachability analysis and informally validated using Monte Carlo simulations and experimental results.

A Convex Approach for Reducing Conservativeness of Kharitonov’s-Based Robustness Analysis

This note shows that conservativeness of former results on stability analysis of uncertain systems based on the Kharitonov’s Theorem can be significantly reduced by means of convex structures which lead to linear matrix inequalities. The relationship between an interval-coefficient family of polynomials and interval matrices is exploited in order to obtain novel LMI tests whose advantages are twofold: feasibility domains (both in ordinary as well as D-stability) are increased while LMIs are efficiently solved via convex optimization techniques. Illustrative examples are provided.

ON THE CHARACTERIZATION OF TASKS MODELED BY INTERVAL DESIGN STRUCTURE MATRIX ON DOMAIN-DRIVEN DESIGN SOFTWARE DEVELOPMENT

Development and design of new products of various kinds often contain a very complex set of relationships among many coupled tasks. Ranking, controlling and redesigning the features of these tasks can be usefully performed by a suitable model based on the design structure matrix in an iteration procedure. The proposed interval approach of design iteration controls and predicts the convergence speed of iteration work on tasks within a project. Interval method is based on Perron-Frobenius theorem and interval linear algebra where intervals and interval matrices are employed instead of real numbers and real matrices. In this way, a more relaxed quantitative estimation of tasks is achieved and the presence of undetermined quantities is allowed to a certain extent. The presented model is demonstrated in the example of simplified domain-driven design process, an approach to software development.

Fast verified computation for stabilizing solutions of discrete-time algebraic Riccati equations

Fast iterative algorithms for computing interval matrices containing solutions of discrete-time algebraic Riccati equations are proposed. These algorithms involve only cubic complexity per iteration. The stabilizability and uniqueness of the contained solution can moreover be verified by these algorithms. Numerical results show the properties of the algorithms.

Extremal eigenvalue intervals of symmetric tridiagonal interval matrices

SummaryComputing the extremal eigenvalue bounds of interval matrices is non‐deterministic polynomial‐time (NP)‐hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices.

Strong regularity of parametric interval matrices

Several verifiable sufficient and necessary conditions for strong regularity of parametric interval matrices are given. The conditions are formulated in terms of the spectral radius, positive definiteness and eigenvalues of certain real matrices and a scaled maximum norm.

Eigenvalue bounds for symmetric matrices with entries in one interval

In this paper, we consider the eigenvalue bounds for real symmetric matrices whose entries are in a given interval. Based on some matrix computation theories, a new algorithm and some theorems are presented which provide ways for solving the left problems in former papers and obtain satisfying results. Furthermore, the ideas are spread to general symmetric interval matrices. Finally, numerical examples are presented which show the validity of the proposed methods.

A multiplicative perturbation method for satellite orbit prediction based on symplectic property

In order to predict artificial satellite orbit accurately, complex dynamic models need to be considered and numerical integration algorithms are used to solve the problem. Desired effect has been obtained for orbit prediction by various integration algorithms such as Runge-Kutta algorithm and Adams-Cowell algorithm. But the symplectic property of the dynamic problem is not taken into consideration, therefore the inherent characteristics of the system are neglected. In this paper, the multiplictive perturbation method based on symplectic propterty, which adapts to orbit prediction, is proposed. Hamiltonian canonical equation, which describes the motion of the satellite, is divided into two sections, including two-body problem and perturbation section. Analytical solution of two-body problem is used. Perturbation section is solved approximately by interval matrices algorithm. Because state transition matrix of the two-body problem is a symplectic matrix, it is a symplectic structure-preserving process. Perturbation factors, including non-spherical Earth, solar-lunar gravitational force, solar radiation pressure and tidal perturbation, are taken into account. GPS satellites are selected to make numerical simulations. The error of orbit prediction is obtained by reference to precise ephemeris of GPS satellites. The results of the proposed perturbation method are compared with that of Runge-Kutta algorithm and Adams-Cowell algorithm. Results show: 3 d prediction errors of proposed algorithm for PRN01 GPS satellite and PRN02 GPS satellite are 4.56 and 10.10 m, respectively. The accuracy of proposed algorithm coincides with that of Runge-Kutta algorithm and Adams-Cowell algorithm. The computational time of Runge-Kutta algorithm is 237.7% of that of proposed algorithm. The computational time of Adams-Cowell algorithm is 71.3% of that of proposed algorithm. So the efficiency of proposed algorithm is obviously higher than that of Runge-Kutta algorithm and is slightly lower than that of Adams-Cowell algorithm.

Rigorous constraint satisfaction for sampled linear systems

We address a specific but recurring problem related to sampled linear systems. In particular, we provide a numerical method for the rigorous verification of constraint satisfaction for linear continuous-time systems between sampling instances. The proposed algorithm combines elements of classical branch and bound schemes from global optimization with a recently published procedure to bound the exponential of interval matrices.

Filtering algorithm for eigenvalue bounds of fuzzy symmetric matrices

Purpose The solution of dynamic problems of structures using finite element method leads to generalised eigenvalue problem. In general, if the material properties are crisp (exact) then we get crisp eigenvalue problem. But in actual practice, instead of crisp material properties we may have only bounds of values as a result of errors in measurements, observations and calculations or it may be due to maintenance induced error etc. Such bounds of values may be considered in terms of interval or fuzzy numbers. The purpose of this paper is to develop a fuzzy filtering procedure for finding real eigenvalue bounds of different structural problems. Design/methodology/approach The proposed fuzzy filtering algorithm has been developed in terms of fuzzy number to solve the fuzzy eigenvalue problem. The initial bounds of fuzzy eigenvalues are filtered to obtain precise eigenvalue bounds which are depicted by fuzzy (Triangular Fuzzy Number) plots using α-cut. Findings Previously, bounds of eigenvalues of interval matrices have been investigated by few authors. But when the structural problem consists of fuzzy material properties, then the interval eigenvalue bounds may be obtained for each interval of the fuzzy number. The proposed algorithm has been applied for standard fuzzy eigenvalue problems which may be extended to generalised fuzzy eigenvalue problems for obtaining filtered fuzzy bounds. Originality/value The developed fuzzy filtering method is found to be efficient for different structural dynamics problems with fuzzy material properties.