Search

UID/MAT/00297/2019 UID/MAT/04721/2019 We consider the class (Formula presented.) of symmetric (Formula presented.) -matrices with zero trace and constant row sums k which can be identified with the class of the adjacency matrices of k-regular undirected graphs. In a previous paper, two partial orders, the Bruhat and the Bruhat-graph order, have been introduced in this class. In fact, when k = 1 or k = 2, it was shown that the two orders coincide, while for (Formula presented.) the two orders are distinct. In this paper we give general properties of minimal and maximal matrices for these orders on (Formula presented.) and study the minimal and maximal matrices when k = 1, 2 or 3. publishersversion published

In this paper, the problem of fuzzy filter design is investigated for a class of nonlinear networked systems on the basis of the interval type-2 (IT2) fuzzy set theory. In the design process, two vital factors, intermittent data packet dropouts and quantization, are taken into consideration. The parameter uncertainties are handled effectively by the IT2 membership functions determined by lower and upper membership functions and relative weighting functions. A novel fuzzy filter is designed to guarantee the error system to be stochastically stable with $\boldsymbol {H_{\infty }}$ performance. Moreover, the filter does not need to share the same membership functions and number of fuzzy rules as those of the plant. Finally, illustrative examples are provided to illustrate the effectiveness of the method proposed in this paper.

Inspired by the paper of Fasbender and Ikramov [H. Fasbender, Kh.D. Ikramov, A note on an unusual type of polar decomposition, Linear Algebra Appl. 429 (2008) 42–49], in this note we introduce an unusual type of generalized polar decomposition for a rectangular matrix A of the form A = GE , where G is a complex symmetric matrix and E is a partial isometric matrix. Following the pattern used in the paper mentioned above, we call this decomposition a symmetric-partial-isometric generalized polar decomposition or an SPIGPD for short. Some properties of this decomposition are presented and results of SPIGPD related to conjugate-normal matrices are also obtained.

This paper is concerned with the distributed linear quadratic optimal control problem. In particular, we consider a suboptimal version of the distributed optimal control problem for undirected multi-agent networks. Given a multi-agent system with identical agent dynamics and an associated global quadratic cost functional, our objective is to design suboptimal distributed control laws that guarantee the controlled network to reach consensus and the associated cost to be smaller than an a priori given upper bound. We first analyze the suboptimality for a given linear system and then apply the results to linear multiagent systems. Two design methods are then provided to compute such suboptimal distributed controllers, involving the solution of a single Riccati inequality of dimension equal to the dimension of the agent dynamics, and the smallest nonzero and the largest eigenvalue of the graph Laplacian. Furthermore, we relax the requirement of exact knowledge of the smallest nonzero and largest eigenvalue of the graph Laplacian by using only lower and upper bounds on these eigenvalues. Finally, a simulation example is provided to illustrate our design method., Comment: 11 pages, 2 figures

This paper considers the problem of estimating a high-dimensional covariance matrix in a low sample support situation where the sample size is smaller, or not much larger, than the dimensionality of the data, which could potentially be very large. We develop a regularized sample covariance matrix (RSCM) estimator which can be applied in commonly occurring high-dimensional data problems. The proposed RSCM estimator is based on estimators of the unknown optimal (oracle) shrinkage parameters that yield the minimum mean squared error between the RSCM and the true covariance matrix when the data is sampled from an unspecified elliptically symmetric distribution. We propose two variants of the RSCM estimator which differ in the approach in which they estimate the underlying sphericity parameter involved in the theoretical optimal shrinkage parameter. The performance of the proposed RSCM estimators are evaluated with numerical simulation studies. In particular, when the sample sizes are low, the proposed RSCM estimators often show a significant improvement over the conventional RSCM estimator by Ledoit and Wolf (2004). We further evaluate the performance of the proposed estimators in a portfolio optimization problem with real data wherein the proposed methods are able to outperform the benchmark methods.

This paper is concerned with the model reduction problem of discrete-time interval type-2 Takagi-Sugeno (T-S) fuzzy systems which represent the discrete-time nonlinear systems subject to uncertainty. With the use of interval type-2 fuzzy sets, the uncertainty of the discrete-time nonlinear system can be captured by the lower and upper membership functions. For a given high-order discrete-time interval type-2 T-S fuzzy system, the purpose is to find a lower dimensional system to approximate the original system. To achieve the approximation performance, an H$\infin$ norm is used to suppress the error between the original system and its simplified system. By introducing a membership-functions-dependent technique and applying a convex linearization method, a membership-functions-dependent condition, which takes the information of membership functions into account, is obtained to reduce the dimensions of system matrices and the number of fuzzy rules of the system. All the obtained theorems are represented as in the form of linear matrix inequalities (LMIs). Finally, simulation results are demonstrated to show the effectiveness of the derived results.

Traffic control of congestion regimes is considered in this paper. A perturbed distributed parameter model is used, and a boundary control is designed to reject the perturbations. More precisely an optimal proportional-integral (PI) feedback control law is computed to maximally reject the disturbances, and to stabilize the traffic in congested regime. The disturbance applies at the boundary of the linearized Aw-Rascle-Zhang (ARZ) model. Therefore the disturbance operator is unbounded, rendering the control problem very challenging. In order to analyze and design the optimal PI controller for this infinite-dimensional system, the ${\mathcal{L}_2}$ ain is computed to estimate the disturbance rejection. Numerically tractable conditions are computed and written with linear matrix inequalities (LMIs). As a result, the estimation of an upper bound of the ${\mathcal{L}_2}$ gain, from the disturbance to the controlled output, can be formulated as an optimization problem with LMI constraints. The validity of this method is checked on simulations of the nonlinear ARZ model in closed-loop with this optimal PI controller.

In this paper, we investigate and describe all the minimal completions and covers of the symmetric, Hankel symmetric, and centrosymmetric doubly substochastic matrices, respectively.

[EN] Given a nonlinear system, the sector-nonlinearity methodology provides a systematic way of transforming it in an equivalent Takagi-Sugeno (T-S) model. However, such transformation is not unique: conservatism of shape-independent performance conditions in the form of linear matrix inequalities results in some models yielding better results than others. This paper provides some guidelines on choosing a sector-nonlinearity T-S model, with provable optimality (in a particular sense) in the case of quadratic nonlinearities. The approach is based on Hessian and restrictions of a function onto a subspace., This work was supported by the following institutions: Project Ciencia Basica SEP-CONACYT CB-168406, Project DPI2016-81002, (Spanish government, MINECO), Grant PROMETEOII/2013/004 (Generalitat Valenciana) and, the Scholarship GRISOLIA/2014/006.

In this paper, we revisit the asymptotic reverse-waterfilling characterization of the nonanticipative rate distortion function (NRDF) derived for a time-invariant multidimensional Gauss-Markov processes with mean-squared error (MSE) distortion in [1]. We show that for certain classes of time-invariant multidimensional Gauss-Markov processes, the specific characterization behaves as a reverse-waterfilling algorithm obtained in matrix form ensuring that the numerical approach of [1, Algorithm 1] is optimal. In addition, we give an equivalent characterization that utilizes the eigenvalues of the involved matrices reminiscent of the well-known reverse-waterfilling algorithm in information theory. For the latter, we also propose a novel numerical approach to solve the algorithm optimally. The efficacy of our proposed iterative scheme compared to similar existing schemes is demonstrated via experiments. Finally, we use our new results to derive an analytical solution of the asymptotic NRDF for a correlated time-invariant two-dimensional Gauss-Markov process. 14 20

International audience; In this paper, we propose a new regularized (penalized) co-variance matrix estimator which encourages grouping of the eigenvalues by penalizing large differences (gaps) between successive eigenvalues. This is referred to as fusing eigenval-ues (eFusion), The proposed penalty function utilizes Tukey's biweight function that is widely used in robust statistics. The main advantage of the proposed method is that it has very small bias for sufficiently large values of penalty parameter. Hence, the method provides accurate grouping of eigenval-ues. Such benefits of the proposed method are illustrated with a numerical example, where the method is shown to perform favorably compared to a state-of-art method.

A map that is defined on the set of all symmetric matrices over a field is rank-one nonincreasing if it maps the matrices of rank one to matrices of rank at most one. In the case of the fields with two or three elements nonstandard examples of such additive maps are known to exist. In this paper, all these maps are characterized. Moreover, the existence of nonstandard maps is understood in a geometric sense since they exist precisely in the two cases, where the whole n-dimensional vector space over can be written as a union of two hyperbolic/parabolic quadrics of low indices. This geometric interpretation enables us to realize that these weird examples represent just a tip of an iceberg that appears in an analogous problem on symmetric tensors, where nonstandard maps exist also over larger fields. A few open problems related to the multilinear analog and to homogeneous polynomials are posed.

Factor analysis aims to describe high dimensional random vectors by means of a small number of unknown common factors. In mathematical terms, it is required to decompose the covariance matrix $\Sigma$ of the random vector as the sum of a diagonal matrix $D$ | accounting for the idiosyncratic noise in the data | and a low rank matrix $R$ | accounting for the variance of the common factors | in such a way that the rank of $R$ is as small as possible so that the number of common factors is minimal. In practice, however, the matrix $\Sigma$ is unknown and must be replaced by its estimate, i.e. the sample covariance, which comes from a finite amount of data. This paper provides a strategy to account for the uncertainty in the estimation of $\Sigma$ in the factor analysis problem., Comment: Draft, the final version will appear in the 56th IEEE Conference on Decision and Control, Melbourne, Australia, 2017

Previous work has shown that perceptual texture similarity and relative attributes cannot be well described by computational features. In this paper, we propose to predict human’s visual perception of texture images by learning a nonlinearmapping from computational feature space to perceptual space. Hand-crafted features and deep features, which were successfully applied in texture classification tasks, were extracted and used to train Random Forest and rankSVM models against perceptual data from psychophysical experiments. Three texture datasets were used to test our proposed method and the experiments show that the predictions of such learnt models are in high correlation with human’s results.

The aim of this paper is to suggest a modification to the usual bounded confidence model of opinion dynamics, so that “changes of opinion” (intended as changes of the sign of the initial state) of an agent are never induced by the dynamics. In order to do so, a bipartite consensus model is used, endowing it with a confidence range. The resulting signed bounded confidence model has a state-dependent connectivity and a behavior similar to its standard counterpart, but in addition it preserves the sign of the opinions by “repelling away” opinions localized near the origin but on different sides with respect to 0. Funding agencies: Swedish Research Council [2015-04390]

The eigenvalues and eigenvectors of a matrix have many applications in engineering and science. For example they are important in studying and solving structural problems, in the treatment of signal or image processing, in the study of quantum mechanics and in certain physical problems. It is therefore essential to analyze methodologies to obtain the eigenvectors and eigenvalues ​​of symmetric and Hermitian matrices. In this paper the authors present a methodology for obtaining the eigenvectors and eigenvalues of a symmetric or Hermitian matrix using a genetic algorithm. Unlike other methodologies, the process is centred in searching the eigenvectors and calculating the eigenvalues afterwards. In the search of the eigenvectors a genetic-based algorithm is used. Genetic algorithms are indicated when the search space is extended, unknown or with an intricate geometry. Also, the target vector space can be either real or complex, allowing in this way a wider field of application for the proposed method. The algorithm is tested comparing the results with those obtained by other methods or with the values previously known. So, seven applications are included: a real symmetric matrix corresponding to a vibrating system, a complex Hermitian matrix and an important application of the diagonalization problem (Coope matrix) corresponding to quantum mechanics examples, a physical problem in which data are analysed to reduce the number of variables, a comparison with the power method and the studies of a degenerate and an ill-conditioned matrix.

International audience; This paper deals with the problem of estimating the state of a linear time-invariant system in the presence of sporadically available measurements. An observer with a continuous intersample injection term is proposed. Such an intersample injection is provided by a linear dynamical system, whose state is reset to the measured output estimation error at each sampling time. The resulting system is augmented with a timer triggering the arrival of a new measurement and analyzed in a hybrid system framework. The design of the observer is performed to achieve global exponential stability of a set wherein the estimation error is equal to zero. Moreover, three computationally efficient procedures are proposed to design the observer. Finally, the effectiveness of the proposed methodology is shown in two examples.

Let R be a Euclidean domain with quotient field F of characteristic not equaling 2. Jacobi showed that every symmetric R-matrix is congruent over R to a matrix in triple diagonal form. Since it is generally not possible to fully diagonalize these matrices, it is of importance to gain as much control as possible of this triple diagonal form. Two different refinements have since been made to Jacobi’s triple diagonal form. This paper works toward combining these refinements.

Let R be a Euclidean domain with quotient field F of characteristic not equaling 2. Jacobi showed that every symmetric R -matrix is congruent over R to a matrix in triple diagonal form. Since it is generally not possible to fully diagonalize these matrices, it is of importance to gain as much control as possible of this triple diagonal form. This paper focuses on controlling the off-diagonal elements.

An important task in connectomics studies is the classification of connectivity graphs coming from healthy and pathological subjects. In this paper, we propose a mathematical framework based on Riemannian geometry and kernel methods that can be applied to connectivity matrices for the classification task. We tested our approach using different real datasets of functional and structural connectivity, evaluating different metrics to describe the similarity between graphs. The empirical results obtained clearly show the superior performance of our approach compared with baseline methods, demonstrating the advantages of our manifold framework and its potential for other applications.

In this paper, Linear-Quadratic (LQ) differential games are studied, focusing on the notion of solution provided by linear feedback Nash equilibria. It is well-known that such strategies are related to the solution of coupled algebraic Riccati equations, associated to each player. Herein, we propose an algorithm that, by borrowing techniques from algebraic geometry, allows to recast the problem of computing all stabilizing Nash strategies into that of finding the zeros of a single polynomial function in a scalar variable, regardless of the number of players and the dimension of the state variable. Moreover, we show that, in the case of a scalar two-player differential game, the proposed approach permits a comprehensive characterization - in terms of number and values - of the set of solutions to the associated game.

This paper investigates the problem of sampled-data stabilization for neural-network-based control systems with an optimal guaranteed cost. Using time-dependent Lyapunov functional approach, some novel conditions are proposed to guarantee the closed-loop systems exponentially stable, which fully use the available information about the actual sampling pattern. Based on the derived conditions, the design methods of the desired sampled-data three-layer fully connected feedforward neural-network-based controller are established to obtain the largest sampling interval and the smallest upper bound of the cost function. A practical example is provided to demonstrate the effectiveness and feasibility of the proposed techniques. Refereed/Peer-reviewed

Let G =( V , E ) be a undirected graph containing n vertices, and let M G be the set of all Hermitian n × n matrices M =( m i , j ) with m i , j ≠0 if i and j are connected by one edge of G , with m i,j ∈ C if i and j are connected by at least two edges, with m i , j =0 if i ≠ j , and i and j are not connected by an edge of G , and with m i , i for i =1,…, n a real number. What is the largest nullity attained by any positive semi-definite matrix M∈ M G ? In this paper we characterize, for t =1 and 2, those graphs G for which the maximum nullity is not greater than t .

We characterize the solution set S of real linear systems Ax=b by a set of inequalities if b lies between some given bounds b̄,b̄ and if the n×n coefficient matrix A varies similarly between two bounds A̱ and Ā. In addition, we restrict A to a particular class of matrices, for instance the class of the symmetric, the skew-symmetric, the persymmetric, the Toeplitz, and the Hankel matrices, respectively. In this way, we generalize the famous Oettli–Prager criterion (Numer. Math. 6 (1964) 405), results by Hartfiel (Numer. Math. 35 (1980) 355) and the contents of the papers (in: R.B. Kearfott, V. Kreinovich (Eds.), Applications of Interval Computations, Kluwer, Boston, MA, 1996, pp. 61–79) and (SIAM J. Matrix Anal. Appl. 18 (1997) 693).

International audience; This paper present a novel approach using symmetric fuzzy numbers to deal with uncertainty in power system analysis. Special attention is put on reducing overestimation due to conservatism inherent in fuzzy arithmetic. The midpoint-radius interval representation and a linear programming optimization problem are used in order to enhance efficiency and accuracy, respectively. The proposed method is tested over the IEEE 14-bus test system and compared with other techniques available in the literature. The case study that the proposed method yields accurate results at a reduced computational cost.

Letr Σn(C) denote the space of all n χ n symmetric matrices over the complex field C. The main objective of this paper is to prove that the maps Φ : Σn(C) -> Σn (C) satisfying for any fixed irre- ducible characters X, X' -SC the condition dx(A +aB) = dχ·(Φ(Α ) + αΦ(Β)) for all matrices A,В ε Σ„(С) and all scalars a ε C are automatically linear and bijective. As a corollary of the above result we characterize all such maps Φ acting on ΣИ(С).

Linear systems of the form Ax = b , where the matrix A is symmetric and positive definite, often arise from the discretization of elliptic partial differential equations. A very successful method for solving these linear systems is the preconditioned conjugate gradient method. In this paper, we study parallel preconditioners for the conjugate gradient method based on the block two-stage iterative methods. Sufficient conditions for the validity of these preconditioners are given. Computational results of these preconditioned conjugate gradient methods on two parallel computing systems are presented.

This paper contains a general characterization for the permutation polynomials of the symmetric matrices over any field. Specific characterizations are for symmetric matrices over algebraically closed fields, principal axis fields, and finite fields. In the latter case enumeration formulas are established as well. More general results concerning permutation polynomials on sets other than the symmetric matrices are sometimes stated.

In this paper we present models for symmetric stochastic matrices showing how to adjust and validate them. We discuss the validation in the general case.

In this paper we develop a reduction technique for the generalised Riccati difference equation arising in optimal control and optimal filtering. This technique relies on a decomposition method for the generalised Riccati difference equation that isolates its nilpotent part, which becomes constant in a number of iteration steps equal to the nilpotency index of the closed-loop, from another part that can be computed by iterating a reduced-order Riccati difference equation.

This paper presents a novel algebraic formulation of the central problem of screw theory, namely the determination of the principal screws of a given system. Using the algebra of dual numbers, it shows that the principal screws can be determined via the solution of a generalised eigenproblem of two real, symmetric matrices. This approach allows the study of the principal screws of the general two-, three-systems associated with a manipulator of arbitrary geometry in terms of closed-form expressions of its architecture and configuration parameters. We also present novel methods for the determination of the principal screws for four-, five-systems which do not require the explicit computation of the reciprocal systems. Principal screws of the systems of different orders are identified from one uniform criterion, namely that the pitches of the principal screws are the extreme values of the pitch. The classical results of screw theory, namely the equations for the cylindroid and the pitch-hyperboloid associated with the two- and three-systems, respectively have been derived within the proposed framework. Algebraic conditions have been derived for some of the special screw systems. The formulation is also illustrated with several examples including two spatial manipulators of serial and parallel architecture, respectively. � 2008 Elsevier Ltd. All rights reserved.