Search

This paper considers the stabilization issue for continuous-time linear parameter varying (LPV) positive systems. The time varying parameters are known and are modeled as belonging to the simplex set. The proposed stabilization approach relies on a parameter dependent Lyapunov function combined with a subtle choice of a slack variable that is not necessary diagonal. In fact, due to the positivity constraint on the closed-loop system the slack variable is chosen to be a Metzler matrix. Indeed, the particular case when the slack matrix is diagonal may work but the resulting stabilization conditions can be conservative. This fact is illustrated by a comparison example.

In this paper, we present a new variations of the popular nonnegative matrix factorization (NMF) approach to extend it to the data with negative values. When a NMF problem is formulated as μ ≈μμ, we try to develop a new method that only allows μ to contain nonnegative values, but allows both μ and μ to have both nonnegative and negative values. In this way, the original NMF is extended to be used for real value data matrix instead restricted to only negative value data matrix. To this end, we develops novel method to factorize the real value data matrix. The method is evaluated experimentally and the results showed its effectiveness.

The resurgence of vector-borne diseases is an increasing public health concern, and there is a need for a better understanding of their dynamics. For a number of diseases, e.g. dengue and chikungunya, this resurgence occurs mostly in urban environments, which are naturally very heterogeneous, particularly due to population circulation. In this scenario, there is an increasing interest in both multi-patch and multi-group models for such diseases. In this work, we study the dynamics of a vector borne disease within a class of multi-group models that extends the classical Bailey–Dietz model. This class includes many of the proposed models in the literature, and it can accommodate various functional forms of the infection force. For such models, the vector-host/host-vector contact network topology gives rise to a bipartite graph which has different properties from the ones usually found in directly transmitted diseases. Under the assumption that the contact network is strongly connected, we can define the basic reproductive number R 0 and show that this system has only two equilibria: the so called disease free equilibrium (DFE); and a unique interior equilibrium—usually termed the endemic equilibrium (EE)—that exists if, and only if, R 0 > 1 . We also show that, if R 0 ≤ 1 , then the DFE equilibrium is globally asymptotically stable, while when R 0 > 1 , we have that the EE is globally asymptotically stable.

A real n × n matrix A is said to be an M -matrix if there exists a nonnegative matrix B with maximal eigenvalue r such that A = α I — B , where α ≥ r . In this work, we present a sufficient condition for the existence of an M -matrix with prescribed complex spectrum.

In this note, we shall consider the sequence of consecutive powers in max algebra of a nonnegative matrix with each element less than or equal to 1. The notion of asymptotic period is defined to study the `limiting behavior' of this sequence. A simple and effective characterization for the asymptotic period of the sequence is given.

An m-by-n matrix A is called totally nonnegative if every minor of A is nonnegative. The Hadamard product of two matrices is simply their entry-wise product. This paper introduces the subclass of totally nonnegative matrices whose Hadamard product with any totally nonnegative matrix is again totally nonnegative. Many properties concerning this class are discussed including: a complete characterization for min {m,n} ; a characterization of the zero–nonzero patterns for which all totally nonnegative matrices lie in this class; and connections to Oppenheim's inequality.

We present a method of calculating the maximal eigenvalue of an indecomposable nonnegative matrix, which is based on ideas of geometric programming. In addition to that, we obtain estimates for elements of an indecomposable nonnegative matrix by its spectral radius. The results make it possible to obtain new necessary conditions for the productivity of the matrix of coefficients in the Leontief input-output model and have the immediate relation to the analysis of M- matrices. Another interesting application of the developed method is given by conditions of stability of the dynamic system of market equilibrium.

We consider n-dimensional positive linear switched systems. A necessary condition for stability under arbitrary switching is that every matrix in the convex hull of the matrices defining the subsystems is Hurwitz. Several researchers conjectured that for positive linear switched systems this condition is also sufficient. Recently, Gurvits, Shorten, and Mason showed that this conjecture is true for the case n = 2, but is not true in general. Their results imply that there exists some minimal integer np such that the conjecture is true for all n < np, but is not true for n = np. We show that np = 3.

Let A be an n X n nonnegative matrix. In this paper we consider the problems of maximizing the spectral radii of (i) A + ,X. and (ii) A + D, where X is a real n X n matrix whose Frobenius norm is restricted to be 1 and where D is as X but is further constrained to be a diagonal matrix. For both problems the maximums occur at nonnegative X and D, and we use tools of nonnegative matrices, most notably due to Levinger and Fiedler, as well as the Kuhn--Tucker criterion for constrained optimization, to find upper and lower bounds on the maximums, and, when A is additionally assumed to be irreducible, to characterize cases of equalities in these bounds. In the case of the first problem, when A is irreducible, we characterize a matrix which gives the global maximum. A matrix which yields a global maximum to the second problem is more complicated to characterize as, depending on A, the problem admits local maximums within the nonnegative diagonal matrices of Frobenius norm 1.

We consider the problem of nonnegative matrix factorization where the typical objective function is altered based on geometrical arguments. A noneuclidean geometry on positive real numbers is used to describe the nonnegative entries of a nonnegative matrix, influencing the factorization model. We design an optimization procedure from a differential geometric point of view for the newly proposed model. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

The real nonegative eigenvalue problem is concerned with determining which n -tuples of real numbers are the spectrum of some n -by- n nonnegative matrix. Some questions are presented whose answer could help to characterize the boundary of this set, i.e., the set of those spectra of nonnegative matrices which are not the spectrum of any positive matrix.

WOS: 000071526500003, We present a sequence of progressively better upper bounds for the Perron root of a nonnegative matrix. Each element in the sequence is a function of the Perron root of the arithmetic symmetrization of a power of the matrix. The results complement those of Szyld based on the geometric symmetrization of a power of the matrix. (C) 1998 Elsevier Science Inc.

This paper shows how the set of nonnegative matrices can be seen as the union of smaller-dimensional sets called slices. For $2\times 2$ matrices a model is drawn and it is shown how the model can be used to study the eigenvector behavior of a perturbed matrix.

This work is the numerical analysis and computational companion of the paper by Kamgang and Sallet [J.C. Kamgang, G. Sallet, Computation of threshold conditions for epidemiological models and global stability of the disease free equilibrium (DFE), Mathematical Biosciences 213 (2008) 1-12] where threshold conditions for epidemiological models and the global stability of the disease-free equilibrium (DFE) are studied. We establish a discrete counterpart of the main continuous result that guarantees the global asymptotic stability (GAS) of the DFE for general epidemiological models. Then, we design nonstandard finite difference (NSFD) schemes in which the Metzler matrix structure of the continuous model is carefully incorporated and both Mickens' rules (World Scientific, Singapore, 1994) on the denominator of the discrete derivative and the nonlocal approximation of nonlinear terms are used in an innovative way. As a result of these strategies, our NSFD schemes are proved to be dynamically consistent with the continuous model, i.e., they replicate their basic features, including the GAS of the DFE, the linear stability of the endemic equilibrium (EE), the positivity of the solutions, the dissipativity of the system, and its inherent conservation law. The general analysis is made detailed for the MSEIR model for which the NSFD theta method is implemented, with emphasis on the computational aspects such as its convergence, or local truncation error. Numerical simulations that illustrate the theory are provided.

In a partial continuation of work by Fiedler, some spectral properties of symmetric nonnegative matrices are extended to general nonnegative matrices.

Let Nn denote the set of those (1, λ2, …, λn) ∈ C n such that there exists a nonnegative matrix with Perron root equal to one and spectrum {1, λ2, …, λn}. We prove that Nn is star-shaped with respect to (1, 0, …, 0) and that (1, λ 2 , …, λ n ) ∈ N n is on the boundary of Nn if and only if {1, λ2, …, λn} is not the spectrum of any positive matrix. As a consequence, attention is given to the problem of determining which nonnegative matrices are similar to positive ones. More precisely, we address the question of which pattern matrices P satisfy that any nonnegative matrix with pattern P is similar to a positive matrix. Some partial results are obtained (among them that any irreducible nonnegative matrix with a positive line is similar to a positive matrix), which allow us to give a complete solution to the case of 3-by-3 matrices.

We describe the structure of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus order.

This Paper considers the problem of constrained stabilization of linear continuous-time systems by state feedback control law. The goal is to solve this problem under positivity constraint which means that the resulting closed-loop systems are not only stable, but also positive. We focus on the class of linear continuous-time positive systems (Metzlerian systems) and use the interesting properties of Metzler matrix to provide the necessary ingredients for the main results of the paper. First, some necessary and sufficient conditions are presented for the existence of controllers satisfying the Metzlerian constraint, and the constrained stabilization is solved using linear programming (LP) or linear matrix inequality (LMI). A major objective is to formulate the constrained stabilization problem with the aim of maximizing the stability radius. We show how to solve this problem with an additional LMI formulation.

In this work we study the dynamics of a vector borne disease on a metapopulation model that accounts for host circulation. For such models, the movement network topology gives rise to a contact network topology, corresponding to a bipartite graph. Under the assumption that the contact network is strongly connected, we can define the basic reproductive number R_0 and show that this system has only two equilibria: the so called disease free equilibrium (DFE); and a unique interior equilibrium that exists if, and only if, the basic reproduction number, R_0, is greater that unity. We are also able to show that the DFE is globally asymptotically stable, if R_0 ≤ 1. If R_0 > 1, the dynamics is uniformly persistent and, with further assumptions on the contact network structure, we also show that the endemic equilibrium (EE) is globally asymptotically stable.; Dans ce travail, nous étudions la dynamique d'une maladie vectorielle (transmise par vecteur) sur un modèle de métapopulation qui tient compte de la circulation humaine. Sous l'hypothèse que le réseau de contacts est fortement connecté, nous définissons le nombre de re- production de base R0 et montrons que ce ce système a seulement deux équilibres: l'équilibre sans maladie (DFE), et un équilibre intérieur unique qui existe si , et seulement si, le nombre de reproduction de base, R0, est plus grand que l'unité. Dans ce cas, le système est également persistant. Nous sommes également en mesure de montrer que le DFE est globalement asympto- tiquement stable, si R0 ≤ 1, et que la dynamique est uniformément persistante si R0 > 1. Avec d'autres hypothèses sur la structure du réseau de contacts, nous montrons alors que l'équilibre endémique (EE) est globalement asymptotiquement stable, si R0 > 1.

This paper is concerned with the analysis of large-scale interconnected systems constructed from positive subsystems and a nonnegative interconnection matrix. We first show that the interconnected system is admissible and stable if and only if a Metzler matrix built from the coefficient matrices of the positive subsystems and the interconnection matrix is Hurwitz stable. By means of this key lemma, we further provide several results that characterize the admissibility and stability of interconnected systems in terms of the weighted L1-induced norm of each positive subsystem and the Frobenius eigenvalue of the interconnection matrix. Moreover, in the case where every subsystem is SISO, we provide explicit conditions under which the interconnected system has the property of persistence, i.e., the state of the interconnected system converges to a unique strictly positive vector (up to a strictly positive constant multiplicative factor) irrespective of nonnegative and nonzero initial states. We illustrate the effectiveness of the persistence results via formation control of multi-agent systems.

The max-algebraic core of a nonnegative matrix is the intersection of column spans of all max-algebraic matrix powers. Here we investigate the action of a matrix on its core. Being closely related to ultimate periodicity of matrix powers, this study leads us to new modifications and geometric characterizations of robust, orbit periodic and weakly stable matrices., 27 pages

We present two versions of the same inequality, relating the maximal diagonal entry of a nonnegative matrix to its eigenvalues. We demonstrate a matrix factorization of a companion matrix, which leads to a solution of the nonnegative inverse eigenvalue problem (denoted the nniep) for 4×4 matrices of trace zero, and we give some sufficient conditions for a solution to the nniep for 5×5 matrices of trace zero. We also give a necessary condition on the eigenvalues of a 5×5 trace zero nonnegative matrix in lower Hessenberg form. Finally, we give a brief discussion of the nniep in restricted cases.

Some new sufficient conditions are found for n real numbers to be the spectrum of some n × n symmetric nonnegative matrix, and for n complex numbers to be the spectrum of some n × n normal nonnegative matrix.

We consider a controlled quantum system whose finite dimensional state is governed by a discrete-time nonlinear Markov process. In open-loop, the measurements are assumed to be quantum non-demolition (QND). The eigenstates of the measured observable are thus the open-loop stationary states: they are used to construct a closed-loop supermartingale playing the role of a strict control Lyapunov function. The parameters of this supermartingale are calculated by inverting a Metzler matrix that characterizes the impact of the control input on the Kraus operators defining the Markov process. The resulting state feedback scheme, taking into account a known constant delay, provides the almost sure convergence of the controlled system to the target state. This convergence is ensured even in the case where the filter equation results from imperfect measurements corrupted by random errors with conditional probabilities given as a left stochastic matrix. Closed-loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box, a cavity quantum electrodynamics system., Comment: submitted (16 pages, 4 figures)

We show that a rank-three symmetric matrix with exactly one negative eigenvalue can have arbitrarily large nonnegative rank., Comment: note, 1 page

Totally nonnegative matrices, i.e., matrices having all their minors nonnegative, and matrix intervals with respect to the checkerboard ordering are considered. It is proven that if the two bound matrices of such a matrix interval are nonsingular and totally nonnegative (and in addition all their zero minors are identical) then all matrices from this interval are also nonsingular and totally nonnegative (with identical zero minors).

We studym×n matrices,m≤n, whose elements are either 1) arbitrary nonnegative numbers or 2) belong to a given finite set of nonnegative numbers that includes zero. In the finite case, we obtain an asymptotic expression, asn → ∞, for the number of matrices with zero permanent. For any nonnegative matrix with zero permanent a standard representation is derived.

The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of complex numbers ¦O to be the spectrum of a nonnegative matrix. In this paper the problem is completely solved in the case when all numbers in the given list except for one (the Perron eigenvalue) are pure image numbers. Lets. Let ( ,bi, bi,,b i, b i)be a list of complex numbers with ¦N,bj > 0 for j =1,2,,k . A simple necessary and sufficient conditions for the existence of an entry wise nonnegative 2k +1 order matrix A with spectrum ¦O are presented , and the proof is elementary.

This paper considers the problem of robust stability and stabilization of uncertain time delay systems. We focus on the class of Metzlerian delay systems and use the interesting properties of Metzler matrix to provide the necessary ingredients for the main results of the paper. It is shown that Metzlerian delay system is asymptotically stable if and only if the Metzlerian system without delay∗ is asymptotically stable. This fact allows one to establish strong results on constrained stabilization for delay systems. The robust stability of Metzlerian delay system is also revisited and, without requiring long derivation, direct formulas for calculating the stability radius of this class of systems are provided. Finally, we consider the problem of constrained robust stabilization of interval delay systems and outline an LMI based solution for it. The paper concludes with two numerical examples to support the theoretical results.

In this paper we present a new method for approximating the matrix exponential of a Metzler matrix. This method is useful in discretising switched positive systems. In particular, the method preserves both linear and quadratic stability of the original continuous time system, as well as positivity of states starting for initial conditions in the positive orthant. The usefulness of the method is highlighted by illustrating some of the drawbacks of Pade approximations when applied to positive linear systems.

Conditions for the existence of positive stable realizations with system Metzler matrices for linear continuous-time systems are established. A procedure for finding a positive stable realization with system Metzler matrix based on similarity transformation of proper transfer matrices is proposed and demonstrated on numerical examples. It is shown that if the poles of stable transfer matrix are real then the classical Gilbert method can be used to find the positive stable realization.

The matrix with centrosymmetric structure is an important kind of structured matrices with many applications in various physics and engineering problems. In this paper, we focus on the spectral radius of a subclass of centrosymmetric matrix, the nonnegative matrix with centrosymmetric structure. We discuss some results about the Perron root of a nonnegative centrosymmetric matrix. Then, the Collatz-Wielandt method for the Perron root of a nonnegative irreducible matrix is introduced, based on which a corresponding new algorithm is constructed. We show that the new algorithm ensures significant savings of computation and time cost.

In this paper, we focus on the spectral radius of nonnegative matrices with generalized K-centrosymmetric structure. We first discuss some results about the Perron root of generalized K-centrosymmetric nonnegative matrices. An algorithm of diagonal transformation for Perron root of nonnegative irreducible matrices is introduced, based on which a new algorithm for the Perron root of a generalized K-centrosymmetric nonnegative matrix is constructed. We show that the new algorithm ensures significant savings of computation and time cost.

Using the relationship between totally nonnegative matrices and directed acyclic weighted planar networks, we show that 2 × 2 minors of minor matrices of totally nonnegative matrices are also nonnegative. We give a combinatorial interpretation for the minors of minor matrices in terms of the weights of families of paths in a network.

We consider partitioned lists of real numbers � = {�1,�2,...,�n}, and give efficient and constructive sufficient conditions for the existence of nonnegative and symmetric nonnegative matrices with spectrum �. Our results extend the ones given in (R.L. Soto and O. Rojo. Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem. Linear Algebra Appl., 416:844- 856, 2006.) and (R.L. Soto, O. Rojo, J. Moro, and A. Borobia. Symmetric nonnegative realization of spectra. Electron. J. Linear Algebra, 16:1-18, 2007.) for the real and symmetric nonnegative inverse eigenvalue problem. We also consider the complex case and show how to construct an r × r nonnegative matrix with prescribed complex eigenvalues and diagonal entries.

The nonnegative definiteness of a symmetric partitioned matrix can be characterized in an interesting way in terms of submatrices. Our experience is that this characterization is a very handy tool in various problems related to the Lowner partial ordering.

For a set σ with n complex numbers, some sufficient conditions are found for σ to be the spectrum of an n ×n normal (entrywise) nonnegative (positive) matrix. After proving a fundamental theorem and introducing the companion set σ′ of σ which consists of real numbers, we prove that if σ′ satisfies any known sufficient conditions for a real set to be the spectrum of a nonnegative matrix introduced by Suleimanova, Perfect, Salzmann and Kellogg respectively, then σ is the spectrum of an n×n normal nonnegative matrix.

The nonnegative rank of a nonnegative matrix is the minimum number of nonnegative rank-one factors needed to reconstruct it exactly. The problem of determining this rank and computing the corresponding nonnegative factors is difficult; however it has many potential applications, e.g., in data mining, graph theory and computational geometry. In particular, it can be used to characterize the minimal size of any extended reformulation of a given combinatorial optimization program. In this paper, we introduce and study a related quantity, called the restricted nonnegative rank. We show that computing this quantity is equivalent to a problem in polyhedral combinatorics, and fully characterize its computational complexity. This in turn sheds new light on the nonnegative rank problem, and in particular allows us to provide new improved lower bounds based on its geometric interpretation. We apply these results to slack matrices and linear Euclidean distance matrices and obtain counter-examples to two conjectures of Beasly and Laffey, namely we show that the nonnegative rank of linear Euclidean distance matrices is not necessarily equal to their dimension, and that the rank of a matrix is not always greater than the nonnegative rank of its square.

Necessary and sufficient conditions are derived in this paper for the existence of a diagonal Lyapunov function for a class of switched systems. The switching is carried out intermittently between the open loop and the closed loop of a feedback system, a procedure encountered frequently in many interconnected systems at the present time. The open loop system considered in the paper is described by a stable Metzler matrix, which is known to be diagonally stable. The closed loop system is stable, but does not retain the Metzler property. The existence of a common diagonal Lyapunov function assures system stability for arbitrary switching. This decouples stability and performance issues in the switched system, and permits switching to be carried out entirely for improving performance. Towards the end of the paper, it is also shown that some recent results in the control literature can be derived as special cases of the principal results derived in the paper.

We consider a list of complex numbers ¤ = {¸1, ¸2, . . . , ¸n} and give a simple and efficient sufficient condition for the existence of an n × n nonnegative matrix with spectrum ¤. Our result extends a previous one for a list of real numbers given in (Linear Algebra Appl., 416:844-856, 2006). In particular, we show how to construct a nonnegative matrix with prescribed complex eigenvalues and diagonal entries. As a by-product, we also construct Hermitian matrices with prescribed spectrum, whose entries have nonnegative real parts.

In this paper we determine the nonnegativity structure of the principal components of an n× nnonnegative matrix Pin terms of the marked reduced graph , the minus M-matrix which can be associated with P. We then apply this result to consider various types of nonnegative bases for the Perron eigenspace of P which can be extracted from a certain nonnegative matrix which is a polynomial in P. We also obtain a characterization for the eigenprojection Pon the Perron eigenspace of Pto be, itself, a nonnegative matrix. Our results provide new proofs and extensions of results of Friedland and Schneider and of Hartwig, Neumann, and Rose.

The problem of determining necessary and sufficient conditions for a set of real numbers to be the eigenvalues of a symmetric nonnegative matrix is called the symmetric nonnegative inverse eigenvalue problem (SNIEP). In this paper we solve SNIEP in the case of trace zero symmetric nonnegative 5x5 matrices., Revised structure, fixed typos, added more details in some proofs, added figures