Search

A simple necessary and sufficient condition ensuring that a real matrix of dimension 3 is similar to a Metzler matrix is exhibited. When this condition is satisfied, a construction of the transfer matrix is given. This construction is used to design an interval observer for a family of continuous-time systems. An example is provided with interval observer design for the so-called love dynamics in the case of limit cycles.

This brief investigates the positive consensus of fractional-order interval networked systems with the order $\alpha $ meeting $0 and $1 , respectively. Based on the theories of fractional stability and properties of Metzler matrix, by virtue of some rigorous theoretical analyses, the distributed control protocol is proposed and sufficient conditions of positive consensus for fractional-order interval networked systems with $0 are given, which have no connection with the order. What’s more, the results can be extended to conditions without using the information of Laplacian matrix with respect to networked topology. Furthermore, sufficient conditions for the systems with $1 to reach positive consensus are presented in terms of linear matrix inequality. Finally, simulation examples are given to verify the effective of results.

A method is proposed for calculation of transition matrices of positive electrical circuits. It is shown that if the transition matrix is presented as finite series of the Metzler matrix with real distinct eigenvalues then the coefficients of the series are nonnegative function of time. The method is applied to positive linear electrical circuits.

The paper introduces block defined matrix constraints on system matrices of uncertain Metzlerian systems, conditioning reduced order interval observers design. Interval block matrix boundaries are projected into the set of linear matrix inequalities, representing Metzler matrix block parameter constraints. By combining the thus defined representation of boundaries with a pair of Lyapunov inequalities it is guarantied that reduced order interval observer system matrices are Metzler and Hurwitz and the observer interval stability is attained. A numerical example is included to assess the feasibility technique.

For linear continuous-time positive systems the paper proposes an approach, reflecting structural constraints and positiveness in the problem of Metzlerian and strictly Metzlerian interval observers design. Every interval matrix boundary is represented by a set of linear matrix inequalities representing Metzler matrix parameter constraints and reflecting potential zero elements in a Metzler matrix structure by structural diagonal matrix variables. Combined with couple of Lyapunov inequalities, the observer Metzler matrix parameters are guaranteed and interval stability is attained. A numerical example is included to assess the feasibility technique.

In this study, a deterministic model for the transmission dynamics of yellow fever (YF) in a human–mosquito setting in the presence of control measures is constructed and rigorously analyzed. In addition to horizontal transmissions, vertical transmission within mosquito population is incorporated. Analysis of the mosquito-only component of the model shows that the reduced model has a mosquito-extinction equilibrium, which is globally-asymptotically stable whenever the basic offspring number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(N_{0})$$\end{document}(N0) is less than unity. The vaccinated and type reproduction numbers of the full-model are computed. Condition for global-asymptotic stability of the disease-free equilibrium of the model when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_{0} > 1$$\end{document}N0>1 is presented. It is shown that, fractional dosing of YF vaccine does not meet YF vaccination requirements. Optimal control theory is applied to the model to characterize the controls parameters. Using Pontryagin’s maximum principle and modified forward–backward sweep technique, the necessary conditions for existence of solutions to the optimal control problem is determined. Numerical simulations of the models to assess the effect of fractional vaccine dosing on the disease dynamics and global sensitivity analysis are presented.

National Bureau of Statistics of China reports that the incidence of schistosomiasis has been increasing in recent years. To study dynamic behaviors of schistosomiasis transmission, based on practical experience of staff in Anhui Institute of Schistosomiasis, a mathematical schistosomiasis model with reinfection of recovered people is established in this paper. Metzler matrix theory and center manifold theorem are used to analyze stability of equilibria. Parameter estimation has been performed by combining model and monitoring data. It is found that the basic reproduction number is different every year. The most concern of Institute of Schistosomiasis is whether or when to kill snails every year. To answer this question, threshold value of snail density can be obtained. Once the snail density exceeds the threshold, the staff will need to kill snails. To find the best control measures, sensitivity analysis is used to find out sensitive parameters, and then control measures can be obtained by optimization control measures. The results show that combination of spraying molluscicide, publicity and education, improving the health facilities, and large-scale treatment of patient groups have the best effect. In additional, it is found that the number of patients does not change much when the reinfection rate of recovered people is very small. However, when the reinfection rate is slightly larger, the number of patients will suddenly increase to a large value.

Complex dynamical networks can often be analyzed as the interconnection of subsystems: This allows us to considerably simplify the model and better understand the global behavior. Some biological networks can be conveniently analyzed as aggregates of monotone subsystems. Yet, monotonicity is a strong requirement; it relies on the knowledge of the state representation and imposes a severe restriction on the Jacobian (which must be a Metzler matrix). Systems with a monotonic step response (MSR), which include input–output monotone systems as a special case, are a broader class and still have interesting features. The property of having a monotonically increasing step response (or, equivalently, in the linear case, a positive impulse response) can be evinced from experimental data, without an explicit model of the system. We consider networks that can be decomposed as aggregates of MSR subsystems and we provide a structural (parameter-free) classification of oscillatory and multistationary behaviors. The classification is based on the exclusive or concurrent presence of negative and positive cycles in the system aggregate graph , whose nodes are the MSR subsystems. The result is analogous to our earlier classification for aggregates of monotone subsystems. Models of biomolecular networks are discussed to demonstrate the applicability of our classification, which helps build synthetic biomolecular circuits that, by design, are well suited to exhibit the desired dynamics.

In this paper, we investigate the problem of finite-time stability (FTS) of linear non-autonomous systems with time-varying delays. By constructing an appropriated function, we derive some explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. Finally, two examples are given to show the effectiveness of the main results.

This paper considers an extension of diagonal stability for continuous fractional positive linear systems (FPLS) with the fractional order 0 α 1 . Based on diagonal stability of Metzler matrix, an extension that involves the combination of a diagonal positive definite matrix and a skew-symmetric anti-diagonal matrix related a Hurwitz and Metzler matrix is established. Combining this extension and the linear matrix inequalities (LMIs) criteria of stability for fractional order systems (FOS), a necessary and sufficient condition for extension diagonal stability of FPLS is presented. A state feedback controller is given, which ensures the stabilization and positivity of the closed-loop systems. Numerical examples are provided to demonstrate the effectiveness and applicability of the proposed methods.

It is known that the stability of a Metzler matrix can be characterized in a Routh–Hurwitz-like fashion based on a recursive application of scalar Schur complements [1] . Our objective in this brief note is to show that recently obtained stability conditions are equivalent statements of this result and can be deduced directly therefrom using only elementary results from linear algebra. Implications of this equivalence are also discussed and several examples are given to illustrate potentially interesting system-theoretic applications of this observation.

Applications of nonnegative matrices have been of immense interest to both social and physical scientists, particularly to economists and statisticians. This paper considers the question as to when a nonnegative singular matrix can be decomposed as a product of nonnegative idempotents analogous to the well-known result for any arbitrary matrix. It is shown that (i) all singular nonnegative matrices of rank , (ii) all singular nonnegative matrices that have a nonnegative von Neumann inverse, (iii) (0-1) nonnegative definite matrices and (iv) periodic matrices have the property that they decompose into a product of nonnegative idempotents. An example is given, in general, showing that this need not be true for singular matrices of rank three or higher, including stochastic and symmetric matrices. Besides computational techniques, a recent result that a singular nonnegative quasi-permutation matrix is a product of nonnegative idempotents plays a key role in the proofs of the results.

International audience; This paper is concerned with the analysis and synthesis of interconnected systems constructed from heterogeneous positive subsystems and a nonnegative interconnection matrix. We first show that admissibility, to be defined in this paper, is an essential requirement in constructing such interconnected systems. Then, we clarify that the interconnected system is admissible and stable if and only if a Metzler matrix, which is built from the coefficient matrices of positive subsystems and the nonnegative interconnection matrix, is Hurwitz stable. By means of this key result, we further provide several results that characterize the admissibility and stability of the interconnected system in terms of the Frobenius eigenvalue of the interconnection matrix and the weighted L1- induced norm of the positive subsystems again to be defined in this paper. 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., its state converges to a unique strictly positive vector (that is known in advance up to a strictly positive constant multiplicative factor) for any nonnegative and nonzero initial state. As an important consequence of this property, we show that the output of the interconnected system converges to a scalar multiple of the right eigenvector of a nonnegative matrix associated with its Frobenius eigenvalue, where the nonnegative matrix is nothing but the interconnection matrix scaled by the steady-stage gains of the positive subsystems. This result is then naturally and effectively applied to formation control of multiagent systems with positive dynamics. This result can be seen as a generalization of a well-known consensus algorithm that has been basically applied to interconnected systems constructed from integrators.

A real matrix is called totally nonnegative if all of its minors are nonnegative. In this paper, the minors are determined from which the maximum allowable entry perturbation of a totally nonnegative matrix can be found, such that the perturbed matrix remains totally nonnegative. Also, the total nonnegativity of the first and second subdirect sum of two totally nonnegative matrices is considered.

Real matrices with non-negative off-diagonal entries play a crucial role for modelling the positive linear dynamical systems. In the literature, these matrices are referred to as Metzler matrices or negated Z-matrices. Finding the closest stable Metzler matrix to an unstable one (and vice versa) is an important issue with many applications. The stability considered here is in the sense of Hurwitz, and the distance between matrices is measured in $$l_\infty ,\,l_1$$, and in the max norm. We provide either explicit solutions or efficient algorithms for obtaining the closest (un)stable matrix. The procedure for finding the closest stable Metzler matrix is based on the recently introduced selective greedy spectral method for optimizing the Perron eigenvalue. Originally intended for non-negative matrices, here is generalized to Metzler matrices. The efficiency of the new algorithms is demonstrated in examples and numerical experiments for the dimension of up to 2000. Applications to dynamical systems, linear switching systems, and sign-matrices are considered.

This paper presents a formal description and analysis of an SIR (involving susceptible- infectious-recovered subpopulations) epidemic model in a patchy environment with vaccination controls being constant and proportional to the susceptible subpopulations. The patchy environment is due to the fact that there is a partial interchange of all the subpopulations considered in the model between the various patches what is modelled through the so-called travel matrices. It is assumed that the vaccination controls are administered at each community health centre of a particular patch while either the total information or a partial information of the total subpopulations, including the interchanging ones, is shared by all the set of health centres of the whole environment under study. In the case that not all the information of the subpopulations distributions at other patches are known by the health centre of each particular patch, the feedback vaccination rule would have a decentralized nature. The paper investigates the existence, allocation (depending on the vaccination control gains) and uniqueness of the disease-free equilibrium point as well as the existence of at least a stable endemic equilibrium point. Such a point coincides with the disease-free equilibrium point if the reproduction number is unity. The stability and instability of the disease-free equilibrium point are ensured under the values of the disease reproduction number guaranteeing, respectively, the un-attainability (the reproduction number being less than unity) and stability (the reproduction number being more than unity) of the endemic equilibrium point. The whole set of the potential endemic equilibrium points is characterized and a particular case is also described related to its uniqueness in the case when the patchy model reduces to a unique patch. Vaccination control laws including feedback are proposed which can take into account shared information between the various patches. It is not assumed that there are in the most general case, symmetry-type constrains on the population fluxes between the various patches or in the associated control gains parameterizations.

This paper investigates the cluster synchronization problem of linearly coupled complex networks via aperiodically intermittent control (AIC). The dynamical behaviors of nodes in different clusters are assumed to be governed by different dynamical functions, while the dynamical behaviors of nodes in the same cluster are the same. Moreover, the original functions of nodes are defined by continuous-time ordinary differential equations with time-varying delays. As for the coupling matrix, we assume it is a Metzler matrix with zero row sums. The main contribution is that we pin some simple AIC to realize the cluster synchronization. Furthermore, as for the pinning control gains, both static and adaptive control cases are considered and some criteria are obtained. We also present some numerical simulations to verify the theoretical results.

Arising from the classification of the matrix-roots of a nonnegative imprimitive irreducible matrix, we present results concerning the Jordan chains of an $h$-cyclic matrix. We also present ancillary results applicable to nonnegative imprimitive irreducible matrices and demonstrate these results via examples., Comment: To appear in the special issue "The Legacy of Hans Schneider" in Linear Algebra and its Applications

The max-times algebra is the set R + of nonnegative reals with operations ⊕ : ( a , b ) → max ⁡ { a , b } and ⊙ : ( a , b ) → a b . We discuss the property of matrices to be squares of max-times or conventional nonnegative matrices. We prove that there exists a matrix having a conventional nonnegative square root but no max-times square root. Also, we present a set S of cardinality three for which there is a nonnegative matrix M with spectrum S, and every such M has both conventional and max-times square roots. These results answer two questions from the recent paper by Tam and Huang.

An extended Kalman-Yakubovich-Popov (KYP) Lemma for positive systems is derived. The main difference compared to earlier versions is that non-strict inequalities are treated. Matrix assumptions are also less restrictive. Moreover, a new equivalence is introduced in terms of linear programming rather than semi-definite programming. As a complement to the KYP lemma, it is also proved that a symmetric Metzler matrix with $m$ nonzero entries above the diagonal is negative semi-definite if and only if it can be written as a sum of $m$ negative semi-definite matrices, each of which has only four nonzero entries. This is useful in the context large-scale optimization.

The Real Nonnegative Inverse Eigenvalue Problem (RNIEP) asks when is a list \[ \sigma=(\lambda_1, \lambda_2,\ldots,\lambda_n)\] consisting of real numbers the spectrum of an $n \times n$ nonnegative matrix $A$. In that case, $\sigma$ is said to be realizable and $A$ is a realizing matrix. In a recent paper dealing with RNIEP, P.~Paparella considered cases of realizable spectra where a realizing matrix can be taken to have a special form, more precisely such that the entries of each row are obtained by permuting the entries of the first row. A matrix of this form is called permutative. Paparella raised the question whether any realizable list $\sigma$ can be realized by a permutative matrix or a direct sum of permutative matrices. In this paper, it is shown that in general the answer is no.

A new method is given to estimate an ultimate state bound on a time-varying linear system with delay and bounded disturbances by using some results on Metzler matrices. The effectiveness of the obtained results is illustrated by a numerical example.

A multi-agent continuous-time nonlinear model of social behaviour allowing for both competition and cooperation is presented and analysed. The state of each agent is represented by its payoff, which the agent aims at maximising. The role of control variables is played by the model parameters, which account for the agents' decisions to either cooperate with or boycott the other agents and can vary in time within assigned intervals. Alliances and enmities can be established at any time, according to either a greedy or a longsighted criterion. The general nonlinear case is first considered. It is proved that, under realistic assumptions, the system evolution is bounded positive (no extinction) and there is a unique globally-stable equilibrium point. As is somehow expected, the optimal decision for all agents corresponds to full cooperation (decision parameters kept at their positive maximum value) in the case of both shortsighted and farsighted criteria. This is not true if some parameters have negative upper bounds (meaning that some agents systematically boycott some others). Then, in the linear case, it is shown that the system is stable for arbitrarily-varying decision parameters, provided that a Metzler matrix associated with full cooperation is Hurwitz. A characterisation of the long-term behaviour of the linear system is also provided. In particular, it is proved that, under stability conditions, a Nash equilibrium exists if a steady strategy is adopted.

This work presents a mathematical model for the spread of lymphatic filariasis in a population. We use the Metzler Matrix Theory and the Kamgang–Sallet [2] (Math Biosci 213, 1–12, 2008) algorithm to compute the threshold conditions and global stability of the disease-free equilibrium. We showed that if \(R_0 1\), the disease-free equilibrium is unstable.

The authors address control system design based on an interval observer for non-linear switched systems with non-linear vector functions that are assumed to satisfy Lipschitz conditions. First, the observer gain satisfying a Metzler matrix can be solved by optimisations of linear matrix inequalities (LMIs). Second, an interval observer is designed to estimate the states of non-linear switched systems with the average dwell time scheme, and sufficient conditions for state estimation are presented in terms of an LMI formulation. Third, based on an interval observer, state feedback matrices are designed to construct an asymptotically stabilising switching controller. Finally, a numerical example is provided to demonstrate the efficiency of the approach.

We study the semialgebraic structure of $D_r$, the set of nonnegative tensors of nonnegative rank not more than $r$, and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for cubical nonnegative tensors and show that the direct sum conjecture is true for nonnegative tensor rank. We show that nonnegative, real, and complex ranks are all equal for a general nonnegative tensor of nonnegative rank strictly less than the complex generic rank. In addition, such nonnegative tensors always have unique nonnegative rank-$r$ decompositions if the real tensor space is $r$-identifiable. We determine conditions under which a best nonnegative rank-$r$ approximation has a unique nonnegative rank-$r$ decomposition: for $r \le 3$, this is always the case; for general $r$, this is the case when the best nonnegative rank-$r$ approximation does not lie on the boundary of $D_r$. Many of our general identifiability results also apply to real tensors and real symmetric tensors., 25 pages, to appear in SIMAX

The purpose of this note is two-fold: (1) to study when quasi-Euclidean rings, regular rings and regular separative rings have the property (∗) that each right (left) singular element is a product of idempotents, and (2) to consider the question: “when is a singular nonnegative square matrix a product of nonnegative idempotent matrices?” The importance of the class of quasi- Euclidean rings in connection with the property (∗) is given by the first three authors and T.Y. Lam [Journal of Algebra, 406:154–170, 2014], where it is shown that every singular matrix over a right and left quasi-Euclidean domain is a product of idempotents, generalizing the results of J. A Erdos [Glasgow Mathematical Journal, 8: 118–122, 1967] for matrices over fields and that of T. J. Laffey [Linear and Multilinear Algebra, 14:309–314, 1983] for matrices over commutative Euclidean domains. We have shown in this paper that quasi-Euclidean rings appear among many interesting classes of rings and hence they are in abundance. We analyze the properties of triangular matrix rings and upper triangular matrices with respect to the decomposition into product of idempotents and show, in particular, that nonnegative nilpotent matrices are products of nonnegative idempotent matrices. We study as to when each singular matrix is a product of idempotents in special classes of rings. Regarding the second question for nonnegative matrices, bounds are obtained for a rank one nonnegative matrix to be a product of two idempotent matrices. It is shown that every nonnegative matrix of rank one is a product of three nonnegative idempotent matrices. For matrices of higher orders, we show that some power of a group monotone matrix is a product of idempotent matrices.

This paper considers the non-convex problem of finding the nearest Metzler matrix to a given possibly unstable matrix. Linear systems whose state vector evolves according to a Metzler matrix have many desirable properties in analysis and control with regard to scalability. This motivates the question, how close (in the Frobenius norm of coefficients) to the nearest Metzler matrix are we? Dropping the Metzler constraint, this problem has recently been studied using the theory of dissipative Hamiltonian (DH) systems, which provide a helpful characterization of the feasible set of stable matrices. This work uses the DH theory to provide a block coordinate descent algorithm consisting of a quadratic program with favourable structural properties and a semidefinite program for which recent diagonal dominance results can be used to improve tractability., To Appear in Proc. of 56th IEEE CDC

Metzler matrices are algebraically characterized by having nonnegative off-diagonal entries and, from the dynamical point of view, define continuous-time positive systems. Our work studies polytopes of Metzler matrices and linear dynamics generated by such sets. The first part of the paper explores the algebraic properties of matrix polytopes, by focusing on the estimation of a right outer bound for all eigenvalues. The second part analyzes the dynamical properties of positive polytopic systems, by revealing connections between the estimated right outer bound and the evolution of trajectories (related to copositive Lyapunov functions and invariant sets).

A new approach to design interval observers for a class of nonlinear systems is proposed in this paper. Based on the matrix theory, a simple elementary operation is given to find a positive realization for the systems with bounded uncertainties. Compared with previous works, the main contribution of this work is to provide a method to guarantee the existence of an observer gain by means of such a positive representation. Finally, a numerical example is provided to demonstrate efficiency of the proposed approach.

This paper considers the stabilization problem of linear dynamical systems with combined structural properties of symmetry and positivity. Such systems arise in various applications including electromechanical systems, aerodynamics, structural vibration, robotics and compartmental systems, in which the stabilization and performance improvement are the main objectives. Although the goal of the paper is to design controller for such systems, we broaden the scope by designing controller for general systems such that the closed-loop systems admits the structural constraints of symmetry and positivity. We concentrate on two different classes of symmetric positive systems. The first class has the state-space symmetric representation A = AT, C = BT with A being a Metzler matrix and the positive pair of matrices (B, C). The stability conditions of this class are used to formulate and solve symmetric positive stabilization by means of state feedback for systems with arbitrary state-space parameters. The second class is defined through the block controllable canonical form in which the block sub-matrices associated with the system matrix A are symmetric Metzlerian. Assuming that such systems are unstable, we design state feedback control law such that the closed-loop system becomes stable and maintains its structure. A generalized symmetric Metzlerian stabilization algorithm is provided through a set of chain equations to achieve this goal. Numerical examples are provided for stabilization of both classes.

The nonnegative inverse eigenvalue problem is the problem of finding necessary and sufficient conditions for the existence of an entrywise nonnegative matrix A with prescribed spectrum. This problem remains open for . If the matrix A is required to be normal, the problem will be called the normal nonnegative inverse eigenvalue problem (NNIEP). Sufficient conditions for a list of complex numbers to be the spectrum of a normal nonnegative matrix were obtained by Xu [Linear Multilinear Algebra. 1993;34:353–364]. In this paper, we give a normal version of a rank-r perturbation result due to Rado and published by Perfect [Duke Math. J. 1955;22:305–311], which allow us to obtain new sufficient conditions for the NNIEP to have a solution. These new conditions significantly improve Xu’s conditions. We also apply our results to construct nonnegative matrices with arbitrarily prescribed elementary divisors.

In this note we consider a supervisory control scheme that achieves either asymptotic stability or consensus for a group of homogenous agents described by a positive state-space model. Each agent is modeled by means of the same SISO positive state-space model, and the supervisory controller, representing the information exchange among the agents, is implemented via a static output feedback. Necessary and sufficient conditions for the asymptotic stability, or the consensus of all agents, are derived under the positivity constraint.

In this note, the problem of state bounding for linear time-varying systems with delay and bounded disturbances input is considered for the first time. By using a novel approach which does not involve the Lyapunov–Krasovskii functional method, new explicit delay-independent conditions are derived for the existence of a ball such that all the state trajectories of the system converge exponentially within it. A numerical example is given to illustrate the effectiveness of the obtained result.

The goal of this paper is to study the switched stochastic control problem of discrete-time linear systems with multiplicative noises. We consider both the quadratic and the H∞ criteria for the performance evaluation. Initially we present a sufficient condition based on some Lyapunov–Metzler inequalities to guarantee the stochastic stability of the switching system. Moreover, we derive a sufficient condition for obtaining a Metzler matrix that will satisfy the Lyapunov–Metzler inequalities by directly solving a set of linear matrix inequalities, and not bilinear matrix inequalities as usual in the literature of switched systems. We believe that this result is an interesting contribution on its own. In the sequel we present sufficient conditions, again based on Lyapunov–Metzler inequalities, to obtain the state feedback gains and the switching rule so that the closed loop system is stochastically stable and the quadratic and H∞ performance costs are bounded above by a constant value. These results are illu...

In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result.

Chattering is an undesirable phenomenon characterised by infinitely fast switching which may cause equipment damage in real systems. To avoid its occurrence, this study proposes a chattering-free switching strategy for continuous-time switched linear systems ensuring global asymptotical stability and a guaranteed cost level associated to the rms gain of a class of input to output signals. The switching function is designed considering a minimum dwell-time constraint in order to avoid chattering and a maximum one to ensure robustness with respect to sampling jitters and implementation imperfections as, for instance, delays in the switching process. The conditions are based on Riccati-Metzler inequalities which take into account an equivalent discrete-time switched linear system obtained from the continuous-time one guided by a sampled switching rule without any kind of approximation. As a new result, for a subclass of Metzler matrices, necessary and sufficient conditions for the existence of a solution for the Riccati-Metzler inequalities are provided. Theoretical aspects are illustrated by some academical examples.

Tridiagonal matrices are considered which are totally nonnegative, i. e., all their mi- nors are nonnegative. The largest amount is given by which the single entries of such a matrix can be perturbed without losing the property of total nonnegativity.

This paper is concerned with the analysis and synthesis of linear positive systems based on linear matrix inequalities (LMIs). We first show that the celebrated Perron–Frobenius theorem can be proved concisely by a duality-based argument. Again by duality, we next clarify a necessary and sufficient condition under which a Hurwitz stable Metzler matrix admits a diagonal Lyapunov matrix with some identical diagonal entries as the solution of the Lyapunov inequality. This new result leads to an alternative proof of the recent result by Tanaka and Langbort on the existence of a diagonal Lyapunov matrix for the LMI characterizing the H ∞ performance of continuous-time positive systems. In addition, we further derive a new LMI for the H ∞ performance analysis where the variable corresponding to the Lyapunov matrix is allowed to be non-symmetric. We readily extend these results to discrete-time positive systems and derive new LMIs for the H ∞ performance analysis and synthesis. We finally illustrate their effectiveness by numerical examples on robust state-feedback H ∞ controller synthesis for discrete-time positive systems affected by parametric uncertainties.

In statistical practice, rectangular tables of numeric data are commonplace, and are often analyzed using dimension-reduction methods like the singular value decomposition and its close cousin, principal component analysis (PCA). This analysis produces score and loading matrices representing the rows and the columns of the original table and these matrices may be used for both prediction purposes and to gain structural understanding of the data. In some tables, the data entries are necessarily nonnegative (apart, perhaps, from some small random noise), and so the matrix factors meant to represent them should arguably also contain only nonnegative elements. This thinking, and the desire for parsimony, underlies such techniques as rotating factors in a search for “simple structure.” These attempts to transform score or loading matrices of mixed sign into nonnegative, parsimonious forms are, however, indirect and at best imperfect. The recent development of nonnegative matrix factorization, or NMF, is an att...

This paper studies the construction problems of five order nonnegative matrices from spectrum data. Let be a list of complex numbers with . Necessary and sufficient conditions for the existence of an entry-wise nonnegative 5×5 matrix with spectrum are presented.

The basin of attraction to the origin in the space is obtained for the m dimensional system of difference equations of the formwhen coefficient is a Metzler matrix. Some information about the dynamics of the solutions outside of the basin is also given.

Given any nonnegative matrix $A \in \mathbb{R}^{m \times n}$ , it is always possible to express A as the sum of a series of nonnegative rank-one matrices. Among the many possible representations of A, the number of terms that contributes the shortest nonnegative rank-one series representation is called the nonnegative rank of A. Computing the exact nonnegative rank and the corresponding factorization are known to be NP-hard. Even if the nonnegative rank is known a priori, no simple procedure exists presently that is able to perform the nonnegative factorization. Based on the Wedderburn rank reduction formula, this paper proposes a heuristic approach to tackle this difficult problem numerically. Starting with A, the idea is to recurrently extrat, whenever possible, a rank-one nonnegative portion from the previous matrix while keeping the residual nonnegative and lowering its rank by one. With a slight modification for symmetry, the method can equally be applied to another important class of completely positive matrices. No convergence can be guaranteed, but repeated restart might help alleviate the difficulty. Extensive numerical testing seems to suggest that the proposed algorithm might serve as a first-step numerical means for exploring the intriguing problem of nonnegative rank factorization.

Nonnegative matrix factorization/approximation (NMF/NMA) is a widely used method for data analysis. So far, many multiplicative update algorithms have been developed for NMF. In this paper, we propose a nonnegative projection based NMF algorithm, which is different from the conventional multiplicative update NMF algorithms and decreases the objective function by performing Procrustes rotation and nonnegative projection alternately. The experiment results demonstrate that the new algorithm converges much faster than traditional ones.

In this paper, we theoretically clarify the phase transition structure of the variational Bayesian nonnegative matrix factorization (VBNMF). By asymptotic analysis of the objective functional in variational inference, we find that the variational posterior distribution of the VBNMF is drastically changed by hyperparameters; we call this phenomenon phase transition of the VBNMF. We also discuss a numerical experiment demonstrating our theoretical results.

In this paper we illustrate the use of Nonnegative Matrix Factorization (NMF) to analyze real data derived from an e-learning context. NMF is a matrix decomposition method which extracts latent information from data in such a way that it can be easily interpreted by humans. Particularly, the NMF of a score matrix can automatically generate the so called Q-matrix. In an e-learning scenario, the Q-matrix describes the abilities to be acquired by students to correctly answer evaluation exams. An example on real response data illustrates the effectiveness of this factorization method as a tool for EDM.

The nonnegative tensor (matrix) factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. To solve the subproblem efficiently, we first exploit a variable regularization term which makes the subproblem far from ill-condition. Second, an augmented Lagrangian alternating direction method is employed to solve this convex and well-conditioned regularized subproblem, and two accelerating skills are also implemented. Some preliminary numerical experiments are performed to show the improvements of the new method.

Nonnegative matrix factorization is a linear dimensionality reduction technique used for decomposing high-dimensional nonnegative data matrices for extracting basic and latent features. This technique plays fundamental roles in music analysis, signal processing, sound separation, and spectral data analysis. Given a time-varying objective function or a nonnegative time-dependent data matrix Y(t), the nonnegative factors of Y(t) can be obtained by taking the limit points of the trajectories of the corresponding ordinary differential equations. When the data are time dependent, it is natural to devise factorization techniques that capture the time dependency. To achieve this, one needs to solve continuous-time dynamical systems derived from iterative optimization schemes and construct nonnegative matrix factorization algorithms based on the solution curves. This article presents continuous nonnegative matrix factorization methods based on the solution of systems of ordinary differential equations associated with time-dependent data. In particular, we propose two new continuous-time algorithms based on the Kullback–Leibler divergence and the Amari \(\alpha \)-divergence.