Positive Diagonal Matrix Research Articles

The Distributed Adaptive Bipartite Consensus Tracking Control of Networked Euler–Lagrange Systems with an Application to Quadrotor Drone Groups

Actuator faults and external disturbances, which are inevitable due to material fatigue, operational wear and tear, and unforeseen environmental impacts, cause significant threats to the control reliability and performance of networked systems. Therefore, this paper primarily focuses on the distributed adaptive bipartite consensus tracking control problem of networked Euler–Lagrange systems (ELSs) subject to actuator faults and external disturbances. A robust distributed control scheme is developed by combining the adaptive distributed observer and neural-network-based tracking controller. On the one hand, a new positive definite diagonal matrix associated with an asymmetric Laplacian matrix is constructed in the distributed observer, which can be used to estimate the leader’s information. On the other hand, neural networks are adopted to approximate the lumped uncertainties composed of unknown matrices and external disturbances in the follower model. The adaptive update laws are designed for the unknown parameters in neural networks and the actuator fault factors to ensure the boundedness of estimation errors. Finally, the proposed control scheme’s effectiveness is validated through numerical simulations using two types of typical ELS models: two-link robot manipulators and quadrotor drones. The simulation results demonstrate the robustness and reliability of the proposed control approach in the presence of actuator faults and external disturbances.

The Computational Solution of Generalized Inverse Eigenvalue Problem for Pseudo‐Jacobi Matrix

This paper is concerned with two generalized inverse eigenvalue problems for a kind of pseudo‐Jacobi matrix. From a non‐Hermite matrix, an r × r Jacobi matrix, two distinct real eigenvalues, and part of the corresponding eigenvectors, an n × n pseudo‐Jacobi matrix is constructed. Furthermore, an n × n pseudo‐Jacobi matrix can be made by two different eigenpairs and a positive definite diagonal matrix. It is shown that a unique pseudo‐Jacobi matrix can be recovered from partial eigenpairs and certain special mixed eigendata. Two algorithms are provided for the reconstruction of such a pseudo‐Jacobi matrix, and illustrative numerical examples are presented to verify the proposed algorithms.

Exact Solutions v.s. Perturbative Calculations of Finite Φ3-Φ4 Hybrid-Matrix-Model

There is a matrix model corresponding to a scalar field theory on noncommutative spaces called Grosse-Wulkenhaar model (Φ4 matrix model), which is renormalizable by adding a harmonic oscillator potential to scalar Φ4 theory on Moyal spaces. There are more unknowns in Φ4 matrix model than in Φ3 matrix model, for example, in terms of integrability. We then construct a one-matrix model (Φ3-Φ4 Hybrid-Matrix-Model) with multiple potentials, which is a combination of a 3-point interaction and a 4-point interaction, where the 3-point interaction of Φ3 is multiplied by some positive definite diagonal matrix M. This model is solvable due to the effect of this M. In particular, the connected ∑i=1BNi-point function G|aN11⋯aN11|⋯|a1B⋯aNBB| of Φ3-Φ4 Hybrid-Matrix-Model is studied in detail. This ∑i=1BNi-point function can be interpreted geometrically and corresponds to the sum over all Feynman diagrams (ribbon graphs) drawn on Riemann surfaces with B boundaries (punctures). Each |a1i⋯aNii| represents Ni external lines coming from the i-th boundary (puncture) in each Feynman diagram. First, we construct Feynman rules for Φ3-Φ4 Hybrid-Matrix-Model and calculate perturbative expansions of some multipoint functions in ordinary methods. Second, we calculate the path integral of the partition function Z[J] and use the result to compute exact solutions for 1-point function G|a| with 1-boundary, 2-point function G|ab| with 1-boundary, 2-point function G|a|b| with 2-boundaries, and n-point function G|a1|a2|⋯|an| with n-boundaries. They include contributions from Feynman diagrams corresponding to nonplanar Feynman diagrams or higher genus surfaces.

Inertias of matrices and sign patterns related to a system of second order ODEs

The inertia of real matrices of the form C=[ADIO] and of the sign pattern matrix C of C are considered, where A is irreducible and D is a positive diagonal matrix. Such matrices C occur in applications that give rise to systems of linear second order differential equations. If D=dI is a positive scalar matrix, then the eigenvalues of C are determined by the magnitude of d and the eigenvalues of A, giving results for the inertia of C when A has certain specified forms. Results for the inertia of C are also given when D is an arbitrary positive diagonal matrix. The set of possible refined inertias of general sign pattern matrices C of the above form are determined, and are completely specified when A is a spectrally arbitrary sign pattern of order 2 or 3.

Hierarchical Estimation Approach for RBF-AR Models With Regression Weights Based on the Increasing Data Length

In the radial basis function-based state-dependent autoregressive (RBF-AR) models with regression weights, the local linear models are included between the hidden layers and the output layers of the networks. The parameter estimation for the RBF-AR models with regression weights is studied in this brief. Considering the separable feature of the models, two criterion functions based on the increasing data length are defined to fit the observation data of the whole dynamical process. Two sub-algorithms are proposed by minimizing the criterion functions. Aiming to overcome the existence of the singular matrix during the Newton search and to make the algorithm more stable, a positive definite diagonal matrix is introduced to the algorithm. Based on the hierarchical principle, a hierarchical Newton recursive algorithm is proposed, which can realize the on-line parameter estimation. Simulation results verify the validity.

Attitude stabilization of rigid spacecraft with angular velocity and torque limitations

An adaptive backstepping control scheme has been put forward to investigate the attitude stabilization problem for a spacecraft with modeling uncertainties, external disturbances, actuator failures, angular velocity limitations and actuator limitations simultaneously. Based on the active disturbance rejection concept (ADRC), a positive definite diagonal matrix is selected by the designer to separate a control item from the system. Then, an extended state observer is constructed to deal with the total uncertain item including actuator failure. In order to deal with the limitation of torques, a saturation coefficient is introduced and its adaptive law is obtained. Meanwhile, a command filter and the compensation signal are constructed to handle the angular velocity limitations. Finally, the stability of the closed-loop system is analyzed. Simulation results are given to verify the effectiveness of the proposed scheme.

On the derivative-free quasi-Newton-type algorithm for separable systems of nonlinear equations

A derivative-free quasi-Newton-type algorithm in which its search direction is a product of a positive definite diagonal matrix and a residual vector is presented. The algorithm is simple to implement and has the ability to solve large-scale nonlinear systems of equations with separable functions. The diagonal matrix is simply obtained in a quasi-Newton manner at each iteration. Under some suitable conditions, the global and R-linear convergence result of the algorithm are presented. Numerical test on some benchmark separable nonlinear equations problems reveal the robustness and efficiency of the algorithm.

Extremal eigenvalues of sample covariance matrices with general population

We consider the eigenvalues of sample covariance matrices of the form Q=(Σ1/2X)(Σ1/2X)∗. The sample X is an M×N rectangular random matrix with real independent entries and the population covariance matrix Σ is a positive definite diagonal matrix independent of X. Assuming that the limiting spectral density of Σ exhibits convex decay at the right edge of the spectrum, in the limit M,N→∞ with N/M→d∈(0,∞), we find a certain threshold d+ such that for d>d+ the limiting spectral distribution of Q also exhibits convex decay at the right edge of the spectrum. In this case, the largest eigenvalues of Q are determined by the order statistics of the eigenvalues of Σ, and in particular, the limiting distribution of the largest eigenvalue of Q is given by a Weibull distribution. In case d<d+, we also prove that the limiting distribution of the largest eigenvalue of Q is Gaussian if the entries of Σ are i.i.d. random variables. While Σ is considered to be random mostly, the results also hold for deterministic Σ with some additional assumptions.

Sums and products of symplectic eigenvalues

For every 2n×2n real positive definite matrix A, there exists a real symplectic matrix M such that MTAM=diag(D,D), where D is the n×n positive diagonal matrix with diagonal entries d1(A)≤⋯≤dn(A). The numbers d1(A),…,dn(A) are called the symplectic eigenvalues of A. We derive analogues of Wielandt's extremal principle and multiplicative Lidskii's inequalities for symplectic eigenvalues.

Generalization of the concept of diagonal dominance with applications to matrix D-stability

In this paper, we introduce the class of diagonally dominant with respect to a given LMI region D⊂C matrices. They are shown to possess the analogues of well-known properties of (classical) diagonally dominant matrices, e.g. their spectra are localized inside the region D. Moreover, we show that in some cases, diagonal D-dominance implies (D,D)-stability (i.e. the preservation of matrix spectra localization under multiplication by a positive diagonal matrix).

General fixed-point method for solving the linear complementarity problem

&lt;abstract&gt;&lt;p&gt;In this paper, we consider numerical methods for the linear complementarity problem (LCP). By introducing a positive diagonal parameter matrix, the LCP is transformed into an equivalent fixed-point equation and the equivalence is proved. Based on such equation, the general fixed-point (GFP) method with two cases are proposed and analyzed when the system matrix is a $ P $-matrix. In addition, we provide several concrete sufficient conditions for the proposed method when the system matrix is a symmetric positive definite matrix or an $ H_{+} $-matrix. Meanwhile, we discuss the optimal case for the proposed method. The numerical experiments show that the GFP method is effective and practical.&lt;/p&gt;&lt;/abstract&gt;

Interval Observer Design for One Class of Uncertain Linear Strictly Metzlerian Time-Delay Systems

The paper is concerned with design requirements when the problem of nonnegative state estimation for one class of uncertain linear Metzlerian time-delay systems with constant delays is tackled, while system states take nonnegative values whenever the initial conditions are nonnegative, the upper and lower system matrix bounds are strictly Metzler matrices, and the upper and lower output matrix bounds are nonnegative matrices. By defining positive definite diagonal matrix variables and introducing an associate structure of linear matrix inequalities, the design conditions are proven, guaranteeing if they are feasible, the resulting observer gain matrix is positive and the reflected observer system matrices are strictly Metzler and Hurwitz. A numerical example illustrates the solvability of the proposed design conditions.

Improvement of finite-time stability for delayed neural networks via a new Lyapunov-Krasovskii functional

The topic of finite-time stability criterion for neural networks with time-varying delays via a new argument Lyapunov-Krasovskii functional (LKF) was proposed and the time-varying delay of the system is without differentiable. For sufficient conditions of this study, a new (LKF) is combined with improved triple integral terms, namely the functionality of finite-time stability, integral inequality, and a positive diagonal matrix without using a free weighting matrix. The improved finite-time sufficient conditions for the neural network with time varying delay are given in terms of linear matrix inequalities (LMIs) and the results show improvement on previous studies. Numerical examples are given to illustrate the effectiveness of the proposed method.

Variational principles for symplectic eigenvalues

AbstractIf A is a real $2n \times 2n$ positive definite matrix, then there exists a symplectic matrix M such that $M^TAM=\text {diag}(D, D),$ where D is a positive diagonal matrix with diagonal entries $d_1(A)\leqslant \cdots \leqslant d_n(A).$ We prove a maxmin principle for $d_k(A)$ akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality $d_{i+j-1}(A+B)\geqslant d_i(A)+d_j(B).$

Convergence of the non-uniform directed Physarum model

The directed Physarum dynamics is known to solve positive linear programs: minimize cTx subject to Ax=b and x≥0 for a positive cost vector c. The directed Physarum dynamics evolves a positive vector x according to the dynamics x˙=q(x)−x. Here q(x) is the solution to Af=b that minimizes the “energy” ∑icifi2/xi.In this paper, we study the non-uniform directed dynamics x˙=D(q(x)−x), where D is a positive diagonal matrix. The non-uniform dynamics is more complex than the uniform dynamics (with D being the identity matrix), as it allows each component of x to react with different speed to the differences between q(x) and x. Our contribution is to show that the non-uniform directed dynamics solves positive linear programs.

An alternating minimization algorithm for Factor Analysis

The problem of decomposing a given covariance matrix as the sum of a positive semi-definite matrix of given rank and a positive semi-definite diagonal matrix, is considered. We present a projection-type algorithm to address this problem. This algorithm appears to perform extremely well and is extremely fast even when the given covariance matrix has a very large dimension. The effectiveness of the algorithm is assessed through simulation studies and by applications to three real datasets that are considered as benchmark for the problem. A local convergence analysis of the algorithm is also presented.

Linear preservers of copositive matrices

ABSTRACT An n-by-n real symmetric matrix is called copositive if its quadratic form is nonnegative on nonnegative vectors. Our interest is in identifying which linear transformations on symmetric matrices preserve copositivity either in the into or onto sense. We conjecture that in the onto case, the map must be congruence by a monomial matrix (a permutation times a positive diagonal matrix). This is proven under each of some additional natural assumptions. Also, the into preservers of standard type are characterized. A general characterization in the into case seems difficult, and examples are given. One of them provides a counterexample to a conjecture about the into preservers.

Criteria for H-Matrices Based on γ−Diagonally Dominant Matrix

In this paper, we present a new practical criteria for H-matrix based on &amp;gamma;-diagonally dominant matrix. In order to make the judgment conditions convenient and effective, we give two new definitions, one is called strong and weak diagonally dominant degree, the other is called the sum of non-principal diagonal element for the matrix. Further, we obtain a new practical method for the determination of the H-matrix by combining the properties of &amp;gamma;-diagonally dominant matrix, constructing positive diagonal matrix, and adding the appropriate parameters. Finally, we offer numerical examples to verify the validity of the judgment conditions, corresponding numerical examples compared the new criteria and the existing results are presented to verify the advantages of the new determination method.

Integral Control of Stable Negative-Imaginary Systems Preceded by Hysteresis Nonlinearity

In this note, constant reference tracking using integral control for negative-imaginary (NI) systems preceded by hysteresis nonlinearity is addressed. The hysteresis nonlinearity is considered to be slope-restricted, though no specific modeling framework is exploited. It is effectively shown that integral tracking controllers are guaranteed to exist when a stable NI system with sign-definite dc-gain is preceded by such hysteresis nonlinearities. When the dc-gain is negative-definite, under positive feedback configuration with positive slope-restricted hysteresis, it is established that the integral controller gain can be any positive diagonal matrix. The development of the proposed model-independent inversion-free scheme is carried out in general state-space setting and is applicable for multiple-input multiple-output systems with decoupled hysteresis nonlinearities. Effectiveness of the results is illustrated with examples.

Metrics transformations preserving the types of one-dimensional minimal fillings

Given a class F of metric spaces and a family of transformations T of a metric, one has to describe a family of transformations T' ? T that transfer F into itself and preserve some types of minimal fillings. The article considers four cases. First, when F is the class of all finite metric spaces, T = {(M,?) ? (M, f??) | f : R&gt;0 ? R&gt;0}, and the elements of T' preserve all non-degenerate types of minimal fillings of four-point metric spaces and finite non-degenerate stars, and we prove that T' = {(M,?) ? (M,?? + a):a &gt; ?a?}. Second, when F is the class of all finite metric spaces, the class T consists of the maps ? ? N?, where the matrix N is the sum of a positive diagonal matrix A and a matrix with the same rows of non-negative elements. The elements of T' preserve all minimal fillings of the type of non-degenerate stars. It has been proven that T0 consists of maps ? ? N? where A is scalar. Third, when F is the class of all finite additive metric spaces, T is the class of all linear mappings given by matrices, and the elements of T' preserve all non-degenerate types of minimal fillings, and we proved that for metric spaces consisting of at least four points T' is the set of transformations given by scalar matrices. Fourth, when F is the class of all finite ultrametric spaces, T is the class of all linear mappings given by matrices, and we proved that for threepoint spaces the matrices have the form A = R(B + ?E), where B is a matrix of identical rows of positive elements, and R is a permutation of the points (1,0,0),(0,1,0) and (0,0,1).